CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 7

the groupH/qand in effect the fundamental dimensions are 4 (M, L, TandH/q)and
there will be 8  4 D4 groups. For the recurring set, the variablesdi,,kand will
be chosen. Thus:

diL, LDdi

M/L^3 MD L^3 D

d^3 i

M/LT, TDM/L D

d^3 i/di D

d^2 i/

kH/q/LT,H/qDkLTDkdi

d^2 i/Dk

d^3 i/

Dimensionless group 1:hL^2 T/H/qDhd^2 i
d^2 i/k
d^3 i/Dhdi/k

Dimensionless group 2:uT/LDu
d^2 i/diDdiu
/

Dimensionless group 3:d 0 /LDd 0 /di

Dimensionless group 4:CpM/H/qDCp
d^3 i/k
d^3 i/DCp/k

∴ hdi/kDfdiu
/,Cp/k,d 0 /diwhich is a form of equation 9.94.

PROBLEM 1.

Obtain by dimensional analysis a functional relationship for the wall heat transfer coef-
ficient for a fluid flowing through a straight pipe of circular cross-section. Assume that
the effects of natural convection may be neglected in comparison with those of forced
convection.
It is found by experiment that, when the flow is turbulent, increasing the flowrate by a
factor of 2 always results in a 50% increase in the coefficient. How would a 50% increase
in density of the fluid be expected to affect the coefficient, all other variables remaining
constant?

Solution

For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is
detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the
wall is given by equations 9.64 and 9.58 which may be written as:

hd/kDfdu

/,Cp/k

or: hd/kDKdu
/nCp/km

∴ h 2 /h 1 Du 2 /u 1 n.

Increasing the flowrate by a factor of 2 results in a 50% increase in the coefficient, or:

1. 5 D 2. 0 nandnDln 1. 5 /ln 2. 0 D 0 .585.

Also: h 2 /h 1 D
2 /
1 ^0.^585

When
2 /
1 D 1 .50,h 2 /h 1 D 1. 50 ^0.^585 D 1 .27 and the coefficient is increased by 27%