Process Dynamics and Control PROJECT: Dynamics and Control of a Chemical Reactor

Consider a jacketed continuously stirred tank reactor (CSTR) shown below:

The following series of reactions takes place in the reactor:

A →B →C

where A to B is a highly exothermic, first-order reaction and B to C is a first-order reaction that generates a negligible amount of heat. The reactions follow an Arrhenius rate law:

r₁ = k₁₀e^-E_1/RTC_A, r₂ = k₂₀e^-E_2/RTCB

where k₁₀ and k₂₀ are the pre-exponential factors (given in units of per time), E₁ and E₂ are activation energies, and R is the gas constant.

The reactor is fed by species A in an inert solvent at temperature T₀, concentration C_A0 and volumetric flow rate F₀. The effluent stream leaves the reactor at concentrations C_A, C_B, C_C, and temperature T. The liquid contents of the reactor are well-mixed (temperature and composition are uniform) and have a constant density (ρ), liquid hold-up volume (V), heat of reactions (?H₁ and ?H₂ for reaction 1 and 2, respectively) and heat capacity (C_p). The jacket is fed by cooling water at temperature T_j0 and flow rate F_j0. The overall heat transfer coefficient and heat transfer area of the jacket are U and A_ht, respectively. The effluent stream leaves the jacket at temperature T_j. The jacket water is assumed to have a uniform temperature.

Process model -

The material and energy balances that describe the dynamic behavior of the process have the following dimensionless form:

 dx₁/dθ = 1 - x₁ - Da₁ exp(E¯₁/x₃)x₁

dx₂/dθ = -x₂ + Da₁ exp(-E¯₁/x₃)x₁ - Da₂ exp(-E¯₂/x₃)x₂                                                  (1)

dx₃/dθ = x₃₀ - x₃ + Da′₁exp(-E¯₁/x₃)x₁ + Da′₂exp(-E¯₂/x₃)x₂ - U¯(x₃ - x₄)

dx₄/dθ = ∈₁∈₂(x₄₀ - x₄) + U¯∈₁∈₃(x₃ - x₄)

where:

x₁ = C_A/C_A0: dimensionless concentration of species A in the reactor

x₂ = C_B/C_A0: dimensionless concentration of species B in the reactor

x₃ = T/T_ref : dimensionless temperature of the reactor

x₄ = T_j/T_ref: dimensionless temperature of the jacket

x₃₀ = T₀/T_ref: dimensionless temperature of reactor inlet stream

x₄₀ = T_j0/T_ref: dimensionless temperature of jacket inlet stream

θ = F₀t/V: dimensionless time

T_ref is the reference temperature and the remaining parameters (D_a1, D_a2, D_a′1, D_a′2, E¯₁, E¯₂, U¯, ∈₁, ∈₂, and ∈₃) are dimensionless process parameters.

Specific Questions -

1. Using mass and energy balances and the assumptions listed in the problem statement above, derive a dynamic model (system of four nonlinear ordinary differential equations for C_A, C_B, T, and T_j ) for the process (no need to express the model in dimensionless variable form). Clearly explain your derivation.

2. For the model of Eq. 1 and for the following set of parameters: D_a1 = 10⁶, D_a2 = 10⁷, Da′₁ = 1.5 × 10⁶, x₃₀ = 0.025, x₄₀ = 0.025, E¯₁ = 1.0066, E¯₂ = 1.0532, U¯ = 8, ∈₁ = 12.5, ∈₂ = 40, ∈₃ = 1, and Da′₂ = 0 (because B → C generates a negligible amount of heat), compute the three steady-states of the nonlinear algebraic system.

3. Derive the linearization of the nonlinear system of Eq. 1 around each of the steady-states and show proof that the CSTR system has two stable steady-states and one unstable steady-state.

4. Using temperature of the reactor as controlled and measured output (i.e., y = x₃ - x_3s where x_3s is the steady-state value), temperature of the jacket inlet stream as manipulated input (i.e., u = x₄₀ - x_40s where x_40s is the steady-state value), and temperature of the reactor inlet stream as disturbance (i.e., d = x₃₀ - x_30s where x30s is the steady-state value), create a block diagram of the closed-loop system.

5. The operation under the unstable steady-state operating condition is preferable. Using the input, output, and disturbance defined in Problem 4, compute the transfer function of the linearized system.

6. On the basis of this transfer function, design a proportional-integral (PI) controller to stabilize the closed-loop system resulting from the linearized system and the controller.

7. Apply the PI controller to the linearized system and the nonlinear system of Eq. 1, for set-point changes in the reactor temperature, y, of magnitude 0.005 and 0.01 (show the plots of the manipulated input and the controlled output for all the simulation runs). Explain your results.

8. Study the disturbance rejection capabilities of the controller, for step changes in the temperature of the reactor inlet stream, d, of magnitude 0.001 and 0.002. Consider both the linearized system and the nonlinear system (show the plots of the manipulated input and the controlled output for all the simulation runs). Explain your results!

9. Apply the PI controller of Problem 6 to the nonlinear process model with the parameters of Problem 1 and Da′₂ = 10⁵. Discuss your results!