INFLUENCE OF PARAMETER DISTRIBUTION ON THE CONTROL PROCESS IN HEAT EXCHANGERS

A. F. Fedorov, P. A. Andriyanov

(Presented by Prof. A. N. Eremin)

At the present time, a fairly large number of works have appeared on the study of the dynamics of heat exchangers. The question now naturally arises concerning the features of temperature-control processes in distributed models of heat-exchange apparatuses.

In the present work an attempt is made to give a qualitative assessment of the influence of parameter distribution on the control process, proceeding from the concept of a lumped mathematical model of a heat exchanger and the residence time of the heated liquid in a heat exchanger [1] with constant temperature of the heating heat carrier.

The initial equation of the heat exchanger, taking into account the distribution of the temperature of the heated heat carrier along the length and certain assumptions [1, 2], will have the following form:

\[ T_{12}T_{23}v\frac{\partial^{2}t_{1}(x,\tau)}{\partial\tau\partial x} + (T_{21}+T_{23})\frac{T_{12}}{T_{21}}v\frac{\partial t_{1}(x,\tau)}{\partial x} + T_{12}T_{23}\frac{\partial^{2}t_{1}(x,\tau)}{\partial\tau^{2}} + \]

\[ + \left( T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23} \right) \frac{\partial t_{1}(x,\tau)}{\partial\tau} + t_{1}(x,\tau) = t_{3}(\tau). \tag{1} \]

We shall evaluate the influence of distribution on the dynamics of the heat exchanger by the quantity

\[ \delta_{\rho} = \exp\left[ - \frac{\tau_{0}T_{21}}{T_{12}(T_{21}+T_{23})} \right]. \tag{2} \]

\(\delta_{\rho}\) is the distribution coefficient, equal to the difference between the steady-state values of the temperature of the heated heat carrier in the lumped and distributed mathematical models of the heat exchanger. As \(\delta_{\rho}\to 0\), the distributed mathematical model of the heat exchanger passes into the lumped mathematical model. Thus, the latter may be regarded as a particular case of the distributed model.

The transfer function may be obtained from (1) after applying the Laplace transform

\[ W(p)=W_{c}(p)[1-W_{p}(p)]. \tag{3} \]

where

\[ W_c(p)= \frac{1}{T_{12}T_{23}p^2+\left(T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23}\right)p+1} \tag{4} \]

is the transfer function of the lumped mathematical model;

\[ W_p(p)=\exp\left(-\frac{\tau_0}{T_{12}}\right)\exp(-\tau_0 p)\exp\left[ \frac{\tau_0}{T_{12}T_{21}\left(p+\frac{T_{21}+T_{23}}{T_{21}T_{23}}\right)} \right] \tag{5} \]

is the transfer function in the channel \(t_0-t_1(x,\tau)\), taking into account the influence of distributedness.

The expression for the frequency response can be obtained from (3) by replacing \(p\) with \(i\omega\).

Because of its bulkiness, the analytical expression for the frequency response \(W(i\omega)\) is not given in the present work [3]. In Fig. 1

Fig. 1. Amplitude–phase frequency responses.
\(1\) — \(\delta_p=0\), \(2\) — \(\delta_p=0.02\), \(3\) — \(\delta_p=0.2\), \(4\) — \(\delta_p=0.5\).

are shown graphs of the frequency responses of a heat exchanger of type TH in the channel \(t_3-t_1(x,\tau)\) for various values of \(\delta_p\), calculated on the “Promin” machine \((T_{12}=7.1\ \text{s},\ T_{21}=2.68\ \text{s},\ T_{23}=1.02\ \text{s})\).

