Conduction with heat sources

Conduction with Heat Source Application of the law of conservation of energy to a one-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equations for steady-state conduction in a flat plate of thickness 2R (b = 1), a cylinder of diameter 2R (b = 2), and a sphere of diameter 2R (b = 3). The parameter b is a measure of the curvature. The thermal conductivity is constant, and there is convection at the surface, with heat-transfer coefficient h and fluid temperature I. ... 

Geometric one-dimensional heat conduction with heat sources... 

The temperature field in the filter is described by the equation of transient heat conduction with heat sources in axisymmetric coordinates ... 

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. 

Typical of the 1,2,5-oxadisilacyclopentane ring is insertion of active short-lived species into its Si—O bond McjSi , 0=Si=0, R2Si=S, RMeSi=0, Me2C=0, RB=0 to give (144)-(149) respectively. The reaction is conducted with the source of the active species under normal heating... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

We will assume that the material properties are constant. The heat conduction equation for planar, transient temperature fields with heat sources has the form... 

Example 4.5. Heat Conduction with a Source Term... 

The steady state heat conduction equation with heat source in cylindrical coordinate system is given by... 

The model was based on the assumption that each polymerization site was a spherical particle of radius ro immersed in a fluid (as shown in Fig. 2.2.3). A steady-state differential energy balance along with Fourier s Law of heat conduction with a source term (from Eq. 1.2.9) gave... 

Example 3.1 1D transient heat conduction with a source term The variables and parameters used in this example are as follows p = density, Cp = heat capacity, q = heat flux, k = thermal conductivity, A = cross-sectional area, S = source strength (J m s ). 

The third characteristic of interest grows directly from the first, ie, the high thermal conductance of the heat pipe can make possible the physical separation of the heat source and the heat consumer (heat sink). Heat pipes >100 m in length have been constmcted and shown to behave predictably (3). Separation of source and sink is especially important in those appHcations in which chemical incompatibilities exist. For example, it may be necessary to inject heat into a reaction vessel. The lowest cost source of heat may be combustion of hydrocarbon fuels. However, contact with an open flame or with the combustion products might jeopardize the desired reaction process. In such a case it might be feasible to carry heat from the flame through the wall of the reaction vessel by use of a heat pipe. 

In all cooled appliances, the heat from the device s heat sources must first arrive via thermal conduction at the surfaces exposed to the cooling fluid before it can be transferred to the coolant. For example, as shown in Fig. 2.2, it must be conducted from the chip through the lid to the heat sink before it can be discharged to the ambient air. As can be seen, thermal interface materials (TIMs) may be used to facilitate this process. In many cases a heat spreader in the form of a flat plate with high thermal conductivity may be placed between the chip and the lid. 

Fig. 2.3 Effect of thickness on heat spreading for different heat source areas, material thermal conductivities, and heat transfer coefficients (A in cm, in W/mK, hinW/m K). Reprinted from Lasance and Simons (2005) with permission...

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

The stationary problem. To avoid misunderstanding, we concentrate primarily on the simplest problem, the statement of which is related to the stationary heat conduction problem with nonlinear sources ... 

For the heat conduction equation with a heat source depending on the temperature in accordance with the law... 

Here Go(a ,t) is a function of the heat source of the Cauchy problem associated with the one-dimensional heat conduction equation... 

Thermal conduction is assumed to take place from the small spherical heat source with a radius, r. This approximation leads to the one-dimensional heat conduction... 

Tests at the boiling point should be conducted with minimum possible heat input, ana boiling chips should be used to avoid excessive turbulence and bubble impingement. In tests conducted below the boiling point, thermal convection generally is the only source of liquid velocity. In test solutions of high viscosities, supplemental controlled stirring with a maraetic stirrer is recommended. 

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

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