Modeling the conjugate problem of heat transfer at hot jet impingement onto a cooled surface

The paper solves the problem of thermal conductivity inside a flat plate under the impact of a hot jet of nitrogen impinging from one side and cooled by a gas flow from the other side. In this formulation of the problem, there may be local maxima and minima of the temperature inside the plate, caused by an uneven distribution of heat fluxes along the plate.

On boundary optimal control of a coefficient in a nonlinear parabolic equation

Работа связана с изучением нелинейных параболических систем, возникающих при моделировании и управлении физико-химическими процессами, в которых происходят изменения внутренних свойств материалов. Исследовано оптимальное управление одной из таких систем, которая включает в себя краевую задачу третьего рода для квазилинейного параболического уравнения с неизвестным коэффициентом при производной по времени, а также уравнение изменения по времени этого коэффициента. Обоснована постановка оптимальной задачи с финальным наблюдением искомого коэффициента, в которой управлением является граничный режим на одной из границ области. Получено явное представление дифференциала минимизируемого функционала через решение сопряженной задачи. Доказаны условия ее однозначной разрешимости в классе гладких функций. Полученные результаты имеют практическое значение для приложений в различных технических областях, медицине, геологии и т.п. Приведены некоторые примеры таких приложений. The work is connected with investigation of nonlinear parabolic systems arising in the mathematical modeling and control of physical-chemical processes in which inner properties of materials are subjected to changes. We consider optimal control in one of such systems that involves a boundary value problem of the third kind for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional equation for a time dependence of this coefficient. The optimal problem with a boundary control regime is justified for the given final observation of the sought coefficient. The exact representation for the differential of the minimization functional in terms of the solutions of the conjugate problem is obtained. The form of this conjugate problem and conditions of unique solvability in a class of smooth functions are shown. The obtained results are important for applications in various technical fields, medicine, geology, etc. Some examples of such applications are discussed.

On the recovering of acoustic attenuation in 2D acoustic tomography

The inverse problem of recovering the acoustic attenuation in the inclusions inside the human tissue is considered. The coefficient inverse problem is formulated for the first-order system of PDE. We reduce the inverse problem to the optimization of the cost functional by gradient method. The gradient of the functional is determined by solving a direct and conjugate problem. Numerical results are presented.

Applying the scalability apparatus to estimate the thermal efficiency of a single finned tube

The paper explores the possibility of scaling the integral parameters of the cooling system operation using the working elements with the length from 0.01 to 0.5 m. Numerical solution of the conjugate problem of external aeromechanics, internal hydrodynamics and heat exchange is carried out. Parametric studies of the cooling process and aerodynamic resistance of finned tubular elements of various lengths are performed. As a result of generalization and unification of the computational experimental data, a numerical coefficient has been obtained to calculate the necessary integral characteristics for a cooling element of the assigned length.

The effect of model material properties on thermal imaging measurements

The temperature distributions arising on the surface of the solid model are investigated. A flow with a shock wave and laminar boundary layer interaction is considered as the main flow. Calculations of the conjugate problem are performed based on RANS approach. The investigated cases correspond to homogeneous material of the solid model and the model consisting of two materials with different thermal conductivity. In this study, the model designs are found to obtain a temperature distribution suitable for thermal imaging measurements. In addition, in the calculations, the effect of forced heating of the model wall on the temperature distribution is investigated.

Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) ⁢ u ′′ ⁢ ( t ) = λ ⁢ f ⁢ ( t , u ⁢ ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) {A((b*u^{q})(1))} appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the ( n - 1 , 1 ) {(n-1,1)} -conjugate problem.

Numerical Simulation of Intrachamber Processes in the Power Plant

This paper considers the issues of numerical modeling of nonstationary spatial gas dynamics in the pre-nozzle volume of the combustion chamber of a power plant with a cylindrical slot channel at the power plant of the mass supply surface. The numerical simulation for spatial objects is based on the solution conjugate problem of heat exchange by the control volume method in the open integrated platform for numerical simulation of continuum mechanics problems (openFoam). The calculation results for gas-dynamic and thermal processes in the power plant with a four-nozzle cover are presented. The analysis of gas-dynamic parameters and thermal flows near the nozzle cover, depending on the canal geometry, is given. The topological features of the flow structure and thermophysical parameters near the nozzle cap were studied. For the first time, the transformation of topological features of the flow structure in the pre-nozzle volume at changes in the mass channel’s geometry is revealed, described, and analyzed. The dependence of the Nusselt number in the central point of stagnation on the time of the power plants operation is revealed.

NUMERICAL AND ANALYTICAL SOLUTION OF THE HEAT EXCHANGE PROBLEM

In this work, a numerical-analytical analysis of the solution of the conjugate problem is carried out, which most fully takes into account the actual conditions of heat transfer in tubular heat exchangers under the condition of a laminar flow of the working fluid in pipes and channels. When solving the conjugate problem, the role of the non-Newtonian properties of the indicated fluid is taken into account. The obtained quantitative estimates of the errors, which are due to the fact that the conjugacy of the problem and the indication on the wall of the boundary conditions of the third kind were not taken into account. The given conditions and criteria, the fulfillment of which reduces the problem of heat transfer under boundary conditions of the third kind to a problem under boundary conditions of the first kind.Key words: heat exchange problem, pipe with a round cross section, boundary conditions of the third and first kind, non-Newtonian fluid, heat exchangers, heat transfer.

NON-STATIONARY BEHAVIOR OF A SOLID PROPELLANT CHARGE FOR NOZZLELESS SOLID ROCKET MOTORS UNDER GAS-DYNAMIC LOAD

The numerical solution to a conjugate problem of an unsteady flow of combustion products in a flow path of the nozzleless solid rocket motor (SRM) and the oscillation of a solid propellant charge under the action of the forces directed from combustion products is considered. The Navier-Stokes equations for a compressible viscous gas are used to mathematically describe the flow of the combustion products. To model the charge oscillations, the equations of solid mechanics are applied, which take into account the propellant hyperelasticity. Pressure distributions and the propellant burning rate along the charge channel are presented for different models of the propellant burning rate. It is revealed that at the stage of SRM design, the use of the burning rate law, determined by pressure in the head of the combustion chamber, is more preferable in order to assess the internal ballistic characteristics. The solution to the conjugate problem shows that in the nozzleless SRM with the propellant having low Young's modulus, resonance can occur, which causes uncontrolled charge oscillations.