(5) transient conduction - lumped cap mtd - s1 2013-2014 (1)

7/27/2019 (5) Transient Conduction - Lumped Cap Mtd - S1 2013-2014 (1)

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Dr. K.C. Leong, 2006

Lecture 2:Radiation & Conservation of Energy Requirement

(5) Trans ient Conduc t ion : Lumped

Capacitance Methodby

Assoc Prof Leong Kai ChoongSchool of Mechanical and Aerospace Engineering

MA3003/MP3003/AE3003 Heat TransferSemester 1, AY 2013-2014

Read Chapter 4: Section 4-1 of the

textbook before these lecture

slides


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2

At the end of these lectures, you should be able to

assess when the spatial variation of temperature of a

solid is negligible and temperature varies nearly

uniformly with time, making the lumped capacitance

analysis applicable,

calculate the time taken by a solid to reach some

temperature Tor the temperature reached by a solid

at some time t,

calculate the amount of heat transferred (in J) during

the transient conduction process, and perform general lumped capacitance calculations.

Learning Objectives


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3

Transient Conduction

Solid body suddenly subjected to change in thermal

conditions at the boundary temperature varies with

space and time until steady-state.

Unsteady, time-dependent ortransient heat

conduction occurs in the interim period.

We can solve general heat conduction equation for

T(x,y,z,t) in solid subject to initial and boundary

conditions.


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4

t

Tce

z

Tk

zy

Tk

yx

Tk

xpgen

For example, solve

or solve

with 6 boundary conditions and 1 initial condition

T(x,y,z,t)

1

2

2

t

T

x

T

with two boundary conditions and one initialcondition T(x,t)

semi-infinite solid


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5

Analytical solutions may be obtained for simple

geometries and conditions.

Such complicated solutions may be obtained

from advanced heat transfer texts such as

Carslaw, H.S. and J.C. Jaeger, Conduction ofHeat in Solids, 2nd ed., Oxford UniversityPress, London, 1959.

Ozisik, M.N., Heat Conduction, 2nd ed., Wiley,New York, 1993.


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6

Lumped Capacitance Method

Sudden change in thermal environment e.g.quenching of hot metal forging in cooler liquid

(Figure 1)

Temperature of solid spatially uniform during

transient process i.e. no temperature gradients in

solid

Approximated if resistance to solid conduction is

small compared to resistance to convectionbetween solid and fluid.


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Fig. 1 Cooling of a hot metal forging

Source: Incropera et al. (2007)

convQ


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8

Temperature of forging cannot be obtained by solving heat

conduction equation. Take overall energy balance on solid

(3)exp

(2)ln

Write

0

tVc

hA

TT

TT

hA

Vct

dtd

hA

Vc

TT

s

ii

i

s

t

si

convQ

dt

dTVcTThA

EE

s

stout

(1)


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9

Fig. 2 Transient temperature response of lumped

capacitance solids for different time constants


1/e

t

Vc

hAs

i

exp

s

i

hA

Vc

texp


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10

Electrical analogy can be used if we define a

thermal time constant

)4(1

tt

st

CRVchA

Analogous to voltage decay,Ewhen charged

capacitor,Cdischarges through a resistor,R in a

RCcircuit (See figure on next slide)

(3)Recall

tVc

hA

TT

TT s

ii

exp

%8.361

At

e

t

i

t


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Equivalent thermal circuit for a lumpedcapacitance solid


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12

0

RC

E

dt

dE

ForRC electric circuit,

Solution with initial conditionE(0) =E0isRCteEE /

0

For transient conduction (lumped capacitancemodel), total energy transferQin J occurring up

to some time tcan be calculated by

ti

tt

sconv

tVc

dthAdtQQ

/exp1

00

ttCRt

ie/

analogous to

s

t hA

Vc

N.B.

f f


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QE

EQ

EEEE

st

st

stoutgin

or00

Alternatively, use conservation of energy over finite

time interval, t= t- 0 = t

tistti

iist

tVcEQ

tVc

VcTTVcE

/exp1Hence,

/exp1

For quenching,Q

is positive (Eout

)

For heating, Q is negative. (Ein

)

tt

ie /

Whi h h l b l b


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Which thermal system below can bemodelled as a lumped capacitance system?

