m 5 Teacher Slides

8/13/2019 m 5 Teacher Slides

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2/20

UNSTEADY 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 theprocessed materials. Annealing (slow cool) can soften

metals and improve ductility. On the other hand,

quenching (rapid cool) can harden the strain boundary andincrease strength. In order to characterize this transient

behavior, the full unsteady equation is needed:

2 21, or

k

where = is the thermal diffusivityc

T Tc k T T t t


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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 thethermal circuit compared to the convective heat transfer coefficient.

Fig.5.1


4/20

Biot No. BiDefined to describe the relative resistance in a thermal circuit

of

the convection compared

Lc

is a characteristic length of the body

Bi0:

No conduction resistance at all. The body is isothermal.

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

be approximated as isothermalLumped capacitance analysis

can be performed.

Large Bi: Conduction resistance is significant. The body cannot be treated as

isothermal.

surfacebodyatresistanceconvectionExternal

solidwithinresistanceconductionInternal

/1

/

hA

kAL

k

hLBi cc


5/20

Transient heat transfer with no internalresistance: Lumped Parameter Analysis

Solid

Valid for Bi


6/20

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 ofspace!

-

To determine the temperature at a given time, or

- To determine the time required for the

temperature to reach a specified value.


7/20

)exp(T0

tcV

hA

TT

TT

tL

BitLLc

kk

hLtcVhA

ccc

c

211

c

k

Thermal diffusivity:

(m

s-1)



8/20


t

L

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

thickness2Lawithwallaplaneissolidthewhen)thickness(halfcL

sphereissolidthewhenradius)third-one(3cL

cylinder.aissolidthewhenradius)-(half2or

cL,examplefor

problemtheininvlovedsolidtheofsizethetorealte:scalelengthsticcharacteriaiscLwhere

L

or


9/20

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

TtLThx

Tk

lx

),(

k

x*=x/L x=+L

T,

h

x= -L

T,

h

T(x, 0) = Ti


10/20

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 thephenomenon, identify parameters governing theprocess, and group them into meaningful non-

dimensional numbers.


11/20

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

nnnn

n

n

tan

2sin2

sin4

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


12/20

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 obtain1C and 1

as a function ofBi.

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


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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

Tkxt

Tc


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PPWW EE

xxww ee

((ddx)x)ww ((ddx)x)ee

tt

xx

Integration over the control volume and over a time intervalgives

tt

t CV

tt

t cv

tt

t CV

dtSdVdtdV

x

Tk

x

dtdV

t

Tc

tt

t

tt

t we

e

w

tt

t

dtVSdt

x

TkA

x

TkAdVdt

t

Tc


15/20

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 weightingparameterf between 0 and 1:

tfftdttt

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,


16/20

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

equation into the standard form

bTafafaTffTaTffTaTa

old

PEWP

old

EEE

old

WWWPP

)1()1(

)1()1(

0

0PEWP aaaa t

xcaP

0

w

wW

x

ka

e

eE

x

ka

xSb

;

;

;

;

The time integration scheme would depend on the choice of the

parameterf. Whenf

= 0, the resulting scheme is explicit; when

0


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t

T

TPold

tTPnew

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 = 0u

old

PEWP

old

EE

old

WWPP STaaaTaTaTa )(0

0PP aa txcaP 0

w

wW

x

ka

e

e

Ex

k

a For stability, all coefficients must be positive in the discretized

equation. Hence, 0)(0 PEWP Saaa


18/20


19/20

For stability, all coefficient must be positive in the discretized

equation, requiring2

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 TaTaTaTa0

PWEPP

Saaaa 0t

xcaP

0

w

wW

x

ka

e

eE

x

ka

;

;

;


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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 t. For theimplicit 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 transientcalculations because of its robustness and unconditional stability.

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