Numerical Bode and Nyquist plots
The above transfer functions can be used to numerically calculate the Bode and Nyquist plots for the plate heat exchanger. The heat exchanger consists of two de-coupled sections placed in series. The first uses hot water from the vacuum condenser and the second uses hot water from the MVR evaporator shell. We will determine the Bode and Nyquist plots for the four transfer
functions, of the second section, acting on the outlet hot milk temperature. The resulting Bode plots are shown in Figure 3-7 and the Nyquist plots in Figures 3-8 and 3-9.
-0.2 : : : o 0 •
---t--r- ---r---t---i ---tl---
� : : : ! : : : ':. • : I : I : I :: .... - - - � - � - - - • - - - ! - - - ! - - - ., - - - - -:. I I I I 1: .� ·0.4 \: : : : !: ;� : I : .� : - - - -t- - - r - - - y - - - y - - - -�- T - - - - : \ : : : ..: : ·S, ·0 6 � . -0.8 I ", I I • 4"" I : "I,,� : : : .l : : .... : : :,
.-
: - - - � - - - � - - - � - - - · - - - · - - - - -I- - - + - - - - • '.. . I . ...
I : " '"'' : : ... : : �.t·l. : I �.... : -1 -- - - -: - - - ... ,: - - - -; - - - -.�."; --- - --; --- - - ---
1
1
....
.-.... , .... �... �
1
o 0 0-1._�.�
0Figure 3-9 : Nyquist plots for transfer functions
T",e3 (s)
andT",e3 (s)
Me(s)
MAs) '
There are some clear similarities between these transfer functions and conventional polynomial transfer functions. The
Tmc3 (s)
transfer function appears to contain a significant pure delay and aT
mc2(s)
third order dynamic. Its magnitude appears to follow a third order dynamics whereas its phase lag continuously falls and therefore suggests a pure delay. The important
Tmc3(S)
transfer functionMc(s)
appears to be third order. Likewise the
Tmc3(S)
transfer function appears to be second order andTeI (
s)
the To",) (s) transfer function first order.
Mj{s)
3.4) Conclusions
In Chapter 2 the entire Evaporator A plant was separated into four sections. These sections were the MVR evaporator section, the TVR evaporator section, the DSI unit preheat section and the preheat plate heat exchanger section. Chapter 2 was primarily concerned with deriving models for the MVR evaporator and TVR evaporator sections. The aim of this Chapter was to derive the models for the remaining DSI preheat section, preheat plate heat exchanger and shell/tube condensers.
In Section 4.2 the DSI preheat section was considered. We firstly gave a brief description of the process and then we derived the energy balances and mass balances for the DSI unit and the flash vessels. An important part of this was the thermal equi librium between the liquid and vapour
Modelling Evaporator Preheat and Heat Exchangers
phases. It was shown that the presence of air causes a temperature difference between the top and bottom of the vessel. The final result was a set of static and differential equations for the temperatures of the DSI unit and the flash vessels.
In Section 4.3 we investigated the preheat plate heat exchanger and the shell/tube condensers. The partial differential equations for these were presented and transformed in the Laplace Domain. For the shell/tube condensers we were able to produce an ordinary differential equation with time delays, whereas the result for the plate heat exchanger was a set of transcendental transfer functions.
The models derived in this and the previous Chapter potentially provide a complete description of the Evaporator A plant. However, they are still in a somewhat 'raw' state. In the next two Chapters we aim to develop and identify these models into a form that can be used for the optimisation and controllability studies. Chapter 4 will develop the steady state and linear dynamic models of the Evaporator A plant.
4
:
Model
4. 1) Introduction
The work discussed in this thesis aims to analyse the operation and control of the Evaporator A plant at Kiwi Co-op Dairies. Optimisation and controllability studies will be made to determine
the 'best' operating conditions and methods for controlling the plant. In Chapters 2 and 3,
models were derived for the entire evaporator plant. These were developed for analysing the Evaporator A plant operation and control, but they must be further developed before they can be used. Specifically, we need a validated steady state model for the optimisation studies and a linear dynamic model for the controllability studies. In this Chapter, we will develop the steady state and linear dynamic models.
Firstly, the steady state model will be developed in Section 4.2. We will separate the Evaporator A plant into four sections and consider each independently before combining these to produce the complete model. These four sections are the MVR evaporator section, the TVR evaporator section, the OSI preheat section and the preheat plate heat exchanger. Typical operating points will be determined for each individual section and also the complete Evaporator A model. The plant geometries are listed in the Appendix E and the process parameters (i.e., heat transfer coefficients) are taken from the identification of Chapter 5 .
In Section 4. 3 , we discuss the methodology for developing a linear dynamic model. Specifically, we investigate the linearisation of dynamic models using truncated Taylor series approximations and the description of pure delays using Pade approximations. This has been included to provide some background to the linearisation work, which will be done in Section 4.4.
The linear dynamic model will be developed in Section 4.4. As with the steady state model development, we separate the plant into four parts and develop each individually. The linear dynamic models are developed using the methods discussed in Section 4 . 3 . However, in order to simpl ify the linearisation, we have made a number of significant assumptions. For example, distribution plate flashing, variable heat transfer coefficients and multiple evaporator passes have all been neglected. These assumptions were not made in the development of the steady state model and therefore the linear dynamic model is not a generalisation of the steady state model.
Model Development
4.2) Steady State
4.2. 1) Introduction