Source: engel, Y.A. and Ghajar, A.J., Heat and Mass Transfer:

Fundamentals and Ap pl ications, 4th Ed. (SI Units), McGraw-Hill, 2011.


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Validity of Lumped Capacitance Method

Assume steady-state conditions of Fig. 3 to developthe conditions under which the lumped capacitance

method may be used.


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Fig. 3 Effect of Biot number on steady-state temperature

distribution in a plane wall with surface convection


convQ

condQ TThATTL

kAsss 2,2,1,

Bi

/1

/

2,

2,1,

khL

R

R

hA

kAL

TT

TT

conv

cond

s

ss


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17

Dimensionless numberBiot(pronounced as Bee-oh)

numbercan be interpreted as

conv

cond

R

R

hA

kAL

k

hL

/1/

Bi

Note that kis the thermal conductivity of the solid, not thefluid as in Nusselt number,

Nu hL/kto be introduced in convection.Bi


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Fig. 4 The Biot number can be viewed as the ratio of

convection (not convection resistance) at the

surface to conduction (not conduction resistance)

within the body

T

L

Ak

ThA

k

hL

or

R

R

hA

kAL

k

hL

conv

cond

Bi

/1

/Bi

Illustration from engel and Ghajar (2011)


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Born 21 April 1774, Paris

Died 3 February 1862, Paris

Nationality

French

physicist, astronomer, and

mathematician.

Fields

Professor of Mathematics at

Beauvais (1797) and

Professor of Physics at

Collge de France, Paris

(1800)

Doctoral student William Ritchie

Known for Biot-Savart law

Influences Louis Pasteur

Jean-Baptiste Biot

Fig 5 Transient temperature distributions for different Biot


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Fig. 5 Transient temperature distributions for different Biot

numbers in a plane wall cooled by convection



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For arbitrarily-shaped bodies, characteristic

length is defined as

s

cA

VL

?lengthaxialof

cylindersizefiniteandsideofcubeaforisWhat

spherefor3/

cylinderlongfor2/

wallplanefor2/

L

LL

rL

rL

LL

c

c

c

c

Show that


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)5(FoBiexp

TTTT

ii

c

kydiffusivitthermal

LtmberFourier nu

L

t

k

hL

Vc

thA

c

c

cs

,and

timeessdimensionlisFo,where

FoBi

2

2

Dimensionless temperature solution of Equation

(3) can be re-written as


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Example 1: Transient Response of a Thermocouple

Thermocouple used to measure temperature of a gas

stream flowing in a duct

Approximate junction as a sphere (D= 1 mm).

Actual thermocouples come in standard wire sizes denoted

as American Wire Gage (a.w.g.), e.g.

a.w.g. 18 denotes 1.024 mm diameter wire




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Thermocouple in this example is Type Kmade of

chromel and alumel wires

c= 0.4 kJ/kgK, k= 30 W/ mK, = 8600 kg/m3

Convection coefficient, h = 280 W/m2K

chromelalumel

Gas

stream


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Find:

(a) time required for thermocouple to reach 98% of

the applied temperature difference, i.e.

TTTtTi

100

2)(

(b) time constant or response time of thermocouple


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Solution:

Must check Bito see whether we can use lumped

capacitance approach.

m1067.1

3100.5

34

3

4

4-

3-

2

3

r

r

r

A

VL

s

c

1.01056.130 1067.1280Bi3-

-4

k

hL c


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s03.8

2

100ln

280

104.01067.18600

lnln

34

ici

sh

cLhAVct

(a) Time required to reach 98% of applied

temperature difference is given by

(b) Time constant is

s05.2

280

104.01067.18600 34

h

cL

hA

Vcc

s


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Magnitude of time constant indicates how sensitive

the thermocouple is to a change in temperature.

A smaller bead diameter implies a more sensitive

thermocouple.


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Example 2: Annealing of Steel Balls

Carbon steel balls ( = 7833 kg/m3, k= 54 W/mK,c =

465 J/kgK, and = 1.474 x 10-6 m2/s) 8 mm indiameter are annealed by heating them first to 900C in

a furnace and then allowing them to cool slowly to

100C in ambient air at 35C.

a) If the average heat transfer coefficient is 75 W/m2K,determine how long the annealing process will take.

b) If 2500 balls are to be annealed per hour, determine

the total rate of heat transfer from the balls to the

ambient air.

