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Ind. Eng. Chem. Res. 1997, 36, 122-129
Experimental Implementation of a Recursive Algorithm To Control the Temperature Trajectory of an Exothermic Batch Reactor Pavan K. Shukla, S. Pushpavanam,* Ashok Khanna, and Jeffrey L. Harmon Department of Chemical Engineering, I.I.T., Kanpur 208016, India
In this paper we demonstrate a recursive algorithm to control the temperature profile of a batch reactor sustaining an exothermic reaction. The algorithm suggested here does not assume any prior knowledge of the detailed process, i.e., kinetics, etc. It, however, is different from generic black-box models as it uses an overall energy balance. This balance takes into account the different macroscopic effects, i.e., accumulation, heat generation, and heat loss. The heat generation rate Hgen(t) is estimated from the temperature trajectory and is used to generate a control law for isothermal operation. To achieve temperature control, the rate of heating or cooling was manipulated. The algorithm was successfully implemented experimentally to achieve the temperature control in the hydrolysis of acetic anhydride in a laboratory scale reactor. Heating of the reactor was done electrically and cooling by a coolant pumped by a refrigerated circulator. A PC-386 was used to measure temperature and send control signals. Isothermal operation was achieved by using this algorithm in a few iterations. We discuss how this control action can be used to design a composite controller. The control action is now taken independently by two controllerssone takes care of smooth but large changes in manipulated variable and the other small but rapid fluctuations (or noise) introduced by on line process perturbations. Introduction Batch reactors are used normally in the manufacture of specialty chemicals. Here it is necessary to have precise control over the quality of the end product. Control of the various processes in these units is hence very important. This can also help in preventing runaway or unsafe operation in exothermically reacting systems. Normally the experience of a skilled operator enables him to select and vary operating parameters so as to maintain the process along a desired trajectory. This, however, is prone to human error. It is, hence, desirable to operate a process with an automatic computer based system. The computer tracks the progress of the process and initiates the necessary action to control the process. Batch reactors are inherently dynamic systems. Here, the process conditions vary continuously with time. In contrast to this, continuous systems are usually operated at a steady state. The control of the trajectory of a batch reactor has been discussed extensively in the literature. Lewin and Lavie (1990) have recently investigated the implementation of a desirable trajectory in a batch reactor. The adaptive automatic control of an exothermically reacting system has been studied by Akesson (1987). They developed a control strategy for a chemical reactor sustaining an exothermic reaction. Adaptive internal model control has been applied on a batch polymerization reactor by Takamatsu et al. (1986). These methods use a black-box model to determine the relation between the manipulated variable u and the control variable x. They do not use any information about the detailed chemical or physical process occurring in the system. They are particularly useful in reactions like cellulose acetylation (used in the manufacture of cellulose triacetate or secondary acetate) where the quality of the raw material cellulose as determined by moisture content, * Author to whom correspondence should be addressed. E-mail address:
[email protected]. S0888-5885(95)00556-2 CCC: $14.00
etc., is heterogeneous in a batch and can also vary from batch to batch. In such systems, the kinetics of the reaction is usually not known and control can be implemented effectively by black-box models. These are on line strategies, and they update the manipulated variable at any instant by using the relationship between u and x from the earlier instants of time. The advantage here is that the empirical relationship determining the control parameter keeps getting updated by using the most recent history of the process. The difference between the predicted and the desired value at any instant is used to reduce the difference at later instants. An inherent limitation of this approach is that the model used to determine the relationship between the control variable and the manipulated variable is linear, whereas in most processes the relationship is nonlinear. The method is successful as the linear model is valid in the immediate neighborhood of every point on the trajectory. Variations in process parameters/ conditions may entail large changes in the manipulated variable. These may be impractical to implement. Jarupintusophon and Cabassud (1994) used a realistic model based predictive and adaptive control of a batch reactor. They show how their strategy can be used to estimate the process parameters like the heat of reaction, etc. In this work we discuss a control strategy for the isothermal operation of a batch reactor sustaining an exothermic reaction. An overall energy balance is used to describe the process. This is again a black-box model, and it does not use any information about the detailed chemical process such as its kinetics or its thermodynamics, like the heat of reaction. It uses a lumped energy balance, but the different parameters occurring here have a physical significance. The algorithm estimates the heat generation parameter at every instant of time from a batch run. This information is used to determine the manipulated variable, i.e., heating or cooling required to maintain the desired temperature profile for the next run. This is continued until we attain convergence. The algorithm was successfully © 1997 American Chemical Society
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applied to experimentally control the temperature in the hydrolysis of acetic anhydride. We also simulated the effect of our iterative control algorithm on system behavior. The iterative control scheme determines the manipulated variable using information from a complete run of the system. The manipulated variable variation generated on convergence is imposed on the system by an “outer” controller. We demonstrate how this controller can be used to reduce the load on a second “inner” controller, which can be used to fine tune the process. Isothermal operation is the preferred mode in many batch systems. Marroquin and Luyben (1973) established that the best isotherm for series reactions gives a yield of the desired product which compares favorably with that obtained by imposing an optimal temperature profile. For a single irreversible exothermic reaction isothermal operation is sufficient and possibly one of the best alternatives to ensure safe operation and to prevent runaway.
