Calculate the number of degrees of freedom for the system and the maximum number

EX40HC – Tutorial 9 Degrees of freedom and generation of control loopsProblem 1. Consider the CSTR below, where an exothermic reaction 𝐴𝐴→𝐵𝐵 takes place. The feed containsspecies A at concentration𝐶𝐶𝐴𝐴𝐴𝐴(𝑡𝑡). The aim of the control system is to maintain the conversion ofspecies A 𝐶𝐶𝐴𝐴𝐴𝐴(𝑡𝑡)and the reaction temperature 𝑇𝑇(𝑡𝑡) at the desired setpoints. The disturbances are 𝐶𝐶𝐴𝐴𝐴𝐴(𝑡𝑡),𝑇𝑇𝐴𝐴(𝑡𝑡),𝑇𝑇𝑐𝑐𝑐𝑐(𝑡𝑡). a) Calculate the number of degrees of freedom for the system and the maximum number ofcontrolled and manipulated variables. b) Generate a suitable control loop that controls the maximum number of controllable variables. SolutionWe have the following equations in the current system: Dynamic total mass balance (assuming the flow rates 𝐹𝐹(𝑡𝑡) are in m3/s and constant density):( )( ) ( )fdV tF t Ftdt=−Dynamic mass balance for species A (assuming concentrations 𝐶𝐶(𝑡𝑡) are in mol/m3 and the reaction rate𝑟𝑟𝐴𝐴(𝑡𝑡) is in mol/m3-s):( ) ( )()( ) ( )( ) ( ) ( )( )( )(),AfAfAAAd C tV tF tC t FtC t V tr Tt C tdt= −−Dynamic heat balance for the reactor, where 𝛥𝛥𝐻𝐻𝑟𝑟 is the reaction heat in J/mol A reacted and it is assumed that the liquid density and heat capacity are not affected by changes in concentration an