Bivariant systems

H2O ratios) of the carrier gas. This Is consistent with the requirements based on the phase rule for a bivariant system. 

The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows ... 

Example calculations for a bivariate system can be found in Marchisio and Fox (2006) and Zucca et al. (2006). We should note that for multivariate systems the choice of the moments used to compute the source terms is more problematic than in the univariate case. For example, in the bivariate case a total of 3 M moments must be chosen to determine am, bm and cm. In most applications, acceptable accuracy can be obtained with 3[Pg.283]

Biyariant systems.—The importance of monovariant systems should not make us forget the not less important Bivariant systems. 

A very simple case of a bivariant system is furnished by a solid salt in presence of an aqueous solution of this salt two independent components, the salt and the water, have two phases, the solid salt and the solution. For every temperature and pressure such a system is in equilibrium the solution is then saturated with the salt the concentration of the saturated solution depends upon the temperature to which it is brought and upon the pressure it supports but it is independent of the masses of salt and water that the S3rstem contains. Also, if to a knowledge of the temperature and pressure we join the knowledge of the total mass of the salt and water in the 83nstem, the masses of the solution and of the undissolved salt are determined. 

Remark on the law of equilibrium of bivariant systems. — Here we must guard against a possible confusion. We have said that, when the temperature and pressure were stated, the concentration of the saturated solution was determined we understand by this that it is impossible, at a given temperature and pressure, to find a series of saturated solutions such that the concentration varies in a continuous manner from one solution to the following but it is not to be understood that the constitution of the saturated solution is determined without amhiquity it may happen, in fact, that to a given temperature and pressure corre-... 

A similar remark may be made relative to the composition of each of the phases of a bivariant system in equilibrium at a given pressure and temperature it is a remark whose importance we shall see while studying in Chap. XI the indifferent states of a bivariant system. 

The water solution of a salt, in the presence of this solid salt gave us a first example of a bivariant system. For another, consider a definite mass of ether into which it poured increasing quantities of water the first quantities of water poured in mix completely with the ether but beyond a certain point the mixture divides into two layers, an upper one richer in ether and a lower richer in water we therefore have to do with a S3rstem formed of two independent components, ether and water, and divided into two phases, the two superposed liquid layers such a system is bivariant and if the temperature and pressure rest constant, the composition of the two liquid layers will remain invariable as water is little by little added to the mixture, we see the upper mass decrease and the lower mass increase, but neither the concentration of the upper nor of the lower layer undergoes any change up to the moment when enough water has been added to cause the upper layer to disappear the S3rstem will then cease to be bivariant. 

A great number of important problems in chemical statics are dependent upon the study of bivariant S3rstems. The theory of the solubility of gases is the theory of a bivariant system for the two independent components, the gas and the solute, exist in two phases, a liquid solution and a gaseous atmosphere, mixed or simple according as the solute is volatile or not. The theory of the... 

When the solution exists in the presence of two solid deposits C, C, the system, formed of three components, is divided into three phases it is therefore bivariant let us show that the results we have just obtained are in accord with the properties of bivariant systems. 

In general, for a bivariant system, every modification which alters the mass of the phases alters at the same time the composition of some among them. 

Take, for instance, a bivariant system consisting of two independent components, water and sodium chloride, divided into two phases, solid sodium chloride and an aqueous solution of this... 

Consider at a gi en pressure and temperature a bivariant system whose twb components are a salt and water and whose two phases are the solid salt and an aqueous solution when the system is in equilibrium, the solution has a definite concentration S it is scUwraJted at the given pressure and temperature. 

Under a given pressure, such as the atmospheric pressure, and at the temperature T, there is stable equilibrium in a bivariant system formed by ice in contact with a salt solution s is the concentration of the solution. i... 

Various types of bivariant systems Solutions and double mixtures.—bivariant system is one divided into a number of phases equal to the number of independent components which form it a S3rstem of two independent components and divided into two phases is the most generally studied type. 

In all cases, by the words is determined is not to be understood a determination which excludes all ambiguity it may happen, and does in certain cases which we shall meet in this chapter, that at a given pressure and temperature a bivariant system formed of the same independent components presents two distinct states of equilibrium corresponding to different compositions of the several phases. 

Consequently, when a bivariant system is thus given, it is impossible, except for the particular case we have just mentioned, to vary the masses of the different phases which are held in equilibrium without causing their composition to vary, so that at constant pressure and temperature the system in equilibrium could not undergo any modification without destroying the equilibrium the exceptional case aside, the equilibrium of a bivariant system is not an indifferent equilibrium this distinguishes the bivariant tems sharply from the monovariant systems. 

It may be shown that the state of equUibrium of a bivariant system is stable except for the special case, when it is indifferent this proposition is of considerable importance, for it shows that in general one may apply to bivariant s3rstems the two laws of displacement of equilibrium by variation of pressure and of the displacement of equilibrium by change of temperature in fact, in the preceding chapter we have borrowed several examples of these laws from the study of bivariant systems. 

When the representative point is elsewhere than at w, it is impossible for our bivariant system to remain in equilibrium it must be transformed until the complete disappearance of one of its phases takes place. What laws govern these transformations To determine these laws, we must distinguish three cases, as follows ... 

BIVARIANT SYSTEMS. TRANSITION AND EUTEXIA. 259 solubility curve of IGa this is a eutectic point for which... 

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