**2. Mathematical Model**

We will here solve a problem of the steady-state natural convection over a vertical heated plate with the temperature *Tw* located in a non-moving gas, whose temperature is also the constant *T*∞. We assume that *Tw* > *T*∞, however, the solution and its results obtained below will be valid for the case of *T*<sup>∞</sup> > *Tw* as well. As a result of heating, a boundary layer of heated gas with a thickness of δ is formed near the plate, is formed near the plate with a vertical lifting movement affecting it. In the coordinate system we have chosen, the origin is at the lower edge of the plate, the *x*-axis is directed longitudinally upward, and the *y*-axis is directed perpendicular to the plate (Figure 1). The problem is solved in a two-dimensional statement under the assumption that the plate is infinite in the *z*-direction.

In the present paper, we investigate the influence of the thermophysical properties of a gas within the framework of the van der Waals equation of state on the characteristics of natural convection in comparison with the case of an ideal gas. Therefore, as a basis for comparison, we take the results for an ideal gas, which are also obtained based on the integral approach.

Within the framework of the adopted model, we assume that the physical properties of the gas, except for the density, are constant. In this regard, we only consider the buoyancy arising from the dependence of density on temperature. The energy dissipation is not considered, since the flow rate with free convection is small. The considered process of free convection is stationary.

As a result, the temperature and velocity fields can be described by the following differential equations in the boundary layer approximation [29]:

$$\frac{\partial \mu}{\partial x} + \frac{\partial \mathbf{v}}{\partial y} = \mathbf{0},\tag{9}$$

$$
\mu \frac{\partial u}{\partial x} + \mathbf{v} \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} + g \left( 1 - \frac{\rho}{\rho\_{\infty}} \right),
\tag{10}
$$

$$
\alpha \frac{\partial T}{\partial \mathbf{x}} + \mathbf{v} \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2} \,. \tag{11}
$$

where *x* and *y* are the Cartesian coordinates, *u* and v are the streamwise (along the *x*coordinate) and normal (along the *y*-coordinate) velocity components, respectively, and *T* is the local temperature, ν is the kinematic viscosity, *g* is the gravitational acceleration, ρ is the density, α is the thermal diffusivity, and the subscript "∞" refers to the parameters outside of the boundary layer.

**Figure 1.** Schematic representation of two-dimensional natural convection boundary layer near a heated vertical plate.

As one can see from Equations (9) to (11), this model considers inertial terms and convective heat transfer. The lifting force is taken into account based on the last term of Equation (10).

In contrast to the simplified problem statement in the existing work [28], the problem statement in the present study considers the complete system of the equations of continuity, momentum, and energy, in which the inertial and convective components are explicitly taken into consideration.

The boundary conditions for system (9)–(11) are set as follows:

$$
\mu = 0, \ T = T\_{w\nu} \text{ at } y = 0,\tag{12}
$$

$$
\mu = 0, \ T = T\_{\infty \prime} \text{ at } y = \delta,\tag{13}
$$

where *δ* is the boundary layer thickness, and the subscript "*w*" refers to the parameters at the wall. The boundary conditions for the *x*-coordinate are described below in Section 3.

This problem statement must be completed with the van der Waals equation of state (3), rewritten using the gas density ρ instead of the specific volume υ:

$$\left(p + a\rho^2\right)\left(\frac{1}{\rho} - b\right) = RT\_\prime \tag{14}$$

where *p* is pressure, *R* is the individual (specific) gas constant, and *a* and *b* are the van der Waals constants.

The last term on the right-hand side of the equation of motion (10) can be transformed using the van der Waals equation of state (14). To do this, we solve Equation (14) with respect to density. The result is a cubic equation that has two complex conjugate roots and one real root.

$$\rho = \frac{1}{6b} \left( \begin{array}{c} 2 + \frac{\sqrt[3]{16}(a - 3b(bp + RT))}{\sqrt[3]{2a^3 + 9a^2b(2bp - RT)} - \sqrt{a^3 \left(a(2a + 9b(2bp - RT))^2 - 4(a - 3b(bp + RT))^3\right)}^3} + \\ + \frac{\sqrt[3]{2a^3 + 9a^2b(2bp - RT) - \sqrt{a^3 \left(a(2a + 9b(2bp - RT))^2 - 4(a - 3b(bp + RT))^3\right)}^3} \end{array} \right), \tag{15}$$

The solution to this equation is rather cumbersome, so it is difficult to use it when integrating the system of Equations (9)–(11). Avramenko et al. [28] showed that in the approximation of small values of the constants *a* and *b*, Equation (15) can be expanded in a Maclaurin series, which, in what follows, considers only the first three terms:

$$
\rho = \frac{p}{RT} - \frac{bp^2}{R^2T^2} + \frac{ap^2}{R^3T^3} \tag{16}
$$

This equation can be represented in the dimensionless form [28]:

$$\mathbf{Z}(\mathbf{1} + \mathbf{W}\mathbf{a}\_a - \mathbf{W}\mathbf{a}\_b) = \mathbf{1},\tag{17}$$

where

$$\text{Wa}\_{a} = \frac{ap}{R^{2}T^{2}}, \quad \text{Wa}\_{b} = \frac{bp}{RT} \,\text{}\,\tag{18}$$

are van der Waals numbers.

Avramenko et al. [28] validated the simplified van der Waals Equation (16) for ethylene. It was shown that Equation (16) is more accurate than the ideal gas Equation (8), and agrees well with the full van der Waals Equation (8) up to the pressure of *p* = 40 bar (or *p*/*pcr* = 0.4) and *T* = *Tcr*. In this case, the relative error of Equation (16) is about 8%, whereas the relative error of the ideal gas Equation (8) is 22.7%.

As it was demonstrated by Avramenko et al. [28], for these pressures and temperatures, the values of both van der Waals numbers for ethylene are Wa*<sup>a</sup>* = 2.78 · <sup>10</sup>−<sup>6</sup> and Wa*<sup>b</sup>* = 9.03 · <sup>10</sup><sup>−</sup>7. This completely justifies the assumption of the small values of the parameters Wa*<sup>a</sup>* and Wa*<sup>b</sup>* that lie in the background of the simplified van der Waals Equation (16) in the series form.

The density ratio ρ/ρ∞ in Equation (10) can be expressed using Equations (16)–(18). As a result, we get:

$$u\frac{\partial u}{\partial x} + \mathbf{v}\frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} + g\left(1 - \frac{Z}{1 + \beta \Delta T \theta} \left(1 - \frac{\text{Wa}\_b}{1 + \beta \Delta T \theta} + \frac{\text{Wa}\_d}{\left(1 + \beta \Delta T \theta\right)^2}\right)\right),\tag{19}$$

Here, the compressibility factor *Z* and van der Waals numbers Wa*<sup>a</sup>* and Wa*<sup>b</sup>* are defined using the parameters outside of the boundary layer (subscript "∞").

The dimensionless local temperature in the boundary layer is defined as:

$$\Theta\_{\parallel} = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}} = \frac{T - T\_{\infty}}{\Delta T}, \ \Delta T \ = T\_w - T\_{\infty \prime} \tag{20}$$

whereas

$$
\beta = \frac{1}{T\_{\infty}} \,\prime \tag{21}
$$

is the volume expansion coefficient [9–11].
