2. A cylindrical tark with radius A(m) and height L(m) is used to mix 2 solutions with concertirations c1(kg/m2) of solule x1(kg) and c2(kg/m2) of solule x2(kg) (Figure 2) These solutions flow in at constant rates f1(m2/s) and f2(m2/s), respectively. Concentration c1 is not constant in time and given by c1=X(1+cos(f)) with X al positive coristant. The solutions entering the tank both hawe a known constant temperature of Tin(K) There is heat loss through the side of the tank (not the bottom and top) to the emironment which has a constant temperature given by Te(K). The heat transfer coeflicients from the tank to the environment is given by ho (Jtin1K1m2). The specilic heat capscity of the fluid is given by cp(/kg1K1). Al heat transfers can be corisidered ideall and can be described using Newton's law of cooling dQ/dt=hAT. The density (kg/m2) can be assumed constamt, the tank is fully filled and perfectly mixed at arty point in fime. Figure 2: Mixing (a) Derive the ODE that describes the time evolution of solute x1 as, dfdx1+x1cos(t)= and determine , and . [b marks] (b) Classify the ODEs in (a) (arder, constant/variable coefliciemts, (non)-homogeneous, (non)-linear). Jusfity your answer [3 marks] question 2 continues on the following page Page 4 of 7 TUPN OVER question 2 continued (c) Find the complementary solution xc for the ODE in (a). [4 markss (d) Find the particular solution xp for the ODE in (a). [5 mairks] (e) Derive the ODE that describes the fime evolution of temperature T inside the tank. [6 marks ] [TOTAL 25 MARKS]