Heat conduction differential equation

General form of three-dimensional non-steady-state heat conduction differential equation in Cartesian coordinate system can be written as ... 

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

Fourier s law is the fundamental differential equation for heat transfer by conduction ... 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

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 heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Transient Heat Conduction. Our next simulation might be used to model the transient temperature history in a slab of material placed suddenly in a heated press, as is frequently done in lamination processing. This is a classical problem with a well known closed solution it is governed by the much-studied differential equation (3T/3x) - q(3 T/3x ), where here a - (k/pc) is the thermal diffuslvity. This analysis is also identical to transient species diffusion or flow near a suddenly accelerated flat plate, if q is suitably interpreted (6). 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

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). 

Chapter 4 eoncerns differential applications, which take place with respect to both time and position and which are normally formulated as partial differential equations. Applications include diffusion and conduction, tubular chemical reactors, differential mass transfer and shell and tube heat exchange. It is shown that such problems can be solved with relative ease, by utilising a finite-differencing solution technique in the simulation approach. 

The heating of the liquid can be approximated by the transient conduction of heat to a slab of finite thickness 8, with the conventional differential equation,... 

Let us consider the semi-infinite (thermally thick) conduction problem for a constant temperature at the surface. The governing partial differential equation comes from the conservation of energy, and is described in standard heat transfer texts (e.g. Reference [13]) ... 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

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 data needed are the rate equation, energy of activation, heat of reaction, densities, heat capacities, thermal conductivity, diffusivity, heat transfer coefficients, and usually the stoichiometry of the process. Simplified numerical examples are given for some of these cases. Item 4 requires the solution of a system of partial differential equations that cannot be made understandable in concise form, but some suggestions as to the procedure are made. 

Many texts, such as Crank s treatise on diffusion [2], contain solutions in terms of simple functions for a variety of conditions—indeed, the number of worked problems is enormous. As demonstrated in Section 4.1, the differential equation for the diffusion of heat by thermal conduction has the same form as the mass diffusion equation, with the concentration replaced by the temperature and the mass diffusivity replaced by the thermal diffusivity, k. Solutions to many heat-flow... 

Allen and Severn (A3, A4) demonstrate how relaxation methods, originally developed for elliptic partial differential equations, can be extended to the heat conduction equation. With elliptic equations, the value of the dependent variable at any mesh point is determined by all... 

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... 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

The two terms of the differential equation correspond precisely to these two parts of the energy release the term —pucpdT/dx represents the energy spent on heating the reacting mixture, the term d(r]dT / dx) / dx—the energy carried away by heat conduction. Because the interval in which the reaction occurs is narrow,... 

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 ... 

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 ... 

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. 

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. 

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... 

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