On the Calculation of Heat Exchanger Dynamics
Abstract Generated abstract
This paper examines how variations in heat exchanger load affect the coefficients in a dynamic model whose parameters depend on the heat transfer coefficient and flow velocity. Using a coil heat exchanger as an example, the authors compare time characteristics calculated for several flow rates in both lumped and distributed mathematical models, including the residence time corresponding to the actual load. The calculations indicate that despite substantial coefficient changes over a broad load range, the time response changes modestly, and for load deviations within approximately 25 percent of nominal the error remains within about 5 percent. The study concludes that nominal dynamic coefficients may be used for practical time characteristic calculations if the appropriate residence time is substituted.
Full Text
On the Calculation of Heat Exchanger Dynamics
A. F. Fedorov, P. A. Andriyanov
(Presented by the scientific seminar of the Department of Processes, Machines, and Apparatuses of Chemical Production)
The coefficients \(T_{12}\), \(T_{21}\), \(T_{23}\) in the equation
\[ T_{12}T_{23}v \frac{\partial^2 t_1(x,\tau)}{\partial x\,\partial \tau} + (T_{21}+T_{23}) \frac{T_{12}}{T_{21}} v \frac{\partial t_1(x,\tau)}{\partial x} + T_{12}T_{23} \frac{\partial^2 t_1(x,\tau)}{\partial \tau^2} + \]
\[ + \left( T_{12} + \frac{T_{12}T_{23}}{T_{21}} + T_{23} \right) \frac{\partial t_1(x,\tau)}{\partial \tau} + t_1(x,\tau) = t_3(x,\tau), \]
which describes the dynamic properties of heat-exchange apparatuses, are variable. Their value is determined by the magnitude of the heat-transfer coefficient \(\alpha_{21}\), which depends on the velocity of motion of the medium inside the tubes. The variability of the coefficients \(T_{12}\), \(T_{21}\), \(T_{23}\) is usually neglected within certain sufficiently small limits of variation of the flow rate of the heated medium relative to the nominal (design) load.
Under production conditions the load may differ from the design load. Then the coefficients \(T_{12}\), \(T_{21}\), \(T_{23}\) must be recalculated.
Practical calculations show that, even with a considerable deviation from the design loads and a substantial change in the coefficients \(T_{12}\), \(T_{21}\), and \(T_{23}\), the ordinates of the time characteristics of the lumped mathematical model (LMM) [1] differ little from one another. Taking into account in the solution [1] the residence time of the medium in the apparatus, corresponding to the load, \(\tau_0\), one can calculate with sufficient accuracy the time characteristic of the distributed mathematical model of the heat exchanger.
Let us show this by an example. For a coil heat exchanger [2], at different loads the values of the coefficients \(T_{12}\), \(T_{21}\), \(T_{23}\) were determined (Table 1), and the time characteristics for the LMM were calculated (Fig. 1).
Despite the wide range of loads \((0.42 \div 1.44\ \text{m}^3/\text{h})\) and the considerable difference in the values of the coefficients, the maximum difference
Table 1
| Parameters | Experiment no. 1 | Experiment no. 2 | Experiment no. 3 | Experiment no. 4 | Experiment no. 5 |
|---|---|---|---|---|---|
| \(G,\ \text{m}^3/\text{h}\) | 0.42 | 0.565 | 0.88 | 1.06 | 1.44 |
| \(T_{12},\ \text{s}\) | 5.16 | 3.90 | 2.93 | 2.57 | 2.03 |
| \(T_{21},\ \text{s}\) | 3.15 | 2.37 | 1.75 | 1.57 | 1.24 |
| \(T_{23},\ \text{s}\) | 1.41 | 1.55 | 1.62 | 1.65 | 1.69 |
| \(\tau_0,\ \text{s}\) | 19.6 | 14.6 | 9.36 | 7.80 | 5.70 |
Fig. 1. Time characteristics of the SMM heat exchanger
the ordinates of the time characteristics does not exceed 11%. If the loads differ from the calculated one by less than $\pm 25\%$, then the difference in ordinates does not exceed 5%. Thus, over the time interval $(0,\tau_0)$ the error in calculating the time characteristic [1] of the distributed mathematical model will not exceed 5%. Over the interval $[\tau_0,\infty]$ the error will likewise be less than 5%. This can be confirmed by analyzing the structure of the solution [1]. As an example, let us calculate the time characteristic for $G_3=0.88\ \text{m}^3/\text{h}$ using the coefficients $T_{12}, T_{21}, T_{23}$ calculated for $G_2=0.565\ \text{m}^3/\text{h}$ and $G_4=1.06\ \text{m}^3/\text{h}$, and compare it with the actual one (Fig. 2). The difference in loads is 24.5% and 20.5%; the maximum error in calculating the time characteristic does not exceed $\pm 4\%$.
Fig. 2. Time characteristics of the distributed mathematical model of the heat exchanger:
1. actual, $G_3=0.88\ \text{m}^3/\text{h}$;
2. calculated using coefficients for $G_2=0.565\ \text{m}^3/\text{h}$ and $\tau_0=9.36\ \text{s}$;
3. calculated using coefficients for $G_4=1.06\ \text{m}^3/\text{h}$ and $\tau_0=9.36\ \text{s}$
Conclusions
The coefficients $T_{12}, T_{21}, T_{23}$ of the heat-exchanger dynamics equation, calculated for the nominal load, can, with accuracy sufficient for practice, be used to calculate time characteristics at loads differing from the nominal by no more than $\pm 25\%$.
In the solution it is necessary to substitute the corresponding residence time of the medium in the apparatus.
Notation: \(t\) — temperature; \(v\) and \(\tau_0\) — velocity of motion and residence time of the heated medium in the apparatus; \(T=\frac{c\gamma V}{\alpha F}\) — time constant; \(c\) — heat capacity; \(\gamma\) — specific weight; \(V\) — volume; \(F\) — heat-transfer surface; \(\alpha\) — heat-transfer coefficient; \(x\) — spatial coordinate chosen in the direction of motion of the heated medium; \(\tau\) — time.
Subscripts: 1 — heated medium; 2 — wall; 3 — heating heat carrier.
REFERENCES
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P. A. Andrianov, A. F. Fedorov. Izv. AN SSSR Energetika i transport, No. 6, 114–129, Moscow, 1967.
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P. A. Andrianov, A. F. Fedorov. In the collection: Algorithmization of Calculations of Processes and Apparatuses of Chemical Production, Technologies of Processing and Transport of Oil and Gas on Digital Computers, issue 3, 114–118, Kiev, 1967.