Summary Sheets Complete Version F2013

8/11/2019 Summary Sheets Complete Version F2013

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MECH-346 Heat Transfer: Summary Sheets Complete Version 1

1

MECH-346 Heat Transfer: Summary Sheets Complete Version

Fouriers law of heat conduction for isotropic materials:

"q k T=

Convection heat loss from an isothermal surface:

Net radiation loss from a flat isothermal surface to surroundings atsurr

T

(or a large enclosure with isothermal walls atw surr

T T= ): assumption is

that surface is gray w.r.t. radiation from surroundings, sosurf surf

=

Three-dimensional unsteady heat conduction in an isotropic material:

General (vector calculus) form of the governing equation:

( ). pT

k T S ct

+ =

Asur ace

Tsur

T

hav

, ( )conv loss surface av surf exposedtoconv

q A h T T =

,( )conv surf th convq T T R

( ), 1th conv surface exposed t o conv avR A h

surr wT or T

, ,surf surf surf T A

4 4

.

net surf surf surf surr radiation abs abssurface surrorsurface encl walls

q A T T

=

8

2 4

Stefan-Boltzmann constant

5.669x10 [ ]W

m K

=

=


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2

Governing equation expressed in Cartesian coordinates:

p

T T T T k k k S c

x x y y z z t

+ + + =

Governing equation expressed in cylindrical coordinates:

2

1 1p

T T T T rk k k S c

r r r r z z t

+ + + =

Governing equation expressed in spherical coordinates:

2

2 2 2 2

1 1 1(sin )

sin sinp

T T T Tr k k k S c

r r r r r t

+ + + =

Steady-State One-Dimensional Heat Conduction [Isotropic Materials]

1. Plane Wall

T1

T2

L

x

Ac.s.

Governing equation (1-D Cartesian):

0d dT

k Sdx dx

+ =

For constantk= and 0S= , the solution is:

2 1 2( ) ( ) 1 ( / )T T T T x L =

. . 1 2 1 2. .

, //

, //. .

( ) ( )( )

Restrictions: 1-D, steady-sta

0, ., // wall

c sx c s

th wall

th wallc s

A k T T T TdTq k A

dx L R

L

R S k const kA

= =

=

= =

1T 2T xq

, //th wallR


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3

T1 T2

qr

Rth,long hollow cylinder

Plane wall of thickness 2L: S.S., 1-D Cartesian, S= constant, k= constant

2. Long Hollow Cylinder [L>> (r2 r1)]

Resistance analogy: Long hollow cylinder ( 0r ), steady-state, 1-D radialheat conduction, k= constant, and 0S= are the restrictions here

L

x

Ac.s.L

S>0(const.)

Tx=L

= TRight

General case: (i) atx= -L, T= TLeftand

(ii) atx=L, T= TRight

22

12 2 2

Right Left Right LeftT T T T S L x xT

k L L

+ = + +

Symmetric case: (i) atx= -L, T= TWand

(ii) atx=L, T= TW

22 2

max 0

1 ;2 2W x W

S L x S LT T T T T

k L k=

= + = = +

T1

T2

r2r1

r

r

L

Steady-state, 1-D radial heat conduction: Governing equation

Tx=-L

= TLeft

1 0d dTrk Sr dr dr

+ =

k = constant, S = 0, with B.C.s;(i) at 1 1,r r T T = = ; and (ii) at 2 2,r r T T= =

Solution for this case is the following:

2 1 2 2 1 2( ) /( ) ln( / ) / ln( / )T T T T r r r r =

2 1,

ln( / )

2th longhollow cylinder

r rR

kL=


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4

T1 T2

qr

Rth,hollow sphere

3. Long solid cylinder (L>> R): steady-state, 1-D radial heat conduction

4. Hollow Sphere

Resistance analogy: Hollow sphere ( 0r ), steady-state, 1-D radial heat

conduction, k= constant, and 0S= are the restrictions here

r2r1

Tw

R

L

Governing equation:

10

d dTrk S

r dr dr

+ =

k= constant, S= constant, with B.C.s;(i) at 0, is finiter T= ; and (ii) at , wr R T T = =

