Mass flux approximation

IOOC What is the low mass flux approximation in mass transfer analysis Can Ihe evaporation of water from a lake be treated as a low mass flux process ... 

In reactive flow analysis the Pick s law for binary systems (2.285) is frequently used as an extremely simple attempt to approximate the multicomponent molecular mass fluxes. This method is based on the hypothesis that the pseudo-binary mass flux approximations are fairly accurate for solute gas species in the particular cases when one of the species in the gas is in excess and acts as a solvent. However, this approach is generally not recommend-able for chemical reactor analysis because reactive mixtures are normally not sufficiently dilute. Nevertheless, many industrial reactor systems can be characterized as convection dominated reactive flows thus the Pickian diffusion model predictions might still look acceptable at first, but this interpretation is usually false because in reality the diffusive fluxes are then neglectable compared to the convective fluxes. 

T] Low M.T. rates. Low mass-flux, constant property systems. Ns, % L local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius approximate solution. 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Bo = q/Gh] Q, where t is the period between successive events, U is the mean velocity of single-phase flow in the micro-channel, Jh is the hydraulic diameter of the channel, q is heat flux, m is mass flux, /zlg is the latent heat of vaporization). The dependence t on Bo can be approximated, with a standard deviation of 16%, by... 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

The interaction of parametric effects of solid mass flux and axial location is illustrated by the data of Dou et al. (1991), shown in Fig. 19. These authors measured the heat transfer coefficient on the surface of a vertical tube suspended within the fast fluidized bed at different elevations. The data of Fig. 19 show that for a given size particle, at a given superficial gas velocity, the heat transfer coefficient consistently decreases with elevation along the bed for any given solid mass flux Gs. At a given elevation position, the heat transfer coefficient consistently increases with increasing solid mass flux at the highest elevation of 6.5 m, where hydrodynamic conditions are most likely to be fully developed, it is seen that the heat transfer coefficient increases by approximately 50% as Gv increased from 30 to 50 kg/rrfs. 

The mass flux GT is given by Equation 9-15 or 9-18, and the venting time is given approximately by... 

A corresponding mass flux of fuel leaving the surface and moving by diffusion to the pure air with 7Fj00 = 0 would be given approximately as... 

Up to now we have presented this example without any regard for consistency, i.e. satisfying thermodynamic and conservation principles. This fuel mass flux must exactly equal the mass flux evaporated, which must depend on q and h(g. Furthermore, the concentration at the surface where fuel vapor and liquid coexist must satisfy thermodynamic equilibrium of the saturated state. This latter fact is consistent with the overall approximation that local thermodynamic equilibrium applies during this evaporation process. 

Finally, we estimate the order of magnitude of the time of the heated solid to achieve Tpy at the surface. This is primarily a problem in heat conduction provided the decomposition and gasification of the solid (or condensed phase) is negligible. We know that typically low fuel concentrations are required for piloted ignition (XL 0.01-0.10) and by low mass flux (mv 1-5 g/m2 s) accordingly. Thus, a pure conduction approximation is satisfactory. A thermal penetration depth for heat conduction can be estimated as... 

Here Af. will be taken to represent the projected external fuel surface area that would experience the direct heating of the compartment. From Equation (11.21), accounting approximately for the effects of oxygen and temperature, and distribution effects in the compartment, the fuel mass flux might be represented for qualitative considerations as... 

The mass flux arising from the evaporation of liquid droplets is significant to fire scaling applications. Such scaling has been demonstrated by Heskestad [11], and specific results will be discussed later. As a first approximation, independent monodispersed droplets of spherical diameter, D, and particle volume density, can be considered. 

Figure 17.8 shows the probe, which consists of a 1-millimeter diameter t3rpe K thermocouple centered between two 1-millimeter diameter pressure taps. Each of the pressure tubes was bent 90° and sheared at the bend. To obtain a measurement, the tube is rotated until the pressure difference between the two taps is maximized. This is the position at which one tube is directed into the oncoming flow and the other is parallel to it. The approach flow thus observes an approximately 1-millimeter thick planar obstruction. The pressure difference and temperature are then recorded. The pressure difference is related to the approach velocity, and the angle determines the tangential and axial velocity components in this case. The local mass flux is then determined from the axial velocity component and the temperature (necessary to compute the flow density), and... 

In a two-component mixture in which p is approximately constant, the average molecular-mass flux is... 

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

Moreover if we apply equation (2) to the deep interior of stars (r=0), eather the velocity or the density should become infinitely large at r=0. Therefore we cannot get any normal stellar structures. This means that equation (2) is inadequate to the interior part of stars. This difficulty comes from the steady-state approximation (1). The mass flux must reduce zero at the center of the stars or the surface of the degenerate stars. The interior flow therefore should be described by another steady states, not by equation (1). Therefore we will present a new steady-state approximation and derive mass-loss equations which is available also to the deep interior of stars. 

An analysis of the driving force of evaporation is necessary, as an increase in gas mass flow yields to a lower outlet humidity. An approximation for the evaporated molar mass flux is given by ... 

Since the air pressure decreases with increasing flight altitude, at constant nozzle diameter, the total thrust increases with increasing flight altitude. This increase can correspond to approximately 10 to 30 % of the total thrust depending on the rocket. The maximum thrust is reached in vacuo. The so-called effective ejection velocity ceff (of the combustion gases) is defined as the ratio between the thrust and the mass flux (dm / dt) ... 

The description of multicomponent gas transport in porous media at arbitrary Knudsen numbers is very complicated, especially in the transition region of Kn 0.1 -10. This problem has never been solved rigorously. The situation remains extremely complex, even if the porous medium is regarded as composed of long, straight, circular channels of equal diameter. In these circumstances a consideration of simplified and limiting situations, which are better understood, is very important. Moreover, the approximate relationship between mass fluxes of the species and concentrations can be found by an appropriate combination of the corresponding relationships for simpler situations. 

With F(rj) known, we may readily calculate u from equation (28) and pj from equation (47). Obtaining profiles in physical coordinates involves first evaluating p(/ )—for example, from pj and the ideal-gas law by introduction of a flame-sheet approximation—and then performing the integration in equation (27) for y. Profiles of the normal mass flux, pv, are given by equation (29), which—for the present problem—reduces to... 

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