Variable, iteration

Ehie to the dilute nature of the stream, we may assume that the latent heat of condensation for MEK is much smaller than the sensible heat removed from the gas. Therefore, we can apply the procedure presented in Section 10.5. Once a value is selected for ATj ", Eq. (10.19) can be used to determine T and Eqs. (10.3b) and (10.7) can be employed to calculate the value of Since the bounds on p (and consequently on AT ) are tight, we will iterate over two values of 0.96 and 1.00 (Ar2 " = 90.0 and 94.5 K). The other iterative variable is AT, . Both variables are used to trade off fixed versus operating costs. As an illustration, consider the following iteration AT," = 5 K and... 

Selection of values for the iteration variables for example, estimated stage temperatures, and liquid and vapour flows (the column temperature and flow profiles). 

A procedure for the selection of new values for the iteration variables for each set of trial calculations. 

The critical data and values used for inert components were those given by Ambrose (24). The interaction parameters between the water and the inert component were found by performing a dew-point calculation as described above but with the interaction parameter k.. rather than P taken as the iteration variable. 

A typical use for this model would be to solve for the number of moles of a gas, given its identity, pressure, volume, and temperature. The iterative solver is used for this purpose. You must decide which variable to choose for iteration and what a reasonable initial guess is. Real gases approach ideal behavior at low pressure and moderate temperatures. Since the compressibility factor z is 1 for an ideal gas, and since knowing z along with P, V, and T allows a calculation of n, we choose z as the iteration variable and 1.0 as the initial guess. 

Because of these problems, most recent methods avoid the distinction between components and derived species and take the moles of all species as iteration variables. 

The resulting C + E equations are nonlinear in unknowns nj, nj, and tt but In nj are iteration variables since nj occur in logarithmic terms. These equations are linearized using first-order Taylor Series (Newton-Raphson method), in the variables An , A (In nj), and ir, and with n nj are reduced to S + 1 + E linear equations in unknowns AN, A (In N), and tt. When extended to include P mixed phases, we nave shown that they are nearly identical to the equations of the RAND Method and have the same coefficient matrix. 

Given specifications other than temperature and pressure, in principle, nonlinear programming algorithms can optimize the appropriate thermodynamic function. When H and P are specified, entropy is maximized. But with compositions as iteration variables, the relation S = H,P,n) is needed. Similarly, given S and P, enthalpy is minimized but with compositions as iteration variables, H = H ,P,n is needed. Since these relations are usually not available, entropy can be maximized using Equation (3) and enthalpy minimized using Equation (5), with temperature and compositions as iteration variables. 

Alternatively all iteration variables are adjusted simultaneously, as demonstrated by Gordon and McBride (27) in the NASA method. They add the specification equation (e.g., (4) or (6)) to the equations that give minimum Gibbs free energy (22). Linearization leads to S + 2 + E equations in the unknowns AN, A(In N), 7tk, and A (In T). A formulation to permit multiple mixed phases is presented by Zeleznik and Gordon (2). 

For calculation of compositions in chemical equilibrium, we do not recommend the use of derived species as iteration variables, as recommended by Brinkley (4, 5), because solution of the mass balances can lead to negative moles of components. This problem does not arise when the moles of all species are taken as iteration variables. 

One unresolved question concerns whether it is possible to use volatility parameters as iteration variables in a nonlinear programming algorithm, with an approach similar to that introduced by Boston and Britt (13) for solution of nonlinear algebraic equations involving K-values. Our conclusion is that volatility parameters apply where K-values are used, and would be awkward to use in minimization of Gibbs free energy. 

Conventional methods for solving the coupled set of describing equations and thermo-physical property models are characterized by taking the primitive variables, or some subset of them, as the main iteration variables, and by working with the equations in essentially their "primitive" forms. Many methods have been proposed which may be regarded as conventional methods in this sense. For purposes of this paper, it is convenient to consider all conventional methods as members of one of two classes based on two fundamentally different approaches. 

Here the main iteration variables are the stage temperatures and inter-stage phase rates. The temperatures are paired with the combined constitutive and phase equilibrium equations, and the phase rates with the enthalpy and total mass balances. Unfortunately, this pairing is effective only for relatively narrowboiling systems, hence the method frequently fails for wide-boiling systems. Further, the computational procedure involves a lag of the K-value composition dependence from iteration to iteration, which makes the method unsuitable when the composition dependence is strong. 

