heat transfer kakac

8/12/2019 Heat Transfer Kakac

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4 HEAT CONDUCTION

group of particles, subsequent to Some initial states of affairs. Such an approach gives

insight into the details of heat conduction; however, as one would predict, itwould

betoo cumbersome for most situations arising in engineering.

Inmost engineering problems, primary interest lies not in the molecular behavior

of a substance, but rather in how the substance behaves as a continuous medium. In OUr

study of heat conduction, we will therefore ignore the molecular structure of the sub-

stance and consider it to be a continuous medium, or continuum, which is fortunately

a valid approach to many practical problems where only macroscopic infonnation is

of interest. Such a model may be used provided that the size and the mean free path

of molecules are small compared with other dimensions existing in the medium, so

that a statistical average is meaningful. This approach, which is also known as the

phenomenological approach in the study of heat conduction, is simpler than micro-scopic approaches and usually gives the answers required in engineering. On the other

hand, to make up for the information lost by the neglect of the molecular structure,

certain parameters, such as thermodynamic state (i.e., thermophysical) and transport

properties, have to be introduced empirically. Parallel to the study of heat conduction

by the continuum approach, molecular considerations can also be used to obtain in-

formation on thermodynamic and transport properties. In this book, we restrict ourdiscussions to the phenomenological heat conduction theory.

1.4 SOME DEFINITIONS AND CONCEPTS

OF THERMODYNAMICS

In this section, we review some definitions and concepts of thennodynamics needed

for the study of heat conduction. The reader, however, is advised to refer to textbooks

on thermodynamics, such as References [4,18], for an in-depth discussion of thesedefinitions and concepts, as well as the laws of thermodynamics.

In thermodynamics, a system is defined as an arbitrary collection of matter of

fixed identity bounded by a closed surface, which can be a real or an imaginary one.

AH other systems that interact with the system under consideration are known as its

surroundings. In the absence of any mass--energy conversion, not only does the mass

of a system remain constant, but the system must be made up of exactly the same

submolecular particles. The four general laws listed in Section 1.1 are always stated

in terms of a system. In fact, On e cannot meaningfully appJy a general law until adefinite system is identified.

A control volume is any defined region in space, across the boundaries of which

matter, energy and momentum may flow, within which matter, energy and momentum

storage may take place, and on which external forces may act. Its position and/or size

may change with time. However, most often we deal with control volumes that arefixed in space and of fixed size and shape.

The dimensions of a system or a control volume may be finite or they may even

be infinitesimal. The complete definition of a system or a control volume must include

at least the implicit definition of a coordinate system, since the system or the control

volume can be stationary or may even be moving with respect to the coordinate system.


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F O U ND A T IO N S O F H E A T TR A N SF E R 5

The characteristic of a system we are most interested in is its thermodynamic state,

which is described by a jist of the values of all its properties. A property of a system

is either a directly or an indirectly observable characteristic of that system which call,

in principle, be quantitatively evaJuated. Volume, mass, pressure, temperature, etc. are

all properties. If all the properties of a system remain unchanged, then the system is

said to be in an equilibrium state. A process is a change of state and is described in

part by the series of states passed through by the system. A cycle is a process whereinthe initial and final states of a system are the same.

If no energy transfer as heat takes place between any two systems when they are

placed in contact with each other, they are said to be in thermal equilibrium. Any twosystems are said to have the same temperature ifthey are in tbermal equilibrium with

each other. Two systems not in thermal equilibrium would have different temperatures,

and energy transfer as heat would take place from one system to the other. Temperature

is, therefore, the property of a system that measures its "thermaJlevel."

The laws of thermodynamjcs deal with interac60ns between a system and its

surroundings as they pass through equilibrium states. These interactions may be

divided into two as (I) work or (2) heat interactions. Heat has already been defined as

the form of energy that is transferred across a system boundary owing to a tempera-

ture difference existing between the system and its surroundings. Work, on the other

hand, is a form of energy that is characterized as follows. When an energy form of one

system (such as kinetic, potential, or internal energy) is transformed into an energy

form of another system or surroundings without the transfer of mass from the

system and not by means of a temperature difference, the energy is said to have beentransferred through the performance of work.

