CHAPTER 2 CONTINUOUS STIRRED TANK REACTOR PROCESS DESCRIPTION

Transcription

1 11 CHAPTER 2 CONTINUOUS STIRRED TANK REACTOR PROCESS DESCRIPTION 2.1 INTRODUCTION This chapter deals with the process description and analysis of CSTR. The process inputs, states and outputs are identified for modeling and control purposes. The I/O characteristics are studied. The dynamic behavior of the CSTR process for parameter variations and noise are analyzed. The nonlinear I/O characteristic of CSTR is linearised by applying Taylor s series. The stability analysis is carried out by determining the eigen values of the system matrix and from the phase plane trajectory. 2.2 PROCESS DESCRIPTION Reactors play a major role in any chemical industry. Chemical reactions in a reactor are either exothermic or endothermic and require that energy can either be removed or added to the reactor to maintain constant temperature. Exothermic reactions are highly nonlinear because of potential safety problems and the possibility of interesting behavior such as multiple steady states. In the present work, the CSTR exhibiting exothermic reaction is considered. The schematic diagram of the CSTR process is shown in figure 2.1. In the CSTR process model, an irreversible exothermic reaction takes place. The heat of the reaction is removed by a coolant medium that flows through a

2 12 jacket around the reactor. A fluid stream is fed to the reactor from the feed tank through a pump. A catalyst is placed inside the reactor which speeds up the reaction and remains chemically unchanged throughout the reaction. The concentration is measured using a conductivity meter. The temperature of the fluid in the tank and the jacket are measured using a thermocouple. Sensor information are fed to the system through a Multi-Input-Multi-Output data acquisition card. The fluid inside the reactor is perfectly mixed and sent out through the exit valve. The jacket has the coolant and is fed from the coolant feed tank through a pump. The jacket is assumed to be at a lower temperature than the reactor in order to make the cooling effective. The actual feed flow rate of the coolant is determined and a variable speed pump is activated to control the manipulated variable. Figure 2.1 Schematic Diagram of CSTR Process

3 13 Using material and energy balance equations, CSTR process dynamics are described as follows d T F i U A k o H E / R T T i T T c T e C d t V V C p C p (2.1) d C F d t V d T d t c i E / R T i o C C k e C (2.2) U A F c T c i T c T c T C p c (2.3) where T and C are the effluent temperature and effluent concentration in the reactor respectively. F i and T i are the input flow rate, considered as disturbances and temperature of the reactant, V is the volume flow rate of the inlet reactants and F c is the coolant flow rate is the manipulated variable/input to the CSTR and T c is the coolant temperature. The process outputs/states are the concentration (C), temperature of the product (T) and coolant temperature (T c ). The proposed work aims to develop linear and nonlinear models and controllers for the CSTR. It is proposed to carry out simulation and verify the performance of the developed models and controllers. Table 2.1 gives the parameters of the CSTR considered in the present work. Table 2.1 Parameters of CSTR Parameter Data F i /V, hr -1 (Flow/Volume) 1 k 0, hr -1 (Frequency factor) 9,703* E, kcal/kgmol (Activation energy) p, kcal/(m 3 K) (Density* molar concentration) 500 T i, K (Feed Temperature) 298 C Ai, kgmol/m 3 (Feed concentration) 10

4 14 UA/V, kcal/(m 3 K hr) (Heat transfer/volume) 150 T c, K (Coolant Temperature) STEADY STATE CHARACTERISTICS The process differential equations are perturbed with various coolant flow rates and the steady state behavior of the CSTR process is obtained. The changes in effluent temperature, effluent concentration and coolant temperature for various coolant flow rates are shown in Figures 2.2, 2.3 and 2.4. Figure 2.2 Temperature vs. Figure 2.3 Concentration vs. Coolant Flow Coolant Flow Figure 2.4 Coolant Temperature vs. Coolant Flow

