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The most important unit operation in a chemical process is generally a chemical reactor. Chemical reactions are either exothermic (release energy) or endothermic (require energy input) and therefore require that energy either be removed or added to the reactor for a constant temperature to be maintained. Exothermic reactions are the most interesting systems to study because of potential safety problems (rapid increases in temperature, sometimes called "ignition" behavior) and the possibility of exotic behavior such as multiple steady-states (for the same value of the input variable there may be several possible values of the output variable).
In this module we consider a perfectly mixed, continuously stirred tank reactor (CSTR), shown in Figure 1. The case of a single, first-order exothermic irreversible reaction, A --> B. We will show that very interesting behavior that can arise in such a simple system.
In Figure 1 we see that a fluid stream is continuously fed to the reactor and another fluid stream is continuously removed from the reactor. Since the reactor is perfectly mixed, the exit stream has the same concentration and temperature as the reactor fluid. Notice that a jacket surrounding the reactor also has feed and exit streams. The jacket is assumed to be perfectly mixed and at a lower temperature than the reactor. Energy then passes through the reactor walls into the jacket, removed the heat generated by reaction.
There are many examples of reactors in industry similar to this one. Examples include various types of polymerization reactors, which produce polymers that are used in plastic products such as polystyrene coolers or plastic bottles. The industrial reactors typically have more complicated kinetics than we study in this module, but the characteristic behavior is similar.
The Modeling Equations
For simplicity we assume that the cooling jacket temperature can be directly manipulated, so that an energy balance around the jacket is not required. We also make the following assumptions
- Perfect mixing (product stream values are the same as the bulk reactor fluid)
- Constant volume
- Constant parameter values
The constant volume and parameter value assumptions can easily be relaxed by the reader, for further study.
Parameters and Variables
A Area for heat exchange
CA Concentration of A in reactor
CAf Concentration of A in feed stream
cp Heat capacity (energy/mass*temperature)
F Volumetric flowrate (volume/time)
k0 Pre-exponential factor (time-1)
R Ideal gas constant (energy/mol*temperature)
r Rate of reaction per unit volume (mol/volume*time)
T Reactor temperature
Tf Feed temperature
Tj Jacket temperature
Tref Reference temperature
U Overall heat transfer coefficient (energy/(time*area*temperature))
V Reactor volume
DE Activation energy (energy/mol)
(-DH) Heat of reaction (energy/mol)
r Density (mass/volume)
The parameters and variables that will appear in the modeling equations are listed
Overall material balance
The rate of accumulation of material in the reactor is equal to the rate of material in by flow-the material out by flow.
Balance on Component A
The balance on component A is
where r is the rate of reaction per unit volume.
The energy balance is
where Tref represents an arbitrary reference temperature for enthalpy.
State Variable form of Dynamic Equations
We can write (1) and (2) in the following state variable form (since dV/dt = 0)
where we have assumed that the volume is constant. The reaction rate per unit volume (Arrhenius expression) is
where we have assumed that the reaction is first-order.
The steady-state solution is obtained when dCA/dt = 0 and dT/dt = 0, that is
To solve these two equations, all parameters and variables except for two (CA and T) must be specified. Given numerical values for all of the parameters and variables we can use Newton's method (chapter 3) to solve for the steady-state values of CA and T. For convenience, we use an ësí subscript to denote a steady-state value (so we solve for CAs and Ts).
We noted in the previous section that were three different steady-state solutions to the case 2 parameter set. Here we wish to study the dynamic behavior under this same parameter set. Recall that numerical integration techniques were presented in chapter 4.
The m-file to integrate the modeling equations iscstr_dyn.m, shown in Appendix 2. The command to integrate the equations is
[t,x] = ode45('cstr_dyn',t0,tf,x0);
wheret0is the initial time (usually 0),tfis the final time,x0is the initial condition vector.tis the time vector andxis the state variable solution vector. Before performing the integration it is necessary to define the global parameter vectorCSTR_PAR. To plot only concentration or temperature as a function of time, useplot(t,x(:,1))andplot(t,x(:,2)), respectively.
Initial condition 1
Here we use initial conditions that are close to the low temperature steady-state. The initial condition vector is [ conc , temp] = [9,300]. The curves plotted in Figure 2 show that the state variables converge to the low temperature steady-state.
Initial condition 2
Here we use initial conditions that are close to the intermediate temperature steady-state. The initial condition vector for the solid curve in Figure 3 is [conc, temp] = [5,350], which converges to the high temperature steady-state. The initial condition vector for the dotted curve in Figure 3 is [conc, temp] = [5,325], which converges to the low temperature steady-state.
