Graphical presentation

Our model consists of the four ordinary differential equations (7.189) to (7.192) in the dynamics case and of the corresponding set of coupled rational equations in the static case. These two sets of equations can be solved and studied via MATLAB in order to find the system s steady states, the fermentor s dynamic behavior and to control it. 

In the design problem, the dilution rate D = q/V is generally unknown and all other input and output variables are known. In simulation, usually D is known and we want to find the output numerically from the steady-state equations. For this we can use the dynamic model to simulate the dynamic behavior of the system output. Specifically, in this section we use the model for simulation purposes to find the static and dynamic output characteristics, i.e., static and dynamic bifurcation diagrams, as well as dynamic time traces. 

As usual in this industrial problems chapter, we do not include actual MATLAB programs for this model. The readers should, however, be well prepared to create such programs and try to verify our included graphical results. 

The bifurcation analysis, i.e., the analysis of the steady states and the dynamic solutions is carried out for the dilution rate D as the bifurcation parameter. We have chosen D as the bifurcation parameter since the flow rate q V/D is directly related to D and q is most easily manipulated during the operation of a fermentor. 

In order to evaluate the performance of the fermentor as an ethanol producer, we calculate the conversion of substrate, the product yield of ethanol and its productivity according to the following simple relations  

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These examples are sufficient to illustrate one of the principal factors which should influence the choice of dimensionless parameters. Neglect of these considerations may lead, and indeed sometimes has led, to graphical presentations which are at best clumsy and at worst valueless. 

K, have been tabulated (2). Also given are data for superheated carbon dioxide vapor from 228 to 923 K at pressures from 7 to 7,000 kPa (1—1,000 psi). A graphical presentation of heat of formation, free energy of formation, heat of vaporization, surface tension, vapor pressure, Hquid and vapor heat capacities, densities, viscosities, and thermal conductivities has been provided (3). CompressibiHty factors of carbon dioxide from 268 to 473 K and 1,400—69,000 kPa (203—10,000 psi) are available (4). 

Experimental evidence presented in a classic paper bv Coghill and De Vaney ("Ball Mill Grinding, U.S. Bur. Mines Tecti. Publ. 581, 1937) indicated that the efficiency of battered reject balls was about 11 percent less than that of new spherical balls. These and other results have been graphically presented (Rose and Sullivan, Ball Rod andTube Mills, Chemical Pubnshing Co., NY, 1958). 

The F-N curve, the risk profile, and the risk contour are the three most commonly used methods of graphically presenting risk results. Normally, you will elect to use more than one of these methods when evaluating risk estimates for decision making. 

Five percent random error was added to the error-free dataset to make the simulation more realistic. Data for kinetic analysis are presented in Table 6.4.3 (Berty 1989), and were given to the participants to develop a kinetic model for design purposes. For a more practical comparison, participants were asked to simulate the performance of a well defined shell and tube reactor of industrial size at well defined process conditions. Participants came from 8 countries and a total of 19 working groups. Some submitted more than one model. The explicit models are listed in loc.cit. and here only those results that can be graphically presented are given. 

To produce this kind of grid map by quickly measuring the concentrations at some points for immediate processing and graphical presentation is a simple and often effective way to communicate the results to persons w ho are not trained to analyze primary results of measurements. The method must, however, be used with care since there is a risk that the sampling of data itself may affect the airflow in the studied area. I he equipment needed is relatively expensive, and the method is therefore of interest when the prerequisites are already available for other reasons. 

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

In practical open circuit gas turbine plants with combustion, real gas effects are present (in particular the changes in specific heats, and their ratio, with temperature), together with combustion and duct pressure losses. We now develop some modifications of the a/s analyses and their graphical presentations for such open gas turbine plants, with and without heat exchangers, as an introduction to more complex computational approaches. 

Figure 8.3. Graphical presentation for sample problem of the radiation heat flux as a function of time.

The most common manifestation of a structure-reactivity correlation in a reaction series of this type is a plot of log k for the reaction against pX of the conjugate acid of the nucleophile. Of course, this is identical with the graphical presentation... 

Pourbaix s pioneering work on the graphical presentation of gas-metal equilibria and the concept of stability zones and their boundaries between the various stable compounds lead to the second type of diagrams. Figure 7.65 shows a Pourbaix plot of the log P02 system against the reciprocal... 

Obviously the regularity expressed in the qualitative form (a) is far less informative than any one of the quantitative presentations, (b), (c), or (d). The relative merits of the expressions (b), (c), and (d) depend upon the use. Table l-II tells in most detail exactly how much is known about the pressure-volume behavior of oxygen gas (from this experiment). In the graphical presentation of Figure 1-8 the trend of the data is shown by the smooth curve drawn to pass near as many points as possible. Uncertainties caused... 

A graph of l/v versus S is plotted and the slope is 1 IK2ke0. There is an intercept in the graphical presentation to identify another constant. From the above equation, K2 can be calculated, which is similar to, Smax, where. S m lx means that the substrate concentration gives the maximum rate. 

Based on die graphical presentation, this is a competitive inhibition. 

The inhibition analyses were examined differently for free lipase in a batch and immobilised lipase in membrane reactor system. Figure 5.14 shows the kinetics plot for substrate inhibition of the free lipase in the batch system, where [5] is the concentration of (S)-ibuprofen ester in isooctane, and v0 is the initial reaction rate for (S)-ester conversion. The data for immobilised lipase are shown in Figure 5.15 that is, the kinetics plot for substrate inhibition for immobilised lipase in the EMR system. The Hanes-Woolf plots in both systems show similar trends for substrate inhibition. The graphical presentation of rate curves for immobilised lipase shows higher values compared with free enzymes. The value for the... 

The graphical presentation of the equation shows a straight line with a negative slope for kA. As the death rate constant follows Arrhenius law,1 the death rate constant is temperature dependent. The value of kA is about 0.02 min 1 at 100 °C, the death rate constant increases by 10-fold at 110 °C and 100-fold at 120 °C.2... 

Graphics In most cases a graphical presentation of each application is necessary. Here, it is possible to store photographs, sketches, CAD drawings, computer simulation results, and experimental results for use in the database. 

In this light, the type II error is a hypothetical entity, but very useful. A graphical presentation of the situation will help (see Figure 1.34). 

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