Algorithms for Calculating the Transient Characteristics of a Distributed Model of a Heat Exchanger with a Constant Temperature of the Heating Heat Transfer Medium
Abstract Generated abstract
The paper develops computational algorithms for transient analysis of heat exchangers in which the heating medium remains at constant temperature, treating the exchanger as a distributed-parameter object. Starting from the governing differential equation and its transfer-function representation, the authors derive inverse-Laplace and convolution formulas for outlet-temperature responses to disturbances in either the heating-medium temperature or the inlet temperature of the heated medium. The resulting expressions cover step, linear, and exponential disturbances and are arranged for digital computer calculation, including series and tabulated-integral forms. Example curves computed on a Minsk-1 computer are compared with experimental transient data, illustrating the use of the proposed algorithms.
Full Text
Algorithms for Computer Calculation of the Transient Characteristics of a Distributed Model of a Heat Exchanger with a Constant Temperature of the Heating Medium
P. A. Andriyanov, A. F. Fedorov
(Presented at the final scientific conference of the Faculty of Chemical Technology)
A group of heat exchangers with a constant temperature of the heating medium (steam heat exchangers, condenser-coolers) is very widely used at enterprises of the chemical industry. Knowledge of the dynamic characteristics of heat exchangers becomes necessary not only under production conditions, but also during the design period for purposes of their optimization; this is possible when mathematical expressions are available that relate the design and technological parameters in transient regimes.
In the present article a generalized algorithm is proposed for calculating transient regimes of heat exchangers on a digital computer under different laws of disturbance from the temperature of the heating medium and the temperature at the inlet of the heated medium. In this case the heat exchanger is considered as an object with distributed parameters.
The original equation has the form [1]
\[ T_{12}T_{23}v\frac{\partial^2 t_1(x,\tau)}{\partial x\,\partial \tau} + (T_{21}+T_{23})\frac{T_{12}}{T_{21}}\,v\frac{\partial t_1(x,\tau)}{\partial x} + T_{12}T_{23}\frac{\partial^2 t_1(x,\tau)}{\partial \tau^2} + \]
\[ + \left( T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23} \right) \frac{\partial t_1(x,\tau)}{\partial \tau} + t_1(x,\tau)=t_3(\tau). \tag{1} \]
The transfer function
\[ W(p)=W_c(p)\,[1-W_p(p)]+W_p(p), \tag{2} \]
where
\[ W_c(p)=\frac{1}{(T'p+1)(T''p+1)} \]
is the transfer function of the lumped model;
\[ W_p(p)= \exp\left(-\frac{\tau_0}{T_{12}}\right) \exp(-\tau_0p) \exp\left[ -\frac{\tau_0}{ T_{12}T_{21} \left( p+\frac{T_{21}+T_{23}}{T_{21}T_{23}} \right)} \right] \]
is the transfer function that accounts for the influence of distributedness; at the same time it is also the transfer function for the channel \(t_0-t_1(l,\tau)\).
Applying the inverse Laplace transform to expression (2) and the theorem on real convolution, we obtain a general expression for the transient characteristic.
- Under a disturbance from the temperature of the heating medium.
In the interval \([0,\tau_0]\)
\[ t_1(l,\tau)=\varphi(\tau). \]
In the interval \((\tau_0,\infty]\)
\[ t_1(l,\tau)=\varphi(\tau)-\exp\left(-\frac{\tau_0}{T_{12}}\right) \left\{\varphi(\tau-\tau_0)+\int_0^{\tau-\tau_0}\varphi(\tau-z-\tau_0)\exp\right. \]
\[ \left.\times\left(-\frac{T_{21}+T_{23}}{T_{21}T_{23}}z\right) \sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!}\,dz\right\}, \tag{3} \]
where \(\varphi(\tau)\) is the response of the lumped model to the disturbance:
a) for a step disturbance
\[ \varphi(\tau)=\left[ 1-\frac{T'}{T'-T''}\exp\left(-\frac{\tau}{T'}\right) +\frac{T'}{T'-T''}\exp\left(-\frac{\tau}{T''}\right) \right]t_3; \tag{4} \]
b) for a linear disturbance
\[ \varphi(\tau)=\left[ \tau-(T'+T'')+\frac{(T')^2}{T'-T''}\exp\left(-\frac{\tau}{T'}\right) -\frac{(T'')^2}{T'-T''}\exp\left(-\frac{\tau}{T''}\right) \right]\frac{t_3'}{\tau_1}; \tag{5} \]
c) for an exponential disturbance
\[ \varphi(\tau)=\left[ \frac{T'T}{(T'-T'')(T'-T)}\exp\left(-\frac{\tau}{T'}\right) -\frac{T''T}{(T'-T'')(T''-T)} \right. \]
\[ \left. \times\exp\left(-\frac{\tau}{T''}\right) +\frac{T^2}{(T'-T)(T''-T)}\exp\left(-\frac{\tau}{T}\right) \right]t_3'. \tag{6} \]
- For a disturbance from the temperature at the inlet of the heated heat-transfer medium. In the interval \([0,\tau_0]\)
\[ t_1(l,\tau)=0. \]
In the interval \([\tau_0,\infty]\)
\[ t_1(l,\tau)=\exp\left(-\frac{\tau_0}{T_{12}}\right) \left\{\varphi_0(\tau-\tau_0)+\int_0^{\tau-\tau_0}\varphi_0(\tau-z-\tau_0)\times\right. \]
\[ \left. \times\exp\left(-\frac{T_{21}+T_{23}}{T_{21}T_{23}}z\right) \sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!}\,dz \right\}, \tag{7} \]
where \(\varphi_0(\tau-\tau_0)\) and \(\varphi_0(\tau-z-\tau_0)\) are the response of the lumped model \((W_c(p)=1)\) to a disturbance according to a specified law, shifted in time by \(\tau_0\) or \(z+\tau_0\), respectively. Their expressions are analogous to the expressions for heat exchangers with internal heat sources.
