Lever rule and

Himmelbau (1995) or any of the general texts on material and energy balances listed at the end of Chapter 2. The Ponchon-Savarit graphical method used in the design of distillation columns, described in Volume 2, Chapter 11, is a further example of the application of the lever rule, and the use of enthalpy-concentration diagrams. 

For solidification described by the lever rule and assuming linear liquidus and solidus lines, the composition of the solid C, as a function of the fraction solid transformed (/,) is given by the equation... 

Equations (2.40) and (2.41) are the lever rule and can be used to determine the relative amounts of each phase in any two-phase region of a binary component phase diagram. For the example under consideration, the amount of liquid present mrns out to be... 

The phase diagrams of two-component systems are represented in the two-dimensional space, where the composition is shown on the x axis (in molar or in mass fractions) in agreement with the lever rule, and the temperature is given on the y axis (in °C or in Kelvin). They are the so-called isobaric diagrams, since the constant pressure, mostly the atmospheric one, is assumed. The Gibbs phase law attains thus the form... 

Since three phases (Aa, Ap, and the melt L) are present, the system has no degree of freedom (fc = 3 - 1 = 2,/= 3, v = 0), which means that its cooling must stop due to the evolution of the transformation heat. The system keeps at the polymorphic transformation temperature until all crystals of the a-modification of the component A transform to the -modification. Below the temperature of the polymorphic transformation, the crystals of the -modification coexist with the melt of the composition x([(B). The amount of the solid phase and of the melt is given by the lever rule and the system is composed of Ap mol Ap and l mol melt with the composition x([(B). 

The final mixture composition is on a straight line connecting the two feed compositions. This is another example of the lever rule, and is merely a result of the mass balances being linear equations. Note also that the composition of the final mixture is found at tworthirds of the distance from the first feed to the second feed in accordance with their relative amounts. This graphical linear relation between the two feeds and the final mixture is the opposite case to that of a single feed that splits into two equilibrium streams, which is the case in liquid-liquid extraction. ... 

This is the familiar lever rule and applies to all three components water... 

Starting from a high-pressure one-phase region, phase separation may be induced by reducing the pressure. The equilibrium compositions of the two phases that form when pressure is reduced from Pi to Pf are given by the binodals ( > i b and n b- These will be the same irrespective of polymer concentration in the initial solution in the one-phase. For pressure quenches that follow paths AB, A B , and A B which start at the same initial pressure and end at the same final pressure, the compositions of the two coexisting phases that will be in equilibrium after phase separation will be the same, even though the concentrations of the initial one-phase solutions are different. However, the phase volumes will be different. The phase volumes are dictated by the lever rule and depend on the initial concentration. 

This problem is concerned with the properties of the triangular diagram used for representing ternary systems. Use the lever rule and notice that if the residue were mixed with the distillate at any moment it would give a mixture represented by point A, Note also that the composition of the distillate is alwa3rs somewhere on XF because Z is not volatile. 

E) Calculate the flow rate of stream 8 and the flow rate of stream 7. You may use the lever rule and/or mass balances. Note that the flow rate of stream 4 may be changed by the addition of the second flash drum. 

It is also observed that the feed point lies on a straight line connecting the distillate and bottoms product concentrations. This is a consequence of the lever rule and is similar to the representation of mixing and... 

The fractions of total a, (both eutectic and primary), and also of total /S, W, are determined by use of the lever rule and a tie line that extends entirely across the a + p phase field. Again, for an alloy having composition C4,... 

Fractions of both total a (eutectoid and proeutectoid) and cementite are determined using the lever rule and a tie line that extends across the entirety of the a + FesC phase region, from 0.022 to 6.70 wt% C. 

Mass fractions of the primary phase and eutectic microconstituent may be computed using the lever rule and a tie Une that extends to the eutectic composition (e.g.. Equations 9.10 and 9.11). 

Systems that exhibit this effect include Sn/Bi, Sn/In, and Sn-Pb. This effect is illustrated in Figure 4 for the Sn/Ag/Bi system with graphs depicting the solid fraction as function of temperature and composition based on the phase diagram lever rule and nonequilibrium solidification. For example, considering the composition Sn-3.5Ag-7.5Bi, the last liquid... 

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