The amount of change in the temperature of a substance with the amount of energy

stored within that substance is expressed in terms of specific heat. Because of the

different ways in which energy can be stored in a substance, the definition of specific

heat depends on the nature of energy addition, The specific heat at constant volume

is the change in internal energy (see Section 1.6) of a unit mass per degree change oftemperature between two equilibrium states of the same volume. The speqlic heat at

constant pressure is the change in enthalpy (see Section 1.6) of a unit mass between

two equilibrium states at the same pressure per degree change of temperature.

1.5 LAW OF CONSERVATION OF MASS

The law of conservation of mass, when referred to a system, simply states that, ill

the absence 0/allY mass-energy conversion, the mass ofthe system remains constant.Thus, for a system,

drn

-=0dt

orm =constant

(I.I)Wherem is the mass of the system.

We now proceed to develop tbe form of this law as it applies to a control vol-

ume. Consider an arbitrary control volume fixed ill space and of fixed shape and size,

as illustrated in Fig. 1.1. Malter flows across the boundaries of this control volume.


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. 1I/J(I+Il.I)-m\(I) ( a m ) a 1lim = - =- pdV

A I---rO ~t a t cv a t cv( 1.4)

6 HEAT CONDUCTION

v

System b o u n d ary

a t t i me t

dA in

'ilQ

C ont ro l s ur f ac e, CS

Figure 1.1 Flow of mutter through a fixed control volume.

Define a system whose boundary at some time Ihappens to correspond exactly to that

of the control volume. By definition, the control volume remains fixed in space, but

the system moves and at some later t ime I + 6.t occupies di f ferent volume in space.The two positions of the system are shown in Fig. 1.1 by dashed lines. Since the mass

of the sys tem is conserved, we can wri te

(1.2)

where mi . 1112. an d 1n 3 represent the instantaneous values of mass contained in the

three regions of space shown in Fig. 1.1. Dividing Eq. (I .2) by Il.t and rearranging we

get

/11\(1 + 1l.1) - "/J(I)

Il.I(1.3)

As Il.I -> 0, the left-hand side of Eq. (1.3) reduces to

where dV is an element of the control volume and p is the local density of that

element. In addition, "cv" designates the control volume bounded by the control

surface, "cs." Equation (1.4) represents the time rate of change of the mass within

the control volume. Furthermore,

. /113(1 + Il.t). 1 .11m =Jt1in=- pV.ndAin

8/-)00 6.1 Am

( 1.5)


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F O U N DA T IO N S O F H E A T T R A NS F ER 7

and

. m2(t + 61) . 1 .1 1m =m O ut= pV.ndAoUl~t--+O 6. r AO t J l

where 'ilin and J i l ou1 are t he m ass f l ow rat es i nt o and out of t he cont rol vol um e,

V denotes the velocity vector, and n is the outward-pointing unit vector normal to

the control surface. Thus, as 61 .... 0, Eq . (1.3)becomes

: 1 1 pdV =- r pVndA;" - r pV'n dAoUl

cv J A m J A OUI(1.70)

which can be rewritten as

~!PdV=-!PV.i1dAa t cv cs

(l.7b)

where dA is an element of the control surface. Eguation(1.7b) states that the lime

rare of increase of mass within a control volume is equal to the net rate of mass flow

into the control volume. Since the control volume is fixed in location, size and shape,Eq, (I.7b) can also be written as

1 ap dv=-l pVndAcv a t cs ( I.7c)

The surface integral on the right-hand side of Eq. (1.7c) can be transformed into a

volume integral by the divergence theorem [6]; that is,

1 pv.ndA=l V.(pV)dVcs cvHence, Eq . (1.7c) can also be written as

1 ap dV=-1 v.(pV) dVcv a t cv (1.9a)or

1 rap +v'(PV)]dV=Ocv a t (1.9b)

Since this result would be valid for all arbitrary control volumes. the integrand must

be zero everywhere, thus yielding

ap-+V'(pV)=oa l

(1.10)

This equation is referred to as thecontinuity equation.

