Differential Equation of Heat Conduction

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

The rate at which the amount of heat changes in the volume A.dx is also  

From the equality of these two expressions of the rate of heat, by recalling the value of the flux deflned in Equation 2.2, by simplifying by the volume A.dx, it becomes  

This equation is fundamental for the monodirectional of heat transfer. 

When the thermal conductivity is constant, this equation simplifies as follows  

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

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 ... 

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. 

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. 

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 ... 

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 ... 

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 ... 

Consider the hot spot (at time zero) to be a tiny sphere of material at a uniform temp greater than that of the surroundings. If the material were inert and produced no heat by reaction, the cooling of the sphere in subsequent periods of time would be represented by the temp distribution curves in Fig 7 (Ref 6), These are derived from the classical treatment of the differential equation for heat conduction and may be found in Carslaw and Jaeger (Ref 5). Time is depicted in terms of a dimensionless parameter crt/a2, where a is the radius of the sphere and a. is its thermal diffusivity [a = X/(pc) j... 

The immediate result of the above discussion is that the diffusion equation can be transformed into the differential equation for heat conduction by substitution of c by T and D by k. This analogy has the consequence that practically all mathematical solutions of the heat conductance equation are applicable to the diffusion equation. The analogy between diffusion and conductance should be kept in mind in the following discussion although the topic here will be mainly the treatment of the diffusion equation, which represents the most important process of mass transport. 

So far we have derived the differential equations for heat conduction in various coordinate systems and discussed the possible boundary conditions. A heat conduction problem can be formulated by specifying the applicable differential equation and a set of proper boundary conditions. 

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 ... 

The thermal diffusivity can also be measured directly by employing transient heat conduction. The basic differential equation (Fourier heat conduction equation) governing heat conduction in isotropic bodies is used in this method. A rectangular copper box filled with grain is placed in an ice bath (0°C), and the temperature at its center is recorded [44]. The solution of the Fourier equation for the temperature at the center of a slab is used ... 

Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m s, which is equal to the thermal conductivity k divided by the density and specific heat capacity. The Laplacian operator is + d dy + d ld-z, where x, y, and z are the space coordinates. 

Equation (3.7) has the same form as the partial differential equation for heat conduction. The mathematical solutions which are known for a great variety of heat-conduction problems can be applied in the theory of diffusion. 

The temperature field can be calculated on the basis of the general differential equation of thermal conduction. The general differential equation of thermal conduction describes the non-steady-state temperature field, with the absorption of the laser radiation being taken into account as an internal heat source d> ... 

Assuming that no energy exchange occurs over the length of the components (direction of the beam path (z-axis see Fig. 1)), the differential equation for heat conduction can be simplified to a two-dimensional heating and cooling... 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

The laser we use in these experiments is an exclmer laser with a pulse width of approximately 20 nsec. In this time regime the laser heating can be treated using the differential equation for heat flow with a well defined value for the thermal diffusivity (k) and the thermal conductivity (K) (4). 

The differential equations of fluid dynamics express conservation of mass, conse rvation of momentum, conservation of energy and an equation of state. For an adiabatic reversible process, viscosity and heat conduction processes are absent and the equations are 2.1.1 to 2.1.13, inclusive... 

The simple boundary value problem describing this experiment is commonly solved in elementary texts on differential equations under the guise of heat conduction in a slab of finite width [30,31]. Differentiation of the concentration profile, C0x t, at the electrode surface immediately affords the chrono-amperometric response... 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

J. Crank and P. Nicolson, A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-conducting Type, Proc. Cambridge. Philos. Soc., 43,50-67 (1947). 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

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