MODELING OF HEAT EXCHANGERS

P. A. Andriyanov, A. F. Fedorov

(Presented by the Department of Machines and Apparatuses of Chemical Production)

Transient processes in a whole series of heat-exchange apparatuses, characterized by constancy of the thermal action along the length (Fig. 1), can be described by differential equations with second-order partial derivatives:

Fig. 1. Diagrams of heat-exchange apparatuses and the established temperature distribution along the length.
I — heat exchanger with the temperature of the heating heat carrier constant with respect to \(x\);
II — heat exchanger with a heat flux constant with respect to \(x\), released in an insulated wall;
III — insulated pipeline;
IV — heat exchanger with a heat flux constant with respect to \(x\), entering from the heating heat carrier;
V — uninsulated pipeline.

a) for heat exchangers with the temperature of the heating heat carrier constant with respect to \(x\)

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

\[ +\left(T_{12}+\frac{T_{12}T_{23}}{T_{21}}+T_{23}\right)\frac{\partial \Delta t_1}{\partial \tau} +\Delta t_1 =\Delta t_3-\kappa_1\left(1-n+\frac{T_{23}}{T_{21}}\right)\Delta D \]
\[ -(1-n)\kappa_1T_{23}\frac{\partial \Delta D}{\partial \tau}; \tag{1} \]

b) for heat exchangers with a heat flow constant with respect to \(x\)

\[ T_{12}T_{21}v\frac{\partial^2\Delta t_1}{\partial \tau\,\partial x} +T_{12}v\frac{\partial\Delta t_1}{\partial x} +T_{12}T_{21}\frac{\partial^2\Delta t_1}{\partial \tau^2} +(T_{12}+T_{21})\frac{\partial\Delta t_1}{\partial \tau} \]
\[ =\kappa_2\Delta q-\kappa_1\Delta D-(1-n)T_{21}\kappa_1\frac{\partial\Delta D}{\partial \tau}, \tag{2} \]

where

\[ T_{12}=\frac{c_1\gamma_1V_1}{\alpha_{21}F_{21}}; \qquad T_{21}=\frac{c_2G_2}{\alpha_{21}F_{21}}; \qquad T_{23}=\frac{c_2G_2}{\alpha_{32}F_{32}}; \]

\[ \kappa_1=\frac{t_{20}-t_{10}}{D_0}\,T_{12}v; \qquad \kappa_2=\frac{1}{\alpha_{21}}. \]

After applying the Laplace transform to equations (1) and (2), the transfer functions for the disturbance channels can be represented in the following form:

a) from equation (1)

\[ W_1(p)=\frac{\Delta t_1(p)}{\Delta t_0(p)} =\exp\left(-\frac{\tau_0}{T_{12}}\right)\exp(-\tau_0p)\exp\left(\frac{\lambda}{p+\beta}\right); \tag{3} \]

\[ W_2(p)=\frac{\Delta t_1(p)}{\Delta t_3(p)} =\frac{1}{(T'p+1)(T''p+1)}\,[1-W_1(p)]; \tag{4} \]

\[ W_3(p)=\frac{\Delta t_1(p)}{\Delta D(p)} =-\kappa_1\left[1-n+\frac{T_{23}}{T_{21}}\right] +(1-n)T_{23}p\,W_2(p); \tag{5} \]

b) from equation (2)

\[ W_4(p)=\frac{\Delta t_1(p)}{\Delta t_0(p)} =\exp\left(-\frac{\tau_0}{T_{12}}\right)\exp(-\tau_0p)\exp\left(\frac{\lambda}{p+\beta}\right); \tag{6} \]

\[ W_5(p)=\frac{\Delta t_1(p)}{\Delta q(p)} =\frac{\kappa_2}{p(T_{12}T_{21}p+T_{12}+T_{21})}\,[1-W_4(p)]; \tag{7} \]

\[ W_6(p)=\frac{\Delta t_1(p)}{\Delta D(p)} =-\frac{\kappa_1}{\kappa_2}\,[1+(1-n)T_{21}p]\,W_5(p). \tag{8} \]

To obtain the transient characteristics, it is necessary to apply once again to expressions (3)—(8) the inverse Laplace transform with respect to \(\tau\). After some transformations we obtain:

for the channel \(\Delta t_0 \to \Delta t_1\)

\[ \Delta t_1= \begin{cases} 0 \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ 0<\tau<\tau_0,\\[6pt] \displaystyle \exp\left(-\frac{\tau_0}{T_{12}}\right) \left\{ \varphi_0(\tau-\tau_0) +\int_0^{\tau-\tau_0} \varphi_0(\tau-\tau_0-z)\times \right.\\[6pt] \displaystyle\left. \times \exp(-\beta z) \sum_{\kappa=0}^{\infty} \frac{\lambda^{\kappa+1}z^\kappa}{\kappa!(\kappa+1)!}\,dz \right\} \ .\ .\ .\ \tau_0<\tau<\infty, \end{cases} \tag{9} \]

where

\(\varphi_0(\tau-\tau_0)\) and \(\varphi_0(\tau-\tau_0-z)\) are the responses of the lumped mathematical model with transfer function \(W_0(p)=1\) to the corresponding disturbance, shifted in time by the amount \(\tau_0\) or \(\tau_0+z\);

\[ \lambda=\frac{\tau_0}{T_{12}T_{21}}; \]

\[ \beta=\frac{T_{21}+T_{23}}{T_{21}T_{23}} \quad \text{for case “a”;} \]

