Rearrangement, integral equations

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). 

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... 

Integrating Equation 4-116 assuming incompressible drilling fluid flow (p is constant) and after simple rearrangements yields the pressure loss across the bit Apj (psi) which is... 

By integrating and rearrange this equation, one obtains the following linear relationship ... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

Thus, the SAD equations of motion can be transformed into (61) aild (62), by formally integrating (21) to obtain a system of integral equations, which can then be put into Inglesfleld s form, by making the change of variables (65), followed by some rearrangement of terms. 

The process of solving for the integrated form of a differential equation, achieved by rearranging the equation into a form where each side of the resulting expression can be integrated with respect to one of the variables. 

Rearranging and integrating Equation 5 between the limits o-f and o-t leads to ... 

If we assume that Lv is a constant, we can rearrange and integrate Equation 2-3,... 

Rearrangement of Equation 2 and integration yields the following expression ... 

Rearranging and integrating Equation 3-95 between the limits with the boundary conditions at time t = 0, CA = CAO, CB = CBO, Cc = Cco, gives ... 

If we rearrange and then integrate equation (27.3) for the addition of 1 mole of material, we have ... 

Since dV/dT)p can be obtained either from P-V-T data, or from a suitable equation of state, it is possible to determine the variation of entropy with pressure at constant temperature. Upon rearrangement of equation (20.15) and integration between the pressure limits of Pi and Pi, at constant temr-perature, the result is... 

Rearranging and integrating Equation 8-150 from t, to tj while the time passes from 0 to 0 gives ... 

The most important variables in the system are the state variables, since it is their evolving behaviour in time that is the basis of the dynamic response of the system. The importance of their role may be brought out further by rearranging the equations in Section 2.1 to eliminate all the algebraic equations and leave just the four state equations, integration of which enables us to trace the response of the system. 

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