This is a predictive step. The temperature profile measured may not be the desired profile. This is rectified in the next step, by modifying m(t). (2) The desired temperature profile is imposed in eq 1. The estimated value of Hgen(t) from the previous step is used to obtain a new m(t). This is the correction step where m(t) is corrected to obtain the desired T(t). (3) This value of m(t) is now employed in step 1 to predict Hgen(t). This is used in step 2 to recalculate m(t). This process is repeated until we attain our desired temperature profile. The implementation of this algorithm will now be discussed. The manipulated variable m(t) can be either positive or negative, signifying that the reactor needs heating or cooling, respectively. The temperature is sampled at a fixed interval of time or a fixed sampling frequency. The entire duration of the process is discretized into time intervals equal to the sampling time ∆t. The manipulated variable is determined for each time interval. The choice of sampling time stems from the following criteria (Seborg et al., 1986):
Model and Control Algorithm The physical processes occurring in a batch reactor can be viewed as consisting of the following steps: (1) the energy accumulation term; (2) the heat generation/ consumption due to reaction; (3) the heat loss to the ambient; (4) heat loss to the coolant or heat gained by the system due to heating. The reactor temperature profile is controlled by manipulating the rate of cooling/heating, i.e., the fourth term. A lumped macroscopic balance on the basis of these processes can be written as
VFCp (dT/dt) ) Hgen(t) - UaAa(T - Ta) + m(t)
(1)
The term on the left denotes the energy accumulation term. The first term on the right signifies the heat generation due to the exothermicity of the reaction, the second term denotes the heat loss to ambient, and third term arises due to control action implemented. The different variables are defined in the nomenclature. In this paper we assume that the thermal capacity VFCp and the heat transfer coefficient Ua are determined experimentally. Hgen(t), however, is unknown as we assume no prior knowledge of chemical kinetics. The manipulated variable m(t) is to be determined such that the desired control objective is met. In this work this objective is the isothermal operation of the reactor. We further assume that the reactor temperature T is the only process parameter that can be experimentally measured. At this stage it is necessary to emphasize that the control algorithm discussed is sufficiently general and can be implemented on general time dependent temperature trajectories. These trajectories are usually determined such that they ensure optimum performance, as in series reactions. Here, the selectivity of the intermediate product is usually optimized. As mentioned earlier, it is possible to operate these systems at a “best” isothermal temperature. Here, the performance is very close to that obtained by imposing an optimal temperature profile. The control algorithm we propose consists of two steps. (1) Estimating Hgen(t) for a fixed m(t). This is done by measuring T(t), for a given m(t). The accumulation term is measured from the experimental temperature profile and Hgen(t) obtained from the energy balance.