The "sum-rates" method of Sujata (2) also uses the temperatures and phase rates as iteration variables, but reverses the pairing. The temperatures are paired with the enthalpy balances, and the phase rates are corrected by summing the component flow rates resulting from solution of the combined component mass balance and phase equilibrium equations. This method is especially effective for wide-boiling systems, such as absorbers, but is not suitable for narrow-boiling systems. 

Class II Methods. The methods of Class II are those that use the simultaneous Newton-Raphson approach, in which all the equations are linearized by a first order Taylor series expansion about some estimate of the primitive variables. In its most general form, this expansion includes terms arising from the dependence of the thermo-physical property models on the primitive variables. The resulting system of linear equations is solved for a set of iteration variable corrections, which are then applied to obtain a new estimate. This procedure is repeated until the magnitudes of the corrections are sufficiently small. 

The challenges presented by these goals are to exploit ones knowledge of how the system behaves and how it is structured to select well-behaved iteration variables, and to recognize ways in which the equations can be beneficically rearranged and transformed to accommodate these variables. 

These simple model parameters become the main (or "outer loop") iteration variables, the role played by the primitive variables temperature, pressure, vapor and liquid composition and phase rates in Class I and Class II methods. 

The new outer loop iteration variables are relatively free of interaction with each other, and are relatively independent of the primitive variables, hence precise initialization is not critical to good algorithm performance. 

As a result of these weak dependencies, A and B are excellent iteration variables. The values of A and B may be determined by evaluating the K s at two temperature levels, while holding x and y constant. 

Modification of Describing Equations. In the simple K-value and enthalpy models just described, there are Ng+6 parameters, a, A, B, C, D, E and F, which have characteristics that make them excellent choices for the iteration variables of an outer iteration loop. This is in fact the essence of the inside-out concept. Its success, however, rests on the ability to transform and rearrange the describing equations to properly accommodate these variables. The result should be an efficient and well-behaved inner iteration loop in which values of the primitive variables - now regarded as dependent variables - are calculated. When the inside loop is converged, the actual K-value and enthalpy models can be employed to calculate new values of the simple model parameters. 

The inner loop iteration variable for each stage is a unique combination of temperature and phase ratio which eliminates the need to make a distinction between wide-and narrow-boiling systems. In certain cases there are additional inner loop variables. 

Single-Stage Inside-Out Algorithm. Table I shows how the transformed and rearranged equations can accommodate the new iteration variables in an integrated algorithm having all six of the features discussed above. 

The difficulties associated with highly nonideal multi-stage systems have been overcome by introducing a simple model for the composition dependence of K-values. In keeping with the spirit of the inside-out concept, the parameters of the simple model become outside-loop iteration variables, and are determined by applying the actual models only in the outer loop. Further, they are as independent as possible of the primitive variables. 

The objective is to substantially reduce the composition dependence of the a s by using a simple model for Y in terms of parameters that will be well-behaved iteration variables. The treatment of Kb is the same as before, therefore a, which is now defined as follows ... 

Application to Simultaneous Phase and Chemical Equilibrium. The single-stage process with simultaneous phase and chemical equilibrium is another application of the inside-out concept where the Newton-Raphson method has been employed in a judicious way in the inside loop. There would appear to be no reaction parameter having characteristics that make it suitable as an outside loop iteration variable in the spirit of the inside-out concept. On the other hand, the chemical equilibrium relationships are simple in form, and do not introduce new thermophysical properties that depend in a complicated way on other variables. Thus it makes sense to include them in the inside loop, and to introduce the reaction extents as a new set of inside loop variables. 

If the equilibrium ratios are functions of phase compositions as occurs in liquid extraction or extractive distillation, it is necessary to include more variables in the iterative process. It was later shown (3) that for liquid extraction problems with known stage temperatures, the minimum number of iteration variables for quadratic convergence is nm, the n vapor flow rates, and n(m — 1) of the phase compositions. The total number of variables is n(2m + 2) because the temperatures are known. The iteration sequence is completely different for this case as compared with the previous case with composition independent equilibrium ratios. 

Before proceeding with the evaluation of Equations 14-16, some useful relations among the variables will be derived by differentiating the defining equations (Equations 1-3 and 6-10) with respect to the iteration variables. These results will be used when the error equations are differentiated implicitly. 

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