HEAT TRANSFER

Many heat transfer problems require the understanding of

the complete time history of the temperature variation. For

example, in metallurgy, the heat treating process can be

controlled to directly affect the characteristics of the

processed materials. Annealing (slow cool) can soften

metals and improve ductility. On the other hand,

quenching (rapid cool) can harden the strain boundary and

increase strength. In order to characterize this transient

behavior, the full unsteady equation is needed:

2 21, or

kwhere = is the thermal diffusivity

c

T Tc k T T

t t

“A heated/cooled body at Ti is suddenly exposed to fluid at T with a

known heat transfer coefficient . Either evaluate the temperature at a

given time, or find time for a given temperature.”

Q: “How good an approximation would it be to say the annular cylinder is

more or less isothermal?”

A: “Depends on the relative importance of the thermal conductivity in the

thermal circuit compared to the convective heat transfer coefficient”.

Fig. 5.1

Biot No. Bi •Defined to describe the relative resistance in a thermal circuit of

the convection compared

Lc is a characteristic length of the body

Bi→0: No conduction resistance at all. The body is isothermal.

Small Bi: Conduction resistance is less important. The body may still

be approximated as isothermal

Lumped capacitance analysis can be performed. Large Bi: Conduction resistance is significant. The body cannot be treated as

isothermal.

surfacebody at resistance convection External

solid withinresistance conduction Internal

/1

/

hA

kAL

k

hLBi cc

Transient heat transfer with no internal resistance: Lumped Parameter Analysis

Solid

Valid for Bi<0.1

Total Resistance= Rexternal + Rinternal

GE: dT

dt

hA

mcpT T BC: T t 0 Ti

Solution: let T T , therefore

d

dt

hA

mcp

Lumped Parameter Analysis

ln

hA

mct

i

tmc

hA

i

pi

ii

p

p

eTT

TT

e

tmc

hA

TT

Note: Temperature function only of time and not of space!

- To determine the temperature at a given time, or

- To determine the time required for the

temperature to reach a specified value.

)exp(T0

tcV

hA

TT

TT

tL

BitLLc

k

k

hLt

cV

hA

ccc

c

2

11

c

k

Thermal diffusivity: (m² s-1)

Lumped Parameter Analysis

Lumped Parameter Analysis

tL

Foc

2

k

hLBi C

T = exp(-Bi*Fo)

Define Fo as the Fourier number (dimensionless time)

and Biot number

The temperature variation can be expressed as

thickness2La with wallaplane is solid the when) thickness(half cL

sphere is solid the whenradius) third-one(3cL

cylinder.a is solid the whenradius)- (half2or

cL, examplefor

problem thein invloved solid theof size the torealte:scale length sticcharacteria is c Lwhere

L

or

Spatial Effects and the Role of Analytical Solutions

The Plane Wall: Solution to the Heat Equation for a Plane Wall with

Symmetrical Convection Conditions

iTxT )0,(

2

21

x

TT

a

00

xx

T

TtLTh

x

Tk

lx

),(

k

x*=x/L x=+L

T∞, h

x= -L

T∞, h

T(x, 0) = Ti

The Plane Wall:

Note: Once spatial variability of temperature is included, there is existence of seven different independent variables.

How may the functional dependence be simplified?

•The answer is Non-dimensionalisation. We first need to understand the physics behind the phenomenon, identify parameters governing the process, and group them into meaningful non-dimensional numbers.

Dimensionless temperature difference:

TT

TT

ii

*

Dimensionless coordinate: L

xx *

Dimensionless time: FoL

tt

2

*

The Biot Number: solidk

hLBi

The solution for temperature will now be a function of the other non-dimensional quantities

),,( ** BiFoxf

Exact Solution:

*

1

2* cosexp xFoC nn

nn

BiC nn

nn

nn

tan

2sin2

sin4

The roots (eigenvalues) of the equation can be obtained from tables given in standard textbooks.

