Transient Conduction:Spatial Effects and the Role of

Analytical Solutions

Chapter 5

Sections 5.4 to 5.8

Plane Wall Solution to the Heat Equation for a Plane Wall withSymmetrical Convection Conditions

• If the lumped capacitance approximation can not be made, consideration must be given to spatial, as well as temporal, variations in temperature during the transient process.

• For a plane wall with symmetrical convection conditions and constant properties, the heat equation and initial/boundary conditions are:

2

2

1T T

x t

(5.26)

,0 iT x T (5.27)

0

0x

T

x

(5.28)

,x L

Tk h T L t T

x

(5.29)

• Existence of seven independent variables:

, , , , , ,iT T x t T T k h (5.30)

How may the functional dependence be simplified?

Plane Wall (cont.)

• Non-dimensionalization of Heat Equation and Initial/Boundary Conditions:

Dimensionless temperature difference: *

i i

T T

T T

*

xx

LDimensionless coordinate:

Dimensionless time: *2

tt Fo

L

Fourier the Nu mberFo

The Biot Number:solid

hLBi

k

* *, ,f x Fo Bi • Exact Solution:

* 2 *

1exp cosn n n

nC Fo x

(5.39a)

4sin

tan2 sin 2

nn n n

n n

C Bi

(5.39b,c)

See Appendix B.3 for first four roots (eigenvalues ) of Eq. (5.39c)1 4,...,

Plane Wall (cont.)

• The One-Term Approximation : 0.2Fo Variation of midplane temperature (x*= 0) with time : Fo

* 2

1 1expoo

i

T TC Fo

T T

(5.41)

1 1Table 5.1 and as a function of C Bi

Variation of temperature with location (x*) and time : Fo

* * *1coso x (5.40b)

Change in thermal energy storage with time:

stE Q (5.43a)

1 *

1

sin1o oQ Q

(5.46)

o iQ c T T (5.44)

Can the foregoing results be used for a plane wall that is well insulated on oneside and convectively heated or cooled on the other?

Can the foregoing results be used if an isothermal condition is instantaneously imposed on both surfaces of a plane wall or on one surface ofa wall whose other surface is well insulated?

s iT T

Heisler Charts

Graphical Representation of the One-Term ApproximationThe Heisler Charts

• Midplane Temperature:

Heisler Charts (cont.)

• Temperature Distribution:

• Change in Thermal Energy Storage:

Radial Systems

Radial Systems• Long Rods or Spheres Heated or Cooled by Convection.

2

/

/o

o

Bi hr k

Fo t r

• One-Term Approximations:Long Rod: Eqs. (5.49) and (5.51) Sphere: Eqs. (5.50) and (5.52)

1 1, Table 5.1C

• Graphical Representations:Long Rod: Figs. D.4 – D.6 Sphere: Figs. D.7 – D.9

Semi-Infinite Solid

The Semi-Infinite Solid• A solid that is initially of uniform temperature Ti and is assumed to extend to infinity from a surface at which thermal conditions are altered.

• Special Cases: Case 1: Change in Surface Temperature (Ts)

0, ,0s iT t T T x T

, xerf

2 ts

i s

T x t T

T T (5.57)

s is

k T Tq

t

(5.58)

Semi-Infinite Solid (cont.)

1

2 22 /, exp

4

erfc2

oi

o

q t xT x t T

k t

q x x

k t

(5.59)

Case 2: Uniform Heat Flux s oq q

0

0,x

Tk h T T t

x

2

2

,erfc

2

exp erfc2

i

i

T x t T x

T T t

hx h t x h t

k k kt

(5.60)

Case 3: Convection Heat Transfer ,h T

Multidimensional Effects

Multidimensional Effects• Solutions for multidimensional transient conduction can often be expressed as a product of related one-dimensional solutions for a plane wall, P(x,t), an infinite cylinder, C(r,t), and/or a semi-infinite solid, S(x,t). See Equations (5.64) to (5.66) and Fig. 5.11.

• Consider superposition of solutions for two-dimensional conduction in a short cylinder:

, ,, ,

,

i

Plane Infinitei iWall Cylinder

T r x t TP x t x C r t

T T

T x t T T r,t Tx

T T T T

Problem: Thermal Energy Storage

Problem 5.66: Charging a thermal energy storage system consisting ofa packed bed of Pyrex spheres.

KNOWN: Diameter, density, specific heat and thermal conductivity of Pyrex spheres in packed bed thermal energy storage system. Convection coefficient and inlet gas temperature

FIND: Time required for sphere to acquire 90% of maximum possible thermal energy and the corresponding center and surface temperatures.