Figure 1 gives the curves of transient processes along the channel \(t_3-t_1\), calculated on the Minsk-1 machine by the algorithms presented.
As for heat exchangers with internal heat sources, the integral in expressions (3) and (7), after substituting the values \(\varphi(\tau-z-\tau_0)\) and \(\varphi_0(\tau-z-\tau_0)\), can be expanded into a sum of tabulated integrals [2]. After transformations, instead of the integral from (3), for a step disturbance we shall have the following expression
\[ \int_0^{\tau-\tau_0} \varphi(\tau-z-\tau_0) \exp\left(-\frac{T_{21}+T_{23}}{T_{21}T_{23}}z\right) \sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}z^k}{k!(k+1)!}\,dz = \]
\[ = t_3\sum_{k=0}^{\infty} \frac{\left(\dfrac{\tau_0}{T_{12}T_{21}}\right)^{k+1}}{k!(k+1)!} \left\{ \left(-\frac{T_{21}+T_{23}}{T_{21}T_{23}}\right)^{-\frac{T_{21}+T_{23}}{T_{21}T_{23}}(\tau-\tau_0)} \left[ -\frac{T_{21}T_{23}}{T_{21}+T_{23}}(\tau-\tau_0)^k+ \right. \]
\[ +\sum_{m=1}^{\kappa}(-1)^m \frac{\kappa(\kappa-1)\cdots(\kappa-m+1)} {\left(-\dfrac{T_{21}+T_{23}}{T_{21}T_{23}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m} \right] -\frac{T'}{T'-T''}\times \]
\[ \times \exp\left(-\frac{\tau-\tau_0}{T'}\right) \left(\frac{1}{T'}-\frac{T_{21}T_{23}}{T_{21}+T_{23}}\right)^{ \left(\frac{1}{T'}-\frac{T_{21}T_{23}}{T_{21}T_{23}}\right)(\tau-\tau_0)} \left[ \frac{(\tau-\tau_0)^\kappa} {\dfrac{1}{T'}-\dfrac{T_{21}T_{23}}{T_{21}+T_{23}}} + \right. \]
\[ \left. +\sum_{m=1}^{\kappa}(-1)^m \frac{\kappa(\kappa-1)\cdots(\kappa-m+1)} {\left(\dfrac{1}{T'}-\dfrac{T_{21}T_{23}}{T_{21}+T_{23}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m} \right] +\frac{T''}{T'-T''}\exp\times \]
\[ \times\left(-\frac{\tau-\tau_0}{T''}\right) \left(\frac{1}{T'}-\frac{T_{21}T_{23}}{T_{21}+T_{23}}\right)^{ \left(\frac{1}{T''}-\frac{T_{21}T_{23}}{T_{21}T_{23}}\right)(\tau-\tau_0)} \left[ \frac{(\tau-\tau_0)^\kappa} {\dfrac{1}{T''}-\dfrac{T_{21}T_{23}}{T_{21}+T_{23}}} + \right. \]
\[ \left. +\sum_{m=1}^{\kappa}(-1)^m \frac{\kappa(\kappa-1)\cdots(\kappa-m+1)} {\left(\dfrac{1}{T''}-\dfrac{T_{21}T_{23}}{T_{21}+T_{23}}\right)^{m+1}} (\tau-\tau_0)^{\kappa-m} \right]\Biggr\}. \]
Fig. 1. Transient functions of the heat exchanger. ○—experimental; — — computed: a) for a step disturbance; b) for a linear disturbance; c) for an exponential disturbance.
REFERENCES
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P. A. Andriyanov, I. M. Maslennikov. Influence of distributed parameters on the dynamic properties of heat exchangers. Chemical Industry, No. 12, 22–28, 1963.
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I. S. Gradshteyn, I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. 106, Fizmatgiz, Moscow, 1962.