ALGORITHMS FOR COMPUTER CALCULATION OF TRANSIENT PROCESSES IN DISTRIBUTED MODELS OF HEAT EXCHANGERS WITH AN INTERNAL HEAT SOURCE

P. A. ANDRIYANOV, A. F. FEDOROV

(Presented by the scientific final conference of the Faculty of Chemical Technology)

Heat exchangers with an internal heat source (electric heaters, nuclear reactors, chemical reactors in which the reaction occurs inside the solid phase), in terms of dynamic properties, can be combined with the group of heat exchangers with a heat flux constant along the length and independent of the temperature of the heated heat carrier (steam superheaters, etc.), whose original equation has the same form [1].

\[ T_{12}T_{21}v \frac{\partial^2 t_1(x,\tau)}{\partial \tau \partial x} + T_{12}v \frac{\partial t_1(x,\tau)}{\partial x} + T_{12}T_{21}\frac{\partial^2 t_1(x,\tau)}{\partial \tau^2} + \]

\[ + (T_{12}+T_{21})\frac{\partial t_1(x,\tau)}{\partial \tau} = \chi_2 q(\tau). \tag{1} \]

The problems of investigating the transient regimes of heat exchangers of this type are being studied by power engineers [2—5,7], since the problems of regulating the temperature of superheated steam have become especially acute in connection with the use, at thermal power stations, of equipment of elevated and superhigh parameters. Subsequently heat exchangers began to be studied in a broader range of applications (chemistry, industrial heat-power engineering) [1, 6, 8—10].

Researchers proceed in two ways:

1. The method of transient characteristics [1—3, 6—10].
2. The method of frequency characteristics [4, 5].

Both methods have their advantages and disadvantages. Recently preference has been given to the first method, since it makes it possible to investigate the dynamics of heat exchangers on electronic computing machines [6, 8—10].

In the present article an attempt is made to give a general form of the algorithm for calculating on an electronic digital computer the transient characteristics of heat exchangers with an internal heat source, with a constant heat flux, under various forms of disturbance.

From expression (1), by applying the Laplace transform, one can obtain the transfer function.

\[ W(p)=W_c(p)\,[1-W_p(p)]+W_p(p), \tag{2} \]

where:

\[ W_c(p)=\frac{\kappa_2}{p\left(T_{12}T_{21}p+T_{12}+T_{21}\right)} \]

is the transfer function of the lumped model of the heat exchanger;

\[ W_p(p)=\exp\left(-\frac{\tau_0}{T_{12}}\right)\exp(-\tau_0p)\exp\left[\frac{\tau_0}{T_{12}T_{21}\left(p+\frac{1}{T_{21}}\right)}\right] \]

is the transfer function that accounts for the effect of distributedness; at the same time, it is also the transfer function in the channel \(t_0-t_1\). Applying the inverse Laplace transform to expression (2) and the convolution theorem in the real domain, we obtain an expression for the transient response in general form for any law of disturbance:

1. From the side of the heat flow. In the interval \([0,\tau_0]\)

\[ t_1(l,\tau)=\varphi(\tau). \tag{3} \]

In the interval \([\tau_0,\infty]\)

\[ t_1(l,\tau)=\varphi(\tau)-\exp\left(-\frac{\tau_0}{T_{12}}\right) \left\{\varphi(\tau-\tau_0)+\int_0^{\tau-\tau_0}\varphi(\tau-z-\tau_0)\times \right. \]

\[ \left. {}\times \exp\left(-\frac{z}{T_{21}}\right) \sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!}\,dz \right\}, \tag{4} \]

where

\(\varphi(\tau)\) is the response of the lumped model to the disturbance:

a). For a step disturbance

\[ \varphi(\tau)= \frac{\kappa_2T_{12}T_{21}}{(T_{12}+T_{21})^2} \left[ \exp\left(-\frac{T_{12}+T_{21}}{T_{12}T_{21}}\tau\right) +\frac{T_{12}+T_{21}}{T_{12}T_{21}}\tau-1 \right]q'; \tag{5} \]

b). For a linear disturbance

\[ \varphi(\tau)= \frac{\kappa_2T_{12}^2T_{21}^2}{(T_{12}+T_{21})^3} \left[ 1-\frac{T_{12}+T_{21}}{T_{12}T_{21}}\tau +\frac{(T_{12}+T_{21})^2}{2T_{12}^2T_{21}^2}\tau^2 -\exp\left(-\frac{T_{12}+T_{21}}{T_{12}T_{21}}\tau\right) \right]\frac{q'}{\tau_1}; \tag{6} \]

c). For an exponential disturbance

\[ \varphi(\tau)= \frac{\kappa_2T}{T_{12}+T_{21}} \left\{ 1+ \frac{T_{12}T_{21}}{T(T_{12}+T_{21})-T_{12}T_{21}} \exp\left(-\frac{T_{12}+T_{21}}{T_{12}T_{21}}\tau\right) -\right. \]

\[ \left. {}-\left[ \frac{T_{12}T_{21}}{T(T_{12}+T_{21})-T_{12}T_{21}}+1 \right]\exp\left(-\frac{\tau}{T}\right) \right\}q'. \tag{7} \]

