Three-Phase Calculations

Similar calculations need to be conducted on any monitored run in three-phase work. A sample set of calculations is given here. Data  

Rinse water rate 1 m /h Bowl speed 3150rpm 

First a total mass balance, followed by a solids mass balance, is conducted. One of the two unknowns, solids rate or effluent rate, is eliminated by substituting from one equation into the other, and the two unknowns are calculated. Thus, from an equation similar to equation (4.11), the total mass balance is  

FeedRate + WaterRate = CakeRate -I- EffluentRate -F OilRate 4 + 1 = CakeRate -f EffluentRate -F 0.8 x 0.85 

This assumes that the densities of effluent, feed, and cake are all unity. This is not quite true, but within the experimental error associated with this type of work, this is acceptable. 

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Fig. 12. Growth of a barrier layer on A1 as the voltage across the film is increased to a maximum of 60 V, The broken line is a three-phase calculation for a film of n = 1.60.

The general multistage compressor model may have several functions polytropic and isentropic compressor, positive displacement compressor, inter-cooler between stages, isentropic turbine. Single phase, as well as two- or three-phase calculations, is possible. 

Pressure drop unit are used to simulate hydraulic operations, as for example the pressure drop in a pipeline or in a distribution network. Adiabatic operation or heat transfer with the surroundings can be treated. In flowsheeting the module is generally tailored to simulate the transport of fluids in process plants. It can cover a large variety of physical conditions, as for example three-phase calculations in different flow regimes. However, special applications are better modelled by dedicated software, as the network of utilities (water and steam), or pipeline systems in oil gas production. 

Applicability of the model to the experiment and feasibility of three-phase calculations in such rather complex configurations has therefore been shown, both qualitatively and quantitatively. Numerical modelling of hydro-mechanical three phase coupling can, as in the presented example, yield useful and probably critical support for the design and the safety assessment in the context of e.g. nuclear waste disposal. 

Figure 3.52 Specifying three-phase calculation inside the column.

Suppose that the line tension for a given three-phase line is 1 x 10 dyn. Calculate ff for drops of radius 0.1, 0.01, and 0.001 cm if the value for a large drop is 56. Assume water at 20°C. 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

For a three-phase electric motor, the horsepower is calculated as... 

The final factor influencing the stabiHty of these three-phase emulsions is probably the most important one. Small changes in emulsifier concentration lead to drastic changes in the amounts of the three phases. As an example, consider the points A to C in Figure 16. At point A, with 2% emulsifier, 49% water, and 49% aqueous phase, 50% oil and 50% aqueous phase are the only phases present. At point B the emulsifier concentration has been increased to 4%. Now the oil phase constitutes 47% of the total and the aqueous phase is reduced to 29% the remaining 24% is a Hquid crystalline phase. The importance of these numbers is best perceived by a calculation of thickness of the protective layer of the emulsifier (point A) and of the Hquid crystal (point B). The added surfactant, which at 2% would add a protective film of only 0.07 p.m to emulsion droplets of 5 p.m if all of it were adsorbed, has now been transformed to 24% of a viscous phase. This phase would form a very viscous film 0.85 p.m thick. The protective coating is more than 10 times thicker than one from the surfactant alone because the thick viscous film contains only 7% emulsifier the rest is 75% water and 18% oil. At point C, the aqueous phase has now disappeared, and the entire emulsion consists of 42.3% oil and 57.5% Hquid crystalline phase. The stabilizing phase is now the principal part of the emulsion. 

Z( = zero phase sequence or residual impedance. This is measured between the three-phase terminals of a star winding shorted together and the neutral (Figure 20.9(b)) and is calculated by... 

Calculation of long-term interference voltages is involved with a multiconductor problem which, in contrast to the short-term interference that derives from a one-pole grounding short circuit, in this case is related to the superposition of alternating magnetic fields of all the conductors of one or several three-phase systems as well as the ground wire. 

Figure 1 shows water content of lean, sweet natural gas. It can be used also for gases that have as much as 10% CO2 and/or HiS if the pressure is below 500psia. Above 500psia, acid gases must be accounted for by rigorous three-phase flash calculations or approximation methods. ... 

