Degree of Freedom Analysis

A degrees-of-freedom analysis separates modeling problems into three categories ... 

The process has four separate subsystems for the degree-of-freedom analysis. Redundant variables and redundant constraints are removed to obtain the net degrees of freedom for the overall process. The 2 added to Nsp refers to the conditions of temperature and pressure in a stream +1 represents the heat transfer Q. 

For a given feed charge (nc specifications), a degree of freedom analysis shows that there are 4nc (state variables) - nc (feed specifications) - nc (mass balance) = 2nc degrees of freedom. 

A degrees-of-freedom analysis indicates that the variables subject to the designer s control are C -H 3 in number. The most common way to use these is to specify the feed rate, composition, and pressure (C -H 1 variables) plus the drum temperature T2 and pressure P2. This operation will give one point on the equU rium flash curve shown in Fig. 13-17. This curve shows the relation at constant pressure between the fraction V/F of the feed flashed and the drum temperature. The temperature at V/F = 0.0 when the first bubble of vapor is about to form (saturated liquid) is the bubble point temperature of the feed mixture, and the v ue at V/F = 1.0 when the first droplet of liquid is about to form (saturated vapor) is the dew point temperature. 

The K j are the equilibrium ratios or K values for species i on stagey. Degrees-of-Freedom Analysis and Problem Formulation... 

Once you are certain that all equations are independent and no equations are missing, then unspecify one of the variables. For example, unspecify the nitrogen concentration at the eonverter inlet, y3,e. Because is now unspecified, correct the degree of freedom analysis for both the mixer and converter. At the mixer and converter the number of variables increases by one as shown in Table 3.5.6. Thus, for the mixer F = 12 - 7 = 5 and for the converter F = 14 - 9 = 5. Because Equations 3.5.27 and 3.5.29 are not independent, the niunber of equations at the condenser-separator combination and the splitter are reduced by one, as shown in Table 3.5.6. Finally, because Z yi, is no longer vahd, it is not a repeated equation. Thus, the repeated equations in line 7 are now zero. The revised calculation for the degrees of freedom in Table 3.5.6 shows that the process degrees of freedom is now zero. 

Given a process description, (a) draw and fully label a flowchart (b) choose a convenient basis of calculation (c) for a multiple-unit process, identify the subsystems for which balances might be written (d) perform the degree-of-freedom analysis for the overall system... 

To perform a degree-of-freedom analysis, draw and completely label a flowchart, count the unknown variables on the chart, then count the independent tq xz.. on relating them, and subtract the second number from the first. The result is the number of degrees of freedom of the process, df (= unknowns rtindep eqns)- T ere are three possibilities ... 

Stoichiometric relations. If chemical reactions occur in a system, the stoichiometric equations of the reactions (e.g., lUi + O2 —2H2O) provide relationships between the quantities of the reactants consumed and of the products generated. We will consider how to incorporate these relationships into a degree-of-freedom analysis in Section 4.7. 

We first do the degree-of-freedom analysis. There are six unknowns on the chart—h through h(,. We are allowed up to three material balances—one for each species. We must therefore find three additional relations to solve for all unknowns. One is the relationship between the volumetric and molar flow rates of the condensate we can determine from the given volumetric flow rate and the... 

The degree-of freedom analysis tells us that there are five unknowns and that we have five equations to solve for them [three mole balances, the density relationship between V2 (= 225 Uh) and hj, and the fractional condensation], hence zero degrees of freedom. Hie problem is therefore solvable in principle. We may now lay out the solution—still before proceeding to any algebraic or numerical calculations—by writing out the equations in an efficient solution order (equations involving only one unknown first, then simultaneous pairs of equations, etc.) and circling the variables for which we would solve each equation or set of simultaneous equations. In this... 

The 8%-92% benzene split between the product streams is not a stream flow rate or composition variable nevertheless, v/e wriic it on the chart to remind ourselves that it is an additional relation among the stream variables and so should be included in the degree-of-freedom analysis. 

The degree-of-freedom analysis begins with the overall system and proceeds as follows ... 

We will not go through the detailed solution but will simply summarize. The degree-of-freedom analysis leads to the results that the overall system has one degree of freedom, the evaporator has zero, and the crystsillizer-filter has one. (Verify these statements,) The strategy is therefore to begin with the evaporator and solve the balance equations for mi and m2. Once m2 is known, the crystallizer has zero degrees of freedom and its three equations may be solved for m3, rhi, and ms. The rate of production of crystals is... 

When we first described degree-of-freedom analysis in Section 4.3d, we said that the maximum number of material balances you can write for a nonreactive process equals the number of independent species involved in the process. It is time to take a closer look at what that means and to see how to extend the analysis to reactive processes. 

Similarly, atomic nitrogen (N) and atomic oxygen (O) are always in the same proportion to each other in the process (again 3.76 1) as are atomic chlorine and atomic carbon (4 mol Cl/1 mol C). Consequently, even though four atomic species are involved in this fm>-cess, you may count only two independent atomic species balances in the degree-of-freedom analysis--one for either 0 or N and one for either C or Cl. (Again, convince yourself that the O and N balances yield the same equation, as do the C and Cl balances.)... 

Finally, when you are using either molecular species balances or extents of reaction to analyze a reactive system, the degree-of-freedom analysis must account for the number of independent chemical reactions among the species entering and leaving the system. Chemical reactions are independent if the stoichiometric equation of any one of them cannot be obtained by adding and subtracting multiples of the stoichiometric equations of the others. 

If molecular species balances are used to determine unknown stream variables for a reactive process, the balances on reactive species must contain generation and/or consumption terms. The degree-of-freedom analysis is as follows ... 

The third way to determine unknown molar flow rates for a reactive process is to write expressions for each product species flow rate (or molar amount) in terms of extents of reaction using Equation 4.6-3 (or Equation 4.6-6 for multiple reactions), substitute known feed and product flow rates, and solve for the extents of reaction and the remaining reactive species flow rates. The degree-of-freedom analysis follows ... 

The feed to the reactor contains 7.80 mole% CHi, 19.4% O2, and 72.8% N2. The percentage conversion of methane is 90.0%, and the gas leaving the reactor contains 8 mol C02/mol CO. Carry out a degree-of-freedom analysis on the process. Then calculate the molar composition of the product stream using molecular species balances, atomic species balances, and extents of reaction. 

Notice that we only subtracted four balances and not one for each of the five species. The reason is that when we labeled the outlet flow of I as 2.0 mol, we implicitly used the balance on I input = output) and so can no longer count it in the degree-of-freedom analysis. We will use the same reasoning in the analysis of the condenser. 

In this analysis we presumed that we knew m, ni, 3, and 4 from the reactor analysis, and since we used the methanol and water balances when we labeled the bottom product stream we only counted three available balances in the degree-of-freedom analysis. 

Every chemical process analysis involves writing and solving material balances to account for all process species in feed and product streams. This chapter outlines and illustrates a systematic approach to material balance calculations. The procedure is to draw and label a flowchart, perform a degree-of-freedom analysis to verify that enough equations can be written to solve for all unknown process variables, and write and solve the equations. 

To perform a degree-of-freedom analysis on a single-unit nonreactive process, count unknown variables on the flowchart, then subtract independent relations among them. The difference, which equals the number of degrees of freedom for the process, must equal zero for a unique solution of the problem to be determinable. Relations include material balances (as many as there are independent species in the feed and product streams), process specifications, density relations between labeled masses and volumes, and physical constraints (e.g., the sum of the component mass or mole fractions of a stream must add up to 1.)... 

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