Performance equations

What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visuaHzable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. 

Gas Turbine Aerothermal Performance Equations for Correction to Standard-Day Conditions... 

Then find the value of the feedback capacitor C. The designer knows the value of the input resistor (R). It is the upper resistor in the voltage divider responsible for the voltage feedback to the error amplifier. One then performs Equation B.15. 

The performance equation of a mixer relates mixer size or mixing time to the input and output of the mixing device. The rate of transfer, r, incorporates the mass transfer coefficient, kL, and interfacial area, a, as calculated above. This rate can be used in conjunction with a material balance to relate concentrations of interest to time or size variables. 

Three basic fluid contacting patterns describe the majority of gas-liquid mixing operations. These are (1) mixed gas/mixed liquid - a stirred tank with continuous in and out gas and liquid flow (2) mixed gas/batch mixed liquid - a stirred tank with continuous in and out gas flow only (3) concurrent plug flow of gas and liquid - an inline mixer with continuous in and out flow. For these cases the material balance/rate expressions and resulting performance equations can be formalized as ... 

From diese various estimates, die total batch cycle time t(, is used in batch reactor design to determine die productivity of die reactor. Batch reactors are used in operations dial are small and when multiproducts are required. Pilot plant trials for sales samples in a new market development are carried out in batch reactors. Use of batch reactors can be seen in pharmaceutical, fine chemicals, biochemical, and dye industries. This is because multi-product, changeable demand often requues a single unit to be used in various production campaigns. However, batch reactors are seldom employed on an industrial scale for gas phase reactions. This is due to die limited quantity produced, aldiough batch reactors can be readily employed for kinetic studies of gas phase reactions. Figure 5-4 illustrates die performance equations for batch reactors. 

Eor constant density system, the performance equation in terms of becomes... 

The heat exchanger effectiveness shows how close the heat exchanger is operating to the maximum heat transfer performance. Equation (9.5) is valid for any type of heat exchanger. 

If lor evei y joule of work done, 2 joules of heat flowed out of the refrigerator then the performance would be 2. Using = Qi + W, the performance equation can be written... 

There are many baffle type impingement separators. The efficiency of operation for entrainment is entirely a function of the contacting action inside the particular unit. There are no general performance equations which will predict performance for this type of unit therefore manufacturers performance data and recommendations should be used. A few of the many available units are shown in Figures 4-28 to 4-31. Many use the Chevron-style verdcal plates as shown in Figures 4-17A and 4-30. 

A = Constant in cooling tower performance equation, Equation 9-129, or in mass transfer =... 

The second factor pertains to the contacting patterns for multiphase systems. Many combinations of contacting patterns are possible, even for a two-phase system. Each of the many ways of contacting two phases has associated with it a specific form of performance equation which must be developed for that particular contacting pattern. 

For the various reactor types this equation simplifies in one way or another, and the resultant expression when integrated provides the basic performance equation for that type of unit. Since in batch reactor or operation, no stream is entering or leaving the reactor,... 

For semibatch or semiflow reactors all four of the terms in the basic material and energy balance relations (equations 8.0.1 and 8.0.3) can be significant. The feed and effluent streams may enter and leave at different rates so as to cause changes in both the composition and volume of the reaction mixture through their interaction with the chemical changes brought about by the reaction. Even in the case where the reactor operates isothermally, numerical methods must often be employed to solve the differential performance equations. 

In this chapter, we first consider uses of batch reactors, and their advantages and disadvantages compared with continuous-flow reactors. After considering what the essential features of process design are, we then develop design or performance equations for both isothermal and nonisothermal operation. The latter requires the energy balance, in addition to the material balance. We continue with an example of optimal performance of a batch reactor, and conclude with a discussion of semibatch and semi-continuous operation. We restrict attention to simple systems, deferring treatment of complex systems to Chapter 18. 

A PFR may take various configurational forms, as indicated in Figure 15.1. The design or performance equation, 15.2-2 or its equivalent, provides a volume, V, which must then be interpreted geometrically for the purpose of fabrication. Thus, for cylindrical shape, the most common, it must be converted to a diameter (D) and length (L) ... 

If the particles in PF are not all of the same size, the performance equation, from equation 22.2-2 in the summation form of equation 22.2-4, is... 

The performance equation for the model is obtained from the continuity (material-balanoe) equations for A over the three main regions (bubble, cloud + wake, and emulsion), as illustrated schematically in Figure 23.7. Since the bed is isothermal, we need use only the continuity equation, which is then uncoupled from the energy equation. The latter is required only to establish the heat transfer aspects (internally and externally) to achieve the desired value of T. 

In Figure 23.7, the bubble, cloud, and emulsion regions are represented by b,c + iv, and e, respectively. The control volume is a thin horizontal strip of height dx through tiie vessel. The overall depth of the bed is Lfl, which is related to the holdup of catalyst, Wcat. The performance equation may be used to determine Wcat for a given conversion /A (and production rate), or the converse. 

The continuity equations for the three main regions lead eventually to the performance equation for the reactor model. 

Equation 23.4-6 is one form of the performance equation for the bubbling-bed reactor model. It can be transformed to determine the amount of solid (e.g., catalyst) holdup to achieve a specified /A or cA ... 

As the pressure is constant and only mechanical work is performed [Equation (4.4)],... 

A6.1 Derivation of the Meiting Performance Equations for a Conventional Channel... 

To find what a reactor is able to do we need to know the kinetics, the contacting pattern and the performance equation. We show this schematically in Fig. 1.2. 

This is called the performance equation. Why is this important Because with this expression we can compare different designs and conditions, find which is best, and then scale up to larger units. 

Such reactions are so different in rates and types that it would be awkward to try to treat them all in one way. So we treat them by type in this book because each type requires developing the appropriate set of performance equations. 

We particularly like these three flow or reacting patterns because they are easy to treat (it is simple to find their performance equations) and because one of them often is the best pattern possible (it will give the most of whatever it is we want). Later we will consider recycle reactors, staged reactors, and other flow pattern combinations, as well as deviations of real reactors from these ideals. 

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