6.25 The staged system model for a three-stage absorber is presented in Eqs. 2-73-2-75, which are in state-space form. A numerical example of the absorber model suggested by Wong and Luus 1 has the following parameters: H=75.72lb,L=40.8lb/min,G=66.7lb/min,a=0.72, and b=0.0. Using the MATLAB function ss2tf, calculate the three transfer functions (Y1/Yf,Y2/Yf,Y3/Yf) for the three state variables and the feed composition deviation Yf as the input. Figure 2.10 A three-stage absorption unit. the liquid cascading down through them. A series of weirs and downcomers typically are used to retain a significant holdup of liquid on each stage while forcing the gas to flow upward through the perforations. Because of intimate mixing, we can assume that the component to be absorbed is in equilibrium between the gas and liquid streams leaving each stage i. For example, a simple linear relation is often assumed. For stage i yi=axi+b where yi and xi denote gas and liquid concentrations of the absorbed component. Assuming constant liquid holdup H and perfect mixing on each stage, and neglecting the holdup of gas, the component material balance for any stage i is Hdtdxi=G(yi1yi)+L(xi+1xi) In Eq. 2-71, we also assume that molar liquid and gas flow rates L and G are unaffected by the absorption, because changes in concentration of the absorbed component are small, and L and G are approximately constant. Substituting Eq. 2-70 into Eq. 2-71 yields Hdtdxi=aGxi1(L+aG)xi+Lxi+1 Dividing by L and substituting =H/L (the stage liquid residence time), =aG/L (the stripping factor), and K=G/L (the gas-to-liquid ratio), the following model is obtained for the three-stage absorber: dtdx1=K(yfb)(1+)x1+x2dtdx2=x1(1+)x2+x3dtdx3=x2(1+)x3+xf In the model of Eqs. 2-73 to 2-75, note that the individual equations are linear but also coupled, meaning that each output variable x1,x2,x3-appears in more than one equation. This feature can make it difficult to convert these three equations into a single higher-order equation in one of the outputs, as was done in Eq. 2-49