Heat equation source term

Here, the temperatures on the left-hand side are the new, unknown values while that on the right is the previous, known value. Note that the heat sink/source term is evaluated at the previous location, — A. The computational template is backwards from that shown in Figure 8.2, and Equation (8.78) cannot be solved directly since there are three unknowns. However, if a version of Equation (8.78) is written for every interior point and if appropriate special forms are written for the centerline and wall, then as many equations are... 

In order to account for the heat loss through the metallic body of the cone, a heat conduction equation, obtained by the elimination of the convection and source terms in Equation (5.25), should also be incorporated in the governing equations. 

In the absence of fluctuations in the heat release rate, dq/dt = 0, Equation 5.1.14 reduces to the standard wave equation for the acoustic pressure. It can be seen that a fluctuating heat release then acts as a source term for the acoustic pressure. 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

The initial condition is that (T — T )/T = Ys = 0 and at solid boundaries (dT/dn = dYs/dn = 0, with no heat or mass flow. Of course, radiation is ignored, but this is justified far away from the source. Note that the dimensionless source terms are Q and m where t = (1/p)12 and l is chosen according to a geometric dimension, or as Equation (12.16) for an unconfined plume depending on whether the flow is confined or unconfined. 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

With the increased computational power of today s computers, more detailed simulations are possible. Thus, complex equations such as the Navier—Stokes equation can be solved in multiple dimensions, yielding accurate descriptions of such phenomena as heat and mass transfer and fluid and two-phase flow throughout the fuel cell. The type of models that do this analysis are based on a finite-element framework and are termed CFD models. CFD models are widely available through commercial packages, some of which include an electrochemistry module. As mentioned above, almost all of the CFD models are based on the Bernardi and Verbrugge model. That is to say that the incorporated electrochemical effects stem from their equations, such as their kinetic source terms in the catalyst layers and the use of Schlogl s equation for water transport in the membrane. 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

Since G and Ch appear only as a product, G = GCh is used as the relevant nondimensional parameter governing heat release in the present studies. The other governing parameters for this flow are Re and Pr, along with the precise strength and distribution of the source term in the energy equation. 

The thermal-energy equation has no explicit source term to describe the heat release associated with chemical reaction. Nevertheless, as stated, the thermal-energy equation does fully accommodate chemical reaction. As is described subsequently, the thermal effects of chemical heat release are captured in the enthalpy term on the left-hand side. 

Similarly, the source term in the energy equation is sum of the electrical resistance heating, heat of formation of water, electrical work, and heat release due to phase change (condensation of water vapor). 

Taking the absorbed optical power density as source term S = a /(pcp) 1 in the heat equation (5), an analytical expression for the normalized heterodyne diffraction efficiency can be derived as a cascaded linear response [88, 89] ... 

The temperature profile evolves according to the heat equation (5) with the heat source supplied by absorption of the focused laser beam. An additional advection term accounts for the influence of convection ... 

The occurrence of demixing morphologies characteristic for the metastable regime between the binodal and the spinodal can be understood from Fig. 17. The red dot marks the initial position of the sample with c = 0.3. Upon laser heating the temperature within the laser focus rises by AT and the distance to the binodal first increases. A stationary temperature distribution is rapidly reached and the Laplacian of the temperature field T(r,t) is obtained from the stationary solution of the heat equation (5) with the power absorbed from the laser as source term ... 

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. 

Example 7.7 Absorption of ammonia vapor by lithium nitrate-ammonia solution The following modeling is from Venegas et al. (2004). For simultaneous heat and mass transfer during the absorption of ammonia vapor by lithium nitrate-ammonia (A) solution droplets, the ammonia concentration profile in the liquid phase can be estimated from the continuity equation without a source term... 

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. ... 

Application of the law of conservation of energy to a three-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equation for unsteady-state conduction in rectangular coordinates. 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

The second part of the source term S is associated to the liquid phase through the drag force and the evaporation. It adds a vector I to the right hand side of the momentum equations, a heat transfer term II on the energy equation and a mass transfer term T on the fuel equation (see Section 10.1). 

In these equations, the momentum and heat phase exchange source terms are split in two parts I = —Fa - - U T that includes both the drag force and a momentum transfer due to the mass transfer, and II = - -Ths F(Ti), where /is,f(2 ) is the fuel vapor enthalpy taken at the interface temperature... 

The temporal evolution of the total (i.e. spatially integrated) power deposited by the spark is presented on Fig. 10.10, along with the total heat release LOT and the spatially averaged temperature. After a heating phase due to the source term on the energy equation (up to approximately 0.08 ms), the temperature is sufficient to initiate the reaction between fuel vapour and air, leading to a sudden increase of the heat release of the exothermic... 

For the enthalpy balance, c becomes H, the enthalpy of unit mass of fluid, which is naturally expressed in terms of the temperature and composition of the fluid. In this case, the source term becomes zero, corresponding with the conservation of enthalpy in the flow. The heat of reaction enters when the enthalpy balance is transformed into an equation involving the temperature. The enthalpy balance is then... 

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