SCHEMATIC:

Pyrex sphere D = 75 m m ,

T = 25 Ci o

Gas

T Cg,i o= 300

h = 75 W /m -K2

= 2225 kg/m 3

k = 1.4 W /m -Kc = 835 J/kg-K

Problem: Thermal Energy Storage

ASSUMPTIONS: (1) One-dimensional radial conduction in sphere, (2) Negligible heat transfer to or from a sphere by radiation or conduction due to contact with adjoining spheres, (3) Constant properties.

ANALYSIS: With Bi h(ro/3)/k = 75 W/m2K (0.0125m)/1.4 W/mK = 0.67, the lumped capacitance method is inappropriate and the approximate (one-term) solution for one-dimensional transient conduction in a sphere is used to obtain the desired results.

To obtain the required time, the specified charging requirement / 0.9oQ Q must first be used to obtain the dimensionless center temperature,

*.o

From Eq. (5.52),

31

oo1 1 1

Q1

Q3 sin cos

With Bi hro/k = 2.01, 1 2.03 and C1 1.48 from Table 5.1. Hence,

3

o0.1 2.03 0.837

0.1555.3863 0.896 2.03 0.443

Problem: Thermal Energy Storage

From Eq. (5.50c), the corresponding time is

2o o

211

rt ln

C

3 7 2k / c 1.4 W / m K / 2225 kg / m 835 J / kg K 7.54 10 m / s,

2

27 2

0.0375m ln 0.155/1.48t 1,020s

7.54 10 m /s 2.03

From the definition of *,o the center temperature is o g,i i g,iT T 0.155 T T 300 C 42.7 C 257.3 C

The surface temperature at the time of interest may be obtained from Eq. (5.50b) with r 1,

o 1s g,i i g,i

1

sin 0.155 0.896T T T T 300 C 275 C 280.9 C

2.03

Is use of the one-term approximation appropriate?

Problem: Thermal Response Firewall

Problem: 5.82: Use of radiation heat transfer from high intensity lamps for a prescribed duration (t=30 min) to assessability of firewall to meet safety standards corresponding tomaximum allowable temperatures at the heated (front) andunheated (back) surfaces.

4 210 W/msq

KNOWN: Thickness, initial temperature and thermophysical properties of concrete firewall. Incident radiant flux and duration of radiant heating. Maximum allowable surface temperatures at the end of heating.

FIND: If maximum allowable temperatures are exceeded.

q s = 10 W /m 24

L = 0.25 m x

Concrete, T = 25ioC

= 2300 kg/m 3

c = 880 J/kg-Kk = 1.4 W /m -Ks = 1.0

T Cm ax o= 325 T Cm ax

o= 25

SCHEMATIC:

Problem: Thermal Response of Firewall

ASSUMPTIONS: (1) One-dimensional conduction in wall, (2) Validity of semi-infinite medium approximation, (3) Negligible convection and radiative exchange with the surroundings at the irradiated surface, (4) Negligible heat transfer from the back surface, (5) Constant properties.

ANALYSIS: The thermal response of the wall is described by Eq. (5.59)

1/ 2 2o o

i2 q t / q xx x

T x, t T exp erfck 4 t k 2 t

where, 7 2pk / c 6.92 10 m / s and for

1 / 2ot 30 min 1800s, 2q t / / k 284.5 K. Hence, at x = 0,

T 0,30 min 25 C 284.5 C 309.5 C 325 C

At

1 / 22ox 0.25m, x / 4 t 12.54, q x / k 1, 786K, and x / 2 t 3.54.

Hence,

6T 0.25m, 30min 25 C 284.5 C 3.58 10 1786 C ~ 0 25 C

Problem: Thermal Response of Firewall

Both requirements are met.

Is the assumption of a semi-infinite solid for a plane wall of finite thickness appropriate under the foregoing conditions?

COMMENTS: The foregoing analysis may or may not be conservative, since heat transfer at the irradiated surface due to convection and net radiation exchange with the environment has been neglected. If the emissivity of the surface and the temperature of the surroundings are assumed to be = 1 and Tsur = 298K, radiation exchange at Ts = 309.5C would be

4 4 2rad s surq T T 6,080 W / m K,

which is significant (~ 60% of the prescribed radiation). However, under actual conditions, the wall would likely be exposed to combustion gases and adjoining walls at elevated temperatures.