Subscript notation

Subscript notations are identified on the sketch. The n are lbmols/hr. Find the flow rates that will result in the minimum operating cost. 

Extension of the model to take into account the bond bending effect and the interactions within the product cluster leads to the following modification of equations (6.209) and (6.210), in which the diabatic character of the surfaces is emphasized in the subscript notations ... 

In the case of a binary alloy containing only A and B atoms there is no need to use a subscript notation Xi becomes the number of A atoms and rji = = t - Then for... 

A system for distribution of fluids such as cooling water in a process plant consists of many interconnecting pipes in series, parallel, or branches. For purposes of analysis, a point at which several lines meet is called a node and each is assigned a number as on the figure of Example 6.6. A flow rate from node i to node / is designated as Qij-, the same subscript notation is used for other characteristics of the line such as /, L, D, and NRc. 

Note Beginning with Problem 7.3 and continuing throughout this chapter, we will be using subscript notation rather than notation in parentheses after a symbol. As an example C(total) becomes Ctotai-... 

Using this expansion for the K(x, yj, Zj) values in Eqn. (1) and the subscript notation where... 

Sillen and Martell 1964), where in this case, m and n are the number of metal ions and protons involved in the reaction, respectively. For proton-cation exchange, where m = 1, the value of m is omitted from the subscript notation for /3 and /3, which is then of the form given above for mononuclear complexation reactions. 

A subscript notation for differentiation is more compact thus we write (19) as... 

It is easy to see by interchanging subscript notations that the first and third terms on the right of (10.146) cancel each other, as do the second and fourth terms. Other consistency issues are discussed in Section 10.4.10. 

Introducing a double-subscript notation, with the second subscript identifying the stage, the inlet velocity to the first stage of a turbine will be essentially zero, i.e. 

The letters in parentheses refers to the subscript notation used for the model formulation. 

Some of the molecular frame R-axis quantum numbers (Sr,Ir,sr and Jr) correspond to quantum numbers C = Ir and S = Sr that have been in widespread use in the literature. There is, however, an ambiguity concerning onto which molecule frame rotation axis (J+,N+,or R), the projection quantum number is defined. The subscript- notation eliminates this ambiguity and we therefore recommend its use. Also, the lower-case forms of Ir and sr conform to the convention that single-electron quantum numbers appear as lower-case letters. 

The differences in notation between Sections II and III (and the primary references on which they are based) are not great, and for the most part they are not gratuitous, but follow from differences in substance and purpose. For example, all the expressions giving e in terms of two-point correlations discussed in Section II can be expressed in terms of the A and D components of correlation introduced there. To quantitatively evaluate these expressions, however, many other components of correlation must be taken into account (in terms of their effect on the A and D components), and more complete rotational-invariant subscript notation is introduced in Section III to accommodate such evaluation. 

As implied by the subscript notation in Eqs. (327)-(328), the translational and coupling diffusivity dyadics vary with choice of origin. This dependence can be quantitatively established. By invoking appropriate kinematic argu-... 

For brevity we have used the subscript notation (y) in Eq. (1.48) to indicate the partial differentials. The last equality in Eq. (1.48) emerges from the extremum conditions = 0. The second derivative we are searching for turns out to be... 

It should be emphasized that the indices in Eq. 7.84 refer to the reps, and to their frequency of occurrence among the entire set of molecular orbitals. The simplifications reduce the determinant Eq. 7.90 into block-diagonal form. Since the determinant will only contain terms with I = k the double subscript notation can be discarded and written for Hi, For example, HJ3 = JC03 ) where and 03 are the first and... 

The definitions of stress and strain developed earlier were for uniform stress states but, often, one has to deal with situations in which the stress and strain are non-uniform and vary from point to point in a body. In these cases, one must consider stresses and strains in a more general way. When a body is loaded in a complex way, different particles of the body will be displaced relative to one another. It is important, therefore, to define both the coordinate of a point and its displacement. The position of a particle P can be defined by its coordinates x, X2, Xj in a set of cartesian axes V, X, Xy which are fixed and independent of the body. As a shorthand, the coordinates can be written as x., where /= 1,2 and 3. Suppose, as shown in Fig. 2.13, that the deformation and movement of the body displaces the particle at P to P, such that the new coordinates are X +M, X2+U2 and X3+M3. The vector u. (same subscript notation) is then the displacement of P. There must, however, be a relationship between the vectors u. and x. because if u.v/as a constant for all the particles in the body, this would only represent a rigid translation of the body and not a deformation. It is the relationship between the two vectors that leads to the concept and precise definition of strain. An essential part of this definition is that u. varies from one particle to another in a body, that is u.=f x.). 

Often, we will use the subscript notation for partial derivatives. For example. 

In the following chapters we use a very familiar form of representation of stresses by a double-subscript notation in which the first subscript represents the direction of the normal of the area across which the stress is acting and the second is the direction of action of the stress. In the Cartesian rectangular system of coordinates, the subscripts i and j stand for x, y, z or xi,X2,X. In the cylindrical, polar coordinate system, the subscripts stand for r, 6, z. 

This equation should be compared with Eqn. 2 in Contact angles and interfacial tension, where the subscript notation is explained. 

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