CHEMICAL ENGINEERING

HEAT TRANSFER 193

Solution

If the heat transfer coefficienthcan be expressed as a product of powers of the variables,
then:
hDk^0
kaCpbcd
ˇgelfTg wherek^0 is a constant.

The dimensions of each variable in terms ofM, L, T, Q,andqare:

heat transfer coefficient, hDQ/L^2 Tq

thermal conductivity, kDQ/LTq

specific heat, CpDQ/Mq

viscosity, DM/LT

density, DM/L^3

the product, ˇgDL/T^2 q^1

length, lDL

temperature difference, TDq

Equating indices:
M: 0 DbCcCd
L:  2 Da 3 cdCeCf
T:  1 Dad 2 e
Q: 1 DaCb
q:  1 DabeCg

Solving in terms ofbandc:

aD 

1 b, dD

bc, eD

c/ 2 , fD 

3 c/ 2  1 , gD

c/ 2

and hence:

hDk^0

(

k

kb

Cpbc

b

c

ˇgc/^2

l^3 c/^2

l

Tc/^2

)

Dk^0

(

k

l

)(

Cp

k

)b(

l^3 /^2 

ˇg^1 /^2 T^1 /^2



)c

or:

hl

k

Dk^0

(

Cp

k

)b(

l^3 ^2 ˇgT

^2

)c/ 2

where
Cp/kis the Prandtl number and
l^3 ^2 ˇgT/^2 the Grashof number. A full
discussion of the significance of this result and the importance of free of natural convection
is presented in Section 9.4.7.

PROBLEM 9.63

A shell-and-tube heat exchanger is used for preheating the feed to an evaporator. The
liquid of specific heat 4.0 kJ/kg K and density 1100 kg/m^3 passes through the inside of
tubes and is heated by steam condensing at 395 K on the outside. The exchanger heats
liquid at 295 K to an outlet temperature of 375 K when the flowrate is 1. 75 ð 10 ^4 m^3 /s
and to 370 K when the flowrate is 3. 25 ð 10 ^4 m^3 /s. What is the heat transfer area and
the value of the overall heat transfer coefficient when the flow rate is 1. 75 ð 10 ^4 m^3 /s?