Assumptions:


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min2.7s163ln

0013.0

validisanalysisecapacitanclumped

1.00018.0Bi

TT

TT

hA

Vct

A

VL

k

hL

i

s

s

c

c

kW543

J/h1,952,500

J/ball781balls/h2500

ballperJ781

QnQ

TTmcEQ

ball

ifst

Assumptions:

(1) Constant properties (2) Constant and uniform ha)

b)

Source: engel and Ghajar (2011)

Example 3: Freezing of Biological Tissue


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Example 3: Freezing of Biological Tissue

A 1-cm cube of biological tissue is to be frozen in a

freezer at 80C. The properties of the tissue has thesame properties as liquid water at 25C. The convective

heat transfer coefficient is 10 W/m2K, and the tissue is

initially at 25C.

Determine the time it takes to freeze the tissuecompletely.

The freezing temperature is taken as 0C. The properties

of liquid water at 25C may be taken as: Thermal

conductivity of 0.6 W/mK, density of 997 kg/m3

, andspecific heat of 4.18 kJ/kgK.

Schematic:


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validisModeleCapacitancLumped

1.0028.06

Bi

k

hak

hLc

ANALYSIS:

min3.15s189ln

i

shA

Vct

66 2

3 a

a

a

A

VL

s

c


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General Lumped Capacitance Analysis

Transient conduction commonly initiated by

convection heat transfer - simple lumped

capacitance analysis.

In general, thermal conditions in solid may besimultaneously influenced by convection, radiation,

surface heat fluxand internal heat generation as

shown in Fig. 6.


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Fig. 6 Control surface for general lumped

capacitance analysis

sq

radq

convq


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Applying conservation of energy at any instant t:

)7(),(44, dtdTVcATTTThEAq rcssurghss

Eq. (7) - non- linear, first-order, non-

homogeneous ODE with no exact solution.

)6(),(,dt

dTVcAqqEAq rcsradconvghss

Simplified cases:


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Simplified cases:

Case 1: Radiation exchange only

dtdT

VcTTA surrs

44

,

T

T

t

rs

i surTT

dTdt

Vc

A44

0

,

(8)

Time required to reach temperature T

isur

isur

sur

sur

rs TT

TT

TT

TT

TA

Vct

sur

lnln4 3,

sur

i

sur T

T

T

T 11

tantan2(9)

Cannot obtain Tin terms oftexplicitly.

For T = 0 i e radiation to deep space


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ForTsur

0 i.e. radiation to deep space,

(10)11

3 33,

iTTA

Vct

rs

Case 2: Convection, surface heat flux and heat

generation without radiation where h is assumedto be independent of time.

Exact solution to the linear, first-order non-

homogeneous differential equation of the form

TT

VcEAqbVchAawhere

badt

d

ghsscs

and

,/

0

,,

tiontransformagintroducinbyityinhomogeneEliminate


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)exp()/(

)/(

,forngSubstituti

).exp(

),(to0fromgintegratinandvariablesSeparating

0

ngSubstituti

tiontransformagintroducinbyityinhomogeneEliminate

'

'

'

''

''

'

atabTT

abTT

at

tot

adt

d

a

b

i

i

i

Hence


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Hence,

)11(exp1/

exp at

TT

abat

TT

TT

ii

When b = 0, Eq. (11) reduces to Eq. (3) introduced

previously and yields T = Tiat t= 0.

)3(exp

tVc

hA

TT

TT s

i

./becomesEq.(11),As abTTt See Example 5.2 of Incropera et al. (2007)

Some Review Questions


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1. For which solid is the lumped capacitance analysis more

likely to be applicable: an actual apple or a golden apple of

the same size? Why?

2. For what kind of bodies made of the same material is the

lumped capacitance analysis more likely to be applicable:

slender ones or well-rounded ones of the same volume?

3. What is the significance of the thermal time constant?

4. To what electrical circuit is the lumped capacitance systemanalogous?

5. Why is it necessary to employ the concept ofcharacteristic

length in lumped capacitance analysis?

6. What are the physical significances of the Biot and Fouriernumbers?

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(5) transient conduction - lumped cap mtd - s1 2013-2014 (1)

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