Solution for this case is the following:22

( ) 14

w

S R rT T

k R

=

; and

2

max 04

r wS RT T T

k== = +

Steady-state, 1-D radial heat conduction: Governing equation

2

2

10

d dTr k S

r dr dr

+ =

k= constant, S= 0, with B.C.s;(i) at 1 1,r r T T = = ; and (ii) at 2 2,r r T T= =

Solution for this case is the following:

2

2 1 21 2

( ) 1 1 1 1

( )

T T

r r r r T T

=

,

1 2

1 1 1

4th hollowsphereR

k r r

=


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5

5. Solid sphere: steady-state, 1-D radial heat conduction

Thermal Contact Resistance

( ) ( ),th contact contact interfaceI I I Iq T T R h A T T + +

Thus, ( ), 1th contact contact interfaceR h A=

Here, contacth is the thermal contact coefficient [ 2 oW

m C]

R

Governing equation:

2

2

10

d dTr k S

r dr dr

+ =

k= constant, S= constant, with B.C.s;(i) at 0, is finiter T= ; and (ii) at , wr R T T = =

Solution for this case is the following:22

( ) 16

w

S R rT T

k R

=

; and

2

max 06

r wS RT T T

k== = +

Tw

Interface

Material Material I

T IT +

q

Note: For unit

contact area, thethermal contact

resistance is

denoted as:"

, 1/th contact contaR h=


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6

Critical Radius of Insulation or Coating of Curved Surfaces

(Conduction-Convection Systems)

Long hollow cylindrical geometryIf { }1 1, , , , ,inslr L k T T h all constant, 2r variable; 0inslS = ; steady-state; and

1-D radial, then when

2insl

critlong hollow cyl

kr r

h= = , maxq q=

Hollow spherical geometry

If { }1 1, , , , ,inslr L k T T h all constant, 2r variable; 0inslS = ; steady-state; and

1-D radial, then when

2

2insl

crithollowsphere

kr r

h= = , maxq q=

Note:For both the cylindrical and spherical geometries, if

{ }1, , , , ,inslr L k h T q all constant, 2r variable; 0inslS = ; steady-state; and 1-D

radial, then when 2 critr r= , 1 1( )minimumT T=


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7

Classical Fin Theory

(fink = constant; finS =0; h= constant; T = constant; steady-state)

For uniform fin cross-sectional area and perimeter, . .c sA = constant and

. .c sPeri = constant, and the fin equation reduces to2

1/ 22

. . . .2( ) 0 ; ( ) /( )

c s fin c s

d Tm T T m h Peri k A

dx

= =

B.C. (i): atx= 0, 0x baseT T= =

For B.C. (ii), the following four cases and solutions are considered:

Case A:at x L= ,

}( / ) ( ) Convection heat loss from the tip surfacefin x L x Lk dT dx h T T = = =

Case A solution:

[ ]cosh ( ) sinh[ ( )]

cosh( ) sinh( )

fin

base

fin

hm L x m L x

m kT T

T T hmL mL

m k

+ =

+

Case A total rate of heat transfer from fin to fluid:

( ). . . .sinh( ) cosh( )

( )

cosh( ) sinh( )

fin

total fin c s c s basefin fluid

fin

hmL mLm k

q k A h Peri T T h

mL mLm k

+

= +

Case B:at x L= ,

}( / ) 0 Adiabatic tip or symmetry surface atx LdT dx x L= = =

Case B solution: [ ]cosh ( )

cosh( )base

m L xT T

T T mL

=

Case B total rate of heat transfer from fin to fluid:

( ). . . . ( ) tanh( )total fin c s c s basefin fluid

q k A h Peri T T mL

=


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8

Case C:Long-fin, at x L= ,x L

T T= =

Case C solution: ( )expbase

T Tmx

T T

=

Case B total rate of heat transfer from fin to fluid:( ). . . . ( )total fin c s c s base

fin fluid

q k A h Peri T T

=

Case D:at x L= , } Specified fin tip temperaturex L LT T= =

Case D solution:

sinh( ) sinh[ ( )]

sinh( )

L

base

base

T Tmx m L x

T TT T

T T mL

+ =

Case D total rate of heat transfer into fin across base:

