A Program for Solving Heat and Mass Transfer Problems on a PC
198
( ( • A PROGRAM FOR SOLVING HEAT AND MASS TRANSFER PROBLEMS ON A PC by Murray J. Brown A Thesis Submitted to the Facultyof Graduate Studies and Research in Partial Fulfilment of the Requirements for the Degree of Muter of Engineering Department of Mining and Metallurgical Engineering McGill University Montreal, Canada SEPTEMBER 1990
description
A Program for Solving Heat and Mass Transfer Problems on a PC
Transcript of A Program for Solving Heat and Mass Transfer Problems on a PC
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•
A PROGRAM FOR SOLVING HEAT AND MASS TRANSFER PROBLEMS ON A PC
by
Murray J. Brown
A Thesis Submitted to the Facultyof Graduate Studies and Research in Partial Fulfilment of the Requirements for the Degree of Muter of Engineering
Department of Mining and Metallurgical Engineering McGill University Montreal, Canada
SEPTEMBER 1990
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Abstract
The thesis describes a computer program. (FASTP), written for the DOS environment and
based on finite dift'erence algorithms, which can be used to solve both transient heat and
mass transfel' problems. Relatively simple geometries can be used as building blocks to
model problems in cartesian, cylindrical, and spherical coordinate systems. The user can
model diffusion behaviour through .:!uy material provided the relevant material properties
are known. A completely menu driven ~vstem allows for the specification of a number of
boundary conditions including convection, conS~~1\t (": 7ero Hux, and radiation. Heat gener
ation or mass accumulation, as wel1 as interboundary resistan(.~ or partition coefficient terms
can also be assigned. The program can also be used to model phase transformations and
the effects of mixing in liquid systems. The results of severa! prohlems run on FASTP have
been documented in this report and are shown to compare favourably with results generated
from mathematically exact solutions.
Résumé
Ce rapport décrit un logiciel (FASTP), écrit pour l'environment DOS et qui utilise
des algorithmes basés sur la méthode de différence fini, qui peut être utiliser pour résoudre
des problèmes de transfert thermique ou massique. Des géométries relativement simple sont
utilisés comme blocs pour construire un modèle d'un problème en coordonnés cartésien,
cylindrique, ou sphérique. L'utilisateur peut modeler la diffusion à travers n'importe quellf'
matière si les propriétés ayant rapport sont connues. Le logiciel est dirigé pa.r menu par laqurl
les paramètres pour les conditions aux limites, comme la convection, un flux constant, ou la
radiation, sont fournies. Des termes de génération de chaleur ou d'accumulation de m<'...-;sc,
et aussi la résistance thermique et les coéfficients de partition peuvent-être accommoder. Le
logiciel peut aussi modeler des systèmes où il y a un changement d'état et peut modeler
les effets du mixage dans un système liquide. Les résultats de plusieurs problèmes résolus
par FASTP sont documenter et sont montrer de comparer favorablement avec les solutions
exactes.
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Acknowledgements
1 would like to make clear that this report documents a project that has been con
tributed to by various other persons, not just myself. FASTP was initiated several years ago
as a class project and has slowly grown into what it is today; a fully iuteractive transient heat
or mass transfer software package capable of being run in the DOS environment. Foremost
on the list of people 1 wish to thank is my supervisoI, Dr. Frank Mucciardi, who has been
the source of many interesting, if not heated discussions on FASTP and many other topics.
As well, 1 would like to thank both Kevin O'Leary and Angelo Gra.ndillo for their cn-going
efforts in this venture, and Ryan Katofsky without who se mastery of ~TEX both this thesis
and the FASTP manual might not have been completed. Finally, 1 would like to thank
NSERC for their financial support.
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Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
1 The FASTP Algorithms
1 The Basis for FASTP
1.1 Control Volume Approach
1.2 Mass Transfer .
1.3 Exact Solutions
2 FASTP Conventions Governing Finite Difference Equations
2.1 Nodes & Nodal Points ............... .
2.2 FASTP's Governing Finite Difference Equations ..
2.3 Half-Node Equations & Boundary Conditions
2.4 Curvilinear Coormnate Configurations ..
2.5 Matelial Properties . . . . . . . . . .
2.6 Manipulation of Variables by FASTP
2.7 Stability Criteria for FASTP Solutions
2.8 Comparis.Jn to Heisler Chart Solutions
2.8.1 Example 2-1: Infinite plate .....
2.8.2 Exampie 2-2: Semi-infinite cylinder
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2.8.3 Example 2-3: Sphere . . . . . . . . . .
2.9 Dynamica.lly Changing Boundary Conditions.
2.9.1 Example 2-4: Heat flow through a reciangular slab
3 Conventions for Composite Configurations
3.1 Finite Difference Equations For Composite Heat Transfer Systems
3.2 Node & Section Progression Conventions for FASTP
3.3 Interboundary Resistances . . . . . . . . . . . .
3.4 illustration of Heat Transfer Across an Interface
3.5 Mass Transfer Across an Interface .
3.6 Stability at an Interface ..... .
4 Conventions for Multidimensional Configurations
4.1 MtÙtidimensional Systems ................... .
4.1.1 Conventions for modelling a mtÙtidimensional problem
4.1.2 The classical product solution method
4.2 FASTP's Product Solution . . . . . . .
4.2.1 The prod",ct solution a.lgorithm
4.2.2 Examples illustrating the product solution
4.2.3 Advantages and limitation., of the product solution
4.3 FASTP's Interaction Solution . . . . ...
4.3.1 The interaction solution a.lgorithm
4.3.2 Example illustrating the interaction solution
4.3.3 Advantages and limitations of the interaction solution .
5 The Phase Transformation and Mixing Algorithms
5.1 Phase Transformations . . . . . . . . . . . . . . . . .
IV
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5.2 Non-eqnilibrium Conditions ..... .
5.3 The Phase Transformation Algorithm .
5.4 Stabilityand the Phase Transformation Fcature
5.5 Phase Transformations at a Single Temperature
5.6 Phase Transformations in Profile Nodes .
5.7 Phase Transformations at an Interface .
5.8 Phase Transformations in Mass Transfer
5.9 The Mixing Feature. . . . . . . . . . . .
5.10 Comparison to an Exact Mixing Solution.
6 Conclusions
II The FASTP Manual
Disclaimer . . . . .
7 Getting Started
7.1 Introductory Remarks
7.2 Backup Copies .... 7.3 Hardware & Software Requirements .
7.4 The Block Security Deviee
7.5 Hard Disk Installation .. 7.6 Running FASTP From Disk Drives
7.7 File Breakdown . . . . . . . . . .
8 Program 1: FASTP [Data entry]
8.1 Introductory Remarks .. 8.2 Preliminary Screens ....
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8.3 Geometry ........ .
8.3.1 Problem definition
8.3.2 System Dimensions screen (1)
8.3.3 System Dimensions screen (2)
8.4 Boundary Conditions ....... .
8.4.1 The boundary condition options.
tlA.2 Batch mode operation
8.5 Material Properties .....
8.6 Starting Temperatures / Concentrations
8.7 Other A vailable Options . . . . . . . . .
8.7.1 Interboundary resistances / partition coefficients
8.7.2 Heat generation / Mass accumulation
8.7.3 Mixing ................. .
8.7.4 Product solution - Interaction solution
8.8 Option Menu: End of Data Entry
8.8.1 Edit the input data .
8.8.2 Execute simulation
8.8.3 Stop execution ..
8.8.4 Summarize input data
8.8.5 Store input data .
8.9 FASTP Database Facility
8.9.1 Accessing the database
8.9.2 Ma.nipulating the databasc
9 Program 2: CALS
9.1 Introductory Remarks
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9.2 Time Parameters ..
9.2.1 Initial values
9.2.2 Itera.tion time increment
9.2.3 Total simulation time .
9.3 Display Options
9.3.1 Print options
9.3.2 Plot options .
10 The Graphies Faeility
10.1 Introductory Remarks
10.2 Running FASTPLT .
10.3 Common Parameters
10.4 Option-specifie Parameters .
10.5 Ending a FASTPLT Session
10.6 Run'.lÏng HPPLOT
Il File Export
Bibliography
Appendix A The Heisler Charts
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List of Figures
1.1 Control volume for 1-0 cartesian coordinate systems. . . . . 5
1.2 Raoultian & Henrian behaviour of two-component mixtures. 9
2.1 Oiscretized bar with seven nodes. . . . . . . . . . . . . . . . 14
2.2 illustration of a three node configuration (node i is internal). 16
2.3 FASTP approximation of temperature distribution across three nodes. . 18
2.4 Nodal point 1 of I-D cartesian coordinate configuration. 24
2.5 1-0 cylindrical cootdinate configuration. 24
2.6 illustration of a stable solution. 27
2.7 Temperature distribution across aluminum bar at ten second intervals. 31
2.8 Cooling of a semi-infinite cylinder at 1.25 cm radius. 33
2.9 Temperature distribution along the sphere radius. . . 34
2.10 Schematic showing relationship between batch mode feature's time, tempera-ture and time-temperature entries. . . . . . . . . . . . . . . . . . . . . . .. 37
2.11 illustration of the effect of time dependent entries on selected nodes in a configuration. . . . . . . . . . . . ............... " ...... 39
3.1 Two-section configuration, I-D carte'iian coordinate configuration. 42
3.2 Section progression and boundary condition conventions for three-section I-D cartesian configuration. . . . . . . . . . . . . . . . . . . . . . . 45
3.3 2-0 cylindrical coordinate configuration (1 sectionjdimensi.)n). 46
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3.4 Air gap at interface (two-section configuration). . ............. " 47
3.5 Heat flow terms at, and surrounding, the interface must he set equal when interhoundary resistance specified.. . . . . . . . . . . . . 49
3.6 Configuration for resistallce vs. no resistance comparison. 51
3.7 No resistance case steady state temperature distribution. 52
3.8 Resistance case steady state temperature distribution. . 53
4.1 2-D cartesian geometry.. . 57
4.2 2-D cylindrical geometry. . 58
4.3 3-D rectangular coordinate node progression conventions. 59
4.4 2-D cylindrical coordinate configuration (1 section/dimension). . 61
4.5 Typical profile node arrangement when product solution requested. 61
4.6 Typical prcfile through 3-D cross section. . . . . . . . 62
4.7 Typical profile through 2-D cylindrical configuration. 63
4.8 Procedure for determining central section. '" . . . 63
4.9 illustration of a 2-D cartesian configuration with three sections pel dimension. 64
4.10 Procedure for determining the centrai node. .. . . . . 65
4.11 Profile node progression in 2-D cartesian configuration. 66
4.12 3-D rectangular slab (1 section per dimension). 67
4.13 2-D cylindrical configuration. . . . . . . . . . . 70
4.14 Full 2-D nodal arrangement (left) vs. product solution arrangement (right). 71
4.15 illustration of interaction solution algorithm. . . . . . . . . 74
4.16 Configuration for illustration of interaction solution method. 76
4.17 Node distribution wh en having selected an outer radius that is haIf the spec-Bied thickness (Equispaced no de progression). . . . . . . . . . . . . . . . .. 76
5.1 Phase diagram for hypothetical binary alloy. . 80
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( 5.2 Variation of enthalpy with temperature of hypothetical alloy. . 81
5.3 Cooling curve of hypothetical 50-50 binary alloy. . 82
5.4 Illustration of state property change at the liquidu3. 83
5.5 Approximation of cooling behaviour of alloy containing eutectic. 85
5.6 Typical temperature profile for a Neumann problem. 88
5.7 Typical temperature profile for a Stefan problem. . . 89
5.8 Typical profile generated from the Stefan simulation. 90
5.9 Plot showing dependence of solidification front position on time. 91
5.10 Flowchart showing how FASTP accurately predicts a profile node's tempera-ture change across a phase transformation (state) boundary. 92
5.11 Mass transfer bounda.ry layer extending from solid's surface. 94
5.12 Illustration of full vs. partial mixing during one iteration. . . 98
5.13 Configuration used by FASTP to model the mixing problem. . 99
8.1 Flowchart showing sequence of informa.tion supplied when connguring a prob-lem with FASTP [Data entry]. . 110
8.2 FASTP logo.. . . . . . . . . 110
8.3 FASTP Main Menu screen. . 112
8.4 Problem Definition screen. . 113
8.5 2-D Slab Configuration (1 section per dimension). 114
8.6 System Dimensions screen (1) - section lengths and number of nodes. 115
8.7 System Dimensions screen (2) - nodal point progression selection. 116
8.8 Sample node report from printer. 118
8.9 Boundary Conditions screen. . . . . 118
8.10 Multiple radiation sources illustration. 121
8.11 Material Properties screen. . . 124
,( 8.12 Starting Temperatures screen. 126 •
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8.13 Interboundary Resistances screen.
8.14 Heat generation screen ..
8.15 Stirring screen. . ....
8.16 Profile grid generated using the product solution technique ..
8.1" Option Menu screen.
8.18 Edit Menu screen ..
8.19 Summary screen. .
8.20 Database Menu screen.
8.21 Create a New Entry screen (with phase change) ..
8.22 Create a New Entry screen (without phase change) ..
8.23 View Database screen.
9.1 Display Option screen.
9.2 Sample temperature and energy content prin tout results.
9.3 Variable vs. Time display option screen.
9.4 Selection of principal axis nodes ...
9.5 Profile node selection screen. . . . . .
9.6 Profile arrangement (3-D cartesian configuration).
9.7 The 3-D profile selection screens. . ..... .
9.8 Temperature vs. Section display option screen.
9.9 Profile section selection screen ..
9.10 Profile selection windows. . ..
9.11 Isotherm display option screen.
10.1 Temperature vs. Time display option screen.
10.2 Temperature VE Section display option screen.
10.3 Tempeu ... ture Isotherm display option screen.
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{ 11.1 File Export screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
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f - . t !I 1
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List of Tables
2.1 Comparison of FASTP and exact solution for the semi-infinite problem de-scribed in Chapter 1 . . . . . . . . . . . 15
2.2 Boundary conditions handled by FASTP 20
4.1 Associated volumes and starting enthalpy values of participating principal axis nodes and profile node (l,l,l). .. . . . . . . . . . . . . . . . . . . . .. 68
4.2 Enthalpy change of the participa.ting principal axis nodes and new enthalpy values for same nodes at t = 60 seconds. .................... 69
4.3 Comparison of FASTP interaction solution method to product solutions generated by Heisler charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
5.1 Solid, liquid, and transforming state properties for aluminum . 89
5.2 Comparison of results generated by FASTP and an exact solution for a perfectly mixed solution in contact with water. ... . . . . . . . . . . . . . .. 97
8.1 Available geometries for model configurations. 112
8.2 Relevant parameters to boundary condition options 119
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Introduction
This thesis documents a software package that predicts both transient heat and mass
trander bchaviour in simple shapes and that can be run in the DOS environment. FASTP, as
it is known, is an acronym for Facility for the Analysis of Systems in Transport Phenomena.
It was bome out of the need to solve both heat and rnass transfer problems quickly, without
constantly rcferring back to charts, or worse yet, untamed exact solutions.
Througbout this software package's evolution, the philosophy has always been to de
velop an interactive, user friendly systcm capable of being run on a personal computer. To
the author'~ knowledge, this type of software is unique in this respect. Certainly, none as
versatile as FASTP has been found. It is the expressed hope that FASTP will not only serve
as an educational tool, but that it will also succeed in finding a niche in industry.
FASTP's algorithms are based on explicit fini te difference methods, which will be de
scribed in the five chapters making up the first part of the thesis. In Chapter 1, the partial
differcntial equations used to describe heat and mass transfer behaviour in cartesian, cylin
drical, and spherical coordinate systems are described. In Chapter 2, the explicit finite
difference equations derived from the partial differential equations are presented along with
sorne very basic terminology associated with the finite difference method used by FASTP.
Chapter 3 discusses heat and mass transfer behaviour in one-dimensional (I-D) compos
ite configurations; as modelled by FASTP. In Chapter 4, the same is discussed, but for
multidimellsiollal systems. Finally, FASTP's phase transformation and stirring features are
described in Chapter 5. Severa! examples are presented throughout Part lof the thesis, many
of which wcre taken from J.P. Holman's Heat TranJfer (5th cd.) [1]. Holman's textbook is
highly rcgarded and is used in many undergraduate courses on heat transfer. This textbook
is a source of independently verified solutions to the problems presented in this text.
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Part II of this thesis is an operator's guide to FASTP. Contained is a description of
the programs that make up version 2.0 of the package. Throughout both parts, examples
are used to illustrate the many options available to the user of this package .
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Part 1
The FASTP AlgorithrnR
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Chapter 1
The Basis for FASTP
In this chapter the control volume approach is used to derive the one-dimension al
forms of Fourier's heat conduction equation and Fick's diffusion equation. These are the
equations upon which FASTP's algorithms are ba.sed. In certain circumstances, which will be
described, the second order differential equations governing heat transfer and mass transfer
are equival.ent and can be solved using the same techniques. This chapter will also examinc
the use of mathematically exact solutions for solving transport phenomena problems. Whcrc
possible, exact solutions have been used to verify the FASTP algorithms. To illustrate how
a software package like FASTP can be more useful than exact solutions, a solution to a
relatively simple heat transfer problem is presented.
1.1 Control Volume Approach
The control volume shown in Figure 1.1 is used to derive Fourier's heat conduction
equation for one-dimensional. cartesian systems. Assume that heat couducts through the
control volume in the x-direction oruy. Fourier's first law of heat conduction states that the
...- rate of heat flow anywhere in the system is directly proportional to the local temperat ure
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gradient
where
// /y /; 1
• A qx - .. - - qx +6X
l/ 6Z
Figure 1.1: Control volume for 1-D cartesian coordinate systems.
4 i$ heat flow,
q = -kA ôe ôz
A is the cross-section al area,
~: is the temperature gradient.
(1.1)
The constant of proportiona.lity k is a material property ca.lled the thermal conductivity.
The negative sign is required beca.use heat flows in the direction of decreasing temperature.
Fourier's equation (Eq. 1.1) can also be expressed as a heat flux incorporating the heat flow
term and the area A
il' = -k ôe ôz
(1.2)
An energy balance is applied to the cOlltrol volume. The net rate of thermal energy
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flowing in minus the net rate of thermal energy flowing out plus the heat generated internally
is equal to the control volume's total accumulation of thermal energy.
qan - qoue + Qgen = Accurr.ûlation ( 1.3)
For 1-D heat conduction, in which heat flows in the left face at x and flows out at
x + ~x, the following substitutions can be made:
1. Heat conducted in
.. kAÔE>1 qin = qz = - -ôx z
2. Heat conducted out
Using Taylor series expansion, ~y.vression (1.5) becomes
Note that the area A has been replaced by the product of the terms 6.y and Az.
3. Heat genera.tion term
where qgen = energy generated per unit volume.
4. Accumula.tion term
where
BE> Accumulation = pep at (Ax ~y Az)
Cp = heat capacity,
p = density.
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( 1.4)
(1.5)
(1.6)
(1.7)
( 1.8)
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The accumulation expression incorporates a differential term in which the temperature
e changes with time t. Substitution of equations (1.4), (1.6), (1.7) and (1.8) gives the
1-0 cartesian form of Fourier's heat conduction equation
( 1.9)
This is a second order partial differential equation showing the dual dependence of
temperature on position and time. The thermal diffusivity Dl incorporates the material's
thermal conductivity, density, and heat capacity.
k Q= -
pc, (1.10)
Equation 1.9 is perhaps the simplest form of Fourier's second law; it can be used as the
basis for solving 1-0 heat conduction problems in cartesian systems. SimHar equations have
been derived for 1-0 heat conduction in cylindrical and spherical systems. In both cases,
curvature effects are taken inta account with the radial term r.
For cylindrical systems
For spherical systems
!~ (r aS) + qgen = !. as rar ar k Qat
(1.11)
(1.12)
The discussion is restricted to the I-D forms of Fourier's second law because these are
the forms upon which FASTP's algorithms are based. Latet chapters will describe how the
1-0 solutions gent'rated by FASTP are used to generate approximate solutions to 2-D and
3-D problems.
The overlying assumption made in deriving these equations is that conduction is the
only mode of thermal transport within the body. The equations are especially suited to
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systems consisting of solids or motioulcss fluids. lu the derivation, constant propcrtics were
assumed. In other words k, p, and Cp were assumed to remaill constant rcgardlcss of tem
perature, direction of flow, or position within the control volume.
1.2 Mass Transfer
An equivalent set of arguments can be prcscnted for mass transfer wbere the diffusion
of one species A through another B results from a concentration gradicnt. The maliS transfcr
equivalent of !ourier's first law of heat conduction (Eq. 1.1) is Fick's first law of lllass
diffusion. . 8C
NA = -DABA-8x
(1.13)
The relation states that the mass flow rate NAis directIy proportiollal to thc concen
tratioI' gradient 8Cj8x. The constant of proportionality DAB is called the mass diffusion
coefficient for material A diffusing through B. Once again, the negative sign is rcquir<.d be
cause mass diffuses in the direction of dccreasing concentration. Note that the ar('as A and
the subscript A bear no relation.
In reality, it is not the solute's concentration that drives the diffusion process, but more
correctly, it's activity. The activity of a species is defilled as the ratio of the partial pressure
exerted by the impure species in the solvent to the partial pressure that could be excrtcd if
the species was pure. The activity of a species A is rclated to the mole fraction of spccics A
in B by the following relation
(1.14)
where 'i'A is the activity coefficient and X A is the mole fraction of spccies A in B. Whcn the
A-A, A-B, and B-B attractive forces are equal, the activity coefficient is cqual to unit y and
the activity of species A is equal to the mole fraction (i.e. aA = X A). This is rdcrrcd to as
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( ideal or Raoultian behaviour (Fig. 1.2). Deviations from ideality occur when the attractive
forces betweell A-A, A-B, and B-B differ: non-ideal or Henrian behaviour is then said to be
observed. Over !>hort mole fraction ranges the activity coefficient of a Henrian solution can
be approx.imated by cl constant value. The activity of species A in soivent Bis then said to
be proportional to its mole fraction. Thus, in both the ideal and non-ideal cases the mole
fraction - and hence concentration of the species - is a measure of the activity. Fick's
laws will apply when the activity coefficient fA is constant throughout the system. Undcr
these circumstances the concentration gradients can be used to predict the rates of mass
diffusion. The advantage to this simplification is that the concentration of a species if! more
easily measured than its activity.
Il 1600'C
os
OK-______________ ~' ______________________ ~I,
05
A mole fraction B
XA ---.
Figure 1.2: Raoultian & Henrian behaviour of two-component mixtures.
The differential equations governing mass diffusion in solids are also derived using the
control volume approach. The 1-0 versions of Fick's second law are listed for cartesian,
cylindrical, and spherical systems.
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t For cartesian systems
[ô2e]. ae
DAB ôx2 + "'flcn = at (1.15)
For cylindrical systems
[1 ô ( ae)]. ae
DAB ;: ôr r Br + '\Iflen = ôt (1.16)
For spherical systems
(1.17)
The similarities between Fick 's and Fourier's second laws are clearly evident. The
mass diffusion coefficient D AB and the thermal diffusivity Q are analogous; as are the con
centration C and teruperature 0, and the mass production Vgen and hcat gcneration qgen
terms.
1.3 Exact Solutions
A great dt·al of wor'k has gone into the derivation and collection of mathematically
exact solutions to heat and mass transfer problems. Carslaw & J aeger [2] have compiled one
such reference. Several solutions are presented for single and multidimensional heat transfer
problems for cylinders, rectangular slabs, and spheres. The book was intended to be used as a
reference, however, the techniques required to arrive at the solutions are involved and require
a strong mathematical backgrounrl. Each solution relates specifically to the applicd boundary
conditions. Orten, computers are required to solve the lengthy summations that are inyolved:
the solutions become even more complex when multidimensional systems are considcrcd.
As a consequence, exact solutions have secn lirnited application. To illustrate, considcr
the problcm of determining the temperature distribution 8 (x, t) in a scmi-infinite plate.
The plate possesses constant k, P, and Cp values and is initially at a uni!orm tempcrature
8 0 , At time t = 0 the end of the bar is exposed to a convective cIlvironm('ut wit,h a
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( constant heat transfer coefficient h and some ambient tdmperature 8 0mb below that of the
bar (i.e. 9 0mb < 9 0 ), The Fourier equation that governs heat conduction in this system is
Equation 1.9. The solution to this problem was worked out by Schneider [3] and is given as
~9_-_e--::· 0::- = 1- erf X _ [exp(_hX + h2o:t)] [1 _ erf(X + _h~_a_t)] 9 0mb - 9 0 k k2 k
(1.18)
where x = xj(2.;at).
The solution satisfies the boundary condition
ÔS] hA(9omb - 8)~=o = -kA ÔX ~=O
(1.19)
The techniques used to obtain the solution are not relevant to this report and are not
presented. However, it is important to note that the solution is specifie to a I-D semi-infinite
heat conduction problem whose end is exposed to a convective environment. The solution
would have to be rederived if, for example, the end was exposed to a radiative environment,
a constant heat flux, or constant surface temperature. Entirely different solutions must
be derived for bars of finite length. Other methods must be used if the problem involves
internai heat generation. The number of solutions required to handle aU of the possible
combinations is cClnsiderable. In many cases, exact solutions simply do not exist. However,
the eJdsting solutions do serve a very useful purpose and one that will be applied in this
report. Exact solutions provide a means of verifying the solutions that are generated by
FASTP's algorithms. This particular exact solution has been solved for a refradory brick
material and given boundary conditions for comparison to FASTP's solution. The results
are given in the next chapter.
