Cheat Sheet_Exam 3

8/10/2019 Cheat Sheet_Exam 3

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Chapter 1: Introduction to Heat Transfer

- Fouriers law ( T ! T ( x ) ): !!q x = " k dT dx

[Wm -2]

- Newtons law of cooling: !!q = h(T s " T # ) [Wm-2

]

- Stefan-Boltzmann law: E b = ! T s4 [Wm -2] where ! = 5.67 ! 10 " 8 Wm -2K -4

- Conservation of energy: E

in ! E

out + E

g = E

st [W]

Chapter 2: Introduction to Conduction

- Fouriers law ( T !

T ( x, y, z) ):!!

q x = "

k

# T # x ,

!!q y

= "k

# T # y ,

!!q z

= "k

# T # z [Wm

-2

]

- Thermal diffusivity: ! =k

" c p [m 2s-1]

- Heat diffusion equation (Cartesian): !! x

k !T ! x

"#$

%&'

+!! y

k !T ! y

"

#$

%

&' +

!! z

k !T ! z

"#$

%&'

+ q

= ! c p! T ! t

Chapter 3: One-Dimensional, Steady-State Conduction

3.1 Plane wall without heat generation

- Heat equation ( T ! T ( x ) ):d

dxk

dT

dx

!"#

$%&

= 0

- Equivalent thermal circuit:

q x =T ! ,1 " T ! ,2

1

h1 A+

LkA

+1

h2 A


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- Composite wall:

3.2 Cylinder without heat generation

- Heat equation ( T ! T (r ) ):d

drkr

dT

dr

!"#

$%&

= 0

- Equivalent thermal circuit:

3.3 Plane wall with heat generation

- Heat equation ( T ! T ( x ) ):

d dx

k dT dx

!"#

$%&

+ q

= 0

- Temperature distribution:

T ( x) =q

L2

2 k 1 ! x

2

L2

"

#$

%

&' +

T s,2 ! T s,12

x

L+

T s,1 + T s,2

2

q x =T ! ,1 " T ! ,4

1

h1 A+

L Ak A A

+ L Bk B A

+ LC k C A

+1

h4 A

qr =T ! ,1 " T ! ,2

1

h1 2 ! r1 L+

ln( r2 / r1 )2 ! kL

+1

h2 2 ! r2 L


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Chapter 5: Transient Conduction

- Biot number: Bi =hL

c

k , where Lc = V / A s

- Lumped capacitance method ( Bi < 0.1):! ! i

=

T ! T "T

i ! T "= exp !

hAs

" Vc#

$%

&

'( t

)

*+

,

-.

where T ! T (t )

- Total energy transfer up to some time t [J]:

Q = ( ! Vc )! i 1! exp ! t

! t

"

#$

%

&'

(

)*

+

,- where ! t =

" Vc

hAs

Chapter 6: Introduction to Convection

- Definition of heat transfer coefficient:

h =! k " T " y

y= 0

T s ! T #

- Reynolds number (flat plate):

Re x

=

! u ! x

=

u!

x

"

- Prandtl number: Pr =!

"

- Local Nusselt number: Nu x = hxk

= f ( x*, Re x, Pr)

- Average Nusselt number: Nu x =hxk

= f (Re x, Pr)


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Chapter 7: External Flow

7.1 Flow over a flat plate

- Critical Reynolds number:

Re x , c = ! u

! x

c

= 5 " 10 5

- Convection correlations (properties evaluated at T f = (T ! + T s ) / 2 ):

Local Nusselt number Average Nusselt numberLaminar flow overan isothermal flatplate

Nu x

= 0.332 Re x

1 2 Pr 1 3

Pr ! 0.6

Nu x = 0.664Re x1 2 Pr 1 3 = 2 Nu

x

Pr ! 0.6

Turbulent flowover an isothermalflat plate

Nu x

= 0.0296 Re x

4 5 Pr 1 3

0.6 ! Pr ! 60

Nu L = (0.037Re L4 5

! A )Pr 1 3

0.6 " Pr " 60

Re x ,c " Re L " 108

A = 0.037Re x ,c4 5 ! 0.664Re x ,c

1 2 Note : A = 0 if fully turbulent

7.2 Cylinder in cross flow

- Critical Reynolds number:

Re D ,cr =! VD

= 2 ! 10 5

- Average Nusselt number:

Nu D =hD

k = C Re D

m Pr 1 3 , Pr ! 0.7


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Chapter 8: Internal Flow

8.1 Hydrodynamic considerations

- Mean velocity (tube of circular cross-

section): u m = 2r

o2

u (r , x )r dr0

r0

!

- Mass flow rate: m

= ! u m A c

- Reynolds number: Re D =! u m D

- Critical Reynolds number: Re D ! 2300 " Laminar flowRe D ! 10, 000 " Turbulent flow

- Hydrodynamic entry length for laminar flow: x fd ,h

D

!