5 15 For coolant flow rate variations between 2 m 3 /hr m 3 /hr, the CSTR is said to operate in the low region and for variations between 13.8 m 3 /hr -20m 3 /hr, the CSTR is said to operate in the high region. The CSTR operates in the middle region for coolant flow variations between 13.6 m 3 /hr and 13.8 m 3 /hr. The process output variation for coolant flow changes in the low and high regions are small, whereas the outputs posses multiplicity problem in the middle region. It is likely to operate the CSTR at the middle unstable steady state because at low temperature, the concentration is of low yield as the temperature is very low and at high temperatures, steady state may be very high causing unsafe conditions, destroying the catalyst in the rector degrading the product. 2.4 OPEN LOOP DYNAMICS The variation in the output concentration, temperature with change in system parameters and disturbance are discussed in this section. The dynamics of CSTR process subject to changes in system parameters like k 0 coolant temperature and feed concentration and the behavior of CSTR process is analysed Parameter Change On investigation of the material balance equation, it is evident that the reaction rate produces large variation on the effluent concentration. The reaction rate is proportional to the concentration of the input solution. Reaction rate = k o e -E/RT C. (2.4)

6 16 Here, k o is the Frequency factor, E is the Activation energy, R is the Gas constant and T is the Temperature k o Change k o is decided by factors like the frequency of collisions and their orientation. It varies slightly with temperature and can be considered constant over small temperature ranges. Figure 2.5 shows the variation of CSTR effluent concentration and temperature with changes in the process parameter k 0 from the specified actual value. The shape of the process response has not changed but there is a slight shift in response either to the left or to the right. A decrease in k 0 causes a reduction in the frequency of collision thereby making the system to respond in a sluggish manner. An increment in k 0 causes more collisions, as a result the system responds at a faster rate. Figure 2.5 Process Output for Parameter Change in k 0

7 H is the accumulation of total enthalpy of the material in the CSTR. From the laws of thermodynamics, the enthalpy of a liquid system is a function of the temperature and its composition. Figure 2.6 shows the CSTR effluent concentration and temperature with changes in the process parameter from the specified actual value response is not changed but a slight shift in response either to the left or to the a small change in the reaction temperature thereby making the system to respond in a sluggish manner, temperature, as a result the system responds at a faster rate. Figure 2.6 Proce Disturbance CSTR process can be disturbed by introducing changes in coolant temperature and feed concentration

8 Disturbance by change in inlet coolant temperature The coolant temperature acts as one of the disturbances. Figure 2.7 shows the CSTR effluent concentration with a change in the coolant temperature. Change in coolant temperature either accelerates or decelerates the process reaction. An increase in the coolant temperature results in fast response, whereas a decrease in the coolant temperature results in a sluggish response. Figure 2.7 Process Output for Disturbance in Coolant Temperature Disturbance by change in feed concentration When the feed concentration is increased, the response is faster, whereas a decrease in the feed concentration causes a slow response without affecting the characteristics. Figure 2.8 shows the CSTR effluent concentration with a change in the feed concentration.

9 19 Figure 2.8 Process Output for Disturbance in Feed Concentration 2.5 LINEARISATION The CSTR process dynamic Equations (2.1), (2.2) and (2.3) are nonlinear, due to the presence of the terms e -E/RT C, F c T c and F c T ci. The nonlinear equations are linearised around the operating point (C 0, T 0 ) to carry out stability studies. Applying Taylor s series, the nonlinear term e -E/RT C is linearised. e C e C e C ( T T ) e ( C C ) E / R T E / R T E 0 E / R T 0 E / R T R T0 (2.5) The nonlinear terms F c T c and F c T ci are also linearised as follows FcT c FcT c0 Fc 0Tc Fc 0T c0 (2.6) FcT ci FcT ci0 Fc 0Tci Fc 0T ci 0 (2.7) Applying the linearised terms, the Equations (2.1), (2.2) and (2.3) are converted into the deviation variable form,