τ/15 < ∆t < τ/4 This criteria is applicable to continuous systems where τ is chosen as 95% of the settling time of the process. (1) Heating Action (m(t) > 0). The reactor was placed on a heating mantle. The heating is done electrically. The heating rate was determined as qh (cal/s) and was maintained constant at this value. The m(t) value determined for an interval can be different from this heating rate. To ensure that the reactor is subjected to the required heating rate of m(t) (cal/s) for a given sampling period, we used a time proportioning method to determine the fraction of the sampling period for which the heating should be on. Here, instead of heating the reactor at a constant rate of m(t) for the entire sampling period, we heat it at a rate of qh (cal/s) for a time fraction th which is determined as
th ) (|m(t)|∆t)/qh
(2)
This ensures that the total amount of heat supplied during a sampling period is equal to that computed by the algorithm. (2) Cooling Action (m(t) < 0). The cooling of the reactor is accomplished by circulating cold water from a refrigerated circulator. This pumps water at a fixed temperature to the reactor cooling coil. For an isothermal reactor operation at Ts, neglecting the rise of the coolant temperature, the cooling rate qc is
qc ) UcAc(Ts - Tc)
(3a)
Here, Uc represents the heat transfer coefficient between the cooling coil and the reactor. Here, again, the flow rate of the coolant and its temperature are assumed as being maintained constant. This ensures a constant rate of cooling. Using a time proportioning algorithm as described for the heating action, the time tc for which the reactor is to be cooled is obtained from
tc ) (|m(t)|∆t)/qc
(3b)
To avoid runaway and ensure safe operation, it is necessary that tc should always be less than ∆t. This is assured if the maximum value of Hgen(t) satisfies
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software sends a signal of 5 V to relay 2. This in turn opens the solenoid valve for tc seconds and removes the required m(t) (cal/s) for that time interval. Experimental Procedure
Figure 1. Experimental setup for control of exothermic batch reactor.
max Hgen(t) < UcAc(Ts - Tc)
(4)
This condition enables us to select Tc such that the cooling provided is sufficient. Equations 2 and 3 are not applicable when th and tc are greater than ∆t. This occurs if the heating and cooling actions are not designed properly. In this case we should supply qh or qc (cal/s) for the entire duration of sampling time. Equations 2 and 3 are directly applicable as long as th and tc are both less than ∆t. Experimental Setup The experimental setup is shown in Figure 1. A glass reactor of volume 1 L is used to carry out the reaction. The reactants were well-stirred with a motor driven stirrer. A K-type thermocouple is used to sense the temperature of the reactor. The analog signal from the thermocouple is fed to a signal conditioning module where the signal is converted into a digital mode via an A/D card. This is fed to a data acquisition system (RTI-815). The temperature from a batch run of the process is sampled, measured, and stored in a file. We choose for our batch system the time constant τ to be the characteristic reaction time tr defined in Shukla and Pushpavanam (1994). For our system tr is approximately 600 s. The criteria from Seborg discussed earlier yields that the sampling time should lie between 40 and 150 s. For our experiments we have used a sampling time of 10 s. We carried out numerical simulation of the algorithm for various sampling times. No significant variation in the calculated m(t) was found when we increased it to 150 s. The manipulated variable showed a drastic change when it was raised beyond this value. The control action taken over each sampling time is either heating or cooling. We now discuss how the control actions were implemented. The PC is used to send the signals to relays through D/A cards. The heating action is initiated by closing relay 1, and the solenoid valve (on the cooling water line) is activated by closing relay 2. The software uses the discretized temperature trajectory for a fixed imposed manipulated variable m(t) to estimate the rate of heat generation Hgen(t) from eq 1. The temperature derivative was approximated numerically using a first order forward finite difference. It also corrects the value of m(t) as discussed in the earlier section. The m(t) generated is also in the form of a time series. The software uses this m(t) to trigger the required control step. For m(t) > 0.0, the software sends a control signal of 5 V to relay 1 for th s. This ensures that the reactor is heated at qh (cal/s) for th seconds. For m(t) < 0.0, the
The heat transfer rate to the ambient UaAa was determined by monitoring the temperature profile of 750 cm3 of water. The water was heated to an initial temperature of 55 °C. It was well-stirred and allowed to cool to the ambient temperature. The heat transfer coefficient term UaAa was estimated from the temperature profile as 0.15 cal/s/K. The average heating rate qh as supplied by the heating mantle was determined by measuring the temperature rise of 750 cm3 of water over a period of 5 min. This was found to be 28.0 cal/s. The heat transfer coefficient to the cooling coil UcAc was determined by heating 750 cm3 water to 60 °C. The solenoid valve was kept fully open, and refrigerated water was circulated. From a plot of temperature profile vs time the UcAc was determined as 2.15 (cal/s)/ K. These experimentally measured