1. From the side of the inlet temperature. In the interval \([0,\tau_0]\)

\[ t_1(l,\tau)=0. \tag{8} \]

In the interval \([\tau_0,\infty]\)

\[ t_1(l,\tau)=\exp\left(-\frac{\tau_0}{T_{12}}\right) \left\{\varphi_0(\tau-\tau_0)+ \int_0^{\tau-\tau_0}\varphi_0(\tau-z-\tau_0)\times \right. \]

\[ \left. {}\times \exp\left(-\frac{z}{T_{21}}\right) \sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!}\,dz \right\}. \tag{9} \]

Expression (9) is obtained from (2) if the transfer function in the channel \(t_0-t_1\) is written in the form \(W(p)=W_c(p)W_p(p)\), under the condition that \(W_c(p)=1\). Then, by analogy with the preceding case, \(\varphi_0(\tau-\tau_0)\) and \(\varphi_0(\tau-z-\tau_0)\) are the response of a lumped model \((W_c(p)=1)\) to a disturbance according to the specified law, shifted in time by \(\tau_0\) or \((z+\tau_0)\), respectively:

a) for a step disturbance,

\[ \varphi_0(\tau)=t_0', \tag{10} \]

b) for a linear disturbance,

\[ \varphi_0(\tau)=\tau \frac{t_0'}{\tau_1}, \tag{11} \]

c) for an exponential disturbance,

\[ \varphi_0(\tau)=t_0'\exp\left(-\frac{\tau}{T}\right). \tag{12} \]

Figure 1 gives the curves of transient processes calculated on the Minsk-1 computer by the algorithms presented.

Expressions (4) and (9) are not optimal from the point of view of using machine time. To calculate one point with the required accuracy on the Minsk-1 computer requires about 1.5 minutes. In order to

Fig. 1. Transient characteristics of a heat exchanger. ○ — experimental [6], — calculated: a) for a step disturbance; b) for a linear disturbance; c) for an exponential disturbance

reduce the computation time, the integral after substituting the value \(\varphi(\tau-z-\tau_0)\) or \(\varphi_0(\tau-z-\tau_0)\) can be decomposed into a sum of integrals, each of which is reduced to the form [11]

\[ \int z^k \exp az\,dz = e^{az} \left( \frac{z^k}{a} + \sum_{m=1}^{k} (-1)^m \frac{k(k-1)\ldots(k-m+1)}{a^{m+1}} z^{k-m} \right). \tag{13} \]

For example, for a step disturbance, after transformation we shall have the following expression:

\[ \int_{0}^{\tau-\tau_0} \varphi(\tau-z-\tau_0) \exp\left(-\frac{z}{T_{21}}\right) \sum_{k=0}^{\infty} \frac{\left(-\frac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!} \,dz = \]

\[ = \frac{q'k_2T_{12}T_{21}}{(T_{12}+T_{21})^2} \sum_{k=0}^{\infty} \frac{\left(\frac{\tau_0}{T_{12}T_{21}}\right)^{k+1}}{k!(k+1)!} \left\{ \exp\left[ -\frac{T_{12}+T_{21}}{T_{12}T_{21}}(\tau-\tau_0) \right] \left(\frac{T_{21}}{T_{12}}\right)^{\frac{T_{21}(\tau-\tau_0)}{T_{12}}} \times \right. \]

\[ \begin{aligned} &\times\left[\frac{T_{12}(\tau-\tau_0)^\kappa}{T_{21}} +\sum_{m=1}^{\kappa}(-1)^m \frac{\kappa(\kappa-1)\ldots(\kappa-m+1)} {\left(\dfrac{T_{21}}{T_{12}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m}\right]+\\ &+\left[\frac{T_{12}+T_{21}}{T_{12}T_{21}}(\tau-\tau_0)-1\right] \left(-\frac{1}{T_{21}}\right)^{-\frac{\tau-\tau_0}{T_{21}}} \left[-T_{21}(\tau-\tau_0)^\kappa+\right.\\ &\left.+\sum_{m=1}^{\kappa}(-1)^m \frac{\kappa(\kappa-1)\ldots(\kappa-m+1)} {\left(-\dfrac{1}{T_{21}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m}\right] -\frac{T_{12}+T_{21}}{T_{12}T_{21}}\times\\ &\times\left(-\frac{1}{T_{21}}\right)^{-\frac{\tau-\tau_0}{T_{21}}} \left[\frac{(\tau-\tau_0)^{\kappa+1}}{\left(-\dfrac{1}{T_{21}}\right)} +\sum_{m=1}^{\kappa+1}(-1)^m \times \right.\\ &\left.\left.\times \frac{(\kappa+1)\kappa(\kappa-1)\ldots(\kappa-m+2)} {\left(-\dfrac{1}{T_{21}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m+1}\right]\right\}. \end{aligned} \tag{14} \]

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