Conventionally RAIRS has been used for both qualitative and quantitative characterization of adsorbed molecules or films on mirror-like (metallic) substrates [4.265]. In the last decade the applicability of RAIRS to the quantitative analysis of adsorbates on non-metallic surfaces (e.g. semiconductors, glasses [4.267], and water [4.273]) has also been proven. The classical three-phase model for a thin isotropic adsorbate layer on a metallic surface was developed by Greenler [4.265, 4.272]. Calculations for the model have been extended to include description of anisotropic layers on dielectric substrates [4.274-4.276]. 

In this equation, AH is not known, but Eucken (as quoted by von Stackelberg48) suggested, AH = 0, and comparison of experimental and calculated heats of hydrate formation30 certainly supports a low value of AH. The variation of the composition of a gas hydrate along the three-phase line ice-hydrate-gas will therefore be small. (The variation along the three-phase line hydrate-aqueous liquid-gas is larger, cf. Section III.C.(l).)... 

Let us first consider the three-phase equilibrium ( -clathrate-gas, for which the values of P and x = 3/( +3) were determined at 25°C. When the temperature is raised the argon content in the clathrate diminishes according to Eq. 27, while the pressure can be calculated from Eq. 38 by taking yA values following from Eq. 27 and the same force constants as used in the calculation of Table III. It is seen that the experimental results at 60°C and 120°C fall on the line so calculated. At a certain temperature and pressure, solid Qa will also be able to coexist with a solution of argon in liquid hydroquinone at this point (R) the three-phase line -clathrate-gas is intersected by the three-phase line -liquid-gas. At the quadruple point R solid a-hydroquinone (Qa), a hydroquinone-rich liquid (L), the clathrate (C), and a gas phase are in equilibrium the composition of the latter lies outside the part of the F-x projection drawn in Fig. 3. The slope of the three-phase line AR must be very steep, because of the low solubility of argon in liquid hydroquinone. 

Figure 2 shows the conversions obtained with the three series studied, as a function of the mechanical mixtures composition, one hour after the beginning of the reaction and at the steady-state. Each series presents a maximum of activity, but at a different composition. SA6 series has a maximum between R , values of 50 and 75, whereas SA12 series has a maximum around = 50, and SA60 series near R , = 75. The dashed lines on the figures represent the sum of the individual contributions of the pure phases, calculated according to Equation 3. A very important synergetic effect is observed in all series, i.e., the activity of the mixtures is... 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

A separator divides a process stream into three phases a liquid organic stream, a liquid aqueous stream, and a gas stream. The feed stream contains three components, all of which are present to some extent in the separated steams. The composition and flowrate of the feed stream are known. All the streams will be at the same temperature and pressure. The phase equilibria for the three phases is available. How many design variables need to be specified in order to calculate the output stream compositions and flow rates ... 

When the SVE technology is applied in a contaminated site, the NAPL is gradually removed. Towards the end of the remediation and when NAPL is no longer present, a three-phase model should be considered to calculate the phase distribution of contaminants (see Table 14.3). In this case, the vapor concentration in pore air (Ca) is calculating using the Henry s Law equation (Equation 14.5), which describes the equilibrium established between gas and aqueous phases ... 

The removal of toluene is assumed to take place in two stages. The first stage corresponds to the removal of free NAPL, which, according to the phase distribution calculations (Step 2 Table 14.4) represents a mass of Mreml = 11.941. The second stage corresponds to the removal of toluene, which is distributed among the other three phases, and represents a mass of Mrem2 = 16.57 - 11.94 = 4.631. 

On the basis of a simple three-phase model (i.e. electrode, monolayer of modifier, solution), the authors calculations suggested that the initially formed layer of SSBipy on the surface was formed without bond cleavage and with the pyridine rings essentially flat on the surface. 

Calculated Recovery and Enrichment in Three-Phase LPME and Simple LLE at Different Ka/d Values... 

The calculated critical lines for the binary pairs are shown in Figure 1. All these lines are discontinuous, indicating high density phase separations. For each binary pair the principal part of the critical line begins at the critical point for the component with the higher critical temperature. There is a second branch of each of the critical lines, beginning at the critical point of the component with the lower critical temperature, which terminates on intersecting a liquid-liquid-vapor three-phase line. 

Two- and Three-Phase Equilibrium Calculations for Coal Gasification and Related Processes... 

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