Given the system of equations

where c = concentration, t = time, x = distance, D = diffusion coefficient, U = fluid velocity, and k = a first-order decay rate. Convert this differential equation to an equivalent system of simultaneous algebraic equations. Given D = 2, U = 1, k = 0.2, c0 = 80 and c10 = 10, solve these equa- tions from x = 0 to 10 and develop a plot of concentration versus distance. 9.13 A stage extraction process is depicted in Fig. P9.13. In such systems, a stream containing a weight fraction y in of a chemical enters from the left at a mass flow rate of F 1 . Simultaneously, a solvent carrying a weight fraction x in of the same chemical enters from the right at a flow rate of F 2 . Thus, for stage i, a mass balance can be represented as F 1 y i − 1 + F 2 x i + 1 = F 1 y i + F 2 x i P9.13a At each stage, an equilibrium is assumed to be established between y i and x i as in K = x i y i P9.13b where K is called a distribution coefficient. Equation P9.13b can be solved for x i and substituted into Eq. P9.13a to yield y i − 1 − 1 + F 2 F 1 K y i + F 2 F 1 K y i + 1 = P9.13c If F 1 = 400 kgh, y in = 0.1, F 2 = 800 kgh, x in = 0, and K = 5, determine the values of y out and x out if a five-stage reactor is used. Note that Eq. P9.13c must be modified to account for the inflow weight fractions when applied to the first and last stages. 9.14 A peristaltic pump delivers a unit flow Q 1 of a highly viscous fluid. The network is depicted in Fig. P9.14. Every pipe section has the same length and diameter. The mass and mechanical energy balance can be simplified to obtain the pits from which these materials can be obtained. The com- position of these pits is Sand Fine Gravel Coarse Gravel Pit1 52 30 18 Pit2 20 50 30 Pit3 25 20 55 How many cubic meters must be hauled from each pit in order to meet the engineer’s needs?

9.11 An electrical engineer supervises the production of three

types of electrical components. Three kinds of material— metal, plastic, and rubber—are required for production. The amounts needed to produce each component are Metal g Plastic g Rubber g Component component component component 1 15 0.25 1.0 2 17 0.33 1.2 3 19 0.42 1.6 If totals of 2.12, 0.0434, and 0.164 kg of metal, plastic, and rubber, respectively, are available each day, how many com- ponents can be produced per day? 9.12 As described in Sec. 9.4, linear algebraic equations can arise in the solution of differential equations. For example, the following differential equation results from a steady-state mass balance for a chemical in a one-dimensional canal: 0 = D d 2 c d x 2 − U dc d x − kc Flow = F 1 Flow = F 2 x 2 x out x 3 x i x i ⫹ 1 x n ⫺ 1 x n x in y 1 y in y 2 y i ⫺ 1 y i y n ⫺ 2 y n ⫺ 1 y out 1 2 n 0i n ⫺ 1 • • • • • • FIGURE P9.13 A stage extraction process.