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In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the ...
Engineering Equation Solver (EES) is a commercial software package used for solution of systems of simultaneous non-linear equations.It provides many useful specialized functions and equations for the solution of thermodynamics and heat transfer problems, making it a useful and widely used program for mechanical engineers working in these fields.
This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation. In order to be concrete, this article focuses on heat flow, an important example where the convection–diffusion equation applies. However, the same mathematical analysis works equally well to ...
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Temperature gradient: No standard symbol K⋅m −1: ΘL −1: Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer
Joule heating (also known as resistive, resistance, or Ohmic heating) is the process by which the passage of an electric current through a conductor produces heat.. Joule's first law (also just Joule's law), also known in countries of the former USSR as the Joule–Lenz law, [1] states that the power of heating generated by an electrical conductor equals the product of its resistance and the ...
This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance ...
Basis functions are linear or higher order polynomials. Applying the differential equation and the boundary conditions of the problem to the basis functions, a system of equations is formulated using either the Ritz or Galerkin method. Finally, a direct or iterative method is employed to solve the system of linear equations. [3]
Combined with the Seebeck equation for , this can be used to solve for the steady-state voltage and temperature profiles in a complicated system. If the material is not in a steady state, a complete description needs to include dynamic effects such as relating to electrical capacitance, inductance and heat capacity.