formula sheet ht

7/23/2019 Formula Sheet HT

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Formula Sheet For Heat Transfer (MSE 321)

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Heat transfer to a substance corresponding

to temperature rise of T 2 1( )P PQ m c T mc T T massm = heat capacityPc =

Heat flux2

s

W

m

Qq

A

=

Heat conduction

(Fouriers law) cond s[W ]

dTQ k A

dx = - 2cond [ W/m ]

dTq k

dx = -

Heat convection

(Newtons law of cooling) ( )conv s s [W ]Q hA T T

= - ( )2

conv s [W/m ]q h T T = -

Radiation heat4 4

rad s s surr ( ) [W]Q A T T = - 4 4 2rad s surr ( ) [W/m ]q T T = -

Thermal resistance totalK

W

TR

Q

=

ortotal

[W]T

QR

=

Conduction resistance walls

K

W

LR

kA

=

( )2 1

cyl

ln / K

2 W

r rR

Lk

=

2 1sph1 2

K

4 W

r rR

r r k

- =

Convection resistance convs

1 K

WR

h A

=

Radiation resistance radrad s

1 K

WR

h A

=

( )

( )( )2 2radrad s s 2s s

W

m K

Qh T T T T

A T T

= = + + -

Critical radius of insulation inscr, cyl [m]krh

= inscr, sph 2 [m]krh

=

Surface temperature of solids with

volumetric heat generation3[W/m ]g s, wall

gLT T

h

= + 0s, cyl

2

grT T

h

= + 0s, sph

3

grT T

h

= +

Maximum temperature difference in solidswith volumetric heat generation

3[W/m ]g ( )

2

max min wall

gLT T

k

- = ( )

2

0max min cyl 4

grT T

k

- = ( )

2

0max min sph 6

grT T

k

- =

Temperature distribution on very long fins c( )

exp /b

T x Tx h p k A

T T

- = - - base temp.bT = perimeterp= c cross-sectional areaA =

Temperature distribution on fins withadiabatic tips

( ) cosh ( )

coshb

T x T m L x

T T mL

- -=

-

c/m h p k A= perimeterp= c cross-sectional areaA =

Heat transfer rate from long fins long fin c ( )bQ h p k A T T

= -

Heat transfer rate from fins with adiabatictips adi. tip c

( ) tanhbQ h p k A T T mL = -

Fin efficiencyfin

fin

fin, max

actual heat transfer rate from the fin

ideal heat transfer rate from the fin (if the entire fin were at base temperature)

Q

Q

= =

Fin effectiveness( )

fin finfin

no fin

heat transfer rate from the fin of base area

heat transfer rate from the surface area

b

b b b

Q Q A

Q h A T T A

= = =-

area of the fin basebA =

Relation between fin efficiency and fin

effectiveness

( )

( )fin finfin fin

fin fin

no fin

b

b b b

h A T T Q A

Q hA T T A

-= = =

-

Transient temperature for lumped systems [ ] s( ) 1

exp withi P

T t T h Abt b

T T c sV

- = - = -

Dimensionless numberssolid

Bi ch L

k

= c

fluid

Nu h L

k

= Re c cV L V L

= = Pr Pc

k

=

3

2Gr s cL

g T T L

3s

2Ra Gr Pr Pr cL L

g T T L

2

Foc

t

L

=

One term approximation formula (valid for

0.2> )

( ) ( )

( )

( ) ( )

( )

( ) ( ) ( )

( )

2

1 1 1wall

2

1 1 0 1 0cyl

1 02

1 1sph1 0

,, exp cos /

,, exp /

, sin /, exp

/

i

i

i

T x t T x t A x L

T T

T r t T r t A J r r

T T

T r t T r r r t A

T T r r

- = = - -

- = = - -

- = = - -

2

2

0

2

0

t

L

t

r

t

r

=

=

=

P

k

c

=

Shear stress2

s

0

1[Pa]

2 f

y

VC V

y

=

= =


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Buoyancy force B fluid body-wetF g= V

