Dynamics and Control of Liquid Level in Annular Conical Tank Process: Modelling and Experimental Validation

Department of Chemical Engineering and Technology, Indian Institute of Technology (Banaras Hindu University) Varanasi – 221005, Uttar Pradesh, India; anupamsrivastav.che17@itbhu.ac.in, parmanandm.che17@itbhu.ac.in, rssingh.che@itbhu.ac.in, dprasad.che@itbhu.ac.in Indian Journal of Science and Technology, Vol 12(8), DOI: 10.17485/ijst/2019/v12i8/140486, February 2019 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645


Introduction
Conical tanks are extensively used in the various process industries as its shape provides better drainage of solid mixtures, slurries and viscous liquids.The process nonlinearity in conical tanks is caused by two factors: (a) its constantly varying cross-section and (b) the nonlinear flow resistance. Various works have been published in literature that address the conical tank level control methodologies based on simulation results of selected process models. While the simulation results may offer satisfactory control, thereare deviations when implemented on practical experimental setups. This may be attributed to the dynamics associated with the various components of the setup and also the process noise associated with the instrumentation. The present work addresses both the simulation studies as well as experimental validation on laboratory scale physical setup.
Many researchers have published works on the level control of conical tank processes. In 1 proposed the PID

Nonlinear Conical Tank Liquid Level Process
The experimental setup designed by Apex Innovations 22 is shown in Figure 1 and the technical specification of setup has been shown in Table 1. The control objective is to maintain the liquid level (as measured by a Level Transmitter) at desired steady-state by manipulating the inlet flowrate to the tank. The setup is interfaced to a computer which records all the desired process variables Proposed Closed-loop automatic tuning of PID controller for nonlinear systems. In 4 proposed presents the synthesis and analysis of optimal tuning of (PID) controller tuning parameters for (FOPTD), (SOPTD) systems.
Apart from using simple PID to various modified PID many model predictive control and fuzzy control technique have been studied and published. In 5 proposed Introduced Design Procedure and Simulation result of Internal Model Controller for a Real Time Nonlinear Process. In 6 proposed Model based Controller Design for nonlinear Conical Tank System. In 7 proposed Introduced Optimal Actuation of PI Controller using Predictive Technique for Level Control of Nonlinear Process. In 8 proposed Introduced design procedure of Internal Model Controller to establish PID rules with a well described approach. In 9 proposed Developed nonlinear inferential control (NLIC as a method for improving control of nonlinear systems. In 10 Proposed Model reference Adaptive Control based on neural network for level control of non-linear process. In 11 developed design of a soft computing based controller for level control of non linear conical tank process. In 12 Proposed Online tuning of fuzzy logic controller using Kalman algorithm. In 13 Devloped the technique for level control of conical tanksystem using fuzzy based model predictive controller (FMPC). In 14 Developed a Smart controller for level control of non linear conical tank system using reinforcement learning algorithm and eliminated the drawbacks of PID and fuzzy controllers. In 15 Proposed level control of non linear conical tank using PID controller and fuzzy logic algorithm. In 16 PG Presented a comparative study of PI controller, model reference adaptive controller and fuzzy logic controller for a coupled tank system.
In 17 developed Takagi-Sugeno fuzzy model for direct inverse control of conical tank system. In the present work, the conventional PID controller and IMC based PID Controller [18][19][20] has been implemented on a non linear annular flow conical tank process. Our work is primarily divided into two sections: (a) Experimental and (b) Modeling. Various sections of our work include process description, Mathematical Modelling of annular flow conical tank process, System Identification andstudy of steady-state and dynamic behaviour of the process (based on experimental runs), Controller design techniques 21 and closed loop responses with experimental validation.  and implements the Direct Digital Control action on the process. The setup consists of a reservoir tank, submersible pump, pneumatic control valve (linear, air to close), I to P converter, annular conical tank and level sensor. The schematic process diagram of the setup is shown below in Figure 2. The setup can be operated in two modes:

Process Variables and Parameters
Consider the annular flow conical tank process model as shown in Figure 3.

Model Equations
Consider the differential volume of cone at any height: The maximum height h max can be evaluated using Equation (2) Total volume of cone can be obtained by integrating equation (1) Mass balance around the annulus: Assuming constant density of fluid, Assume Replacing V a From Equation (10), Since dV dh Substitute equation (19) into the mass balance equation (15), where, Equation (20) is the nonlinear differential equation that can be solved for steady-state as well as used for obtaining the dynamic response. From a control perspective, in terms of input-output model, the system can be viewed as a SISO system. The State space model of the system can be represented as: Here the steady state condition is denotes by subscripts s.
The elements of A and B Matrices of state space model can be obtained by linearization of nonlinear model (based on Taylor Series approximation at desired steady state operating point) Evaluating the individual terms separately: From Equation (21), Evaluating value of (a 1 h s + a 2 ) From Equation (25), Using Equation (29B) Equation (28) reduced to, Now, evaluated the second term of (27), Finally, substituted Equations (30) & (33) into Equation (27), From Equation (20), For SISO system, where, (37) can be re-written as, where, the process gain and time constant are defined respectively as: The process has a variable gain and variable time constant and since the Eigen value of A matrix (as described by Equation 34) is negative, the system is stable. Equation (38) suggests that the conical tank system is first order system, characterized by its Capacitance and Resistance defined as η(h s ) and 2 h s b respectively.