If we perform many simulations with initial conditions close to the intermediate temperature steady-state, we find that the temperature always converges to either the low temperature or high temperature steady-states, but not the intermediate temperature steady-state. This indicates to us that the intermediate temperature steady-state isunstable. This will be shown clearly by the stability analysis in section 5.
Initial condition 3
Here we use initial conditions that are close to the high temperature steady-state. The initial condition vector is [conc, temp] = [1,400]. The curves plotted in Figure 4 show that the state variables converge to the high temperature steady-state.
In this section we have performed several simulations and presented several plots. In section 6 we will show how these solutions can be compared on the same ìphase planeî plot.
Linearization of Dynamic Equations
The stability of the nonlinear equations can be determined by finding the following state-space form and determining the eigenvalues of theA(state-space) matrix.
The nonlinear dynamic state equations (1a) and (2a) are
let the state, and input variables be defined in deviation variable form
Performing the linearization, we obtain the following elements forA
where we define the following parameters for more compact representation
From the analysis presented above, the state-space A matrix is
The stability characteristics are determined by the eigenvalues ofA, which are obtained by solving det (lI-A) = 0.
the Eigen values are the solution to the second-order polynomial
l2-(trA)l+det (A) =0(13)
The stability of a particular operating point is determined by finding theAmatrix for that particular operating point, and finding the Eigen values of the A matrix.
Here we show the Eigen values for each of the three case 2 steady-state operating points.
Input / Output Transfer Function Analysis
The input-output transfer functions can be found from
where the elements of theBmatrix corresponding to the first input (u1 = Tj-Tjs) are
the reader should find the elements of the B matrix that correspond to the second and third input variables (see exercise 8)
Here we show only the transfer functions for the low temperature steady-state for case 2. The input/output transfer function relating jacket temperature to reactor concentration (state 1) is
and the input/output transfer function relating jacket temperature to reactor temperature (state 2) is
Notice that the transfer function for concentration is a pure second-order system (no numerator polynomial) while the transfer function for temperature has a first-order numerator and second-order denominator. This indicates that there is a greater ìlagî between jacket temperature and concentration than between jacket temperature and reactor temperature. This makes physical sense, because a change in jacket temperature must first affect the reactor temperature before affecting the reactor concentration.
In section 4 we provided the results of a few dynamic simulations, noting that different initial conditions caused the system to converge to different steady-state operating points. In this section we construct a phase-plane plot by performing simulations for a large number of initial conditions.
The phase-plane plot shown in Figure 6 was generated usingcstr_run.mandcstr.mfrom the appendix. Three steady-state values are clearly shown; 2 are stable (the high and low temperature steady-states, shown as ëoí), while one is unstable (the intermediate temperature steady-state, shown as ë+í). Notice that initial conditions of low concentration (0.5 kgmol/m3) and relatively low-to-intermediate temperatures (300 to 365 K) all converge to the low temperature steady-state. When the initial temperature is increased above 365 K, convergence to the high temperature steady-state is achieved.
Now, consider initial conditions with a high concentration (9.5 kgmol/m3) and low temperature (300 to 325 K); these converge to the low temperature steady-state. Once the initial temperature is increased to above 325 K, convergence to the high temperature steady-state is achieved. Also notice that, once the initial temperature is increased to around 340 K, a very high overshoot to above 425 K occurs, before the system settles down to the high temperature steady-state. Although not shown on this phase-plane plot, higher initial temperatures can have overshoot to over 500 K before settling to the high temperature steady-state. This could cause potential safety problems if, for example, secondary decomposition reactions occur at high temperatures. The phase plane analysis then, is able to ìpoint-outî problem initial conditions.
Also notice that no initial conditions have converged to the intermediate temperature steady-state, since it is unstable. The reader should perform an eigenvalue/eigenvector analysis for theAmatrix at each steady-state (low, intermediate and high temperature) (see exercise 3). You will find that the low, intermediate and high temperature steady-states have stable node, saddle point (unstable) and stable focus behavior (see chapter 13), respectively.
It should be noted that feedback control can be used to operate at the unstable intermediate temperature steady-state. The feedback controller would measure the reactor temperature and manipulate the cooling jacket temperature (or flowrate) to maintain the intermediate temperature steady-state. Also, a feedback controller could be used to make certain that the large overshoot to high temperatures does not occur from certain initial conditions.
