Constant-Rate Filtration

The scale-up of conventional cake filtration uses the basic filtration equation (eq. 4). Solutions of this equation exist for any kind of operation, eg, constant pressure, constant rate, variable pressure—variable rate operations (2). The problems encountered with scale-up in cake filtration are in estabHshing the effective values of the medium resistance and the specific cake resistance. 

Filtration. In many mineral processing operations, filtration follows thickening and it is used primarily to produce a soHd product that is very low in moisture. Filtration equipment can be either continuous or batch type and either constant pressure (vacuum) or constant rate. In the constant pressure type, filtration rate decreases gradually as the cake builds up, whereas in the constant rate type the pressure is increased gradually to maintain a certain filtration rate as the cake resistance builds. The size of the device is specified by the required filter surface area. 

By operating cycle. Filtration may be intermittent (batch) or continuous. Batch filters may be operated with constant-pressure driving force, at constant rate, or in cycles that are variable with respect to both pressure and rate. Batch cycle can vary greatly, depending on filter area and sohds loading. 

Constant-rate filtration. Positive-displacement pumps of various types are employed. 

Constant-Rate Filtration For substantially incompressible cakes, Eq. (18-51) may be integrated for a constant rate of slurry feed to the filter to give the following equations, in which filter-medium resistance is treated as the equivalent constant-pressure component to be deducted from the rising total pressure drop to... 

In the filtration of small amounts of fine particles from liquid by means of bulky filter media (such as absorbent cotton or felt) it has been found that the preceding equations based upon the resistance of a cake of solids do not hold, since no cake is formed. For these cases, in which filtration takes place on the surface or within the interstices of a medium, analogous equations have been developed [Hermans and Bredee, J. Soc. Chem. Ind., 55T, 1 (1936)]. These are usefully summarized, for both constant-pressure and constant-rate conditions, by Grace [Am. In.st. Chem. Eng. J., 2, 323 (1956)]. These equations often apply to the clarification of such materials as sugar solutions, viscose and other spinning solutions, and film-casting dopes. 

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 ... 

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 ... 

Filtration experiments in a prototype machine at constant pressure or constant rate permit determination of ax , as well as s and Rf, for a given sludge and filtering medium. Consequently, it is possible to predict the time required for the pressure drop to reach the desired level for a specified set of operating conditions. In the initial stages of filtration, the filter medium has no cake. Furthermore, AP is not zero but has a certain value corresponding to the filter medium resistance for a given rate. This initial condition is ... 

As follows for the filtration of incompressible sediment (at a constant rate), the pressure increases in a direct proportion to time. However, the above equation... 

We now consider a second case, in which the filtration rate changes from one cycle to another however, a constant rate is maintained during each cycle. The filtration is terminated when the pressure difference reaches a maximum allowable value. The amount of filtrate and cake thickness for each cycle will be different, as in Case 1, because the pressure difference depends not only on the cake thickness, but on the filtration rate as well. The following set of equations apply to this case ... 

To apply these equations, let s consider the following example. Determine a constant rate of filtration and the time of operation corresponding to the maximum capacity of a batch filter having the following conditions maximum permissible pressure difference AP = 9x10 N/m sludge viscosity /r = 10 N-s/m filter plate resistance Rf = 56x 10 ° m specific cake resistance r = 3 X 10 m ° x = 0.333 auxiliary time = 600 s maximum permissible cake thickness h = 0.025 m. The solution is as follows ... 

The ratios in parentheses express the constant volume rate per unit filter area. Hence, Equation 24 is the relationship between time i and pressure drop Ap. For incompressible cakes, rg is constant and independent of pressure. For compressible cakes, the relationship between time and pressure at constant-rate filtration is ... 

Using the filtration constants determined from the above question, determine the filtration time of a 5 m CaCOj suspension containing 5 % solids on the filter with 10 m area. The cake wetness is 40% and the solids density is 2,200 kg/m The density of the liquid is 1,000 kg/m Also determine the final filtration rate after 2 hrs. of operation. 

Assume constant flowrate 0-4 bar. Integrated filtration equation (constant rate)... 

You want to select a rotary drum filter to filter a coal slurry at a rate of 100,000 gal of filtrate per day. The filter operates at a differential pressure of 12 psi, and 30% of the surface is submerged in the slurry at all times. A sample of the slurry is filtered in the lab through a 6 in diameter sample of the filter medium at a constant rate of 1 gpm. After 1 min the pressure drop across this filter is 3 psi, and after 5 min it is 10 psi. If the drum rotates at a rate of 3 rpm, what total filter area is required ... 

In the filtration of a sludge, the initial period is effected at a constant rate with the feed pump at full capacity, until the pressure differences reaches 400 kN/m2. The pressure is then maintained at this value for a remainder of the filtration. The constant rate operation requires 900 s and one-third of the total filtrate is obtained during this period. 

Filtration is carried out in a plate and frame filter press, with 20 frames 0.3 m square and 50 mm thick, and the rate of filtration is maintained constant for the first 300 s. During this period, the pressure is raised to 350 kN/m2, and one-quarter of the total filtrate per cycle is obtained. At the end of the constant rate period, filtration is continued at a constant pressure of 350 kN/m2 for a further 1800 s, after which the frames are full. The total volume of filtrate per cycle is 0.7 m3 and dismantling and refitting of the press takes 500 s. It is decided to use a rotary drum filter, 1.5 m long and 2.2 m in diameter, in place of the filter press. Assuming that the resistance of the cloth is the same in the two plants and that the filter cake is incompressible, calculate the speed of rotation of the drum which will result in the same overall rate of filtration as was obtained with the filter press. The filtration in the rotary filter is carried out at a constant pressure difference of 70 kN/m2, and the filter operates with 25 per cent of the drum submerged in the slurry at any instant. 

For constant rate filtration through the filter leaf ... 

A continuous rotary filter is required for an industrial process for the filtration of a suspension to produce 0.002 m3/s of filtrate. A sample was tested on a small laboratory filter of area 0.023 m2 to which it was fed by means of a slurry pump to give filtrate at a constant rate of 0.0125 m3/s. The pressure difference across the test filter increased from 14 kN/m2 after 300 s filtration to 28 kN/m2 after 900 s, at which time the cake thickness had reached 38 mm. What are suitable dimensions and operating conditions for the rotary filter, assuming that the resistance of the cloth used is one-half that on the test filter,... 

A laboratory filtration study is to be carried out at constant rate. The basic equation (Cook, 1984) comes from the relation... 

Various rules of thumb exist for standard water filtration rates and cycle time before backwashing. Higher filtration rates may appear to be economically justified, however, when the filter loading is within conventional limits. In this example, we examine the issues involved for constant-rate filtration for a dual-media bed. Dual- and mixed-media beds result in increased production of water in a filter for two reasons. First, the larger grains (say charcoal approximately 1-mm size) as a top layer help reduce cake formation and deposition within the small (150-mm) top layer of the bed. Second, the head loss in the region of significant filtration is reduced. 

In practice, it is sometimes possible to incorporate moving blades in the filter equipment so that the thickness of the cake is limited to the clearance between the filter medium and the blades. Filtrate then flows through the cake at an approximately constant rate and the solids are retained in suspension. Thus the solids concentration in the feed vessel increases until the particles are in permanent physical contact with one another. At this stage the boundary between the slurry and the cake becomes ill-defined, and a significant resistance to the flow of liquid develops within the slurry itself with a consequent reduction in the flowrate of filtrate. 

In the filter press, a volume V) of filtrate is obtained under constant rate conditions in time t, and filtration is then carried out at constant pressure. 

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