System Identification
To begin with, the experimental setup was operated in Manual Mode.The parameters of process transfer function have been identified from experimental results of open loop responses. Since the controller output signal (measured as percentage of the maximum value) happens to be the input signal to the final control element (Pneumatic valve), the linear range of operation of the control valve was obtained in the controller output range of 42% to 80%. The correlation between controller output and liquid inlet flowrate is obtained from the experimental data as shown in Figure 4 as: An analogous correlation of liquid inlet flowrate with the Pneumatic valve stem pressure (input) has been obtained and shown in Figure 5 as: The correlation between the Pneumatic valve stem pressure and the controller output is obtained and shown in Figure 6 as:      Figure 7. The value of beta estimated is 6.03674 LPH/cm^0.5.
Based on Equation 41, the inlet liquid flowrates corresponding to the controller output range of 42% to 80% have been calculated to be in the range of 95.17 LPH to 41.97 LPH.

Steady-state and Dynamic Open Loop Responses
The physical setup was operated at various steady-state operating points by changing the controller output in the range of 42% to 80%. The corresponding steady-state flowrates were calculated and the steady-state heights were measured.
In order to obtain the transient open loop responses, the process was subjected to a small step change in the inlet flowrate by manually varying the controller output from 42% to 44%, 44% to 46% and so on. The transient responses between a pair of initial and final steady-states were used for the identification of Process parameters (time constant and steady-state gain) corresponding to various steady states, as shown in Table 2, using the following methods: (a) Although the process is inherently first order, but experimental results revealed that there is a delay of approximately 5 seconds in the process output. This may be attributed to the dynamics associated with the other components in the setup. Using experimental data, the time constant is calculated by initial slope method and ultimate gain is obtained by the final value theorem based on the equation for a standard first order response: Order plusDead time (FOPDT) model was therefore fitted to the experimental data using the The nonlinear behaviour of the process can be examined from the variation of process gain and time constant at different steady-state operating points, as shown in Table 2. In addition, the process parameters computed by the Mathematical Model (Equations 39 and 40) show significant variation from the other methods. This may be attributed to the fact that the outlet resistance (Beta) calculated by Equation (14A) and Figure 7 may also be varying in real sense.
The variation in process time constant and gain with respect to the steady-state heights (as computed by the above three methods) has been shown in Figures 9 and  10 respectively.

Controller Tuning based on Cohen and Coon Settings
The open loop experimental process reaction curve (data as obtained by operating the setup in Manual Mode) was used to tune the conventional PID controller based on Cohen and Coon settings as described below: Vol 12 (8) | Febraury 2019 | www.indjst.org where, K C , τ 1 and τ D represent the controller parameters and K,τ and θ represent the gain, time constant and dead time of the process transfer function. The PID controller was tuned at four different steady-state heights namely 6.15 cm, 10.62 cm, 14.71 cm and 20.59 cm cor-responding to 25%, 43%, 60% and 84% respectively of the total tank height, as shown in Table 3. The transfer function parameters were taken from the FOPDT model identified earlier.In order to study the servo problem, the setup was operated in Auto mode. The closed loop performance of PID controller was studied at four different steady-states, in terms of quantitative performance indices such as ISE, IAE, ITAE, rise time, settling time and percentage overshoot. Table 4 shows the comparison of controller performance for servo problem, when the process was subjected to step changes in set point. Since the controller was tuned based on a fixed gain and time constant at a specific steady-state operating point, the PID controller performance was not satisfactory for significant step changes in setpoint.

Tuning of IMC PID based Controller
An IMC based PID controller has been designed to overcome the limitations of conventional PID controller. Figure 11 shows the closed loop block diagram of an IMC PID control system.
The parameters of IMC PID controller can be obtained based on the following equations: where, the parameter τ C has been set to the minimum possible value which is equal to the process time constant corresponding to the specified steady-state. The closed loop transfer function of an IMC PID based controller for servo problem is shown below:   Corresponding to the four steady-states described in the section above, the tuning parameters of IMC PID controller and the closed loop transfer functions have been shown in Tables 5 and 6 respectively.

Experimental Validation of Closed Loop Response
In order to compare the performance of IMC PID and conventional PID controllers, the physical setup was      Figure 12 show that the IMC PID controller provides more stabilized and superior performance. In addition to experimental validation, the simulated closed loop responses of the IMC PID and the PID controllers were also obtained using the closed loop block diagram of Figure 11 and based on the FOPDT process transfer function. Figures 13 and 14 show the comparison of simulated and experimental results from the PID and IMC PID controllers respectively.

Conclusion
In this work, the dynamics and control of annular conical tank liquid level nonlinear process has been studied. Mathematical model of the process was developed and the process parameters were validated using the open loop experimental data obtained from the physical setup. The superior performance of IMC PID controller has been verified from both the closed loop simulations and experimental runs.