Gradient equation

The electron current density J has units of A/cm and in a semiconductor results from drift and diffusion. In the absence of concentration gradients, equation 7 reduces to Ohm s law, = nqp E = [Pg.346]

At the onset of constitutional supercooling, the melting-point gradient exceeds the temperature gradient. Equating these gradients leads to the criterion for constitutional supercooling ... 

For the case under consideration, where the value of a — 1 is quite small, it follows that everywhere in the cascade, except possibly at the extreme ends, the stage upflow is many times greater than the product withdrawal rate. Thus L/ L — P) can be set equal to unity. Furthermore when the value of a — 1 is small, the stage enrichment can be approximated by the differential ratio dx/dn without appreciable error. The gradient equation for the... 

Similarly, one obtains a gradient equation for the stripping section that has the form... 

The preceding two equations may be used to write the gradient equation for the countercurrent gas centrifuge in an alternative form. If the ratio of the actual gas circulation rate in the centrifuge to the circulation rate that rninimizes the stage length T/Tq is designated by m, then equation 74 may be rewritten... 

For the stripping section of the centrifuge, that is, the section between the point at which the feed is introduced and the end at which the waste stream is withdrawn, the gradient equation has the corresponding form... 

The flow pattern efficiency depends solely upon the shape of the velocity profile in the circulating gas. In terms of the integrals appearing in the gradient equation, the flow pattern efficiency is given by equation 86. 

The simplifying assumptions that make Tick s law useful for other processes are not vaHd for pei vaporation. The activity gradient across the membrane is far more important than the pressure gradient. Equation (22-110) is generally used to describe the pei vaporation process ... 

Now, we would like to attach a variance to the estimates of a and P that make c2 minimum. Given the complex and non-Unear nature of the gradient equations (5.4.67) and (5.4.68), we assume for simplicity that a and / are normally distributed and resort to linear propagation in order to retrieve an estimate for the covariance matrix of a and p. The covariance matrix Saj, of the vector (aT, ft) is approximated by... 

The fluxes of charged solutes depend on the diffusion potential arising from differences in the mobihties of individual ions, as well as on an ion s own concentration gradient (Equation 2.21). The effect of diffusion potentials will be important if the carbonate species are a large part of the total ion concentration, as they often will be. Therefore we have for the net flux of ion B... 

FIGURE 5.10 Comparison of isocratic (P, Equation 5.27) and gradient Equation 5.24) peak capacity for... 

Plots of In i as a function of q are shown in Fig. 2.15. By extending the linear portions of these curves to 77 = 0, a value of the exchange current is given, while the a coefficients may be evaluated from their gradients. Equation (2.62) may be recast in the form ... 

When using the dusty gas model we obtain the following gradient equation for the total diffusive flux... 

A solution of equation (7.25) for a steady case at small velocity gradients can be easily found as an expansion in series in powers of velocity gradient. Up to the second-order terms with respect to velocity gradients, equation (7.26) immediately gives... 

In the Fig. 7, we have reported the flux as function of the transmembrane pressure (AP) for a NaCI solution at a concentration of 10-1molL-1 (6gL 1) which is typical of a synthetic brackish water. The linearity observed suggests that this salty solution follows the Kedem-Katchalscky model (i.e. Spiegler-Ke-dem model, with pressure and osmotic linear gradients). For linear gradients, equation (1) amounts to... 

Small step and without creating significant concentration gradients Equation (10.59) gives... 

Consequently, to determine the overall rate of reaction, which is given by Equation (12-17), we first solve Equation (12-11) for C, differentiate with All the reactant respect to r, and then substitute the resulting expression into Equation (12-17). thatdi uses jnjQ Differentiating the concentration gradient, Equation (12-15), yields... 

Away from solid surfaces, the electrolyte can be said to be electrically neutral, and, in the absence of concentration gradients, equation (5.62) is satisfied exactly. If... 

At a certain time during the evolution of a sedimentary basin the secondary separate phase hydrocarbon migration is driven by hydrocarbon potential gradients (Equation 4.11 Sections 4.1.1 and 5.2), which in turn are controlled by... 

Fick s Second Law shows that the driving force for extraction is the concentration gradient within the particles, i.e., the difference between the phytochemical concentration inside the particle and the concentration at the particle surface. The rate of extraction, SC/St, increases with a larger concentration gradient. Equation 11.2 also shows that the rate of extraction can be altered by increasing the diffusion coefficient, D, or reducing the particle diameter, x. 

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