( 1.6)

(1.8)


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When a system undergoes a cyclic process, the first law of thermodynamics can be

ex pressed as

(Lll)

8 HEATCONDUCfION

1.6 FIRST LAW OF THERMODYNAMICS

where the cyclic integral 0Q represents the net heat transferred to the system, andthe cyclic integral oW is the net work done by the system during the cyclic process.

Both heat and work are path functions; that is, the net amount of heat transferred

to, and the net amount of work done by, a system when the system undergoes a change

of state depend on the path that the system follows during the change of state. This is

why the differentials of heat and work in Eq. (J .11) are inexact differentials, denoted

by the symbols 0Q and 0 W , respectively.

For a process that involves an infinitesimal change of state during a time interval

dt, the first law of thermodynamics is given by

dE =oQ -oW ( 1.12)

where 0Q and 8 Ware the differential amounts of heat added to the system and the

work done by the system, respectively, and dE is the corresponding increase in

the energy of the system during the time interval dt. The energy E is a property

of the system and, like all other properties, is apoint function. That is, dE depends on

the initial and final states only, and not on the path followed between the two states.

For a more complete discussion of point and path functions, see References [4, 18].The property E represents the total energy contained within the system and, in the ab-

sence of any mass-energy conversion and chemical reactions, is customarily separated

in to three parts as bulk kinetic energy, bulk potential energy, an d internal energy; that

IS,

E=KE+PE+U (1.J 3)

The internal energy, U, represents the energy associated with molecular and atomic

behavior of the system.

Equation (1.J2) can also be written as a rate equation:

dE

dt

oQ sw

---dt dt (Ll4a)

ordE .-=q-Wdt

(Ll4b)

where q = oQ/dt represents the rate of heat transferto the system and I V = 0W[dtis the rate of work done (power) by the system.

Following the approach used in the previous section, we now proceed to develop

the control-volume form of the first law of thermodynamics. Referring to Fig. 1.J, the

first law for the system under consideration can be written as

t>.E =IlQ - IlW ( 1.15)


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F O UN D AT IO N S O F H E A T TR A NS F ER 9

where VQ is the amount of heat transferred to the system, VW is the work done by

the system, and 6 is the corresponding increase in the energy of the system during

the time interval zxr. Dividing Eq. (1.15) by /),1 we get

/),

/),t

VQ VW~---

/),t /), 1

The left-hand side of this equal ion can be written as

, (I + /),t) + 2(t + /),1)- 3(t + /),1) - 1(I)

/),t

/),

/),t(1.17a)

or(1.I7b)

where E I, E2, an d 3 are the instantaneous values of energy contained in the three

regions of space shown in Fig. l.l.

As /),t - - - > 0, the first term on the right-hand side ofEq, (1.17b) becomes

. I(I+/),t)-I(t) ( a ) a 1lim = - =- epdV

.6 . 1 - - - ,l > 0 /).,t a t cv a t cv

where e is the local specific energy (i.e., energy per unit mass). Equation (1.18) repre-

sentsthe time rate of change of energy with in the contro l volume. Furthermore,

. 2(t + /),1) /, ,lim = epV.o dAoUI

.6.1-+0 6 . . 1 A O U I

and3(t + /),1) /, ,

lim =- epV.n dAio.6. / . . . . . . , .0 6.( A in

which represent, respect ively, the rates of energy leaving and enter ing the contro l

volume attune I. Thus, as /),t - - - > 0, Eq. (J.J7b) becomes

d /),- = lim ~dt 81-+0!'!.t

=: 1 1 epdV + / , epV.n dAou,+ ! epV.n dAioc v A 0I J 1 A m

which can also be written as

d a l l ' - =~ epdV + epV.o dAdt a t cv cs

The first term on the right-hand side of Eq. (1.16) is the rate of heat transferto the

system, which is also the rate of heat transfer across the control surface as ! ; ; . . t -- + 0;

that is,

lim VQ =( 8Q) =qcs6(--+0 fj.t dt cs

( 1.16)

