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ME 259Heat Transfer

Lecture Slides III

Dr. Gregory A. Kallio

Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology

California State University, Chico

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Transient Conduction

Reading: Incropera & DeWitt

Chapter 5

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Introduction

Transient = Unsteady (time-dependent ) Examples

– Very short time scale: hot wire anemometry (< 1 ms)

– Short time scale: quenching of metallic parts (seconds)

– Intermediate time scale: baking cookies (minutes)

– Long time scale – daily heating/cooling of atmosphere (hours)

– Very long time scale: seasonal heating/cooling of the earth’s surface (months)

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Governing Heat Conduction Equation

Assuming k = constant,

For 1-D conduction (x) and no internal heat generation,

Solution, T(x,t) , requires two BCs and an initial condition

t

T

k

qT

12

t

T

x

T

1

2

2

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The Biot Number

The Biot number is a dimensionless parameter that indicates the relative importance of of conduction and convection heat transfer processes:

Practical implications of Bi << 1: objects may heat/cool in an isothermal manner if– they are small and metallic– they are cooled/heated by natural

convection in a gas (e.g., air)

convt

condt

R

R

hA

kALk

hLBi

,

, 1

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The Lumped Capacitance Method (LCM)

LCM is a viable approach for solving transient conduction problems when Bi<<1

Treat system as an isothermal, homogeneous mass (V) with uniform specific heat (cp)

For suddenly imposed, uniform convection (h, T) boundary conditions, solution yields:

where

ti

tTT

TtT

exp)(

s

pt hA

c V

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General LCM

LCM can be applied to systems with convection, radiation, and heat flux boundary conditions and internal heat generation (see eqn. 5.15 for ODE)

– Forced convection, h = constant– Natural convection, h = C(T-T)1/4

– Radiation (eqn. 5.18)– Convection and radiation (requires

numerical integration)– Forced convection with constant surface

heat flux or internal heat generation (eqn. 5.25)

(Note that the Biot number must be redefined when effects other than convection are included)

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Spatial Effects (Bi 0.1)

LCM not valid since temperature gradient within solid is significant

Need to solve heat conduction equation with applied boundary conditions

1-D transient conduction “family” of solutions:

– Uniform, symmetric convection applied to plane wall, long cylinder, sphere (sections 5.4-5.6)

– Semi-infinite solid with various BCs (section 5.7)

– Superposition of 1-D solutions for multidimensional conduction (section 5.8)

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Convection Heat Transfer: Fundamentals

Reading: Incropera & DeWitt

Chapter 6

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The Convection Problem

Newton’s law of cooling (convection)

“h” is often the controlling parameter in heat transfer problems involving fluids; knowing its value accurately is important

h can be obtained by– theoretical derivation (difficult)– direct measurement (time-consuming)– empirical correlation (most common)

Theoretical derivation is difficult because– h is dependent upon many parameters– it involves solving several PDEs– it usually involves turbulent fluid flow, for which no

unified modeling approach exists

)( TThq ss

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Average Convection Coefficient

Consider fluid flow over a surface:

Total heat rate:

If h = h(x), then

)(

1)(

)(

TTAh

hdAA

ATT

hdATTdAqq

ss

A

ss

ss

A A

sss

s

s s

Ldxh

Lh

0

1

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The Defining Equation for h

The thermal boundary layer:

From Fourier’s law of conduction:

Setting qconv = qcond and solving for h:0

0

y

fluidfluid

y

solidsolids

y

Tk

y

Tkq

TT

yT

k

hs

y 0

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Determination of

Need to find T(x,y) within fluid boundary layer, then differentiate and evaluate at y = 0

T(x,y) is obtained by solving the following conservation equations:– Mass (continuity) ……. Eqn (6.25)– Momentum (x,y) ……. Eqn (6.26, 6.27)– Thermal Energy …….. Eqn (6.28a,b)

Solution of these four coupled PDEs yields the velocity components (u,v), pressure (p), and temperature (T)

Exact solution is only possible for laminar flow in simple geometries

0

yyT

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Boundary Layer Approximations

Because the velocity and thermal boundary layer thicknesses are typically very small, the following BL approximations apply:

The BL approximations with the additional assumptions of steady-state, incompressible flow, constant properties, negligible body forces, and zero energy generation yield a simpler set of conservation equations given by eqns. (6.31-6.33).

x

T

y

T

x

v

y

v

x

u

y

u

vu

,,

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Boundary Layer Similarity Parameters

BL equations are normalized by defining the following dimensionless parameters:

where L is the characteristic length of the surface and V is the velocity upstream of the surface

The resulting normalized BL equations (Table 6.1) produce the following unique dimensionless groups:

Note: all properties are those of the fluid

2,

/,/

/,/

V

pp

TT

TTT

VvvVuu

LyyLxx

s

s

pc

k

VL

and where

number Prandtl

number Reynolds

(Pr)

(Re)

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Boundary Layer Similarity Parameters, cont.

