$4\mathrm{.}20$ The drag force, $F_D,$ is defined as the interfacial transfer of momentum from the fluid to the solid. In Chapter $3,$ power, $W^{},$ is given by Eq$.(3\mathrm{.}1-11)$ as

$W^{}=F_D{v}_{ch}$

For flow in conduits, power is also expressed by Eq$.(4\mathrm{.}5-2)$ in the form

$W^{}=Q|P|$

a$)$ For flow in a circular pipe, the characteristic velocity is taken as the average velocity. For this case, use Eqs. $(1)$ and $(2)$ to show that

$F_D=A|P|$

where $A$ is the cross$-$sectional area of the pipe.

b$)$ For flow through packed beds, the characteristic velocity is taken as the actual average velocity or interstitial velocity, i$.$e$.,$

Problems

$115$

$v_{ch}=\frac{v_o}{lon}$

in which $v_o$ is the superficial velocity and $lon$ is the porosity of the bed. Show that

$F_D=lonA|P|$

where $A$ is the cross$-$sectional area of the packed bed.

c$)$ In fluidization, the drag force on each particle should support its effective weight, i$.$e$.,$ weight minus buoyancy. Show that the drag force is given by

$F_D=AL(1-lon)({}_P-)glon$

where $L$ is the length of the bed, and $$ and ${}_P$ are the densities of the fluid and solid particle, respectively. Note that in the calculation of the buoyancy force the volume occupied by solid particles should be multiplied by the density of suspension, i$.$e$.,lon+(1-lon){}_P,$ instead of by $.$

Combine Eqs. $(5)$ and $(6)$ to get

$\frac{|P|}{L}=g(1-lon)({}_P-)$

which is a well$-$known equation in fluidization.