Heat conduction equation boundary conditions

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

A concentrated heat capacity. We now consider the boundary-value problem for the heat conduction equation with some unusual condition placing the concentrated heat capacity Co on the boundary, say at a single point X = 0. The traditional way of covering this is to impose at the point a = 0 an unusual boundary condition such as... 

To study the effects due to droplet heating, one must determine the temperature distribution T(r, t) within the droplet. In the absence of any internal motion, the unsteady heat transfer process within the droplet is simply described by the heat conduction equation and its boundary conditions... 

This is called the partially reflecting or radiation boundary condition by analogy to the heat conduction equation analyses. It is also variously called a mixed, an inhomogeneous or the Robbins boundary condition because it mixes the value of the dependent variable p and the first... 

The simplest technique is to use separate numerical solvers for the fluid and solid phases and to exchange information through the boundary conditions. The use of separate solvers allows a flexible gridding inside the solid phase, which is required because of the three orders of magnitude difference in thermal conductivities between the solid and gas. It is also easy to include various physical phenomena such as charring and moisture transfer. Quite often, ID solution of the heat conduction equation on each wall cell is sufficiently accurate. This technique is implemented as an internal subroutine in FDS. 

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. 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

Under these conditions, the variation of the temperature profile (during irradiation with constant flux) could be computed from the heat-conduction equation in terms of the thermal conductivity, k the absorbance, a the incident flux, H the specific heat capacity, c and the thermal diffusivity, a. With the boundary conditions ... 

The resulting solution is a function of two dimensionless parameters, AT kpcy laH(ty, and xliat). In reality, the nonsteady-state temperature distribution in a cellulosic fuel is not accurately represented by the above solution, since the boundary conditions are not perfectly matched with those of the experiment, and the partial differential does not include the effects of heats of reaction and of phase change. However, Martin and Ramstad, " in their study of ignition, have demonstrated that the actual temperature profiles can be expressed as functions of the same dimensionless parameters derived from the solution of the heat-conduction equation,... 

The heat conduction equations above were developed using an energy balance on a differential element inside the medium, and they remain the same regardless of the thermal conditions on tlie surfaces of the medium. That is, the differential equations do not incorporate any information related to the conditions on the surfaces such as the surface temperature or a specified heat flux. Yet we know that the heat flux and the temperature distribution in a medium depend on the conditions at the surfaces, and (he description of a heat transfer problem in a medium is not complete without a full description of the thermal conditions at the bounding surfaces of the medium. The mathematical expressions of the thermal conditions at the boundaries are called the boundat7 conditions. 

The physical argument presented above is consistent with the mathematical nature of the problem since tlie heat conduction equation is second order (i.e., involves second derivative.s with respect to the space variables) in all directions along which heat conduction is significant, and the general solution of a second-order linear differential equation involves two surbitrary constants for each direction. That is, the number of boundary conditions that needs to be specified in a direction is equal to the order of the differential equation in that direction. 

The heat conduction equation is first order in time, and thus the initial condition cannot involve any derivatives (it is limited to a specified temperature). However, the heal conduction equation is second order in space coordinates, and thus a boundary condition may involve first derivalives at the boundaries as well as specified values of temperature. Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

The heat conduction equation only determines the temperature inside the body. To completely establish the temperature field several boundary conditions must be introduced and fulfilled by the solution of the differential equation. These boundary conditions include an initial-value condition with respect to time and different local conditions, which are to be obeyed at the surfaces of the body. The temperature field is determined by the differential equation and the boundary conditions. 

The Laplace transformation has proved an effective tool for the solution of the linear heat conduction equation (2.110) with linear boundary conditions. It follows a prescribed solution path and makes it possible to obtain special solutions, for example for small times or at a certain position in the thermally conductive body, without having to determine the complete time and spatial dependence of its temperature field. An introductory illustration of the Laplace transformation and its application to heat conduction problems has been given by H.D. Baehr [2.25]. An extensive representation is offered in the book by H. Tautz [2.26]. The Laplace transformation has a special importance for one-dimensional heat flow, as in this case the solution of the partial differential equation leads back to the solution of a linear ordinary differential equation. In the following introduction we will limit ourselves to this case. 