Understanding Multiple Steady-state Behavior
In previous sections we found that there were three steady-state solutions for case 2 parameters. The objective of this section is to determine how multiple steady-states might arise. Also, we show how to generate steady-state input-output curves that show, for example, how the steady-state reactor temperature varies as a function of the steady-state jacket temperature.
Heat generation and heat removal curves
In section 3 we used numerical methods to solve for the steady-states, by solving 2 equations with 2 unknowns. In this section we show that it is easy to reduce the 2 equations in 2 unknowns to a single equation with one unknown. This will give us physical insight about the possible occurance of multiple steady-states.
Solving for Concentration of A as a function of Temperature
The steady-state concentration solution (dCA/dt) = 0) for concentration is
We can rearrange this equation to find the steady-state concentration for any given steady-state reactor temperature, Ts
Solving for Temperature
The steady-state temperature solution (dT/dt = 0) is
The terms in (17) are related to the energy removed and generated. If we multiply (17) by VrCp we find that
Energy Removed by flow and heat exchange Heat Generated by reaction
Note the form of Qrem
Notice that this is an equation for a line, where the independent variable is reactor temperature (Ts). The slope of the lineis and the intercept is. Changes in jacket or feed temperature shift the intercept, but not the slope. Changes in UA or F effect both the slope and intercept.
Now, consider the Q gen term
Substituting (16) into (20), we find that
Equation (21) has a characteristic S shape for Q gen as a function of reactor temperature.
From equation (18) we see that a steady-state solution exists when there is an intersection of the Q rem and Q gen curves.
Effect of Design Parameters
In Figure 6 we show different possible intersections of the heat removal and heat generation curves. If the slope of the heat removal curve is greater than the maximum slope of the heat generation curve, there is only one possible intersection (see Figure 6a). As the jacket or feed temperature is changed, the heat removal lines shifts to the left or right, so the intersection can be at a high or low temperature depending on the value of jacket or feed temperature.
Notice that as long as the slope of the heat removal curve is less than the maximum slope of the heat generation curve, there will always be the possibility of three intersections (see Figure 6b) with proper adjustment of the jacket or feed temperature (intercept). If the jacket or feed temperature is changed, the removal line shifts to the right or left, where only one intersection occurs (either low or high temperature). This case is analyzed in more detail in section 7.3.
Multiple Steady-State Behavior
In Figure 7 we superimpose several possible linear heat removal curves with the S-shaped heat generation curve. Curve A intersects the heat generation curve at a low temperature; curve B intersects at a low temperature and is tangent at a high temperature; curve C intersects at low, intermediate and high temperatures; curve D is tangent to a low temperature and intersects at a high temperature; curve E has only a high temperature intersection. Curves A, B, C, D and E are all based on the same system parameters, except that the jacket temperature increases as we move from curve A to E (from equation (7) we see that changing the jacket temperature changes the intercept but not the slope of the heat removal curve). We can use Figure 7 to construct the steady-state input-ouput diagram shown in Figure 8, where jacket temperature is the input and reactor temperature is the output. Note that Figure 8 exhibits hysteresis behavior, which was first discussed in chapter 15.
The term hysteresis is used to indicate that the behavior is different depending on the direction that the inputs are moved. For example, if we start at a low jacket temperature the reactor operates at a low temperature (point 1). As the jacket temperature is increased, the reactor temperature increases (points 2 and 3) until the low temperature limit point(point 4) is reached. If the jacket temperature is slightly increased further, the reactor temperature jumps (ignites) to a high temperature (point 8); further jacket temperature increases result in slight reactor temperature increases.
Contrast the input-output behavior discussed in the previous paragraph (starting at a low jacket temperature) with that of the case of starting at a high jacket temperture. If one starts at a high jacket temperature (point 9) there is a single high reactor temperature, which decreases as the jacket temperature is decreased (points 8 and 7). As we move slighly lower than the high temperature limit point (point 6), the reactor temperature drops (also known asextinction) to a low temperature (point 2). Further decreases in jacket temperature lead to small decreases in reactor temperature.
The hysteresis behavior discussed above is also known asignition-extinctionbehavior, for obvious reasons. Notice that region between points 4 and 6 appears to be unstable, because the reactor does not appear to operate in this region (at least in a steady-state sense). Physical reasoning for stability is discussed in the following section.
Conclusion and future work
Finally the conclusion is that a small study on the continuous stirred tank reactor and its model equation after going through we come to know its importance in the chemical engineering field and also its significance as a chemical reactor
The future work is that we have to calculate and prove the equation of the continuous stirred tank reactor using Laplace transformation and check it using the MATLAB he equation of the continuous stirred tank reactor using Laplace transformation and check it using the MATLAB