transient or unsteady state heat conduction

8/12/2019 Transient or Unsteady State Heat Conduction

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Chapter 3: Unsteady State [ Transient ] Heat

Conduction

3.1 . Introduction

3.2 . Biot and Fourier Number

3.3 . Lumped heat capacity analysis

3.4 . Time constant and response of a thermocouple

3.5 . Transient heat conduction in solids with finite conduction and

convective resistances [ 0 < Bi< 100 ]


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3.1 Introduction

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.

The temperature of such a body varies with time as well as position.

T(x,y,z,t)

(x,y,z)- Variation in thex, yandzdirections andt - Variation with time

Temperature will vary with location within a system and with time.

NOTE:Temperature and rate heat transfer variation of a system are dependent

on itsinternal resistanceandsurface resistance

Steady-state conduction:

temperature do not change with time;equilibrium condition

unsteady-state conduction:temperature change with time


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3.2 Biot and Fourier Number

Rr

cond

conv

dr

trdTkA

TtRThABi

,

),(

bodyewithin thrateferheat transConduction

surfaceat therateferheat trans/radiationConvection

Bi

R

TTkA

TThABi

cond

conv

0

0

solid

sticcharacteri

sticcharacteri

cond

conv

k

hL

L

kA

hABi

Biot Number,


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Note: Whenever the Biot number is small, the internal temperature gradients are

also small and a transient problem can be treated by the lumped thermal capacity

approach. The lumped capacity assumption implies that the object for analysis isconsidered to have a single mass-averaged temperature.

conv

cond

conv

cond

sticcharacteri

sticcharacteri

cond

conv

RR

hA

kA

L

L

kAhABi 1

Biot and Fourier Number (continue.)

The Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a

dimensionless number that characterizes transient behavior of a system.

Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy

storage. It is defined as:

22

sticcharacteristiccharacteri

sticcharacteri

oL

t

CL

kt

TCV

tL

TkA

F


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3.3 Lumped heat capacity analysis

The simplest situation in an unsteady heat transfer process is to use the lumped

capacity analysis.Note:Neglect the temperature distribution inside the solid and only deal with the

heat transfer between the solid and the ambient fluids.

70o

C

70oC

70o

C

70

o

C

70o

C

70o

C70

o

C75

o

C

60o

C

65o

C

60o

C

Temperature of the metal ball

changes with time, but it does

not change with position at any

given time. Temperature of the ball

remains uniform at all times

Large potato put in a vessel with boiling water.

After few minutes, if you take out the potato,

temperature distribution within the potato is not

even close to being uniform.

Thus, lumped system analysis is not applicablein this case.

Note:The first step in the application of lumped system analysis is the calculation of the Biot

number, and the assessment of the applicability of this approach. One may still wish to use

lumped system analysis even when the criterion Bi 0.1 is not satisfied, if high accuracy is not

a major concern.


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3.4 Time constant and response of a thermocouple

Thermocouple: When two dissimilar metals are joined together at two points toform a closed loop and temperature difference exists between junctions, an

electrical potential is set up between the junctions. Such an arrangement is known

as thermocouple and is frequently used for the temperature measurement and

used in lumped parameter analysis.

Response of a thermocouple: Time required for the thermocouple to reach thesource temperature when it is exposed to it.

Sensitivity:Time required by the thermocouple to reach 63.2% of its steady state

value.

expa

i a

t t hA

t t Vc

exp 1 0.632 0.368 exp[ 1]a

i a

t t hA

t t Vc


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1h AV c

The parameter Vc/hA has units of time and is called time constant of the

system and denoted by*.

Using time constant, the temperature distribution in the solid can be

expressed as

* Vc k V

hA h A

exp*

a

i i a

t tt t

Time constant and response of a thermocouple (continue.)

Note: Low value of time constant can be achieved by (i) decreasing the wire

diameter (ii) using light metals of low density and low specific heat (iii) increasing

the heat transfer coefficient


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3.5 Transient heat conduction in solids with finite

conduction and convective resistances [ 0 < Bi < 100 ]

Consider the heating and cooling of a plane wall of thickness l = 2and extendingto infinity in the y and z direction. Initially the wall is at uniform temperature tiand

suddenly both surfaces + and are exposed to and maintained at the ambient

temperature ta. The controlling equation for the transient heat conduction is:

Boundary conditions are:

(i) t = ti at=0

(ii) dt/dx=0 at x=0; symmetric nature of the

temperature profile within the plane wall;

(i) Conduction = convection

kA(dt/dx)=hA(t-ta) at x= + and x=

The solution obtained after mathematical analysis that

2

2

1d t dt

dx d

2, ,a

i a

t t x hlf

t t l k l


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Obvious when conduction resistance is not negligible, the temperature history

becomes a function of Biot number hl/k, Fourier number /l2 and the

dimensionless parameter x/l which indicates the location of point within the plate

where temperature is to be obtained.

Note:In case of cylinder and sphere x/l is replaced by r/R.

Graphical charts have been prepared for the above equation in a variety of forms.

The Hiesler chartsdepict the dimensionless temperature (to-ta)/(ti-ta) versus Fo

for various values of 1/Bi for solids of different geometrical shapes such as a plate,cylinders and spheres.

Note:Hiesler charts give the temperature history of the solid at its mid plane, x=0.

Temperatures at other locations are worked out by multiplying the mid plane

temperature by the correction factors read from charts which is given. Use is made

of the following relationship.

0

0

a a a

i a i a a

t t t t t t X

t t t t t t


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transient or unsteady state heat conduction

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