Pasteurization is an important method to sterilize water, milk, fruit juices etc. With milk, it is typically a three-stage process. In the first stage (1st pasteurization), the milk is heated to 70 C for 10-30 s, which kills all bacteria without destroying the nutrients. However, this process does not kill the bacterial spores. To increase the efficiency of the pasteurization, as a second stage, the milk is left to age at room temperature for a certain time to allow the spores to germinate. The process during this stage can be modelled as two consecutive "reactions": production of bacteria B from spores s: production of spores from bacteria: S 1 B, B 2 > B+S, r = k[s]; r2 = K2[B]. The third stage must takes place after the spores turn to bacteria, but not before the new- born bacteria produce spores: this stage is a second pasteurization, killing all bacteria again, and leaving the milk with a significantly lower concentration of spores. a) Assume that the second stage of the process takes place in a plug flow bioreactor; initial concentrations are [s]o for the spores and 0 for the bacteria. Express the concentrations [B] and [s] through the yields and & of processes 1 and 2 above. Use these to formulate the mole balances (differential equations for [B] and [s] vs. t). [6 marks] [B] = 1 - - [sl], [exp (K6)-exp(- 11k.)]; 10 b) Let k2= 11k1/100. For this value of k2, the solution to the equations formulated in a) reads: k,t [s] =[s], exp p(41)+1+ -11exp 12 10 exp(-11K)]. 10 Substitute these formulae in the mole balances from a) to verify that they are indeed a correct solution. Show also that they fulfil the initial conditions. Sketch schematically the solution (no need for precise values). [5 marks] c) The concentration of spores [s] passes through a minimum as time advances. Find the optimal time between the two pasteurizations that will result in highest sterility and shelf life of the milk. Find "the conversion", i.e. the fraction of spores that will be destroyed under optimal conditions. Parameters: k = 0.1 min-. [5 marks] d) Based on the above, draw a schematic for a production line for pasteurization of milk using isothermal plug flow reactors and heat exchangers only (with values of temperatures and space-times indicated). [4 marks]