Analyzing the frequency responses, one may suppose that the quality of the control process should change as \(\delta_p\) increases. The same can also be said from an analysis of the analytical expression for the control process of a system with a PI controller, the one most often used in practice. In this case the equations of the automatic control system (ACS) will have the form:

\[ \left\{ \begin{aligned} &T_{12}T_{23}v\,\frac{\partial^2 t_1(x,\tau)}{\partial\tau\,\partial x} +\left(T_{21}+T_{23}\right)\frac{T_{12}}{T_{21}}v\,\frac{\partial t_1(x,\tau)}{\partial x} +T_{12}T_{23}\frac{\partial^2 t_1(x,\tau)}{\partial\tau^2} \\ &\quad+\left(T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23}\right)\frac{\partial t_1(x,\tau)}{\partial\tau} +t_1(x,\tau)=t_3(\tau)-t(x,\tau), \\[6pt] &t(x,\tau)=\kappa_p+\frac{\kappa_p}{T_u}\int t_1(x,\tau)\,d\tau . \end{aligned} \right. \tag{6} \]

After applying the Laplace transform to (6), we obtain an expression for the transfer function of the closed-loop ACS

\[ W_{\text{cl}}(p)=W_{\text{cl}}^{c}(p)\left\{1-\exp\left(-\frac{\tau_0}{T_{12}}\right)\exp(-\tau_0 p)\exp\left(-\frac{1}{cp}\right)\exp\left[\frac{1}{a(p+b)}\right]\right\}, \tag{7} \]

where

\[ W_{\text{cl}}^{c}(p)= \frac{p}{ T_{12}T_{23}p^3+\left(T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23}\right)p^2+(1+\kappa_p)p+\frac{\kappa_p}{T_i} } \tag{8} \]

is the transfer function of the closed-loop ACS for the lumped mathematical model of the heat exchanger;

\[ a= \frac{T_{12}T_{21}}{ \left[ \frac{T_{23}}{T_{21}}+ \frac{\kappa_p T_{12}^{2}}{(T_{21}+T_{23})T_i} -\kappa_p \right]\tau_0 }; \qquad b=\frac{T_{21}+T_{23}}{T_{21}^{2}}, \qquad c=\frac{T_iT_{12}(T_{21}+T_{23})}{\kappa_pT_{21}\tau_0}. \]

For a stepwise change in \(t_3(\tau)\), we obtain, for \(0\leq \tau<\tau_0\),

\[ t_1(l,\tau)=F(\tau), \tag{9} \]

and for \(\tau_0\leq \tau<\infty\),

\[ \begin{aligned} t_1(l,\tau)=F(\tau) &-\exp\left(-\frac{\tau_0}{T_{12}}\right) \Bigg\{ F(\tau-\tau_0) +\int_{0}^{\tau-\tau_0} F(\tau-\tau_0-z)\times \\ &\times \left[ \frac{1}{\sqrt{az}}\exp(-bz)I_1\left(2\sqrt{\frac{z}{a}}\right) - \frac{1}{\sqrt{cz}}I_1\left(2\sqrt{\frac{z}{c}}\right) \right]dz \\ &+\int_{0}^{\tau-\tau_0}\exp(-bz)\frac{1}{\sqrt{az}}I_1\left(2\sqrt{\frac{z}{a}}\right)\times \\ &\times \left[ \int_{0}^{\tau-\tau_0-z} F(\tau-\tau_0-z-\beta) \left(-\frac{1}{\sqrt{c\beta}}\right) I_1\left(2\sqrt{\frac{\beta}{c}}\right)d\beta \right]dz \Bigg\}, \end{aligned} \tag{10} \]

where \(F(\tau)\) is the response of the lumped ACS \(\bigl(W_{\text{cl}}^{c}(p)\bigr)\) to a step input from \(t_3(\tau)\).