( ). . . .cosh( )

( )sinh( )

L

base

total fin c s c s basebase in

T TmL

T Tq k A h Peri T T

mL

=

Case D total rate of heat transfer from lateral surface of fin to fluid:

. .lateral surface total out total fin c sfin fluid base in fin tip base in tipx L

dTq q q q k Adx =

= = +

Fin Efficiency:,

Entire fin at

Same fin geometry and tip condition

base

actual fin fluid

fin T

fin fluid

q

q

For Case B (insulated tip): ( ), tanhfin Case B mL mL =

Compensated length:c

L may be used to approximate the total rate of heat

loss from the fin to the fluid in Case A using the results for Case B: in general,

( ). . . .c c s tip c s tipL L L L A Peri= + + ; for a straight fin of uniformrectangular cross-section with W >> t, / 2cL L t= + ; for a straight fin ofuniform circular cross-section, with diameter d, / 4cL L d= +


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9

Fin Thermal Resistance: , ,1 [ ]th fin fin surface total fin fluid R Area h ; note that this

fin thermal resistance accounts for both conduction through the fin and

convection at its surface

Fin Effectiveness:,

, ( )

actual fin fluid

fin

base fin w

q

Area h T T

Fin Design Charts and Related Procedures

Basis: Solutions based on adiabatic fin tip assumption (akin to Case B

solution for a fin of uniform rectangular cross-section)

Thus, when using fin design charts, use Lc instead of L if the fin tip

looses heat by convection (but if the actual fin tip is adiabatic, then use

the actual fin length,L).

3/ 2 1/ 2[ { /( )} , ]fin c fin m

fnc L h k A geometric paramters = ;Amis the profile area

If the fin efficiency,fin

, is obtained from fin design charts, then the fin

thermal resistance is ,,

,

1/( )c

th fin fin surface lateral totalfin fluid withL if needed

R Area h=

If

fin is obtained from fin design charts, then use the following

expressions:,

,,

( )( )

c

base

actual total fin surface lateral basefin fluid total fin

th finwith L if needed

T Tq Area h T T

R

= =

Design charts for uniform fins of triangular cross-section, uniform fins

of rectangular cross-section, and circumferential (or annular) fins of

rectangular cross-section are given on the following page: again, use the

compensated fin length,Lc, if needed (that is, if the fin tip looses heat by

convection); but if the actual fin tip is adiabatic, then use the actual fin

length,L.


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10

Fin efficiency charts: Fin

of uniform rectangular

and triangular cross-

sections

rectangular fi2

triangular fin

rectangular

triangular fi2

c

c

m

c

tL

L

L

tL

A tL

+

=

=

Figures taken from Heat Transfer by J.P. Holman, 7th

Edition, 1990 Fin efficiency

chart:

Circumferential or

annular fins ofuniform

rectangular cross-

section

2 1

2 1

2

( )

c

c c

m c

tL L

r r L

A t r r

= +

= +

=


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11

Heat Conduction Shape Factor

A heat conduction shape factor, S, can be defined in problems that allow the

following approximations or restrictions: isotropic and homogeneous material;

steady-state conditions prevail; k= constant; volumetric source term S = 0; and

only two different uniform boundary temperatures.

1 2

1 2

1 2

( )( )

totalT T

th cond

T Tq k T T

R

= =S where 1/( )

th condR k= S


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12

Unsteady Heat Conduction (Isotropic Materials)

Governing equation: ( ) /

pdiv k T S c T t + =

Properties and source data ( , , ,p

k c S ), B.C.s ,and I.C. are needed to

complete the mathematical model: these are problem specific.

