The heat conduction equation for a straight fin with an arbitrary profile in the presence of energy release in the fin is obtained in the article. The resulting equation differs from the approximate equation given in the literature by the presence of energy release and a more accurate determination of the length of the arc element. The equation is solved for the fin of a rectangular profile with continuously operating heat sources. The efficiency of the fin and the heat flow through the base of the fin are determined. It is shown that energy release in the fin increases its efficiency in comparison with the efficiency of the fin in the absence of energy release. There is also a decrease in the heat flow in the presence of energy release in the fin. The restriction on the values of energy release in the fin is found as condition for the applicability of the finning. The fin efficiency must be less than one. If the efficiency exceeds one, the fin plays the opposite role: the flow is directed in the reverse side. To increase the build-up coefficient of the surface, tend to reduce the distance between the fins. There is a limit to such reduction. Theoretically, the distance between the fins should be at least double the maximum thickness of the boundary layer. As experience shows, this distance can be reduced to about one thickness. An approach to achieve the largest build-up coefficient at finning is described in the article.
The problem of determining a non-stationary three-dimensional temperature field in a k-layer cylinder of length is solved. There is a symmetrically located cylindrical cavity in the center of this body. The absence of a cavity is a special case of the problem. In each layer, there are heat sources, depending on the coordinates and time. The initial temperature of the layers is a function of the coordinates. In the center of the body the symmetry condition is fulfilled. At the boundary of contact of the layers - ideal thermal contact: continuity of temperatures and heat flows. On the inner and outer side surfaces and ends, heat exchange occurs according to Newton's law with environments whose temperatures change over time according to an arbitrary law. The periodicity condition is set for the angle φ. The problem in this statement is solved for the first time. For the solution of the problem the following approach is used: by means of the method of finite integral transformations differential operations on longitudinal coordinate, angle and transverse coordinate are sequentially excluded, and the determination of time dependence of temperature is reduced to the solution of the ordinary differential equation of the first order.
The heat conduction equation for an annular fin with an arbitrary profile in the presence of energy release in the fin is obtained in the article. The resulting equation differs from the approximate equation given in the literature by the presence of energy release and a more accurate determination of the length of the arc element. As boundary conditions, the temperature of the base of the fin is set, and at the end of the fin, heat exchange occurs according to the Newton - Richmann law with the environment. The equation for the fin of a rectangular profile is an inhomogeneous modified Bessel equation. Its solution contains the Bessel functions of the imaginary argument of the first and second kind of zero order. The efficiency of the fin and the heat flow through the base of the fin are determined. The energy release in the fin increases its efficiency compared to the efficiency of the fin in the absence of energy release, and also reduces the heat flow. The restriction by the values of energy release in the fin is found as condition for the applicability of the finning. The fin efficiency must be less than one. If the efficiency exceeds one, the fin plays the opposite role: the flow is directed in the reverse side. In the article, an expression is obtained for the surface build-up coefficient kh. When calculating the heating (cooling) of a body with a finned surface, the heat transfer coefficient should be increased by kh times.
Safety of the ″Master″ reactor plant in conditions of blocking flow section of a technological channel in the form of a Field’s tube is considered. The possibility of cooling a separate technological channel in the passive mode, considering heat-conducting zirconium matrix, is evaluated. It is noted that the project of the power unit with the ″Master″ reactor with 30 MW power has all the prerequisites for a successful application. General description of the power unit is presented. The mathematical formulation of the problem is formulated. Temperatures in fuel assembly with averaged properties and zirconium matrix (E-110 alloy) are described by differential equations of thermal conductivity with appropriate initial and boundary conditions. Temperatures in external and internal channels of the Field’s tube are calculated using a lumped parameters model and depend only on time. The mathematical formulation also includes an equation for determination of the flow rate of the coolant in the natural circulation circuit. Temperatures in assembly and zirconium matrix are determined by the tridiagonal matrix algorithm. System of equations for determination of temperatures in channels is solved analytically in a quasi-stationary approximation using the Laplace transformation. The developed mathematical model is implemented in the form of TK computer code. Calculations showed that the fuel temperature remains relatively low, and there is no melting of fuel elements.
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