Overall Heat Transfer Coefficient PDF

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Derivation of Overall Heat Transfer Coefficient for double pipe or shell-&-tube exchangers Consider a pipe of length L with an inner radius of ri and an outer radius of ro . At some radius r , where r is between ri and ro , the rate of heat transfer is given by: dT (1) dr where T is the temperature, kw the thermal conductivity of the pipe material, and A the heat transfer area. The negative sign is necessary to make the heat duty, Q, positive if the sign convention adopted is that the positive direction is the direction of heat flow. In this direction as r increases T decreases making the temperature gradient negative. The surface area available for heat transfer at this point, A, is given by: Q = −kw A

A = 2πrL

(2)

Substituting equation 2 into equation 1. Q = −kw 2πrL

dT dr

(3)

Rearranging to separate the variables. dr = −2πkw LdT r This can be integrated using the following boundary conditions: If r = ri then T = Ti If r = ro then T = To Where Ti is the inner wall surface temperature and To is the outer wall surface temperature. Q

Q

Z ro dr

r

ri

= −2πkw L

Z To

dT

(4)

(5)

Ti

ro ) = 2πkw L(Ti − To ) ri 2πkw L Q= (Ti − To ) ln( rroi )

Q ln(

(6) (7)

In order to be able to eliminate the unknown wall surface temperatures it is necessary to look at the heat transfer across the fluid films at the walls. For the inner film: Q = hi Ai (T1 − Ti )

(8)

Q = 2πhi ri L(T1 − Ti )

(9)

Where T1 is the bulk fluid temperature inside the pipe. For the outer film: Q = ho Ao (To − T2 )

(10)

Q = 2πho ro L(To − T2 )

(11)

Where T2 is the bulk fluid temperature outside the pipe. Equation 9 can be rearranged to give an expression for Ti Q 2πhi ri L Equation 11 can be rearranged to give an expression for To Ti = T1 −

To = T2 + 1

Q 2πho ro L

(12)

(13)

Equations 12 and 13 can be subsituted into equation 7 to eliminate the wall surface temperatures. Q=

2πkw L  Q T1 − ln( ro ) 2πhi ri L ri



Q − T2 + 2πho ro L 



(14)

Rearranging to make Q the subject: Q= 

wL [ 2πk ln( ro ) ](T1 − T2 ) ri

1+

kw ri hi ln( ro ) r

+

kw ro ho ln( rro ) i

i

(15)



This can be simplified to: Q = 2πL(T1 − T2 )

"

ln( rroi ) kw

1 1 + + ri hi ro ho

#−1

(16)

The overall heat transfer coefficient, U , is defined by: U=

Q A(T1 − T2 )

(17)

The inner area in given by: A = 2πri L

(18)

Therefore the overall heat transfer coefficient based on the inner area is given by: Ui =

Q 2πri L(T1 − T2 )

(19)

Substituting in for Q using equation 16 gives: r 1 ln( rio ) 1 1 Ui = + + ri kw ri hi ro ho

"

#−1

(20)

In the more usual notation this becomes: ri ln( rroi ) ri 1 1 + + = hi kw Ui ro ho

(21)

Performing a similar analysis based on the outside area of the pipe gives: ro ln( rroi ) 1 1 ro = + + ho kw ri hi Uo Adding fouling factors would give: ri ln( rrio ) 1 1 ri ri fo = + + fi + + Ui hi kw ro ho ro and an equivalent expression for the overall heat transfer coefficient based on outer area of the pipe.

2

(22)...