CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 15

The viscosity of the liquid is given by:

D 0  1  0. 015 T 273 

where is the viscosity at a temperatureTKand 0 is the viscosity at 273 K.

Solution

The volume flow of filtrate per unit area,um^3 /m^2 s, will depend on the fluid density, ;
fluid viscosity, ; particle size,d; pressure difference across the bed,P, and the voidage
of the cake,eor:vDf
,,d,P,e. The dimensions of each of these variables are
uDL/T, DM/L^3 , DM/LT,dDL,PDM/LT^2 andeDdimensionless. There
are 6 variables and 3 fundamental dimensions and hence  6  3 D3 dimensionless
groups. Taking,d,
and as the recurring set, then:

dL, LDd

M/L^3 , MD L^3 D

d^3

M/LT, TDM/L D

d^3 /dD

d^2 /

Thus: dimensionless group 1:uT/LDu
d^2 /dDdu
/

dimensionless group 2:PLT^2 /MDPd

d^2 /^2 /

d^3 DP

d^2 /^2

and the function is: P
d^2 /^2 Dfdu
/

This may be written as: P
d^2 /^2 DKdu
/n

Since the filtration rate is doubled when the pressure difference is doubled, then:

u/PandnD 1 ,P

d^2 /^2 DKdu

/

and: uD 1 /KPd/,oru/ 1 /

 293 / 313 D 0  1  0. 015  293  273 / 0  1  0. 015  313  273 D 0. 7 / 0. 4 D 1. 75

∴ v 313 /v 293 D 293 / 313 D 1. 75

and the filtration rate will increase by 75%.