Use MATLAB to determine the spectral condition num-

c 1 c 2 c 3 c 4 Upper Colorado River Lake Mead Lake Mohave Lake Havasu Lake Powell FIGURE P11.12 The Lower Colorado River. FIGURE P11.15 11.14 Polynomial interpolation consists of determining the unique n – 1th-order polynomial that fits n data points. Such polynomials have the general form, f x = p 1 x n− 1 + p 2 x n− 2 + · · · + p n− 1 x + p n P11.14 where the p’s are constant coefficients. A straightforward way for computing the coefficients is to generate n linear algebraic equations that we can solve simultaneously for the coefficients. Suppose that we want to determine the coefficients of the fourth-order polynomial f x = p 1 x 4 + p 2 x 3 + p 3 x 2 + p 4 x + p 5 that passes through the following five points: 200, 0.746, 250, 0.675, 300, 0.616, 400, 0.525, and 500, 0.457. Each of these pairs can be substituted into Eq. P11.14 to yield a system of five equations with five unknowns the p’s. Use this approach to solve for the coef- ficients. In addition, determine and interpret the condition number.

11.15 A chemical constituent flows between three reactors

as depicted in Fig. P11.15. Steady-state mass balances can be written for a substance that reacts with first-order kinet- ics. For example, the mass balance for reactor 1 is Q 1,in c 1,in − Q 1,2 c 1 − Q 1,3 c 1 + Q 2,1 c 2 − kV 1 c 1 = P11.15 where Q 1,in = the volumetric inflow to reactor 1 m 3 min, c 1,in = the inflow concentration to reactor 1 gm 3 , Q i,j = the flow from reactor i to reactor j m 3 min, c i = the concentra- tion of reactor i gm 3 , k = a first-order decay rate min, and V i = the volume of reactor i m 3 . a Write the mass balances for reactors 2 and 3. b If k = 0.1min, write the mass balances for all three reactors as a system of linear algebraic equations. c Compute the LU decomposition for this system. d Use the LU decomposition to compute the matrix inverse. e Use the matrix inverse to answer the following ques- tions: i What are the steady-state concentrations for the three reactors? ii If the inflow concentration to the second reactor is set to zero, what is the resulting reduction in concentration of reactor 1? iii If the in- flow concentration to reactor 1 is doubled, and the V 2 ⫽ 50 Q 2,in ⫽ 10 c 2,in ⫽ 200 Q 1,in ⫽ 100 Q 1,3 ⫽ 117 Q 3,2 ⫽ 7 Q 1,2 ⫽ 5 Q 2,1 ⫽ 22 Q 3,out ⫽ 110 c 1,in ⫽ 10 V 1 ⫽ 100 V 3 ⫽ 150