# Solution of the two-dimensional problem of non-stationary thermal conductivity in a k-layer plate and cylinder

3/19/2020 2020 - #01 Thermal physics and thermal hydraulics

Levchenko V.A. Kascheev M.V. Dorokhovich S.L. Zaytsev A.A.

https://doi.org/10.26583/npe.2020.1.06

### UDC: 536.21

The problem of determining a two-dimensional non-stationary temperature field in a k-layer cylinder and plate of length l is solved. There is a symmetrically located gap (plate) or cylindrical cavity (cylinder) in the center of these bodies. The absence of a gap or 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 bodies the symmetry condition is fulfilled. At the boundary of contact of the layers – ideal thermal contact: continuity of temperatures and heat flows. On the outer side surface and ends, heat exchange occurs according to Newton’s law with environments whose temperatures change over time according to an arbitrary law. With the help of the geometric parameter Γ in the mathematical formulation of the problem, one differential equation for both multilayer bodies is written. The problem in this statement is solved for the first time.

To solve the problem, a combined method is used: first, by using the method of finite integral transformations, differential operations on the longitudinal coordinate are excluded, and then the resulting equation in the images is solved by the Fourier method (separation of variables), bringing the determination of the time dependence of the temperature to the solution of an ordinary differential equation of first order.

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two-dimensional non-stationary heat conduction problem geometrical parameter k-layer plate and cylinder heat sources method of finite integral transformations Fourier method characteristic equation kernel of transformation environment