PFTR mass balance

This expression can be inserted into the CSTR and PFTR mass-balance equations to yield... 

For a PFTR, the substitution dt —> dz/u transforms the batch reactor equation into the PFTR mass-balance equation. 

These are batch reactor or PFTR mass-balance equations on all seven species we identified in our mechanism. The reader should verify each term, particularly the signs and the factor of 2 in some terms. 

We have to solve the PFTR mass-balance equation to find Cj (z), but the first issue is exactly what form we should use for r or rj in the mass-balance equations. This is a major topic of this chapter. 

The mass transfer coefficient is usually much lower in the Hquid phase, and therefore is a function of R, the distance from the wall to the interface. One would have to solve for the steady-state profile C iR), and find its average CX(z) to insert into the PFTR mass-balance equations simultaneously to find Ca(L) in each phase. 

This equation is not appropriate if all five of these conditions are not met. We can relax the third and fourth restrictions for the PFTR by considering the differential element of volume dV = At dz rather than the differential element of length dz. The mass-balance equation at a position where the fluid has moved from volume V to volume V + d V then becomes... 

For the PFTR we return to the original mass-balance equation... 

For a PFTR followed by a CSTR we solve each reactor mass balance sequentially to find Ca2(Cao t). For first-order kinetics this gives... 

Transients in a PFTR require solution of a partial differential equation. The transient version of the derivation of the mass balance in a PFTR is... 

These look similar to series reactions, but the solution is quite different. In a PFTR or batch reactor the mass-balance equations are... 

For tliese reactions in a PFTR the mass-balance equation becomes... 

All these arguments require a single reactant A on which to base the calculation of selectivity. For more complex situations we can stiU determine how the selectivity varies with conversion in PFTR and CSTR, but calculation of the selectivity requires complete solution of the mass-balance equations. 

For the PFTR we wrote the mass balance on species j by writing a shell balance between position z and z + dz. After letting ds —> 0 we obtained the expressions... 

For a single reaction in a steady-state PFTR the mass-balance equations on species A the temperature are... 

We have used the first-order irreversible reaction as an example, but this is easy to generalize for any reaction, irreversible or reversible, with any kinetics. In a PFTR the mass-balance equation for an arbitrary reaction becomes... 

If there is an internal cooling coil in a CSTR, the coil can exhibit a coolant temperature that is a function of position in the coil. Thus we should write a mass balance on the coolant, which is that of a PFTR without chemical reaction,... 

The fluid in a packed bed reactor flows from one end of the reactor to the other so our first approximation will be to assume no mixing, and therefore the mass balance on the fluid win be a PFTR,... 

Thus in the RTR the PFTR section sees (1) a diluted feed, (2) a higher flow rate, and (3) a shorter residence time if the recycle ratio R is greater than zero. However, the mass balance on the PFTR itself is unchanged. 

We can of course write the six mass balances on each species and solve them in batch or continuous reactors to find the species concentrations as a function of residence time [A](t), [CH3 ](t), [CH4](t), etc. The mass-balance equations in a PFTR (or in a batch reactor by replacing r by t) are... 

We begin by writing the mass balance on aU species in batch (t) or PFTR (t). For monomer [M] we obtain... 

Again we have an infinite set of coupled equations. For the PFTR or batch reactor we can write these mass balances for condensation polymerizatioa The monomer A[ is lost by reacting with all the polymeis,... 

Let us next try to find the total concentration of these functional groups in the solution in a PFTR or batch reactor. The mass balance gives... 

As before, we consider only two ideal continuous reactors the PFTR and the CSTR, because any other reactor involves detailed consideration of the fluid mechanics. For a phase a the mass balance if the fluid is unmixed is... 

For the PFTR the mass transfer area is simply the total interfacial area between the phase in question and the other fluid, while for the PFTR, where Cj is a function of position, [areaWolume] is the area per unit volume of that phase at position Z- The steady-state mass balance is therefore... 

One of the simplest models used to describe the performance of tubular reactors is the well-known isothermal one-dimensional plug flow tubular reactor (PFTR) model. The mass balance of this model for steady-state conditions, the simultaneous occurrence of M reactions and a constant volumetric flow rate V is ... 

If only one reaction needs to be considered, and conversion Xa is chosen to be the state variable of interest, using Eqs. (5) and (7), the mass balance of the PFTR, Eq. 8, can be expressed as follows ... 

Integration from 0 to the reactor length L and from 0 to Xa(L) leads to the well-known dimensionless mass balance of the PFTR ... 

The selectivity and the yield with respect to D depend strongly on the ratio kn/ku. Typical results obtained solving numerically the mass balance equations of the PFTR model (Eq. (8)) are shown in Fig. 12.3. 

The resulting equation is completely identical with the steady-state mass balance of the PFTR. This is an important finding for process safety. It allows the experimental characterization of a tube reactor process to be performed batch-wise on laboratory scale without violating the boundary conditions to be observed when applying model reduction. The demand of a parameter determination under closest proximity to plant conditions remains fulfilled. The sequence along the length of the tube reactor is completely equivalent to the time sequence of a batch process. 

In Table 4-4 sample solutions for the steady-state mass balances of CSTR and PFTR and the obtainable conversion in a SBR at the time of stoichiometric input amount are compiled. Isothermal, operation is assumed. The calculations refer to the following simple formal kinetic rate laws ... 

The equation given for the calculation of the conversion at the stoichiometric input point in a SBR for a second order reaction is an approximation derived by Hugo [47]. The solutions for the BR and the sfeady-state operated PFTR are identical for the reasons given when deducing the different mass balances. 

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