The One-Term Approximation 2.0Fo

Variation of mid-plane temperature with time

)0( * xFo

FoCTT

TT

i

2

11

*

0 exp

From tables given in standard textbooks, one can obtain 1C and 1

as a function of Bi. Variation of temperature with location )( *x and time ( Fo ):

*

1

*

0

* cos x

Change in thermal energy storage with time:

TTcVQ

QQ

QE

i

st

0

*

0

1

10

sin1

Numerical Methods for Unsteady Heat Transfer

Unsteady heat transfer equation, no generation, constant k, one-

dimensional in Cartesian coordinate:

The term on the left hand side of above eq. is the storage term,

arising out of accumulation/depletion of heat in the domain under

consideration. Note that the eq. is a partial differential equation as a

result of an extra independent variable, time (t). The corresponding

grid system is shown in fig. on next slide.

Sx

Tk

xt

Tc

PP WW EE

xx ww ee

(d(dx)x)ww (d(dx)x)ee

tt

∆∆xx

Integration over the control volume and over a time interval

gives

tt

t CV

tt

t cv

tt

t CV

dtSdVdtdVx

Tk

xdtdV

t

Tc

tt

t

tt

t we

e

w

tt

t

dtVSdtx

TkA

x

TkAdVdt

t

Tc

If the temperature at a node is assumed to prevail over the whole

control volume, applying the central differencing scheme, one obtains:

tt

t

tt

t w

WPw

e

PEe

old

PPnew dtVSdt

x

TTAk

x

TTAkVTTc

Now, an assumption is made about the variation of TP, TE and Tw with

time. By generalizing the approach by means of a weighting

parameter f between 0 and 1:

tfftdt

tt

t

old

P

new

PPP

1

xSx

TTk

x

TTkf

x

TTk

x

TTkfx

t

TTc

w

old

W

old

Pw

e

old

P

old

Ee

w

new

W

new

Pw

e

new

P

new

Ee

old

P

new

P

)1(

Repeating the same operation for points E and W,

Upon re-arranging, dropping the superscript “new”, and casting the

equation into the standard form

bTafafa

TffTaTffTaTa

old

PEWP

old

EEE

old

WWWPP

)1()1(

)1()1(

0

0

PEWP aaaa t

xcaP

0

w

wW

x

ka

e

eE

x

ka

xSb

;

;

;

;

The time integration scheme would depend on the choice of the

parameter f. When f = 0, the resulting scheme is “explicit”; when

0 < f ≤ 1, the resulting scheme is “implicit”; when f = 1, the

resulting scheme is “fully implicit”, when f = 1/2, the resulting

scheme is “Crank-Nicolson”.

t

T

TP old

t TP

new t+Dt

f=0

f=1

f=0.5

Variation of T within the time interval ∆t for different schemes

Explicit scheme

Linearizing the source term as and setting f = 0

u

old

PEWP

old

EE

old

WWPP STaaaTaTaTa )(0

0

PP aa t

xcaP

0

w

wW

x

ka

e

eE

x

ka

For stability, all coefficients must be positive in the discretized

equation. Hence, 0)(0 PEWP Saaa

0)(

e

e

w

w

x

k

x

k

t

xc

x

k

t

xc

2

k

xct

2

)( 2

The above limitation on time step suggests that the explicit

scheme becomes very expensive to improve spatial accuracy.

Hence, this method is generally not recommended for general

transient problems.

Crank-Nicolson scheme

Setting f = 0.5, the Crank-Nicolson discretisation becomes:

bTaa

aTT

aTT

aTa PWE

P

old

WWW

old

EEEPP

00

2222

PPWEP Saaaa2

1)(

2

1 0 t

xcaP

0

w

wW

x

ka

e

eE

x

ka

old

ppu TSSb2

1

;

;

;

;

For stability, all coefficient must be positive in the discretized

equation, requiring

2

0 WEP

aaa

k

xct

2)(

The Crank-Nicolson scheme only slightly less restrictive than the

explicit method. It is based on central differencing and hence it is

second-order accurate in time.

The fully implicit scheme

Setting f = 1, the fully implicit discretisation becomes:

old

PPWWEEPP TaTaTaTa 0

PWEPP Saaaa 0

t

xcaP

0

w

wW

x

ka

e

eE

x

ka

;

;

;

General remarks:

A system of algebraic equations must be solved at each time

level. The accuracy of the scheme is first-order in time. The time

marching procedure starts with a given initial field of the scalar

0. The system is solved after selecting time step Δt. For the

implicit scheme, all coefficients are positive, which makes it

unconditionally stable for any size of time step. Hence, the

implicit method is recommended for general purpose transient

calculations because of its robustness and unconditional stability.