CHEMICAL ENGINEERING

MOMENTUM, HEAT AND MASS TRANSFER 307

as fully developed. The ratio of the velocity at the edge of the laminar sub-layer to the
mean velocity of flow may be taken as 2Re^0.^125 ,whereReis the Reynolds number in
the pipeline.
If the tube walls are at an approximately constant temperature of 393 K and the inlet
temperature of the water is 293 K, estimate the outlet temperature. Physical properties of
water: densityD1000 kg/m^3 , viscosityD1mNs/m^2 , thermal conductivityD 0 .6W/mK,
specific heat capacityD 4 .2kJ/kgK.

Solution

The Taylor – Prandtl modification of the Reynolds analogy to heat transfer, discussed in
Section 12.8.3, leads to:

StDh/CpuDR/u^2 /[1C ̨Pr 1 ] (equation 12.117) (i)

From equation 3.23:

R/u^2 Re^2 DPfd^3 / 4 l^2 

D 5600 ð 40 / 1000 ^3 ð 1000 / 4 ð 10 ð 1 ð 10 ^3 ^2 D 8 , 960 , 000
From Fig. 3.8:ReD 62 ,000 andR/u^2 D 8 , 960 , 000 / 62 , 000 ^2 D 0. 0023
The ratio of the velocity at the edge of the laminar sub-layer to the mean velocity of
flow is:
̨D 2 Re^0.^125 D 2 / 62000 ^0.^125 D 0. 5035

The Prandtl group,PrDCp/kD 4200 ð 1 ð 10 ^3 / 0. 6 D 7. 0

and from equation (i), the Stanton group,StDR/u^2 /[1C ̨Pr 1 ]

D 0. 0023 /[1C 0. 5035  7. 0  1 ]D 0. 000572

∴ hD 0. 000572 CpuD 0. 000572 CpR e/d

D 0. 000572 ð 4200 ð 62000 ð 1 ð 10 ^3 / 40 / 1000 D3724 W/m^2 K.

The area for heat transfer per unit length of pipeD 40 / 1000 D 0. 04 D
0 .126 m^2 /m and making a heat balance over unit length of pipe dl:

∴ hAdlTwTDuCpAcdTDR e/dCpAcdT

∴ 3724 ð 0 .126 dl 393 TD 62000 ð 1 ð 10 ^3 / 40 / 1000 

ð 4200 ð/ 4  40 / 1000 ^2 dT

∴ 0 .0572 dlDdT/ 393 T

Integrating:

0. 0572

∫ 10

0

dlD

∫T 0

293

dT/ 393 T

 0. 0572 ð 10 Dln[ 393  293 / 393 T 0 ]

∴ 1. 772 D 100 / 393 T 0 andT 0 D 336 .6K