Problem Output Variations

All material balance problems are variations on a single theme given values of some input and output stream variables, derive and solve equations for others. Solving the equations is usually a matter of simple algebra, but deriving them from a description of a process and a collection of process data may present considerable difficulties. It may not be obvious from the problem statement just what is known and what is required, for example, and it is not uncommon to... 

The multivari chart is used to classify the family into which the red X or pink Xs fall. A parameter that is indicative of the problem, and can be measured, is chosen for study. Sets of samples are then taken and the variation noted. The categories used to distinguish the parameter output variation are (1) variation within sample sets (cyclical variation) is larger than variation within samples or variation over time, (2) time variation (temporal variation) between sample sets is larger than variation within sample sets or variation of the samples, and (3) variations within samples (positional variation) are larger than variation of sample sets over time or variation within the sample sets. These are demonstrated in Fig. 23.21. 

There are two general weaknesses associated with capacitance systems. First, because it is dependent on a process medium with a stable dielectric, variations in the dielectric can cause instabiUty in the system. Simple alarm appHcations can be caUbrated to negate this effect by cahbrating for the lowest possible dielectric. Multipoint and continuous output appHcations, however, can be drastically affected. This is particularly tme if the dielectric value is less than 10. Secondly, buildup of conductive media on the probe can cause the system to read a higher level than is present. Various circuits have been devised to minimize this problem, but the error cannot be totally eliminated. 

This problem can be cast in linear programming form in which the coefficients are functions of time. In fact, many linear programming problems occurring in applications may be cast in this parametric form. For example, in the petroleum industry it has been found useful to parameterize the outputs as functions of time. In Leontieff models, this dependence of the coefficients on time is an essential part of the problem. Of special interest is the general case where the inputs, the outputs, and the costs all vary with time. When the variation of the coefficients with time is known, it is then desirable to obtain the solution as a function of time, avoiding repetitions for specific values. Here, we give by means of an example, a method of evaluating the extreme value of the parameterized problem based on the simplex process. We show how to set up a correspondence between intervals of parameter values and solutions. In that case the solution, which is a function of time, would apply to the values of the parameter in an interval. For each value in an interval, the solution vector and the extreme value may be evaluated as functions of the parameter. 

Using the formula C, + Vt + S], values can be calculated for each department. In Department 1, for example, the total value of output is 120 in Department 2, the total value is 60. In our previous analysis of Marx s reproduction schema, based on the second volume of Capital, it was assumed that these values are also the total prices of each department. However, in the third volume Marx focuses on the organic composition of capital, which measures the ratio (Ct/Vt) between constant and variable capital.2 These ratios vary between 4 and 0.4 in this example. And it is this variation that leads Marx to argue that values cannot be sustained as indicators of price for each department of production. The problem is that the rate of profit (.SVC, + V) for each department is calculated as a ratio between total surplus value and total capital. Yet, for each department its own mass of surplus value is calculated from the variable capital employed. 

It seems inevitable in view of our discussion on variations of anatomy and of heart outputs that normal individuals should have circulatory peculiarities. An extreme case of what may be a circulatory peculiarity has been called to my attention. This individual continually has a problem of cold feet he uses a heating pad under his working desk, carries one around with him and on social occasions sits near an electric outlet, plugs it in, and attempts to be comfortable. It seems likely that this individual suffers because of unrecognized... 

The local production of natural gas in Georgia is limited, and the gas supplied from the transit systems has an almost constant output. This represents a major managerial problem as the supply does not correspond to the domestic seasonal demand (see Figure 11). The dependence on seasonal variations shows periods with significant imbalances between supply and demand (see Figure 12) [36]. That s why Georgia has to develop some sound strategy and associated infrastructure to utilize natural gas rationally. 

Sensitivity analysis can and should be used both iteratively and proactively during the course of developing an exposure model or a particular analysis. For example, sensitivity analysis can be used early in model development to determine which inputs contribute the most to variation in model outputs, to enable data collection to be prioritized to characterize such inputs. Furthermore, analysis of the model response to changes in inputs is a useful way to evaluate the appropriateness of the model formulation and to aid in diagnosing possible problems with a model. Thus, sensitivity analysis can be used to guide model development. 

Figure 2.13 illustrates the variation of the economic potential during flowsheet synthesis at different stages as a function of the dominant variable, reactor conversion. EPmin is necessary to ensure the economic viability of the process. At the input/output level EP2 sets the upper limit of the reactor conversion. On the other hand, the lower bound is set at the reactor/separation/recycle level by EP3, which accounts for the cost of reactor and recycles, and eventually of the separations. In this way, the range of optimal conversion can be determined. This problem may be handled conveniently by means of standard optimization capabilities of simulation packages, as demonstrated by the case study of a HDA plant [3]. 

Although there is a great deal of individual variation, on average the increase in maximum cardiac output appears to account for only about 50% of the rise in maximum 02 uptake rates occurring in response to training (see Brooks et al., 1996). The other 50% or so of the increase is accounted for by improved extraction of 02 by the working muscles, which is reflected in larger AV P02 differences and lower 02 tensions in venous blood. How improved 02 extraction is achieved, however, is a problem area in need of further research. 

Calculations of the relations between the input and output amounts and compositions and the number of extraction stages are based on material balances and equilibrium relations. Knowledge of efficiencies and capacities of the equipment then is applied to find its actual size and configuration. Since extraction processes usually are performed under adiabatic and isothermal conditions, in this respect the design problem is simpler than for thermal separations where enthalpy balances also are involved. On the other hand, the design is complicated by the fact that extraction is feasible only of nonideal liquid mixtures. Consequently, the activity coefficient behaviors of two liquid phases must be taken into account or direct equilibrium data must be available. In countercurrent extraction, critical physical properties such as interfacial tension and viscosities can change dramatically through the extraction system. The variation in physical properties must be evaluated carefully. 

Although collection of data at the unit process level solves much of the input allocation problem there is still room for variation. Most input inventories are allocated on a mass basis because whatever the input units, the inputs can be accurately converted to mass units, and mass balances between mill inputs and outputs can be easily checked for accuracy. The mass balance between input and output can be used as a sound vaUdity check on the data a difference of a few percent is generally considered acceptable. 

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