11
Chapter 2
FASTP Conventions Governing Finite Difference Equations
In Chapter 1 an exact solution to a simple heat transfer problem was given and it was
explained that although exact solutions to many heat transfer problems exist, they are of
limited practical value because they are so complicated. The traJitional methods used to
solve heat or mass transfer problems have remained primarily an academic eJCercise. Fortu
nately, numerical methods for solving transport phenomena problems have been developed,
which are well suited for solution with computers.
FASTP has been built around such a method. The program's algorithms are based
on explicit finite difference equations derived from Fourier's and Fick's second laws. With
FASTP, heat or mass transfer behaviour can be predicted in simple shapes that have been
combined in a manner that approximates a real system. In this chapter these building blocks
are described and the finite difference equations governing their behaviour are presented. The
concept of the node, nodal points, boundary conditions, material properties, and the neees
sary time parameters are introduced. As well, the mathematical stability of the equations
is discussed. Three examples are presented in which a comparison is made between results
''"'' generated by FASTP and Heisler chart [4] solutions. Finally, an advanced {eature that allows
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the user to model a system with dynamically changing boundary conditions is described.
2.1 Nodes &; Nodal Points
In the previous chapter the illustration of a semi-infinite rectangular shaped bar with
one end exposed to a convective environment was presented. An exact solution to this
pl'oblem was also given. In this chapter the same example is used to introduce some basic
concepts associated with FASTP's algorithms. In a FASTP ana.lysis a bar or section of finite
length would be discretized into subsections called nodes (Fig. 2.1). These nodes possess both
volume and mass. Located at the geometric center of each node is the nodal point. Nodal
points are positions in the bar where FASTP will iteratively predid the bar's temperature
or concentration with time. Half·nodes are located at either end of the bar section. They
are called half·nodes because their nodal points are located at the section '8 boundary rather
than the node's center.
When a configuration is set up like the one shown, each nodal point in the section is
assigned a starting temperature. Boundary conditions are also assigned to the haJf-nodes. In
this configuration, node 1 will be exposed to convective conditions and node 7 will be assigned
a zero ft1J.X or adiabatic boundary condition. A bar of fini te length with the given setup
can model a semi-infinite problem provided the temperature at node 7 does not fall below
the starting temperature. The constraint is similar to that for the error function solution
illustrated in Chapter t when it is applied to a system of finite length. In this example
FASTP will request that a heat transfer coefficient h and an ambient temperature 9amb be
input for Dode 1. In this example, the same ambient value is assigned to node 7. A simple
calculation will revea.l the enthalpy that each node possesses relative to its surroundings.
(2.1)
13
where subscript i corresponds to the node number and mis the node's mass.
langth ,. L
"4
node (shadedl
ù< 1
1"4 ~ 1
T 1
x'''''< '><1 1 ~,~ 1 1
1 i • • • :x<. .-:<,<: • • • 1
Il 2 3~ n-1 n
1
1.. Unit helght 1 Unit depth 1
NOdes are 8(JJally spaced 1
nodal potnt half-node
t.x • Iength (L)
(. of nodes) - 1
Figure 2.1: Discretized bar with seven nodes.
FASTP is now used to solve the semi-infinite problem described in Chapter 1 when
the following conditions are applied
Material properties
k = 3.0 W/m-oC, p = 2000 kg/rn3, Cp = 1200 J / kg-OC
Boundary conditions (exposed surface)
h = 100 W/m2_oC, eomb = 20 oC
Starting conditions eo = 500 oC
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The temperature is to be predicted at a point five centimetres below the surface of
the exposed brick. To ensure that the temperature at the adiabatic boundary does not faU
below the starting temperature, the length of the configuration is made to be one metre.
This length will be ample for simulating the semi-infinite problem given the time frame and
heat 10ss characteristics of the material.
FASTP simulated the problem for a period of 1 1/3 hours, during which time sever al
temperature profiles were generated. Using the exact solution in Chapter 1 (Eq.1.18), tem
perature values were calculated for comparison. The results are given in Table 2.1 and show
that the two solutions compare favourably.
Table 2.1: Comparison of FASTP and exact solution for the semi-infinite problem described in Chapter 1
1 Time (s) 1 Exact Solution 1 FASTP 1
0 500.0 500.0
60 500.0 499.3
120 499.7 497.5
240 495.4 491.4 480 477.9 473.1
600 463.5 462.8
900 435.5 436.7 1200 409.9 412.6
2400 342.6 341.8
4800 268.9 269.0
2.2 FASTP's Governing Finite Difference Equations
Heat conduction through the section is simulated by the incremental transfer of thermal
energy from one node to another. The interaction between the nodal points is governed by
an algorithm that uses explicit finite diffetence equations, which are derived from Fourier's
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-
-
or Fick's second laws. Figure 2.2 shows an arbitrary Dode i bounded by two other nodes; i-l
and ifl. In a 1-D cartesian configuration, the cross-sectional area. of the boundary hetwecn
two nodes is unit y because FASTP sets the depth and height of the configuration to ouc.
In curvilinear systems the eross-sectional area associated with the node boundaries depends
on the radius.
e. 1
___ --.:1+1
•..... .... ......... .•. • (i - 1) (i + 1) • •
Figure 2.2: illustration of a three node configuration (node i is internaI).
A temperature profile across the three nodes is shown for some time t. The nodal points
are separated by an increwentallength Llx. Because the flow of heat is proportional to the
temperature gradient (Eq. 1.1), an energy balance on node i will show that the temperature
at nodal point i will rise, sinee the average temperature gradient between nodal point i-l and
i is greater than the one between nodaI point i and i+lj i.e. the rate of heat conducted to
node i is greater than the rate conducted away. The change in temperature at nodal point i
over a time bat is illustrated in the figure's inset. The one-dimensionaI energy balance for
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internal llodc i is sltOWll below.
r "*' oondut1Id !rom nodIl·' ] [ Hel, oonducIId !rom nodIl ] [ L to nodt IInAIiIIII - ID nodt 1+ 1 ln Al II1II + HMI gIIWIkrlat nodeIInA' .... ] [
Amcurt of la1t aunUalld ln ] - nodt I_AI lImI
In FASTP's analysis the temperature distribution between the nodal points is assumed
to be linear (Fig. 2.3). For short space increments this approximation is reasonable because
the slope at the midpoint between two nodal points - the average temperature gradient -
is equal to the slope of a straight line drawn between the temperature values of both nodal
points.
The same argument can be drawn for the change in temperature at any node over a
time increment ~t. The differential terms in Fourier's and Fick's second law expressions are
effectively replaced by ~ terms. The following substitutions are made in the energy balance
to arrive at a finite difference expression applicable to aU internal nodes:
1. Heat conducted from nodal point i-l to i
2. Heat conducted from nodal poÏllt i to i+1
3. Heat generation term
Q _ q'" tJ: gen - gen Yi
17
(2.2)
(2.3)
(2.4)
--------------------~
t.x
•......... . .....................• (1 - 1) (i + 1)
• • i'
Figure 2.3: FASTP approximation of temperature distribution across three nodes .
....
18
( 4. Accumulation term
(2.5)
With sOllle manipulation the following finite difference expression is obtained for any
internal node i
S~ = 8, + kil.+! A'liH ~t (Si+! _ ai) _ k,-lli Ai-Iii ~t (ai _ 8i-l) + qi"gen ~t (2.6) 1 pep V. 6.x pc" Yï 6.x pc"
where Ai-III = x-section al area between no de i-1 and i,
AiitH = x-section al area. between no de i a.nd i+1,
V. = volume of node i,
q;~n = heat generated (per unit volume) ,
8, = temperature at node i at time t,
8~ = temperature at node i at time t + ~t.
The only unknown in the expression is e~. Thus, the new temperature at i can he
predicted explicitly using the existing values at i-1, i, and i+1. At the start of the simulation,
FASTP uses the values that were initially assigned to calculate valUf'S for a time to + 6.t.
When the new values have been generated the time is incremented by 6.t and the values for
the next time are predicted. This iterative process will continue until the total simulation
time, a user specified parameter, has elapsed.
2.3 Half-Node Equations & Boundary Conditions
Since the half-nodes are not bound on both sides by other nodes of the same section,
the finite difference cquation developed for the internal nodes does not apply. Inste?.d, new
expressions must be dcveloped for the surface nodes that take into account the possibility
of exposure to any number of boundary conditions, as well as contact with half-nodes from
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-other sections. This subject is dealt with in Chapter 3. The boundary conditions handled
by FASTP are listed in Table 2.2.
Table 2.2: Boundary conditions handled by FASTP
1 Heat Transfer 1 Mass Transier
Fixed Surface Temperature Fixed Surface Concentration
Convection Convection
Radiation --Zero Flux Zero Flux
Constant Surface Flux Constant Surface Flux
(Combinations are possible)
Figure 2.4 is an illustration of nodal point 1 in a 1-0 cartesian configuration. The
half-node is bounded on one side by node 2 and is open to the environment on the other.
The area of the exposed surface is unit Y because the height and depth of the node are set to
one for 1-D cartesian systems.
One equation has been developed to handle all of the possible boundary conditions.
The energy balance on node 1 is given as
! Heat added tOJ
1
SlI'face node 1 il ~t seconds
L [
Heat ConâJcted tO] node 2 from node 1
in Dt seconds li Heat generated 8
J + surface node 1
1 in 6t seconds L
• [
AmoU1t ilf haat acct.mU8ted 1
in node 1 in 6t seconds
The following substitutions are made into the balance to a.rrive at the finite difference
expression for surface node 1:
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:(
:[ ""
1. Heat conductcd from nodal point 1\ to nodal point 2 (loss term)
2.(i) Heat gain term (convection)
qconu = hA(0amb - 0 1)
2.(ii) Hcat gain term (radiation sources (max. 5))
5
qrad = E [fi q Fl .... j A(0' - e1>] j=l
S. Heat generation term
Q '11 A gen = qgen 1
4. Accumulation term
where
(0' - 0 1) pep V ...;:.....:1"-7----'-b.t
kl l2 = thermal conductivity existing between nodal point 1 and
nodal point 2,
h = heat bansfer coefficient,
0amb = ambient temperature,
fi = emissivity of surface no de 1,
(J = Stefan-Boltzmann constant,
Fl .... J = vicw factor for surface node 1 viewing radiation source j,
0, = tempcrature of radiating surface j,
0 1 = tempcrature of nodal point 1 at time t,
0~ = t~mperature of nodal point 1 at time t = t + b.t,
q;en = heat generatcd pel unit surface area.
21
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
-
With some manipulation an expression similar to Equation 2.6 is obtained for surface
node 1
Some additional simplifications are required to reduce the expression to the form ap
peanng as FORTRAN code in the program. However, al! of the parameters on the right
of the expression are known quantities. The new temperature e: at time t + 6.t cau be"
calculated using the existing temperatures at nodal points 1 and 2 from time t.
Similarly, for surface no de n
Once again, some additional simplifications are required to redllce the equation to the
form appeanng as FORTRAN code in the program.
At the start of a FASTP session aIl of the boundary condition variables in the exprcssion
are set to zero. When assigning the boundary conditions, the program will only request valucs
for the parameters that apply to the selected option. The uncalled parameters will remain
zero and will not contribute to the expression throughout the simulation. For example, if
the constant surface flux option is requested, a value for 4;en wOlùd have to be supplicd.
The constant flux term is treated like a heat generation term applied to the surface. If
no other boundary conditions are requested in combination with this option, both the hcat
transfer coefficient h and the emissivity € would remain zero and would play no part in the
calculations for that bOllndary.
Note that to derive the equations, heat was assumed to flow through the bar from
surface node 1 to surface node n. This does not mean that the algorithm will not work if
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the flow of heat is in the opposite direction. The resulting negative signs would disappear
in the algebra.
When a boundary is exposed to a fixed surface temperature the nodal point at the
surface is he Id fixed at the specified value throughout the simulation. For example, if a fixed
temperature of 500 oC is specified at surface node 1, then nodal point 1 will remain fixed at
that value for the duration of the simulation. To maintain an energy balance, nodal point 2
will either gain or lose thermal energy from node 1 depending on the temperature differential
existing between the two nodal points. This boutldary condition simulates the idealized
situation where the applied heat flux maintains the surface at a constant temperature.
2.4 Curvilinear Coordinate Configurations
One-dimensional cylindrical and spherical node configurations make use of the same
equations developed for cartesian systems. The only important difference is that the cross
section al area at the boundaries between two nodes is dependent on the radius. Consequently,
each !i.od~: in a curvilinear configuration has an associated inner and outer boundary area.
Figure 2.5 illustrates a progression of nodes in a 1-D cylindrical system. When configuring
a problem with either of the curvilinear coordinate systems, an addition al parameter, the
outer radius, must be specified. The outer radius, combined with the section's thickness,
set the dimensions of the system. FASTP uses the convention whereby nodes progress from
1 to n inwards. The boundary conditions are applied uniformly around the inner and outer
radial surfaces of the cylinder.
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• 2
.. - Flow of heat
eamb
Figure 2.4: Nodal point 1 of 1-0 cartesian coordinate configuration.
\ ." n-1 ... 3 2 1 1 x·:············ · ''- .... '1.
\
.,'
\
/ /
"~ Outer Radus
Figure 2.5: 1-D cylindrical coordinate configura.tion.
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2.5 Material Properties
In a heat transfer problem, three material properties are assigned to a section; the
thermal conductivity k, the heat capacity Cp, and the density p. The assigned values are
discrete and cannot be made to vary with temperature unIess the phase change option,
described in Chapter 5, is selecte<!.
The units specified for the material properties must be consistent and ultimately deter
mine the dimensions of the system being configured. For example, ifthe thermal conductivity
is expressed in W/m-oC and the density is expressed in kg/m3 , the heat capacity must be
expressed in J/kg. In this example the value specified for the section's length would have to
be input in meters.
In a mass transfer problem, only the mass diffusion coefficient D AB for material A
diffusing in B must be supplied. Recall from Chapter 1 that the thermal diffllsivity Q
and mas .. diffusion coefficient D AB are analogous. Consequently, the same finite difference
equations can be used to model diffusion problems. In Equations (2.6), (2.12), and (2.13)
the mass diffusion coefficieut is assigned to the thermal conductivity parameter and the heat
capacity and density are set to one.
2.6 Manipulation of Variables by FASTP
In the discussion thus far, many variables and parameters have been introduced. Each
node possesses specifie dimensions. Each nodal point must be assigned a set of material
propertie:; and initial starting conditions. Their positiou relative to one another must be
stored. Areas, volumes, radii, boundary conditions, etc ... must aU be easily accessible 50
that the finite difference equations can operate on the nodal points in sequential order. The
25
task becomes even more complicated when configurations made up of multiple sections and
multiple dimensions are considered (Chapters 3 and 4).
In the program, aIl of these variables and parameters are stored in arrays. Values
that are specifie to a nodal point, such as temperature or concentration, are stored in tluee
dimension al arrays that locate the nodal point in a specifie section and dimension of tbe
configuration. Values that apply to an entire section, such as the material properties, are
F, ~ored in two-dimensional arrays. The subprogram performing the iterative calculations 011
the nodes is written so that the m",thema.tical operations are performed within a nested loop.
The loop sequentially works through aIl of the nodes in a section. The same calculations are
then performed on th~ nodes in any adjacent sections. Once every section in the dimension
has been addressed, the next dimension - if another exists - is treated. When new values
have been determined for aIl of the nodes in the configuration the time is incremented by ~t
and the nested calculations are repeated. Many new COnfl!pts have been introduced here,
but will be clarified over the next two chapters.
2 ;-. , Stability Criteria for FASTP Solutions
Because FASTP uses an explicit finite difference approach, certain stability require
ments must be met. The stahility of the solution is ensured through the user's selection of
an appropriate iteration time increment. A series of calculations are performed on the nodes
prior to the start of the simulation to determine the maximum iteration time increment al
lowed, ahove which the solution will he unstahle. Before presenting the stability equations,
the distinction will he made hetween a stable and an unstable solution.
If a stable solution is being generated, each node in the system will tend towards
some steady state value. The drive towards steady state is greatest at the beginning of
the simulation when the system is least stable in relation to its surroundings. Figure 2.6
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shows the characteristic distribution across an aluminum bar a~ different times. The ends
of the bar are exposed to two different convective environments. One end is hot (200°C)
and the other is exposed to éI.Il ambient temperature, which is the same as the bar's uniform
starting temperature (20°C). The bar changes temperature rapidly at first, but then slows
as the system reaches equilibrium. Each nodal point in a stable solution displays asymptotic
behaviour. An unstable solution is characterized by fiuctuating nodal point valU('~ that never
reach a steady state.
500
45.0
400
35.0 ! ::1 'i6 Qi ~ 30.0
~
250
200
000 0.25 0.50 075 1 0 Alumlnum bar· 1 m length
Figure 2.6: illustration of a stable solution.
Considering, once again, the three-node progression made up of nodes i, i-l, and i+l,
a stable solution will be obtained if, during each iteration, the new enthalpy at nodal point i
does not exceed the weighted average of the previous enthalpy values at i-l and i+1. In
other words, the nodal point is limited by the amount it can cha.nge to its projected steady
state value of the previous iteration. Physically, the restriction ensures that the second
law of thermodynamics is complied with. The average is weighted because the nodes may
27
-possess different volumes; as in the case of unevenly distributed nodes or curvilinear systems.
Mathematically, the constra.int is such that
aAt <! (Ax)2 - 2 (2.14 )
The node spacing Âx is set early when configuring the problem. Consequently, it is the
iteration time At that is chosen to ensure stability. In the algorithm the weighted a.verage
is taken into account with the following
At = pc" V. (Ax, * Ax.-d . k(A.Ax. + A'-l ~X'-l)
(2.15)
Prior to the simulation this calculation is performed on every internal node in the
configuration and a similar calculation is performed on the half-nodes. Because the drive
towards steady state is greatest at the start, the calculation will pro duce the lowest Ât values
expected during the simulation. The lowest At value calculated from all of the nodes rep
resents the maximum iteration time that can be specified before an unstable solution might
result. FASTP's stability criteria algorithm uses the information from the configuration and
assesses all of the situations that might arise during the ana.lysis. The stability value that
is ultimately used is based on the worst case scenario. Whether or not this scenario do es in
tact arise is not known until the simulation is run. The discussion on solution stability is
continued in Chapter 3, where the stability at a.n interface between two sections is considered
and in Chapter 5, where it will be shown how the phase of a material can affect the stability
of a solution.
The stability calculation performed for the half-nodes at a surface is a modified version
of Equation 2.15, but follows the same reasoning. To illustrate, consider a ha.lf-nllde that is
exposed to a convection boundary condition. The finite difference equation for this no de is
(2.16)
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Because, by definition, the heat transfer coefficient h is defined as k/ tl.x, it can be
easily incorporated into the stability equation
At = pc" V tl.x hAAx+kA
(2.17)
By applying a similar argument for radiation boundary conditions the stability equation
for surface nodes becomes
A pc"VAx ~t = -------'-==---:::-:--~-~-
h A Ax + f q Fa-13 e~az Ax + k A (2.18)
If a radiation boundary condition is specified, the user must supply a value for emaz
before the stability calculations can be completed: 9 maz is the maximum surface temperature
likely to be reached at the boundary during the simulation.
2.8 Comparison to Heisler Chart Solutions
Three examples are now presented in support of the algorithm presented so far. The
examples show how the program can predict transient heat tra.nsfer behaviour in al! three
of the coordinate systems discussed. In the first example, heat transfer in an infinite plate
- two sides of which are exposed to a convective environment - is examined. The exact
solution to this problem is similar to the one given in Chapter 1 (Eq. 1.18). Fortunately,
the results from the exact solution have been expressed in chart form. The collection of
charts are referred to as the Heisler charts. The charts, which are appended, are expressed
in terms of œmensionless numbers. They are used to predict temperatures and heat loss in
systems that are exposed to a convective enviroc.ment. Heisler charts have also been derived
for cylindrical and spherica1 configurations and can be used to predict the temperature of a
point located anywhere within a body at any given time t. The one restriction is that the
29
Fourier modulus must be greater than 0.2, where
where
or Fo=
S2
o = thermal diffusivity,
T = time,
s = characteristic length.
2.8.1 Example 2-1: Infinite plate
(2.19)
An infinite plate of aluminum 5.0 cm thick and initially at ai = 200 oC, is exposed
to a convective environment where h = 5e5 W/m2.OC and 9 0mb =70 oC. Calculate the
temperature at a depth of 1.25 cm /rom one of the faces one minute after the plate has been
exposed to the environment. How much energy has been removed per umt area from the plate
in this time ~
This example was taken directly from Holman. If the flow is &long the x-axis only,
the plate is assumed to extend infinitely &long the y and z.a.xes. The reported answers were
147.7 oC and 6.48 * 106 J/m2 respectively. The properties and conditions used in both the
Heisler chart and FASTP a.nalyses are listed as follows:
k = 215 W/m-K, p = 2700 kg/m3,
Total simulation time = 1 min = 60 sec,
9 i = 200 oC (starting temperature),
9 0mb = 70 oC (ambient temperature),
h = 525 W/m2_oC.
cp = 900 J/kg
The results from a FASTP analysis can be presented in many ways. In this example
30
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the Temperature vs. Section plot option was used. This option allows the user to generate
snapshots of the temperature profile across a section at specified times. Figure 2.7 shows the
temperature profiles predicted by FASTP at ten second intervals. Note that as the simulation
proceeds, the drop in temperature for each su'.:cessive time interval decreases. This is to
be expected, as all of the nodes in the configuration are moving towards equilibrium or
steady state relative to the environment. Although this may not be a particularly impressive
example, due to the high thermal conductivity of the material, FASTP has correctly predicted
the temperature at the 1.25 cm depth. Using the Temperature print option a temperature
of 147.5 oC was predicted. Cooling of an alum.num slab
200 o Osee • • .
190 -..... 190 -.....
9 170 -"L
-GI 160 ~ 500
"";1
ca 150
1 )40 1-
~60.0 --"L
-.. ~
130 100 0
120 ~ -) 10 .. -100 _J • •
0.00 0.01 0.02 0.03 0.04 o.os
Length (m)
Figure 2.7: Tempera.ture distribution actoss a.luminum bar at ten second interva.ls.
The second part of the problem was to determine the amount of energy removed per
unit area in this space of time. FASTP keeps track of the change in enthalpy with time for
the system as a whole and for each individuaI node in the system. At t = 60 seconds an
enthalpy change of 6.30 *106 Jlm2 was recorded. The small discrepancy between FASTP 's
31
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and the Heisler chart prediction can he attributed to the inaccuracies that come with trying
to read the Heisler charts. Please refer to charts (Al), (A4), and (A7) in Appendix A.
2.8.2 Example 2-2: Semi-inflnite cylinder
A long aluminum cylinder 5.0 cm in diameter and initially at 200 oC is suddenly
exposed to a convective environment where h == 525 W/m2 .OC and Samb = 70 oC . Calculate
the temperature at a radius of 1.25 cm and the heat loss per unit length one minute after the
cylinder is exposed to the environ ment.
Since the cylinder is long, the heat los ses from the ends are assumed to he negligihle.
The Heisler chart method gives a solution of 118.4 oC for a radius of 1.25 cm after one minute
of exposure. The predicted heat lost per unit length Îs 4.032 * 105J/m.
With FASTP, a 1·D cylindrical configuration was set up using eleven evenly spaced
nodes. The convention used places node 1 at the outer radius, while node 11 is placed on
the cylinder's central axis. Node 6 is located 1.25 cm below the surface. In a 1·D cylindrical
configuration the cylinder length is set to one. Zero heat loss occurs at the cylinder's fiat
ends, so the configuration models a cylinder that extends to ± infinity. Figure 2.8 shows the
change in the node 6 temperature with time. Note that the asymptotic solution is correctly
predicted. Using the Temperature fi energy content print option, the temperature at no de 6
after one minute was 119.7 oC. The enthalpy change after one minute was 4.056 * 105 J / m.
Once again, FASTP 's predictions and the Heisler chart predictions are comparable. Plcase
refer to Heisler charts (A2), (A5) and (A8) in Appendix A.
32
( 200
175
150
-.0 - 125 (1)
~ '"tU
100 ~
f .... 15
50
25
a
0 100
Cooling of an aluminum cylinder
-.00 () II () - -00
1 .25 cm below I\rlace
200 300 400
lime (seconds)
500
Figure 2.8: Cooling of a semi·infinite cylinder at 1.25 cm radius.
2.8.3 Example 2-3: Sphere
A 5.0 cm diameter sphere, made of a refractory mate rial and initially at a u.niform
temperature of 1~00 oC, i3 ezpo,ed to a convective environment where h = 5~5 W/m2.oC
and 9 0mb = 20 oC. In this illustration, the sphere is allowed to cool for five minutes.
FASTP was ,et up to generate temperature profiles along the radius at one minute intervals.
Uaing the Heisler chart8 the temperature profile along the radius at T = 2~0 seconds was
calculated and compared. The propertiu for the refractory material are as followa:
k = 3.0 W/m.K, p = 2000 kg/m3 , Cp = 1200 J / kg.
This example illustrates a. disadvantage associated with the use of Heisler charts. The
( properties used in this example correspond to those of a refractory material. Restrictions
33
-
arise because tbe value of Q is somewhat smalier for this matcrial tban for a matcrial snch
as aluminum. Because of the Fourier modulus restriction (Eq. 2.19), the Hcisler cbarts
can only be used to predict temperatures after 100 seconds eXIlosure. FASTP, on the otber
band, has no such restriction. The temperature profiles along the sphcre's radius a.re sbown
in Figure 2.9. The Heisler chart results for T = 240 seconds have been superimposed. Again,
the results are comparable. Please refer to Heisler charts (A3) and (A6) in Appendix A.