Biot number (conduction-convection systems):

c

solid

hLBi

k ; where

[Total volume of solid]

[Surface area of solid exposed to convection]c

L

Lumped Parameter Analysis [LPA; valid if 0.1Bi ]

Key idea

Governing equation: ( ) ( / )solid surface solid p solid

exposed to convection

SV A h T T c V dT dt =

where Sis the volumetric source term

LPA solution for 0S= , [ , , , ]p

c h T all constant,0t ini

T T uniform= = = :

( )exp

( )

surface exposedto convection

ini ini p solid

h AT T

tT T c Vol

= =

Solid

Fluid, T, h( , , , ) ( )T T x y z t T t =

when

0.1c

solid

h LBi

k

; with

( )solidc

surface solid exposed to convection

VolumeL

Area=

( )p solid surface exposedto convection

c Vol h A

=

time constant of conduction-convection

system


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13

When LPA solution for 0S= , [ , , , ]pc h T all constant, and

0t iniT T uniform= = = applies; and the time constant of the conduction-convec

system is ( )p solid surface solid exposedto convection

c Vol h A =

:

1 2

1 2

1 2( ) ( ) exp exptotal loss solid p t t t t solid p init t t

t tQ Vol c T T Vol c T T

= =

= =

Transient heat conduction in semi-infinite solids

Mathematical model:with the thermal diffusivity /( )p

k c =

Governing equation:p

T Tk c

x x t

=

or2

2

1T T

x t

=

x

k= const.; S= 0;

= const.; cp= const.

uniformi

T T= = ,for 0t

Isotropic materials

Common B.C.s

(imposed) on the

surface: for 0t> a)Constant temperature,

o iT T

b)Constant heat flux,0q

c)Convection boundary

condition, ,h T


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14

I.C.: at t= 0, T= Ti= constant; 0 x

B.C.s

a)0, .

0, .

o

i

at x T T const for all t

at x T T const

= = = > = =

b) 00, .

0

, .

o

x

i

Tat x q k const

x for all t

at x T T const

=

= = =

> = =

c)0

0

0, ( )

0, .

x

x

i

Tat x h T T k

x for all t

at x T T const

=

=

= =

> = =

Solutions:

a)( , )

erf2

o

i o

T x t T x

T T t

=

Notes:

1) Error function: ( )2

0

2erf e d

; values in Table 8-1 (attached)

2) Complementary error function: ( ) ( )erfc 1 erf

b)22 / -

( , ) exp - erfc4 2

o oi

q t q xx xT x t T

k t k t

=

c)2

2

( , )erfc exp erfc

2 2

i

i

T x t T x hx h t x h t

T T k k k t t

= + +

Penetration depth: ( )t

When ( )x t= ,

0( , ) 0.992i o

T t Terf

T T t

= =


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15

Table 8.1: Values of error function


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16

One-dimensional unsteady heat conduction in solids with

convection B.C. [S =0; k, , cp, h, T all constant; cLBi > 0.1;

uniform initial temperature,i

T; and /( )pk c = ]

Case B:

L>> ro

ro

h, T

Long solid cylinder

oM

solid

hrBi

k= ; *

2o

tt

r

= ; *

o

rr

r=

Case A:

T

h

T

h

LL

t 0, T=Ti

Symmetrically cooled/heated

plane wall

M

solid

hLBi

k= ; *

2

tt

L

= ; *

xx

L=

Case C:

ro

h, T

Solid sphere

oM

solid

hrBi

k= ; *

2o

tt

r

= ; *

o

rr

r=

Notes:

1.One-dimensional

transient heat conduction

in these three cases can

be predicted analytically:

solutions are in the form

of infinite series;

2.

However, these series

are rapidly convergent;

3.For * 0.2t , one-termapproximation of infinite

series is excellent:

[Error 2%]


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17

For * 0.2t

, use the following one-term approximation:

Case A:Symmetrically cooled/heated plane wall (total thickness = 2L)

* * * 2 * * * * 21 1 1

( , )( , ) exp[ ]cos( ) ; / ; /

i

T x t T x t A t x x x L t t L

T T

= = = =

* * * 2 *1 1

( 0, )( 0, ) exp[ ]

i

T x t T x t A t

T T

= = = =

*

* * * 1

, 101

sin( )1 [ ( 0, )] ; 2 ; ( )

t

o total p iloss xo

Qx t Q q dt Q Vol c T T

Q

=

= = = =

Case B:Long solid cylinder (radiuso

r ; cylinder length >> 2or)

