Convective flux, defined

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

In electrochemical cells we often find convective transport of reaction components toward (or away from) the electrode surface. In this case the balance equation describing the supply and escape of the components should be written in the general form (1.38). However, this equation needs further explanation. At any current density during current flow, the migration and diffusion fluxes (or field strength and concentration gradients) will spontaneously settle at values such that condition (4.14) is satisfied. The convective flux, on the other hand, depends on the arbitrary values selected for the flow velocity v and for the component concentrations (i.e., is determined by factors independent of the values selected for the current density). Hence, in the balance equation (1.38), it is not the total convective flux that should appear, only the part that corresponds to the true consumption of reactants from the flux or true product release into the flux. This fraction is defined as tfie difference between the fluxes away from and to the electrode ... 

The second process of water vapor removal down the channel can be described by the convective flux, Q(pw,Sat PvMet)KRTA), representing the maximum amount of water vapor removed with the purge gas when the exit purge gas is fully saturated with vapor. In the above definition Q is the purge gas volumetric flow rate and A the active area of the fuel cell. Both parameters defined above have the unit of mol/s per unit of the fuel cell active area. It follows that... 

In order to obtain Xer and aw from Equations 2 and 3, the pseudohomogeneous temperature of model I-T must first be defined. Equal axial heat convection flux will be obtained if ... 

The concentration profile is different as a result of the finite convective flux when compared to the profile produced for vanishing convective flux h0 —> 0. The greater the convective flux the larger the deviation in the concentration profile. When the convection flux is in the direction of the wall, h0 < 0, such as when a vapour is condensing out of a mixture, the concentration profile will be steeper as a result of the convective flux, Fig. 1.47. In conjunction with this the material transported out of the phase by diffusion will increase, whilst in the case of mass transport into the phase, Fig. 1.46, the diffusional transport decreases. The mass transfer coefficient for vanishing convection flux (h0 —> 0), defined in (1.180) is therefore different from that with finite convection flux, and so it is valid that... 

Vertical Tubes. The flow regime chart for vertical tube flow shown in Fig. 4.51 was prepared by Metais and Eckert [190] for either a uniform-heat-flux or uniform-wall-temperature boundary condition. The two boundaries of the mixed convection are defined in such a way... 

High Pressure Plasmas and Their Application to Ceramic Technology The convective flux of species j is defined by ... 

This step certainly makes sense in the analogy with a person s body. Pick s law is applied in a reference frame (her body) with no convection, which allows use of the procedures of Section 15.2.3. Since we are doing a steady-state analysis the total fluxes, and Ng [mol/(m s)] are constant (e.g., not functions of z) however, the diffusive and convective fluxes do depend on z. To use this separation of terms, choose a reference velocity Vj-ef(z) so that there is no convection in the reference frame. This is always possible in a steady-state system The convective flux is defined in terms of the reference or basis velocity Vj.gf,... 

Students can find the development in this section separating the diffusive and convective fluxes confusing and even distressing. The choice of the reference velocity based on volume, mole, or mass is arbitrary and may result in different values for the convective flux. This in turn will result in different values for the diffusive flux. Even though the total fluxes of A and B will be the same, how can this be correct The method works because the diffusive flux is defined with respect to a reference frame in which the convective flux is zero. If we change Vj-gf, we change the convective and diffusive fluxes by the exact amounts required to keep the total fluxes unchanged. If we report the diffusive flux and the convective flux, we need to be clear whether a volume, mole, or mass basis was used. Another way to think about this calculation is the total fluxes of A and B are state functions—they do not depend on the calculation path chosen. On the other hand, the diffusive and convective fluxes are path functions and depend on how the calculations are done. 

The quantities Ap , Arj, and A/j are defined in a maimer analogous to Ap and Ap in Equation 1.33. The term involving the unit dyadic (or tensor) I employed in Equation 6.5 ensures that includes only the tangential portion of the surface excess flux. Also, the convective flux in the integral over Sq has been neglected, as in the previous derivation of the overall mass balance, because can be made small in comparison to S by reducing pillbox thickness. 

