Inequality

We list here some of the important inequalities followed by an example showing the use of one of the inequalities in the context of a chemical reaction. Let and [bi ti be sequences of real numbers andp,q .  

Greater than , which is represented in symbols by Thus we might have a statement snch as X 3, which means that x can take any valne as long as it is greater than 3. Notice that x can t be 3 exactly, bnt it conld be very close to 3, for example 3.000 0001. 

Greater than or eqnal to , which is represented in symbols by . This time the statement x 3 does allow x to be exactly 3 or any value above. 

To solve an ineqnality snch as x + 2 9, we manipnlate the expression in exactly the same way as an eqnation, retaining the ineqnality sign and performing the same operations on either side of the ineqnality. In this case, snbtracting 2 from either side gives 

In a polyhedron, we may denote the number of faces surrounded by three edges, as /3, the number of faces surrounded by four edges, say /4, etc. The total number of faces is 

Now every edge shares two faces, therefore the number of edges is 

We repeat the same considerations with the vertices. A vertex connect with three edges is p3, a vertex connected with four edges is p, etc. So the total number of 

---

The CP inequality for individual matches. Figure 16.2a shows the temperature profile for an individual exchanger at the pinch, above the pinch.Moving away from the pinch, temperature differences must increase. Figure 16.2a shows a match between a hot stream and a cold stream which has a CP smaller than the hot stream. At the pinch, the match starts with a temperature difference equal to The relative slopes of the temperature-enthalpy... 

Note that the CP inequalities given by Eqs. (16.1) and (16.2) apply only at the pinch when both ends of the match are at pinch conditions. 

It is not only the stream number that creates the need to split streams at the pinch. Sometimes the CP inequality criteria [Eqs. (16.1) and (16.2)] CEmnot be met at the pinch without a stream split. Consider the above-pinch part of a problem in Fig. 16.13a. The number of hot streams is less than the number of cold, and hence Eq. (16.3) is satisfied. However, the CP inequality also must be satisfied, i.e., Eq. (16.1). Neither of the two cold streams has a large enough CP. The hot stream can be made smaller by splitting it into two parallel branches (Fig. 16.136). 

Clearly, in designs different from those in Figs. 16.13 and 16.14 when streams are split to satisfy the CP inequality, this might create a problem with the number of streams at the pinch such that Eqs. (16.3) and (16.4) are no longer satisfied. This would then require further stream splits to satisfy the stream number criterion. Figure 16.15 presents algorithms for the overall approach. ... 

Figure 16.13 The CP inequality rules can necessitate stream splitting above the pinch.

The network can now be designed using the pinch design method.The philosophy of the pinch design method is to start at the pinch and move away. At the pinch, the rules for the CP inequality and the number of streams must be obeyed. Above the utility pinch and below the process pinch in Fig. 16.17, there is no problem in applying this philosophy. However, between the two pinches, there is a problem, since designing away from both pinches could lead to a clash where both meet. 

Figure 16.215 shows an alternative match for stream 1 which also obeys the CP inequality. The tick-off" heuristic also fixes its duty to be 12 MW. The area for this match is 5087 m , and the target for the remaining problem above the pinch is 3788 m . Tlius the match in Fig. 16.216 causes the overall target to be exceeded by 16 m (0.2 percent). This seems to be a better match and therefore is accepted. 

The cold-utility target for the problem shown in Fig. 16.22 is 4 MW. If the design is started at the pinch with stream 3, then stream 3 must be split to satisfy the CP inequality (Fig. 16.22a). Matching one of the branches against stream 1 and ticking off stream 1 results in a duty of 8 MW. 

Of course, it is well known that because of the duality of the time and frequency parameters, there is no universal time-frequency technique. Indeed the two parameters cannot be simultaneously known with arbitrarily high resolution. This is expressed by the Heisenberg - Gabor inequality ... 

Inequality Re > H corresponds to the other case, when only a part of a penetrant is extracted by a developer and can form crack s indication. Such a situation can take place when one use kaolin powder as the developer. We measured experimentally the values Rj for some kaolin powders. For the developer s layer of kaolin powder, applied on tested surface. Re = 8 - 20 pm depending on powder s quality. 

Defect s indication of linear size W (and larger) are forming above such the cracks, the width of which satisfies the inequality... 

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

There is no reason to doubt that the inequalities of section A2.5.4.5(e) are other than equalities. The equalities are assumed in most of the theoretical calculations of exponents, but they are confmned (within experimental error) by the experiments. 

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

This inequality indicates the amphiphile adopts a shape essentially equivalent to that of a cone with basal area <3. Such cones self-assemble to fonn spheroidal micelles in solution or spheroidal hemimicelles on surfaces (see section C2.3.15). Single-chain surfactants with bulky headgroups, such as SDS, typify surfactants in this category. 

---