102 M.J. Lewis Thus Vc=(V-dv)(c-dc)+c(l-r)dv (Note: d vdc is assumed to be negligible.) Integration between the final and initial conditions gives In(VFVc)=l/R In(cc/cF) If In(Va/Vc) is plotted against In(cc/ce), the gradient is 1/R vE/vc=f=(cc/cF) (4.5) From eqs.(4.2)and(4.3), it can be shown that Substitution into eq (4.5)gives VE/Vc =(f 1/R Therefore =(y/R. This simplifies to Y=fR-I. Therefore the yield r=f <s However, this equation applies only if the rejection remains constant. Nevertheless,it extremely useful, as it gives an insight into the features of the separation process, Let us consider the two extreme values of rejection IfR= l, then yield 1; all the material is recovered in the concentrate i.e. it is not possible to remove all of a component from a feed by ultrafiltration alone Diafiltration may be more useful in helping to achieve this objective(see Section 4.4). However, for most components being concentrated, the rejection values are close to 1.0, typically 0.9-1.0, whereas for those being removed the values would be between 0 Table 4.3 shows a range of yield values for some different concentration factors. One interesting point is that losses can be quite high, even though the rejection value appears good; e.g. for R=0.95 and a concentration factor of 20, the yield is 0.86. Therefore 14%0102 M. J. Lewis Thus cp=c(l -R) Eliminating cp gives VC= (V-dV) (c-dc)+c(l-R)dV - Vdc = cR dV (Note: dVdc is assumed to be negligible.) dV cc dc 1 - -- I v -IcF -zi Integration between the final and initial conditions gives: In (vF/vC)=l/R In (cC/cF) (4.4) vF/Vc = f = (cC/cF)l'R If In (VF/Vc) is plotted against In (cc/cF), the gradient is 1/R. (4.5) From eqs. (4.2) and (4.3), it can be shown that CC/CF = yf Substitution into eq. (4.5) gives vF/vC =(fY)'lR Therefore f = (fY)'IR. This simplifies to Y = f '-'. Therefore the yield y= fR-1 (4.6) However, this equation applies only if the rejection remains constant. Nevertheless, it is extremely useful, as it gives an insight into the features of the separation process. Let us consider the two extreme values of rejection: If R = 1, then yield = 1; all the material is recovered in the concentrate. If R = 0, then yield = l/J in this case the yield is determined by the concentration factor. As the concentration factor is finite (typically 2-20), the yield can never be zero; i.e. it is not possible to remove all of a component from a feed by ultrafiltration alone. Diafiltration may be more useful in helping to achieve this objective (see Section 4.4). However, for most components being concentrated, the rejection values are close to 1.0, typically 0.9-1.0, whereas for those being removed the values would be between 0 and 0.1, Table 4.3 shows a range of yield values for some different concentration factors. One interesting point is that losses can be quite high, even though the rejection value appears good; e.g. for R = 0.95 and a concentration factor of 20, the yield is 0.86. Therefore 14%