Mole balances pressure drop

When using an ordinary differential equation (ODE) solver such as POLYMATH or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation as we did to obtain an analytical solution. Writing She equations separately leaves it to the computer to combine them and produce a solution. The formulations for a packed-bed reactor with pressure drop and a semibatch reactor are given below for two elementary reactions. 

We now must determine the ratio F/Fq a function of volume V or the catalyst weight, W to account for pressure drop. We then can combine the concentration, rate law, and design equation. However, whenever accounting for the effects of pressure drop, the differential form of the mole balance (design equation) must be used. 

Rather than rederive everything starting ffom the mole balance, rate law, stoichiometry, and pressure drop equations, we will use the equations developerl in Example 4-6. Combining Equations (E4-6.I) and (B4-6.8) gives... 

Combining mole balance, rate laws, and stoichiometry for species 1 through species y in the gas phase and for isothermal operation with no pressure drop gives us... 

It is customary to use the mole fraction Xg of the gas phase instead of Cg, the two being related by Cg = XgCt, where Ct is the total molar concentration. Here, Xg is the value of Xg space-averaged over the x- and y-coordinates in the same manner as Cg is developed from Cg. If the gas pressure drop along the packed bed is very small, Ct may be considered constant along the bed since Ct is proportional to the total gas pressure. The mass balance equation (7.1.25) may now be written in terms of Xg as (Ruthven, 1984)... 

Consider a gas mixture consisting of two components with mole fractions yio and y2o at pressure pi and temperature Ti. At some point in time, taken to be the initial time, the pressure and temperature change abmptly and become equal to p2 and T2, which then remain constant. Thus, the thermodynamic balance that existed in the system at pi and Ti will be disturbed, and at the new values p2 and T2 drops of liquid phase with average radius Rq = Rn will appear in the mixture due to homogeneous condensation. These drops grow in time until a new equilibrium appropriate to the values pz and Tz is established. 

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