流体力学与传热 ：3-4-2 Flow Rate-Pressure Drop Relationships

1、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.,As the pores in the medium are normally small and rate of flow of filtrate is low, laminar flow conditions are almost i

2、nvariably obtained.,So, the relationship between the superficial velocity u and the pressure drop can be described by Kozeny-Carman equation,3.4-2,The linear velocity u is given by the equation,3.4-3,The sphericity shape factor of a particle is the ratio of the surface area of this sphere having the

3、 same volume as the particle to the actual surface area of the particle.,Substituting u from Eq. (3.4-3 ) into Eq. (3.4-2 ) gives,3.4-4,The filtration rate is,Let,3.4-5,3.4-6,Where k is a constant referred to as the permeability of the bed and substitutes equation (3.4-6 ) into equation (3.4-5 ),(3.

4、4-7 ),This equation relating the flow rate of a filtrate with viscosity through a bed of thickness L and face area A to the driving pressure p is called Darcys basic filtration equation.,Equation (3.4-7) is often written in the form.,3.4-8,Where R is called the resistance (and is equal to L/k, the t

5、hickness divided by the permeability of the bed),3.4-8,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:,3.4-9,In pr

6、actice, 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 d

7、eposited (only true for incompressible cakes) it follows that for a given filtration area A.,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 equation (

8、3.4-11 ) for Rm in equation (3.4-9 ) gives,Equation (3.4-12) relates the flow rate of filtrate to the pressure drop,3.4-12,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 p may be consta

9、nt or variable with time depending on the characteristics of the pump used or on the driving force applied.,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 m

10、edium or a rotary drum.,3. Liquid viscosity The liquid viscosity is constant provided that the temperature remains constant during the filtration cycle and that the liquid is Newtonian.,4. Specific cake resistance The specific cake resistance r should be constant for incompressible cakes but it may

11、change with time in the case of variable rate filtration, because of variable approach velocity.,3.4-13,Most cakes, however, are compressible and their specific resistance changes with the pressure drop across the cake.,5. Volume of the cake deposited per unit volume of filtrate obtained,The volume

12、of the particles deposited per unit volume of filtrate is defined by c.,Where c is constant, which depends on the concentration of solids in the slurry and the porosity of cake.,c can be related to the cumulative volume of filtrate V and the width of cake L,3.4-14,6. Medium resistance The medium res

13、istance Rm should normally be constant but it may be vary with time as a result of some penetration of solids into the medium and sometimes it may also change with applied pressure because of the compression of fibres in the medium.,problem,Fluid passes through a packed bed in laminar flow, the basi

14、c equation applied to packed bed for relating the pressure drop with the width of bed, average interstitial velocity, diameter of tortuous channels, and the properties of fluids is,Hagen-Poiseuille modified by substituting equivalent diameter of channel for the diameter, and superficial velocity for

15、 the interstitial velocity gives a( ) equation,Basic filtration equation comes from ( ) equation, and constant k depends on the ( ) of packed bed and ( ) of particle, and specific cake resistance r is ( ) k,For an incompressible cake When the filtrate increases The rate of filtration will ( ) When the filtration area increases The rate of filtration will ( ) When the pressu