Design equation variable volume

Rewrite the design equation in terms of the measured variabte. When there is a net increase or decrease in the totai number of moles in a gas phase reaction, the reaction order may be determined from experiments performed with a constant-volume batch reactor by monitoring the total pressure as a function of time. The total pressure data should not be converted to conversion and then analyzed as conversion-time data just because the design equations are written in terms of the variable conversions. Rather, transform the design equation to the measured variable, which in this case is pressure. Consequently, we need to express the concentration in terms of total pressure and then substitute for the concemtation of A in Equation (E5-I.1),... 

It is usually most convenient to have the reactor volume V rather than the reactor length y as the independent variable. Accordingly, we shall change variables using the relation dV = A riy to obtain one form of the design equation for a tubular leactor ... 

This procedure follows the same pattern as that for stepwise solutions -of design, equations for simple batch and flow reactors. In each case-all the quantities in the differential mass balance are functions of two variables, conversion and reactor volume for the flow case and conversion and time for the batch case. 

The reaction-based design equations derived in the previous section are expressed in terms of extensive quantities such as reaction extents, reactor volume, molar flow rates, and the like. To describe flie generic behavior of chemical reactors, we would like to express the design equations in terms of intensive, dimensionless variables. This is done in two steps ... 

Below, we analyze the operation of variable-volume, gaseous batch reactors and describe how to apply the design equation for different cases. 

First, we discuss the application of the design equation for a variable-volume batch reactor when the reaction rate is provided in the form of an algebraic expression. We start by considering chemical reactions of the form... 

The design equation for gaseous variable-volume batch reactors was derived under two assumptions (i) AU the species are gaseous, and (ii) the mixture behaves as an ideal gas. In some operations, one or more of the species (especially heavier products generated by the reaction) may be saturated vapor. In this case, any additional amount generated will be in a condensed phase (liquid). While the ideal gas relation provides a reasonable approximation for the volume of species in the vapor phase, it cannot be applied for dieir volume in the liquid phase. Below, we modify the design equations for a variable-volume batch reactor with saturated vapors. 

Consequently, we see that Equation (1-11) applies equally well to our model of tubular reactors of variable and constant cross-sectional area, although it is doubtful that one would find a reactor of the shape shown in Figure 1-11 unless it were designed by Pablo Picasso. The conclusion drawn from the application of the design equation to Picasso s reactor is an imponant one the degree of completion of a reaction achieved in an ideal plug-flow reactor tPFR) does not depend on its shape, only on its total volume. 

Experimental design includes specifying what variables to measure and how best to measure them. Included in the list of variables are reactor volume, inlet flowrates, temperature, inlet (initial.) concentrations of one or more components, and effluent (final) concentrations of one or more components. Concentration or molar flowrate are the dependent composition variables in the design equations, and reaction rates are generally specified in terms of component concentrations. Whether the reaction is homogeneous or heterogeneous, solution of the material -balance requires knowledge of the fluid-phase concentrations, so... 

The procedure for variable-volume reactions is the same as described for a PFR. Thus we account for a volume change through the term (1 -I- eA- A)- With this modification, the design equation for reaction 4.1, with a first-order rate law, for instance, is... 

The concentration of one particular species, usually the limiting reactant, is a very convenient variable to use for constant-volume (constant-density) systems. Note that Eqn. (4-7) could have been obtained by substituting the rate equation, Eqn. (4-5), into the design equation for a constant-volume batch reactor, Eqn. (3-8). However, the stoichiometric table still would have been required to relate the various concentrations in the rate equation. 

Solving the Design Equation Equations (4-7)-(4-9) are equivalent forms of the design equation for an ideal, constant-volume, batch reactor. The only difference between them is that three different variables, Ca, xa, and have been used to describe the composition of the system at any time. 

When sludge is fed to a filter by a positive-displacement pump, the rate of filtration is nearly constant (i.e., dV/dr = constant). During constant-rate filtration, the pressure increases with an increase in cake thickness. Therefore, the principal variables are pressure and filtrate volume, or pressure and filtration time. Equation 9 is the principal design relation, which may be integrated for a constant-rate process. The derivative, dV/dr, may be replaced simply by V/t ... 

It is also possible to circumvent these mechanical design decisions at the early process design stage by rewriting the differential equations (4.9) in terms of V, the volume of the reactor, instead of in terms of the length /. This is our chosen approach here. In it the independent variable V will vary from V = 0 to V = Vr = Vend = 1.0 m3. 

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