( 1.18)

( 1.19)

(1.20)

(l.21)

( l.22)

( 1.23)


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10 HEAT CONDUCTION

As 1',.1 --+ 0, the second term on the right-hand side of Eg. (l.16) becomes

VW o W .

lim

- - = - = Wt ..(--" f O /: ).t dt (1.24)

which is the rate of work done (power) by the matter in the control volume (i.e., the

system) on its surroundings at time I. Hence, as I',.t --+ 0, Eg. (1.16) becomes

a l l ' .~ epdV + epV,n dA =qcs - Wa t cv cs

/ (1.25)

Tbe surface integral on the left-hand side of Eg. (1.25) can be transformed into a

volume integral by the divergence theorem [6] to yield

1 epV,n dA =1V,(epV)dVcs cv= 1 [eV.(pV)+pV.Ve]dV

cv

(1.26)

Substituting this result into Eg. (1.25) and then rearranging the tenms on the left-hand

side we obtain [

1 p [a e + v. v e ] dV =qcs - I Vcv a t (1.27)where we have made use of the continuity relation (1.10).

Equation (1.25), or Eg. (1.27), is the control-volume form of the first law of

thermodynamics. However, a final form can be obtained after further consideration

of the power term IV .

Work can be done by the system in a variety of ways. In this analysis we consider

the work done against normal stresses (pressure) and tangential stresses (shear), the

work done by the system that could cause a shaft to rotate (shaft work), and the work

done on the system due to power drawn from an external electric circuit. We neglect

capillary and magnetic effects.

Consider now the work done by the system against pressure, which is also calledflow work. The work done against the pressure p acting at a surface element dAou t

during the time interval M is pdAoull',.n. where I',.n = (V.n) 1',.1 is the distance

moved normal to dAou t. Hence, the rate a/work done by the system against pressure

at dAou t is P dAoutV.n. The rate of work done against pressure over Aou t is, then,

IEquation ( J .27) can also be written as

1 D, .p- dV =qcs - W

cv Dt

w her e t he derivativeDe 8e-=-+VVeDr a t

i s c o m m o n ly r e fe r re d to as the substantia! derivative o r the material derivative.


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FOUNDATIONS OF HEAT TRANSFER 11

given by fAoul pv.n dAout. Similarly, rate of work done on the system by pressure

acting on Ai" would be given by - fA pV fidAi"' Thus, the net rate of work doneby the system against pressure will be m

Wnormal = [ pV.n dAout+ ( pVn dAinJ AOUI J A i n

(1.28a)

which can also be rewritten as

W"orrnal =[ pV.ii dAi:

(1.28b)

Let Wshaft represent the rate at which the system does shaft work, and Mishear be

the rate at which the system does work against shear stresses. The rate of work done

on the system due to power drawn from an external electric circuit, on the other hand,

can be written as fe v q e dV, where q e is the rate of internal energy generation per unitvolume due to the power drawn to the system from the external circuit.

Hence, the control-volume form of the first law thermodynamics, Eq. (1.25) or

Eq,(1.27), can also be written as

~1epdV + 1epV.ii dAa t cv C$=qes-1 pVn dA - ""shear - ""shaft +1 qedV

es ev

(1.29)

or

1p[ae+V.veJdv

cv a t

=qcs - { pv.n dA - Wshear - ""shaft + { q edV h s h v (1.30)

The specific energy is given by e = u + V2/2 + gz, where u is the internalenergy per unit mass, V2/2 is the bulk kinetic energy per unit mass, and gz is the bulk

potential energy per unit mass. Hence, Eq. (1.29) can also be written as

~1ePdv+1 (n+ ~V2+gZ)pV.ii dAa t cv cs 2

=qes - W sh ea r - Wshafl + 1q e dV

c ,

(1.31)

Where n is the enthalpy per unit mass defined as

n= u+ Ep

Eguation(1.3I) is an alternative expression for the control-volume form of the first

law of thermodynamics.

