Tear variables

A temperature profile plus a vapor-rate profile through the column must be assumed to start the procedure. These variables are referred to as tear variables and must be iterated on until convergence is achieved in which their values no longer change from iteration to iteration and all equations are satisfied to an acceptable degree of tolerance. Each iteration down and then up through the column is referred to as a column iteration. A set of assumed values of the tear variables consistent with the specifications, plus the component K values at the assumed temperatures, is as follows, using assumed end and middle temperatures and K values from Fig. 13-14. ... 

Convergence to the final solution is rapid with the TG method for narrowboiling feeds but may be slow for wide-boiling feeds. Generally, at least five column iterations are required. Convergence is obtained when successive sets of tear variables are identical to approximately four significant digits. This is accompaniedby 0 = 1.0, a. = normahzed x, and nearly identical successive values of ft as well as Q,.. 

The NC equations (13-75) are linearized in terms of the NC unknowns Xjj by selecting unknowns Vj and Tj as tear variables and using values of vectors andy from the previous iteration to compute values of Kij for the current iteration. In this manner all values of A, Bj j, and Cjj can be estimated. Values of Djj are fixed by feed specifications. Furthermore, the NC equations (13-75) can be partitioned... 

Compute a new set of values of tear variables by computing, one at a time, the bubble-point temperature at each stage based on the specified stage pressure and corresponding normalized values. The equation used is obtained by combining Eqs. (13-69) and (13-70) to eliminate yj j to give... 

Check to determine if the new sets of tear variables and are within some prescribed tolerance of sets and... 

Initial estimates provided for the tear variables were as follows compared with final converged values (after 23 iterations), where numbers in parentheses are consistent with specifications. 

By employing successive substitution of the tear variables and the criterion of Eq. (13-83), convergence was achieved slowly, but without oscillation, in 23 iterations. Computed products are. 

Compute a corresponding new set of Vj tear variables from the following total material balance, which is obtained by combining Eq. (13-74) with an overall material balance around the column ... 

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

Pe and f6 -> q56. Notice that in this procedure we have left out the model equation/56 = 0, which would be satisfied, if the assumed value of ql2 is correct. If it is not, then a process of iteration will be required. The variable q12 in the above example is the tear variable and the equation fS6 = 0 is sometimes referred to as the tear equation. 

The effect of tearing is to delete the tear variables and tear equations from the original set and to solve them iteratively external to the remaining set of equations and variables. In order for tearing to be a viable strategy, the number of tear variables required must be small and the tear equations must not be too difficult to solve. In this example, after tearing the iteration will involve only one equation, assuming the model equations are pressure explicit. 

If the objective in selecting streams to tear is to minimize the number of the tear variables (Pho and Lapidus, 1973) subject to the constraint that each loop be broken at least once, this problem is an integer programming problem known as the covering set problem. Refer to Biegler et al. (1997) and Section 8.4. 

One at a time, perturb the elements of the tear variable xTi. Calculate the dependent variables, and evaluate the tear equations. Calculate the gradients of/, g, h, and h with respect to each xT i by a forward difference equation in which the xD are the perturbed values and x7 are the unperturbed values. 

After values of the variables 1) and Vp called tear variables, are specified, Eqs. (182)ff become a linear set in the x9 variables. Initial estimates of the vapor flows are made by assuming constant molal overflow modified by taking account of external inputs and outputs, and those of the temperatures by assuming a linear gradient between estimated top and bottom temperatures. Initially, also, the Kji are taken as ideal values, independent of composition, and for later iterations the compositions derived from the preceding one may be used to evaluate corrected values of With appropriate substitutions,... 

The concept of "tearing11 has been developed in connection with the iterative methods. First an output set for the system of equations is chosen. Then one or more tearing variables are selected. These variables are the iterates that need to be chosen to obtain a solution of the system. The number of tearing variables is usually much smaller than the number of the equations. An accepted criteria for selecting tearing variables is the minimum number of such variables which will make it possible to solve the whole system. The ordered set of equations that results is then solved using an iterative method. 

The m variables of x- can be regarded as tearing variables and clearly the smaller m is, the more effective the algorithm. Since computer programs for selection of minimal sets are readily available (Ledet and Himnelblau (5)), this problem will not be discussed. 

In some cases, the rearrangement of matrix A for improving accuracy is undesirable or even impossible, since it increases the number of tearing variables. In such cases, an iterative method can be used to improve the accuracy of the solution. 

Many flow-sheeting programs perform the partitioning, solution ordering, and tearing functions discussed above and present the user with one or more choices of solution sequence and tear variables. FLOWTRAN, however, does not do this. The user must identify the recycle loops, the calculation sequence, and the tear streams. The preceding example illustrated their identification and selection. 

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