Cross-derivative equality requirement

The derivative with respect to x of the derivative of F with respect to y is equal to the derivative with respect to y of the derivative of F with respect to x. If this is the case, then the original differential dF in equation 4.28 satisfies one requirement of an exact differential The value of the multiple differential does not depend on the order of differentiation. Equation 4.29 is known as the cross-derivative equality requirement of exact differentials. In the application of the double derivatives in equation 4.29 to real thermodynamic equations, the partial derivatives may have some other expression, as the following example shows. 

Equation (8) indicates that the current measured at the SECM tip, it(z = 0), is directly proportional to ft. At steady state, mass continuity requires that the rate of transport from the pore into the receptor compartment be equal to the rate of transport within the pore. Thus, it(z = 0) is also proportional to the rate of transport at any point within the pore. The molecular flux in the pore, N, is obtained by simply dividing ft by the cross-sectional area of the pore. Note that in deriving Eq. (8), no restrictions have been placed on the mechanism of transport within the pore. Thus, it(z = 0) is proportional to the flux in the pore, independent of whether the flux is due to diffusion, migration, or convection. It can be shown that the tip current at any arbitrary separation distance, z, is also proportional to the flux in the pore. 

Equation (14.111) for the minimum power of 0.0923 kW to produce 1 kg of separative work per year in uranium isotope separation was derived for cross flow on the low-pressure side of the barrier, with the composition of gas on that side y equal to the composition of the net flow u. The purpose of this section is to show that the minimum power requirement could be reduced further by having v greater than y by an appropriate amount and to derive an expression for the optimum difference between v and y and the corresponding power consumption per unit separative capacity. For this minimum wer case, pressures on the high-pressure and low-pressure sides of the barrier must be so low the only flow through the barrier is of the separating, molecular type, and the mixing efficiency on each side of the barrier is unity. 

A very simple, but also coarse, method of detertniriitig the minimally required gas volume flow rate is to define it as the flow rate creating conditions of minimal fluidization at the upper surface of a stagnant bed of solids placed in the equipment under consideratiort This derivation will be illustrated for the ProCell unit depicted schematically in Fig. 4.9. The superficial gas flow velocity at the free surface of the respective stagnant bed is the ratio of the gas volume flow rate Vg to the respective cross-sectional area, which is twofold the product of the width Wo and the length L of the apparatus, perpendicular to the plane of Fig. 4.9. By setting the superficial gas flow velocity equal to the minimal fluidization velocity u f and the gas volume... 

The above derivations are concerned with the relation between the strain measures in the beam and in the wall description. Correspondingly, such a relation is also required for the electric field strength. The latter is accessed with the aid of electrodes that necessarily connect certain areas and thus induce an equalization therein. To achieve unified behavior for the beam description, a complete connection in parallel needs to be introduced along the cross-sectional coordinate s ... 

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