\[ \beta=\frac{1}{T_{21}} \quad \text{for case “b”;} \]

for the channels \(\Delta t_3-\Delta t_1\), \(\Delta q-\Delta t_1\), \(\Delta D-\Delta t_1\)

\[ \Delta t_1= \begin{cases} \varphi_c(\tau)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ 0<\tau<\tau_0,\\[6pt] \varphi_c(\tau)-\exp\left(-\dfrac{\tau_0}{T_{12}}\right) \left\{\varphi_c(\tau-\tau_0)+\right. \end{cases} \tag{10} \]

\[ \left. +\int_{0}^{\tau-\tau_0} \varphi_c(\tau-\tau_0-z)\exp(-\beta z) \sum_{\kappa=0}^{\infty} \frac{\lambda^{\kappa+1}z^\kappa}{\kappa!(\kappa+1)!}\,dz \right\} \ .\ .\ .\ \tau_0<\tau<\infty, \]

where \(\varphi_c(\tau)\) is the response of the corresponding lumped mathematical model

\[ W_{c1}(p)=\frac{1}{(T'p+1)(T''p+1)}; \tag{11} \]

\[ W_{c2}(p)=- \frac{ \kappa_1\left[1-n+\dfrac{T_{23}}{T_{21}}+(1-n)T_{23}p\right] }{ (T'p+1)(T''p+1) }; \tag{12} \]

\[ W_{c3}(p)= \frac{\kappa_2}{p\,(T_{12}T_{21}p+T_{12}+T_{21})}; \tag{13} \]

\[ W_{c4}(p)=- \frac{\kappa_1\left[1+(1-n)T_{21}p\right]} {p\,(T_{12}T_{21}p+T_{12}+T_{21})} \tag{14} \]

to the specified disturbance, which can be obtained quite simply.

Thus, expressions (9) and (10) are general for any of the heat exchangers considered, under any specified disturbances, and are convenient for calculation on a digital computer.

The transient characteristics can be obtained more simply with the aid of analog computing machines, since the transfer functions along the disturbance channels consist of transfer functions of elementary links.

Figure 2 presents the scheme for assembling expressions (3)—(8) on an analog computer. The modeling schemes for cases “a” and “b” differ only in the modeling schemes of the lumped model. The greatest difficulty is posed by modeling expression (3), which contains the function:

\[ \exp\frac{\lambda}{p+\beta}. \tag{15} \]

In [1] function (15) is expanded in a Taylor series. This method is expedient to use for \(\lambda/\beta < 3\), since the number of terms of the expansion increases sharply for \(\lambda/\beta > 3\).

In [2], expressions (9) and (10) are approximated by fractional-rational functions, which are then set up on the machine. In this case the structure of the transfer functions along the channels \(\Delta t_3 — \Delta t_1\), \(\Delta q — \Delta t_1\), and \(\Delta D — \Delta t_1\) is violated, which distorts the physical picture of the nonstationary heat-exchange process. This can be avoided if only characteristic (9) is approximated and substituted into (4), (5), (7), (8).

Fig. 2. Setup diagram of the mathematical model of a heat exchanger on an analog computer.
a) setup diagram for the lumped model of equations (4) and (5);
b) setup diagram for the lumped model of equations (7) and (8)

It follows from [2] and [3] that, for \(\lambda/\beta < 2\), the transient characteristic along the channel \(\Delta t_0 — \Delta t_1\) differs little from an exponential in the interval \(\tau_0 < \tau < \infty\).

Then function (15) can be represented in the form

\[ \exp \frac{\lambda}{p+\beta} = 1 + \frac{\lambda_1}{p+\beta}, \tag{16} \]

where

\[ \lambda_1 = \beta \left[\exp \left(\frac{\lambda}{\beta}\right) - 1\right]. \]

For \(\lambda/\beta > 2\), the heat exchanger must be divided into \(n\) sections with \(\lambda/\beta \leq 2\); then the model of expression (15) will consist of \(n\) series-connected functions (16). Figure 2 shows diagrams for modeling heat exchangers for \(\lambda/\beta = 2\).

The proposed modeling methods make it possible to obtain accurate values of the transient characteristics in the initial interval \(0 < \tau < \tau_0\), which is especially important in the study of control systems; to obtain models with a minimum number of operational amplifiers; to pre-

compare the structure of the transfer functions (4), (5), (7), (8), which are specific to objects with distributed parameters.

Notation: \(c\)—heat capacity; \(\gamma\)—density; \(G\)—mass; \(D\)—flow rate; \(F\)—heat-transfer surface; \(\tau_0\)—time of residence of the liquid in the heating process; \(v\)—velocity of motion of the liquid; \(x\)—current coordinate; \(\tau\)—current time; \(t\)—temperature; \(q\)—specific heat flux referred to a unit of heat-transfer surface; \(p\)—Laplace-transform variable; \(n\)—exponent in the function

\[ \alpha = \alpha_0 \left( \frac{\Delta D}{D} \right)^n . \]

Subscripts: 1—heated liquid; 2—wall; 3—heating heat carrier.

REFERENCES

1. P. A. Andrianov, I. M. Maslennikov. Mathematical modeling of heat exchangers as control objects with distributed parameters, IFZh, No. 4, 1964.

2. B. P. Korolkov. Approximation of transcendental transfer functions of heat exchangers with independent heating, Teploenergetika, No. 7, 1966.

3. V. I. Sen’kin. Transient process in a pipeline in the presence of heat release in the pipe wall, Proceedings of TsKTI, Boiler-Turbine Engineering, No. 45, 1964.