Cooling of a ceramic sphere 1400
o Osee
1200 • He'a'" çhal1 IOMion Q FASTP IOluIlon
1000 -~ Q) 800 ~ -;
1 1500 120.0
.... 400
180.0
200 240.0
a
0.000 0.005 0.010 0.015 0.020 0.025
Distance along ractus (ml
Figure 2.9: Temperature distribution along the sphere radius.
A result that is fully expected, and that FASTP has predicted, is the asymptotic be
haviour of the nodes. 'l'he change in temperature with time is greatcst at the start of the
simulation because this is when the system is least stable with respect to its new environmcnt.
The rate of temperature decrease progressively slows as the sphcre cstablishcs equilibril1m
with the ambient conditions. As well, take note that FASTP has corrcctly predicted symme
try about no de 7, which is located at the sphere's center. Although only half the diamcter is
shown, the temperature gradient at node 7 is zero. The net heat flux at no de 7 is thus also
34
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f
zero. This result is to be expected because heat is flowing outwards equally in all directions.
2.9 Dynamically Changing Boundary Conditions
To complete this chapter a feature, known as the batch mode feature, which allows
FASTP to predict transient behaviour in systems with dynamically changing boundary con
ditions, is described. This feature allows the user to program FASTP to change the boundary
conditions at a designated time during the simulation, or when the surface node has reached
a specified temperature or concentration.
When using this feature, any surface node can be exposed to a sequence of boundary
conditions. When entering the parameters, the boundary conditions are designated as eitherj
(i) time, (ii) temperature, or (iü) time-temperature dependent. A time dependent entry is
one that will come into effect at the specified time during the simulation and will remain in
effect until the next entry is implemented or the simulation is discontinued. A temperature
dependent entry is one that will be set in place if specified conditions exist at the surface.
Typically, severa! temperature dependent entries - covering a range of surface conditions
- will be entered. For example, consider a situation where a hot metal surface is exposed
to the convective environment of a water spray. Because the heat transfer coefficient can
vary considerably with the surface temperature the use of just one value for h throughout
the simulation may be inac,curate. Using the batch mode feature a range of coefficients can
be enteredj each correspondhlg to a difIerent surface temperature. While the simulation is
running the algorithm will decide which of the entries is the most suit able for the existing
surface temperature. With each iteration, the algorithm will check to see if the surface
temperature has changed sufliciently to ·Narrant the implementation of a new temperature
dependent entry.
When using the batch mode feature in an analysis, as manyas twenty different time
35
..
dependent boundary conditions can be assigned to any given surface. For any one timc,
ten temperatuIe dependent entries can be specified. Note that any time dependent eutry
will always take precedence over the temperature dependent entries. In other words, when a
designated time has been reached during the simulation the existing boundary conditions arc
rejected t,,~ the new conditions regardless of the surface conditions. Figure 2.10 illustrates
the concept of time and temperature dependent entries. When setting up the model, the
time dependent boundary conditions are entered in ascending chronologica.l order and the
temperature dependent boundary condit.ions, if any, are entered in descending temperature
order.
A special form of the temperature dependent entry is the time-temperature dependent
boundary condition. In the water spray on hot metal illustration, assume that the water
spray is not applied until ten seconds alter the analysis has begun. This implies that the
range of temperature dependent heat transfer coefficient values are not applied until ten
seconds into the analysis. When entering the parameters, the fiIst entry - i.e. highest
temperature entry - is designated as a time, but also temperature dependent entry. AU
subsequent entries in that range are simply designated as temperature dependent. Entry of
the boundary conditions when using the batch mode feature id covered in the second half of
this thesis .
36
(
(
t .0.0 (start)
(end)
i ,
starting conditions
tlme dependent entry (1)
tlme dependent entry (2)
(highest) time & temperature dependent entry
, (Iowest)
temperature dependant entry (2) temperature dependant entry (3)
temperature c1ependent entry (10)
time dependent entry (20)
Figure 2.10: Schematic showing relationship between batch mode feature's time, temperature and time-temperature entries.
2.9.1 Example 2-4: Beat flow through a rectangular slab
In this example, the batch mode feature is used to model the transient behaviour of
a l·D slab configuration exposed to a variety of boundary conditions. The slab possesses
a thickness of 4.5 cm and is initially at 20 oC. Ten equally spaced nodes are used in the
analysis, which lasts a total of six minutes. Throughout the analysis, surface node 1 is
exposed to the following conditions:
9 0mb = 20 oC, h = 50 Wjm?'.oC, No radiation
37
"
.'
The conditions at no de 10 are to be varied every two minutes as follows:
Parameter First 2 minutes 2 to 4 minutes Last 2 minutes
h (W/m2 _0 C) 100 5000 100
E 0.5 0.0 0.5
Fa ..... b 1 1 1 --8 amb (OC) 1200 1600 1200
Although the steps taken te enter the pa":lmeter~ of tbis problcm a.re described in Part
II, a series of time dependent entries are input. The first corresponds to the initial conditions
existing for the first two minutes of the analysis. The second and third clltries correspond
to the conditions that conle into effect after two and {our minutes respectively.
The iesults are presented as Temperature vs. Time plots for surface node 1, node 6,
and surface HO de 10 (Fig. 2.11). As expected, the results predict tbat the node tempera
tures increase with time. The nodes that are furthest from the convective heat source are
least affected by sudden fluctuations in the environ ment wber,~as the temperature at surface
node 10 changes dramatically as soon as the new conditions are imposed (i.e. after two and
four minutes). Surface node 1 and node 6 both show the tendency towards an asymptotic
or steady state solution as they approach the six minute mark.
A special note should be made with regards to the batch mode feature and the require
ments for solution stability. The algorithm calculates the maximum allowable iteration time
based on the boundary conditions that exist initially and not on those that are upcoming
in the batch. If large variations in the boundary conditions exist, thcn differences in the
maximum allowable time increment may result. In this examplc the stahility rcquirclllcnt
was such that the time increment !:l.tmax must be less than 6.770 seconds. For cOllvcnicnce,
a time increment of one ~econd was chosen. The simulation was run for 121 seconds and
stopped. The maximum allowable itcration time was then rccalculatcd and a vaIue of 1.071 1
seconds was obtained. This new ·,aIue corresponds to the couditions cxisting at two min-
38
(
1600
1400
1200
.6 - IlIOO
BOO
600
400
200
o
o
Heat flow in rectangular slab btration of Batch Mode Operation
· . . . . . . . . . . . ... " " " " " " " " " " _. " " " " " " , " ,,_ ... " " . " " " . · . · .
" " ., " ...................... , ........... , ........... , .. " " " "
· . . """,,""" " " .:.,,"""" " " """ i ".""""" .. " i """"""""." .:."""""""""" .:'"'''''''''''''''' · . · . · . ........... ; ........... : ............ : ......... . · .
. . . '." .......... ~ ........... ! ..... "."" .. :.".".:.;. ·:J·::A.·IHNI'8~iiM...,.
node1: •
:0 node6: • node 10 •••••••••• ~ ••
· . . . " .. " . " ..... " " . " ._" " .. ". " ..... · .
60 120 180 240 300 360
rme (seconds)
Figure 2.11: illustration of the effect of time dependent entries on selected nodes in a. configuration.
39
, .
utes. Thus, it was fortuitous thl\t a one second increment was chosen. At 241 seconds ~tma.r
reverted back to 6.770 seconds. Users are advised that simulations making use of the batch
mode feature should be checked for stability at every stage where a new boundary condition
is implemented to prevent FASTP from generating unstable solutions.
40
(
(
Chapter 3
Conventions for Composite Configurations
In Chapter 2, the building blocks used by FASTP to model problems were presented.
This chapter looks at configurations made up of blocks or sections placed in series. Situa
tions often arise where I-D conduction or diffusion occurs through a series of materials. The
options discussed in this chapter were incorporated into FASTP's algorithms to model these
types of problems. This cha.pter will begin by explaining the changes made to the finite
difference equations so that transient behaviour can be modelled through two adjacent sec
tions. Throughout, heat transfer will be emphasized with references to mass transfer made
only later in the chapter. The manner in which FASTP models a resistance hetween two
sections will he presented, followed by an explanation of how partition coefficients are used
to model a. similar effect in mass transfer. Solution stability is considered at the very end of
the chapter.
41
....
3.1 Finite Difference Equations For Composite Heat Transfer Systems
FASTP can model heat transfer behaviour through up to five sections placed in series;
each is assigned its own materia! properties. Since a material's thermal diffusivity will
affect transient behaviour, the thermal profiles across any two adjacent sections can differ
cons~jerably depending on their relative property values.
The simplest example of a composite 1-D system is a two-section cartesian configuration
(Fig. 3.1). The half-nodes at the contacting boundaries are not exposed to the convention al
boundary conditions such as convection or radiation and are consequently not handled usillg
the same equations derived for the ordinary half-nodes (Eqs. 2.1'2,2.13). Instead, they are
treated as one whole node and handled using a finite difference expression similar to the one
used for interna! nodes (Eq. 2.6).
Material X Material Y • Section 1 -.' .... Section 2 .. T
1
1 l 1 1
l 1
1 • • • L. • .--. 1 11 2 n - 1 ni 1 2 n - 1 n'
1 .. LX ~ J __
Figure 3.1: Two-section configuration, 1-D cartesian coordinate configuration.
The combined node concept is based on the idea that the temperature at an interface
42
(
(
will be the same on both sides of the interface provided an interbounda.ry resistance does not
exist. The nodal properties of a combined node are calculated from a weighted average of
the two contributing half-nodes. The energy balance at the combined nodal point, denoted
as nodal point 0, is given as follows
[
Helt concllcted irto] nodal point 0
ln lit secondl [
Helt concllcted Iwa
J tram nodal poW 0 in M. eecondl [
Helt generlted J + et the i'It.-face
ln lit aecondI [
Heat aCCU'lUated Jl • In nodal point 0
ln 6t eeconds
The following terms are substituted into the energy balance to arrive at the finite
difference expression for internai boundary nodes:
1. Heat conducted from nodal point n - 1 to nodal point 0
. k A (9n - t - 9 0 ) q,n = - n-1ln n-tln ~Z
(3.1)
2. Heat conducted from nodal point 0 to nodal point 2
(3.2)
3. Interfacial heat generation term
Q '/1 Ao te" = qgen (3.3)
4. Accumulation term
(3.4)
43
Substitution of these terms into the energy balance gives
9' At [k A (9n - 1 - 9 0 ) k A (90 - 9 l ) ." A 1 - 0 = ( V.) + ( ") - n-lln n-lln A + III III A + qgen 112 + 90
pep n ~ pep VI " L.J.X x (3.5)
As with the finite difference equations that were derived in the previous cbapter, the only
unknown is the temperature 0ti at time t + At. With each iteration a new value for 9ti is calculated. This value is assigned to both half-nodes that make up node O.
When assigning the initial conditions to a configuration, the starting tClllperaturcs
for two adjacent sections do not neccssarily have to he the same. A simple cxample of a
prohlem where such a configuration would be used is one where a hot bar is placcd in direct
contact with one that is relatively cooler. Assuming pcdect contact the tcmpcratures at the
two contacting surfaces will hecome equal almost immediatcly, while the remainder of tbe
two bars achieve equilibrium more slowly. FASTP simulates this situation by setting the
starting temperatures of the half-nodes to the same value prior to performing any iterative
calculations. The assigned value is calculated from a. wcighted average of the half-node
enthalpies. The enthalpy rela.tive to some ambient temperature is first calculated for each of
the half-nodes. The two values are then summed, giving the total enthalpy for the combined
Dode O. The enthalpy expression is then used to calculate the adjusted temperature for
Dode O. This value is assigned to both half-nodes.
The steps FASTP follows are:
(i) Calculate the combined no de enthalpy
(3.6)
(ii) Calculate the equilibrium temperature for the two surface nodes
(3.7)
44
(
(
where
(3.8)
3.2 Node & Section Progression Conventions for FASTP
Figure 3.2 illustrates how the nodes and sections would be laid out in a one-dimensional
multiple section configuration in cartesian coordinates. Up to five sections may be placed
in series. Boundary condition 1 corresponds to node 1 in section 1. Boundary condition 2
corresponds to node n of section nt or the last no de in the last section of the configuration.
8.C.l 1 ~ .. n 1 n 1 ... n ~--. • • • • • • • 8.C.2
section 1 section 2 section 3
Figure 3.2: Section progression and boundary condition conventions for three-section 1-D cartesian configuration.
Adjacent sections along the radial component in a cylindrical configuration can he
viewed as a succession of concentric rings (Fig. 3.3). The outermost ring corresponds to
section 1 and the innermost corresponds to section n. When specifying the thicknesses
associated with ea.ch section an additional parameter, the outer radius, must be assigned.
This parameter detines whether FASTP is modelling a solid cylinder or an annulus. In each
section of the radial dimension the nodes progress from 1 to n inwards. Boundary condition 1
is outermost and houndary condition 2 is innermost.
Multiple sections in a spherical configuration are viewed as a succession of shells. The
same conventions outlined for cylinders also apply to spheres.
45
, '<
\ ,
\ \ B C.2 ----Tj------e---\" --.-
• 1
B C.1 ! \ \. J. \1/
• • • • \
\ \ 1 •
B C.1
B C 2 e---•
n
Figure 3.3: 2-D cylindrical coordinate configuration (1 section/dimension).
The next section will describe how interboundary or contact resistances are handlcd
by FASTP. Examples will follow that illustrate how FASTP handles heat traIlsfer across an
interface both with and without a resistance specified.
3.3 Interboundary Resistances
Interboundary resistances between two adjacent sections will result whenever less thall
perfect contact occurs between two surfaces. The presence of oil, grit, an oxide layer, or aIl
air gap will contribute to a resistance to heat trander across an interface. The algorithm
handling interboundary resistances in FASTP is based 011 the assumption that aIl air gap
exists at the interface, across which heat is transferred by convection.
A composite configuration with two sections separated by an air gap is shown in Fig
ure 3.4. The gap measures D.y across and has a cross-sectional area that is equal to the
surface area of the sections' common boundary. For I-D cartesian configurations the surface
area is equal to unity. The air inside the gap has material properties kgapl P, and cp' Sincc
46
(
(
/j.y is small, and because the heat capacity of the air in the gap is assumed small compared
to that of the surrounding materials, steady state is assumed to develop almost immediately
between the gap.
Material X Material Y
air gap
Figure 3.4: Air gap at interface (two-section configuration).
The energy balance for node n is given as
r- Helt conclJcted to l 1 node n trom node n-1 , in 6 t seconds ~ Heat Iost tOJ
ai'gap in 6t seconds
+ [
Helt generationj at the nterflce in 6t seconds [
Arncurt of heat acclll'Uated~ • at node n
in 6t seconds _.J
Substituting using the appropriate terms gives
From boundary layer theory [5], the heat transfer coefficient he existing for the system
47
-between the gap is related to the thermal conductivity (kgop )
h _ kgop
e - Ay (3.10)
By substituting into Equation 3.9, and with some manipulation, the {ollowing fi ni tt·
difference expression is developed for node n
(3.11 )
Similarly for node 1
8, ~t [h (0 a) k A (91 - 9 2 ) '/1 1 8 -1 = pc"lt} eAgoP\l Oomb - ,-"'1 - 1\2 1\2 ~x + qgenAI + -1 (3.12)
When an interboundary resistance is specified FASTP will request that a thermal
resistance Pt.e be input. The thermal resistance is denned as the inverse of he.
1 6y Re=·_=-he k gop
(3.13 )
The limits for Re are 0 and +00. A thermal resistance of zero would imply that
he -+ +00 and heat transfer across the interface would take place without resistance. If
Re -+ +00, the interface acts as a perfect insulator.
In the previous section the algorithm used by FASTP to set the temperatures at ail
interface was discussed. It was pointed out that the temperatures of both half-nodes at the
interface must be the same when perfect contact exists. When an interboundary resistance
is present and two adjacent sections have been assigned different starting temperatures the
adjustment of the internal half-node values depends on the simultaneous solution of three
equations and three unknowns. The equations come from the expressions {or heat fiow from
nodal point n-l to n, across the air gap, and from nodal point 1 to 2. Figure 3.5 shows the
heat fiow terms that must be set equal prior to beginning the iterative calculations.
48
(
section 1 gap
• • q = q
(n-11 n) gap
section 2
• = q
(1i 2)
t'-i '"'2 •
Figure 3.5: Hea.t flow terms at, and surrounding, the interface must he set equal when interhoundary resistance specified.
49
--
'-'"
where
. k A (0n - 0 n - l ) qn-lln = - n-lln n-lln àx (3.14)
(3.15)
(3.16)
Since no accumulation of thermal energy is assumed to take place in the air gap
(i.e. steady state) a.ll three heat flow expressions must be equivalent. The temperatures
0 n and 0 1 must be adjusted to satisfy the condition.
3.4 Illustration of Heat Transfer Across an Interface
Two examples are now illustrated where the heat transfer at an interface takes place
with and without au interboundary resistance specified. Heat tra.nsfer across two contacting
bars of steel has been modelled. The first simulation assumes perfect contact between the
bars, while the second incorporates a contact resistance. Carslaw & Jaeger present only a
general a.pproa.ch to obtaining exact transient solutions to problems involving an interbound
ary resistance. Consequently, wha.t is proposed is tha.t the results generated by FASTP be
compared to a steady sta.te solution; if only to show how the algorithm works and tha.t the
program can indeed arrive at a solution for steady state. Once again, the example is taken
from Holma.n.
Two 10 cm long, 90.4 stainless steel bars (3.0 cm diameter) W'ith a surface roughness
of 1 Jl.m are placed in contact at the ends under 50 atm pressure. The bars are msulated
so that heat conducts m the axial direction only. The boundary conditions are Buch that a
50
(
temperature d'fferential of 100 oC 18 set up along the configuration's length.
Holman's example uses the resistance analogy for solving the steady state problem.
From tables, the resistance across the interface Re was found to be 5.28 ... 10-4 m 2 -oC /W.
For the cross section of the given bar, Re was 0.747 oC/W. The total resistance across the
interface and two bar sections was calculated to be 18.105 oC/W. The overall heat flow
was 5.52 W (7809 W/m2). Tbe temperature difference at the interface at steady state was
4.13 oC.
The configuration used by FASTP to solve the problem is shawn in Figure 3.6. Since
heat conduction through the cylindricaI bars is axial 1-0, a rectangular configuration is
equally applicable. The left and right boundaries of the composite configuration are exposed
to fixed surface temperatures of 100 oC a.nd 0 oC respectively. Section 1 is assigned a uniform
starting temperature of 100 oC, while section 2 is set to 0 oC. The material properties of
both sections are given as follows
p = 7933 kgfm3 , Cp = 460 J / 'cg_OC, k = 16.3 W/m-oC.
t • • • • • • t • • • • • • ~
~ 10 cm .-1..- 10cm -~
(stainless steel> (stainless steel)
Figure 3.6: Configuration fOI resistance vs. no resistance companson.
The simulation was run twice. Once, assuming perlect contact and once with a contact
resistance specified. In both cases the temperatures of the internai boundary nodes were
51
-
readjusted to reflect conditions of perfect contact or less-than-perfect contact. The adjusted
half-node values at the interface prior to the start of the analysis are given as follows:
• No resistance case
section 1 (node 8): 50.0 oC section 2 (node 1): 50.0 oC
• Resistance case (Re = 5.28 * 10-4 m2-oC jW)
section 1 (node 8): 61.575 oC section 2 (node 1): 38.425 oC
Shown in Figures 3.7 and 3.8 are the steady state temperature distributions for the
no resistance and resistance cases. In both, the straight line behaviour expected for steady
state cartesian systems is observed.
Steady state temperature dlstnbutlor.
(no reslstance case)
100 ~~~ _____ ~ ___ ~~ __ .~ _____ ,-_____ ~ ___ ~ ______ ~ ______ ~ ___ ~
00 ~ __ ~ ______ ~~ ___ ~ ___ ~ _____ ~ _____ ~ __ ~ _____ ~ ______ ~~~
10em • --_ .. _ .. -- 10em --- ... Sechon 1 SecllOl1 2
Figme 3.7: No resistance case steady state tempera.ture distribution.
Note that with resistance, the temperature differential that is initially set up hy
FASTP at the interface is approximately 28 oC. As the simulation proceeds FASTP predicts
that the differential will decrease until a steady state difference of 4.12 oC is attained. The
heat flow across the gap was found to be 7803 W/m2. These values correspond weIl to the
52
(
(
û III 5 ~500 QJ
~ 1-
SectlOIl 1
Steady state temperature dlstnbutlon
(resistance Case)
Section 2
Figure 3.8: Resistance case steady state temperature distribution.
resistance analogy used in. the Holman illustration. The values were arrived at using the
Temperature vs. Time prlnt option and not the Temperature vs. Section plots shown.
3.5 Mass Transfer Across an Interface
When dealing with mass transfer across an interface, FASTP readjusts the starting
concentration vl\lues at the internai boundary nodes on the assumption that equilibrium is
reached instantaneously at the interface. ln mass hansfer, however, it is the activities of
a species that are set equal at an interface a.nd not necessa.ri1y the concentrations. It was
pointed out in Chapter 1 that the activity of a species depends on the mole fraction of the
species in the solution and its activity coefficient (Eq. 1.14). Consider the scenario in which
species A is transferring from mixture B to mixture C. il the activities of species A reach
equilibrium instantaneously at the interface the following will hold true for aU time t
• X· • X· "YA-B A8 = "YA-C AC (3.17)
53
where * implies at the interface,
"IA-B = activity coefficient of A in B (assumed constant throughout nùxture),
XAB = mole fraction of A in mixture AB.
The assumption that "1 is constant throughout a mixture applies to ideal solutions
(Raoult's Law) and non-idea.l solutions where the mole fraction of specics A does Dot varv
appreciably throughout the system (Henry's Law). If the situation arises where "YA-B = "YA-C
- i.e. species A has the same activity coefficient in both mixtures - then the activities and
the concentrations will be equal on both sides of the interface. The situation is analogous
to a heat transfer problem where a resistance does Dot exist at the interface. However,
if "IÂ-B "# "IÂ-c the interfacial concentrations will not be equaI and a partition is said
to exist. The partition is the discrepancy between the concentrations on both sides of
the interface. The degree of the discrepancy is determined by the part~tzon coefficient, dn
additiona.l parameter specified for mass trusfer configurations, which is simply the ratio of
the activity coefficients.
X• - X· AB - m AC
where the partition coefficient m = "IÂ-C/"YÂ-B
3.6 Stability at an Interface
(3.18)
Stabilityat the interface must be ensured. Without an interboundary resistance spec
ified the stability equation is similar to the o~e used for an internaI no de (Eq. 2.15). When
the program is evaluating the stability criteria for an interface, the two participating half
nodes are treated as a whole and a weighted value for the thermal diffusivity a is developed.
54
-{ The maximum iteration thne increment is calculated as follows
{
Ât = [pepVn + (m. pc"v.)]. ÂXn ÂXl
kn Vn + m • k1 V.
For heat transfer problems the partition coefficient m is assigned a. value of one.
(3.19)
Since the intcrboundary resistance Re is defined as the inverse of the heat tra.nsfer
coefficient existing between the gap, and because he is defined a.s kgBP/ Ay, the maximum
iteration time increment of a stable solution for an interface with resistance assigned is given
as
(3.20)
Equation 3.20 is evaluated for both half-nodes comprising the inteda.ce. The lowcst
value for At of the two is then compared to At values calculated for the rest of the con
figuration to arrive at an overall maximum iteration time increment below which stability
is ensured. SOllle additional comments Mt) made regarding stability at an interface when
discussing stability and phase transformations in Chapter 5.
55
-
Chapter 4
Conventions for Multidimensional Configurations
So far, discussion of the FASTP algorithms has been restricted to one~dimensional
problems. In this chapter the options available for modelling mtÙtidimensiona.l problems a.rt'
described. Two methods, the product solution and interaction solution methods, are available
for modelling 2-D and 3-D problems. The product method uses independently worked out
1-D solutions to approximate a mtÙtidimensional problem. The technique is conceptually
similar to the traditional product method for deriving multidimensiona.l solutions. Tbe
product solution option can be used to generate a temperature or concentration profile of a
surface or cross section of a configuration. The default option, called the interaction solution,
generates a full 2-D or 3-D solution using a simple cross-like nodal arrangement. A set of
conventions necessary for visualizing a multidimensional problem will be prt'sented a.long
with a discussion of the restrictions associated with each of the solution methods. Examples
are used to illustrate the internai workings of both algorithms and their results are compared
to product solutions generated from the Heisler charts.
56
(
( .
4.1 Multidimensional Systems
In this section the multidimensional systems that can he modelled with FASTP are
presented. Multidimensional transport phenomena occur when conditions are such that the
process cannot he descrihed with just one axial or radial component of a coordinate system.
For example, in rectangular shaped slahs this situation will arise when only one or fewer
planes of the configuration can he assumed to extend to infinity. A 2-D cartesian geometry
is illustrated in Figure 4.1. Similarly, a multidimensional cylindrical system is said to exist if
the axial component cannot he assumed to extend to infinity (Fig. 4.2). Extension to infinity
implies that zero losses occur at the ends (± 00). The same effect would he observed in a
finite arrangement if the ends were insulated.