quantities are necessary to obtain the different “macroscopic” parameters characterizing the system. They can be measured easily and do not require any detailed knowledge of the reaction as they describe macroscopic processes occurring in the system. They are used to obtain the control variables th and tc (see eqs 2 and 3). The reaction system chosen for study is the uncatalyzed hydrolysis of acetic anhydride. This reaction is highly exothermic and exhibits runaway characteristics. It is sufficiently slow, however, and it is therefore an ideal system from the point of control studies. The choice of the glass reactor ensures us that the ambient heat loss is sufficiently low and the reactor exhibits a significant temperature rise when no control action is taken. Since it is an irreversible reaction, no criteria exists which allows us to determine optimum temperature trajectory. It is necessary to ensure, however, that the temperature does not rise significantly, leading to runaway behavior. The desired temperature profile is chosen as a constant temperature of 40 °C. Every reaction run consisted of mixing 187.5 mL of acetic anhydride and 562.5 mL of water. The temperature T was measured using the thermocouple for a predetermined m(t). The software was used to estimate Hgen(t) and recalculate m(t) as described. This m(t) was then imposed and the new temperature trajectory determined experimentally. This process was continued iteratively until we attain isothermal operation. Results and Discussion Figure 2 depicts the temperature trajectory of the batch reactor from an initial temperature of 28 °C, when no control action is taken, i.e., m(t) ) 0.0. The temperature rises to a maximum of 60 °C in approximately 16 min. Hgen(t) is estimated from this profile, and we determine an m(t). The required m(t) for isothermal operation at 40 °C is shown in Figure 2 as a dashed curve (the plot is of -m(t) with t). The control signal generated by m(t) was now imposed on the reactor. The temperature profile obtained with this m(t) and an initial temperature of 28 °C is shown in Figure 3. The control signal m(t) generated by this temperature profile is shown in Figure 3 as a broken curve.
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Figure 2. Temperature trajectory (solid line) for m(t) ) 0.0 and the calculated m(t) (broken line) for isothermal operation from it.
Figure 3. Temperature trajectory (solid line) for m(t) used from Figure 2. The broken line is m(t) calculated for isothermal operation using this trajectory.
To compensate for the initial transient period where the temperature rises 28-40 °C over about 8.5 min, we preheated the reaction mixture to 40 °C. This also eliminates the singularity which exists at t ) 0 for isothermal operation at 40 °C, when the initial temperature is distinct from 40 °C. The m(t) shown in Figure 3 is used, and the resulting temperature profile shows an overshoot from 40 °C (Figure 4), because the reaction rate drastically increases when the initial temperature is changed from 28 to 40 °C. The m(t) calculated using 28 °C as the initial value from Figure 3 is unable to account for this large increase in reaction rate. The m(t) generated from this temperature profile is shown in Figure 4 as a broken curve. Using this m(t), we obtain an almost isothermal batch operation. This is shown in Figure 5, where the initial temperature was 37 °C. From the temperature profile in Figure 5, Hgen(t) was estimated from m(t). This is shown in Figure 6 as a solid line. The kinetics of this reaction was studied by Shukla and Pushpavanam (1994). Using this kinetics, Hgen(t) for isothermal operation can be determined theoretically. This is plotted in Figure 6 as a broken line. The two curves agree well to within experimental accuracy. This confirms the successful implementation of the control algorithm.
Figure 4. Temperature trajectory (solid line) for m(t) imposed from Figure 3. The broken line is for isothermal operation calculated from this trajectory.
Figure 5. Isothermal operation using m(t) as calculated in Figure 4.
The iterative algorithm was also simulated numerically on a digital computer. The temperature trajectories generated iteratively are depicted in Figure 7a,b. The figure establishes the convergence of the temperature trajectory to the desired trajectory. The temperature trajectories we see approach the set point trajectory by oscillating around it as we saw experimentally. Parts a and b of Figure 8 represent the manipulated variable profile imposed on each run as generated by the iterative algorithm. We have plotted the manipulated variable profiles generated by the first five temperature trajectories. The manipulated variable profile has converged, so we do not show the sixth and seventh m(t) curves. We have also studied the action of a proportional controller on the batch system. The evolution of the temperature is now governed by
VFCp (dT/dt) ) Hgen(t) - UaAa(T - Ta) + Kc(Tsp - T) (5) where Kc is the controller gain. The controlled temperature trajectory and the corresponding manipulated variable are depicted in Figure 9a,b for Kc ) 0.1. It can
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Figure 6. Comparison of the experimentally measured heat generation rate (solid line) with theoretically predicted rate (broken line).
Figure 8. Manipulated variable determined by numerical simulation of the outer loop control only generated by temperature profiles in Figure 7 represented by iterations (a, top) 1-3 and (b, bottom) 4 and 5.
