Let's work through **Problem 1** step-by-step, as a helpful tutor, focusing on the derivation, linearization transfer function formulation, and stability analysis for a non-isothermal CSTR (Continuous Stirred Tank Reactor). --- ## **Step 1: Write Nonlinear Dynamic Equations** We need two mass and energy balances for a perfectly mixed CSTR: **Component Balance (A):** \[ \frac{dC_A}{dt} = \frac{F}{V}(C_{A,in} - C_A) - k_ e^{-\frac{E}{RT}} C_A \] where: - \( C_A \) = Concentration of A in reactor - \( C_{A,in} \) = Inlet concentration of A - \( F \) = Volumetric flow rate - \( V \) = Reactor volume - \( k_ \) = Pre-exponential factor - \( E \) = Activation energy - \( R \) = Gas constant - \( T \) = Reactor temperature **Energy Balance:** \[ \frac{dT}{dt} = \frac{F}{V}(T_{in} - T) + \frac{(-\Delta H_{rxn})}{\rho C_p} k_ e^{-\frac{E}{RT}} C_A - \frac{U A}{\rho C_p V}(T - T_{C,in}) \] where: - \( T \) = Reactor temperature - \( T_{in} \) = Inlet temperature - \( T_{C,in} \) = Inlet cooling water temperature - \( U \) = Overall heat transfer coefficient - \( A \) = Heat exchange area - \( \rho \) = Density - \( C_p \) = Heat capacity - \( \Delta H_{rxn} \) = Heat of reaction (exothermic, so negative) --- ## **Step 2: Linearize Around Steady-State** Let steady-state values be \( C_{A,s}, T_s \), etc. Define deviations: \[ c_A = C_A - C_{A,s} \] \[ t = T - T_s \] Linearize each equation using Taylor expansion, keeping only first-order terms. The Jacobian matrix is constructed by evaluating partial derivatives of right-hand sides with respect to \( C_A \) and \( T \) at steady-state. This gives a set of linear ODEs: \[ \frac{d}{dt} \begin{bmatrix} c_A \\ t \end{bmatrix} = J \begin{bmatrix} c_A \\ t \end{bmatrix} + B \cdot u \] where \( J \) is the Jacobian and \( u \) is the input (change in \( T_{C,in} \) or \( F_C \)). --- ## **Step 3: Obtain Transfer Functions** Take Laplace transforms of the linearized equations (assuming zero initial deviations): \[ sC_A(s) = a_{11}C_A(s) + a_{12}T(s) + b_1 U(s) \] \[ sT(s) = a_{21}C_A(s) + a_{22}T(s) + b_2 U(s) \] Solve these algebraic equations to obtain the transfer function from cooling flow rate \( F_C \) (or cooling temperature) to the reactor temperature \( T \): \[ \frac{T(s)}{U(s)} = \frac{\text{Numerator}(s)}{\text{Denominator}(s)} \] where the denominator is the **characteristic equation**. --- ## **Step 4: Stability Analysis using Routh-Hurwitz** The system is stable if all roots of the characteristic equation have negative real parts. The Routh-Hurwitz criterion provides a systematic way to check this by constructing the Routh array and ensuring all first-column elements are positive. --- ## **Final Answer:** The step-by-step solution involves: - Writing the nonlinear mass and energy balance equations for concentration and temperature. - Linearizing these equations around a steady-state to obtain the Jacobian matrix. - Using Laplace transforms to derive the transfer functions relating reactor temperature to the cooling flow rate or jacket temperature. - Formulating the characteristic equation and applying the Routh-Hurwitz criterion to determine conditions for local stability. This analytical sequence allows you to assess the reactor's dynamic behavior and stability in the presence of disturbances or control actions.