/ -,:0
0in.2
~~., Din.3
Figure 4.1: 2-D cartesian geometry.
The additiona.l degrees of freedom introduced in a m,ùtidimensional system add to
the complexity of Fourier's or Fick's second law expressions. The expression used for 3-D
57
-
Dim.2
• i i
Dim 1
Figure 4.2: 2-D cylindrical geometry.
unsteady state heat conduction in cartesian systems, for example, would he written as follows
(4.1)
For 2-D cylindrical systems
!~ ('r as) . .1. a2e = 1. ae r Br Br 1 ôz2 ct at (4.2)
4.1.1 Conventions for modelling a multidimensional problem
The multidimensional configurations descrihed set the stage for the types of problems
handled by FASTP. Multidimensional configurations can be composed of multiple sections,
interbounda.ry resistances or partition coefficients can also he requested, and heat generation
or mass accumulation tenns ~an he assigned, To keep track of the many components that
m::a.ke up a configuration. a set of conventions has heen devised for working with the package.
In cartesian configurations, the first dimension (Dim.1) is seen as lying horizontally on
58
(
(
the page, the second dimension (Dim.2) lies yertically, and the third (Dim.3) lies perpendic
ulu to both (Fig. 4.3). The nodes, sections, and boundary conditions progress from 1 to n
following a left to right, bot tom to top, and front to back convention. Boundary condition 1
would apply to node lof section 1. Boundary condition 2 would apply to node n of section n,
or the last llode of the last section in the configuration .
• 1
N
.'
~k~~3 Dm 1
Figure 4.3: 3-D rectangular coordinate node progression conventions.
In cylindrical systems the mst dimension (Dim.l) corresponds to the radial coordinate,
while the second (Dim.2) corresponds to the axial coordinate (Fig. 4.4). In spherical sys
tems, FASTP can only mode! heat or mass trander along the radial component and is thus
restricted to modelling I-D problems in this coordinate system.
4.1.2 Tbe classical product solution metbod
One of the traditional methods used to solve multidimensional heat transfer problems
involves the separation of variables technique for solving differential equations. For a 3-D
59
cartesian problem a solutioll of the following form is proposed
9(x,y,z, t) = X(x) Y(y) Z(z) T(t) (4.3)
The solution assumes that the variables are fully separable. The expressions X(x),
Y(y), Z(z), and T(t) are aU solutions to simpler 1-D problems, which when multiplied
generate the product solution. The problem then becomes one of finding solutions to sen-ral
simpler 1-0 problems. The upcoming section will show how FASTP uses a similar approach
when the product solution option is requested.
4.2 FASTP's Product Solution
The product solution is one of two methods FASTP can use to model a multidimen
sional problem. The method combines the 1-0 solutions described in the previous chap
ters to generate an approximate solution to a multidimensional pro!>lem. To illustrate how
this method works a 2-0 cartesian problem with one section per dimension is considered
(Fig. 4.5). Before showing how the method works, however, some new terms must first be
defined.
Two types of nodes are present in configurations where a product solution has been
requested. These are referred to as principal ax~ nodes and profile nodes. The principal
axis nodes are the nodes upon wmch the iterative calculations are performed. When a
product solution is requested the iterative calculations on the two sets of principal axis
nodes of both Oim.1 and Oim.2 are carried out independently. In effect, two I-D solutions
are generated simultaneously. The iterative procedure continues until such a time that a
profile is requested. When a profile is requested a temperature or concentration for each
profile node in the configuration is generated. Each profile node value is calculated from
the coupling (2-D case) of the principal axis node values; one node from each principal axis.
60
------------------------
( -----------
B C 2 aXIs . -. n
B C 1 B C 2 ~ • • • • • • • 1 n
" 1 < •
B C.1
~Dim.1
___ o. Dim 2
Figure 4.4: 2-D cylindrical coordinate configuration (1 section/dimension).
0 0 G -e---
0 0 0 G ~
t 1
1
• • • • • •
t 1
0 0 i €t- E> t
t 0 i c- e tl
1 C\I (~ -, 0 • ~
0 ) E Â 0 -, ! é5
1 1 1 1
Profile node Principal Axis node 1 •
Dim.1
( Figure 4.5: Typical profile node arrangement when product solution requested.
61
The set of all profile node values is said to form the profile, and is an approximation of the
temperature or concentration distribution across the configuration. Ouring the simulatioll.
several profile snapshots can be generated for comparison.
Profiles can also be generated from 3-D configurations. In the 3-D case the profile
nodes are formed from a grouping of three principal axis nodesj one from each principal
axis. The manner in wruch this is done is described in Section 4.2.1. FASTP can generate il.
temperature or concentration profile of any of the body's surfaces or through a cross section
of the 3-D body. The constraint is that the profile must intersect one node of a principal
axis and be parallel to the plane formed by the other two axes (Fig. 4.6) .
--
• • ,) .-
1
0 0
n •
• n
• i
1 .~
1 •
• --.
n Dm 2
~3 Dm 1
Profil nt_secta nodI 1 of DmInIion 2: ... PIf .... to plane cr •• ted by the iI'It«sectlOf'l of DmenIIon 1 and 0inenIi0n 2
Figure 4.6: Typical profile through 3-D cross section.
Profil,~s for 2-D cylinders are generated from the 1-D axial and 1-D radial solutions. A
typical profile for cylindrical systems is shown in Figure 4.7.
The above cases considered multidimensional systems with only one section per dimen-
62
l
(
B.e.2 ~I ---.-n -----0------ - - ,~---- - ':r
B.e.1 ___.---._---4e-----'e---._ e 1 1 \
/ 1 • 0/ 0 o o c B.e.1
B.e 2 • • n
c cr
~Dim.1
\ \
----~
Dim 2
Figure 4.7: Typical profile through 2-D cylindrical configuration,
sion. When multiple section 2-D or 3-D configurations are analyzed, profiles can only be
generated from what are termed central sections, Each principal axis in the configuration
has a cen tral section. To determine which section is central the thicknesses of the sections
making up the configuration along a dimension are summed. The overall thickness is then
divided by two and the section located at the halfway mark is assigned the central section
status (Fig. 4.8), The procedure is repeated for all dimensions in the configuration,
i 1/2 ~------
lir
• 1 1 1 n 1 1 1
- n - - - n - - n -- -- - -- -- -- -.- • • e
1
: sect. 1 sect. 2 sect.3 sect.4
, 1
._--~--._------_.
central section
Figure 4.8: Procedure for determining central section.
The central sections are common to all of the dimensions of the configuration. The ma-
63
..... -terial properties assigned to the central sections must therefore be the same. FASTP cnsures
this by copying the properties of central section 1 - dimension l, to the central sections of th ..
other two dimensions. The sections on the principal axes that are not central are complett>ly
unrelated to the other dimensions. Figure 4.9 illustrates a 2-D cartesian configuration with
three sections per dimension. Note that the materials used in the unrelated sections all
differ.
:NI • :
1 • 1 Sect 3 1
1 ,
• _" ~ ------l' . , AI ,1 1 Fe Cu • .\ +
~
1 1
~------ • • • • .-- .- • 1
f ,
0 · .) ~
Sect 1 1
1 Sect 2 1 Sect 3 l 0 • G "-----:-----\ Asbestos
1 1
~ i 1 Dim.2
1 • ! 1
1 Sect. 1
• Dim.1
Figure 4.9: illustration of a 2-D cartesian configuration with three sections per dimension.
When a profile is generated the principal axes can be visualized as lining up 50 that
they intersect at right angles. The central sections intersect at what are tcrmed the central
nodes. Each principal axis has a central node. The principal axis no de located closest to
the midpoint of the central section, but is also nearest to node 1 of that section is given the
central node status (Fig. 4.10).
The product method uses the central nodes as a reference point from which the profile
64
( 1/2
----------
~2 n • • • sect.3
central node
Figure 4.10: Procedure for determining the central node.
is generated. No interaction takes place between the central nodes of the different axes when
the product method is used. It should also be noted that when this method is used, the
principal axis node values bear no relation to the actual 2-D system other than to generate
the profile node values. Qnly the profile node values reilect what is taking place in the
multidimensional system.
4.2.1 The product solution algorithm
The algorithm that generates the profile uses a dimensionless enthalpy or dimensionless
mass concentration approach. To illusbate how the algorithm works, consider once agi.n
the 2-D cartesian configuration for a heat transfer problem (Fig. 4.11). A convective en
vironment is set up where the same ambient temperature exists at ea.ch face. The nodes
along both principal axes of the configuration have initia.lly been assigned a uniform start
ing temperature. The conditions are such that the system will lose energy with time (i.e.
9 0mb < 9bod,,). The enthalpy of every principal axÎR uode, calculated relative to the ambient
environment, can easily be determined using the equation for enthalpy (Eq. 2.1).
65
"""" <Ur-
.......
hconv • ~
t:" 8mb
'"' '" ,
..r ., c • .., "'> ~
i ...,
1 1
Q 0 0 • 0 0 ,~ 1
1
, 1
~ 1
1 0 (':1 • 0 0 ,"'1\ ~ .., '7)
1
'" 1 '--' 1 -.1.' • > 1
! 5
r >
(J
~ 8
oC 0 -0 0 (J
0') oC
J 1
0 0 0 0 'f 0 J "', ,;;
c hconv • 8 8mb L __ ~
Dim 1
Figure 4.11: Profile node progression in 2-D cartesian configuration.
If the principal axis nodes are assigned a uniform starting temperature, then all profile
nodes will also start at this temperature. The enthalpy of each profile node relative to its
surroundings can also be calculated for the starting conditions. Assume that with time the
enthalpyof every principal axis node drops to a fraction of its original value. Dimensionless
enthalpies ior the pljncipal axis nodes can be calculated by dividing the current enthalpies of
the principal axis nodes by their original values. The dimensionless values for each coupling
of principal axis nodes that make up a profile node are multiplied to form a product. This
l-,roduct is the profile node's dimensionless enthalpy. In turn, this value is multiplied by the
profile node's starting enthalpy to obtain the actual enthalpy for the profile node for that
time. Using the equation for enthalpy, the current profile node temperature can then be
back calculated. The same procedure applies if the body is being heated .
66
(
(
4.2.2 Examples illustrat\ng the product solution
Two examples are illustrated to show how the product method works. In the first
example, a 3-D rectangular slab is cooled. In the second, the cooling of a 2-D cylindrical
shape is examined. Both examples are subject to convection boundary conditions and are
suitable for solution using the Heisler charts. The Heisler chart method for obtaining the
trc..ditional product solution has been weIl illustrated in Holman.
Example 4-1: Cooling of a 3-D rectangular slab.
In tht.S ezample a three-dimensional slab of aluminum (6.e*9 cm) is exposed at time
t=O to a convechve environment of h = 5e5 W/m2 .oC and 9 0mb = 70 oC (Fig. 4.le).
The unifo'Mn starting temperature of the slab is eoo oC. The objective is to determine the
temperature of the block at the corner after t = 60 seconds.
FOI aluminum
k = 215 W/m- "C, p = 2700 kg/m3 ,
n , • n • •• n
• t· • -e
~ .. ; 1 •
1 .1----~.~--~--~J----~----! , 2cm
1
.1_ .hL ____________ --v
( 1.1.1)
6em ~3cm
cp ::- 948 J /kg- oC.
Oim 2 l Oim.3
: .~ 1 / • 1/ ~ Olm.1
Figure 4.12: 3-D rectangular sI ail (1 section peI dimension).
67
Using the product solution method and with the a.id of Heisler charts, the predicted
corner temperature of the block was 82.1 oC after sixt Y seconds expo~ure to the convcctiw
environment. A 3-D cartesian configuration set up with FASTP predicted a corn('r tcmper
ature of 80.9 oC. The Heisler charts used in this example can be found in the appendix (A 1
and A4). The calculations performed by FASTP to arrive at the predicted temperature a.re
now illustrated.
Every principal axis node in the configuration was assigned a uniforrn starting tcmper
ature of 200 oC. Consequently, cach profile node would also possess a starting temperat urt>
of 200 oC. The enthalpy of each no de relative to its surroundings can be calculated USlllg the
enthalpyequation (Eq. 2.1). The associated volumes and starting enthalpy values for each
of the three participating principal axis nodes and profile node (1,1,1) are given in Table 4.1.
Table 4.1: Associate<.l volumes and starting enthalpy values of participating principal axis nodes and profile node (1,1,1).
Node Associated volume (m3 ) Entha.lpy (t = 0) (J) 1
Dim.l - node 1 0.50000 * 10-2 1.664 * 106
Dim.2 - node 1 0.16667 * 10-2 5.546 * 105
Dim.3 - node 1 0.25000 * 10-2 8.319 * 105
Profile node (1,1,1) 0.2083 * 10-7 6.931
Based on a 7.7*7 cquispaced node progression.
The analysis was carried out for a simulation time of sixt Y seconds. During the simula
tion the herative calculations on the 1-D nodal progressions wele perfolmed independently
of each other. Using FASTP's Temperature & energy content print option, the enthalpy of
each principal axis node was recorded for t = 60 seconds. The res'llts are given in Table 4.2',
along with the new enthalpy of each node relative to the ambient environrnent.
With this information a dimensionless enthalpy can be calculated for each of the par-
68
( Table 4.2: Enthalpy change of the participating principal axis nodes and new enthalpy values for sarne nodes at t = 60 seconds.
Node Change ln enthalpy (0 - 60) (J) Enthalpy (t = 60) (J) 1
Dim.l - no de 1
Dim.2 - node 1
Dim.3 - node 1
.0.5754 • 106
-0.3924 • 106
-0.4665 • 106
ticipating principal axis nodes:
For Dim.l 6.H1(60) _ 1.089 • 106
6. HI(o) - 1.664 • 106 = 0.654
For Dim.2 6.H2(60) _ 1.622 • 105
6.H2(o) - 5.546 • 105 = 0.293
For Dim.3
6.H3(60) = 3.654 • 105
= 0.439 6.H3(o) 8.319. 105
1.089. 106
1.622 • 105
3.654. 105
(4.4)
(4.5)
(4.6)
'l'he dimensionless enthalpy for profile no de (1,1,1) at t = 60 seconds;,s calculated from
the product of (4.4), (4.5), and (4.6)
6.H(l.l.l)(60) = 0.654 * 0.293 * 0.439 = 0.084 AH(l.l.l )(0)
(4.7)
The a.ctual enthalpy of profile node (1,1,1) is calculated by multiplying the dimension
less value by the node's original enthalpy
AH(l.l.l)(60) = 0.084 * 6.931 = 0.583 (4.8)
Inserting this value into the enthalpy equation and solving for the temperature gives
( e = 80.92 oC. The result is compara.ble to the Heisler chart rc;sult.
69
- --~~-----------~
-Example 4-2: Cooling of a cylinder.
In the second example a cylinder is exposed to the same convective environment as
Ezample 4-1 (Fig . . ;.13). However, the objective in this case is to determine the temperature
of a point within the cylinder. This point is located 0.625 cm from onc of the ends of the
cylinder and 1.25 cm along the cylinder's mdius.
T 5em 1
1
l
~------- 10 cm
n
Din 1
(2.3)
Figure 4.13: 2-D cylindrical configuration.
1
~
Din.2
Once again with the aid of the Heisler charts and using the product solution mcthod
outlined by Holman, the predicted temperature at this location after sixt Y seconds exposurc
was E> = 104.5 oC. FA3TP predicted a temperature of e = 108.2 oC using il 5 by 17
equispaced node progression so that the (2,3) profile node lay precisely in the desired location.
Again, the results are comparable. See Heisler charts (Al), (A2), (A4) and (A5) in Appendix
A.
4.2.3 Advantages and limitations of the product solution
The product method offers many advantages, not the least of which is that relatively
70
( few nodes are required by the algorithm to generate useful results .. Compare a fully two
dimensional cartesian configuration consisting of a nine by nine matrix of nodes (Fig 4.14).
In total, 81 nodes ale required in the configuration. By contrast, FAS'l'P's nine by nine
configuration is made up of 18 principal axis nodes. The net reduction in the computation
time required to perform a simulation is significant. The reduction becomes even more
significant when the comparison is made for 3-D cOLfigurations. The drawback to this method
is that only one bpundary condition can he assigned pel boundary. Complex boundary
conditions, which vary across the houndary surface, cannot be modelled.
1 .----.----.-- .- T til S ~ • ;
i • • • • • • t v 1) ., • ,) .:.
~ t ,
• • • • • • () 0 0 • .:. .) . 1 1 , • • • • • • • • :; -0 ~ ., .:. ., .;:> r
1 1 1 • • • - • • '~--4 1 1--4-
~ 1
• 4t • • • • 4» -0 -0 1) , ., ., :. ;:
• • • .~ t 0 0 1) + 0 ., :)
1
• • • • • • ~ 0 ~ .j • <>- ., ) t:
1 1
Î Î Î f e e e r El ~ ') !'
varying boundary condtions one bounclary concition
Figure 4.14: Full 2-D nodal arrangement (left) vs. product solution arrangement (right).
The overriding assumption when using Heisler charts is that the material properties
are constant. This is because the technique is based on a dimensionless temperature ap
proach and not one of dimensionless enthalpy. The advantage of the dimensionless enthalpy
approach used by FASTP is that t1ùs limitation is removed. Consequently, 2-D and 3-D
configurations involving phase transformations can be modelledj as will be shown in the next
71
l
-.
chapter.
Also, Heisler charts can only be used when the ambient temperature 0'II11b is constant
on a1l faces of the configuration. The product method used by FASTP is Ilot restrictcd
in this sense, although errors can be generated. The problem becomes morc }>ronoullccd
when a greater differential exists between the ambient conditions or when thc tcmpcratures
or concentrations within the body are nearly at par with the ambient(s). Ta illu:,tratc the
second point consider two systems. The first (System 1) has a uui{orm tempera.tuIe of
1500 oC with ambients of 20 and 30 oC on the various boundaries. Thc srcolld system
(System 2) is virtually the same with the exception that the body temperature is 35 oC.
System 2 can much less accurately be modelled because the potential drop to the ambient(s)
is relatively small. Of course as time progresses and System 1 gets doser to equilibrium, the
results become less accurate for the same reasons. The inaccuracies will be most notable at
corners exposed to both ambient conditions.
4.3 FASTP's Interaction Solution
The interaction solution is the default method available {or modelling multidimensional
problems. The difference between the two methods is that whereas the product method uses
independently worked out 1-D solutions to approximate a multidimensional problem, the
interaction solution makes use o{ another, albeit simple, algorithm to model the problem
from a truly 2-D or 3-D perspective.
The term simple is used becausl! of the simple nature in which the nodes are laid out
in the configuration. Like the product solution, the prhcipal axes of the configura.tion will
converge at a central node in the central section. Unlike the product solution, however, the
central node on the two (three) principal axes is common. While it may appear as mally
times in the printout as there are axes in the configuration, it is the same node. As such the
72
( energy or mass balance on the central node must account for los ses or gains to or from the
adjacent nodes in every dimension. The role of the central node in the interaction solution
is now discussed along with a description of the algorithm.
4.3.1 The interaction solution algorithm
The Iole of the central node in the central section is the key to understanding how
the interaction solution works. With every iteration the calculations are carried out on the
nodes as if separate I-D solutions were being generated. With each complete iteration the
incremental amount of energy or mass contributed by each dimension to the central node
is determined. FASTP then readjusts the other nodes in each dimension's central section
by adding to tbem the same quantity of energy or mass contributed to the central node by
the other dimensions. The algorithm's internal workings are illustrated for a 2-D cartesian
configuration in Fi~,ure 4.15.
As shown in the figure, the enthalpy added to the central nod~ by dimension 2 is added
to all of the nodes in the central section of dimension 1 in the first pass. Similarly, the
enthalpyadded to the central node by dimension 1 is added to a.ll of the nodes in the central
section of dimension 2 in the second pass. The argument can naturally be extended to 3-D
systems. The node values in the sections not central to the configuration do not undergo the
readjustment. The adjusted tempcratures for each node in the central section are calculated
using the following
where
a~~.i: [aH; + aH. + pc,,"Ha: - 9.lJ C~"J + ai
i,j, k are the principal axis subscripts,
S~IÜ.i is the new, adjusted temperature,
Si is the old temperature,
73
(4.9)
, . -- .... ---- - ---- -----. ---- -n first pass n second pass
,-.. - • ---+- •
1 + J • -.-.--.__e--e---t
1 n· -.... +- • ~ • , , ________ ._1 __
Dim 2
.l-e , • ,
• '.E: • • central node i • central node
Contribution from DIm. 1 Contrbutlon from Oim 2
Contributions to the central node from the other clmension(s) are added to the other nodes ln the central section
Figure 4.15: illustration of interaction solution algorithm.
e~ is the new temperature based only on the finite difference calculation,
AH, and AHk are the enthalpy contributions by the other dimensions to the central 110 de
in dimension i.
4.3.2 Example illustrating the interaction solution
The heating by convpdion of a cylindrical object is to be stmulated by FASTP usmg
the interaction solution method. The result is to be compared to a product solution generated
from Heisler charls. The cylinder is 20 cm long and 10 cm tn dtameter. Usmg the material
properties for magnesium, the following boundary conditions and initial condltwns, the tem
perature at the midpoint along the central axis and 2.5 cm below the rounded surf'Lce is to be
74
,...-------------------------------
predicted at two minute intervals for a period of ten minu.tes. As ezplained, the interaction
solution method is restricted in the locations where values can be iteratively predicted to the
locations of the principal axis 7/.odes (Fig.4.16).
Material properties
k = 171 W/m.oC, p = 1746 kg/m3, Cp = 1013 J / kg.oC
Boundary conditions
h = 500 W/m2.oC, 9 0mb = 500 oC
Starting conditions
Node 2 of dimension 1 was positioned 2.5 cm below the cylinder's rounded surface by
specifying a thickness of 10 cm and outer radius of 5 cm with 5 nodes in the configuration.
The algorithm positions the first and second nodes in negative coordinates. The nodes are
symmetric about node 3 (FigA.17).
The results of the simulation are shown in Table 4.3. The predictions made using
FASTP and the Heisler charts are comparable although there is ~<omewhat of a discrepancy
in the values at two minutes. See Heisler charts (Al), (A2) and (A5) in Appendix A.
4.3.3 Advantages and limitations of the interaction solution
The advantage of the interaction solution method is that more types of problems can
he modelled without the constraints that exist for the product solution. Using this method,
accurate predictions can be made ev en if a boundary has been assigned a fixed surface
75
h = 500 W/m2.t;
& = 500°C
, :' central axis
---··------·····-7··········~········ ....•...........
,"4- 2. 5 cm
o
:~ 20 cm
10cm
J
Figure 4.16: Configuration for illustration of interaction solution method.
10em
Dm 1
20em
Figure 4.17: Node distribution when having selected an outer radius tha.t is half the specified thickness (Equispaced node progression).
76
(
(
Ta.ble 4.3: Comparison of FASTP interaction solution method to product solutions gr.nerated by Heisler charts.
Time Heisler Charts FASTP
(min) 8(OC) 8(OC)
0 20.0 20.0
2 387.0 397.7
4 480.6 479.9
6 495.9 496.1
8 499.2 499.2
10 499.8 499.8
condition or if different ambient conditions were assigned to the different boundarles. The
disadvantage is that, unlike the product solution, there are no profiles nodes and profiles
cannot be generated.
77
, ,
-~---~
r
.'
-..
Chapter 5
The Phase Transformation and Mixing Aigorithms
The two remaining features of FASTP, which are used to model phase transformations
and mixing, are described in this chapter. The phase transformation feature can be used to
model such phenomena as solidification, melting, and dissolution. A simple algorithm is u~ed
where different sets of material properties, corresponding to different physical sta.tes, can he
substituted into a specifie node depending on the conditions at the node. FASTP modcls
phase transformations from a ma.cro energy or mass balance viewpoint. The concept and
algorithm are presented. As weil, the exact solution for a. l-D solidification problem, the
Stefan problem, is compared to a solution generated by FASTP. The stability criteria algo
rithm is further expanded to include systems incorporating a phase transformation. Finaily,
the use of this feature in a dissolution type problem is described.
The second half of this chapter describes FASTP's mixing feature. With this feature,
the program can model the redistributive effects of mixing in liquids, or among nodes with
state {3 properties assigned. The degree of mixing is a parameter that can be controlled
and that ultimately con troIs the degree to which convection can influence the system. The
algorithm is describ~J. and FASTP's results are compared to an exact solution that models a.
78
(
(
perfectly mixed system. The difference between full versus partial mixing is also explained.
5.1 Phase Transformations
A solidification illt'stration is used to introduce the phase transformation tetminology.
Consider the liquid-to-solid phase transformation of a hypothetical binary alloy whose phase
diagram is shown in Figure 5.1. Because the system has no eutectic point, the phase diagram
is t:haraderized by smooth, continuous liquidus and soli dus lines. Under equilibrium con
ditions a molten alloyof some known composition will begin to solidify as its temperature
drops below the liquidus. Solidification will be complete once the temperature drops below
the solidus.
The change in enthalpy with temperature for a 50-50 binary alloy is shown in Figure 5.2.
The heat capacity values for this alloy, for both the liquid and solid states, are assumed to
remain relatively constant with temperature and hence the straight !ine behaviour.
