The Model Again

Actually, we will note in the next chapter that the internal energy C/ is in fact the ena-gy we need to predict which way reactions will go under certain unusual conditions, but it is rarely used in this sense. 

The most general term for a change in H is simply A//. This refers to any change in the enthalpy of any system between two equilibrium states (stable or metastable), not necessarily associated with a chemical reaction. A special case is the A// between the products and reactants of a chemical reaction, called A,.H, so this represents a subset of the more general term LH. A special kind of chemical reaction involves only pure compounds, whose thermodynamic parameters can be found in tables, and so a subset of all A,.// values can be called A,.//°, to indicate that all products and reactants are in their pure reference states. A special case of is the reaction in which a compound is formed from its elements, all in their pure reference states, and this is called fH°. Finally, we found that there are two conventions for defining the enthalpy of formation from the elements, one being the traditional or common sense method, where the compound and all its elements are at 

Later on (Chapter 8) we will find that strictly speaking superscript ° refers to a more general standard state, and that pure reference states are just one kind of standard state. 

When we consider by what means the energy content of systems can change, we find that there are only two - we can heat/cool the system, or we can do work on the system/have the system do work. There are several ways of doing work on systems, depending on the forces we choose to consider (magnetic, electrostatic, surface tension, etc.), and so we start out by choosing the most common, pressure-volume work. The others are all handled in the same way and can be brought in when tlie situation calls for them. 

Finally, we went into some detail on one special kind of energy transfer -the heat absorbed or released during a chemical reaction, where the initial and final states have the same pressure, generally referred to as a constant pressure reaction. This quantity of heat is the enthalpy, and it is one of the fundamental building blocks of our model. The fact that there are no absolute values for // is a decided nuisance, but a very simple way around this is by using the formation from the elements convention. This means that for every compound, we measure LH for the reaction in which a compound is formed from its elements, each in its most stable form, and these quantities are given the symbol where the subscript / stands for formation from the 

As we said in Chapter 3, reversible processes are represented by mathematical functions that have physical parameters as variables. The fact that they are continuous functions means that the processes they represent cannot be carried out in the real world. 

Calculate the value of R, the molar gas constant, from the ideal gas equation PV = nRT). See Appendix A for the answer. 

Show that Jbar is a volume term, and calculate the conversion factor from Jbar to cm. See Appendix A. 

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We now discuss the theory and some of the numerical aspects of the model. Again, this model is based upon the theoretical formulation presented in (12, 13, 14). The theoretical... 

Because Fc = 87.4639 >Ft = 4.21, we can conclude that the X2 predictor variable is significant and should be retained in the model. Again, measures the contribution of X2 to the sum of squares regression, given that Xj is held constant. 

Figures 7 and 8 show the operational interdependence amongst the systems identified at recursion 1 (see Figure 2). In particular. Figure 8 and Table 4 describe the characteristics ofthese interdependencies. On the other hand. Figure 9 shows these systems in the format of the structural organization of the model (again, the managerial interdependence has not been discussed here).

A modified version of the DEM was developed for modeling deteriorating systems (Cifuentes Iwan 1989). The model again consists of a collection of linear springs and slip elements, however, in this case an element is allowed to break if a certain maximum displacement is exceeded, defined as fiXy i, where Xy i is the yield displacement of the fth element and /t is the breaking ductility ratio, which for simplicity is assumed to be the same for all elements. 

Unfortunately, there is some debate over the actual number of degrees of freedom to use for spectral residuals. Some values have been suggested that seem to work well in practice 1 for the numerator (vj) and —/— 1 for the denominator (vj), where / is the number of factors in the model. Again, samples that exhibit probabilities of 0.99 (a = 0.01) are considered outliers and should be removed from the training set before calculating the final calibration model. 

Let us now turn to the time-dependent properties of the model. Again we restrict ourselves to differential fluctuations 6S. = around the steady state. From... 

The predicted phase equilibrium is a strong function of the binary interaction parameters (BIPs). Process simulators have regression options to determine these parameters from experimental phase-equilibrium data. The fit gives a first-order approximation for the accuracy of the equation of state. This information should always be considered in estimating the accuracy of the simulatioa Additional simulations should be run with perturbed model parameters to get a feel for the uncertainty, and the user should realize that even this approach gives an optimistic approximation of the error introduced by the model. If BIPs are provided in the simulator and the user has no evidence that one equation of state is better than another, then a separate, conplete simulation should be performed for each of these equations of state. The difference between the simulations is a crude measure of the uncertainty introduced into the simulation by the uncertainty in the models. Again, the inferred uncertainty will be on the low side. 

Step 14 Check the results in Analysis sheet. Repeat step 14 to assign new lower and upper bounds to calibrate the model again if the model results are not good enough. 

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