Pinch equations

A further insight into the pinch equations is afforded by the following example. 

If there is heat flow across the Pinch, then the energy consumption is higher than minimum necessary. Both hot and cold utility consumption will increase with the same amount XP above the minimum targets. The Pinch equation is ... 

Limiting Equations If the height of a foam-fraciionation column is increased sufficiently, a concentration pinch will develop between the counterflowing interstitial streams (Brunner and Lem-lich, loc. cit.). For an enricher, the separation attained will then approach the predictions of Eq. (22-51) and, interestingly enough, Eq. (22-46). 

The data for S3 are given in Table 3.3. The absorber sizing equations and fixed cost were given in Example 2.2. Using the graphical pinch approach, synthesize a cost-effective MEN that can be used to remove benzene from the gaseous waste (Fig. 3.9). 

From Equation 8-131 expressing 9 and q evaluate 9 by trial and error, noting that 9 will have a value between the a of the heavy key and the a of the light key evaluated at or near pinch temperatures, or at a avg. Suggested tabulation, starting with an assumed 9 t alue, 9a ... 

The Underwood Equation is based on the assumption that the relative volatilities and molar overflow are constant between the pinches. Given that the relative volatilities change throughout the column, which are the most appropriate values to use in the Underwood Equations The relative volatilities could be averaged according to Equations 9.47 or 9.49. However, it is generally better to use the ones based on the feed conditions rather than the average values based on the distillate and bottoms compositions. This is because the location of the pinches is often close to the feed. 

The Underwood Equations tend to underestimate the true value of the minimum reflux ratio. The most important reason for this is the assumption of constant molar overflow. As mentioned previously, the Underwood Equations assumed constant molar overflow between the pinches. So far, in order to determine the reflux ratio of the column, this assumption has been extended to the whole column. However, some compensation can be made for the variation in molar overflow by carrying out an energy balance around the top pinch for the column, as shown in Figure 9.16. Thus... 

Solution Calculate the liquid composition at the rectifying pinch using Equation 9.57. But first, the root of the second Underwood Equation associated with the heavy key component must be calculated. From Table 9.1, this lies in the range ... 

This is a useful result since, if the network is assumed to be loop-free and has a single component, the minimum number of units can be predicted simply by knowing the number of streams. If the problem does not have a pinch, then Equation 17.2 predicts the minimum number of units. If the problem has a pinch, then Equation 17.2 is applied on each side of the pinch separately2 ... 

Solution Figure 17.2 shows the stream grid with the pinch in place dividing the process into two parts. Above the pinch, there are five streams, including the steam. Below the pinch, there are four streams, including the cooling water. Applying Equation 17.3 ... 

In practice, the integer number of shells is evaluated from Equation 17.8 for each side of the pinch. This maintains consistency between achieving maximum energy recovery and the corresponding minimum number of units target Nunits In summary, the number-of-shells target can be calculated from the basic stream data and an assumed value of XP (or equivalently, FTmin). 

Note that the CP inequalities given by Equations 18.1 and 18.2 only apply at the pinch and when both ends of the match are at pinch conditions. 

It is not only the number of streams that creates the need to split streams at the pinch. Sometimes the CP inequality criteria, Equations 18.1 and 18.2, cannot be met at the pinch without a stream split. Consider the above-pinch part of a problem in Figure 18.14a. The number of hot streams is less than the number of cold streams, and hence Equation 18.3 is satisfied. However, the CP inequality, Equation 18.1, must be satisfied. 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 (Figure 18.14b). 

Clearly, in designs different from those in Figures 18.14 and 18.15, when streams are split to satisfy the CP inequality, this might create a problem with the number of streams at the pinch such that Equations 18.3 and 18.4 are no longer satisfied. This would then require further stream splits to satisfy the stream number criterion. Figures 18.16a and 18.16b present algorithms for the overall approach1,2. 

The difficulty in using this equation is that the values of xnA and xnB are known only for special cases where the pinch coincides with the feed composition. Colburn has suggested that an approximate value for xnA is given by ... 

In the saturation region, at and above the pinch-off point, neglecting the effective channel length change due to the Vds value, due to the condition OIds/OVds = 0(thusVDs = Vgs—Vt), this equation becomes ... 

Using this approximate value for Rm, equation 11.109 may be rearranged to give the concentrations of all the light components in the upper pinch as ... 

Since there is no component lighter than light key Ebnxn = 0. In the lower pinch, (from equation 11.112) ... 

The governing equations for the heat available in the gas down to the pinch point (Tg o to Tg,2), and the corresponding heat absorbed by the superheated and saturated steam are presented below. 

Obviously, in order for the pinching points to dominate the dissociation of top terrace, they must, reach sufficient density. Equation (14) suggests that the critical density pc = 1/d is given by the condition... 

Equation (16) stops to be valid when x reaches zero, i.e., the two lines collide. Beyond this point, the line breaks up into a number of segments, each segment ending at a pair of pinching points. The velocity of the pinching points should be of the order, of... 

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