Dummy index

Subscript i identifies species, and J is a dummy index all summations are over all species. Note that Xp however, when i = J, then Xu = = 1. In these equations / (a relative molecular volume) and (a relative molecular surface area) are pure-species parameters. The influence of temperature on g enters through the interaction parameters Xp of Eq. (4-261), which are temperature dependent ... 

The range convention and the summation convention will be used. The range convention is any subscript that appears only once on one side of an expression takes on values 1, 2, and 3. The summation convention is any subscript that appears twice on one side of an expression is summed from 1 to 3. The repeated index is called the dummy index. 

If we replace the dummy index i by a, where a = i + j, then this expression becomes... 

If we omit the first two terms (they vanish according to equations (G.12) and (G.13)) in the first summation on the left-hand side of (G.l 1) and replace the dummy index by + 2 in that summation, we obtain... 

The repeated appearance of an index (j say) indicates that the lhs in an expression like x[ = aijXj (i = 1,2,3) is a sum over the dummy index j for all possible values (here j = 1,2,3). Whenever an index occurs two or more times in a term, it is implied, without any further symbols, that the terms are to be summed over all possible values of the index. 

Thus in Eq. (98), to consider the modes mentioned above, p = X — 0 and v, p = 0,1,2,3 need to considered, where 0 represents the density, 1 the longitudinal current, 2 the transverse current, and 3 the temperature. Please note that the subscript p is a dummy index and is not to be confused with the density. C 1 for different values of v are... 

Subscript i identifies species, and j is a dummy index running over all species. Subscript k identifies subgroups, and m is a dummy index running over all subgroups. The quantity v(k) is the number of subgroups of type k in a molecule of species i. Values of the subgroup parameters Rk and Qk and of the group... 

The attribute DUMMY INDEX (D I) defines a scalar for use as an index. D I s can never be assigned values and can only appear as array indices their sole function is to imply loops. The programmer is free to distinguish D I s typographically by means of the DEFAULT statement. 

The Universal Quasi-chemical Theory or UNIQUAC method of Abrams and Prausnitz divides the excess Gibbs free energy into two parts. The dominant entropic contribution is described by a combinatorial part ( ). Intermolecular forces responsible for the enthalpy of mixing are described by a residual part ( ). The sizes and shapes of the molecule determine the combinatorial part, which is thus dependent on the compositions and requires only pure component data. Since the residual part depends on the intermolecular forces, two adjustable binary parameters are used to better describe the intermolecular forces. As the UNIQUAC equations are about as simple for multi-component solutions as for binary solutions, the UNIQUAC equations for multicomponent solutions are given below. Species are identified by subscript i, subscript j is a dummy index. Here, is a relative molecular surface area and r, is a relative molecular volume. Both of these quantities are pure-species parameters. 

Here, the formula is just 2 raised to a power, the value of which is defined by each element of the domain. Notice that the use of r as a counting index is arbitrary any other appropriate letter (with the exception of u which we have used already) would do. A counting index such as r is often termed a dummy index.. An alternative way of generating this sequence is accomplished using a recurrence relation as the prescription, where each successive term is obtained from the previous term. For example, the sequence given in equation (1.8) can alternatively be expressed as ... 

We can also form an infinite series from a sequence by extending the range of the dummy index to an infinite number of terms ... 

To arrive at a size-extensive expression for W, and at the same time to ensure that our SS-MRCC theory using an IMS is intruder-free, it is necessary to write explicitly the projector P, and change the label of the dummy index /r on the right side of Eq. (61) appropriately. We thus have... 

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