Recall the defining equation for h:

The normalized temperature gradient at the surface is defined as the Nusselt number, which provides a dimensionless measure of the convection heat transfer:

00

yys y

T

L

k

y

T

TT

kh

k

hLNu

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The Nusselt Number

The normalized BL equations indicate that

Since the Nusselt number is a normalized surface temperature gradient, its functional dependence for a prescribed geometry is

Recalling that the average convection coefficient results from an integration over the entire surface, the x* dependence disappears and we have:

dx

dpyxfT Pr,Re,,,

Pr)Re,,( xfNu

Pr)(Re,fk

LhNu

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Physical Significance of the Reynolds and Prandtl Numbers

Reynolds number (VL/) represents a ratio of fluid inertia forces to fluid viscous shear forces– small Re: low velocity, low density, small

objects, high viscosity– large Re: high velocity, high density, large

objects, low viscosity Prandtl number (cp/k) represents the ratio of

momentum diffusion to heat diffusion in the boundary layer.– small Pr: relatively large thermal boundary

layer thickness (t > )– large Pr: relatively small thermal boundary

layer thickness (t < ) Note that Pr is a (composite) fluid property that

is commonly tabulated in property tables

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The Reynolds Analogy

The BL equations for momentum and energy are similar mathematically and indicate analogous behavior for the transport of momentum and heat

This analogy allows one to determine thermal parameters from velocity parameters and vice-versa (e.g., h-values can be found from viscous drag values)

The heat-momentum analogy is applicable in BLs when dp*/dx* 0 (turbulent flow is less sensitive to this)

The modified Reynolds analogy states

– where Cf is the skin friction coefficient

60Pr6.0PrRe2

1 3/1 for fCNu

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Turbulence

Turbulence represents an unsteady flow, characterized by random velocity, pressure, and temperature fluctuations in the fluid due to small-scale eddies

Turbulence occurs when the Reynolds number reaches some critical value, determined by the particular flow geometry

Turbulence is typically modeled with eddy diffusivities for mass, momentum, and heat

Turbulent flow increases viscous drag, but may actually reduce form drag in some instances

Turbulent flow is advantageous in the sense of providing higher h-values, leading to higher convection heat transfer rates

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Forced Convection Heat Transfer – External Flow

Reading: Incropera & DeWitt

Chapter 7

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The Empirical Approach

Design an experiment to measure h– steady-state heating– transient cooling, Bi<<1

Perform experiment over a wide range of test conditions, varying:– freestream velocity (u )

– type of fluid (, )– object size (L)

Reduce data in terms of Reynolds, Nusselt, and Prandtl

Pr, Nu , Re

k

LhLuLL

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The Empirical Approach, cont.

Plot reduced data, NuL vs. ReL, for each fluid (Pr):

Develop an equation curve-fit to the data; a common form is

typically: < m < 1 < n < 0.01 < C < 1

nmLL C PrReNu

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Flat Plate in Parallel, Laminar Flow (the Blasius Solution)

Assumptions:– steady, incompressible flow– constant fluid properties– negligible viscous dissipation– zero pressure gradient (dp/dx = 0)

Governing equations

(energy)

(momentum)

y)(continuit 0

2

2

2

2

y

T

y

Tv

x

Tu

y

u

y

uv

x

uu

y

v

x

u

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Blasius Solution, cont.

Similarity solution, where a similarity variable () is found which transforms the PDEs to ODEs:

Blasius derived:

x

uy

3/1

3/12/1

2/1,

Pr

PrRe332.0Nu

Re664.0

Re

5

/

0.5

t

xx

x

xxf

x

k

xh

C

x

xu

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Empirical Correlations

Flat Plate in parallel flow: Section 7.2

Cylinder in crossflow: Section 7.4

Sphere: Section 7.5

Flow over banks of tubes: Section 7.6

Impinging jets: Section 7.7

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Forced Convection Heat Transfer – Internal Flow

Reading: Incropera & DeWitt

Chapter 8

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Review of Internal (Pipe) Flow Fluid Mechanics

Flow characteristics Reynolds number Laminar vs. turbulent flow Mean velocity Hydrodynamic entry region Fully-developed conditions Velocity profiles Friction factor and pressure drop

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Pipe Reynolds Number

um is mean velocity, given by

Reynolds number for circular pipes:

Reynolds number determines flow condition:– ReD < 2300 : laminar

– 2300 < ReD < 10,000 : turbulent transition

– ReD > 10,000 : fully turbulent

DumRe

pipescircular for

4

2D

m

A

mu

cm

Dm4

Re

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Pipe Friction Factor

Darcy friction factor, f, is a dimensionless parameter related to the pressure drop:

For laminar flow in smooth pipes,

For turbulent flow in smooth pipes,

– good for 3000 < ReD < 5x106

For rough pipes, use Moody diagram (Figure 8.3) or Colebrook formula (given in most fluid mechanics texts)

