流体力学与传热课件Flow Rate-Pressure Drop Relationships

Equation (3.4-7) is often written in the form.
dV Ap
3.4-8
dt R
dV Ap
3.4-8
dt R
Where R is called the resistance (and is equal to L/k, the thickness divided by the permeability of the bed)
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The thickness of cake deposited and other parameters, some of which can , in certain circumstances, be assumed to be constant.
1. Pressure drop
The pressure drop Δp may be constant or variable with time depending on the characteristics of the pump used or on the driving force applied.
As the pores in the medium are normally small and rate of flow of filtrate is low, laminar flow conditions are almost invariably obtained.
Substituting u from Eq. (3.4-3 ) into Eq. (3.4-2 ) gives
dV k Ap
dt L
(3.4-7 )
This equation relating the flow rate of a filtrate with viscosity μ through a bed of thickness L and face area A to the driving pressure Δp is called Darcy’s basic filtration equation.
dV
Ap
3.4-9
dt (R Rm )
In practice, however, the assumption made above that the medium resistance is constant is rarely true because some penetration and blocking of the medium inevitably occurs when particles impinge on the medium
As the resistance of the cake may be assumed to be directly proportional to the amount of cake deposited (only true for incompressible cakes) it follows that for a given filtration area A.
3.4.2 Flow Rate-Pressure Drop Relationships
At the beginning of batch cake filtration, the whole pressure drop available is across the medium itself since as yet no cake is formed.
Filter medium resistance
In the cake filtration, two resistances are presented in series, one of which, the cake resistance R increases and other, the medium resistance Rm may be assumed constant with time. Equation (3.4-8 ) becomes:
3. Liquid viscosity
equation (3.4-11 ) for Rm in equation (3.4-9 ) gives
dV
Ap
dt dV Ap
dt r(LLm)
r(L Lm )
3.4-12
Equation (3.4-12) relates the flow rate of filtrate to the pressure drop,
2. Face area of the filter medium
The face area of the medium A is usually constant, but with a few exceptions such as in the case of equipment with an appreciable cake build-up on a tubular medium or a rotary drum.
R=rL
3.4-10
Where L is the width of cake deposited and r is the specific cake resistance.
Similarly for the filtration medium of width Lm
Rm=rLm
3.4-11
Substitution of equation (3.4-10 ) for R and
u
dV Adt
3
4.17
sp vp
2
1
2
p
L
3.4-4
The filtration rate is
dV
3
Ap
dt
4.17
sp vp
2
1
2
L
3.4-5
Let
3
k
4.17
sp vp
2
1
2
3.4-6
Where k is a constant referred to as the permeability of the bed and substitutes equation (3.4-6 ) into equation (3.4-5 )