Volume expansion coefficient1 1 1

KPT T

ideal gas1

T

Correlations for Forced Convection

Frictional force fF for a plate and drag

force DF for a blunt body

2

s

1

2f f

F C A V= 21

2D D N

F C A V= frontal surface areaNA =

Localthickness of boundary layer, frictionfactor, and Nu number for laminarflow

over a flat plate with isothermalcondition1/ 2

4.91

Rex

x

x = , 1/2

0.664

Ref x

x

C = 1/2 1/3Nu 0.332Re Pr Pr 0.6xx xh x

k= = >

Averagefriction factor and Nu number for

laminarflow over a flat plate withisothermalcondition

1/ 2

1.33

Ref

L

C = 1/2 1/3Nu 0.664Re PrLhL

k= =

Localthickness of boundary layer, friction

factor, and Nu number for turbulentflowover a flat plate with isothermalcondition

1/5

0.38

Rex

x

x = , 1/5

0.059

Ref x

x

C =

4/5 1/3Nu 0.0296Re Pr 0.6 Pr 60xx x

h x

k= =

Averagefriction factor and Nu number forturbulentflow over a flat plate with

isothermalcondition1/5

0.074

Ref

L

C = 4/5 1/3Nu 0.037Re PrLhL

k= =

Averagefriction factor and Nu number forcombined laminar and turbulentflow

over a flat plate with isothermalcondition

5 7

1/5

0.074 1742

5 10 Re 10Re Ref LL LC = -

( )4/5 1/3 5 7Nu 0.037Re 871 Pr 0.6 Pr 60 5 10 Re 10L LhL

k= = -

AverageNu number for laminarand

turbulentflows over a flat plate with

isofluxcondition

1/2 1/3

4/5 1/3

Nu 0.453Re Pr Laminar (isoflux plate)

Nu 0.0308Re Pr Turbulent (isoflux plate)

xx x

xx x

h x

k

h x

k

= =

= =

AverageNu number for cross-flow over a

cylinder

4/ 55/81/ 2 1/3

cyl 1/42/ 3

0.62Re Pr ReNu 0.3 1 Re Pr 0.2

282,0001 0.4 / Pr

h D

k

All fluid properties are to be evaluated at the film temperature film s / 2T T T

AverageNu number for cross-flow over a

sphere

1/41/2 2/3 2/5

sph

s

Nu 2 0.4Re 0.06Re Pr 3.5 Re 80,000 0.7 Pr 380h D

k

All fluid properties are to be evaluated at the flow temperature T , except s which is evaluated at the surface

temperature sT

Correlations for Natural Convection from Finned Surfaces

Vertical isothermalplate of lengthL

For the averageNusselt number for vertical isothermalparallel plates

1/ 2 3

s

2 1/2 2

fluid

576 2.873Nu with Ra Gr Pr Pr

(Ra / ) (Ra / ) S S

S S

g T T Sh S

k S L S L

The optimum fin spacing and the corresponding heat transfer coefficient

opt 1/ 42.714

RaL

LS fluid

opt

1.307khS

All fluid properties are to be evaluated at the film temperature film s / 2T T T

Correlations for Natural Convection in Rectangular Enclosures


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Horizontal enclosure

1/38

L

1708 RaNu 1 1.44 1 1 Ra 10

Ra 18

L

L

The notation [ ] indicates that if the quantity in the bracket is negative, it should be set equal to zero

All fluid properties are to be evaluated at the average temperature avg 1 2 / 2T T T

Vertical enclosure

0.29

3

0.28 1/ 4

10

0.3

1/4 0.012 4 7 4

Pr Pr Nu 0.18 Ra 1 2 Ra 10 any Pr number

0.2 Pr 0.2 Pr

PrNu 0.22 Ra 2 10 Ra 10 any Pr number

0.2 Pr

Nu 0.42Ra Pr 10 40 10 Ra 10 1 Pr 2 10

L L

L L

L L

H

L

H H

L L

H H

L L

All fluid properties are to be evaluated at the average temperature avg 1 2 / 2T T T


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