Heat flux density balances

The equations of heat flux density balances between the disk and the rod are ... 

Figure 10 presents the interface shape of the rivulet for wall superheat as 0.5 K and Re = 2.5. Here also presented the data on pressure in liquid and heat flux density in rivulet cross-section. The intensive liquid evaporation in near contact line region causes the interface deformation. As a result the transversal pressure gradient creates the capillarity induced liquid cross flow in direction to contact line. Finally the balance of evaporated liquid and been bring by capillarity is established. This balance defines the interface shape and apparent contact angle value.For the inertia flow model, the solution is obtained from a non-stationary system of equations, i.e., it is time-dependable. In this case the disturbances in flow interface can create the wave flow patterns. The solutions of unsteady state liquid spreading on heat transfer surface without and with evaporation are presented on Fig. 11. When the evaporation is not included (for zero wall superheat) the wave pattern appears on the interface. When the evaporation includes, the apparent contact angle increase immediately and deform the interface. It causes the wave suppression due to increasing of the film curvature.

The composition and rate of formation of all the products have been studied as a function of flash times. The mass balances are excellent. The results clearly show that the reaction begins with depolymerization processes giving rise to a short life time intermediate confound (ILC) that is liquid at reaction terrq)erature but solid at room temperature. This product partially decomposes into condensible vapours. A fraction of them undergoes a thermal cracking producing gases inside or in the vicinity of the sample. For heat flux densities higher than approximately 9 x 10 W m, no char is observed. 

This formulation assumes that the process is performed under a constant ambient pressure. Strictly considered Equ. 4-43 should therefore be called an enthalpy balance. But in daily routine the established terminology is heat balance. The vector stands for the heat flux density. In setting up a heat balance many more sources and sinks have to be accounted for than in the case of the mass balance. This is indicated by the box for additional sources and sinks enclosed by the dotted line which is enumerated in addition to the main heat fluxes, such as the heat flow exchanged through the heat transfer area. An example for such other heat sources may be the power input of the agitator in a highly viscous system. 

The flux density of heat conducted into the soil can be calculated from the overall energy balance (see Eq. 7.2) ... 

If the heat flux qw at the surface (r = R) of the three bodies is given instead of the power density W0, then from the energy balance,... 

In the experiments water spaays were used to drain heat from the slab, therefore the associated heat transfer coefficient depends very much on the flow rate of the cooling medium, as shown in figs. 4 a-b. Here the heat flux was evaluated according to the lumped temperature energy balance on a slab of volume V=Sx2h, having a heat capacity Cp and density p. 

Fig. 2.9 Energy balance of thermal processes inside a piece of burning plastics ris temperature t is time Q is the heat flux is the mass of gaseous decomposition products hft is the enthalpy of gaseous decomposition products h is the enthalpy of the original material kc is the thermal conductivity of the char layer is the thermal conductivity of the original material p is density c, is specific heat Reprinted from Ref. 10 by permission of the Plastics and Rubber Institute...

This is the first output quantity according to Figure 28.3. The mass flux densities, Hj, can be obtained from Faraday s law combined with the component mass balance [Eqs. (28.68) and (28.69)]. The heat source density, q, is obtained from an enthalpy balance around the electrode pore, similar to Eq. (28.70). Corresponding equations were given by Wolf and Wilemski [2, 3], although not aU of them are formulated as part of the electrode model. 

The output quantities required from this model are obtained in a similar approach as in the film model the current density from the electrode is proportional to the potential gradient at the bottom of the agglomerate, the mass fluxes are obtained from mass balances in the macropores, and the heat flux is obtained from an enthalpy balance over the complete pore, ffowever, in the pubUcations listed in Table 28.3, these equations are not mentioned expHcitly. 

The electric current density is calculated from the gradient of the overpotential at the bottom of the electrode, similar to Eq. (28.74). An additional enthalpy balance, which is not given here in detail, is necessary to quantify the heat flux from the electrode. 

In order to identify EPHs of the cell or electrode reactions from the experimental information, there had been two principal approaches of treatments. One was based on the heat balance under the steady state or quasi-stationary conditions [6,11, 31]. This treatment considered all heat effects including the characteristic Peltier heat and the heat dissipation due to polarization or irreversibility of electrode processes such as the so-call heats of transfer of ions and electron, the Joule heat, the heat conductivity and the convection. Another was to apply the irreversible thermodynamics and the Onsager s reciprocal relations [8, 32, 33], on which the heat flux due to temperature gradient, the component fluxes due to concentration gradient and the electric current density due to potential gradient and some active components transfer are simply assumed to be directly proportional to these driving forces. Of course, there also were other methods, for instance, the numerical simulation with a finite element program for the complex heat and mass flow at the heated electrode was also used [34]. 

To investigate the thermo-mechanical interactions, the laws of thermodynamics are applied. The first law of thermodynamics can be considered as a balance equation with the internal energy density m and the heat fluxes q and r due to close-range and long-range effects, respectively. Considering the direct energy production density f one can obtain... 

Solving this flow model for the velocity the pressure is calculated from the ideal gas law. The temperature therein is obtained from the heat balance and the mixture density is estimated from the sum of the species densities. It is noted that if an inconsistent diffusive flux closure like the Wilke equation is adopted (i.e., the sum of the diffusive mass fluxes is not necessarily equal to zero) instead, the sum of the species mass balances does not exactly coincide with the mixture continuity equation. 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

Solving this flow model for the velocity the pressure is calculated from the ideal gas law. The temperature therein is obtained from the heat balance and the mixture density is estimated from the sum of the species densities. It is noted that the viscous velocity is normally computed from the pressure gradient by use of a phenomenologically derived constitutive correlation, known as Darcy s law, which is based on laminar shear flow theory [139]. Laminar shear flow theory assumes no slip condition at the solid wall, inducing viscous shear in the fluid. Knudsen diffusion and slip flow at the solid matrix separate the gas flow behavior from Darcy-type flow. Whenever the mean free path of the gas molecules approaches the dimensions of pore diameter, the individual gas molecules are in motion at the interface and contribute an additional flux. This phenomena is called slip flow. In slip flow, the layer of gas next to the surface is in motion with respect to the solid surface. Strictly, the Darcy s law is valid only when the flow regime is laminar and dominated by viscous forces. The theoretical foundation of the dusty gas model considers that the model is applied to a transition regime between Knudsen and continuum bulk diffusion. To estimate the combined flux, the model is based on the assumption that the combined flux can be expressed as a linear sum of the Knudsen flux and the convective flux due to laminar flow. 

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