Design equation integrated forms

Reactor type Design equation Integrated forms ... 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

Equations of gas dynamics in integral form are aimed at designing conservative difference schemes by means of the integro-interpolation method ... 

Though packed absorption and stripping columns can also be designed as staged process, it is usually more convenient to use the integrated form of the differential equations set up by considering the rates of mass transfer at a point in the column. The derivation of these equations is given in Volume 2, Chapter 12. 

Gilmore (1963) gives an integrated form of equation 12.57, which can be used for the approximate design of partial condensers... 

With a complex rate equation which does not lead to a form of the design equation that can be integrated analytically, graphical integration may be employed (see Fig. 2). This procedure may also be used for a non-isothermal reaction when k is not constant. 

The integrated form of the design equation gives the reaction time needed to achieve the conversion Xa of an initial number of moles of reactant, Wao- At the end of the reaction, the products must be removed... 

If the density is constant (e = 0), then the integrated forms of the design equation reduce to... 

Now, for a constant-density first-order reaction, the integrated form of the design equation for a plug-flow reactor, eqn. (66), may be rewritten... 

Here N designates the normalization factor. Clearly this equation in integrated form is the product of Gaussian and Lorentzian distribution functions 0mg and 0m define the line-widths of the two components, respectively. Here, the former represents Eq. (17) to a sufficient approximation for 0m 2 G and the latter was introduced to express the coupled rotational and/or the translational motion of proton pairs in the polymer, discussed by Pechhold53. ... 

If the rate equation is to be employed in its integrated form, the problem of determining kinetic constants from experimental data from batch or tubular reactors is in many ways equivalent to taking the design equations and working backwards. Thus, for a batch reactor with constant volume of reaction mixture at constant temperature, the equations listed in Table 1.1 apply. For example, if a reaction is suspected of being second order overall, the experimental results are plotted in the form ... 

This is generally obtained by use of the integrated form of the mechanical energy equation with the frictional energy loss calculated by Eq. (65). Thus, the basic problem facing a design engineer is how to obtain numerical values for the friction factor /. 

Equation (2-6) is the differential form of the design equation, and Equations (2-8) and (2-9) are the integral forms for a batch reactor. The differential form is generally used in the interpretation of laboratory rate data. 

Packed-Bed Reactor. The derivation of the differential and integral forms of the design equations for a packed-bed reactor are analogous to those for a PFR [cf. Equations (2-15) and (2-16)). That is, substituting for Fa Equation (1-11) gives... 

The differential form of the design equation [i.e,. Equation (2-17)] must be u d when analyzing reactors that have a pressure drop along the length of the reactor. We discuss pressure drop in packed-bed reactors in Chapter 4. Integrating with the limits W = 0 at 2( = 0 gives... 

In terms of the conversion, the differential and integral forms of the reactor design equations become ... 

When pressure drop through the reactor (see Section 4.4) and catalyst decay (see Section 10.7) are neglected, the integral form of the packed-cata-lyst-bed design equation can be used to calculate the catalyst weight... 

Unfortunately for most reactor systems, a scale-up process cannot be achieved simply from a knowledge of as a function of J. In Chapter 5 we present elementary forms of the kinetic rate law from which the design equations can be evaluated, either by graphical or numerical integration or with the aid of a table of integrals. 

The designation = 0 refers to the initially established steady state concentration of radicals. We note that in Equation 5a, the variations of C and P operate in opposite directions. Up to conversions of 2Q%, e and a may safely be regarded as constants. Equations 4a,b may be used in the differential or integrated forms (21), to yield expressions for P in terms of the observed rate,... 

Comparison of Eqs. (4-2) and (4-5) shows that the form of the design equations for ideal batch and tubular-flow reactors are identical if the realtime variable in the batch reactor is considered as the residence time in the flow case. The important point is that the integral c/C/r is the same in both reactors. If this integral is evaluated for a given rate equation for an ideal batch reactor, the result is applicable for an ideal tubular-flow reactor this... 

Equation 4.1.3 is the integral form of the general species-based design equation of chemical reactors, written for species j. 

Equation 4.2.2 is the integral form of the species-based design equation for an ideal batch reactor, written for species j. It provides a relation between the operating time, t, the amount of the species in the reactor, Nj(t) and Nj(0), the species formation rate, (rj), and the reactor volume, V. Note that when the reaetor volume does not change during the operation, Eq. 4.2.2 reduces to... 

Equation 4.2.13 is the species-based differential design equation of a plug-flow reactor, expressed in terms of the conversion of reactant A. To obtain the integral form of the design equation, we separate the variables and integrate Eq. 4.2.13 ... 

Equation 6.2.5 is the integral form of the design equation for an ideal, constant-volume batch reactor. Figure 6.3 shows the graphical presentation of this design equation. To solve the design equations, we have to express the reaction rate r in... 

Once the form of the rate expression is known, we can determine the reaction rate constant, k(J), by using the integral form of the design equation. Consider file integral form of the design equation (Eq. 6.2.5) ... 

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