The system cao be divided into three regions. The first lies below the soli dus and
corresponds to the solid SGate (state 1). The second state (state 2) corresponds to the
region located between the solidus and the liquidus. State 2 is also referred to as the
transforming state. The third region is located ab ove the liquidus and corresponds to the
liquid state (state 3). Note that a sharp increase in enthalpy takes place between the soli dus
and liquidus temperatures. This is attributed to the latent heat of fusion. By incorporating
the latent heat term into the heat capacity of the transforming state, the calculation of the
system 's enthalpy at any given tcmperature becomes a piecewise integration.
(5.1)
'T9
-[
{T2 1 [l'olidU' 1 [J.'iQU1dU, 1 [lT2 1 J7. cpde = Cpl de + . Cp2de + cp3de Tl Tl .ol.du, "quldu.
.6 -
PHASE DIAGRAM FOR ALLOY A-B
, . . liquid, _ .. _ ... _--~---_ .. _ .. _~._---_ ... _~---.- ---_J __ . ___ .~ __ J ________ •• J _________ _ 1 1 1 l , 1
1 1 : nQuidus: : ,
__________ ~ _ __ _ _ _ _ _ L _trÂI')~tQ~jf)g, ____ ~ _______ _ . . , , , ,
, , , . ,
__________ • _______ ~~~i~~-~ ____ . __ ~ _____ . ___ j _______ .~ __ ::-.. ;:: .. :-: __ -::-: __ ~ __ ~ .. :-:c .. !-:.::,,:, .. = .. ..; __ ~. ~ 1 1 1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 1 t 1 t l ,
: : : ; solid : 1 1 1 l , 1 ---_._----,----._----,---------.,._--- ----,----------~------_ ... ,--- ------. 1 1 1 1 • 1
: ! : (00;.50) A-B al~y : 1 1 l , 1 1 1 1 • 1 l ,
o 100
A Atomic percent of B B
Figure 5.1: Phase diagram for hypothetical binary alloy.
5.2 Non-equilibrium Conditions
(5.2)
Non-equilibrium processes are generally the rule rather than the exception. If a ther
mocouple is placed in the same molten alloy and solidification is allowed to proceed, a cooling
curve similar to the one shown in Figure 5.3 will be generated. The inflection points on the
curve correspond to the beginning and end of solidification. Closer inspection will reveal
that the fi.rst intlection point is not at the same temperature as the corresponding liquidlls
value on the phase diagram (Fig. 5.1). Because this is a non-equilibrium process, the system
will not begin to solidify immediately upon reaching the liquidus. Instead, the temperature
80
state 1
a ---.i o
o o
state 2 state 3
c
IiqlJidus ;.-o o o
Temperature CC)
d
Figure 5.2: Varicltion of eni.halpy with temperature of hypothetical alloy.
cODtinues to drop until the kinetics are favourable for nucleation to begin.
FlOm the figure, three distinct cooling rates are observed. The cooling behaviollr ahove
the first inflection point corresponds to the cooling of the molten alloy (state 3). Bctwccn the
two inflection points latent heat is released as the liquid transforms to solid. The cooling rat/'
associated with the transforming state (state 2) is considerably slower bccallse the ~y~t(,lll
is releasing the excess latent heat. Below the second inflection point the cooling bchaviour
shown is that of the solidifie'1. alloy (state 1). Since the latent heat is no longer heing given
off, the cooling rate increases once again.
5.3 The Phase Transformation Algorithm
When assigning material property data using FASTP's database fcature, the 11ser is
81
.6 -~
i ... CD a. E
{!.
(50-50) A-B Afloy
liquid
sohd
Time (s)
Figure 5.3: Cooling curve of hypothetical 50-50 binary alloy.
given the choice of either entering one set or three sets of propeJ ties. If the user specifies
that the material involves a phase change, then material property values will be requested for
each of the states (1, 2, and 3). Two additional parameters, the solidus and liqutdus, must
also be specified. These two parameters set the temperature or concentration ranges through
which each set of state properties will apply. When anode temperature or concentration
falls below the liquidus, for example, st;ate 2 properties are substituted for state 3. Similarly,
the soli dus is the transition poj~t kt,ween states 1 and 2. It will be shown later how a latent
heat term can be incorpor~~ed into th'" state 2 heat capacity.
To illustrate how the algorithm works, suppose that some arbitrary node i is losing
energy to its surroundings. lnitially, the nodc's temperature lies above the liquidus and state
tllIee properties are assigned. With t>ach iteration, the node's enthalpy continues to drop
until the predicted tt:mperatnre 0: {alls below the liquidus (Fig. 5.4). When this happens
82
the algorithm williower the node's enthalpy only by an amount that wO\ùd bring the no!lt'\
temperature to the liquidus. The material's state 2 properties are thcll substituted and
the remaining energy that would have lowered the no de temperature bdow the liquidus IS
then removed, but from anode with state 2 properties. Since the latent heat of fusion i~
incorporated into the state 2 heat capacity the drop in temperature is much less.
e li(JJidls Liquidus
/::, Hstate 2
6. Hstate 3 j
, e t +llt
{:; Hstate 3 • è.. Hstate 2
Figure 5.4: illustration of state property change at the liquidus.
The transition at the solidus follows more or less the same pattern. The only differcncc
is that state 1 properties are now substituted for state 2. Because the state 1 heat capacity
does not incorporate a latent heat term the ccoling rate increases once again. The saIne
algorithm applies for melting. At the transition points, however, the state 2 properties are
substituted for state 1, and state 3 for state 2. There are many rc{erences, from which
material property data for the soUd and Uquid states (i.e. states 1 and 3) of a particular
material can be obtained. The recommended approach for assigning state 2 properties is to
use the average of the state 1 and state 3 thermal conductivity and density values. The heat
83
--- -- -- -----------
, •
capacity for state 2 incorporates the latent heat of fusion, which FASTP dissipates linearly
over the liquidus.solidus range. The method for calculating the state 2 heat capacity is given
as follows
if
and
then
liquidus - solidus = 40 oC,
latent heat = 400 J / g,
Cp = 400/40 = 10 J/g.oC
+ the average heat capacity of state 1 and state 3.
It is important to note that the assigned liquidus and soli dus values do not necessarily
correspond to the equilibrium values found on the phase diagram. The solidus and liquidus
are usua.lly chosen from experimental data or from knowledge of the behaviour of a similar
system. In the case of the molten alloy whose cooling curve was shawn in Figure 5.3 the
soli dus and liquidus would correspond to the inflection points on the curve.
Solidification behaviour of alloys containing a eutectic, or any material whose transfor
mation from liquid to solid involves more than one step, cannot be modelled with FASTP.
This behaviour can only be approximated by an overallla.tent heat term that has been
incorporated into the state 2 heat capacity. Figure 5.5 illustrates how the transformatior
hehaviour of such a material would be approximated by the algorithm.
The phase transforma.tion feature can also be used to model changes in the properties of
one state of a. material as it changes with temperature. This can be particularly advantageous
when the use of one value for Q becomes too much of an approximation in the problem.
Instead, three values (an he specifiedj each one applying to a specified range.
84
,
1
Alloy containing eutectic
liquidus , eutectlc temp
state 3
state 2
solidus
state 1
--actual
F ASTP approx
Time (s)
Figure 5.5: Approximation of cooling behaviour of alloy containing eutectic.
5.4 Stability and the Phase Transformation Feature
Calculation of the maximum allowable iteration time increment for stability must takr
into account the fact that anode can exist in any of the three possible states. F ASTP first
de termines which of the three sets of state properties generates the largest value for thf'
thermal diffusivity and then uses them in the subsequent stability calc\ùations for that
section. Reca.ll that for stability Q~t 1 --<-(~X)2 - 2
( 5.3)
Calculations for the stability at an interface must take into account all possi.ble com
binations that might arise. For example, astate 1 property no de in section Amay be in
contact with astate 3 node in section B or vice-versa' all combinations mu!)t be accountcd
for. Recall that the value for Q used at an interface without thermal resistance specified i!)
85
1 a welghted average of the two contnbuting half-node values (Sect. 3.6). To calculate a mini
mum value for ~t, the combination of half-node properties producing a maximum weighted
value for Cl dre used III the stabihty calculations (Eq. 3.19). Similarly, for an mterface wlth
resistance specified, the properties that generate the largest values for Q are used in the
calculation of the maximum iteration time increment (Eq. 3.20).
5.5 Phase Transformations at a Single Temperature
FASTP's algorithm requires that the !iquidus value not equal the solidus. For pure
metals and eutectic alloys, one temperature exists, over which the transformation takes
place. To model phase transformations in such materials a very small temperature range
between liquidus and solidus is specified (e.g. solidus: 650 oC, liquidus: 651°C).
In a review of the available exact solutions for melting and solidification problems.
Carslaw & Jaeger offer little encouragement; especially where systems transform over a
range of temperatures. Instead, approximate solutions are presented and the use of numerical
methods is recommended. One problem that is examined is the Neumann problem (Fig. 5.6).
In a ~eumann problem, one temp~rature exists at which the phase transformation begins
and is completed. The liquid is at some temperature e above the melting point E>m, and is
unidirectionally solidified from one end held fixed at some temperature E>w below the melting
point. The fundamental relations to be satlsfied are
(i) The temperature of the adjacent phases should be equal (E>mte,.Ja:e = 0melhng),
(ii) An energy balance must be satisfted at the interface.
86
The energy balance at the solid-liquid interface would be givcn a.s follows
Heat flux ln the ove x-dlrecbon through the sohd phase
1 Heat flux ln the
- love x-dlrecbo., through the hquld phase
Rate of heat hberated dunng sohdlflcabon per unit area of Interface
Expressed mathematically
[ ." '''] AH!!É.lli '( ) - q, - q, = P Ll Jlmon dt at x = v t
where 4~ = heat flux (solid phase) = -k,~1 8r Ir=6(1)
41' = heat flux (liquid phase) = -kl2.8~1 r r=6(t)
When P, = PI, there are no density variations across the transformation boundary ,Uld
hence, there is no convection. With denslty variation the energy balance would includt> il.
convection term for the liquid phase; making the analysis more difficult.
It has been shown that the solidification front at x = b is a functlon of the ~quare root
of time. The temperature profile shown in Figure 5.6 is representahve of the profile exi~tinl!;
at the interface at any given time. The phase transformation takes place at one tt>rnperaturc
and the liquid is assumed to extend to infinity. Although Simple with rC!lpect to dny rt·al
system, the problem is non-tinear due to the temperature gradient through the liq111d portion
and requires al' iterative approach to arrive at a solution.
One of the solutions that has been derived and that does not require iterative mcthods
is referred to as the Stefan problem (Fig. 5.7). The simplifying assumption is that the system
is initially at the trd.nsformation temperature em • The Neumann problem is simplified thus,
87
6
LU
~ ~ c(
~ LU ~
m
. , (x,l)
1
( l' w
~ , (1)
!
~ Soliciflcation front
POSITION X
. , (x,l)
Neumann Problem
Figure 5.6: Typical temperature profile for a Neumann problem.
by crcating a zero heat flux environment in the liquid portion of the system. Begmrung at
time t = 0 and at x = 0 the melt is held at a fixed temperature ew below the melting point.
The Stefan solution also predicts that the position of the solidification front 8 is proportional
to the square root of time.
ln the following example, a semi-infinite system of molten aluminum is unidirecti(mally
solidified by a boundary held at some temperature below the solid.us. The properties for
alunùnum are given in Table 5.1. Note that the latent heat term h~, been incorporated mto
the state 2 heat capa.àty. In this example, the other state 2 properties were borrowed from
the state 3 column.
The object of this illustration is to show that FASTP can predict the solidification
front's growth dependence on the square root of time. A Stefan problem is to be modelled
so the system will initially st art at the liquidus temperature.
88
1
1 t m
Soidficatlon front
POSITION X
Stefan Problem
Figure 5.7: Typical temperature profile for a Stefan problem.
Table 5.1: Solid, liquid, and transforming state properties for aluminum
1 Property 1 state 1 1 state 2 1 state 3
k (WjmGC) 166 91 91
p (kg/ml) 2700 2300 2300
Cp (J/kg) 920 395000 790
soli dus = 650°C liquidus = 651°C
89
L
To model this problem with FASTP, a. 1-D L'onfiguration comprised of one section
and twenty nodes is used. The section measures 10 cm across and at tlme t = 0 the
boundary at x = 0 îs held fixed at 100 oC. The boundary at x = 10 cm 1S assigned a zero
flux boundary condition. Although the Stefan problem describes a selll1-illfinite system the
FASTP configuration is valid provided the temperature at the adiabatic boundary does not
fail below the initial temperature. At t = a the system 's temperature is uniformly at 651°C.
The simulation was allowed to run for eighty 'ièconds. Because th~ nodes start at
651°C, the heat extracted from the liquid is solely attributed to the system's latent heat.
The temperature profile at any given instant would consist of severa! state 3 nodes at 651°C,
one node betwe~n the liquidus and solidus, and the remaining as state 1 nodes (Fig. 5.8).
700
600
.6 500
~ ~ 400
i ~ 4) a. 300 E {!
200
100
0
o 00
1-0 Solidification of Alumnum
solid
o 02
.. solid-liquid
interface
liquid
o 04 o 06 o 08
Length (m)
o 10
Figure 5.8: Typical profile generated from the Stefan simulation.
A temperature printout of the nodes was generated at one second intervals. The
position ol the translorming no de was noted and the inlormation was used to plot the
90
l posItion of the solidIfication front \'ersus time (Fig. 5.9). The pO~ltlOn n'r~us tUllt' d.iLt
was proce~sed using a curve fitting algonthm and the square root of tllllt' ilq>t'udt'Ill (' \\ .I~
confirmed.
l-D Solidificaliun of AlulUilllUH 111e Slefan Problem
010.------------------------- ------------
... § 006 .!: ... B 006 ;':l i:l
":1
~ 004 (1]
.... c
§ 002 ... :n (,
0..
J .J
O.OO+----,------,-------,-----,.------r-----,----,.---o 10 20 30 40 50 60 70 80
Time (seconds)
Figure 5.9: Plot showing dependence of solidification front position on time.
5.6 Phase Transformations in Profile Nodes
The algorithm for genenting a profile through a cross section of a multidimension.ù
configuration was described in Chapter 4. Because the algorithm uses a dimensionless en
thalpy or mass to calculate temperatures or concentrations for the profile nodes, phtl.!'le
transformations in multidimensional systems can be modelled.
Consider the coupling of two principal axis nodes to form a profile node in a hl'dt
transfer problem. The enthalpy of the profile node at any given time is calcuIated by mul-
91
tiplying the d.imen~lOnless enthalpies of the participating prIncipal aXIS nodes by the profile
uode \ InItIal enthalpy If the temperat ure calculated for the profile no de does not faU wlthin
the range corrcspundlllg to ItS cuuent state, a phase transformatlOn wIll take place. The
enthalpy would be removed or added in stages along with the necessary substItutIOns bemg
made for the matenal properties. The procedure is the same as that outlined fûr the prinCI
pal a.xlS nodes. Figure 5.10 illustrates how the algonthm handles a phase transformation of
il. profile node.
M~ ct 1 iI!IlSiOI _5
~dpa'ed
pTq)aI ads nodes
no
P,dle nodI t.,.....laI.Ie ,BQ.IIS ro Iuttw
~s
(il change the erthaPt to !he pen thIt t iS btcu;tt to !he still!! bo.rdaty
(~~er1l!lClst"e propertles rd ,emcve rerraT1g~
Figure 5.10: Flowcha.rt. showing how FASTP a.ccurately preructs a. profile node's temperature change across a phase transformation (state) boundary.
5.7 Phase Transformations at an Interface
When a hot liquid is brought into contact with a relatively cooler solid, situations may
arise where the liquid at the interface will solidify instaD taneously. Recall from Chapter 3 that
92
prior ta beginrung a simulation. readjustments must take place d.t a.n illtt'rfa.ce hl,twt'cn !w\)
sechons with different starting temperatures to ensure an energy balance The rc.\djustuH'uts
are performed using the same procedure of lllcremen tal enthalpy H'mova} or additiou W1 th
the a.ppropnate material properties substitutions being made.
5.8 Phase Transformations in ~1ass Transfer
The phase transformation feature can also he used ta model dissolution type probll'lll~
in mass transfer. Ta illustrate how this feature would he used, the dissolution of the moditi,'r
strontium in aluminum-sllicon alloys \\'111 be consldered. The illustration makes use of hoth
the phase transformation and mixing features, the latter heing descnbed 1Il the UpUHluug
sections.
Strontium, with a melting temperature of ï69 oC, does not melt at the opt'r.ülIl).!;
temperatures for molten hypoeutectic aluminum-silicon allays (680 - ï30 OC), but disi>Olvl''>
instead. Typically, the strontium must be held submerged and stirred mto the melt to bptt"r
distribute the dissolved species. It is assumed that the system is well rruxed and th,lt tIti'
concentration of strontium in the aluminum, once dissolved, is uniform. Extending from tl1l'
solid surfa.ce, however, is a hypothetical boundary layer that is not mfiuenced by the bllik
fluid motion (Fig. 5.11).
FASTP would use a 1-D, single-section configuration to model this problem. The Il()(k
spacing would he chosen so that the convective conditions of the meIt are Illodellcd. Th,!
soli dus concentration would correspond to the concentration of the solute at the ~olid ~llr
face. In the case of a pure species the soli dus parameter would he assigncd the clensity (Jf
the pure species. The liquidus corresponds to the concentration of strontium dt the hypo
thetical houndary layer edge. As an approximation this value would he choscn as the hulk
concentration of the solute in the liqwd. State 3 properties exist beyond the boundary laY{'r,
93
~--------------------------------------------------------------
t
(
state 1 state 2 state 3 (weil stlrred)
~ c* ~- hypothetlcal boundary layer
1 n . . - ____ . - _____ . __________ . __________ . _______ . 1
1
1 sofich9
Figure 5.11: Mass transfer boundary layer extending hom solid's surface.
state 2 properties exist between the solidus and liquidus, and state 1 properties correspond
to the solid piece of strontium being dissolved. In heat transfer, by definition, the soli dus
tempeI'ature must always be below the liquidus. In mass transfer, two scena..;os can exist.
The first is that the local concentration is greater than the bulk concentration. In this case
the solidus is greater than the liquidus. The second is that the local concentration is less
than the bulk concentration. In this case the soli dus is less than the liquidus.
5.9 The Mixing Feature
Without any form of mixing specified, FASTP assumes the system to be stagnant
and that thermal or mass traJlsport takes place by conduction or diffusion only. The mixing
feature can be used to simuli\te the redistributive effects of bulk fluid motion on heat or mass
94
1
-.
transfer systems. The algorithm works in such a manner that only those Ilodes possessing
state 3 status are stirred.
The concept of mixing tune must first be addressed. The mixing time is the tiuH'
required to fully redistribute the energy or mass in a system such that all nodes posses:,ing
state 3 status in a section possess the same enthalpy or mass per unit volume. The wlxmg
tirne is chosen with the iteration time in mind. Recall that the iteration timc ultimatl'ly
controls how often the temperature or concentration of the nodes is updated in an anal)':,,:,.
If the mixing time is less than or equal to the Iteration time then, during the course of OIH'
iteration, the state 3 nodes will be mixed fully and aU will possess the same enthalpy or ma:,s
per unit volume. The degree of mixing can be controlled by choosing a mixing time grf.'ater
than the iteration time. Under su ch circumstances mixing is only partiaUy completed beforc
the next set of iterative calculations updates the nodes. Thus, a degree of control is rendered
over the extent to which convective processes can influence the system. Figure 5.12 furth"r
illustrates this point.
5.10 Comparison to an Exact Mixing Solution
To demoDstrate that the mixing algor' thm does work, the results of a faidy simple ~illl
ulation are compared to those obtained from an exact solution taken from Carslaw & Jaeger.
The configuration for the problem, shown in Figure 5.13, consists of a 1-D multiple
section arrangement where a solid plate is in contact with a fluid. Both sections of the
configuration are initially at a uniform temperature 0 •. At time t = 0 the face of the plate
at x = 0 is exposed to a fixed temperature boundary condition. Both the plate and the fiuid
heat up as a result. Because the fluid is perfectly mixed it heats up uniformly.
95
1 -- --
1
"
The exact solution describing this prohlem is given as folIows
where
and
0%,1 - e, = 1 _ f: 2(a~ + h2) exp( -aa~t) sin(a"x)
e· - e, n=1 an[l(a~ + h2 ) + hl
h = (pcp)plate
(pCpW)Jluid
(5.4)
(5.5)
(5.6)
The solution expresses the temperature of the plate in terms of a dimensionless value.
The solution requires that successive roots an first he found. The summation in Equation 5.4
is then carried out for successive roots until a stable dimensionless value is obtained. The
exact solution can be used to determine the temperature at any point in the plate.
With this solution, two nearly identical problems were modelIed. The only difference
between the two was the fluid used in the problemj water versus mercury. At time t = 0
the plate surface is exposed to a constant temperature boundary condition of 100 oC. In
both cases, the exact solution was used to determine the temperature of the plate at the
plate/fluid interface.
For the plate, the following were used
k = 363 W/m-oC, p = 8954 kg/m3 ,
section thickness = 0.03 m
starting temperature: node 1 = 100 oC, ail other nodes = 20 oC
The fluid properties, system dimensions, and starting conditions were as fol1ows
96
water:
mercury:
k = 0.6 W/m-oC
k = 11.0 W/m-oC
section thickness = 0.05 m
p = 995 kg/m3
p = 13475 kg 1m3
starting temperature = 20 oC
Cp = 4174 Jlkg- oC
cp = 138.4 JI kg-OC
Note the contrast in the fluid properties. For each of the problems the roots to Equa
tion 5.6 were determined using the Newton-Raphson method [6]. For the conditions dcscrilJ('d
in the problem the temperature at the platejfluid interface was found to change over the
first 100 seconds in the manner shown in Table 5.2. Thus, for both fluids the exact solution
and the FASTP solution were comparable.
Table 5.2: Comparison of results generated by FASTP and an exact solution for a perfectly mixed solution in contact with water.
WATER MERCURY
Time (s) (exact) (FASTP) (exact) (FASTP)
0 20 20 20 20
2 23.08 24.21 25.97 26.61
5 33.30 34.27 43.48 43.81
10 47.99 48.56 64.56 64.63
20 68.38 68.51 86.07 85.99
50 92.90 92.77 99.15 99.13
100 99.41 99.38 99.99 99.99
97
100 Conditions prior
+ to mixing ,C,)
~ 50
+
~ 25 +
• • • 2 3
/ ~ Full Redstribution Partial Redistribution
.C,) 58 3 58 3 583 . (.) .. ~
• • • ~ +
~ t-
• • • • • • 1
-t-- 1 2 3 --r- 1 2 3 1
1
Mixing time c rteration time 1 Mixing time , Iteration tine 1
Figure 5.12: mustration of full vs. partial mixing during one iteration.
(
98
e* 100 'c
plate
sect. 1
4 nodes
fluid
sect.2
5 nodes .- ---.-----.----.- --- .------. - -- -. • q = 0
zero flux
,
'8 = 20 'c : 1 e = 20 'c
1
x=o 1 = 0.03 m I+w = 0.08 m
Figure 5.13: Configuration used by FASTP ta model the mixing problem_
99
Chapter 6
Conclusions
F ASTP is an interactive, user friendly software package designed to solve both tran
sient heat and mass transfer problems. It is the result of over five years of development.
The program contains the relevant partial differential equations pertaining to both types of
transport phenomenon, which are solved using an explicit finite difference technique. The
package can be used quickly and economically to simulate a process. Moreover, it can be
used to quantify the effect of operating parameters on the overall process.
This thesis has described the many features that are available to anyone wishing to use
FASTP. To summarize, FASTP can handle cartesian, cylindrical, and spherical configura
tions. The program can handle a. wide range of boundary conditions, includingj convection,
radiation, zero flux, and constant flux. It can handle both heat and mass generation terms
and can incorporate interboundary resistances and partition coefficients. It can simulate
stirring and moving boundaries, ami can simulate phase changes. The results generated by
FASTP have been thoroughly checked against exact solutions, rather than experimental
data, because they are accepted as correct. This verification procedure has validated the
integrity and accuracy of FASTP's algorithms.
The use of FASTP is primarily oriented towards chemical, mechanical, and metallur-
100
gical engineers in the fields of research and process development, but is by no meallS limitt'd
to them. It can also be used as a teaching aid at the university level. The ouly prereq\lll>ite
for using FASTP is that the user must have a general knowledge of heat 1 mass transft>r.
FASTP can be run on any PC-compatible machine. The menu screellS guidt> t IH'
user through ail aspects of problem configuration and simulation, ail the wlule exteuslVt'ly
checking and validating the input. The program is also equipped with il facility (or lll,ÙIl
taining "l. database and also ailows for the storage of input data {or future use In addition,
FASTP can generate several types of graphs. The second half of this tbesis is an opcratoù
manual, which describes the program's features in detail.
101
.---------~--~ - , (
Part II
The FASTP Manual
102
--~~~-~=-- il
-- - -- - -- - -• - - - -- - -- - - - -- - - - - -- - - - - -- - -FaC11ity for the Analysis of Systems ln Transport Phenomena
lb=====================-=-=_--==~~~
A Heat and Mass Transfer Software Package Developed by:
Frank Mueeiardi &t. Murray J. Brown
McGill University
Department of Mining and Metallurgieal Engineering
3450 University Street Montreal, Quebec, Canada
H3A 2A7
103
Disclaimer
Processes involving heat and mass transfer are complex in nature. Since simulations
using FASTP are highly dependent on the user's input values and their interpretation of
the resulting data generated by FASTP, the author cannot be held responsihle {or any
consequences resulting from the use of FASTP. However, the author assures the user that
through years of development and extensive verification of the algorithms against analytlcal
solutions, FASTP has been found to be very reliable and accurate. FASTP is a unique and
extremely powerful software package that can he of great benefit to engineers, scientists, and
operations personnel. As far as the author knows, no other software of this kind, designed
specifically for use on a PC, is available on the market.
