Diabatic column

Thermodynamic cost analysis relates the thermodynamic limits of separation systems to finite rate processes, and considers the environmental impact through the depletion of natural resources within the exergy loss concept. Still, economic analysis and thermodynamic analysis approaches may not be parallel. For example, it is estimated that a diabatic column has a lower exergy loss (39%) than that of adiabatic distillation however, this may not lead to a gain in the economic sense, yet it is certainly a gain in the thermodynamic sense. The minimization of entropy production is not always an economic criterion sometimes, existing separation equipment may be modified for an even distribution of forces or an even distribution of entropy production. Thermodynamic analysis requires careful interpretation and application. 

The equipartition principle is mainly used to investigate binary distillation columns, and should be extended to multicomponent and nonideal mixtures. One should also account for the coupling between driving forces since heat and mass transfer coupling may be considerable and should not be neglected especially in diabatic columns. 

A reduction of 38% in the entropy production rate was obtained for a binary separation of benzene and toluene in an equimolar mixture. The heat added on each tray in the optimal diabatic column and the corresponding vapour flows are shown in Figure 7. 

Figure 6 Examples of operating lines of an adiabatic and a diabatic column...

Figure 7 The vapour flow (mol s) and heat exchanger load (kW) in an optimised diabatic column and an adiabatic column. Both columns separate an equirrwlar mixture of toluene and benzene...

Figure 7 shows the temperature profiles for the adiabatic, diabatic thermally optimal, and diabatic one side-reboiler VCM column design. For the diabatic thermally optimal column design, the temperature or concentration (not shown) profiles are almost linear between the pinch points at both ends of the column. For the diabatic column with one side-reboiler the profiles are also closer to being linear than for the adiabatic design. The critical factors for obtaining such "near linear" profiles are the location and size of the side-reboiler as well as... 

The net saving in entropy is most apparent in a graphic comparison of the entropy change produced in a traditional Adiabatic column and an optimized Diabatic column, that is, one with heat exchangers along the column. The reversible limit is still clearly lower than the finite-time system, but the separation part of that entropy is very similar for the optimized realistic and reversible columns the difference is almost entirely in the heat exchange. This is shown in Fig. 14.5. 

The aim in performing what is termed the adiabatic-to-diabatic transfonnation is to eliminate from Eq. (16) the eventually problematic matiix, Xm, which is done by replacing the column matrix in Eq. (16) by another column matrix where the two are related as follows ... 

The C matrix, the columns ofwhich, Cj(, are the eigenvectors of H, is normally not too different from the matrix defined above. However, the QDPT treatment, applied either to an adiabatic or to a diabatic zeroth-order basis, is necessary in order to prevent serious artefacts, especially in the case of avoided crossings [27]. The preliminary diabatisation makes it easier to interpolate the matrix elements of the hamiltonian and of other operators as functions of the nuclear coordinates and to calculate the nonadiabatic coupling matrix elements ... 

The transformation T we adopt is induced by the wave function normalization condition which, in terms of the weights, reads w + W3 = 1. From (3.5), it is apparent that if T sends the vvm set into a new set wm with ivi = vvi + iv3 = 1 as one of its elements, then both the first row and the first column of the transformed polarization component of the solvent force constant matrix K, "/ = T. Kp°r. T (T = T) are zero, since the derivatives of wi are zero. Given the normalization condition and the orthogonality requirement — with the latter conserving the original gauge of the solvent coordinates framework — one can calculate T for any number of diabatic states [42], The transformation for the two state case is... 

Example 5.13 Optimal distillation column Diabatic configuration Consider a distillation column made of N distinct elements. The heat and mass transfer area can be defined separately. Defining an investment cost Q for element i as a linear function of the size. 1,. we have... 

In the diabatic distillation column, each element is small enough that equipartition of entropy production may be approximately achieved by adjusting the heat flows and thus the liquid and vapor flow rates. We assume that each element performs a specified duty of Ji0. The total cost function Ct for all N elements is... 

In this expression, W 1)ad(R) (k = i, j) is the X X N matrix whose row n and column n element is the k element of the W(n1, 1ad(R) vector, i.e., [W ad(R)], and the brackets in its right-hand side denote the commutator of the two matrices within. When n and n are allowed to span the complete infinite set of adiabatic electronic quantum numbers, condition (102) is satisfied [24,26], (99) has a solution, and the resulting A(q) leads to the q-independent diabatic electronic basis set mentioned in connection with (83). For the small values of X case being considered here, (102) is in general not satisfied and (98) does not have a solution. On the other hand, the equation obtained by replacing in (99) W(1)ad(R) by its longitudinal part VRd><1)ad(q) [see remark after (98)], namely... 

Example 5.8 Optimal distillation column—Diabatic configuration... 

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