Some Features of Control of Distributed Heat Exchanger Models
Abstract Generated abstract
This paper examines how distributed parameters affect automatic control of outlet temperature in first-order heat exchanger models with PI regulation under disturbances from external heating. It derives closed-loop transfer functions for heat exchangers with either constant heating-medium temperature or constant heat flux along the spatial coordinate, relates the distributed response to the corresponding lumped-parameter response, and evaluates the role of residence time through a distributedness coefficient. Model calculations indicate that increasing distributedness can intensify oscillations and produce self-oscillatory or unstable regimes when residence time approaches characteristic times of the lumped response. The authors propose tuning relations for PI controllers in both cases and suggest their use in assessing whether single-loop PI control is suitable for designed or existing heat exchangers.
Full Text
SOME FEATURES OF THE CONTROL OF DISTRIBUTED HEAT EXCHANGER MODELS
A. F. FEDOROV, A. G. KOKIN, B. S. KOSTIN
(Presented by the scientific seminar of VLPI)
Heat exchangers (HE) are typical control objects with distributed parameters and have become widely used in the chemical industry both as independent apparatuses and as component parts of chemical reactors. Therefore, studying the influence of parameter distribution on the process of controlling the temperature of the heated medium at the outlet of the HE is an important task contributing to a correct understanding of processes in automatic control systems (ACS). As a rule, the study of HE control processes is carried out with the aid of fractional-rational functions obtained as a result of approximating transcendental transfer functions. This may conceal features of the distributed HE model and lead to an incorrect result [1].
In the present work an attempt is made to give a qualitative assessment of the influence of parameter distribution on the process of controlling the temperature of the heated medium at the outlet of the HE under disturbance from the side of the regulating action—external heating. Mathematical models of first-order HEs with a PI controller, most often used in the practice of HE control, are investigated. Under the usual assumptions [2], the initial equations will have the form:
a) for an HE with a heating medium temperature constant with respect to \(x\)
\[ T_{12}v\frac{\partial \Delta t_1}{\partial x} + T_{12}\frac{\partial \Delta t_1}{\partial \tau} + \Delta t_1 = \Delta t_2 - x_p; \tag{1} \]
b) for an HE with a heat flux constant with respect to \(x\)
\[ T_{12}v\frac{\partial \Delta t_1}{\partial x} + T_{12}\frac{\partial \Delta t_1}{\partial \tau} = k_1(\Delta q - x_p); \tag{2} \]
c) for a PI controller
\[ x_p = k_p\left( \Delta t_1 + \frac{1}{T_u} \int \Delta t_1\,d\tau \right). \tag{3} \]
The simultaneous solution of equations (1)--(3) and (2)--(3) in the Laplace-transform domain (the transfer function of the closed-loop system) has the following form:
\[ W_{\mathrm{cs}}(p)=W_{cc}(p)\left\{1-\exp(-a\tau_0)\exp(-\tau_0p)\exp\left(-\frac{b}{p}\right)\right\}, \tag{4} \]
where \(W_{cc}(p)\) is the transfer function of the closed-loop system with a lumped heat-exchanger model, which can be obtained from (1) or (2) if the term containing the derivative with respect to the spatial coordinate is discarded.
For case \(a\),
\[ a=\frac{1+\kappa_p}{T_{12}};\qquad b=\frac{\kappa_p\tau_0}{T_uT_{12}}. \tag{5} \]
For case \(b\),
\[ a=\frac{\kappa_1\kappa_p}{T_{12}};\qquad b=\frac{\kappa_1\kappa_p\tau_0}{T_{12}T_u}. \tag{6} \]
Applying the inverse Laplace transform to (4) and using the representation of the Bessel function \(I_1\left(2\sqrt{\frac{z}{\theta}}\right)\) in the form of a series, we obtain the general expression for the control process in cases \(a\) and \(b\):
\[ \Delta t_1=F(\tau)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ 0<\tau\leq\tau_0 \]
\[ \Delta t_1=F(\tau)-\exp(-a\tau_0)\left\{F(\tau-\tau_0)- \int_0^{z-\tau_0}F(\tau-\tau_0-z)\sum_{\kappa=0}^{\infty}(-1)^\kappa \times \frac{b^{-(1+\kappa)}z^\kappa}{\kappa!(\kappa+1)!}\,dz\right\} \ .\ .\ .\ \tau_0\leq\tau\leq\infty, \tag{7} \]
where \(F(\tau)\) is the response of the automatic control system with a lumped heat-exchanger model to a disturbance from the heating side.
Expression (7) is suitable for calculation on an electronic digital computer.
It follows from (7) that the basis of the control process of the distributed mathematical heat-exchanger model is the control process of the lumped heat-exchanger model, from which, after the time instant \(\tau=\tau_0\), a certain fraction of this process is subtracted, shifted in time by the amount \(\tau_0\) or by \(\tau_0+z\). From what has been said, one may conclude that the control process of the distributed mathematical heat-exchanger model depends on the properties of the lumped model and on the residence time of the heated medium in the apparatus.