10 20 ' d C F i E / R T0 ' E E / R T0 ' F i ' k 0 e C k 0 e C 2 0T C i d t V R T0 V (2.8) ' dt E / RT0 ' E E / RT UA Fi 0 ' UA ' Fi ' Jkoe C Jko e C 2 0 T Tc Ti dt RT0 V Cp V V Cp V (2.9) ' dtc ' UA ' UA Fc Tci Tco Fco Tc T dt Cpc Cpc ' (2.10) where J = - H C p (2.11) Here C = C C 0, C i = C i C i0, T = T T 0, T c = T c T c0, T i = T i T i0 F c = F c F c0 where C 0, C i0, T 0, T i0, T c0, F c0 are the states at the operating conditions, C, T, T c are the deviation state variables and C i, T i, F c are the deviation input variables. The normal feed concentration of inlet feed is 10 kgmol/m 3 and the effluent concentration in the reactor is in the range 0 kgmol/m 3 < C < 10 kgmol/m 3. The lower bound for temperature is 298 K, which occurs if there is no reaction. There is a correlation between concentration and temperature. If the concentration of feed is high reaction rate decreases. Hence a small amount of energy is released by reaction and therefore the temperature will not be much different than the feed and jacket temperatures. The linearised I/O characteristics of concentration Vs coolant flow is shown in Figure 2.9.

11 21 Figure 2.9 Linearised I/O Characteristics In order to apply linear control techniques, the operating range is divided into low, middle and high regions. The operating points considered for each linear region are given in Table 2.2. In the middle and high region the amount of heat transferred to the coolant is reduced drastically by the increased flow rate of the coolant Table 2.2 Operating Points in Different Regions Operating Regions F c0 C 0 T 0 Tc 0 Low Region Middle Region High Region The state equations for each linear region applying the linear operating conditions are then formed. The state space matrices found for each region are given in Table 2.3. Table 2.3 State Space Matrices for the Different Regions

12 22 Operating Regions A Matrix B Matrix Low Region Middle Region High Region STABILITY ANALYSIS This section investigates the stability of the three regions by analyzing the eigen values and phase plane trajectory Analysis using Eigen Values The stability analysis of CSTR process is carried out by determining the eigen values of A matrix. The eigen values obtained for the three operating regions are tabulated in Table 2.4. Table 2.4 Eigen Values for the Different Regions of the CSTR Operating Regions Low Region Eigen Values j j

13 23 Middle Region High Region By investigating the eigen values, it is inferred that the low and high regions are stable whereas the middle region is unstable Analysis by Phase Plane Trajectory A phase-plane plot is constructed by performing simulations for a large number of initial conditions. The phase-plane plots generated for different initial conditions are shown in Figure The singular points in the low and high temperature regions are stable while the middle one is unstable. All the initial conditions with low concentration ( kgmol/m 3 ) and relatively low-to-intermediate temperatures (315 K to 365 K) 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. For initial conditions with high concentration (8 kgmol/m 3 ) and low temperature (315 K to 325 K), the phase plane trajectories converges to the low temperature steady-state. When the initial temperature is increased above 325 K, phase portrait converges to the high temperature steady-state. For initial temperatures around 340 K, a very high overshoot in temperature to 425 K occurs, before the system settles down to the high temperature steady-

14 24 state. No initial conditions converge to the intermediate temperature steadystate, since it is unstable. Table 2.5 shows the singularity for various regions. Figure 2.10 Phase Portrait of the CSTR Table 2.5 Singularity for Different Regions Region Low T Intermediate T High T Singularity Stable Node Saddle point Stable Focus 2.7 CONCLUSION On analyzing the open loop I/O characteristics, it is evident that the process has three operating regions: the low region, middle region and high region. Also investigation of the open loop characteristics for changes in parameter and disturbance reveals that the dynamics is preserved with a slight shift in the output on either direction making the response sluggish or fast. The generalized state space model is formulated and the model parameters for the three regions are calculated separately. The eigen values of the state space

15 25 model for each region are found. The eigen values of the low and high region have negative real parts, specifying that those regions are stable and the eigen value of middle region have a positive real part specifying that the region is unstable. The phase plane trajectory is plotted for the CSTR. Analyzing the phase trajectories, it is inferred that the singular points for the low temperature region is a stable node, intermediate temperature is a saddle point and the high temperature region is a stable focus.