"#

$

%&lam' 0.05Re

D

- Hydrodynamic entry length for turbulent flow: x fd ,h D

!"#

$%&

turb

' 10

8.2 Thermal considerations

- Mean temperature (tube of circular

cross-section): T m =2

um

ro

2 uTr dr0

r0

!

- Thermal entry length for laminar flow: x fd , t D

!

"#

$

%&

lam

' 0.05Re D Pr

- Thermal entry length for turbulent flow: x fd , t D

!"#

$%&

turb

' 10

- Fully developed thermal conditions:dT m

dx! 0 and h ! f ( x)


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- Total heat rate in the tube [W]: q = m

c p (T m,o ! T m,i ) where T m,o and T m,i are the meantemperatures at the outlet and the inlet of the tube, respectively.

8.3 Boundary condition 1: Constant surface heat flux

- Total heat rate in the tube:

q = m

c p (T m,o ! T m,i ) =

""q s PL , where!!

q s is

the known surface heat flux, P is the perimeter of the tube, and L is the lengthof the tube.

- Mean temperature as a function of x: T m ( x) = T m,i +P

m

c p

!!

q s x

8.4 Boundary condition 2: Constant surface temperature

- Mean temperature as a function of x:

T s ! T m ( x)T s ! T m,i

= exp ! Px

m

c ph

"

#

$$

%

&

''

- Total heat rate in the tube: q = m

c p (T m,o ! T m,i ) = hAs" T lm

where ! T lm =! T

o " ! T

i

ln( ! T o / ! T i ) with ! T o = T s " T m

,o and ! T i = T s " T m

,i


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8.5 Convection correlations (fully developed region)

- All properties are evaluated at the average mean temperature: T m =T

m ,i + T

m ,o

2

Nusselt number ( Nu D = hD / k ) Laminar flow,constant surfaceheat flux

Nu D

= 4.36

Laminar flow,constant surfacetemperature

Nu D

= 3.66

Turbulent flow,constant surfaceheat flux andconstant surfacetemperature

Nu D

= 0.023Re D4 5 Pr n

0.6 ! Pr ! 160

Re D " 10, 000

L / D " 10

n = 0.4 if T s > T m and n = 0.3 ifT s < T m

Chapter 11: Heat Exchangers

11.1 Overall heat transfer coefficient and energy balance

- Overall heat transfer coefficient U [Wm -2K -1]:1

UA= R

tot =

1

(hA )c+ R

w +

1

(hA )h

where the subscripts c and h refer to cold and hot side, respectively, while Rw is the wallthermal resistance by conduction.

- Total heat rate [W]:

q = m

h c p,h (T h,i ! T h,o ) = m

c c p,c (T c ,o ! T c ,i )

- Heat capacity rates [WK -1]:

C h = m

h c p, h and C c = m

c c p , c


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11.2 LMTD method: Parallel-flow heat exchanger

- Total heat rate: q = UA! T lm where

! T lm

=

! T 2

" ! T 1

ln( ! T 2 / ! T 1 )

! T 1

= T h ,1

" T c ,1

= T h ,i

" T c , i

! T 2

= T h ,2

" T c ,2

= T h ,o

" T c ,o

11.3 LMTD method: Counterflow heat exchanger

- Total heat rate: q = UA! T lm where

! T lm

=

! T 2 " ! T 1

ln( ! T 2 / ! T 1 )

! T 1

= T h ,1

" T c ,1

= T h , i

" T c ,o

! T 2

= T h ,2

" T c ,2

= T h ,o

" T c ,i

11.4 NTU method

- NTU (number of transfer units): NTU = UAC

min

- Effectiveness of a heat exchanger: ! = qqmax

=

qC min (T h,i ! T c,i )

- Heat capacity ratio: C r =C

min

C max


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- Effectiveness-NTU relations: ! = f ( NTU , C r )

- Effectiveness-NTU relations: NTU = f (! , C r )


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- Effectiveness-NTU relations in graphical forms:

Parallel-flow heat exchanger (Eq. (11.28)) Counterflow heat exchanger (Eq. (11.29))

Shell-and-tube heat exchanger with one shell andany multiple of two tube passes (Eq. (11.30)) Shell-and-tube heat exchanger with two shell

passes and any multiple of four tube passes (Eq.(11.31) with n = 2).


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Single-pass, cross-flow heat exchanger with bothfluids unmixed (Eq. (11.32)).

Single-pass, cross-flow heat exchanger with onefluid mixed and the other unmixed (Eqs. (11.33)and (11.34)).

