**1. Introduction**

Nowadays, the need for knowledge of the physical and chemical properties of gases, the features of their behavior in real production conditions, and during their transportation and storage has significantly increased. The rational and efficient design of various technological processes involving gases can significantly reduce the cost of industrial production.

In technical thermodynamics, it is customary to refer to real gases as the gaseous state of any substance in the entire range of its existence, that is, at any pressures and temperatures. Under appropriate conditions, real gas can be liquefied or converted into a solid state. It is expected that the development of small-scale energy in the coming years will be associated with the widespread use of liquefied natural gas, which is recognized as one of the most promising types of energy carriers. The liquefaction of gases is important when storing and transporting this type of fuel. In its liquefied form, natural gas occupies only about 1/600 of its gaseous volume; therefore, it is easier and more economical to transport it [1–3].

The vapors of various substances, such as water, ammonia, methyl chloride, sulfur dioxide, and others are widely used in technological applications. The most widely used is water vapor, which is the main working medium of steam engines, heating, and other devices [4–7].

It is known that the ideal gas model allows a satisfactory description of the state of real gases only in a relatively small region of the variation of the state parameters. To design and determine the optimal operating conditions for heat exchange equipment, calculations must take into account the real thermophysical properties of the gas. The use of the ideal gas equation can lead to a significant error in determining the parameters of the gas state, up to 100%. The authors in [8] emphasized that for simple, mono-, and diatomic gases, such as air, helium, nitrogen, etc., under certain conditions, the model for ideal gases is well-suited. For gases with a more complex organic structure, the effects of real gases are more significant.

Taking air as an example, as a mixture of mono- and diatomic gases, the limits of the applicability of the ideal gas model for air are low densities (large specific volumes ν), low pressures (<10 bar), and moderate/high temperatures (up to 600 ◦C) [9,10].

From the point of view of the molecular theory of the structure of matter, a real gas is a gas, the properties of which depend on the interaction and size of the molecules. To date, more than 150 equations of state for real gases are known. One of the most widely known approximate equations of state for real gases is the van der Waals equation [9,10]. Some of the other equations of state for real gases are refinements of the van der Waals equation, while others were obtained experimentally.

Three forms of the van der Waals equation are used by different authors [9–13]:

$$\left(p + n^2 \frac{a\_\*}{V^2}\right)(V - nb\_\*) = nR\_mT\_\prime \tag{1}$$

$$(p + \frac{a\_\*}{V\_m^2})(V\_m - b\_\*) = R\_m T \tag{2}$$

$$(p + \frac{a}{\upsilon^2})(v - b) = RT\_\prime \tag{3}$$

where *Vm* = *V*/*m=Mi*/ρ is molar volume, *n* = *m*/*Mi* is the number of moles, whereas *R* = *Rm*/*Mi* and *Rm* = 8314 J/(mol·K).

Because the first and the second derivatives of *p* with respect to ν at the critical point must be zero, one can obtain the following relations, which link the empirical constants *a* and *b* (or *a*<sup>∗</sup> and *b*∗) in the Equations (1)–(3) with the parameters in the critical point [11–13]

$$p\_R = \frac{a\_\*}{27b\_\*^{2'}} \; \; T\_R = \frac{8a\_\*}{27R\_mb\_\*} \; \; \tag{4}$$

$$a = 3p\_{cr}v\_{cr}^2 \; b = \frac{v\_{cr}}{3} \; , \; Z\_{cr} = \frac{p\_{cr}v\_{cr}}{RT\_{cr}} = \frac{3}{8} \; . \tag{5}$$

$$a = \frac{a\_\*}{M\_i^{2'}}, \ b = \frac{b\_\*}{M\_i}. \tag{6}$$

Here the compressibility factor *Z* accounts for the deviation of a real gas from ideal-gas behavior at a given temperature and pressure.

$$Z = \frac{p}{RT\rho} = \frac{pv}{RT\rho} = \frac{pV}{mRT}.\tag{7}$$

The experimentally measured values of the compressibility factor *Zcr* for real gases vary over the range *Zcr* = 0.2 ... 0.3. To make the van der Waals equation of state more accurate for each particular real gas, the constants *a* and *b* are determined via comparisons with precise experiments over a wider range of the parameters instead of from a single point [11–13].

For ideal gases, *Z* = 1 by definition, which results in the known ideal-gas equation [9,10]:

$$pV = m\frac{R\_m}{M\_i}T,\ pV = nR\_mT,\ p\upsilon = \frac{R\_m}{M\_i}T = RT.\tag{8}$$

where *n* = *m*/*Mi* is the number of moles.

For example, in recent works [14–16] on battery design, the van der Waals equation was used as a real gas model versus an ideal gas model. In these studies, various trends in the behavior of a real gas during its compression and expansion are presented and analyzed in both isothermal and adiabatic processes.

In the studies [16,17], several equations of state of a real gas are used, taking into account the interaction of water vapor molecules. The actual physical properties of humid air have been determined, and their influence on the conjugate heat and mass transfer for various conditions has been assessed. This provided an increase in the accuracy of

predictions of heat and mass transfer processes when designing contact heat exchangers, convective drying plants, hygroscopic desalination plants, compressors with the injection of water or steam, as well as combustion chambers where flue gases are mixed with steam.

Heat transfer during natural and mixed convection has been studied extensively. In particular, some works [18–24] investigated the cases of natural and mixed convection in different geometries for the different ranges of physical parameters in the frames of the ideal gas model. The cases of real gases were considered in the studies [25–27].

Avramenko et al. [28] analytically solved the problem of natural convection in van der Waals gases near a heated vertical plate. Based on the use of the simplified van der Waals equation, analytical solutions were obtained for the profiles of velocity, temperature, and normalized Nusselt numbers. The limits of the applicability of the simplified van der Waals equation were determined. The data obtained were compared with the ideal gas model.

The objective of this work is to further study the influence of the thermophysical properties of a real gas in the framework of the above-mentioned simplified van der Waals model on heat transfer during natural convection near a heated vertical plate. A novel solution to the problem will be obtained for the first time using the integral method and compared with the previously obtained analytical solution. To authors' knowledge, such a solution has not been published in the literature yet.
