Module 5 Teacher Slides

8/10/2019 Module 5 Teacher Slides

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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 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

k where = is the thermal diffusivity

c

T T c k T T t t



“A heated/cooled body at T i is suddenly exposed to f luid at T with a

known heat transfer coeff icient . 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.

surface bodyatresistanceconvectionExternal

solidwithinresistanceconductionInternal

/1

/

hA

kA L

k

hL Bi 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

mc p

T T BC: T t 0 T i

Solution: let T T , therefore

d

dt

hA

mc p



Lumped Parameter Analysis

ln

hA

mct

i

t mchA

i

pi

ii

p

p

eT T

T T

e

t mc

hA

T T

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

t cV

hA

T T

T T

t L

Bit L Lc

k

k

hLt

cV

hA

ccc

c

2

11

c

k

Thermal diffusivity: (m² s-1)





t L

Foc

2

k

hL Bi 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(half cL

sphereissolidthewhenradius)third-one(3cL

cylinder.aissolidthewhenradius)-(half 2or

cL,examplefor

problemtheininvlovedsolidtheof sizethetorealte:scalelengthsticcharacteriaiscLwhere

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

iT xT )0,(

2

21

x

T T

a

00

x x

T

T t LT h

x

T k

l x

),(

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 independentvariables.

How may the functional dependence be simplified?

•The answer is Non-dimensionalisation. We first needto understand the physics behind the phenomenon,

identify parameters governing the process, and groupthem into meaningful non-dimensional numbers.



Dimensionless temperature difference:

T T

T T

ii

*

Dimensionless coordinate: L

x x *

Dimensionless time: Fo L

t t

2

*

The Biot Number: solid k

hL

Bi

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

quantities

),,( ** Bi Fo x f

Exact Solution: *

1

2* cosexp x FoC 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 2.0 Fo

Variation of mid-plane temperature with time )0(*

x Fo

FoC T T

T T

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:

T T cV Q

QQ

Q E

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.

S x

T k

xt

T c



PW E

xw e

(dx)w (dx)e

t

∆x

Integration over the control volume and over a time interval

gives

t t

t CV

t t

t cv

t t

t CV

dt SdV dt dV

x

T k

x

dt dV

t

T c

t t

t

t t

t we

e

w

t t

t

dt V S dt x

T kA

x

T kAdV dt

t

T c



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

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

t t

t

t t

t w

W P w

e

P E e

old

P P new

dt V S dt x

T T Ak x

T T Ak V T T c

Now, an assumption is made about the variation of T P , T E and T w with

time. By generalizing the approach by means of a weighting

parameter f between 0 and 1:

t f f t dt

t t

t

old

P

new

P P P

1

xS x

T T

k x

T T

k f

x

T T k

x

T T k f x

t

T T c

w

old

W

old

P w

e

old

P

old

E e

w

new

W

new

P w

e

new

P

new

E e

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

bT a f a f a

T f fT aT f fT aT a

old

P E W P

old

E E E

old

W W W P P

)1()1(

)1()1(

0

0 P E W P aaaa

t

xca P

0

w

wW

x

k a

e

e E

x

k a

xS b

;

;

;

;

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 resultingscheme is “Crank - Nicolson”.



t

T

TP

old

t TPnew

t+Dt

f=0

f=1

f=0.5

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

Explicit schemeLinearizing the source term as and setting f = 0

u

old

P E W P

old

E E

old

W W P P S T aaaT aT aT a )(0

0 P P aa t xca P 0

w

wW

x

k a

e

e

E x

k

a

For stability, all coefficients must be positive in the discretized

equation. Hence,0)(0 P E W P S aaa



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 explicitscheme 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:

bT aa

aT T

aT T

aT a P W E

P

old

W W W

old

E E E P P

00

2222

P P W E P S aaaa2

1)(

2

1 0 t

xca P

0

w

wW

x

k a

e

e E

x

k a

old

p pu T S S b2

1

;

;

;

;



For stability, all coefficient must be positive in the discretized

equation, requiring

20 W E P 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 P P W W E E P P T aT aT aT a 0

P W E P P S aaaa 0

t

xca P

0

w

wW

x

k a

e

e E

x

k a

;

;

;



General remarks:

A system of algebraic equations must be solved at each timelevel. 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, theimplicit method is recommended for general purpose transient

calculations because of its robustness and unconditional stability.

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