2/

)/(2mu

Ddxdpf

Df Re64

264.1)ln(Re790.0 Df

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Pressure Drop and Power

For fully-developed flow, dp/dx is constant; so the pressure drop ( p) in a pipe of length L is

The pump or fan power (Wp) required to overcome this pressure drop is

– where p is the pump or fan efficiency

52

22 8

2

D

fLm

D

fLup m

, p

p

pmW

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Thermal Characteristics of Pipe Flow

Thermal entry region Thermally fully-developed conditions Mean temperature Newton’s law of cooling

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Newton’s Law of Cooling & Mean Temperature

The absence of a fixed freestream temperature necessitates the use of a mean temperature, Tm, in Newton’s law of cooling:

Tm is the average fluid temperature at a particular cross-section based upon the transport of thermal (internal) energy, Et:

Unlike T , Tm is not a constant in the flow direction; it will increase in a heating situation and decrease in a cooling situation.

)( mss TThq

cA

cvmvt uTdAcTcmE

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Fully-Developed Thermal Conditions

While the fluid temperature T(x,r) and mean temperature Tm(x) never reach constant values in internal flows, a dimensionless temperature difference does - and this is used to define the fully-developed condition:

the following must also be true:

therefore, from the defining equation for h:

i.e., h = constant in the fully-developed region

0

ms

s

TT

TT

x

)( xfTT

rT

TT

TT

r msms

s

)( 0 xf

TT

rTkh

ms

r

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Pipe Surface Conditions

There are two special cases of interest, that are used to approximate many real situations:

1) Constant surface temperature (Ts = const)

2) Constant surface heat flux (q”s = const)

)(xqq ss

)( ),( xTTxTT mmss

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Review of Energy Balance Results

For any incompressible fluid or ideal gas pipe flow,

1) For constant surface heat flux,

2) For constant surface temperature,

3) For constant ambient fluid temperature,

)( mimopconv TTcmq

p

smimo cm

PLqTT

pmis

mos

cm

hPL

TT

TT

exp

tottpmi

mo

RcmTT

TT

,

1exp

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Convection Correlations for Laminar Flow in Circular Pipes

Fully-developed conditions with Pr > 0.6:

– Note that these correlations are independent of Reynolds number !

– All properties evaluated at (Tmi+Tmo)/2

Entry region with Ts = const and thermal entry length only (i.e., Pr >> 1 or unheated starting length):

– combined entry length correlation given by eqn. (8.57) in text

constant) ( 66.3

constant) ( 36.4

s

sD

T

qk

hDNu

8.56) (eqn. PrRe)(04.01

PrRe)(0668.0 66.3 3/2

D

DD

LD

LDNu

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Convection Correlations for Turbulent Flow in Circular Pipes

Fully-developed conditions, smooth wall, q”s = const or Ts = const

– Dittus-Boelter equation - fully turbulent flow only (ReD > 10,000) and 0.7 < Pr < 160

– NOTE: should not be used for transitional flow or flows with large property variation

heating) (fluid for 0.4

cooling) (fluid for 0.3

where

8.60) (eqn. PrRe023.0 5/4

ms

ms

nD

TTn

TTn

Nu

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Convection Correlations for Turbulent Flow in Circular Pipes, cont.

– Sieder-Tate equation - fully tubulent flow (ReD > 10,000),very wide range of Pr (0.7 - 16,700), large property variations

– Gnielinski equation - transitional and fully-turbulent flow (3000 < ReD < 5x106), wide range of Pr (0.5 - 2000)

– f given on a previous slide (eqn. 8.21)

8.61) (eqn. PrRe027.0 14.03/15/4sDNu

8.63) (eqn. )1(Pr)8(7.12 1

Pr)1000)(Re8(3/22/1

f

fNu D

D

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Convection Correlations for Turbulent Flow in Circular Pipes, cont.

Entry Region– Recall that the thermal entry region for

turbulent flow is relatively short, I.e.,only 10 to 60D

– Thus, fully-developed correlations are generally valid if L/D > 60

– For 20 < L/D < 60, Molki & Sparrow suggest

Liquid Metals (Pr << 1)– Correlations for fully-developed turbulent

pipe flow are given by eqns. (8.65), (8.66)

DLNu

Nu

fdD

D 6 1

,

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Convection Correlations for Noncircular Pipes

Hydraulic Diameter, Dh

– where: Ac = flow cross-sectional area

P = “wetted” perimeter

Laminar Flows (ReDh < 2300)

– use special NuDh relations– rectangular & triangular pipes: Table 8.1– annuli: Table 8.2, 8.3

Turbulent Flows (ReDh > 2300)

– use regular pipe flow correlations with ReD and NuD based upon Dh

P

AD ch

4

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Heat Transfer Enhancement

A variety of methods can be used to enhance the convection heat transfer in internal flows; this can be achieved by

– increasing h, and/or– increasing the convection surface area

Methods include– surface roughening – coil spring insert– longitudinal fins– twisted tape insert– helical ribs