CHEMICAL ENGINEERING

300 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 12.5

Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal
diameter and the pressure drop over a 4 m length is found to be 1.5kN/m^2 .Usingthe
Reynolds analogy, by how much would the air temperature be expected to fall over the
first metre of pipe length if the wall temperature there is kept constant at 295 K?
Viscosity of airD 0 .018 mN s/m^2. Specific heat capacity of airD 1 .05 kJ/kg K.

Solution

(Essentially, this is the same as Problem 9.40 though, here, an alternative solution is
presented.)

From equations 3.18 and 12.102:PD 4 h/Cpul/du^2.

For a length of 4 m: 1500D 4 h/Cpu 4. 0 / 0. 050 u^2

∴ hu/CpD 4 .69 kg/ms^2 (i)

The rate of heat transferDhdlTmTw, which for a length of 1 m is:

hð 0. 050 ð 1. 0  0. 5  320 CT 2  295 D 0. 157 h 0. 5 T 2  135 

The heat lost by the airDud^2 / 4 CpT 1 T 2 

Duð 0. 0502 / 4 Cp 320 T 2 D 0. 00196 uCp 320 T 2 

∴ 80. 1 h/CpuD 320 T 2 / 0. 5 T 2  135 

Substituting from equation (i):

80. 1  4. 69 /u^2 D 320 T 2 / 0. 5 T 2  135  (ii)

From equation 12.139:

h/CpuD 0. 032 du/^0.^25 D 0. 032 /du^0.^25

At 320 K and 101.3kN/m^2 ,D 28. 9 / 22. 4  273 / 320 D 1 .10 kg/m^3

∴ h/CpuDhu/Cp/u^2 D 4. 69 / 1. 10 u^2 

∴ 4. 69 / 1. 10 u^2 D 0 .032[0. 018 ð 10 ^3 / 0. 050 ð 1. 10 u]^0.^25 anduD 51 .4m/s

Substituting, in equation (ii):

 80. 1 ð 4. 69 / 1. 10 ð 51. 42 D 320 T 2 / 0. 5 T 2  135 andT 2 D316 K

The temperature drop over the first metre is therefore 4 deg Kwhich agrees with the
solution to Problem 9.40.
(It may be noted that, in those problems, an arithmetic mean temperature difference
is used rather than a logarithmic value for ease of solution. This is probably justified in
view of the small temperature changes involved and also the approximate nature of the
Reynolds analogy.)