Rearrangement, integral

Substituting Eq. (9-1) for the density into Eq. (9-15), rearranging, integrating from the inlet of the pipe (point 1) to the outlet (point 2), and solving the result for G gives... 

Measuring at a wavelength at which the absorbance is much lower than that at the wavelength of irradiation, eq. (5.106) can be rearranged, integrated, and reduced to ( (r) = a(t))... 

This equation can be rearranged, integrated, and evaluated from an initial condition where when t = 0 to when t = t. 

The rearranging entity has been shown to be the bivalent cation the adjacent charges may so weaken the N—N link that charges of nearly integral size may be built up in the 4 and 4 positions. In the bent, but strainless, cation the minimum separation of the two p-positions would suffice for the establishment of a lai ely electrostatic bond, which could pass smoothly into the covalent rearrangement product (benzidine). 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

The derivation of this result may be found in various texts (27). Rearranging and integrating equation 13 yields... 

Assuming that U, and are invariant with respect to temperature and space, one can integrate equation 14 subject to equation 19, and obtain, after rearrangement, a basic heat-transfer equation for a parallel-flow heat exchanger (4). 

The most common ways of evaluating the constants are from linear rearrangements of the rate equations or their integrals. Figure 7-1 examines power law and Arrhenius equations, and Fig. 7-2 has some more complex cases. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

Reaction Rate Expression 119 Rearranging and integrating Equation 3-28 between the limits gives... 

Substituting Equation 3-185 into Equation 3-181, rearranging and integrating gives... 

Rearranging Equation 7-117 and integrating from tj to tj while the processing time passes from 0 to 6 gives ... 

Rearranging Equation 13-87 and integrating between the boundary conditions, 6 = 0—>6, T = Tj—> Tj, gives... 

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