A variational method in the problem of convective heat transfer near the vertical plane of the outer building fence in terms of free air movement

An unconventional method for solving the heat transfer problem for free convection near the vertical surfaces of the enclosing structures is proposed. It is based on the laws of thermodynamics of irreversible processes. A variational principle is used to find the unknown functions that analytically express the velocity fields and other flow quantities in the boundary layer near the vertical walls. A variational formulation of the hydrodynamic and thermal boundary layer near a vertical plane surface in conditions of free convection is given. It is given an analytical solution of the problem of the distribution of the vertical component of velocity and temperature in the wall boundary layer in the laminar flow. Theoretical formulas for the thickness of the boundary layer are obtained, as well as for the local and mean values of convective heat transfer coefficient.


Introduction
Most of the phenomena occurring in nature and technology are associated with nonequilibrium states and irreversible processes, and therefore they are described by classical thermodynamics only in the first approximation. The transition from thermodynamics (more correctly from thermostatic) to equilibrium states to the thermodynamics of irreversible processes marks a serious progress in the development of a number of sciences. This section of thermodynamics finds an increasing application in various fields of modern physics, chemistry and technology.
In this paper we propose a non-traditional modern method for solving the heat transfer problem for free convection near a vertical wall, based on the laws of thermodynamics of irreversible processes. A variational principle is used to find the unknown functions that analytically express the velocity and flow temperature fields in the boundary layers of the air flow near the vertical plane.
Let us consider the process of nonisothermal flow of a viscous liquid near a vertical wall having a constant temperature tw. The flow temperature far from the wall is the same and is equal to t0. We set tw> t0, where t0 is the air temperature in the unperturbed flow.
As it nears the wall, the stream temperature rises, and its density decreases. As a result of the motion of the heated flow layers near the wall, the hydrodynamic and thermal boundary layers δ(x) are formed. The velocity Wx(y) changes from a zero value on the wall to a maximum value Wm(ym), and then again decreases to zero at the outer boundary δ(x). The air temperature decreases from t = tw to a value of t0 in the unperturbed medium when y>δ (x). The profiles of the velocity and temperature fields are shown in the Fig. 1. The dependence of the flow density  on the temperature will be taken as linear air density at t0; β -coefficient of volumetric expansion of air. Other physical parameters of air will be considered as constant.
In the boundary layer let us single out an elementary volume of liquid bounded by the planes АА 1 and ВВ 1 , the wall plane and parallel plane CD at an arbitrary distance from the wall surface. The width of the layer in the direction perpendicular to the plane of the drawing will be equal to one. Consequently, the value of the allocated volume is At an arbitrary distance y from the wall surface, we draw a plane CD parallel to it. The frictional force Ffr acting in the indicated plane on the selected element is expressed by the formula whereτ -the shear stress or the frictional stress in the indicated plane. The second work of friction will be equal to the product of this force by the rate of liquid deformation in the plane under consideration: Formula (4) expresses the amount of energy that is dissipated (passes into thermal energy and disperces). The flow of this energy is directed toward lower temperatures to the outer boundary of the wall layer, i.e. in the element C1,D1,B1,A1. It should be noted that, in addition to the indicated energy (4), there will be losses associated with changes in the velocities х However, because of the smallness of these quantities compared with у w х   they can be neglected.
The second change in the energy density of Accordingly, the integral characteristic of viscous friction in the entire selected element AA1 BB1 will have the form For a laminar flow in a boundary layer, the quantity  can be determined according to Consequently, the integral (6) can be rewritten in the following form: From the theory of the boundary layer it is known that the distribution of velocity х w along the coordinate у is given by .
where     d d ' .
The second change in the density of the potential energy of the selected element of the boundary layer will be determined by the work of the lifting forces: where From the thermodynamic point of view, in the flow of a viscous liquid, degradation occurs -energy depreciation, as in all other real processes taking place in nature. However, nature is not "wasteful". Internal nonequilibrium processes always act in a direction that causes a decrease in the rate of entropy increase. This determines the stability of stationary states.
In the thermodynamics of irreversible processes, this law is formulated as follows: as the system moves to a stationary-non-equilibrium state, the value of the incremental increase in entropy decreases and, when the stationary-nonequilibrium state is reached, it assumes the smallest value compatible with external constraints [1][2][3]. Thus, the fields of physical values for stationary nonequilibrium states can be investigated on the basis of the extremal principle. If we neglect the change in the kinetic energy of the flow, then following the well-known in mechanics Helmholtz principle, we can assert that the integral in the steady-state process will have a minimum.
Taking into account expressions (9) and (10), it will be min 1 in the boundary layer can be reduced to the problem of the variations calculus [4][5][6][7][8], namely, to the study of the functional (12) for an extremum under the following boundary conditions 1) With respect to the required function     these conditions can be rewritten in the following form: The fourth boundary condition follows from the equation of a viscous fluid motion in the boundary layer: The necessary condition for the extremum of the integral (12) (the Euler-Lagrange equation) for the case under consideration has the following form: where   Performing the operations of differentiation corresponding to the expression (14), we obtain the following differential equation: (16) In the first approximation = (1 − ) 2 , consequently, equation (15) takes the following form: Denote ′ = ( )φ, then we obtain Equation (18) has the following general solution: Since = ′, integrating this equation once again with respect to the variable η, we obtain = с 1 (1 − Let us now clarify the temperature distribution in the boundary layer. To do this, we consider the differential equation of the layer energy: Taking into account the continuity equation it will be where = ∫ 0 . We write the obvious relations: where ′ = ; " = 2 2 . Taking into account expressions (24), (25), the energy equation (22) can be represented in the following form: We denote: Then " + ′ = 0; We seek the solution of (28) from We find the derivatives ψ 'and ψ "and substitute them in equation (28). Then we recieve: (2 2 + 6 3 + 12 4 2 + ⋯ ) + 3 ( After multiplication of series, it will be: As it nears the wall, the accuracy of the solution increases. This is especially important, since the heat transfer under consideration is mainly determined by the value of the flow temperature gradient at the wall surface. Substitution of this expression into the differential equation (15) leads to a solution that differs little from the solution (21) for φ found earlier, so we will not further refine the function ψ (2). Let us find the ordinate value corresponding to the maximum velocity in the boundary layer. According to the formula (21) ′ ( ) = 3 (