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 kinetics. For example, the mass balance for reactor 1 is Q_1,inc1,in − Q_1,2c1 − Q_1,3c1 + Q_2,1c2 − kV₁c₁ = 0 where Q_1,in = the volumetric inflow to reactor 1 (m³/min), c_1,in = the inflow concentration to reactor 1 (g/m³), Q_i,j = the flow from reactor i to reactor j (m³/min), c_i = the concentration of reactor i (g/m³), k = a first-order decay rate (/min), and V_i = the volume of reactor i (m³).

(a) Write the mass balances for reactors 2 and 3.
(b) If k = 0.1/min, 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 questions:
(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 inflow concentration to reactor 1 is doubled, and the inflow concentration to reactor 2 is halved, what is the concentration of reactor 3?

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22,1 = 22 21.in = 100 C1,in = 10 V₁ = 100 22,in 10 €2,in = 200 V₂ = 50 Q1,2 = 5 21,3 = 117 = 3,2 = 7 V3 = 150 23,out = 110