The functions i(x, t) and Si(t) can be recursively determined from the exact formulation of the problem with the heat conduction equation and its associated boundary conditions. Thereby o(x,t) and so(t) correspond to the quasi-steady approximation with Ph — oo. 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

As a result of this many solutions to the heat conduction equation can be transferred to the analogous mass diffusion problems, provided that not only the differential equations but also the initial and boundary conditions agree. Numerous solutions of the differential equation (2.342) can be found in Crank s book [2.78]. Analogous to heat conduction, the initial condition prescribes a concentration at every position in the body at a certain time. Timekeeping begins with this time, such that... 

Another feature of the present theory is that it provides a formalism for deducing a complete mathematical representation of a phenomenon. Such a representation consists, typically, of (1) Balance equations for extensive properties (such as the "equations of change" for mass, energy and entropy) (2) Thermokinematic functions of state (such as pv = RT, for simple perfect gases) (3) Thermokinetic functions of state (such as the Fourier heat conduction equation = -k(T,p)VT) and (4) The auxiliary conditions (i.e., boundary and/or initial conditions). The balances are pertinent to all problems covered by the theory, although their formulation may differ from one problem to another. Any set of... 

Precise solution of the multidimensional problem of heat conductivity by analytical methods is very complicated and laborious. Therefore, an approximate finite difference method was developed based on the differential heat conductivity equation and boundary conditions. In this method, the temperature of the vulcanized section of the covering fragment was subdivided into elementary volumes of unit thickness because it is necessary to define the temperature field of the vulcanized. 

In the experimental study by Zhu et al. (1998), the heating pattern induced by a microwave antenna was quantified by solving the inverse problem of heat conduction in a tissue equivalent gel. In this approach, detailed temperature distribution in the gel is required and predicted by solving a two- or three-dimensional heat conduction equation in the gel. In the experimental study, all the temperature probes were not required to be placed in the near field of the catheter. Experiments were first performed in the gel to measure the temperature elevation induced by the applicator. An expression with several unknown parameters was proposed for the SAR distribution. Then, a theoretical heat transfer model was developed with appropriate boundary conditions and initial condition of the experiment to study the temperature distribution in the gel. The values of those unknown parameters in the proposed SAR expression were initially assumed and the temperatiue field in the gel was calculated by the model. The parameters were then adjusted to minimize the square error of the deviations theoretically predict from the experimentally measured temperatures at all temperature sensor locations. 

Except for this section and Section 18.7. the solutions of the unsteady diffusion equation in one to three dimensions are beyond the scope of this book. Solutions to Eqs. (15-12c. d, e), the corresponding two-and three-dimension equations, and the equivalent heat conduction equations have been extensively studied for a variety of boundary conditions (e.g., Crank. 197S Cussler. 2009 Incropera et al 2011). Readers interested in unsteady-state diffusion problems should refer to these or other sources on diffusion. 

The goal of the conduction heat transfer is to determine the temperature field in a medium (such as fuel rod) and the rate of heat transfer to and from the medium. Typically, the media is subjected to nonimiform temperature distribution which is a result of either a heat source within the medium or heat flux from the boundary of the medium. In this section various forms of the heat conduction equation that govern the temperature field in a medium and its associated boimdary conditions are given. Some examples of the heat conduction in nuclear fuel rod and other components are presented. 

The strong form of a parabolic PDE, that is, the governing equation (for example, the classical heat conduction equation with constant heat c, variable thermal conductivity k and heat production / see, e.g., Carslaw and Jaeger 1959 Selvadurai 2000a), the boundary conditions (BC) and the initial conditions (IC) are given as... 

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