DOI: https://doi.org/10.20535/1810-0546.2018.5.147067

### Amplitude-Frequency Characteristics of the System “Mixing–Mixing” Heat Exchanger + Heat Transfer”

Igor D. Lucheyko, Roman V. Kotsiurko

#### Abstract

Background. Mathematical modeling of continuous technological processes in non-stationary conditions caused by actions of various destabilizing factors is an actual problem. At the same time, analytical solutions have significant advantages over numerical ones, since they allow us to find out the nature of causation links in the analyzed properties of the modeling object and, as a practical result, to give physically well-founded recommendations for improving the efficiency of its functioning. The possible fluctuations in the temperature of heat carrier (HC) at the inputs of the flow heat exchanger (HE) cause instabilities for heat exchange processes. This usually leads to the efficiency decrease of the HE. Therefore, the calculation of its amplitude-frequency characteristics (AFC) has a significant (weighty) value.

Objective. The aim of the paper is to find out the behavior of the ideal “mixing–mixing” HE in stationary mode and harmonic fluctuations of temperature at the inputs of the apparatus and calculation of its AFC.

Methods. Known mathematical model in the form of a system of linear differential equations (reduced summed to dimensionless form) to calculate the effect of harmonic fluctuations of temperature on the stationary mode of the system “HE + heat transfer” was used.

Results. It is shown that the efficiency of the stationary mode of operation of the ideal “mixing – mixing” HE can be estimated by the efficiency indicator , where – transfer number (k – heat transfer coefficient across the surface area F – the volumetric flow rates of HC and their heat capacities). When , when (the process of heat transfer is much more intense compared with the process of heat outlet by flows from HE). AFC of the system is calculated, where – the simplex of the amplitudes of temperature fluctuations at the inputs and outputs; – frequency complex (w – cyclic frequency, t02 – average time of stay of cold HC in HE).

Conclusions. In the case of common-phase fluctuations of temperatures at the inputs of HE AFC don’t depend on Ai, i.e. they are equal to the AFC of HE as two flow devices of the ideal mixing. In the case of antiphase fluctuations of temperature the values are equal when the formal rearrangement of values Ai: , which is a reflection of the thermal “equality” of HC. When ; when at . In the case (the intense heat transfer reduces the amplitudes of temperature fluctuations at the outputs of the HC). At relatively high frequencies, stationary mode is practically not violated at any values of Ai: . Therefore, in order to ensure practically stationary mode of HE operation, it is necessary to get rid of low harmonics of possible temperature fluctuations at the inputs.

#### Keywords

Mathematical modeling; “Mixing–mixing” heat exchanger; Performance indicator; Harmonic temperature fluctuations; Amplitude-frequency characteristic

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