Equation of heat conduction

Example The differential equation of heat conduction in a moving fluid with velocity components is... 

The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. 

We transform the equation of heat conduction for the steady regime,... 

A simultaneous analysis of the equation of heat conduction, taking into account the release of heat resulting from the chemical reaction, and the diffusion equation which governs the concentrations of the reacting substances and reaction products, and also allowing for variation of these concentrations due to the chemical reaction, leads to conclusions which are very important for subsequent study.1... 

On the basis of this we can express the concentrations which enter into the equation of the chemical reaction rate in terms of the temperature, which will permit us to confine ourselves to consideration of the equation of heat conduction alone. 

Studying anew the differential equations of heat conduction, diffusion, gas motion and chemical kinetics under the conditions of a chemical reaction (flame) propagating in a tube, through a narrow slit or under similar conditions, using the methods of the theory of similarity we find the following dimensionless governing criteria ... 

The equation of heat conduction has an analogous form.1 The thermal... 

With the help of these coefficients all the differential equations of the diffusion of the different substances and the equations of heat conduction for the chemical reaction take exactly the same form, with identical L and F in all the formulas ... 

It does not appear possible to consider in general form, discarding the assumptions of steady propagation of the regime with constant velocity, the differential equations of heat conduction and diffusion in a medium in which a chemical reaction is also running. 

Fick s first law of diffusion (analogous with the equation of heat conduction) states that the mass of substance dm diffusing in the x direction in a time df across an area A is proportional to the concentration gradient dc/dx at the plane in question ... 

Assuming some simplifications, analytical solutions for the transport equation may be inferred from arguments by analogy with the basic equations of heat conduction and diffusion (e.g. Lau et al. (1959), Sauty (1980), Kinzelbach (1983), and Kinzelbach (1987)). 

To continue the analogy with heat conduction, it can be seen that this equation is analogous with the steady-state equation of heat conduction in the presence of heat sources. 

B Understand mullidimensionality and time dependence of heat transfer, and the conditions under which a heal transfer problem can be approximated as being one-dirnensional, B Obtain the differential equation of heat conduction in various coordinate systems, and simplify it for steady one-dimensional case,... 

Note that the boundary conditions have no effect on the finite difference formulation of interior nodes of the medium. This is not surprising since the control volume used in the development of the formulation does not involve any part of the boundary. You may recall that the boundary conditions had no effect on the differential equation of heat conduction in the medium either. 

Because the empty cylinder of FGM is placed in steady state temperature field, two ends are heat insulation, the temperature distribution T is no relation with z, 6 and time t. One dimensional equation of heat conduction in r direction is given as ... 

In the region ou tside the boundary layer, where the fluid may be assumed to have no viscosity, the mathematical solution takes on the form known as potential flow. This flow is analogous to the flow of heat in a temperature field or to the flow of charge in an electrostatic field. The basic equations of heat conduction (Fourier s law) are... 

The differential equation of heat conduction through a thin sheet of thickness dx is established as follows ... 

When there are two axes of heat transfer, x and y, with the two values of the thermal conductivity dependent on the axis, the differential equation of heat conduction ... 

Assuming that the variables are separable is the basic method of obtaining a solution of the partial differential equation. For the equation of heat conduction ... 

As shown in Chapter 2, transfer of heat by conduction is due to random molecular motions, and there is an obvious analogy between the two processes of heat conduction and of mass transport by diffusion. Pick in 1855 recognized this fact, and put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Pourier (1822). 

After regarded U-vertical pipe as an equivalent pipe, soil temperature field which around it is a cylinder temperature field, in the circular direction there is no temperature gradient, the temperature distribution of the concrete around can be regarded as an axisymmetric problems. The styles of differential equations of heat conductivity are axisymmetric and unsteady state ... 

Consider the equation of heat conduction in a solid with the laser energy absorbed on the irradiated surface as a heat source. By judiciously selecting the beam geometry, dwell time, and sample configuration, the problem may be reduced to solvable one- and two-dimensional heat flow analyses. Phase transitions can be included and the temperature distributions that are produced can be calculated. Examples will be selected to provide specific guidance in the choice of lasers and materials. The result of all this will be an idea of the effects that one may produce by laser heating of solids. 

The first quantitative diffusion equation was proposed by Pick (1855). He adopted the mathematical equation of heat conduction derived by Fourier (1822). The basic hypothesis is that the transfer rate of a diffusing substance through unit area of a section is proportional to the concentration gradient normal to the section. This can be expressed by ... 

Example 3.2 (Heat conduction problem, Fourier s law and heat capacity). The dissipative energy equation (3.50) defines the governing equation of heat conduction. If force effects are not accounted for, we can ignore the first term of the r.h.s. of (3.50) to obtain... 

A more refined analysis is required for hear conduction, conductance and insulation thickness in 3D space. Assuming 9 represent and T[, then at any point the temperature 9 satisfies an equation of heat conduction, provided initial and boundary conditions are given... 

In the case when the heat conductivity coefficient is constant Eq. (3) can lead to the equation of heat conductivity ... 

In this case, diffusion can be presented in a quantitative manner by adapting the mathematical equations of heat conduction derived by Fourier in 1822, as shown by Fick in 1855. The... 

The governing equation of heat conduction in steady state is the Fourier s equation given... 

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