104
Chapter 7
Getting Started
7.1 Introductory Remarks
FASTP (ver 2.0) consists of four wsk/~ttes, a user's manual, and a BLOCK sccunty
device. The package can he run frOIl1 disk drives or a hard wsk. Unless otherwise .,tated,
this manual will assume that the files from all wskettes reside on a hard wsk. This chaptt'r
descrihes the procedure required tu set up the package prior to running any ~i1mulations.
7.2 Backup Copies
It is strongly advised that backup copies of the entire FASTP package he made. ThcM~
hackups should he used as the working copy of the package and the originals should be storecl
in a safe place. Copies should he made even if the user plans to run the package from a hard
wsk. Consult a DOS manual on how to make hackup copies.
105
1 7.3 Hardware & Software Requirements
• IB~! PC\ XT\ AT\ 386 or IB)'-! compatible
• BOxB7 math coprocessor chip
.380 K memory (minimum)
• Parallel prin ter port
• DOS v3.0 or above
• CGA, EGA, VGA, or Olivetti graphies adaptor.
* The Hercules graplucs adaptor does not
currently run FASTP's graphies facility.
7.4 The Block Security Deviee
Prior to running the FASTP package, the BLOCK security device must be plugged into
the computer's parallel prin ter port. The package will not Iun if this device is not in place.
If you have a parallel printer, the connector that originally plugged into the parallel
port will now plug onto the back of the BLaCK device. The BLaCK will not hinder the
operation of the parallel printer.
The BLaCK is a trademark of:
Software Security, 870 High rudge Rd. Stamford, CT 06905
106
7.5 Hard Disk Installation
Setup of version 2.0 of this package has been simplified with the incorporation of
the setup program; FPSETUP.EXE. To run the setup program, tnsert disk -1 and tyP('
FPSETlJP. You will then be asked a series of questions about the computer on wruch you
intend to run the program. The setup program explains the remaming steps that must hl'
taken to install the package.
7.6 Running FASTP From Disk Drives
• Diskettes 1,2, and 3 are alternately placed in drive <A> as outlined by the program's main menu .
• Diskette 4 remains in drive <B> for the duration of a FASTP session.
Please note that the program HP PLOT .EXE resides on diskette 4.
7.7 File Breakdown
Diskette 1 Diskette 2 Diskette 3 Diskette 4
FASTPl.EXE CALSl.EXE FASTPLTl.EXE FAST3.FIL
EXPORTl.EXE DBASE.TRM
DBASE.MOL
HPI'LOT l.EXE
HPPLOT.PRO
FP.EXE
FPSETUP .EXE
TEST.PLT
Data entry, Caleulations Graphies Support editing and ASCII diskette
file export facility
107
, "
r
HPPLOT.EXE and the plotting subroutines used in FASTPLT.EXE were provided by
the McGill University Geophysics Department. For more information write to the following
address
McGill University Geophysics Department, Attn. Dr. David J. Crossley Plotting subroutines for Microsoft FORTRAN 77 F.D.A. Bldg., rm. 238 3450 University St., Montreal, P.Q. H3A 2A7
108
Chapter 8
Program 1: FASTP [Data entry]
8.1 Introductory Remarks
AIl FASTP simulations hegin with FASTP [Data entry). It is with this program that
the user configures a model of the prohlem to he studied. This section will outline the options
ava.ilable for configuring a FASTP model. The discussion will follow the sequential order in
which the screens appear while running FASTP [Data entry). The flowchart (Fig. 8.1) is
intended to provide an overviewof the ava.ilable options. FASTP [Data entry] also enables
access to the datahase facility. This facility will also he descrihed in this chapter.
To run FASTP [Data entry), FASTP must he selected from the Main Menu screen.
A ... Loading FASTP message will appear on screen. While this message is displayed, the
program FASTP1.EXE is initiated. The program is ready to run as soon as the FASTP logo
is displayed (Fig. 8.2)
8.2 Preliminary Screens
Before the availableoptions for configuring a problem are presented, severa! preliminary
109
StlWt
Ert. tlctf ...... Il sedkIns
Figure 8.1: Flowchart showing sequence of informa.tion supplied when configuring a problem with FASTP [Data entry].
-- - -- - -- - -• - - - -- - -- -- - -- - - - - -- - - - - - -- - -Facility for the Analysls of Systems in Transport Phenomena
Figure 8.2: FASTP logo.
110
screens must be discussed. Pressing [F6 ContI twice will cause the following to appear
Choos8 an option
1 FASTP <Data entry> 1
Database
Choose FASTP [Data entry]. The database facility is described in Section 8.9. The
next question will then appear
Do you vant to input data from an 8xisting file? y8S Inol
The defatÙt [no] is selected. The [yes] option is discussed in Section 8.8.5. Following
the file question, the Problem Description screen will appear in which a title and description
of the configuration are requested.
Enter a title for your analysis :
Enter a description of the analysis
A title and description can be entered, but are not required. Input for this screen can
be ignored by pressing [F6 Cont] to continue.
Note, two keys are of special importance. They are [F10 Page-back] and [Esc]. The
[FlO] key allows the user to backstep through the screens. This key is useftÙ when a mistakc
has been made while configuring a problem. Pressing the [Esc1 key cancels execution of any
of the programs at any time without saving the input. When the [Esc] key is pressed the
program will terminate and return the computer to the main menu (Fig. 8.3), but not before
requesting confirmation that the user did indeed wish to terminate the program. Please note
as weIl that the [Enter] key performs the same function as the [F61 key.
r
-- - -- - -- - - -• -- - -- - - -- - - - -- - - - - -- - - - - - -- - -".clllty for the Analye1. of sy.ta •• ln Transport Ph.na ... na
,------- ----------------'
IFASTP CALS FASTPLT HPPLOT EXPORT SETUpl __ 1
Figure 8.3: FASTP Main Menu screen.
8.3 Geometry
Part lof this thesis described how FASTP can model transient heat or mass transfer
behaviour in cylinders, slabs, and spheres in up to three dimensions. Table 8.1 reviews the
geometries that can be handled by FASTP.
Table 8.1: Available geometries {or model configurations.
1 Geometry Imax.no.dim·1 I-D 2-D 3-D
Cylinder 2 circle cylinder -
Slab 3 line rectangle cube
Sphere 1 sphere - -
The first step in setting up a simulation is to choose the appropriate coordinate system,
physical dimensions, and nodal progression(s). The procedure and the available options are
herein described.
112
8.3.1 Problem definition
In the Problem Definihon screen shown in Figure 8.4, the three questions in the top
window will appear first. Selection of the appropriate option descrihing the prohlcm I11ll~t
he highlighted using the cursor keys [-,-+] followed by [F6 Select].
PROBLEM DEFINIrrON
Choose problem type: H~at transfer Mass transfer
The unit convent~on to be used: S.I. Br~tlsh
Coordinate system to be used: Cylinder Slab Sphere
Enter the number of dlmenslons? 2 1 How many sections ln each dimenslon ?
Dimenslon 1: 1 Dlmenslon 2: 1
Figure 8.4: Prohlem Definition sereen.
The selection will remain lit and the cursor bar will jump to the next question. The
same procedure is carried out. Following the u.nits question the request will he made whether
or not prompting for units can he suppressed.
SuppreS8 prompting for uni ta: 1 yesl no
The feature was implemented as a means of checking the units for energy, temperature,
length, and time while the configuration is heing input. The program will check the unit~
that were input against those that were specified in the datahase. Should the uni ts not
match, a warning will he displayed with a request to take appIOpriate action. It is the
113
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responsibility of the user to maintain a consistent set of units. After the coordinate system
has been specified, a second window will appear requesting the number of dimensions and
llumber of sections in each dimension of the configuration. A maximum of five (5) sections
are allowed per dimension. Figure 8.5 shows how a 2-D slab configuration, po&sessing one
section per dimension, can he visualized. Once an parameters have heen. entered ta this
screen, press [F2 Re-ntr] to re-enter the screen parameters or [F6 Gont] ta continue with
data entry. BC2
•• dom 1 - sect 1
BC1 - - BC2
t ~ .. 0
N
.5
BC1
Figure 8.5: 2-D Slah Configuration (1 section per dimension).
Note, the unit convention question instructs the program to convert an temperature
entries from oC to Keh;n (S.I.) or oF ta Rankine (British). Mass transfer simulations are
not affected by the unit convention question.
8.3.2 System Dimensions screen (1)
ln the System Dimensions screen shown in Figure 8.6 the thickness and number of
nodes in each section of each dimension are assigned. An additional parameter, the outer
radius, will be requested when either the cylindrical OI spherical configuration options have
been selected. The outer radius always corresponds t<J dimension 1 of the configuration.
Many illustrations of other configurations are given in Part 1 of the thesis.
114
Sectlon 1:
Sectlon 2:
D.l.menS10n 1
Sectlon Thlckness
0,05
0.1
DlmenSlon 2
# of Sectlon nodes Thlckness
7 2.0
14 1.5
# of nodes
12
12
S '{STEM D n'E~J'; : ~ "
Figure 8.6: System Dimensions screen (1) - section lengths and numher of nodes.
A minimum of three (3) and maximum of twenty (20) nodes can he assigned to a.
section. It is advised that while an increased number of nodes will add to the accuracy of
the simulation, it will also add to the execution time required to perform the analysis. Note
that the units corresponding to a section's thickness and outer radius tùtimately depcnd un
the units used to quantify that section's thermal or molectÙar properties. Thesc propcrties
are selected from the database and their units should be kept in mind when assigning the
physical dimensions of the sections. For example, a configuration making use of a thermal
conductivity with units WJm-oC wotÙd require that any corresponding lengths he specifil'd
in terms of metres (m). Once aU parameters have been entered to the System Dtmens'Lons
screen the option is presented to [F2 Re-ntr] or [F6 Cont J.
8.3.3 System Dimensions screen (2)
ln the second System Dimensions screen the spacing between the nodes in each section
of each dimension is assigned. The nodal spacing options are shown in Figure 8.7. Pa.rt
1 of the thesis explains that in FASTP's finite difference scheme nodal points rcpresent
115
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disrrete locations within the section at which values for either the temperature or mass
concentration are calculated on an iterat~ve basis. Nodal points are located at the geometric
center of a suhsection except for nodal points 1 and n, which are located on houndaries of
the corresponding sections.
SECTION l OF DIMENSION l
NODE PROGRESSION Equlspaeed Arlthmetlc Geometrie Comblnatlon '{our own
Select Progression Type
SYSTEM DI~EtIS l 0:15
RESULTS
jNode Node tI Position
1 • 00000 2 .83333E-02 3 .16667E-Ol 4 • 25000E-0 l 5 .33333E-Ol 6 .4166 7E-O l 7 .50000E-Ol
Figure 8.7: System Dimensions screen (2) - nodal point progression selection.
To select the nodal progression, highlight the choice using the [T, ~] cursor keys and
press [F6 Select]. Additional pa.rameters must he supplied with ail progression options with
the exception of the equ1.8paced option. Once these parameters have been supplied the SCIeen
will display the nodal point numbers and their corresponding posithns relative to no de 1 of
that section in the results box. The user can then [Fe Re-ntr] re-enter the nodal progression
for the given section or [F6 Cont] continue with data entry. Once the nodal progressions for
all sections ha.ve been specified, a node report can be sent to the printer.
Bode report to printer? yes ~
Figure 8.8 shows a sample node report. In addition to the relative nodal positions, the
areas and volumes associated with each no de are also printed.
116
8.4 Boundary Conditions
FASTP allows for the application of several types of boundary conditions to the configuration
The options are listed in Table 8.2. Boundary conditions are assigned to the sltrface nodt'S
of the configuration. A surface node is defined as anode that is bounded by oruy one utlH'r
node and that is not bounded by another section. Figure 8.9 shows the Boundary Conditw1/.~
screen that appears for heat transfer problems.
Only one dimension is treated per screen. Each dimension has two surface nod<,:.;
node 1 and node n. Even surface node n of a solid cylinder or sphere must be assigned cl
boundary condition. Typically, the zero heat flux option would be selected. The Bounda11}
Conditions screen will renew itself as many times as there are dimensions speafied for the
configuration. Window [A] corresponds to node 1 of section l, while window [BI corresponds
to the last node of section n. The [Make a selection] message appears in the window for
which the boundary conditions are to be assigned. The [j, tJ cursor keys are used to highlight
the boundary condition( s) that are to be applied to the surface. The selection is made hy
pressing [F6 Cont J.
Once a boundary condition option has been selected, a series of questions relating to
that particular boundary environment will be asked. A complete list of the relevant questions
is a.lso given in Table 8.2.
8.4.1 The boundary condition options
1. Fixed surface temperature or concentr,tion boundary condition
With this option a fixed temperature or concentration can be assigned to the boundary
node. The value that is assigned will remain for the duration of the analysis. This i!)
analogous to having a surface energy or mass flux sufficient to maintain the node at the
111
r
SET-OP OF !IODES
T!':':..E CESCt:(IPT:::l·
SEc,:,rO/l l Of' OIMEl/SIOII
:IODE :IODE :10 POSITION 1 00000 2 833JJE-Ol 3 16667 4 25000 5 .33333 6 41667 7 SOOOO
l
RADIUS OF BOUl/CARY
SECTION l Of' DIMENSION 2
NOCE NODE RADIUS OF No POSITION BOUNCARY 1 00000 ;; .66667E-Ol 3 13333 4 .20000
DIMEIISION CENTRAL SECTION" DIME/lSIC'N CENTRAL SECTION-
NODE SPACI:IG .S3333E-Ol .S3JJ3E-Ol .S3333E-Ol .S3333E-Ol .S3333E-Ol .S3333E-Ol .00000
NODE SPACING .66667E-Ol .66667E-Ol .66667E-Ol .00000
CENTRAL NODE" CENTRAL NODE"
AREA
1. 0000 1. 0000 1. 0000 1.0000 1.0000 1.0000 1. 0000
AREA
1. 0000 1. 0000 1. 0000 1. 0000
.41667E-01 83333E-01
.833331:-01 • S3333E-Cl
83333E-Ol .8333JE-·Jl .41667E-·)l
VOLCME
.J3333E-Ol
.66667E-01 66667E-01
.33333E-01
Figure 8.8: Sample node report from printer.
DlmenS10n 1 - Surface Node 1: Enter convective he~t transfo coef. 50 Enter convectlve ambient tempo 20
DlmenSlon 1 - Surface Nod~ N:
1 MAKE SELECTION
BOUNDARY CONDITIONS
i
1 OPTIONS
Fixed Surface Temp. Convectlve Radiative Conv. + Rad. Zero flux Surface Heat Flux
Surface Heat Flux (+ Conv. and/or Rad
Figure 8.9: Boundary Conditions screen.
118
Table 8.2: Relevant parameters to boundary condition options
1. Fixed surface tempo
2. Convective
3. Radiative
Heat Transfer
Enter a fixed surface temperature
Enter convective heat transfo coef. Enter convective ambient tempo
Emissivity of surface: # of surrounding surfaces Enter view factors: Enter temperatures:
4. Convective & Radiative Combination of (2) & (3) above
5. Adiabatic
6. Constant Heat Flux (adiabatic )
7. Constant Heat Flux (with surface losses)
1. Fixed Surface Cone.
2. Convective
3. Constant Mass Flux (adiabatic)
7. Constant Mass Flux (with surface los ses )
No questions asked
Enter surface flux (per unit area):
combination of (6) & (2,3,4 or 5)
Mass Transfer
Enter a fixed surface concentration
Enter convective mass trans. coef. Enter convective ambient conc.
Enter surface flux (per unit area):
combination of (4) & (2)
119
------------ .---
1-
(
specificd temperature (concentration).
2. Convection boundary condition
Convection conditions are applied to the surface node when this option is selected.
Values must he assigned for the heat or mass transfer coefficient and the amhient temperature
or concentration.
Note that when a 1-D or 2-D problem has heen configured the length of the miss
ing dimensions are assigned unity. For example, if a 1-D slab configuration is assigned
material properties in '.vhich the unit of length is metres (m), the missing dimensions
(i.e. Dims. 2 and 3) will each he assigned a length of one metre. The cross-sectional area
associated with the surface no de will be 1 m2•
3. Radiation boundary condition
For heat transfer problems, radiation boundary conditions can be specified when this
option is chosen. Up to five (5) radiation sources (sinks) can be specified at anyone sudace
boundary. Each reqUÎres a view factor and temperature. The view factor can he assigned any
value between zero and one. An emissivity f must also be specified for the surface houndary.
No option analogous to radiation exists for mass transfer. Figure 8.10 illustrates the use of
the multiple radiation source boundary condition option.
4. Combined Convection &l. Radiation boundary condition
This option allows for combination of options 2 and 3.
5. Zero Flux boundary condition
The zero flux option should be selected whenever no net heat or mass transfer across
the surface boundary is desired. Selection of this option effectively insulates the boundary.
This option requires that an ambient temperature or concentration be specified even though
120
1 V F. =0.2 ----~ - ( 1 )
T ... SO C
V F .. 0.3 ~-y ----
( 2 )
T.50 'c
F .0.2
( 3 )
T.100 C
80undary Surface
Emissivity = 0 . 8 # of radiating surfaces = 3
Figure 8.10: Multiple radiation sources illustration.
it does not figure in the calculations. When assigning an amhient value, enter the same value
assigned to the other boundaries.
6. - 7. Constant Flux (with/without surface losses)
\Vith either of the constant flux options, FASTP allows for the application of a constant
flux to the boundary surface. The surface node's associated cross-sectional area must he kept
in mind when assigning a constant flux value.
An additional boundary condition can be selected in combination with option 7. Hav
ing selected option 7, a second cursor will appear that will be used to highlight the comple
mentaryoption. Options 2, 3, or 4 can he selected. Values for the constant flux and the
parameters for the second option will be then requested.
8.4.2 Batch mode operation
", Batch mode operation is an advanced feature allowing the user to set up a. simulation in
121
,.
which the applied boundary conditions change as the simulation progresses. Dynamically
changing environments can thus be modelled with FASTP. The boundary conditions can be
set to change at predetermined times or pending the attainment of a specifie temperature
or concentration at the boundary surface. In an analysis, up to twenty (20) time dependent
boundary conditions can be assigned for each surface. For each time a maximum of ten (10)
temperature dependent boundary conditions can be put into place. In Part l (Chapter 2) an
illustration of the batch mode feature was given. The steps required to enter the boundary
condition parameters are now presented to give the reader a feel for how to work with this
feature.
Conftguring a problem using the bat ch mode feature
Example 2-4 in Section 2.9.1 describes a problem whereby a 4.5 cm thick slab is exposed
on one side to a variety of boundary conditions at two minute intervals. The properties and
boundary conditions were therein described. This section focuses only on the batch entries
made from the Boundary Conditions screen.
From the Boundary Conditions screen:
Pre lUI <F7 SlTCB>
Question: Enter file n ... and path where batch will b. saved
Make boundary condition selection * Conv. + R~d. *
Enter convection and radiation
Make boundary condition selection * Convection *
122
~ Creates a . BCH file.
~ Choose initial boundary condition option <ENTER>.
~ Select <F2 RE-NTR> if value was incorrectly typed.
~ Choose boundary condition corresponding to the second time interval <ENTER>.
Enter time 2: < 120 >
Is there also a temperature
Enter convection parametera correBponding
Make boundary condition selection * Convection -
Enter time 3: < 240 >
la there a180 a tamperature
Enter convection and radiation
-+ Time 1S entered in seconds.
-+ Boundary condi tion remains fixed for duration.
-+ Select <F2 RE-NTR> if value was incorrectly typed.
-+ Choose boundary con di tion corresponding to the third timc interval <ENTER>.
-+ Time is en tered in seconds.
-+ Boundary condition remains fixed for duration.
-+ Select <F2 RE-NTR> if value was incorrectly typed.
Press <F7 ODB!T> -+This will terminate batch programming for surface node 1 of this configuration. Surface node 10 is exposed to one boundary condition throughout and hence, does not require the batch mode feature.
123
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8.5 Material Properties
In the Material Properties screen shown in Figure 8.11 the material properties to be
assigned to the nodes in each section of the configuration are selected. The phase change
status is also assigned.
Section: 1
sectlon:2
Dimension l Dimension 2
Material Code
Al
test
Phase Material Change Cude
y test
n
Phi\se change
N
Figure 8.11: Material Properties screen.
MATERIAL PROPERTIES
The material properties are stored under a user defined material code in FASTP's
database. The code, which can contain a maximum of eight characters, is used to transfer
the materia.l properties from the database to the configuration. The database differen tiates
between upper and lower case letters. A full description of how to use the database facility
is given in the Section 7.9.
Two slots are associated with each section of each dimension. For any section the
material properties are assigned by typing the material code into the first slot. Once a
material code has been entered the database will be se:uched. If the material <:ode matches
one in the database, the associated properties will be transferred and assigned to every node
124
in that section. If no match can be found a warning message will he displayed.
Material code not found in file
Two options are then presented. The [F2 Re-ntr] option is selected if the material code
was incorrectly typed and the user wishes simply to retype the correct code. The [F3 Fastptlb]
option enables temporary access to the database facility to verify or edit the entries in the
datahase file. To exit the database press [F9 Quit] repeatedly until the Matenal Propertte.~
screen reappears. Data entry can then continue from this point.
Once the material has been selected - and accepted - the phase change status is
assigned for that section. This is done by typing a yes [y] or no ln] answer in the second
slot. A [y] should be specified if the three sets of material properties are to he used during
the simulation (i.e. state 1, state 2, and state 3). A [n] should be specified if only ont> set
of material properties is to apply for the duration of the analysis. When [n] is specified, the
execution time required for the simulation is reduced. The phase change option has been
used to model solidification hehaviour in met aIs and alloys, hut may also he used to model
systems where a change in the thermal properties over a wide temperature range. The reader
is referred to Chapter 5 for a description of the phase transformation algorithm.
8.6 Starting Temperatures / Concentrations
Figure 8.12 shows the Starting Temperatures (Concentrations) screen. In it, the starting
temperatures or concentrations are assigned to the individual nodes in each section of the
configuration.
The program will sequentiaUy display the configuration, section by section, until initial
values have been assigned to aU of the nodes. The dimension, section, number of nodes,
125
c iDlmenSlon: 11
! Iisectlon: l
1 Nades: 7
Materlal: Al
State: transfor.
STARTING TEMP. PROFILES
Startlng temperatures:
l 2 3 4 5
6 7
options: Î Unlform profile specify individual nades
Figure 8.12: Starting Temperatures screen.
values have been assigned to all of the nodes. The dimension, section, number of nodes,
material code, a.nd phase change status are also displayed i.n the windows for reference. One
of two options ca.n be selected when assigning the initial. temperatures or concentrations.
The uniform profile option is selected if the nodes in a section all possess the same starting
value. When this option is selected, the starting value entered for node 1 of a section will
be assigned to the entire section. The specify individu al nodes option is used if the nodes
in a section are to be assigned non-uniform values. When this option is selected, the initial
values for nodes 1 through n are requested in sequence. After the initial values have been
assigned to the section, the option is given to [F2 Re-ntr] re-enter or [F6 Cont] continue
with data entry.
8.7 Other A vailable Options
The rema.inder of this chapter focuses on those configuration options that need not be
( specified, but are available.
126
., ,
·,
----.----------------------------------
8. T.1 Interboundary resistances / partition coefficients
Interboundary resistances due to less than pedect contact between two sections can
be modelled using FASTP. The reader is referred Chapter 4 for more information on how
the resistance algorithm works. The analogous feature in mass transfer is the partition
coefficient. When a configuration possesses more than one section in a given dimension, the
following question will be asked
Does your problem involve an Interboundary Resistance yes Inol
The Interboundary Resistances (Partition Coefficients) screen shown in Figure 8.13 will
appear if the [yes] option is selected. The screen will display aU of the interboundaries in tbe
configuration. The cursor is initially placed at the first interboundary. An interboundary is
targeted for a resistance or partition coefficient by moving the cursor to the desired position
and pressing [F5 Toggle]. The selection will bp. highlighted and the following will appear at
the base of the screen
What is the interboundary resistance value? :~
Note that pressing [F5 Toggle] at the location where an interboundary resistance or
partition coefficient has already been specified will cancel the value that was entered. If no
further values are to be assigned, the screen can be exited by pressing [F9 Qutt].
Currently, oilly fixed partition coefficients can be applied. However, modification to
the algorithm to accommodate fioating partition coefficients is being undertaken. A fioating
partition coefficient will change with surface concentration to refiect the solute's changing
activity coefficient. An intended application of this feature will be to model the transfer of
species at a slag-metal interface.
127
.. _---._-------------------------------
l
\
INTERBOUNDARY RESISTANCES
D1menS1on 1 DimenS10n 2
/sect.l Sect.2 Sect.l Sect.2
1
Sect.l Sect.2 Al test
phase change yes no
Use cursor keys, press F5 where 1nterboundary resistance 1S located
Figure 8.13: Interboundary Resista.nces screen.
8.7.2 Heat generation / Mass accumulation
The heat generation (mass accumulation) option is used to target individual nodes as
heat generation or mass accumulation sources. The following question will appear while
configuring a problem
100e8 your problam involve a ht. gen. tarm(s)? ya8 Inol!