* * * 2 * * * * 21 1 0 1

( , )( , ) exp[ ]J ( ) ; / ; /

o o

i

T r t T r t A t r r r r t t r

T T

= = = =

* * * 2 *1 1

( 0, )( 0, ) exp[ ]

i

T r t T r t A t

T T

= = = =

*

* * * 1 1

, 101

J ( )1 2[ ( 0, )] ; ; ( )

t

o total p iloss ro

Qr t Q q dt Q Vol c T T

Q

=

= = = =

Case C:Solid sphere (radiusor)

** * * 2 * * * 21

1 1 *

1

( , ) sin( )( , ) exp[ ] ; / ; /

o o

i

T r t T r r t A t r r r t t r

T T r

= = = =

* * * 2 *1 1

( 0, )( 0, ) exp[ ]

i

T r t T r t A t

T T

= = = =

*

* * * 1 1 13 , 10

1

sin( ) cos( )1 3[ ( 0, )] ; ; (

t

o total p iloss ro

Qr t Q q dt Q Vol c T T

Q

=

= = = =

Notes:

For cases A, B, and C, values of 1 1,A as functions of MBi are givenin Table 9.1 (see page 18)

Values of 0 1J ( ) and J ( ) as functions of are given in Table 9.2

(see page 18). Note: o 1J ( ) / J ( )d d =


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18

Table 9.1 Table 9.2

Values of 1and 1A for different values of Zeroth- and

MBi and Cases A, B, and C first-order

Bessel functionsof the first kind

Notes:

( ) ; ( ) ; ( )o oM PlaneWall M Long Solid Cylinder M Solid Sphere

solid solid solid

hr hr hLBi Bi Bi

k k k= = =

( / 2) ( /3)( ) ; ( ) ; ( )

c c c

o oL PlaneWall L Long Solid Cylinder L Solid Sphere

solid solid solid

h r h r hLBi Bi Bi

k k k= = =

Case A Case B Case C


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19

Product solution approach: temperature distribution for

multidimensional unsteady heat conduction in solids with convection B.C.

[S=0; k, , cp, h, T all constant; cLBi > 0.1; and iTuniform]

Notation for one-dimensional unsteady solutions:

Plane-wall: ( , )( , )i

T x t T P x tT T

=

Long-cylinder:( , )

( , )i

T r t T C r t

T T

=

Semi-infinite solid:( , )( , )

( , ) 1 i

i i

T x t T T x t T x t

T T T T

= =

S

In general, for three-dimensional unsteady

problems:3

13 ISi iFull D solid Intersecting So

T T T T

T T T T

=

=

P= S 1 2P P=

1 2 3P P= S 1 2 3P P P=

C= S P C=


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20

20

Total heat transfer for multidimensional unsteady heatconduction in solids with convection B.C. [S=0; k, , cp, h, T all

constant; iTuniform; and cLBi > 0.1]

Work of Langston

For solids that can be constructed by the intersection of two objects (1

and 2) for which the 1-D solutions (discussed earlier) apply:

1 2 1

1o o o ototal

Q Q Q Q

Q Q Q Q

= +

For solids that can be constructed by the intersection of three objects

(1, 2, and 3) for which the 1-D solutions (discussed earlier) apply:

1 2 1

3 2 1

1

1 1

o o o ototal

o o o

Q Q Q Q

Q Q Q Q

Q Q Q

Q Q Q

= +

+

If the temperature distribution and/or the total heat transferafter a given time into the heating/cooling process is desired, the

solution is straightforward

Obtain the appropriate one-dimensional solutions and combine

them suitably, as discussed above

If the time needed to obtain a desired temperature distribution or

total heat transfer is required, then:

Explore options offered by symmetry surfaces

Keep your thinking cap on


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21

21

Uniform Newtonian Fluid Flow and Heat Transfer over aFlat Plate with a Sharp Leading Edge and Zero Angle of Attack

Local or running Reynolds number, /

xRe U x

Transition region:

5 61 x 10 1 x 10xRe

In engineering analyses:

5( / ) 5 x 10critx crit

Re U x = = for flow over

a flat plate at zero angle of attack

At , ( / ) 0.99y u U = =

At , [( ) /( )] 0.99T w wy T T T T = =

Fully Turbulent Layer:turb >>

Buffer Layer:turb

Viscous sublayer:turb

Summary Sheets Complete Version F2013

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