Figure 7. Temperature trajectories determined by numerical simulations of outer loop control: (a, top) iterations 1-4; (b, bottom) iterations 5-7.
be seen that this classic controller is unable to control the trajectory accurately in the initial few minutes of the run. We now discuss how a composite controller can be designed which is based on the iterative algorithm described here as well as on the direct action of the proportional controller. The iterative algorithm can be used to estimate the macroscopic changes in the ma-
nipulated variable required to meet the control objective for a process. These changes, though large, are smooth. The proportional controller helps to control disturbances introduced into the process during a particular batch run. The iterative control algorithm essentially generates an off line manipulated variable profile. So, this can be used to design the final control elements (the heating and the cooling elements) so that the control action is satisfactory. The proportional controller can be viewed as an inner loop control, and the iterative controller, as an outer loop control. The control action in this composite controller is split between these two controllers. The outer controller action is predetermined by the previous batch run and is not influenced by the current run. The inner loop control, on the other hand, is on line and takes care of minor disturbances in a process. This could be due to fluctuation or noise or variations in process conditions which cannot be avoided. This kind of composite control is best suited for example in agro based industries, where the quality of raw material is prone to variations and their effect on the process cannot be quantified accurately. In such situations, the inner proportional controller being on line takes into account these deviations in a particular run and maintains the system along the desired trajectory. The outer loop, on the other hand, imposes a manipulated variable variation which takes into account the system information from the previous run. A block diagram demonstrating the design of the composite controller is shown in Figure
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Figure 9. Control action of proportional controller with Kc ) 0.1: (a, top) temperature profile; (b, bottom) manipulated variable.
Figure 10. Block diagram of composite controller.
10. The composite control can be designed such that the outer loop control action gets modified and updated after each batch run. Here, the action of the inner controller can be transferred to the outer controller. This would be helpful, especially when there is an intentional smooth change in the operating parameters of a batch run. The updating of the outer controller keeps the load on the inner controller at a minimum. We have simulated the combined effect of the two controllers on the system. This is again an iterative step. Here, at the end of each iteration, we transfer the action of inner loop control to the outer loop. The outer control loop keeps getting modified at the end of each batch run, in a cumulative manner. The performance of this composite controller will be discussed now. In the first run the inner loop is switched off. The temperature profile of the adiabatic batch run is used to determine the control action of the manipulated variable. This is the same as that depicted in Figure 8a and is reproduced in Figure 11a. For the next run the outer loop controller imposes this manipulated variable on the system. The inner controller is now activated and takes into account minor deviations from
Figure 11. Manipulated variable determined by outer and inner controller during composite action imposed on (a, top) second run and (b, bottom) third run.
the desired set point trajectory. The manipulated variable determined by the proportional control for the entire run is shown in Figure 11a along with the manipulated variable imposed by the outer loop. The control action (manipulated variable) of the inner loop is transferred (added on) to the existing action of the outer loop for the next batch run. During this transfer and updating, we also take into account deviations from the set trajectory. The control scheme is repeated. The outer loop imposes this updated variation of the manipulated variable on the process. For every successive run both inner and outer loops are active. The manipulated variables determined by each controller are shown in Figure 11b. The set of temperature trajectories generated by the composite scheme is shown in Figure 12. The control scheme of the composite control system converges in fewer iterations. The manipulated variable profile imposed on the second and third temperature trajectories are shown in Figure 11a,b. There are different alternatives available by which the action of this composite controller can be started. For example, the first run can be such that the outer loop is switched off and only the inner loop is active. The second run would then be such that the outer loop imposes the m(t) as determined by the inner controller and will also incorporate the effect of the temperature deviation in the first run. The inner controller in the second run would rectify any further errors in the temperature trajectory. For the next run we can transfer the inner controller action to the outer control and continue the iterations. We have implemented this
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variable can be used to design the control system, the cooling coil, etc. The inner loop control design is not difficult as the manipulated variable changes are negligible. A more challenging task would be of scale up. How does one determine the manipulated variable imposed by the outer loop on a full scale plant? We would have to include additional thermal capacitances to make the model more realistic. Nomenclature
Figure 12. Temperature trajectories obtained by composite control action.
