Classical Solution of Non-stationary Heat Equation with
Nonlinear Logarithmic Source Using Generalized
Functional Separable Method (GFSM)

A physical model of the thermal process in the roll caliber during the rolling of the tape on a two-roll rolling mill was constructed. A mathematical model of the temperature field of a rolling hollow roll of a rolling state of a cylindrical shape rotating about its axis with constant angular velocity is proposed. The mathematical model takes into account different conditions of heat exchange of the inner and outer surfaces of the roll with the belt and its surrounding environment. The temperature field of a hollow roll of a rolling mill is considered as an initial boundary-value problem for a homogeneous non-stationary heat equation with inhomogeneous, nonlinear boundary conditions, which also depend on the angle of rotation of the roll around its axis. The equation describes the temperature field of the rolls during uncontrolled heat transfer during rolling. It significantly depends on the time and number of revolutions around its axis. With a large number of revolutions of the roll around its axis, a quasi-stationary temperature distribution occurs. Therefore, the simplified problem of determining a quasistationary temperature field, which is associated with a thermal process that is time-independent, is considered further in the work. In this case, the temperature field is described using the boundary value problem in a ring for a homogeneous stationary heat equation with inhomogeneous boundary conditions and heat transfer conditions outside the ring, which lie from the angular coordinate. After the averaging operation, the solution of this problem is reduced to solving the equivalent integral equation of Hammerstein type with a kernel in the form of the Green's function. The Mathcad computer mathematical system builds the temperature distribution of the roll surface. An algorithm for solving a inhomogeneous problem was developed and the temperature distribution of the roll was constructed.

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Semi-discrete numeric solution for the non-stationary heat equation using mimetic techniques

European Journal of Physics ◽

10.1088/0143-0807/35/6/065013 ◽

2014 ◽

Vol 35 (6) ◽

pp. 065013

Author(s):

R Hernández-Walls ◽

J Castillo ◽

E M Rojas-Mayoral ◽

J A Girón-Nava

Keyword(s):

Heat Equation ◽

Stationary Heat ◽

Numeric Solution

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Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

Journal of Heat Transfer ◽

10.1115/1.4028082 ◽

2014 ◽

Vol 136 (11) ◽

Cited By ~ 3

Author(s):

Antonio Campo ◽

José Garza

Keyword(s):

Heat Flux ◽

Differential Equations ◽

Heat Equation ◽

Ordinary Differential Equations ◽

Surface Heat Flux ◽

Method Of Lines ◽

Surface Heat ◽

Second Order ◽

Partial Differential ◽

Stationary Heat

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

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