If (yes] is selected, FASTP will display the Heat Generation (Mas! Accumulation) screen
shown in Figure 8.14. The window on the left ha.nd side of the screen displays a section by
section breakdown of the configuration; one dimension at a. time. The cursor indicates the
section currently being considered for a heat generation or mass accumulation term. The
cursor is initially positioned at section 1, dimension 1. The central window displays the
material code of the material properties a.nd the state assigned to the current section. The
right ha.nd window displays the central sections of each dimension.
By repeatedly pressing the (~l key the cursor will move past the lowermost section
number on the screen a.nd cause the next dimension's sections, if a.ny, to be displayed. To
128
, f
IIEt\ r GE:H.Ft\ t 1 .',
DlmenSlon l Materlal: 1
Centr~l Sc~tl0ns Al
ISectlon Nodes(#) Distrlbutlon State:
l 7 nomal
2 14 nomal
changlng l ~
2
Central M,~tert,ll test
Use cursor keys and enter (Y) if section contains heat generatlon term
il============================ __ =.=_
Figure 8.14: Heat generation screen.
exit the screen continue to press the [ll key until the active cursor has moved past the last
section of the last dimension. Press [F6 ContI to exit from the screen. When [yI is entf~f(·<l
for a section a follow-up question will appear requesting the numher of nodes in that section
to he assigned heat gcneration or mass accumulation terms
IHo. many node •• ill you be apeeifying ? c=J1
The question is followed by
Enter the node number and heat generation term
This question will reappear as many times as was specified in the preceding question.
Note that a warning will appear if a heat generation or mass accumulation term h~
been specified on a central section of a multi-dimensional configuration. The warning takes
the form of a suggestion that the term be distrihuted among nodes in the other central
sections. The warning cao he disregarded by entering [F6 Cont}.
129
*Warning* Vou are selecting a
node(s) in a central section.
You may vant to consider distributing
tel~ ameng nodes in other cen. sections
Note that surface nodes cannot be targeted for heat generation or mass accumulation
terms. Entering a consta.nt flux from the Boundary Conditions screen would he equivalent
to specifying such a term.
8.7.3 Mixing
The mixing feature is used to model the redistrihutive effect of mixing in a fluid system.
Without mixing specified, FASTP will assume that the system is stagnant and that heat or
mass transfer will he due solely to the material's diffusivity properties. The reader is referred
to Chapter 5 for more information on the mixing algorithm.
FASTP will allow for the specification of a maximum of six stirred sections. Of course,
the maximum is further limited hy the numher of sections in the given configuration. The
following question will appear while configuring cl. prohlem
Ho. many sections are to be conaidered as stirred ?
1 nOilel
one
tvo
If the answer is one or greater, the Stimng screen shown in Figure 8.15 will he displayed
in which the dimension and section number of the stirred section are to he supplied, as well
as the first and last nodes considered stirred in that section. Only those nodes having been
130
1 assigned state 3 properties can be subjected to stirring. Entering zeroes (0) for the first and
last nodes being stirred will result in the stirring of all state 3 nodes lU that section. Onet'
anode is transformed from state 3 it is no longer subjected to stirring. Unless of course tIlt'
no de reverts back to its state 3 status.
rr=========================----- --
Enter the fOllowing: (nodes are first and last stlrred nodes)
Dim: 1 Sect: l First node: 0 Last node: 0
Dlm: Sect: First node: Last node:
IEnter mlxlng tlme for Dlm: 1 Sect: l 1.0
Figure 8.15: Stirring screen.
Having entered the dimension, section, and first and last stirred nodes, one of two
messages will appear:
1. A request will be made for a mixing time for the given section.
Enter mixing time for Dim: 1 Sect: 1 c::::J
2. A message will be displayed indicating that no mixing time is required. This message
will appear if one of the boundaries was specified as stirred and had been assigned a
fixed boundary condition. (i.e. fixed surface temperature / concentration)
Sect: 1 Dim: 1 is vell stirred. no mixing time necessary
131
l
The mixing time is the time required to fully mix designated state 3 nodes. What is
implied by the mixing time is that aU stirred nodes will possess the same enthalpy or mass
concentration at the end of the mixing time. Recall that FASTP's finite difference algorithm
works on an iterative hasis. The temperatures or concentrations of the nodes are updated
at a specified time increment, which can he assigned a value that is greater than, equal to,
or less than, the mixing time. Time increments that are greater than or equal to the mixing
time will result in full mixing over the course of one iteration. When the time increment is
less than the mixing time, only partial mixing will result.
8.7.4 Product solution - Interaction solution
Two methods of solution are availahle to the user when a configuration involving more
than one dimension is being analyzed. These are the product solution and interaction solu
tion. The following will appear when configuring a 2-D or 3-D problem
Do you vant to analyze thia problem
uaing the product solution technique?: Iyesl no
The product solution option is used to generate approximations of temperature profiles
through the central section of a configuration. To better understand this feature se~eral terms
are reviewed from Chapter 3.
Prmc'&pal a:ns nodes are those nodes to which all properties are assigned in the screens
described in the previous sections of this chapter. Profile nodes are nodes whose positions
have been derived from the extrapolation of two ('2-D) or three (3-D) principal axis nodes.
Figure 8.16 illustrates the relationship between principal axis nodes and profile nodes. The
collection of profile nodes for ms what is referred to as a profile. The nine by nine profile in this
132
1 illustration is created from a projection of two nine-node principal axes. The configuration
depicted is 2-D cartesian with one section per dimension.
------'
.-, • '. "
.' - - • • • • ._-..... --e---iII.r---< • .--.__--- •
1
1 0
1 , ,. 1
Profile node
.:. · .) 1 1
• •
Principal Axis node
C\J
E Â cS
----- .. Dlm 1
Figure 8.16: Profile grid generated using the product solution technique.
When a product solution is requested the algorithm is capable of using the values
calculated from the principal axes to generate an entire profile of values. In this example,
calculations on 18 principal axis nodes are used in effect to generate values for 81 profile
nodes. The results can be considered accurate provided the ambient temperatures or con
centrations specified at aU of the boundaries are the same. The one drawback with titis
scheme is that only one boundary condition can be applied to any one boundary at any
biven time. The advantage is the reduction in computation time required to generate values
for 18 nodes rather than 81. Section 9.2 describes the different ways in which the information
obtained from the profile can be presented.
The product solution is aptly named because the approach used to generate the profiles
is similar ta the classical product solution. In the analysis, the iterative calclliations arc
independently carried out on each of the principal axes in the configuration. Wh en a profile
is requested, the algorithm uses the two or three 1-D analyses to approximate a 2-D or
133
•
,(
3-D solution. Unlike the classical product solution, however, the algorithm is capable of
generating profiles that take multiple phases into account (i.e. states l, 2 and 3).
There are limitations to the applicability of the product solution. Non-representative
profiles will be geuerated un der two circumstancesj (i) a boundary has been assigned a fixed
boundary temperature or concentration, and (ii) the specified ambient conditions on all
surface boundaries are not equal. In the former case, the Collowing warning message will
appear
NOTE: Because one of yaur BOUNDARY temperatures is fixed --> PROFILE wlll NOT necessarlly correspond ta the stated problem. PLEASE CHECK THIS
It is important to note one other thing. The principal axis node values that are gener
ated are the result of 1-0 analyses. They cannot be used for direct comparison with results
generated from a 2-D or 3-D analysisj only the profile nodes cano
By default, if the product solution is not requested, the program will be set up for
generating an interaction solution. The interaction solution differs from the former in that
calculations on the principal axes depend on one another. Rather than a series of indepen
dently carried out 1-0 solutions that are combined to generate a 2-D or 3-D approximation,
the algorithm will generate a true multidimensiona.l solution. The locations within the con
figuration that can be studied are restricted to the positions of the principal axis nodes.
Profiles cannot be generated when an interaction solution has been requested.
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---- ---------------
8.8 Option Menu: End of Data Entry
Data entry will be complete once the question(s) regarding stirIlng hal> becn answererl
(1-D analyses) or once the solution type has been requested (2-D and 3-D analyses). The
menu screen shown in Figure 8.17 will then appear. The desired option is chosen u!>ing the
cursor keys followed by [F6 Select].
F===========================~OPTr':"
Ed~t the lnput data
Execute simulation
stop e)(ecutlon
Summar'lze lnput data
Store lnput data
Figure 8.17: Option Menu screen.
8.8.1 Edit the input data
The edit option is used to modify or replace an existing configuration without retyping
the entire problem. Having chosen to edit existing data a new screen will appearj displaying
the edit menu, which is shown in Figure 8.18.
The cursor keys, followed by [F5 Toggle] are used to highlight the appropriate edit
options. Press [F5 Toggle] to instruct the program that a particular option is to be edited.
135
EDIT OPTIONS:
Section thlckness
Surface boundary conditions
Amblent condltions
Materlal propertles
Interboundary reslstances
Startinq temperatures l Heat qeneration
stl.rring
Product solutlon
No changes - proceed
1
1
1
1
1
i
Figure 8.18: Edit Menu screen.
EDIT r::Pl T
Pressing [F5 Toggle] a second time at the same location cancels the edit request and returns
the option to regular display mode.
Once all of the options to be edited have been highlighted press [F6 Cont] to continue.
The program will then page through only those screens selected for editing. Changes to the
configuration can subsequently be made.
8.8.2 Execute simulation
Selection of the execute option causes the program to write an data in memory to the
file FAST2.CFG. FASTP [Data entry} then terminates and the main menu reappears. CALS
is then selected from the menu to continue with the analysis.
8.8.3 Stop execution
The stop execution option terminates the program without storing the configuration.
136
This is a somewhat redundant option as pressing [Esc] will achieve the same desired cf{cct
at any point during the execution of FASTP, CALS, FASTPLT, or EXPORT.
8.8.4 Summarize input data
The summa7'1.ze data option is used to review ail configuration data entered into mcm
ory. Upon selection of this option the Summary screen will be displayed (Fig. 8.19). Cursor
keys are used to highlight the data to be viewed followed by [F6 Select]. A summary of the
data will be presented in the left hand window. Pressing [F9 Quit] will return the program
to the Option Menu screen.
F=============================~=================-,
IInput data ln memo~
Problem type: Heat transfer
Unlt conventlon: 5.1.
Geometry: Slab
Num. of Dlmenslons: 2
Solutlon type: Product
1 Resume options,
Sectlons / nodes
Boundary condltions
Startlng conditic~S
Materlal props. 1 State
Interbnd. reslstances
Heat generatlon
Prlnt report
Figure 8.19: Summary screen.
8.8.5 Store input data
The store option is used to store the configuration under a filename other than FAST2.CFG.
Following the prompt, a filename - including valid path - is requested. Warning mes!)ages
137
(
will appear if either the path is not round or the file already exists. Filenames created with
this option are also gi ven the . CFG extension.
If the configuration has been stored, it can be retrieved at a later date with relative
ease. To retrieve the file select the [yes] option to the file question of Section 8.2:
Do you vant to input data from an existing file? ~ no
The rollowing request will then appear
Enter the file name and path from vhich data is to be retrieved:
The path refers to any valid drive and subdirectory. The .CFG extension need not be
included. The program will search for and read the configuration file and then display the
Option Menu screen. The user is then free to edit, execute, or summarize the configuration.
This feature is especially useful when running a series of simulations with only slightly
ddfering parameters. The stored configuration can be retrieved as many times as is required
and modified using the edit the input data option. The modified configuration can also be
stored under another filename for later reference.
8.9 FASTP Database Facility
The database facility is designed as a timesaving fea.tuIe to be used in conjunction
with FASTP [Data entryJ. From the Material Properties screen a. material code is specified
for each section in the configuration. FASTP [Data entry] will use these codes to retrieve
the material property data. nom the database. This section will describe how to access and
manipulate the database.
138
, i
----------------------------------------
8.9.1 Accessing the database
From FASTP [Data entry] the database can be accessed from two points; at start-up
or from the Mate1'1.al Propert1.es screen. At the start of any FASTP session, the user will be
asked to make the following choice
Choose an option
FASTP (Data entry]
1 Database 1
To access the database highlight the database option and press [F6 Select]. The program
will then request which database is to be viewed in this session. The database menu shown
in Figure 8.20 will appear once the selection has been made.
Select database type
1 Thermal 1 Molecular
In the event that a material code is specified in the Matenal Propert,es screen (Fig. 8.11)
that does not match one in the database, the Material code not found message will appear.
Pressing [F9 Fastpdb] accesses the database menu directly. Press [F9 Qu,t ] to exit th"
database !rom either point of access. The program will return to the original point of exit
from FASTP [Data entry].
8.9.2 Manipulating the database
When access to the database is granted the first screen to appear will be the one shown
139
in Figure 8.20. Each of the options is described separately in the upcoming pages.
DATABASE i"E:::"
Create a new entry
Edit an entry
View data base
Delete/Rename an entry
Figure 8.20: Database Menu screen.
1. Creating a new entry
An entry can be added to the database when the create a new entry option is selected.
The program will first ask whether or not this new entry will involve a phase change. In
other words, will the material being entered undergo a. change of state during the simulation?
Will your entry involve a phase change ? ~ no
Depending on the a.nswer, the Create a New Entry screen will appear with or without
entry slots for the material's solidus, liquidus, state 2, a.nd state 3 properties (Figs. 8.21, 8.22).
The user would begin by assigning a materia.l code to the set of properties being
entered. This is the code specified when the materia.l properties are to be transferred from
the database to the program via the Material Propertles screen. A code can only be used for
one set of entries. Any attempt to assign an existing code to a new set of entries will result
in a request that a different code be used to specify the material.
140
CREATE ; N~W ENTR\
Enter the followlng:
Material code: Exampl
Solidus temp.: 1050 Ll.quidus temp.: 1060
state l State 2 State J
Thermal conductlvlty 45 38 )0
Density 7250 7150 7080
Heat capacl.ty 525 6109 754
Comment: no comment
Unl.ts kg m sec J
Figure 8.21: Create a New Entry screen (with phase change).
CREATE A 1<El-1 EtIT?Y
Enter the followlng: l Material code: Examp2
State l
Thermal conductivlty 45
Densl.ty 7250
Heat capaclty 525
Comment: no comment
Units kg cm sec J
Figure 8.22: Create a New Entry screen (without phase change).
141
(
For entries with a phase change, the liquidus and soli dus values refer to a temperature
or concentration range over which the state 2 properties will apply. The three states can
also be u~ed to simulate a solid whose thermal diffusivity changes over a wide temperature
range. This was discussed briefly in Chapter 5. Once the data has been entered the property
set will be written to one of two database files. These are DBASE.TRM for heat transfer or
DBASE.MOL for mass transfer.
IThermal data vritten to filel
To exit the menu, press [F6 Cont]. Press [F9 Q1nt] at any stage to prematurely exit
the screen without ad ding a new entry to the database. In either case, the database menu
will reappear.
2. Edit an entry
The edit an entry option is used to modify an existing set of properties in the database.
The material code of the property set to be edited is entered from the Edit an Entry screen.
The database will then be searched and the properties will he displayed. If the material code
cannot be found a warning message will appear
IMaterial code not found in file.
The cursor keys are used to move the cursor to the slot to he edited. Once the correc
tions have been made, pressing [F2 Done] will overwrite the existing material property data
in the datahase: [F6 Cont1 returns the program to the data.base menu. The Edit an Entry
screen can he exited at any sta.ge without modifying the property set by pressing [F9 Qu.it1.
142
3. View database
The m.ew database option is used to survey the material codes and properties already
resident in the database. Having selected this option the Vtew Database !lcreen will app<,ar
(Fig. 8.23).
Total records 14 Number of records vlewed 15
lAl 7cement lJtestl
2Alumlnum 8cube l4test22
3eu 9refract
4Exampl lOstain
SExamp2 llstalnl
6FE-STEL 12test
Contl.nue view View a particular materlal
Figure 8.23: View Database screen.
Each siot will display a material code accompanied by a number. Up to thirty material
codes can be dispIayed at the same time. The continue vaew option is used to page through
the next thirty entries in the database. To view the properties of a pa.rticular ruaterial select
the mew a partacular mate rial option and enter the associated number.
IEnter numb.r corresponding to mat.rial code:
AlI of the properties and comments associated with that code will be displayed in the
same format shown for the create an entry option. Pressing [F2 A nother] allows the user to
view another set of properties, thus allowing an unlimited number of entries to he viewed.
Pressing [F9 Quit] returns the program to the first Vaew Database screen: [F9 QUIt] again
will return the program to the database menu.
143
(
4. Delete / Rename an entry
Selection of the delete frename an entry option is used to remove an entry from the
database or rename the material code under which a material's properties are stored. The
standard database screen format will he displayed upon selection of this option. A message
requesting that the material code of the material to he deleted f renamed he entered will
also appear. The program willl.hen find and display the material properties associated with
the material code.
Pressing [F2 Delete] deletes the entry from the database. Internally, the datahase is
reorganized and compacted. Oatabases that contain sever al entIÏes may require a moment
or two to reorganize. Once complete, the program will automatically return to the database
menu.
Pressing [F9 Rename] causes the program to request a new material code to he as
signed to the property set. Once the new material code has been assigned, the program will
reposition the entry in the database in the correct alphabetical order.
Pressing [F9 QUIt] at any stage returns the program to the database menu without
changing the existing entries.
144
l 1
Chapter 9
Program 2: CALS
9.1 Introductory Remarks
CALS1.EXE simulates the prohlems configured with FASTP [Data entry]. The pro
gram's second function is to store selected data in files for later plotting using the Plottmg
Facûity programSj FASTPLT.EXE and HPPLOT .EXE. The configuration data is transferred
from FASTP1.EXE to CALSl.EXE via the file FAST2.CFG.
To run CALS1.EXE, select CALS from the main menu. The program will display the
title screen while reading FAST2.CFG. Once aU data has heen read, the One moment ple(L,~e
... message will be replaced with a Ready. Two options are availahle. Press [F5 Menu] to
access the same option menu that was descrihed in Section 7.8. Press [F6 Cont] to proce(·d
directly with the simulation without saving, editing, etc ... the data in FAST2.CFG.
Note that if the ed~t the ex~stmg data option is selected from the option menu in
CALS1.EXE, the configuration will he rewritten ta FAST2.CFG and the program will ter
minate. The main menu will then reappear. To edit the data, rerun FASTP [Data entnJ].
The following message will then he displayed instead of the FASTP logo.
145
10id you request an edit? 11es 1 no
If [no] is selected, the program will run as if a new configuration was to be entered. If
[yes] is selected, the Edit Menu screen will be displayed directly. Once the correctiol\S have
been made, re-select the execute s~mulation option and rerun CALS.
9.2 Time Parameters
Before running any simulation, CALS requires sever al time parameters. These are now
presented.
9.2.1 Initial values
The following will be requested
00 you vant to execute the program vith
The last calculated temperatures +- tilDe • 10.00
1 The original starting tamperatures +- tilDe· 0 .001
Both options are useful for re-running simulations for either different periods of time
or incremental periods of time. The selection is made by highlighting with the cursor keys
and pressing [F6 Select). For mass transfer simulations, temperatures are replaced by con
centrations.
9.2.2 Iteration time increment
Following the request to specify the simulation's starting point. the program will then
146
request the LteratlOn tlme increment
The t ime lncrement for dim. no. 1 must be < 11.948 unit s
Enter time increment for dim. no. 1 [==::J
To ensure stability of the solution, CALS will evaluate a maximum allowable iteration
time increment for each dimension in the configuration and will request that au iteration
time be input. The iteration time can be any value less than the maximum stated.
~ote that if radiation was specified as a boundary condition at a surface node, the
program will request that an approximate value for the maximum temperature at that surface
be input. This value is required for the program to complete the stability calc\ùations. The
request is made as follows
1 Enter appr. max. tempo for surface node 1 dim.ll.o.1 sect .no.l :c:::I1
9.2.3 Total simulation time
The total simulation time should also be taken into consideration when assigning an
iteration time increment. Intuitively, small iteration time intervals will yield more accurate
results. However, they can also lead to excessive computation times. The total ~1Ill\ùatiou
time is requested as follows
Enter maximum time for the simulation: c:=J
Once the total simulation time has been entered, the number of iterations req\ùrcd to
perform the simulation will he displayed. Press [F1:? Re-ntr] to re-enter the time parameh'rc;
or [F6 Cont] to continue. CALS will display the number of iterations required to complete
the simulation in the following manner
147
About 9800. total node iterations must be performed
The total number of iterations serves as the only indication of the computation time
required to run a simulation. Since F ASTP will undoubtedly be run on comput ers with
different dock speeds, standard realtime indicators are difficult.
9.3 Display Options
The final option to he selected prior to running a simulation is the display option.
The Display Options screen is shown in Figure 9.1 and appears alter the time parameters
have been entered. The display options are hroadly classified as either print options or plot
options.
1 print options
Temperature Temperature and energy content
1 Plot options 1
Temperature vs Tlme Temperature vs Section Temperature Isotherms
Energy content vs Time Energy content vs section Energy content Isotherms
Figure 9.1: Display Option screen.
148
DISPLAY OPTIC·tiS
9.3.1 Print options
When any of the print options are selected, a section by section display of the iterativdy
calculated node values will he displayed. The results can he routed to a printer for hard
copy. One and possihly two additional time parameter( s) must he supplied prior to r\\Illlinp;
the simulation when a print option has been requested. These are as follows
Enter the time interval betveen printouts: c:::J
At vhat time is the profile to be computed? c:::J
The first question refers ta the time interval between the display of prmclpal ax~s node
values. The profile question relers to when a profile is to be generated. The second question
is asked only il a product solution was requested. The option to senrl the output to the
printer is also given.
Send output to printar: yas 1 no 1
The analysis will bcgin immedia.tely alter the prin ter option has heen specified. A
sample printout is shown in Figure 9.2. Output to the printer can also he turned on or off
by pressing [F9] during the simulation. Two other {unction keys are also active while th",
transient analysis is in progress. Press [F2] to generate unscheduled output and [FIO] to
prematurely terminate the s.mulation.
The energy content values represent a net gain or loss of enthalpy (for each no de )
relative ta the initial enthalpy at the start of a simulation.
Regardless of the dis play option the program will signal the end of the simulation and
present the following
149
9.3.2
OIM- l SECTIOII #:
20.00 20.00
OIM- 2 SECTIOIi #:
20 00 20.00 .OOOOE+OO .OOOOE+OO
TIME~ .000
20.00 20.00 20.00
TIME= .000
20.00 20.00 .OOOOE+OO OOOOE+OO
TOTAl.. EIITHAI..P'l CHAlIGE OF SECTlotl: OOOOOOOE+OO 00001:+00
.OOOOOOOE+OO . OOOOE+OO • OOOOE+OO .OOOOE+OO
TOTAl.. EIITHALP'l CHAl. GE OF SECTIOII:
OlM- i TIME= 5.000 SECTION .:
l8.80 l6 35 34.88 34.39 34.88
OIM2 2 TIME= 5.000 SECTICII ,:
60.02 58.52 58 52 bO.02
20.00 20.00
OOOOE+OO .OOOOE-OO .DOOOE
36 35 JB.ao
.2304E+06 .4008E+06 .3647E+06 3526<:+06 .3647E+06 .400BE+06 2304E
TOTAl.. EIITHALP'l CHANGE OF SECTIOll: .l923E+06 .7553E+06 .7553E+06
TOTAl.. EIITHAI..P'l CHANGE OF SECTIOII
.2344382E+07 .3923E+06
.2295237E+07
Figure 9.2: Sam pIe temperature a.nd energy content prin tout results.
IExecute th. program one more time: yes 1 no if
Plot options
The plot options are divided into three categories. These are (i) Variable vs. Time, (ii)
Variable vs. Section, and (ili) Isovariable. The variable can be either temperature, energy
content, or concentration. The three cbtegories offer a considerable degree of flexibility for
obtaining information from the generated. data. The parameters that must be supplied are
presented herein. Some of the parameters common to ail of the categories are discussed once
in the earlier sections and are tben only referred to in the later sections.
[Al Variable vs. Time
The Variable vs. Time display option ca.n be used to track the behaviour of up to three
150
l nodes in the configuration with time. The data coUected is written to a file and can 1)('
plotted using FASTPLT.EXE. Both principal axis nodes and profile nodes can be tracked
with the display option. When a Van.able vs. T,me display option is selected the !'>Cfl'('n
shown in Figure 9.3, or one similar to it, will appear.
Il
GRAPHICS OPTION (temp. vs. t 1 :'1'_" 1
Data fllename: fastp
Calculatlons update nodt?s l.OOO tlme(s) unIt tLI'
Headings
tltle )-0 RECT. ANALYSIS 1 value ln 1 ~111 be stored for plottlng
x-aX1S TIME y-axls TEMPERATURE
Number of nodes ta be tracked : 1 2 )
1 Dlmenslon: 1
\ Node l 2
Sectlon: 1 Selectlon: lst of 2
) .; 5 6 7
1
Figure 9.3: Variable vs. Time display option screen.
A.1 Data Filename: [ ]
The filename under which data will be stored for plotting usmg FASTPLT.EXE IS
requested. The .FPL extension is added to aU plot data files. An optional path can he
specified.
A.2 Title: [ ]
A string of up to 25 characters can be entered for the title. The title will serve as a
heading for the plots generated by FASTPLT.EXE.
A.3 Axis Labels :
Default labels for the x and y axes of the plots are provided. However, they can he
151
replaccd by moving the cursor to the desired slot and typing in a new label. If no changes
are desired, the title and labels can be skipped by pressing [F6 Gont].
