Independent balances

Now suppose some of the solvent is evaporated ia the crystallizer, as is shown ia Figure 6. Independent balances can be written on total mass and... 

If Ne independent balance equations can be written, then the number of variables, Nd, that must be specified for a unique solution, is given by ... 

One difference between conducting and nonconducting media is that in the former case a charge balance may take the place of one of the material balances. For the same number of solution species, however, the number of independent balance equations is the same in the two cases. 

Assume for the moment that it is the concentration changes for Pj+ and (P A) as solid S reacts that are sought. Material balance equations can be written for P-j+ and (P A) as well as a charge balance but trial and error shows that it is impossible to do so without introducing nR+ and n -. P2 need not appear, however. A set of three independent balances is ... 

We need 2 independent balances, and will pick the total balance plus the N SC>4 balance. 

The H and O balances are not independent balances from what we have so far. We need one more equation. 

Only one independent balance can be written for the splitting point because the streams entering and leaving this subsystem are labeled as having identical compositions, so that the dry air/water mixture in all three streams behaves like a single species. (Convince yourself write a total mole balance and a dry air or water balance around the splitting point and observe that you gel the same equation each time.)... 

Overall system. 7 unknowns nQ,XQc,n >n4,np,X5c> 5H) + 1 reaction - 5 independent balances (C02,H2,I,CH30H, H2O) 3 degrees of freedom. Since we do not have enough equations to solve for the number of unknowns in the overall system, we check subsystems to see if one can be found with zero degrees of freedom. 

Recycle-fresh feed mixing point. 5 unknowns (wo, oc, n- 5c,- 5H) " 3 independent balances... 

Reactor. 4 unknowns ( 1, 21 3i 4) + 1 reaction - 4 independent balances (CO2, H2. CH3OH, H2 O) -1 single-pass conversion => 0 degrees of freedom. We will therefore be able to determine i, 2, 3, and rt4 and proceed from there. 

Purge-recycle splitting point. 2 unknowns (ni,np) - 1 independent balance 1 degree of freedom. Since the labeled component mole fractions are the same in all three streams in this subsystem, balances on all three species reduce to the same equation (try it and see). 

Recycle-fresh feed mixing point (revisited). 3 unknowns (no,xoc, r) 3 independent balances => 0 degrees of freedom. We can now determine (no, xqc, and nr). 

Purge-recycle splitting point (revisited). 1 unknown (np)-l independent balance 0 degrees of freedom. The final unknown variable can now be calculated. 

Your first attempt is likely to yield D.F. = 1. meaning that either (i) you counted one too many independent balances, (ii) you forgot to label one unknown variable, or (iii) the problem statement contains one redundant— and possibly inconsistent—process variable specification. Prove that (i) is actually the case, (Review the definition of independent balances in Section 4.7b.)... 

Three unknown process variables remain—and and since a total of three independent balances can be written, the system is determinate. 

In labeling the flowchart, we have taken advantage of the fact that CO and H2 remain in stoichiometric proportion throughout the process, which cuts down on the number of unknown variables that must be defined if the flow rate of CO is n, the flow rate of H2 must be 2h,. However, this also means that we cannot write independent balances on these two species. We must therefore allow for no more than two independent material balances on each process unit when we do the degree-of-freedom analysis, rather than the three that would normally be allowed when three species are involved in a process. 

We normally use a full set of NS material balances, even though the number of independent balances is limited to the rank NK of the stoichiometric matrix u for the given reaction scheme as shown in Section 2.1. cmd in Aris (1969). Exceptions must be made, however, when one or more constraints are imposed, such as quasi-equilibrium for some reactions or pseudo-steady state (better called quasi-conservation) for some chemical species then each active constraint will replace a mass balance. By these procedures, we avoid catastrophic cancellations that might occur in subtractions performed to reduce the number of species variables from NS to NK. 

This problem has 4m variables and m independent balance equations. There will be a unique solution to the problem if we specify 3m variables, for example, the molar flows of all m components in the three feeds or in two feeds and the product. 

Because you can make four independent balances for the process shown in Fig. 2.7, you can solve for four variables whose values are unknown in the process, such as tico2, n2, tto2> and nn, where n is the number of moles of each species in P P would be the sum of the four n s ... 

List the number of independent balances that can be written ascertain that a unique solution is possible. If not, look for more information or check your assumptions. 

Step 6 Two independent balances can be vn itten so that a unique solution exists. 

Other sets are possible. Write down a different set. Did you note that the set we have used has been chosen so as to include as few of the unknown variables as possible in a given equation that is, we made component balances on KCl and not on H2O. Keep in mind that the number of degrees of freedom summed for all the subsystems must be equal to zero to have a unique solution for the equations. In general, you can make the same number of independent balances as there are components or atomic species in each subsystem you define, but not always. Read the following example. 

Can you show that one or more component balances around systems I and II, or n and III in Example 2.19, or the entire set of three units, will add no additional independent balances to the set of component material balances made on each individual unit Can you show that a total balance on each unit, or on units I + II, or II + in, or around the entire system of three units, will add no additional independent balances Can you substitute one of the indicated alternative material balances for an independent component balance Yes (as long as the precision of the balance is about the same). 

Steps 5, 6, and 8 We have three elements, hence can make three independent balances. Because we have three unknown compositions, a unique solution exists, a solution that can be obtained by direct addition or subtraction. 

It is also of note that any process has a limited number of independent balances you can perform. This is not as much of an issue with a relatively simple problem like this, but will become an issue with more complex problems. Therefore, a step-by-step method exists to tell you exactly how many independent mass balances you can write on any given process, and therefore how many total independent equations... 

This represents two equations in three unknowns. We can pick the value of one stoichiometric coefficient, and since ammonia is the desired product, we set V31 = 1 (recall, products have v > 0). Then (7.4.26) gives = -1/2 and V21 = -3/2. Consequently, the one independent balanced reaction is... 

The first category includes systems in which the number of independent balance relationships is equal to the number of elements or basic species and where, moreover, conversion of the initial to the equilibrium mixture involves a change in the overall number of moles. The two conditions are usually satisfied when the initial mixture includes at least two constituents and the equilibrium mixture is rather complex. In such cases, there is usually at least one reaction among the many possible ones, the sum of stoichiometric coefficients of which is non-zero. For a system thus defined, the set (5.121) may be used unchanged. 

Repeat Example 15.8 for the system consisting of CO, CO2, H2, H2O, and CH4 (a system of great industrial interest in the production of hydrogen). How many independent balanced chemical equations exist between these 5 species ... 

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