Heat conduction equation initial condition

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... 

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 ... 

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 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. 

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... 

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. 

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... 

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... 

The initial and boundary conditions associated with the partial differential equations must be specified in order to obtain unique numerical solutions to these equations. In general, boundary conditions for partial differential equations are divided into three categories. These are demonstrated below, using the one-dimensional unsteady-state heat conduction equation... 

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. 

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. 

The initial conditions are at t = 0, T = To, andp = 0. The parameter n characterizes the dimensions of the volume for a parallel plate reactor n = 0 for a cylindrical reactor n = 1 and for a spherical reactor n = 2. In these equations, x is a space coordinate A. is the coefficient of thermal conductivity r is the characteristic size of the reactor k is the heat transfer coefficient and To is the initial temperature of the initial medium. 

We can illustrate this technique with a transient one-dimensional cooling (or heating) problem. Let s assume that the initial condition is a constant temperature across the thickness of To- In addition, we assume the physical properties such as density, p, specific heat, Cp, thermal conductivity, k, remain constant during the thermal process. This results in the following governing equation... 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

Reconsider the brick wall already discussed. The temperature at any point on the wall at a specified time also depends on the condition of the wall at the beginning of the heat conduction process. Such a condition, which is usually specified at time t = 0, is called the initial condition, which is a mathematical expression for the temperature distribution of the medium initially. Note that we need only one initial condition for a heat conduction problem regardless of the dimension since the conduction equation is first order in time (it involves the first derivative of temperature with respect to time). 

A spherical metal ball of radius r is heated in an oven to a lempefature of Tj throughout an is then taken out of the oven and dropped into a large body of water at T. where it is coole by convection with an average convection heat transfer coeflicieiit of h. Assuming constant Ihermal conduclivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve. 

Consider a semi-inlinite solid with constant thermophysical properties, no internal heat generation, uniform theimal cnnditinn.s on its exposed surface, and initially a uniform temperature of Tj throughout. Heat tfansfec in this case occurs only in the direction uormal to the surface (the x direction), and thus it is one-dimensional. Differential equations are independent of the boundary or initial conditions, and thus Eq. 4—lOa for one-dimensional transient conduction in Cartesian coordinates applies. The depth of the solid is large (x expressed mathematically as a boundary condition as T x —> , 0 = T,. 

As we already know, the solutions for the heat conduction problems in section 2.3.3 can be transferred to the mass diffusion problem, due to the similarity of the differential equations, initial and boundary conditions. The corresponding quantities to heat conduction for mass diffusion are shown in the following table. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

Consider heat conduction in a slab with radiation at both ends. [6] The dimensionless governing equations and boundary/initial conditions are ... 

Carslaw, H. S and Jaeger, J. C. (1959) Cottduciion of Heat in Solids, Oxford University Press. Oxford, 2nd ed. This work includes a compilation of solutions to the equation of unsteady heat conduction in the absence of flow for many different geometries, initial and boundaiy conditions. The basic equation is of the same form as the diffusion equation with the thermal diffusivity. K/pC, in place of the diffusion coefficient. (Here k, p, and C are the thermal conductivity, density, and specific heatof the continuous fluid.) Like D, the thcnnal diffusivity has cgs- dimensions of cm"/sec,... 

The inequality (50) in general is a function parameters Tl, Tc, and ro, which determine the external and initial conditions of the borehole grouting, and also is dependent of parameters that characterize the properties of the cement slurry, namely, parameters q, b, D, and K. Hence, inequality (50) represents the criteria that allows for the specified conditions (Ti, Tc, and ro.) to choose the appropriate recipes of the cement slurries (with different q, b, D, and fC) that may prevent from overheating and thawing of the frozen formation. Due to suggested decomposition of the general solution T of the heat conduction problem (l)-(6) into 3 auxiliary problems for X, Yj and Z, (see equation (8)), condition (50) can be presented in the following form ... 

For certain boundary and initial conditions analytical solutions to Eq. 4 can be obtained. The majority of diffusivity measurement methods are all based on such. solutions. The experimental conditions arc matched to these mathematical conditions as closely as possible, and the appropriate solution is used to give a value for the diffusivity. The e.xpcri-ment can be repeated at different temperatures in order to obtain the temperature dependence of the diffusivity. This type of experimental procedure has been criticized [43] because if the diffusivity changes with temperature then almost invariably the conductivity is also temperature dependent, and Eq. 3, which would not have given an analytical solution, should have been used instead of Eq, 4, However. Hands and Horsfall [44] have shown that, except near melting transitions, thermocouples. sensitive to 0.002 C would be needed to detect the effect of the conductivity term in Eq. 3. Hence, generally speaking, the simpler equation is adequate for diffusivity measurement and for the majority of heat flow calculations. 

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