Expression constants, pressure filters

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. 

A suspension of aluminum hydroxide in water is to be filtered imder constant pressure in a batch Nutsch filter having a filtering area of 1 m. Each filter cycle is estimated to separate out 0.5 m of suspension. The operating temperature is 25° C. The following expression for the cake resistance was empirically determined from pilot tests ... 

Capillarity of the paper with respect to thickness is called the filtering capacity and expressed, according to Herzberg, as the time in seconds needed for 100 ml of water at 20-21° C to filter through a surface of 10 cm2 under a constant pressure of 5 cm of water. The corresponding value may be given for a filter with a diameter of 15 cm. 

A slurry is filtered, and the filter cake is washed by use of a plate-and-frame filter press operated at a constant pressure drop of 40 psi throughout the entire run. Experimental tests have been carried out on this equipment, and the results for the slurry mixture used can be expressed as follows for any one pressure drop ... 

It has been observed experimentally, and explained theoretically (Serrano, 1981), that the filtration behavior of a not very concentrated suspension, such as wine (particle content less than 1%), obeys different physical laws according to the type of porous material used to remove the solids. A mathematical model expresses the variations in volume filtered over time, at constant pressure, for each of these laws. The behavior of a given product in industrial filtration may be predicted on the basis of laboratory tests, by applying the corresponding equation. 

For constant-pressure filtration and washing, such as occur on vacuum filters, the filtration time and washing time can be related through the wash ratio if the wash liquor and filtrate are assumed to have similar physical properties. From the equation for incompressible filtration at constant pressure expressed in terms of a, ecffic resistance ... 

A tube press comprises two concentric cylinders where a permeable tube covered with a filter cloth is positioned centrally within a solid outer tube lined by an elastomer diaphragm (see Figure 1.37). The filter cycle is initiated by pumping the feed suspension into the annular space between the inner tube and the diaphragm. With sufficient suspension in the press, pressure is applied to induce radial filtration. This process is most often performed at constant pressure via the diaphragm in two stages where a lower pressure is used initially to promote more even cake formation. When filtration is complete the elastomer diaphragm is further inflated (hydraulically) to deliquor the cake via mechanical expression. 

The principal objective of an expression test is to determine the compression deliquoring characteristics of a cake. However, the nature of the test allows both filtration and compression characteristics to be determined when the starting mixture is a suspension (i.e. where the solids are not networked or they are interacting to a significant extent). Cake formation rate, specific resistance and solids volume fraction data can be determined for the filtration phase while analysis of a subsequent consolidation phase allows the calculation of parameters such as consolidation coefficient, consolidation index and ultimate solids concentration in the cake. Repeated use of the expression test over a range of constant pressures allows the evaluation of scale-up coefficients for filter sizing and simulation as described in Section 4.7. 

The specific cake resistance is the most troublesome parameter ideally constant, its value is needed to calculate the resistance to flow when the amount of cake deposited on the filter is known. In practice, it depends on the approach velocity of the suspension, the degree of flow consoHdation that the cake undergoes with time, the feed soHds concentration, and, most importantly, the appHed pressure drop Ap. This changes due to the compressibiHty of most cakes in practice. often decreases with the velocity and the feed concentration. It may sometimes go through a maximum when it is plotted against soHds concentration. The strongest effect on is due to pressure, conventionally expressed as ... 

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 ... 

Constants K and C can be readily obtained from experiments conducted on a prototype machine, from whence the volume of filtrate obtained for a definite time interval (for a specified filter, at the same pressure and temperature) can be calculated. If process parameters are varied, new constants K and C can be estimated from the above expressions. The last expression can be further modified by denoting the constant r as = CVK, and substituting ... 

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 ... 

A realistic boundary condition must account for the solubility of the gas in the mucus layer. Because ambient and most experimental concentrations of pollutant gases are very low, Henry s law (y Hx) can be used to relate the gas- and liquid-phase concentrations of the pollutant gas at equilibrium. Here y is the partial pressure of the pollutant in the gas phase expressed as a mole fraction at a total pressure of 1 atm x is the mole fraction of absorbed gas in the liquid and H is the Henry s law constant. Gases with high solubilities have low H value. When experimental data for solubility in lung fluid are unavailable, the Henry s law constant for the gas in water at 37 C can be used (see Table 7-1). Gas-absorption experiments in airway models lined with water-saturated filter paper gave results for the general sites of uptake of sulfur dioxide... 

K is Si constant, proportional to the pore diameter multiplied by a power of 4 and the number of pores per unit area, but inversely proportional to the viscosity of the liquid, S is the surface of the filter layer, E is the thickness of the filter layer and P is the filtration pressure. This law simply expresses the proportionality between the flow rate and surface area, on the one hand, and pressure, on the other hand. It also shows that the flow rate is inversely proportional to the thickness of the filter layer. 

Flow through packed beds of solids is usually analyzed by considering such characteristics as the porosity of the bed and the sphericity of the particles, and Section 7.5.4.1 shows that the analysis of a filter is helped by considering how the deposit of precipitated solids changes those characteristics. In the other filters, the solids deposit as a cake on the filter medium. The resistances of the filter cake and medium are then additive. When the resistivity, or the resistance per unit thickness, of the cake remains constant throughout operation, the specific resistance increases linearly with the amount of solid deposited. Analytical solutions for the filtration rate are then possible. In the constant-rate case, the pressure drop encountered can be expressed as a function of time (Section 7.5.4.2). 

---