Heat transfer

Heat transfer

momentum transfer; result of molecular transfer of momentum conduction; 1) by molecular interaction, 2) by free electrons

; primarily a molecular phenomenon

dv

   dy

(

^T

)



   

τ v v

qx dT A  k dx

Aq   k T

heat conduction in a hollow cylinder

(2 )

r

dT dT

q kA k rL

dr  dr

   

2 ( )

ln /

r i o

o i

q kL T T

r r

  

r lm

q kA t

r

 



2 1

2 1

( )

ln( / )

lm

A A

A A A

 

heat conduction in a hollow sphere

r gm

q kA t

r

 

 A

_gm

 A

¹

A

²

convection; involves energy exchange between a surface and an adjacent fluid

; forced convection, natural convection

radiation; no medium is required, depends on temperature /

q A h T

q 4

A T

2

0

p : u

u u u p

t

c T u T k T u

t

 

 

 

        

  

 

        

  

 

h

_i

h

₀

0 0 5

4

, 4

3

, 3

2

2 1

1 1

A q h

t t

A k

q r t

t

A k

q r t

t

A q h

t t

lm c c

c lm b b

b i i





 



 







Individual temperature drops

Overall temperature drops

 





 



  

 





 

 













0 0 ,

,

0 0 ,

, 5

1

5 1

1 1

1 1

A h A

k

r A

k

r A

q h

A q h

A k q r A

k q r A

q h t

t

t t

t

lm c c

c lm

b b

b i

i

lm c c

c lm

b b

b i

i overall

 





 



  

 



 

0 0 ,

,

5 1

1 1

A h A

k

r A

k

r A

h

t q t

lm c c

c lm

b b

b i

i

R q t^overall



 

 





 



     

 

0 ,

0 ,

0 0

0

1 h A

k

r A

A k

r A

A h

A

t q A

lm c c

c lm

b b

b i

i

overall

Overall coefficients of heat transfer based on the outside area U 0

0 ,

0 ,

0 0

0

1 1

h A

k

r A

A k

r A

A h

A

U

_{c c lm}

c lm

b b

b i

i

 

 





overall

t A

U

q  0 0 

 





 



     

 

0 0 ,

,

1

A h

A A

k

r A

A k

r A

h

t q A

i lm

c c

c i

lm b b

b i

i

overall i

Overall coefficients of heat transfer based on the inside area U i

i i lm

c c

c i

lm b b

b i

i

i

h A

A A

k

r A

A k

r A

h

U

_{, , 0}

1

1  

 





overall i

i A t

U

q  

Overall coefficients of heat transfer

overall i

i

overall

t A

U q

t A

U q







 0 0

R q t overall



 

Overall coefficients of heat transfer

i i A U

A

R U 1 1

0 0







i i A U

A

U 0 0 

Fouling in heat exchanger

Fouling Coefficients

o do

lm c

o c

c lm

b o b

b i

di o i

i o o

scale d

h h

A A k

r A

A k

r A

h A A

h U A

t A h q

1 1

1

, ,



 

 











Outside fluid Outside

fluid Inside

fluid

Outside fluid

k_c k_b

Inside fluid

Inside fouling

h_o

Outside fluid

h_i

h_di

outside fouling

h_do

h_di h_d0

0 ,

0 ,

0 0

0

1 1

h A

k r A A

k r A A

h A

U _{c clm}

c lm

b b

b i

i

 

 





Example

A reflux condenser contains ¾ in. 16-gauge copper tubes in which cooling water circulates. Hydrocarbon vapors condense on the exterior surfaces of the tubes. Find the overall heat-transfer coefficient U

_o

. The inside convective coefficient can be taken as 4500 W/m

²

·K, and the outside coefficient as 1500 W/m

²

·K.

Approximately fouling coefficients from Table are h

_d0

= 5700 W/m

²

·K

h

_di

= 2840 W/m

²

·K

Overhead vapors from crude-oil distillation 1000

Water, Great Lakes, over 125^oF 500

Btu/(h)(ft²)(^oF) ,

1

1 1

o

o o b o

i i di i b b lm do o

U A A r A

h A h A k A h h

     

Outside fluid Outside

fluid Cooling

water

Outside fluid

k

Inside fluid

inside fouling

h_o=1500

Condensing vapor

h_i =4500

h_di =2840

outside

fouling h_do =5700

Example Find U o

¾” 16-gauge copper tube r

_i

=0.0157

r

_o

=0.0191

unit: W/m²·K

,

1

1 1

o

o o b o

i i di i b b lm do o

U A A r A

h A h A k A h h

     

Example

00067 .

1500 0 1 , 1

00018 .

5700 0 1 1

0000047 .

) 0 0175 .

0 )(

380 (

) 0191 .

0 )(

00165 .

0 (

00043 .

) 0 0157 .

0 )(

2840 (

0191 .

0

00027 .

) 0 0157 .

0 )(

4500 (

0191 .

0

0175 .

0 0157

. 0

0191 .

ln 0

0157 .

0 0191 .

0 ln

0191 .

0 ,

0157 .

0 ,

/ 380

/ 2840 678

. 5 500

/ 5700 678

. 5 1000

/ 1500 ,

/ 4500

0 0

0 0

0 0

0 2

2

2 0

2

0 0











 











 

 



































h h

r k

r r

r h

r r h

r

m r

r r r r

m r

m r

K m W k

K m

W h

K m

W h

K m

W h

K m

W h

d lm b

i di

i i

i i l

i d

d i

m i

0 0

0 0

0 1 1

1

0 h

h r

k r r r

h r r

h U r

d lm

b i

d i

i _i



 

 





h_i h_di h_do

k h_o

Example

K m

W U

A U A

U

K m

W

h h

r k

r r r

h r r

h U r

i

o o i

i

d lm

b i

d i

i _i





















 



 

 





2 2

0 0

0 0

0

/ 785

) 0157 .

0 /((

) 0191 .

0 )(

645 (

/ 00155 645

. 0

1

00067 .

0 00018 .

0 0000047

. 0 00043 .

0 00027 .

0

1 1 1

1

0

h_i h_di h_do

k h_o

Resistance of metal wall is negligible

Fouling Resistance is significant~39%

i i

A U A

R U 1 1

0 0





