Pressure drop across cakes

Pc = pressure drop across cake W = dry cake weight/unit area/cycle D = dry time per cycle p = viscosity of liquid phase... 

As Ap increases indefinitely in Equation (22.44), (22.45), the term with Ap in the denominator approaches zero. Plots of calculated values of flow rate and average solidosity as functions of pressure drop across cakes with co = 0.02 m /m are shown in Figures 22.28 and 22.29 for the three materials listed in Table 22.6. Whereas the flow rate q increases linearly with Ap for... 

Deep Bed Filters. Deep bed filtration is fundamentally different from cake filtration both in principle and appHcation. The filter medium (Fig. 4) is a deep bed with pore size much greater than the particles it is meant to remove. No cake should form on the face of the medium. Particles penetrate into the medium where they separate due to gravity settling, diffusion, and inertial forces attachment to the medium is due to molecular and electrostatic forces. Sand is the most common medium and multimedia filters also use garnet and anthracite. The filtration process is cycHc, ie, when the bed is full of sohds and the pressure drop across the bed is excessive, the flow is intermpted and solids are backwashed from the bed, sometimes aided by air scouring or wash jets. 

There is an additional pressure drop across the cake, developed by electroosmosis, which leads to increased flow rates through the cake and further dewatering at the end of the filtration cycle. The filtration theory proposed for electrofiltration assumes the simple superposition of electroosmotic pressure on the hydraulic pressure drop. 

The most important feature of the pressure filters which use hydrauHc pressure to drive the process is that they can generate a pressure drop across the medium of more than 1 x 10 Pa which is the theoretical limit of vacuum filters. While the use of a high pressure drop is often advantageous, lea ding to higher outputs, drier cakes, or greater clarity of the overflow, this is not necessarily the case. Eor compressible cakes, an increase in pressure drop leads to a decrease in permeabiUty of the cake and hence to a lower filtration rate relative to a given pressure drop. 

Sepa.ra.tlon, Sodium carbonate (soda ash) is recovered from a brine by first contacting the brine with carbon dioxide to form sodium bicarbonate. Sodium bicarbonate has a lower solubiUty than sodium carbonate, and it can be readily crystallized. The primary function of crystallization in this process is separation a high percentage of sodium bicarbonate is soHdified in a form that makes subsequent separation of the crystals from the mother hquor economical. With the available pressure drop across filters that separate Hquid and soHd, the capacity of the process is determined by the rate at which hquor flows through the filter cake. That rate is set by the crystal size distribution produced in the crystallizer. 

Vacuum pump capacity is conventionally based on the total cycle and expressed as mVh-m" (cfi7i/ft ) of filter area measured at pump inlet conditions. Thus, the gas volumes per unit area passing during each dry period in the cycle are totaled and divided by the cycle time to arrive at the design air rate. Since air rate measurements in the test program are based on pressure drop across the cake and filter medium only, allowance must be made For additional expansion due to pressure drop within the filter and auxiliary piping system in arriving at vacuum pump inlet conditions. 

For incompressible cake, the pressure distribution and the rate depend on the resistance of the filter medium and the permeability of the cake. Figure L8-150 shows several possible pressure profiles in the cake with increasing filtration rates through the cake. It is assumed that r /i i = 0.8 and /p//i = 0.6. The pressure at / = ri, corresponds to pressure drop across the filter medium Ap, with the ambient pressure taken to be zero. The filtration rate as well as the pressure distribution depend on the medium resistance and that of the cake. High medium resistance or blinding of the medium results in greater penalty on filtration rate. 

The pressure drop through the filter is a function of two separate effects. The clean filter has some initial pressure drop. This is a function of filter material, depth of the filter, the superficial gas velocity, which is the gas velocity perpendicular to the filter face, and the viscosity of the gas. Added to the clean filter resistance is the resistance that occurs when the adhering particles form a cake on the filter surface. This cake increases in thickness as approximately a linear function of time, and the pressure difference necessary to cause the same gas flow also becomes a linear function with time. Usually, the pressure available at the filter is limited so that as the cake builds up the flow decreases. Filter cleaning can be based, therefore, on (1) increased pressure drop across the filter, (2) decreased volume of gas flow, or (3) time elapsed since the last cleaning. 

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. 

Figure 4. Flowrate/area versus pressure drop across the cake.

Assume laminar flow of filtrate of liquid through the filter cake. Rate of filtration is usually measured as the rate at which liquid filtrate is collected. The filtration rate depends on the area of the filter cloth, the viscosity of the liquid, the pressure drop across the filter and filter cake resistance. At any instant during filtration, the rate of filtration is given by the equation ... 

If the filter cloth and the initial layers of cake are together equivalent to a thickness L of cake as deposited at a later stage in the process, and if —AP is the pressure drop across the cake and cloth combined, then ... 

Thus the total pressure drop across the filter cake and the cloth (—AP), say, is given by ... 

Pressure drop across the cake and the filter medium... 

A rotary-drum filter filters 20 m /h (706 ft /h) of a calcium carbonate slurry at 20 °C (68 °F). The pressure drop across the cake is 0.658 bar (9.541 psi). If the slimy contains 0.15 mass fraction of calcium carbonate, and the filter cake contains 0.40 mass fraction of water, estimate the surface area of the rotary-drum filter. 

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