A.4 Frequency of Datapoints Written to File:
The following message will be displayed
Calculations update nodes
110.0001 times / unit time
1 value in 11001 vill be
stored for plotting.
The frequency must be chosen keeping both the iteration time and the total simulation
time in mind. Although a high frequency of datapoints written to file will add to the accuracy
of a plot, a larger data file will also result.
A.5 Number of Nodes to be Tracked DU 2 3
U P to three nodes can be tr acked and t heir values wri t ten to a f1 le d uring one simulation.
The data can subsequently be plotted on the same graph with FASTPLT.EXE. Highlight
the number of nodes desired and press [F6 Gont].
The following option is given if a product solution was requested for a 2-D or 3-D
configuration
Are you tracking (1)
IData from node. on a principal axis (axes) 1
Data generated from a profile
Dnly principal axis nodes can be tracked when an interaction solution has been re-
152
quested or a 1-D problem has been configured.
:\.5.1 Data From Nodes on a Principal Axis (Axes)
\Vhen principal axis nodes are to be tracked the window shown III Figure 9,4 will
appear.
Number of nodes to be tracked 1 2 3
1
DimenSl.On: 1 Sectlon: 1 Selection: Ist of 2
; !,ode l 2 ) 4 5 6 7
Figure 9.4: Selection of principal axis nodes.
Pressing [F5 Toggle] will select or cancel the principal axis nodes to be tracked. Pr('!'!s
[F6 Cont] to page through each section of the configuration until a.ll sections have hef'll
displayed. Once the nodes have b~en selected a summary is displayed. Press [F2 Re-ntr] to
re-enter the selection or [F6 Gont] to begin the simulation. The iteratively calculated values
of the selectt:J nodes will be displayed on screen and written to file.
A.5.2 Data Generated From a Profile
If the Trackmg profile nodes option is selected the screen shown in Figure 9.5 WIll
appear. Each profile no de in the configuration is represen ted by a zero (0). The relative
node positions on the screen are not intended to correspond to the relative positions of the
nodes in the configuration. The cursor is initially positioned at profile node (1,1) and can he
repositioned using the cursor keys. The cursor position is displayed in the window beneath
the Profile posItion heading.
Pressing [F6 Select] selects the profile nodes to be tracked. A record of the selection is
153
:0000000 0000000 o ') 0 ° 0 0 0 ,~OOOOOO
0000000 '0000000 0000000
F2 RE-NTR F6 CONT
Horlzontal Dlm.: 1
Vertlcal Dlm.: 2
Proflle Posltlon ( H,V ) ( l, 4)
Node spaclng may not correspond to actual set-up
i ~~ 1.---
IProflle (C: x D2 )
1 IPOS. ( l, .. ) lS p1.ct:':e::
ESCAPE
Figure 9.5: Profile node selection screen.
kcpt in the box(es) underneath the ll.esume heading. Once selection is complete, [F2 Re-ntr]
is presscd to re-enter the selection or [F6 Cont] is pressed to continue. Once the selection
has bcen approved the simulation w"li begin. The profile node values will be displayed on
screeu and writtcn to file.
For 3- D configurations, the tracking of profile nodes requires two additional parameters;
(i) the principal axis intersected by the profile, (ii) the node on the intersected principal
axis through which the profile will pass. Figure 9.6 illustrates the concept. The windows
appearillg when the parameters are requested are shown in Figure 9.7. The remainder of the
procedure is the same as that outlined for 2-D configurations.
[B] Variable vs. Section display option
The Variable us. Sechon display option is used to track the behaviour of an entire
section of Hodes in a configuration. These values can then be plotted on a frame by frame
or time lapse basis usillg FASTPLT.EXE. \Vhen the Variable vs. Section display option is
selected, the screen shown in Figure 9.8, or one similar to it, will be displayed.
154
• 1
n e
e n •
e e---e
• e '. 1
e
•
• n
Profile Intersects no de 1 of Dimension 2, hes parallel to plane created by the Intersection of Dimension 1 and Dimension 2
Figure 9.6: Profile arrangement (3-D cartesian configuration).
Thraugh WhlCh aX1S lS the proflle ta pass?
Olmo 1 Dlm. 2 Olmo )
central sectlon
IThraUgh which node ln the lS the prof lIe ta pass ~ Node: l 2 ) 4 5 6 7
1
Olmo : Sect, :
Figure 9.7: The 3-D profile selection screens,
155
GRAPHIeS OPTION (temp. vs. sectIOn)
Data fIlename: fastp
Headlngs
tltle Test.2
x-aXIS SECTION y-aXIS TEMPERATURE
Are you tracklng (7) Data from nodes on a princIpal axis (axes) Data generated from a proflle
Calculatlons update nodes 1.000 tlme(s) jUnlt tl~e
1 value ln 10 wIll ce stored for plotting
Figure 9.8: Temperature vs. Section display option screen.
The data filename, plot heading, ax,s labels, and frequency of data questions are the
same as those covered for the Variable vs. Time display option.
As with the Vanable vs. Time option, if a product solution was specified the option to
track either principal axis nodes or profile nodes will he presented. If an interaction solution
was requested or a 1-0 prohlem was configured only principal axis sections can be tracked.
B.1 Data from nodes on a principal axis (axes)
The dimension number and section number of the section to he tracked are entered.
Corrections can he made by pressing [F2 Re-ntr]. Press [F6 Cont] to hegin the simulation.
The selected data will he displayed on screen and written to file.
3.2 Data generated from a profile
When analyzing a 2-D configuration the screen shown in Figure 9.9 will appear imme
diately. The zeroes (0) symbolize profil,.. nodes in the same manner as that explained for
the Vanable vs. T17ne display option. The cursor spans across a row of nodes. The section
156
, !
to be tracked is highlighted using the cursor keys followed by [F6 Select]. Press [F2 Re
ntr] to re-enter the selection or [F6 Gont] to start the simulation. Note that [F.1 Rever."1c]
switches the orientation of two principal axes, allowing a sectIon from another dimf'llsion tn
be viewed.
Ta identify the section to be tracked in a 3-D configuration, two additional items lllu~t
be specified; (i) the principal axis intersected by the profile, (ii) the node on the inter1)ected
principal axis through which the profile will pass. The windows that appear are shown in
Figure 9.10. Selection of the profile section then follows the same procedure as for the 2-D
case.
o 0 [0 0
[0 0 o 0 o 0 o 0 o 0
00000 00000 00000 00000 00000 o 0 0 0 0 00000
Horizontal Olm.: 2
Vertical Olm.: J
Proflle Position ( H,V )
Node spaclng may not correspond ta actual set-up
Figure 9.9: Profile section selection screen.
[Cl Isovariable Display Option
The IsovaT1able display option is used to track an entire profile of values with time.
The profile data can then be used to generate time lapse contour plots using FASTPLT.EXE.
The Isovariable display options are restricted to 2-D and 3-D configurations with product
solution specified.
157
When an Isovartable display option is selected the screen shown in Figure 9.11, or one
.,imilar to it, will appear. The data filename, plot headmg, axts labels, and frequency of data
questions are the same as those covered for the Variable vs. Time display option.
C.I Tracking a Profile
For 2-D configurations, no additional information is required once the frequency of
data WT1.tten to file parameter has heen assigned. The simulation will s~art immediately
after pressing [F6 Gont]. The profile data will be displayed on screen and written to file.
Two additional items are required for 3-D configurations. These pinpoint the profile in the
3-D system to he ohserved. The procedure is the same as that described for both the Vanable
vs. Time and Vanable vs. Section display options.
158
Through WhlCh aX1S 15 the proflle to pass ?
Dlm. l Olmo 2
1 Through Wh1Ch node ln the
t 15 the proflle to pass ? Node l 2 ) 4 5 6
1
Olmo )
central sectlon
7 Oim. : 1 Sect.: l
Figure 9.10: Profile selection windows.
Head1ngs
tlUe
x-aX1s y-ax1s
Axes
Test .3
DIMENSION X DIMENSION 'i
x-tlcks 5 y-t1cks 5
Plot gr1d: yes no Dlsplay to : screen plotter both
DISPLAY OPTION (temp. lsotherns)
Sectlon llmlts .,-\,.>
mlnlmum : ,) maXlmum :
Sectlon ltr.llts 1,-1. ._.,
mlnlmun maXlmum
Isotherms
lst 2nd Jrd 4th 5th
8UU lOUr; 12 'JO 1400 600
tk========================~~:_~-- ----
Figure 9.11: Isotherm display option screen.
159
Chapter 10
The Graphies Facility
10.1 Introductory Remarks
With FASTPLT.EXE severa! types of plots can be generated using the data calculated
by CALS. A second plotting program, HPPLOT.EXE, is used to generate hard copies of
the plots with an HP-compatible piotter. Both programs are described in this chapter. The
plotting facility programs will work with CGA, EGA, VGA, and Olivetti graphics adaptors.
Hercules adaptors currently do not run the plotting programs.
10.2 Running FASTPLT
FASTPLT.EXE is initiated by selecting FASTPLT from the main menu. The title
screen will tirst appear indicating that the program is ready to run. Press [F6 Cont] to
start. The program will request the name of the data file containing the data to be plotted.
Enter the name of the flle contalnlng the data to be plotted
160
The data files were created in CALS (.FPL extensIOns). When spenfying a filt'nalllt"
the extension need not be included. The program will fiud and read the file or e\se di:.play
a file not found message. Depending on the type of data present on the file bt'lIlg rl'MI, Ollt'
of three screens (Figs. 10.1, 10.2, 10.3), or ones similar to them. will be displayed.
DIS PLAY OPTION (temp. vs. tl~el
Headlngs
tltle Test 111
x-aXls TIME y-axlS TEMPERATURE
Axes
x-tlcks 5 y-tlcks 5
Plo~ grld: yes no
Legend X 2
o corner . : mldface
Dlsplay to : screen plotter both
y 2
Tlme llmlts
mlnlmum . Ù
maXlmum . 11) 1) •• ' )
Temperature llmlts
maXlmum ",OU
Plot sy:nbo 1.; no s'y'nl:;o 1"
Data POlnts . 1 ln 1
wlll be plottej
Figure 10.1: Temperature vs. Time display option screen.
10.3 Common Parameters
~lany pa.rameters a.re common ta aU of the screens shawn in Figures 10.1, 10.2, and 10.3
and will he discussed in this section. The headmgs and labels will appear as they had bet'Il
specified in CALS. They may be re-entered or left as is by pressing [F6 ContI to conti1l111'.
The number of tickmarks on both the x and y-axes are speàfied by assigning a value for
x-ticks and y-t1cks. A follow-up question will appea.r requesting whether or not the decimal
point on the numhers lahelling the axes is ta be suppressed.
Suppress decimal y88 Inol
161
(' ...
Heddlngs
tltle
.o(-aXls y-axls
A-<.es
Test 1/2
SECTION TEMPERATURE
x-tlcks 5 y-tlcks
Suppress declmal: yes no Enter no. of deClmal places 2
DISPLAY OPTION (temp. vs. sect~on)
Sect~on l~mlts
ml.nlmUm : 0 maXlmum : .10
Temperature llmlts
m~nimum o max~mum 500
Figure 10.2: Temperature vs. Section display option screen.
Data f llename: fastp
Headlngs
tltle Test n
x-aXlS DIMENSION y-axls DIMENSIO'J
X 'i
GRAPHICS OPTION (temp. lsotherms)
Calculatlons update nodes 1.000 tlme(s) /unlt tlme
l value in l wlll be stored for plottlng
Figure 10.3: Temperature Isotherm display option screen.
162
•
1 If the answer is [no], the program will request the number of places past the dccim.ù
point to be displayed.
Enter no. of decimal places: ~
To plot a grid answer [yes] to the following question.
Plot grid 1 yes 1 no.
The grid option will considerahly slow plotting both on the screen and pIotter and is
generaUy not recommended. When generating Variable vs. Section or [sovar.able plots the
grid line po&itions mater.. the positions of the principal axis nodal points.
The one remaining common option is that of whether the plot will he sent to the scref'U,
pIotter, or both.
Output to 1 screen 1 pIotter bath
If the option to send the output to screen is chosen, no further questions common to
aU display options will be asked. If the option chosen is piotter or both, the program will
request the following additional information
Pagelength : [!!] 17 (paper length in inches),
Enter a filename
More is written on the enter a filename request in subsequent sections. However, when
the option to send to pIotter or both is chosen the data. required to drive the pIotter is writtcn
to this file ( .PLT extension). This file is in turn used by HPPLOT.EXE to generate a hard
copy of the plot. Note that the plot will not appear on screen if the piotter option is cho~en.
163
10.4 Option-specifie Parameters
The remaining parameters are specifie to one of the three display options only. They
are discussed accordingly in the sections that follow.
[Al Variable vs. Time Display Option
A.1 Horizontal and Vertical Axis Limits
The vertical d.xis limits are lelt blank and values must be supplied. The entered limits
would normally be chosen to conform wi th the limits of the data to be plotted. The hori·
zontal axis limits are comprised of the minimum and maximum time values (i.e. simulation
duration). Do not attempt to specify limits that zoom in on portions of the data as this may
have unpredictable consequences.
A.2 Legend x [ 1 y [ 1
A legend will be placed on tbe plot. The x and y correspond to physical coordinates in
inches on the screen measured from the bottom, left hand corner. rhese limits are
X-aJQs
y-axis
2 - 6 10.
1 - 4 10.
Some combinat ions of x-y values will result in an incomplete legend display to screen.
The legend labels may be retyped to better identify the data points being plotted. The
default legend labels are Data Point 1, Data Point 2, ...
A.3 Symbols and Data Point Frequency
The option to plot with or without symbols is available. The user may not wish to plot
164
symbols if many hundreds of data points are to be plotted. Another available option i~ to
plot only one in [x] elata points. Increasing [x] Crom the default of one Will result in ,\ mun'
quickly drawn, but less accurate plot.
A.4 Generating the Plot
This completes the list of questions specifie to the VaTi.able us. Tlme di~play optlOll.
Once all of the information has been supplied, the data will be plotted.
After the graph has been drawn the [F3 PTi.nt screen] option IS given (lower nght hallel
corl.er of screen). The plot can be reproduced on a printer provided the DOS progr,lIll,
GRAPHICS.EXE. was run prior to running F:\STPLT.EXE. Note. ~ome prillters are Ilot
capable of reproducing the graphies generated 00 screeo Please reCer to your pnnter ma.nll,ù.
If a printout is not desired, press [F6 Cont] and the option to [F3 R(~plotl willlH' ~IVt'lI
Due to a problem w:th program code, [F3 Replot] must be pressed twice befort· any vI~lhlt'
response occurs. The original display option screen will then reappear with allllCading~ ,wei
labels appearing as they had been specified in CALS1.EXE. If a replot is Ilot desired, prc'~~
[F6 ContI to continue. The following message will appear
Do you vish to plot another file?
Iyesl no
If [no] is chosen, the program will end and the main menu will reappear. If [ye.~] i~
chosen the program will request a llew .FPL file.
[B] Variable vs. Section Display Option
The Variable vs. Section display option is used to track the behaviour of an entire
section of nodes on a frame by frame or time lapse basis. The display option screen for the
Variabie vs. Section option was shown in Figure 9.8. Fewer parameters are rcquired prior to
165
generatiug Va.nable UI'J ScctlOn l'lots. In fact, only one remains that had not been discussed
.1mong the common options. The request will be made whether or not to superimpose the
gcnerated data
Do you vish to superimpos8
the generated data? yes Ino 1
The question appears after having assigned the x-t1.cks, y-t1.cks parameters. If the [yes]
option is sclected all plots will be supenmposed onto one picture. If [no] is selected the plots.
cach corresponding to a different time, w1ll appear individually as outlined in the {ollowing
manner. Having selected [no] {or the above and having chosen to display to either screen,
pIotter, or both, the first Vanable vs. Sectlon plot will bp drawn. This plot corresponds to
the initial time of the simulation. Press [F2 Next] to view subsequent time frames. The
associated times are displayed in the top right hand corner of each plot.
Press [PB Prmt screen] if a prin ter copy of the screen display is desir~d. A printer copy
can be obtained provided the printer is capable of reproducing graphicl. and the GRAPH
ICS.EXE program was run prior to running FASTPLT.EXE.
If the option to send the drawing to pIotter or both was cho5~D, a .PLT filename will
be requested for each plotted frame. HPPLOT .EXE would subsequently be used to generate
the hard copy of the plot.
Ta skip the remaining Variable vs. Section plots in a series, press [F6 Cont] iustead of
[F2 Next]. The option to [F9 Replot) will then he made available. Select [F9 Replot) if the
axis labels, headings, etc ... are to be changed. Due to a problem with the program coding
[FS Replot] must be plessed twice prior to the reappearance of the original display option
screen with labels and headings written as they had initially been specified in CALS.
If no replot is desired, [F6 Cont] should be pressed repeatedly until the plot another
166
'.
file question appears.
[e] Isovariahle Display Option
The Isovanable display option is used to plot the data generated Crom au cntH!' prufil!'
as contours (e.g. isotherms). The plots are generated on a frame hy frame or timc 1.\\>..,,,
basis.
All parameters entered on the IsotJanable display option screen (Fig. 10.3) Wl'rt' (li~
cussed in Section 9.3. The one exception is that values for the contour lilles to lw plot tt'd
must be !!pecified. e p to five contour lines can he drawn in each frame to be vlewt'ù. Tht'
actual value of the contour line is entered beside the lst, 2nd, etc ... on the \.li~play optIOn
scre~n. The function keys [F2 Next], [F3 Prmt screen], etc ... perfonn the !>ame t~ks th.lt
were described in the previous sections.
10.5 Ending a FASTPLT Session
As with all programs making up the FASTP package, FASTPLT wIll end if [E.ou:] i~
pressed. The [Esc] key is active at ail times except when the plots are dü.playcd. The
program will also terminate if [no] is replied to the plot another file question.
Do you vieh to plot another file?
yes Inol
This question appears when [F6 Cont] is pressed after ail plots have been displayeù or
if [F6] is pressed instead of [F3 Replot1 between plots.
167
l 10.6 Running HPPLOT
HPPLOT.EXE is used to generate hard copies of the plots produced by FASTPLT.EXE.
The program Îs of no use without an HP-compatible pIotter. To run the program select HP·
PLOT {rom the main menu. The menu for HPPLOT.EXE will then appear, where four
options are avrulable.
S Sereen Previev
P Plotter
U Update profile
Q Quit to DOS
Tbe selection is made by entering the first let ter of the option. The screen prevtew
option is used to view the plot prior to sending it for hard copy. The piotter option sends the
file directly to the pIotter without first displaying it to screen. The update profile option sets
the seria.! port communications parameters and qu.it to DOS returns the user to the main
menu.
Prior to sending anything to the pIotter, the communications parameters must he set.
Ta do tbis the u.pdate profile option must first be selected. A list of profile parameters will
appear. Note that the word profile in this context hears no relation to the profiles discussed
in earlier chapters. The profile features would appear as follows
PROFILE PARAMETERS
Monitor type
Comm port
Baud rate
168
[!] [)
196001
Parity W Number of data bits W Number of stop blts ru Number of files plotted ru
The suggested settings are shown for an EG A graphes adaptor. Plea~(' t'mure t hlll
the correct graphies adaptor has been specified. For CGA adaptors type [cl Onet' tht'
parameters have been entered they should be saved for subsequent reuse. The coufigur<ltlOlI
will be stored on the file HPPLOT.PRO.
Save these parameters? [YIN] --) Y
Of course, these parameters must match the bit-switeh settings on the plott('r i bplf.
Please refer to the piotter manual to make the necessary adjustments.
The screen preview and pIotter options perform the same function with the exception
that the first option is used to preview the plot on the screen prior to sending it to the plottf'r.
To demonstrate how the two options work a sample fùe, TEST.PLT has been provlded uu
diskette 4. Select th" screen prev,ew option and type [test] when the following is a..,ked
Enter Plot Filename [.PLT] --) test
The HP PLOT menu screen will disappear and the plot will be drawn. Note that ~ome
of the words will appear as rectangles. This is due simply to the limitations of the screen.
Once the plot is complete the following will be asked
Send to Plotter? [Y/H]
169
If [n 1 is selected the option to plot another is given. If [y] is selected the program
will immediate1y skip ta the pIotter option without returning ta the HPPLOT menu. The
following will appear
PLOTTER STATUS
Adjust paper size on front panel
and load paper
1 Plotting 1
AlI Done !
FILE STATUS
Page size
Line number
The remaining steps should be fairly self-evident. If any problems do arise, check to
make certain that the communications pa.rameters are properly set. To exit the program.
type [q] from the HPPLOT menu screen.
170
- -~ ---- ----
Chapter Il
File Export
The file export facility can be used to rewrite the Vartable vs. Ttme dis play option
data into a form that can be read and used by some of the commercially avruldoblc graphic~
packages. EXPORT.EXE will generate an ASCII file haVlng the {ollowing format
I~ ~, .. II: ..
11 spaces 11 spaces
Record format
To begin an export session select EXPORT from the main menu. The screen shown ln
Figure 11.1 will then appear. Source files have .FPL extensions and are created by CALS.
When specifying a source file, the extension need not be included. Target files can be glvf'n
optional pathnames and extensions. EXPORT will first check to see if the file exists before
rewriting the data. to the target file. The remainder of the program becomes self-evidellt
while being run.
171
',' ~
FILE EXPORT FACILI~,
Enter FASTP data flle to be exported
Source flle 1
1
ITarget flle 1
Records wrl.tten
fastp
fastp
ITRANSFER ANOTHER yes no
101 Flelds/Record J Fleld wldth
Figure 11.1: File Export screen.
172
llx
Bibliography
[1] Holman, J.P.: Heat Transfer, 5th ed., McGraw-Hill Book Company, New York, 1981.
[2] CaIslaw, H.S., and Jaeger, J.C.: Conduction of Heat in Soltds, 2d cd., Oxford UnivNMty
Press, Fair Lawn, N.J., 1959.
[3] Schneider, P.J. : Conduction Heat Trans/er, Addison-Wesley Publishillg Company, lue.,
Reading, Mass., 1955.
[4] Heisler, M.P.: Temperature Charts for Induction and Constant TemperatlL1'e Healtn!l,
Trans. ASME, vo1.69, pp.227-236, 1947.
[5] Schlichting, H. : Boundary Layer Theory, 7th ed., McGraw-Hill Book COlllpa.lly, Nl'w
York, 1979.
[6] O'Neill, P.V. : Advanced Engineering Ma the maties, 1st ed., Wadsworth Publi!>hillg
Company, Belmont, California, 1983.
173
l
1' 1 •
. ' . ,
Appendix A
The Heisler charts
AO
1
___ M
Al
.......
~ 80
S;
0.01
0007
• • • ·illi ... . .,. .•. • .~ •• 1\11."
ar/,l
Figure A2: Axis tempera.ture for an infinite cylinder of ra.dius ro. (From J.P. Holman, "Heat Trans/er 5th ed.} " McGraw-Hill Book Company, New York} 1981
jJ..' •
.. .of
> Co)
(10
(1.
or/, 2 ro
Figure A3: Center temperature for a sphere of radius ro. (From J.P. Holman, " Heat Traf'l_:fer 5th ed., " McGraw-Hill Book Company, New York, 1981
l
0.9
0.8
0.7
0.6 .
o~--~~~--~~~~~~~~~~~
0.01 0 02 0.05 0.1 0.2 0.5 1.0 le
hL
Figure A4: Temperature as a function of center temperature in an infinite plate of thickncss 2L. (From J.P. Holman, "Beat Transfer 5th ed., " McGraw-Hili Book Company, New York, 1981
A4
(
0.4
0.2
0.1 ; .
o~~~~~~~~~~----~-------0.01 0.02 0.05 0.1 0.2 0.5 1.0 2 3 5 10 20 50 100
k hro
Fjgure A5: Temperature as a function of a.xis temperature in 8...1\ infinite cylinder pf radius ro. (From J.P. Holman, Il Heat Trans/er 5th ed., " McGraw-Hill Book Company, New York, 1981
AS
,' .. -
'.'
0.9
0.8 .1'':;~mMlll 0.7 ~"I·II.1i11l1
0.6
0.3
0.2
0.1
o~~~~~~,~~~~~uu~~~~
0,01 0.02 0.05 0.1 0.2 0.5 1.0 50 100 le
hro
Figure A6: Temperature as a function of center tempeY'ature for a sphere of radius ro. (From J.P. Holman, il Heat Transfer 5th ed., " McGraw-Hill Book Company, New York, 1981
A6
(
(
Figure A7: Dimensionles8 heat los8 Q/Qo of an infinite plane of thickness 2L with time. (From J.P. Holman, U Heat Trans/er 5th ed., n McGraw-Hill Book Company, New York, 1981
A7
1.0
0.9
O.B~~~m!!
0.7
0.4
Figure A8: Dimensionless heat lOfS Q/Qo of an infinite cylinder of radius ro with time. (From J.P. Holman, « Heat 'IransJer 5th ed., " McGraw-Hill Book Company, New York, 1981
A8
(
1.0
0.9
0.8
0.7
0.6 Q
0.5 (Ïg 0.4
03
02
0.1
0 10 -1 10'" 10 -3 10-2 10-1
h2Otf'
k'
Figure A9: Dimensionless heat 10ss Q/Qo of a sphere of rat\ius ro with time. (From J.P. Holman, /1 Heat Transfer 5th ed., n McGraw-Hill Book Company, New York, 1981
A9
