Sequential solutions mass balance

However, if he selects an outlet stream composition (say the liquid stream) instead of a flow-rate, then the simultaneous solution of the mass balance and v-l-e relationships would not be necessary. The stream compositions could be calculated by the following step-by-step (sequential) procedure ... 

Now we have to try to solve the mass-balance equations for each of these species to obtain [M](t), [AMj] r), and [P](r), remembering that j and n can go to infinity. We obviously have to solve an infinite number of equations sequentially in the general case. However, we can make the problem quite simple (but obtain a solution that is not entirely accurate) by setting all kpS and aU k S equal. Now we can play games with summations and make solution of this reaction set a rather straightforward problem. 

Step 1 One way to solve this problem is to do a sequential solution. You start with the feed, and solve the mass balance for the mixer, labeled MIXR. Since you do not yet know the amount of A in stream S5, assume it is zero and go on. Then there is one mole of A fed to the reactor, 40 percent of it reacts, and the unreacted part is recycled into stream S5. 

Results of this study as illustrated in Figures 7 and 8 show that (1) the sequential ultrafiltration-mass balance approach is valid for known solutions (2) cadmium, copper, and lead are found in the soluble phase... 

As is true in the design of many separation techniques, the choice of specified design variables controls the choice of the design method. For the flash chamber, we can use either a sequential solution method or a simultaneous solution method. In the sequential procedure, we solve the mass balances and equilibrium relationships first and then solve the energy balances and enthalpy equations. In the simultaneous solution method, all equations must be solved at the same time. In both cases, we solve for flow rates, compositions, and temperatures before we size the flash drum. 

In the sequential solution procedure, we first solve the mass balance and equilibrium relationships, and then we solve the energy balance and enthalpy equations. In other words, the two sets of equations are uncoupled. The sequential solution procedure is applicable when the last degree of freedom is used to specify a variable that relates to the conditions in the flash drum Possible choices are ... 

If the temperature of the feed to the drum, Tp, is the specified variable, the mass and energy balances and the equilibrium equations must be solved simultaneously. You can see from the energy balance, Eq. (2-7 why this is true. The feed enthalpy, hp, can be calculated, but the vapor and liquid enthalpies, and depend upon x, which are unknown. Thus a sequential solution is not possible. 

Simulation of S5mgas processes are performed using flow sheeting [82], where the individual unit operations are connected by streams and the heat and mass balances for the entire process are solved sequentially. Well-proven commercial tools and solution methods for many processes are available as described in [82], but to design catal c processes both simplified and rigorous reactor models are key items and for this purpose a dedicated tool such as the one described in [116] can be used. 

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

The relations between the dilute/dense concentrations and the mean concentration can be resolved with sequential analysis of the balance between mass transfer and reactions, as is in Levenspiel (1999). Thereon, the structure-dependent correction for mass transfer and reaction rate can be determined. Here we exemplify the solution with reference to a reactive system where a gas species is catalyzed by sohd particles. The gas concentrations are therefore of our major concern. In greater details, the mean molar concentration of species j (C ) in a control volume relates with the dense-phase and dilute-phase concentrations by... 

The physical properties of water may be assumed for the impregnating solution. Typical nonuniform catalyst distributions (Chen and Anderson 1973) shown in Figure 1.5 suggest that there definitely exists a resistance to solute removal from the solution since otherwise the profiles should show stepwise distributions if enough solutes are present. Solute removal from the impregnating solution onto the pore wall can be modeled as the sequential events of mass transfer at the liquid-solid interface followed by adsorption onto the pore wall. In many cases, the adsorption/ desorption step is the controlling step. The solute balance for the solid phase in such cases is ... 

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