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3.4.3 Filtration Operations-Basic Equations The general filtration equation,equation(3.4-12),after substitution of L from equation(3.4-14),becomes 3.4-15 where Vm is the medium resistance,which is equal toEquation(3.4-15)can be rewritten 3.4-16 3.4.4 Constant pressure filtration For incompressible cakes r is independent of the pressure drop and of position in the cake.If p is constant,the equation(3.4-16)can be integrated 3.4-17 Using equation(3.4-17)either t or V can be calculated from the value of the other variable.3.4-18It is useful for the mathematical simplicity of the final equations to define a constant K.Substitution of K from equation(3.4-18)in equation(3.4-17)gives an equation for constant pressure filtration:3.4-19The equation of the constant pressure filtration(3.4-19)can also be in the other form 3.4-19 a Where the volume of filtrate per unit filtration areaFor experimental determination of filtration constant K and specific resistance r,equation(3.4-19)is often put in the form 3.4-20 Which gives a straight line if t/V is plotted against V,shown in figure.The equation(3.4-20)and the figure are used to predict the filtration constants.3.4.5 Constant rate filtration If the flow rate Q=dV/dt is kept constant and the pressure p varied,equation(3.4-15)becomes Where V is simply V=Q t 3.4-21 thus(3.4-22)A plot of p against t as in figure3.22,will,from equation(3.4-22),be a straight line.3.4.6 Constant rate followed by constant pressure operation In many cases,the early stages of filtration are conducted at a nearly constant rate.As the cake becomes thicker and offers more resistance to the flow,the pressure developed by the pump becomes a limiting factor and the filtration proceeds at a nearly constant pressure.For such a combined operation,the plot p versus time is as shown in figure3.23.For constant rate filtrationThen p=constant for tts for tts will be as shown in figure The plot of The equations for constant rate followed by constant pressure filtrationV=Qt 3.4-21 and 3.4.7 Vacuum Filtration-Drum Continuous Filtration Drum Continuous filtrationIn a continuous filter,the feed,filtrate,and cake move at steady constant rates.For any particular element of the filter surface,however,conditions are not steady but transient.The process consists of several steps in series cake formation,washing,drying and discharging.The pressure drop across the filter during cake formation is,however,held constant.Automatic valve septum cake washingslurry.Connections from the periphery to an automatic filter valve Thus the foregoing equations for discontinuous constant-pressure filtration may,with some modification,be applied to continuous filters.If t is the actual filtering time(i.e.,the time any filter element is immersed in the slurry),then from Eq.(3.4-19)(3.4-23)where V is the volume of filtrate collected during time t.Solving Eq.(3.4-23)for V,as a quadratic equation,gives(3.4-24)Where V is the volume of filtrate collected during time t that it is the actual filtration time.If the fraction of the drum submerged is,and drum speed is n(3.4-25)Substituting t from Eq.(3.4-25)in Eq.(3.4-24)for the volume of filtrate(3.4-26)The capacity of drum filter Q in m3/s can be derived by equation(3.4-26)(3.4-27)In continuous filtration,the resistance of the filter medium is generally negligible compared with the cake resistance.So in Eq.(3.4-27),Vm=0.(3.4-28)3.4.8 Washing Filter Cakes To wash soluble material that may be retained by the filter cake after a filtration,a solvent miscible with the filtrate may be used as a wash.Water is the most common wash liquid.In most filters the wash liquid follows the same path as that of the filtrate.But in a filter press the wash passes through the entire thickness of the cake.The last filtrate passes through only one half the final cake For a shell-and leaf filter the flow rate of the wash liquid,in principle,equal to that of the last the filtrate,provided the pressure drop remains unchanged.So the flow rate of washing can be obtained from the equation Eq.(3.4-19)The equation(3.4-19)can be written by taking derivative Thus,the relationship between the washing rate ,and the final rate of filtration(3.4-30)(3.4-32)For a filter press the flow rate of washing where the cross-section of the frame A=A/2,then,substitution of A=A/2 in equation(3.4-31)gives the washing rate for a plate and frame filter press(3.4-33)Hence the relationship between the flow rate of washing and that of last filtration in a filter press is(3.4-32)(3.4-33)ProblemFor constant pressure filtration,if the filtering medium(septum)resistance is ignored,the filtering time is double and other operating conditions keep unchanged,the filtrate volumewill be()（A)double;(B)half of original;(C)21/2 of original;(D)uncertainProblemWhich the statement is correct in following()A)Filtration rate is proportional to filtration area(A).B)Filtration rate is proportional to the square of filtration area(A2).C)Filtration rate is proportional to the volume of filtrate(V).D)Filtration rate is inversely proportional to the resistance of filter medium.Problem A standard cyclone separator,0.5m in diameter and inlet width B=D/4,is used to remove the particles from dust gas.If the gas enters at 15m/s and the numbers of spirals N in the cyclone takes as 5,at what critical diameter of particle will occur?(viscosity of air=0.018mPa and density=1.3 kg/m3;density of particle=2700 kg/m3.)solutionFrom equation A filter press of 0.1m2 filtering area is used for filtering a sample of the slurry.The filtration is carried out at constant pressure with a vacuum 350mmHg.The volume of filtrate collected in the first 5min was one liter.(1)How much filtrate will be obtained when the filtration has been carried out for 10min on assuming the cake to be incompressible and the resistance of filter medium is negligible?(2)How much filtrate will be collected for filtrating time of 10 min if the filter press operates at vacuum 700mmHg.ProblemProblemA slurry is filtered in a plate and frame press containing 12 frames,each 0.3m square and 25mm thick.During the first 200s,the filtration pressure is slowly raised to the final value of 400kN/m2,and during this period the rate of filtration is maintained constant,and volume of filtrate collected was0.01937m3.After the initial period;filtration is carried out at constant pressure and the cakes are completely formed after a further 900s.The cake are then washed at 400kN/m2 for 600s.What is the volume of filtrate collected per cycle and how much wash water is used?A sample of the slurry had previously been tested,using a vacuum leaf filter of 0.05m2 filtering surface and a vacuum 71.3kN/m2.the volume of filtrate collected in the first 300s was 250cm3 and,after a further 300s,an additional 150cm3 was collected.Assume the cake to be incompressible and the cloth resistance to be the same in the leaf as in the filter press.solutionIn the leaf filter,filtration is at constant pressurefrom start,thusFor the leaf filterWhen t=300s,V=250cm3And when t=600s,V=400cm3A=0.05m2,and p=71.3kPaThus(0.025)2+20.025Vm=K0.052300And(0.04)2+20.04Vm=K0.052600HenceVm=0.0175,and K=0.002For the filter pressA=2.16m2,p=400kPa In the filter press,a volume V1 of filtrate is obtained under constant rate conditions in time t1,and filtration is then carried out at constant pressureThe total volume of filtrate collected is therefore given byK=Kp/p=0.002400/71.3=0.0112The total volume of filtrate is V=6.84m3Relation between Final rate of filtration and washing rate is given bywash water usedTherefore the washing rate isProblemFiltration is carried out in a plate and frame filter press,with 20 frames 0.3 m square and 50mm thick.At a constant pressure difference of 248.7kN/m2,one-quarter of the total filtrate per cycle is obtained for the first 300s.Filtration is continued at a constantpressure for a further 1800s,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 500sIt is decided to use a rotary drum filter,1.5m long and 2.2m in diameter,in place of the filter press.Assuming that the resistance of the cloth is the same 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 70kN/m2 and with 25%of the drum submerged
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