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Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function q varying in space and time. [5]
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 ...
The problem of heat transfer in the presence of liquid flowing around the body was first formulated and solved as a coupled problem by Theodore L. Perelman in 1961, [1] who also coined the term conjugate problem of heat transfer. Later T. L. Perelman, in collaboration with A.V. Luikov, [2] developed this approach further.
is the temperature of the environment; i.e., the temperature suitably far from the surface (SI unit: K), is the coefficient of heat transfer (SI unit: second). Solving the initial-value problem using separation of variables gives
An explicit scheme of FDM has been considered and stability criteria are formulated. In this scheme, temperature is totally dependent on the old temperature (the initial conditions) and θ, a weighting parameter between 0 and 1. Substitution of θ = 0 gives the explicit discretization of the unsteady conductive heat transfer equation.
Each aforementioned geometry can be analyzed by three charts which show the midplane temperature, temperature distribution, and heat transfer. [1] Although Heisler–Gröber charts are a faster and simpler alternative to the exact solutions of these problems, there are some limitations. First, the body must be at uniform temperature initially.
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
In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. [1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.