Mathematical Modelling Of Variable Delay System

Acknowledgement

We take this opportunity to express our profound gratitude and deep regards to our guide Prof. SnehSoni for his exemplary guidance, monitoring and constant encouragement throughout the course of this project. A special gratitude we give to the panels for stimulating suggestions especially in our project presentation that has improved our presentation skills thanks to their comment and advices. Furthermore I would also like to acknowledge with much appreciation the crucial role of the staff of the Instrumentation and Control Department, who gave the permission to use all required equipment and the necessary materials to complete our project.

Abstract

This work aims at researching on a cylindrical tank and the variable time delay system to detect temperatures along the long length of the output pipeline. Such model is intended to increase the level of optimisation and preservation of set point by taking suitable measure in various processes and industries to counter the effect of delay caused due to time. A Matlab SIMULINK model is created and is taken as a replica of the actual industrial model.

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A real time transfer function is derived and the values are input accordingly. Correct results could end up in optimisation of the process in the sense by increasing efficiency, monitoring and decreasing the response time of the system which is a positive for the applicant. Time delay and Transport delay are two major delays that have been addressed by the simulating model created with the help of MATLAB.

Contents

Chapter 1: derivation of transfer function

Deriving the transfer function of a cylindrical tank with a heater.

There are two pipelines one for each inlet and outlet.

Volumetric tank-v

Height-h

Radius-r

c=specific heat

w=constant

Tref=ref. temp.

d=density of liquid

[temperature inside tank]-[temperature outside tank]+[thermal energy(heater)] = rate of accumulation inside the tank.

Chapter 2: mathematical modelling using Simulink

Chapter 3: variable time delay

The Variable Transport Delay and Variable Time Delay blocks appear as two blocks in the Simulink block library. However, they are the same Simulink block with different settings for the Select delay type parameter. Use this parameter to specify the mode in which the block operates.

Variable Time Delay

In this mode, the block has a data input, a time delay input, and a data output. (See Port Location After Rotating or Flipping for a description of the port order for various block orientations.) The output at the current time step equals the value of its data input at a previous time step. This time step is the current simulation time minus a delay time specified by the time delay input.

y(t)=u(t−t0)=u(t−τ(t))

During the simulation, the block stores time and input value pairs in an internal buffer. At the start of simulation, the block outputs the value of the Initial output parameter until the simulation time exceeds the time delay input. Then, at each simulation step, the block outputs the signal at the time that corresponds to the current simulation time minus the delay time.

If you want the output at a time between input storing times and the solver is a continuous solver, the block interpolates linearly between points.

Variable Transport Delay

In this mode, the block output at the current time step is equal to the value of its data (top, or left) input at an earlier time step equal to the current time minus a transportation delay.

y(t)=u(t−td(t))

Suppose that you want to use this block to model the fluid flow through a pipe where the fluid speed varies with time. In this case, the time delay input to the block is

ti(t)=L/vi(t)

where L is the length of the pipe and vi(t) is the speed of the fluid.

References

1. https://www.geeksforgeeks.org/introductiontomathematicalmodelling/
2. https://towardssimulink.com/mathematicalmodelling/time/delay
3. https://in.mathworks.com/help/physmod/sps/powersys/ref/discretevariabletimedelay.html
4. https://in.mathworks.com/help/control/ug/time-delay-approximation.html?searchHighlight=time%20delay&s_tid=doc_srchtitle