Outlet Boundary Conditions Convection dominates the heat transport at the outlet side and thus we assume no convective heat flux defined as... 

The flow is convection dominated and an essential feature of the method is to "upwind" the convected flux. It is assumed that the temperature of the flux crossing a boundary and "flowing" from node to node is defined by the temperature at the node which is upstream, and not by (say) an average value of the temperature at the two nodes. This gives excellent stability to the iterative solution of the equations for temperature, at some expense of increased truncation error. The method is robust and can accommodate reverse flow at any position in the film (which might occur from, say, recesses or hydrostatic supply ports). The domain can be set up and swept in the same way as an elliptic problem and, as observed, is strongly convergent. 

Considering diffusion of a specie j through the diffusion layer of thickness <5, the mass flux according to Pick s modified first law of diffusion and that for convection are defined by the following expression... 

At a convection heat transfer surface the heat flux (heat transfer rate per unit area) is related to the temperature difference between fluid and surface by a heat transfer coefficient. Newton s law of cooling defines this ... 

As shown in Fig. 21, in this case, the entire system is composed of an open vessel with a flat bottom, containing a thin layer of liquid. Steady heat conduction from the flat bottom to the upper hquid/air interface is maintained by heating the bottom constantly. Then as the temperature of the heat plate is increased, after the critical temperature is passed, the liquid suddenly starts to move to form steady convection cells. Therefore in this case, the critical temperature is assumed to be a bifurcation point. The important point is the existence of the standard state defined by the nonzero heat flux without any fluctuations. Below the critical temperature, even though some disturbances cause the liquid to fluctuate, the fluctuations receive only small energy from the heat flux, so that they cannot develop, and continuously decay to zero. Above the critical temperature, on the other hand, the energy received by the fluctuations increases steeply, so that they grow with time this is the origin of the convection cell. From this example, it can be said that the pattern formation requires both a certain nonzero flux and complementary fluctuations of physical quantities. 

Having defined the balance regions, the next task is to identify all the relevant inputs and outputs to the system (Fig. 1.10). These may be well-defined physical flow rates (convective streams), diffusive fluxes, but may also include interphase transfer rates. 

The technique offers a known interfacial area under convective flow conditions that are quite well-defined, with mass transport rates that are enhanced compared to the Lewis cell and its analogs. However, in common with many other approaches, interfacial fluxes must be determined indirectly from bulk solution measurements. 

The wall heat flux qmi cannot be evaluated as in Section II,B. Numerous experimental studies on heat transfer in this two-phase forced-convection region have been carried out, and the results of these investigations are usually presented in the form of a correlation for the wall heat-transfer coefficient hmi, which is defined as in Eq. (32b). Most of these correlations fit one of two generalized forms. The first is... 

For laminar conditions of slow flow, as in candle flames, the heat transfer between a fluid and a surface is predominately conductive. In general, conduction always prevails, but in the unsteadiness of turbulent flow, the time-averaged conductive heat flux between a fluid and a stationary surface is called convection. Convection depends on the flow field that is responsible for the fluid temperature gradient near the surface. This dependence is contained in the convection heat transfer coefficient hc defined by... 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

This equation along with Equation (9.97) allows us to eliminate the convective flame heat flux to develop an equation for the flame temperature. This equation will still contain the burning rate in terms of the effective heat of gasification, Lm. From Equation (9.97), we define Lm as the modified heat of gasification by the following ... 

Velocity and temperature gradients are confined to the surface layer defined by z < I-. Above L the wind velocity and potential temperature are virtually uniform with height. Venkatram (1978) has presented a method to estimate the value of the convective velocity scale w,. On the basis of this method, he showed that convective conditions in the planetary boundary layer are a common occurrence (Venkatram, 1980). In particular, the planetary boundary layer is convective during the daytime hours for a substantial fraction of each year ( 7 months). For example, for a wind speed of 5 m sec , a kinematic heat flux Qo as small as O.PC sec can drive the planetary boundary layer into a convective state. 

---