(1.32)


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12 HEAT CONDUCTION

1.7 SECOND LAW OF THERMODYNAMICS

The first law of thermodynamics, which embodies the idea of conservation of energy,gives means for quantitative calculation of changes in the state of a system due to

interactions between the system and its surroundings, but it tells us nothing about

the direction a process might take. In other words, physical observations such as the

following cannot be explained by the first law:

A cup of hot coffee placed in a cool room will always tend to cool to the

temperature of the room, and once it is at room temperature it will never

return spontaneously to its original hot state.

Air w ill rush into a vacuum chamber spontaneously.

The conversion of heat into work cannot be carried out on a continuous basis

with a conversion efficiency of 100%.

Water and salt will mix to form a solution, but separation of such a solution

cannot be made without some external means.

A vibrating spring will eventually come to rest all by itself, etc.

These observations concerning unidirectionality of naturally occurring processes

have led to the formulation of the second law of thermodynamics. Over the years,

many statements of the second law have been made. Here we give the following

Clausius statement: It is impossible for a self-acting system unaided by an external

agency to move heat from one system to another at a higher temperature.

The second law leads to the thermodynamic property of entropy. For anyreversible process that a system undergoes during a time interval dt, the change in

the entropy S of the system is given by

dS_(8Q

)T rev

(1.330)

For an irreversible process, the change, however, is

(J.33b)

where 8 Q is the small amount of heat transferred to the system during the time interval

dt, and T is the temperature of the system at the time of the heat transfer. Equations

(1.33a) and (1.33b) together may be taken as the mathematical statement of the second

law, and they can also be written in rate form as

dS 18Q->--

dt - T dt

The control-volume form of the second law can be developed also by following a

procedure similar to the one used in the previous section. Rather than going through

the entire development, here we give the result [16]:

(1.34)

a l l ' 1 18Q

- spdV+ spVndA", ---a t cv cs cs T dt

(1.35)


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FOUNDATIONS OP HEAT TRANSFER 13

where s is the entropy per unit mass, and the equality applies to reversible processes

and the inequality to irreversible processes.

1.8 TEMPERATURE DISTRIBUTION

Since heat transfer takes place whenever there is a temperature gradient in a medium, a

knowledge of the values of temperature at all points of the medium is essential in heat

transfer studies. The instantaneous values of temperature at all points of the medium

of interest is called the temperature distribution or temperature field.

An unsteady (or transient) temperature distribution is one in which temperature

not only varies from point to point in the medium, but also with time. When the

temperature at various points in a medium changes, the internal energy also changes

accordingly at the same points. The following represents an unsteady temperature

distribution:

T=T(r, t) (1.360)

where r = Ix +jy + k z, and i, I. and k are the unit vectors in the x, y, and Zdirections, respectively, in the rectangular coordinate system.

A steady temperature distribution is one in which the temperature at a given pointnever varies with time; that is, it is a function of space coordinates only. The following

represents a steady temperature distribution:

a T-=0a t

T=T(r), (1.36b)

Since the temperature distributions governed by Eqs, (1.36a) and (1.36b) are func-

tions of three space coordinates (r =Ix +jY+k z), they are called three-dimensional.

When a temperature distribution is a function of two space coordinates, it is calledtwo-dimensional. For example, in the rectangular coordinate system,

a T-=0a z

T =rex, y, t), (1.36c)

represents an unsteady two-dimensional temperature distribution.

When a temperature distribution is a function of one space coordinate only, it is

called one-dimensional. For example, in the rectangular coordinate system,

s r e r -=-=0a y a z

T =T(x, I), (1.36d)

represents an unsteady one-dimensional temperature distribution.

. If the points of a medium with equal temperatures are connected, then the result-

II1gsurfaces are caJled isothermal surfaces. The intersection of isothermal surfaces

with a plane yields a family of isotherms on the plane surface. Itis important to note

that two isothermal surfaces never cut each other, since no part of the medium can

have two different temperatures at the same time.

heat transfer kakac

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