scheme and other alternatives, have simulated the behavior of the control algorithm numerically, and have found it to converge to the desired trajectory. The results of these simulations are not presented as we fear they would be repetitive and would not contain any new insight. Conclusions The control algorithm described in this paper is very versatile as it does not require any detailed information about the process occurring in the system. It employs only a macroscopic energy balance from a batch run to determine the manipulated variable. The method of successive substitution is a classical method used to solve nonlinear equations iteratively. The outer control algorithm proposed here is based on the same principle except that we now iterate on a parameter to obtain a desired solution (rather than iterate on the dependent variable for a fixed parameter). The iterative algorithm in this respect is similar to classical adaptive control. The latter is an on line algorithm. Here, the m(t) value is determined on the basis of the immediate history of the problem in a batch run. Our iterative algorithm generates realistic system information from a run and uses it to control the process in a desired manner in the next run. Our algorithm is not based on any linearization of a model and hence can be viewed as a nonlinear control. The outer loop control is essentially off line in its action. The incorporation of the inner and outer loop control makes the implementation on line. The inner control is a feedback control which uses the information in a run to influence the progress of that run. The outer control is also a feedback control and uses the information from a run to influence the progress of the next run. The structure of the controller is such that the inner control takes into account minor deviations of the trajectory and the outer control imposes significant variations of the manipulated variable. The algorithm can be easily used to deal with more complex systems like reactors sustaining multiple reactions. It is also possible to implement a time-varying trajectory in a reactor using this algorithm. We have discussed the issue of the convergence of the iterative control scheme in the Appendix of this paper. The proof is based on the principle of contraction mapping. The iterative control algorithm proposed is essentially an off line process. The estimate of the manipulated
Cp ) heat capacity of reactants (cal/(g °C)) Hgen(t) ) rate of heat generation (cal/s) m(t) ) manipulated variable (cal/s) qh ) heating rate of heater (cal/s) T ) temperature of reactants (°C) Tc ) inlet temperature of cooling water (°C) Ta ) ambient temperature (°C) Ts ) set point temperature (°C) t ) time (s) tc ) cooling time (s) th ) heating time (s) UaAa ) overall heat transfer rate to ambient (cal/(s °C)) UcAc ) overall heat transfer rate to cooling coil (cal/(s °C)) V ) volume of reactor (cm3) Greek Letters F ) density of reactants (g/cm3)
Appendix In this appendix we establish the convergence of the iterative scheme proposed in this work. The proof is based on the principle of contraction mapping. We restrict ourselves to analyzing a one-dimensional system. The iterative algorithm can be represented as
T˙ n + f(Tn) ) un-1 un ) f(Tn)
(A1)
This implies
T˙ n+1 + f(Tn+1) ) f(Tn) T˙ n + f(Tn) ) f(Tn-1)
(A2)
Let
νn ) Tn+1 - Tn represent the difference between two successive iterates. Clearly
ν˘ n + (f(Tn+1) - f(Tn)) ) (f(Tn) - f(Tn-1)) Multiplying throughout by νn exp(-2ηt), we obtain 2 1 d(νn exp(-2ηt)) + ηνn2 exp(-2ηt) + 2 dt νn exp(-2ηt)(f(Tn+1) - f(Tn)) ) νn exp(-2ηt)(f(Tn) - f(Tn-1))
Now we assume that f satisfies the Lipschitz condition, with Lipschitz constant L,
|f(Tn+1) - f(Tn)| < L|Tn+1 - Tn|
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|f(Tn+1) - f(Tn)| < L|νn|
∫0∞(Tn+m - Tn+m-1)2 exp(-2ηt) dt e Sm∫0∞(Tn Tn-1)2 exp(-2ηt) dt
Clearly
|f(Tn+1) - f(Tn)| > -L|νn| Using these inequalities and invoking the CauchySchwarz inequality, we obtain
This implies that we have a contraction, i.e., the difference in the successive elements of {Tn} decreases. This establishes that the sequence converges to the fixed point of (A2). Hence, the control scheme described can maintain an isothermal trajectory. Literature Cited
2
L 1 d(νn exp(-2ηt)) + (η - L)νn2 exp(-2ηt) e (νn2 + 2 dt 2 2 νn-1 ) exp(-2ηt) or 2 1 d(νn exp(-2ηt)) 3L 2 + ην exp(-2ηt) e 2 dt 2 n L ν 2 exp(-2ηt) 2 n-1
(
)
Integrating both sides with respect to “t” from 0 to ∞, we obtain
∫0∞νn2 exp(-2ηt) dt e ∫0∞νn-12 exp(-2ηt) dt
L 2(η - (3L/2)) For η sufficiently large
S) clearly
L
