Differential forms and equations

If work done in a system is distributed over an area, for example, pressure P is acting through volume v, then in specific notation and in differential form the Equation 2.44 results. 

Under these conditions, the differential forms of equation for NA (10.4, 10.18and 10.19) may be simply integrated, for constant temperature and pressure, to give respectively ... 

This equation can be treated in differential form and in integral form. In differential form it becomes... 

The differential form of equation 6.50 with u expressed in terms of G and V is... 

The solution has been worked out in Section 1.5 and given in its differential form as equation (1.5.3)... 

Pseudo-first-order kinetic model (Lagergren s rate equation) In this model, the kinetic rate in differential form and its analytical solution can be expressed as... 

In the experiment the differentiated absorption curves are obtained. Therefore, we begin with the differentiated form of Equation 8. The differentiation of the field strength H is marked here by a comma. Accordingly, the intensity of the lateral extremes for the differentiated absorption curves is marked (Ii )l/2 and the intensity of the inner extremes (I0 )l/2. The linewidths resulting from the distance of the extremes are marked (Affi)i/2 for the lateral gaussion curves and (AHg)i/2 for the inner curve. In this way the areas under the different gaussian curves defined by Equation 8 are determinable. 

As mentioned above an increase of the linewidth occurs by the superposition of the lateral and central gaussian curves or by a shift of the extremes of the differentiated absorption curves, which is essentially the same. Again the real linewidths can be obtained iteratively only. For this purpose we begin with the twice-differentiated form of Equation 8 because the question after the shift of the extremes for the differentiated curves is identical to the question after the shift of the zeros for the twice-differentiated absorption curves. There the twice-differentiated Equation 8 is developed in a Taylor series at the positions of the extremes breaking off after the second term. At these positions the function is... 

In many problems, all terms in equations (55)-(58) involving lim. o are zero for example, there is usually no excess mass at the surface so that lim. o j p d V = 0 in equation (55). If it is also assumed that the interface is not moving dxjdi 0), that viscosity is negligible, and that approximations 1-4 of Section 1.3 are valid, then equations (55)-(58) simplify considerably. Using equations (5) and (6) and the identity h = u (p/p), we find by passing to the limit of small that the differential forms of equations (55)-(58) reduce to... 

The subscript s and d refer to the elastic spring and viscous dashpot, respectively. The differential equation for a Maxwell body is obtained by first differentiating the sum of the strains Equation (35). This allows the viscous Equation (33) to be directly inserted in the differentiated form of Equation (35). Then by differentiating elastic Equation (32) it can also be inserted into the differentiated form of Equation (35) giving the final result ... 

The heat capacity of activation for proton transfer under conditions where the tunnel effect can be neglected (i.e. at the higher temperatures) is probably small (see Section IVCl) and the available results are un-fortimately not sufficiently accurate to allow the evaluation of this parameter. For similar reasons it cannot yet be shown that the temperature-dependent values of AC predicted by the differential form of equation (32) are observed in practice. 

Cumulative values for the chain lengths are calculated as a function of position down the tube using differential forms for Equations 13.14 and 13.15 ... 

Step 3. Use the first law of thermodynamics in differential form, the equation of continuity, and the mass transfer equation to develop an equation of change for internal energy. 

In blowers and compressors pressure changes are large and compressible flow occurs. Since the density changes markedly, the mechanical-energy-balance equation must be written in differential form and then integrated to obtain the work of compression. In... 

Since this is a case of A diffusing through stagnant, nondiffusing 5, Eq. (6.2-18) will be used in its differential form and will be equated to Eq. (6.2-28), giving... 

Equations 7.22 and 7.24 are valid as long as the heat capacities can be assumed to be independent of temperature, which is often a good approximation if the temperature change is small, say 50 K or less. For calculations in which the temperature dependence of the heat capacity cannot be ignored, we must use the differential forms of Equations 7.22 and 7.24 ... 

Equation 15.46 represents the incremental increase in the induced comjwessive str in the inner core. Since the increments are small, in., and tiie numl r of shells is large, F)<. 15.46 will be written in differential form and will la formally inlegraled, a-... 

Equation 15.75 is written in differential form and must be integrated to give the cumulative radial stress, frc, at, any radius, r, when a given tensile winding stress, / , is used. The equation can be rearranged as follows to permit integration ... 

The flux through a membrane segment of thickness dx is written in analogy to Equation 5.16. The equation is written in a differential form, and subscript B is dropped, since we are dealing with pervaporation of a single-component system. Then... 

Because x and y are related through the equation g x,y) = constant, they are not independent variables. To satisfy both Equations (5.14) and (5.15) simultaneously, put the constraint Equation (5.15) into differential form and combine it with the total differential Equation (5.14). Since g = constant, the total differential dg is... 

Equation (3-5) is referred to as the design equation for an ideal batch reactor, in differential form. This equation is valid no matter how many reactions are taking place, provided that Eqn. (1-17) is used to express r/, and provided that all of the reactions are homogeneous. 

Equation 3.33 is obviously of the second-order differential form and, hence, when simulated will give a typical second-order response. To understand better what type of response these second-order systems will display, we will examine another common system and generalise its equations. The closed-form solution is best illustrated by an example that is familiar, namely the spring, mass, and damper system presented in Figure 3.26. Note that this is really just a simplification of the control valve example just cited. 

Strictly speaking, the Ergun equation is valid for isothermal systems, incompressible fluids, and for beds which are spatially uniform. When the system is nonisothermal or when the pressure drop across the bed is comparable to the absolute pressure at the inlet, the use of Eq. (7.3.2) may introduce quite serious errors. When there are large variations in the absolute temperature and pressure within the system, the Ergun equation has to be put in a differential form and then integrated along the appropriate temperature path [12]. 

---