Page 894 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix G: Dimensionless Numbers 849

TABLE G.2
Selection of Dimensionless Numbers Applicable to Water Treatment Process Design
Name Grouping Variables Ratio and Description
2F D g c F D ¼ drag force on object (N) inertia forces
Euler (McCabe et al., 1993)
2
3
E ¼
rv A r ¼ density of fluid (kg=m ) pressure forces
o
v o ¼ velocity in core of jet (m=s)
3
A ¼ area of object (m )
g c ¼ conversion factor (kg m=Ns)
DpDg c inertia forces
Friction factor (McCabe et al., 1993) f ¼ Dp ¼ pressure loss in pipe due to friction in length,
2r L friction shear forces
v
2
2
L (N=m )
D ¼ diameter of pipe (m)
g c ¼ conversion factor (kg m=Ns)
3
r ¼ density of fluid (kg=m )
v ¼ average velocity in pipe (m=s)
L ¼ length of pipe (m)
v 2 inertia forces
Froude (Rouse, 1946; McCabe et al., v ¼ velocity (m=s)
gL gravity forces
F ¼
2
1993) g ¼ gravitational constant (9.18 m=s )
L ¼ characteristic length (m)
v v inertia forces
Mach (Rouse, 1946, p. 328) M ¼ p ffiffiffiffiffiffiffiffi ¼ v ¼ velocity of object or fluid (m=s)
E=r c elastic forces
2
E ¼ bulk modulus of elasticity (N=m )
3
r ¼ density of fluid (kg=m )
c ¼ velocity of sound (m=s)
vL convection mass transfer rate
Peclet (McCabe et al., 1993) P e ¼ v ¼ velocity of fluid (m=s)
D v diffusion mass transfer rate
L ¼ characteristic length (m)
a b (Weber and DiGiano, 1996,
c c
¼ R S 2
D v ¼ molecular diffusivity in liquid (m =s)
Omega, 1997)
a ¼ empirical exponent
b ¼ empirical exponent
Pg c inertia forces
Power (McCabe et al., 1993) P ¼ P ¼ power dissipated by turbulence (N m=s)
3
rn D 5 drag forces
g c ¼ conversion factor (kg m=Ns)
3
r ¼ density of fluid (kg=m )
n ¼ rotational velocity of impeller (rev=s)
D ¼ diameter of impeller (m)

rQ momentum dispersion
2
Prandtl P r ¼ n v H n v ¼ kinematic viscosity (m =s)
k c 3 heat diffusion
r ¼ density of fluid (kg=m ) (Weber and DiGiano, 1996, p. 212)
2
2
2
2
Q ¼ specific heat (kg m =s K) or (mol m =s K)
H
3
k c ¼ thermal conductivity (kg m=s K)
rvD inertia forces
3
Reynolds (Rouse, 1946; McCabe R ¼ r ¼ density of fluid (kg=m )
m viscous forces
et al., 1993) v ¼ velocity (m=s)
D ¼ characteristic length (m)
2
m ¼ dynamic viscosity (kg=ms) or(Ns=m )
m momentum dispersion
2
Schmidt S c ¼ m ¼ dynamic viscosity (kg=ms) or(Ns=m )
D v r 2 mass diffusion
D v ¼ molecular diffusivity in liquid (m =s)
(Weber and DiGiano, 1996, p. 212)
r ¼ density of fluid (kg=m 3
k f d interfacial mass transfer impedance
Sherwood S h ¼ k f ¼ mass transfer coefficient (m=s)
D v molecular diffusion impedance
d ¼ characteristic length, which is the boundary layer
(Weber and DiGiano, 1996, p. 212)
thickness (m)
2
D v ¼ molecular diffusivity in liquid (m =s)
(continued)