Representing the Bessel functions in the form of series, we obtain an expression for the control process in a form suitable for calculation on an electronic digital computer in the interval \(\tau_0\leq \tau<\infty\):

\[ \begin{aligned} t_1(l,\tau)=F(\tau) &-\exp\left(-\frac{\tau_0}{T_{12}}\right) \Bigg\{ F(\tau-\tau_0) +\int_{0}^{\tau-\tau_0}F(\tau-\tau_0-z)\times \\ &\times \left[ \exp(-bz)\sum_{k=0}^{\infty} \frac{\left(\frac{1}{a}\right)^{k+1}z^k}{k!(k+1)!} + \sum_{k=0}^{\infty}(-1)^{k+1} \frac{\left(\frac{1}{c}\right)^{k+1}z^k}{k!(k+1)!} \right]dz \\ &+\int_{0}^{\tau-\tau_0}\exp(-bz) \left[ \int_{0}^{\tau-\tau_0-z} F(\tau-\tau_0-z-\beta) \sum_{k=0}^{\infty}(-1)^{k+1} \frac{\left(\frac{1}{c}\right)^{k+1}\beta^k}{k!(k+1)!} \,d\beta \right]\times \\ &\times \sum_{k=0}^{\infty} \frac{\left(\frac{1}{a}\right)^{k+1}z^k}{k!(k+1)!} \,dz \Bigg\}. \end{aligned} \tag{11} \]

From (9) and (11) it is evident that the basis of the control process of the distributed model is the control process of the lumped mo-

shares \(F(\tau)\), from which, after \(\tau=\tau_0\), the same function \(F(\tau)\), shifted in time by the amount \(\tau_0\) and changed by a factor of \(\exp\left(-\dfrac{\tau_0}{T_{12}}\right)\), is subtracted, and the convolution of the functions.

Figure 2 gives plots of the curves of the control process for different values of \(\delta_p\), obtained on the EMU-10 analog computer.

Fig. 2. Curves of the control processes

Fig. 3

As is evident from Fig. 2, the influence of distributedness begins to make itself felt only when \(\delta_p > 0.2\).

In the interval \(0.2 \leq \delta_p \leq 0.75\), the object has the best dynamic properties; during this period the influence of resonance is manifested. With increasing \(\delta_p\), the oscillatory nature of the process increases. If the value \(\tau_0\) (\(\delta_p \simeq 0.53\)) numerically coincides with the duration of the first half-wave of the control process, then the process will be undamped, with a considerable amplitude of oscillation. With a further increase in \(\delta_p\), the degree of damping increases up to \(\psi = 1\).

Conclusions and Recommendations

1. The dynamic properties of a control system with a distributed mathematical model of a heat exchanger are determined by the dynamic properties of the system with a lumped mathematical model and by the residence time of the heated liquid in the apparatus.

2. The influence of parameter distribution may be neglected when \(\delta_p < 0.2\).

3. When \(\delta_p > 0.2\), the tuning parameters of the PI controller \(K_p^p\) and \(T_i^p\) can be determined from the nomogram (Fig. 3), after first determining the tuning parameters of the PI controller \(K_p^c\) and \(T_i^c\) for the lumped mathematical model at \(\psi = 0.97\). The control time will always be greater than \(\tau_0\) (\(\simeq 3\tau_0\)), and the degree of damping \(\psi \simeq 0.9\).

4. The control system may enter a self-oscillatory regime if \(\tau_0\) is equal to the time of the first half-period of the control process of the system with the lumped model of the heat exchanger. This must be taken into account when the heat exchanger operates under variable load.

Notation: \(T\) — time constant; \(v\) — velocity of motion of the heated liquid; \(\tau_0\) — residence time of the heated liquid in the apparatus; \(x\) — current coordinate; \(\tau\) — current time; \(p\) — Laplace transform variable with respect to \(\tau\); \(t\) — temperature deviation; \(K_p\) — gain coefficient of the PI controller; \(T_i\) — reset time.

Subscripts: 1 — heated liquid; 2 — wall; 3 — heating heat-transfer medium.

REFERENCES

1. P. A. Andriyanov, I. M. Maslennikov. Influence of the distributedness of parameters on the dynamic properties of heat exchangers. Khimicheskaya promyshlennost’, 1963, No. 12, pp. 22–28.

2. B. N. Devyatov. Theory of transient processes in technological apparatuses from the standpoint of control problems, Novosibirsk, Publishing House of the Siberian Branch of the USSR Academy of Sciences, 1964, 71–95.

3. P. A. Andriyanov. Candidate dissertation, Moscow, 1964.