Flux coefficient

Components are commonly represented as nodes in the metabolic network. These nodes can act as branch points if the number of input and output fluxes is not equivalent. Non-essential reactions around a node can be collected into reaction groups the coefficients of their fluxes, in general termed the metabolic flux coefficients (in analogy to rate coefficients), can be rearranged as group control coefficients. 

One operating concern for a rich combustor is the occurrence of high combustor wall temperatures. In a fuel-rich combustor, air cannot be used to film-cool the walls and other techniques (e.g., fin cooling) must be employed. The temperature rise of the primary combustor coolant was measured and normalized to form a heat flux coefficient which included both convective and radiative heat loads. Figure 7 displays the dependence of this heat flux coefficient on primary combustor equivalence ratio. These data were acquired in tests in which the combustor airflow was kept constant. If convective heat transfer were the dominant mechanism a constant heat flux coefficient of approximately 25 Btu/ft -hr-deg F would be expected. The higher values of heat flux and its convex character indicate that radiative heat transfer was an important mechanism. 

The average heat flux coefficient to the primary combustor wall is plotted for the fuels in Figure 9. The results in general displayed a convex character as was observed with NO. 2 fuel. The level of the heat transfer coefficient and its convex trend indicates the importance of radiative heat transfer for these fuels. The maximum value of the coefficient for SCR-II fuel exceeded the maximum for other fuels by 30%. The hydrogen content of SCR-II was less than that for the other fuels tested which apparently resulted in a more intense radiating medium. 

Let us now turn to the more difficult problem of explaining why the limits -> 0, N2 0 appear in Eq. 7.1.3. During the actual mass transfer process itself the composition (and velocity) profiles are distorted by the flow (diffusion) of 1 and 2 across the interface. The mass transfer coefficient defined in Eq. 7.1.3 corresponds to conditions of vanishingly small mass transfer rates, when such distortions are not present. These low-flux or zero-flux coefficients are the ones that are usually available from correlations of mass transfer data. These correlations usually are obtained under conditions where the mass transfer rates are low. For the actual situation under conditions of finite transfer rates, we may write... 

In order to calculate the flux we need the finite flux coefficient kl this coefficient usually is related to the zero-flux coefficient by a general relation of the form... 

The major part of the next few chapters are devoted to methods of estimating the low flux mass transfer coefficients k and [A ] and of calculating the high flux coefficients k and [/c ]. In practical applications we will need these coefficients to calculate the diffusion fluxes 7, and the all important molar fluxes N. The are needed because it is these fluxes that appear in the material balance equations for particular processes (Chapters 12-14). Thus, even if we know (or have an estimate of) the diffusion fluxes 7 we cannot immediately calculate the molar fluxes because all n of these fluxes are independent, whereas only n — 1 of the J I are independent. We need one other piece of information if we are to calculate the N. Usually, the form of this additional relationship is dictated by the context of the particular mass transfer process. The problem of determining the knowing the 7 has been called the bootstrap problem. Here, we consider its solution by considering some particular cases of practical importance. 

The binary mass transfer coefficients estimated from these correlations and analogies are the low flux coefficients and, therefore, need to be corrected for the effects of finite transfer rates before use in design calculations. 

These values are used in the calculation of the mass transfer rate factors form Eqs. 9.3.38, the high flux correction factors from Eqs. 9.3.39 and, hence, new values of the high flux mass transfer coefficients. The cycle of flux-coefficient calculations is repeated and after 10 iterations we obtain the following converged values ... 

For long contact times and/or short distances between the interface and the core 5, the solution given above for the zero-flux coefficient does not apply. This situation may arise for mass transfer inside liquid droplets that stay sufficiently long in contact with the surrounding gas or liquid. For long contact times, the diffusing species will penetrate deep into the heart of the bubble (or drop), and it is important in such cases to define the mass transfer coefficient in terms of the driving forces Ax, = — , where represents... 

This example also can be used to illustrate a point often overlooked, namely the distinction between a phenomenological (experimental) rate constant and a flux coefficient [8]. It is very tempting to say that 4 k is the number of molecules making the transition A C and that 4 C is the number of C molecules making the transition C A per unit time so that the rate equation is simply the difference between two opposing fluxes. However, this is not the case clearly 4 and 4 individually are combinations... 

The flux coefficients are not uniquely defined. If the boundary is put between A and B, then r = / i and / = / i(B/C). The flux coefficients must satisfy two conditions firstly, rA — r C must be equal to — ifl which is the overall rate of the reaction, —dkldt secondly, at equilibrium r/r must be equal to the equilibrium constant. For this simple example, it can be readily shown that the two conditions are satisfied. Use of the steady-state approximation for B gives... 

They probably are of the same order of magnitude, but calculations with simple models suggest the differences are not negligibly small. Although we will, because of common practice, use the term rate constant to describe both the theoretical and experimental coefficients, the conceptual difference between a rate constant and a flux coefficient should be kept in mind. 

In order to assess resuspension at a shallow (0.25 m deep) site in the lake (Fig. 27.1(a)), Eq. (27.32) must be solved for the time-dependent concentration profile, with appropriately calibrated expressions for the erosion flux coefficients and the settling velocity. Pertinent measurements and analysis are described elsewhere. Parameters for simulation of concentration profile are summarized in Table 27.9. The organic-rich bottom is conveniently treated as bed (as opposed to fluid mud) in this analysis. 

B moisture flux coefficient P moisture vapor pressure... 

If, in addition, the flux coefficient A is much larger than the time scale for diffusion L Dj, (5.55) is further simplified to read... 

Figure 21 illustrates the ability of the mechanistic model to match complicated suspended solids density profiles from the 152 mm x 152 mm higher temperature pilot plant. Increas suspension densities at the top of the unit are due to considerable internal inertial separation at the exit. The profiles are used to find best fit values of the scale independent "wall-to-core flux coefficient" and "wall-layer disturbance factor". The model effectively predicts the variation of suspension density with height, solids circulation rate and gas velocity using these best fit values. 

Figure 21. Best fit curves of the Senior and Brereton model to density profiles in the UBC CFB pilot combustor. The roof reflection coefficient, wall layer disturbance factor, and core-to-wall flux coefficient were adjusted to obtain the fit. (Gs = solids circulation flux, U = superficial gas velocity), (Senior and Brereton, 1992).

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