Cyclic rule

MATHEMATICAL INTERLUDE. MORE PROPERTIES OF EXACT DIFFERENTIALS. THE CYCLIC RULE... 

Another useful relation between partial derivatives is the cyclic rule. The total differential of a function z(x, y) is written... 

Suppose that the three variables are pressure, temperature, and volume. We write the cyclic rule using the variables p,T,V ... 

From Eq. (9.10) and the cyclic rule for partial differentials it follows that dqi/dCj)j = (dq /dCj), and this implies that all concentrations can be expressed as functions of z / / rather than as functions of z and t separately. 

Physical chemistry tutorials reinforce conceptual understanding. Over 460 tutorials are available in MasteringChemistry for Physical Chemistry, including new ones on The Cyclic Rule and Thermodynamic Relation of Proofs. 

Various rules about partial derivatives are expressed using the general variables A, B, C, D,. . . instead of variables we know. It will be our job to apply these expressions to the state variables of interest. The two rules of particular interest are the chain rule and the cyclic rule for partial derivatives. 

In the cases ofp, V, and T, we can use equation 1.24 to develop the cyclic rule. For a given amount of gas, pressure depends on V and T, volume depends on p and T, and temperature depends on p and Y. For any general state variable of a gas F, its total derivative (which is ultimately based on equation 1.12) with respect to temperature at constant p would be... 

This is the cyclic rule for partial derivatives. Notice that each term involves p, V, and T. This expression is independent of the equation of state. Knowing any two derivatives, one can use equation 1.25 to determine the third, no matter what the equation of state of the gaseous system is. 

The cyclic rule is sometimes rewritten in a different form that may be easier to remember, by bringing two of the three terms to one side of the equation and expressing the equality in fractional form by taking the reciprocal of one partial derivative. One way to write it would be... 

This might look more complicated, but consider the mnemonic in Figure 1.11. There is a systematic way of constructing the fractional form of the cyclic rule that might be useful. The mnemonic in Figure 1.11 works for any partial derivative in terms of p, V, and T. 

FIGURE 1.11 A mnemonic for remembering the fraction form of the cyclic rule. The arrows show the ordering of the variables in each partial derivative in the numerator and denominator. The only other thing to remember to include in the expression is the negative sign. 

Because both of these definitions use p, V, and T, we can use the cyclic rule to show that, for example. 

Write two other forms of the cyclic rule in equation 1.26, using the mnemonic in Figure 1,11. 

Use Figure 1.11 to construct the cyclic rule equivalent of (8p/8p)j. Does the answer make sense in light of the original partial derivative ... 

Write the fraction a/K in a different form using the cyclic rule of partial derivatives. 

For an ideal gas, /jlj-j- is exactly zero, because enthalpy depends only on temperature (that is, at constant enthalpy, temperature is also constant). However, for real gases, the Joule-Thomson coefficient is not zero, and the gas will change temperature for the isenthalpic process. Remembering from the cyclic rule of partial derivatives that... 

Starting from the cyclic rule involving the Joule-Thomson coefficient, derive equation 2.35. 

Starting with the natural variable equation for dU, derive an expression for the isothermal volume dependence of the internal energy, (dUldV)j, in terms of measurable properties (T, V, or p) and a and/or k. Hint You will have to invoke the cyclic rule of partial derivatives (see Chapter 1). 

Now we invoke the hint. The definitions for a, k, and the partial derivative (dpldT)y all use p, T, and V. The cyclic rule for partial derivatives relates the three possible independent partial derivatives of any three variables A, B, C ... 

Recall that in nature (P, V, T) variables are (seemingly) independent of each other for a given quantity of gas n. Thus we could plot the three variables on an (x, y, z) grid. A useful cyclic rule can be derived with some thought and a trick using the differential quantities. Although we can plot independent variables on a grid, there may actually be some state function that relates... 

Of course there may be situations which are more complicated than this example, but as long as it is possible to produce all the demand that is present at the beginning of the production interval, if necessary by doing extra work, the analysis will only be as difficult as in this example. The situations with the fixed production cycle we have considered in this chapter are on the one hand illustrations of cyclic production rules, which seem to us both useful and interesting to analyse, because cyclic rules are often used in practical situations and on the other hand they can serve very well as some kind of measure for the comparison of non-cyclic production rules in the following chapters. 

First, we shall consider a cyclic production rule in combination with a cyclic rule for the lead times and determine the average profit of this approach. Then we will give a description of a dynamic programming rule (DPR) for the lead times. This DPR is based te reduction and the decision rule itself is determined by means of dynamic programming. We will conclude this section with the description of a simplified version of the DPR, which will be denoted as the SDPR. Then some examples will be given to compare the performance of the different decision rules. 

By using these two properties we can avoid much of the work for determining the values of Ch(y,t,j). In the next subsection we will compare this rule with the DPR and with the cyclic rule n. 

The fixed cycle production rule can be used directly in the multi-type capacitated situation. However, in this rule we assume that we have a production interval of a fixed length for every type of product. Since the length of a period is also fixed, this may lead to problems. Usually it makes sense to produce more than one type of product in one period. This cyclic rule in which we can produce different types of products during one period with a given length for the period will be called a semifixed cycle rule. The performance of this rule, the extended (x,7)-mle and the two-step approach will be compared in a few examples at the end of this subsection. 

For the cyclic rules Hi, i=l,2,3, and the corresponding time between set-ups 7,- and average costs g, which we use in the salvage function L,(a,) in the first step of the two-step rule, we find the following values ... 

In the Examples 5.1 and 5.2 the capacity restrictions were very tight. In order to see whether the differences in average costs also appear in the situation in which the capacity is rather loose, we will consider a third example. Therefore we take again Example 5.1, but now with an available capacity of 15 instead of 12. With one set-up every period, about 72 percent (10.75 out of 15) of the available capacity will be used. The cyclic rules w,- are not affected by the change in capacity, therefore (5.2.18) still holds and we can also use the same semi-fixed cycle. For the extended (x,7>-mle we used the same set of values for the pairs iXi,Ti) as in (5.2.20). Because it will not be very profitable to produce orders with a longer residual lead time, the penalty points for these orders have been decreased and after some trials we have found the following set of values for the penalty points ... 

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