Chapter 12: Thermal Radiation: Processes and Properties

12.1 Solid angle, emissive power, irradiation and radiosity

- Electromagnetic spectrum:

- Net radiative heat flux from a surface(outgoing energy incoming energy)

[Wm-2

]:!!qrad = J " G


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- Solid angle [sr]: d ! =dA

n

r 2 = sin " d " d #

- Intensity: I ! (! ," , # )=

dqdA1 cos " d $ d ! [Wm

-2

sr -1

m-1

]

- Total, hemispherical emissive power[Wm -2]:

E = I ! ,e (! ," , # )cos " sin " d " d # d !

" = 0

$ 2

! # = 0

2 $

! ! = 0

"

!

- Diffuse emitter: I ! ,e ! f (" , # )

- Total, hemispherical irradiation[Wm -2]:

G = I ! ,i (! ," , # )cos " sin " d " d # d !

" = 0

$ 2

! # = 0

2 $

! ! = 0

"

!

- Diffuse incident radiation: I ! ,i ! f (" , # )

- Total, hemispherical radiosity [Wm -2]:

J = I ! ,e + r (! ," , # )cos " sin " d " d # d !

" = 0

$ 2

! # = 0

2 $

! ! = 0

"

!


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- Diffuse reflector and diffuse emitter: I ! ,e+ r ! f (" , # )

12.2 Blackbody radiation

- Blackbody spectral emissive power

[Wm-2

m-1

]:

E ! ,b ( ! , T ) = " I ! , b (! , T )

- Wiens displacement law:!

maxT = 2898 m !K

- Stefan-Boltzmann law [Wm -2]:

E b (T ) = E ! ,b ( ! , T )

! = 0

!

" d ! = " T 4

12.3 Band emission

- Fraction of blackbody emissive power contained between 0 and ! :

F (0 ! ! )

=

E ! ,b

d !

0

!

"

E ! ,b d ! 0

#

"

- Fraction of blackbody emissive power contained between ! 1 and ! 2:

F ( ! 1! ! 2 )

=

E ! ,b

d !

0

! 2

" # E ! ,b d ! 0

! 1

"

E ! ,b

d !

0

$

" = F

(0 ! ! 2 ) # F (0 ! ! 1 )


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12.4 Emissivity

- Calculated at the surface temperature

- Spectral, directional emissivity:

! " ,# (" ,# , $ , T ) = I

" ,e (" ,# , $ ,T

) I

" ,b (" , T )

- Spectral, hemispherical emissivity:

! " (" , T ) = E

" (" , T ) E

" ,b (" , T )=

1

# ! " ,$ (" ,$ , % , T )cos $ sin $ d $ d %

$ = 0

# 2

! % = 0

2 #

!

- Total, hemispherical emissivity:

! (T )=

E (T ) E

b (T )=

! " (" , T ) E " ,b (" , T ) d "

" = 0

!

" E

b (T )

12.5 Absorptivity

- Calculated using the temperature of the source

- Spectral, directional absorptivity:

! " ,# (" ,# , $ ) = I

" ,i ,abs (" ,# , $ ) I

" ,i (" ,# , $ )

- Spectral, hemispherical absorptivity:

! " (" ) =G

" , abs (" )G

" (" )=

! " ,# (" ,# , $ ) I " ,i (" ,# , $ )cos # sin # d # d $ # = 0

% 2

! $ = 0

2 %

!

I " ,i (" ,# , $ )cos # sin # d # d $

# = 0

% 2

! $ = 0

2 %

!

- Total, hemispherical asborptivity:

! =G

abs

G

=

! " (" )G " (" ) d "

" = 0

!

"

G" (" ) d "

" = 0

!

"


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12.6 Kirchhoffs law

- Equality always applicable: ! " ,#

= $ "

,#

- ! " = # " if either of the following conditions is satisfied:

(1) The irradiation is diffuse ( I ! ,i ! f (" , # ) )(2) The surface is diffuse

- ! = " if condition (1) or (2) is satisfied, and if either of the following conditions issatisfied:

(3) The irradiation corresponds to emission from a blackbody at the surfacetemperature(4) The surface is gray

Chapter 13: Radiation Exchange between Surfaces

13.1 View factors

F ij =1

Ai

cos ! i

cos ! j

" R2 A j ! dA j dAi

Ai

!

F ji =1

A j

cos ! i

cos ! j

" R2 A j ! dA j dAi

Ai

!

- Reciprocity relation: AiF ij = A j F ji

- Summation rule (enclosure of N surfaces): F ij j = 1

N

! = 1


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- View factors for common geometries:


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View factor for aligned parallel rectangles. View factor for coaxial parallel disk.

View factor for perpendicular rectangles with a common edge.


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13.2 Blackbody radiation exchange

- Net heat rate between two surfaces i and j:qij = qi! j " q j ! i = AiF ij ! (T i

4" T j

4 )

- Net heat rate from surface i due toexchange with N surfaces within anenclosure of N black surfaces:

qi = AiF ij ! (T i4 ! T j

4 ) j = 1

N

"

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Cheat Sheet_Exam 3