Filtrate volume

This equation is the basis of cake filtration analysis. Feed Hquid flow rate and filtrate volume Dare usually assumed to be related as... 

Traditionally, the average specific cake and medium resistances have been deterrnined from constant pressure experiments and the solution of the basic filtration equation for constant pressure which relates filtrate volume to time. This relationship is, in theory, paraboHc but deviations occur in practice. 

Solvent system CMC concentration, g/350 cm Minimum viscometer dial reading at 600 rpm Maximum filtrate volume, 3 cm... 

The back of the leaf assembly and the joint where the dam overlaps must be sealed with some suitable material so that the filtrate volume collected accurately represents the liquid associated with the deposited cake solids. 

At the end of the run, measure and record the filtrate volume (and weight, if appropriate), cake thickness, final cake temperature (if appropriate), wet cake weight, and note the cake discharge characteristics (roU, sticks to media, etc.). 

Dry Solids or Filtrate Rate Filtration rate, expressed either in terms of diy solids or filtrate volume, may be plotted as a function of time on log-log paper. However, it is more convenient to delavthe rate calculation until the complete cycle of operations has been defined. 

Defining the ratio of cake volume to filtrate volume as x , then the cake volume is x V, The cake volume also may be expressed by the product h,A, where h,. is the cake height in m. Hence,... 

Parameter x can be expressed in terms of the ratio of the mass of solid particles settled on the filter plate to the filtrate volume, x, and, instead of r , a specific mass cake resistance, r , is used. That is, r, is the resistance to the flow presented by a uniformly distributed cake in the amount of 1 kg/m. Replacing units of volume by mass, the term r x into the above expression changes to r x,j,. Neglecting the filter plate resistance (i.e., R, = 0), then ... 

This expression shows the relationship between filtration time and filtrate volume. The equation is applicable to both incompressible or compressible calces, because at constant AP the values and x are constant. For constant AP, an increase in the filtrate volume results in a reduction in the filtration rate. If we assume a definite filtering apparatus and set up a constant temperature and filtration pressure, then the values of Rf, r , fi and AP will be constant. We now take note of the well-known filtration constants K and C, which are derived from the above expressions ... 

A plot of this expression is provided in Figure 32, where the slope of the straight line is 2/K, and the intercept value is C. Experimental determination of dr/dV is straightforward. Filtrate volumes V, and V2 are measured at time intervals Tj and t2, respectively. From the linear equation of filtration, the quotient - "tO/fVj -V,) is ... 

Constants C and K can be determined from several measurements of filtrate volumes taken at different time intervals. There are some doubts as to the actual constancy of C and K during constant pressure filtration. Constants C and K depend on r (specific volumetric cake resistance), which, in turn, depends on the pressure drop across the cake. This AP causes some changes in the cake, especially during the initial stages of filtration. When the cake is very thin, the main portion of the total pressure drop is exerted on the filter medium. As the cake becomes thicker, the pressure drop through the cake increases rapidly but then levels off to a constant value. Isobaric filtration shows insignificant deviation from the expressions developed. For approximate calculations, it is possible to neglect the resistance of the filter plate, provided the cake is not too thin. Then the filter plate resistance, Rf, is equal to zero, C = 0, and r = 0. Hence, a simplified equation is = Kr. 

We now turn attention towards the ease of eonstant-rate filtration. When sludge is fed to a filter by means of a positive displaeement pump, the rate of filtration is nearly constant, i.e., dV/dx = constant. During constant-rate filtration, pressure increases with cake thickness. As sueh, the principal filtration variables are pressure ind filtrate volume, or pressure and filtration time. Integrating the filtration equation for a constant-rate process, we find that the derivative dV/dx ean simply be replaeed by V/x, and we obtain ... 

This expression can be represented graphically in dimensionless form to simplify its use. It is generally expressed as the so-called filtration number , defined as follows E, = /iR, / 2APT3 jr x . The filtration number, E, is dimensionless and varies from zero at Rf = 0 to a large value when there is an increase in the viscosity of the sludge and Rf or a decrease in pressure drop, auxiliary time, specific cake resistance and the ratio of cake volume to filtrate volume. It may be assumed in practice that F(, = 0 to 10. If washing and drying times are constant and independent of filtration time, they may be added directly to the auxiliary time. In... 

The filtrate volume is obtained, assuming which corresponds to the... 

The permeability relative to a pure liquid, usually water, may be determined with the help of different devices that operate on the principle of measurement of filtrate volume obtained over a definite time interval at known pressure drop and filtration area. The permeability is usually expressed in terms of the hydraulic resistance of the filter medium. This value is found from ... 

We denote the ratio of cake volume to filtrate volume as Xq. Hence, the cake volume is XoV. An alternative expression for the cake volume is hcA where he is tire cake height in meters. Consequently ... 

Equation 18 defmes a parabolic relationship between filtrate volume and time. The expression is valid for any type of cake (i.e., compressible and incompressible). From a plot of V + C versus (t+Tq), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 5. Moving the axes to distances C and Tq provides the characteristic filtration curve for the system in terms of volume versus time. Because the parabola s apex is not located at the origin of this new system, it is clear why the filtration rate at the beginning of the process will have a finite value, which corresponds to actual practice. 

This form of the equation provides a linear relation like the plot in Figure 6. The expression is that of a straight line having slope 2/K, with intercept C. The experimental determination of dr/dV is made simple by the functional form of this expression. Filtrate volumes V, and Vj should be measured for time intervals r, and Tj. Then, according to Equation 16 ... 

In examining the right side of this expression, we note that the quotient is equal to the inverse value of the rate at the moment of obtaining the filtrate volume, which is equal to the mean arithmetic value of volumes V, and Vj ... 

Filtration constants C and K can be determined on the basis of several measurements of filtrate volumes for different time intervals. 

The concentration of solids in the feed sludge is expressed by weight fraction c. It is also possible to evaluate experimentally the weight ratio of wet cake to its dry content m. Hence, a unit weight of sludge contains me of wet cake. We denote y as the specific weight of feed sludge. This quantity contains c amount of solids hence, the ratio of the mass of solids in the cake to the filtrate volume is ... 

Equation 46 states that when complete pore blockage occurs, the intensity of the increase in the total resistance with increasing filtrate volume is proportional to the square of the flow resistance. 

Note that for constant Wj, parameter K is proportional to the ratio of the settled volume of cake in the pores to the filtrate volume obtained, and is inversely proportional to total pore volume for a unit area of filter medium. 

Finally, for the case of intermediate filtration, the intensity of increase in total resistance with increasing filtrate volume is less than that occurring in the case of gradual pore blocking, but greater than that occurring with cake filtration. It may be assumed that the intensity of increase in total resistance is directly proportional to this resistance ... 

The numerator of Equation 79 characterizes the cake resistance. The denominator contains information on the driving force of the operation. Constant K (sec/m ) characterizes tile intensity at which the filtration rate decreases as a function of increasing filtrate volume. 

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