Variable-volume systems

The rate is defined as an intensive variable. Note that the reciprocal of system volume is outside the derivative term. This consideration is important in treating variable volume systems. 

In variable volume systems the dV/dt term is significant. Although equation 3.0.9 is a valid one arrived at by legitimate mathematical operations, its use in the analysis of rate data is extremely limited because of the awkward nature of the equations to which it leads. Equation 3.0.1 is preferred. 

In the analysis of experimental rate data taken in variable volume systems it is possible to develop another expression for the rate of formation of species i that is more convenient to use than equation 3.1.39. This alternative approach has been popularized by Levenspiel (2). It involves the use of fraction conversion rather than concentration as a primary variable and is applicable only to systems in which the volume varies linearly with the fraction conversion. 

Equations 3.1.47 and 3.1.50 express the relation between system concentrations and the fraction conversion for variable-volume systems that satisfy the linearity assumption of equation 3.1.40. This assumption is a reasonably unre-strictive one that is valid for all practical purposes in isothermal constant pressure systems in... 

To develop expressions for the reaction rate in variable volume systems, one need only return to the fundamental definition of the reaction rate (3.0.1) and combine this relation with equations 3.1.40 and 3.1.48. 

For variable volume systems equation 3.1.51 is much simpler in form and easier to use than equation 3.0.9. 

In experimental kinetics studies one measures (directly, or indirectly) the concentration of one or more of the reactant and/or product species as a function of time. If these concentrations are plotted against time, smooth curves should be obtained. For constant volume systems the reaction rate may be obtained by graphical or numerical differentiation of the data. For variable volume systems, additional numerical manipulations are necessary, but the process of determining the reaction rate still involves differentiation of some form of the data. For example,... 

If one combines the definition of the reaction rate in variable volume systems with a general nth-order rate expression, he finds that the time necessary to achieve a specified fraction conversion is given by... 

Note that the material balances for fixed beds are valid for die case of constant-density (constant volume) systems. The important term here is the one including the fluid velocity, i.e. the term uJdCIdz. For a variable volume system,... 

Thus, the term usc)Cldz comes from the derivative dFIdV, which should be used instead in eq. (3.287) for a variable-volume system. For catalytic fixed beds, in the typical case of a steady-state operation, eq. (3.288) takes the familiar form... 

For a variable volume system, / R / 0 by analogy to batch reactor. 

The integrated form of the BCST material balance for the case of a variable-volume system becomes (Levenspiel, 1972)... 

In case of a variable-volume system, the following form of the material balance should be used (eq. (3.293)) ... 

Figure 7.12 shows a minimal variable-volume system. or some substantial fraction of it, is a volume that depends on the position of a piston, ideally realised with a computer-operated motor drive ... 

TLC plate with different coatings Titration cell Variable volume system... 

In the analysis of experimental rate data taken in variable volume systems, it is possible to develop another expression for the rate of formation of species i that... 

In the analytical approach, the components are equilibrated and die compositions of the co-existing phases are determined either through sampling and off-line analysis or on-line speclroscopic techniques. A typical system for determination of vapor/liq-uid/liquid phase behavior is shown in Figure 21.1.10. The system is a double re-circulation variable volume system. A dynamic variant of this method is for the determination of the solubihty of solids in a supercritical fluid. The stationary solid phase is contacted with a continuous supply of... 

Note that for a variable volume system, the total change in the nnmber of moles should be considered. Thus, for example, in a reaction of the type like A + 36 26, the total number of moles of reactants and prodncts are 4 and 2, respectively. Therefore, there is a volume contraction. In this case, to express the concentrations in terms of the conversion, it is necessary to take into acconnt the volnme change. Using the ideal gas law, the total number of moles can be determined ... 

The local molar flow in relation to component A is defined as Fa = Cav (moles/h), where f is the volumetric flowrate (L/h). The molar inflow rate will bepAO = CaoVo- For any other component, reactant, or product, we analogously obtain Fb, Fr. However, note that the local volumetric flow for a variable volume system is not constant and varies with the factor of contraction or expansion. In this case, for a system at constant... 

The level transmitter should be a flush-diaphragm d/p cell with wetted parts of Hastelloy C. This is one of the vessels in the optional variable volume system (Fig. 11.2). 

Equation (4-9) is valid for both constant- and variable-volume systems. However, if the volume of the system varies, V must be known as a function of either t or If Vis constant, Eqn. (4-9) can be integrated. 

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