= Two tanks are connected in series, as shown in the schematic figure below. Water at a temperature Tin = 60°C is fed in

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Two Tanks Are Connected In Series As Shown In The Schematic Figure Below Water At A Temperature Tin 60 C Is Fed In 1 (221.66 KiB) Viewed 70 times

= Two tanks are connected in series, as shown in the schematic figure below. Water at a temperature Tin = 60°C is fed into tank 1 at a mass flow rate of = 100 kg min-1. Water exits the system from tank 2 at a flow rate of ^. In addition, some of the water from tank 2 is recycled at a rate m = 10 kg min-1 back into tank 1. Water flows out of tank 1 and into tank 2 at a mass flow rate of À + m. Tank 1 initially contains M1 = 1000 kg of water at 20°C. It loses heat to the surrounding environment, which is at temperature To = 20°C; the overall heat transfer coefficient is UA = 1000 J min-K-1. Tank 2 initially contains M2 = 500 kg of water at 20°C. It has an electrical heating element, which has a power of = 1000 J min-1, but is otherwise perfectly insulated. The contents of each of the two tanks are perfectly mixed. The heat capacity of water is C = 1000 J kg. = Tank 1 Tank 2 UA ů M2 M Ti M + m Ti M T2 Tin T2 To m T2 (a) Perform an energy balance around tank 1 to develop a differential equation that governs the rate of change of the liquid temperature in tank 1. [5 marks] (b) Perform an energy balance around tank 2 to determine a differential equation that governs the rate of change of the liquid temperature in the tank. [5 marks] (C) Write these differential equations in matrix/vector form: d dt as (7.)--( A1 A2 Ti A) (1) (3) 91 91 by providing values for the elements in the matrix A and vector q. [2 marks] (d) In this and the remaining questions, use the values A11 = 0.1 min-1, A12 = -0.01 min-!, A21 = -0.2 min-1, A22 = 0.2 min?, ,91 = 5 °C min-1 and q2 = 0.1 °C min-!. Determine the steady state temperature T1,00 of the liquid in tank 1 and the steady state temperature T2,00 of the liquid in tank 2. [5 marks] (e) Determine the eigenvalues and eigenvectors of the matrix A. What is the matrix that diagonalizes A (i.e. the matrix D such that D-'AD is diagonal)? [5 marks] (f) Solve the differential equation to determine the variation with time of the temperature Tį of the liquid in tank 1 and the temperature T2 of the liquid in tank 2. Plot the variation of the two tanks with time. [8 marks]