We shall agree to define the degree of distributedness by the distributedness coefficient
\[ \delta=\exp\left(-\frac{\tau_0}{T_{12}}\right), \]
which varies from 0 to 1. As \(\delta\to0\), the distributed mathematical heat-exchanger model tends to the lumped model.
The automatic control system of the distributed mathematical model was set up on an EMU-10 modeling machine [3].
Using the method of V. Ya. Rotach [4], the controller tuning parameters for the lumped heat-exchanger model were determined so that the damping ratio was \(\psi=0.9\). With these parameters, control processes were obtained for different \(\delta\). Figure 1 presents the control processes for case \(a\), and Fig. 2 for case \(b\). Despite the common form of the mathematical expression (7), cases \(a\) and \(b\) have their own features and will be considered separately. It is clear from Fig. 1 that with
as \(\delta\) increases, the oscillatory nature of the control process increases. In the case when \(\tau_0=\tau_n\)—the times of the first half-period—the system passes into a self-oscillatory regime. For \(\tau_0<\tau_n\) the oscillatory nature decreases sharply.
Fig. 1. Curves of control processes (case a)
In case \(b\), for \(\tau_0>\tau_n\) the system is always unstable (except for \(\delta=0\)). When \(\tau_0=\tau_n\), the system passes into a self-oscillatory regime, and for \(\tau_0<\tau_n\) the system is stable.
Fig. 2. Curves of control processes (case b)
It follows from the above that the tuning parameters of the automatic-control-system regulator with a distributed mathematical model must be chosen so that the condition \(\tau_0<\tau_m\) is always satisfied. This makes it possible to eliminate the undesirable influence of the resonance phenomenon.
For \(\tau_0=\tau_m\)—the time of the maximum—the degree of damping \(\psi=1\).
Consequently, to obtain a process with $\psi = 0.9$, the residence time must lie in the interval $\tau_m < \tau_0 < \tau_n$.
The values $\tau_m$ and $\tau_n$ are easily obtained from the solution for the ACS with a lumped HE model [4].
For case $a$,
\[ \frac{\Delta t_1}{\Delta t_3} = \frac{2m}{1+\kappa_p} \cdot \exp\left(-\frac{1+\kappa_p}{2T_{12}}\tau\right) \sin \frac{1+\kappa_p}{2T_{12}m}\tau; \tag{8} \]
\[ \tau_n=\frac{2\pi mT_{12}}{1+\kappa_p}; \qquad \tau_m=\frac{2mT_{12}}{1+\kappa_p}\arctan\frac{1}{m}. \tag{9} \]
For case $b$,
\[ \frac{\Delta t_1}{\Delta q} = \frac{2m}{\kappa_p} \exp\left(-\frac{\kappa_1\kappa_p}{2T_{12}}\tau\right) \sin \frac{\kappa_1\kappa_p}{2mT_{12}}\tau; \tag{10} \]
\[ \tau_n=\frac{2\pi mT_{12}}{\kappa_1\kappa_p}, \qquad \tau_m=\frac{2mT_{12}}{\kappa_1\kappa_p}\arctan\frac{1}{m}. \tag{11} \]
Setting $m=1$, since an ACS with a distributed HE model has lower stability than an ACS with a lumped HE model, and using the condition
\[ \tau_0=0.5(\tau_n-\tau_m)+\tau_m \quad \text{for case } a, \]
\[ \tau_0=0.25(\tau_n-\tau_m)+\tau_m \quad \text{for case } b, \]
we obtain expressions for calculating the tuning parameters:
for case $a$,
\[ \kappa_p=\frac{4T_{12}}{\tau_0}-1; \qquad T_{\mathrm{и}}=\frac{2\kappa_p T_{12}}{(1+\kappa_p)^2}; \]
\[ \left(\frac{\Delta t_1}{\Delta t_3}\right)_{\max} = \frac{0.646}{1+\kappa_p}. \tag{12} \]
for case $b$,
\[ \kappa_p=\frac{4T_{12}}{\kappa_1\tau_0}; \qquad T_{\mathrm{и}}=\frac{2T_{12}}{\kappa_1\kappa_p}; \]
\[ \left(\frac{\Delta t_1}{\Delta q}\right)_{\max} = \frac{0.646}{\kappa_p}. \tag{13} \]
The control time in both cases is always greater than $\tau_0$.
For case $a$, the use of formulas (12) is advisable for $\delta > 0.3$, since for $\delta < 0.3$ the dynamic error and the control time increase sharply.
For $\delta < 0.3$, the parameters selected for the lumped HE model with $m=1$ may be considered optimal.
All arguments concerning a qualitative assessment of the influence of distributedness are also valid for second-order objects.
The results of this work can be applied to assess the possibility of using PI controllers in a single-loop ACS during the design of HEs and for existing apparatuses.
Notation: $t$ — temperature; $v$ — velocity of the heated medium; $T$ — time constant; $\tau_0$ — residence time of the heated medium in the apparatus; $m$ — degree of oscillation of the control process; $x$ — spatial coordinate.
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