Walstra equation

Numerous applications of multiple emulsions in various fields have been reported. More applications need to be realized if multiple emulsions stability is to be fully understood and approaches to stabilize multiple emulsions fully rationalized. The stability of multiple emulsions is influenced by numerous formulation and process variables. As demonstrated in this chapter, long-term multiple emulsion stability is dependent on the osmotic and Laplace pressures of the inner droplets as well as on the pressure balance between them described by the Walstra equation. Stability also equally, in some cases even more, depends on the strength of the interfacial film formed on the interface of droplets of multiple emulsions. This property can be characterized by interfacial rheology. 

The principles involved in these equilibria are presented in detail in numerous works and will not be repeated here. For a thorough treatment, including the definition of pH scales and methods of measurement, the reader is referred to such works as those of Edsall and Wyman (1958) and Bates (1964). Applications to milk are discussed by Walstra and Jenness (1984). It should suffice here to present some of the basic relationships in equation form. 

Fractionation of milk and titration of the fractions have been of considerable value. Rice and Markley (1924) made an attempt to assign contributions of the various milk components to titratable acidity. One scheme utilizes oxalate to precipitate calcium and rennet to remove the calcium caseinate phosphate micelles (Horst 1947 Ling 1936 Pyne and Ryan 1950). As formulated by Ling, the scheme involves titrations of milk, oxalated milk, rennet whey, and oxalated rennet whey to the phenolphthalein endpoint. From such titrations, Ling calculated that the caseinate contributed about 0.8 mEq of the total titer of 2.2 mEq/100 ml (0.19% lactic acid) in certain milks that he analyzed. These data are consistent with calculations based on the concentrations of phosphate and proteins present (Walstra and Jenness 1984). The casein, serum proteins, colloidal inorganic phosphorus, and dissolved inorganic phosphorus were accounted for by van der Have et al (1979) in their equation relating the titratable acidity of individual cow s milks to the composition. The casein and phosphates account for the major part of the titratable acidity of fresh milk. 

If fat globules are present as separate particles, the fat content is <40% and the milk fat completely molten, milk and cream behave as Newtonian fluids at intermediate and high shear rates (Phipps, 1969 McCarthy, 2003), i.e., its viscosity is not influenced by shear rate (t = rj / y, where r is the shear stress [Pa], 17 is the viscosity [Pa s] and y is the shear rate[l/s]). For a Newtonian fluid, Eilers equation (Eilers, 1941) is generally obeyed (Walstra, 1995) ... 

Much more effort has gone into relating hardness value to the yield stress of fats than to their elastic properties. For example, the International Dairy Federation proposed (Walstra, 1980) that penetration depth be converted to an apparent yield stress (AYS) for sharp-ended cones according to the equation ... 

It was found that is a function of temperature but the model was found to give a better fit than analytical expressions like the Avrami model or the modified Gompertz model (Kloek, Walstra and Van Vliet 2000). The main advantage of this model is that as it is formulated as a differential equation, it can be used to predict isothermal as well as dynamic crystallization. However, this model does not consider the polymorphism of the material which is a critical point in the crystallization of cocoa butter. Another contribution is the model of Fessas et al. (Fessas, Signorelli and Schiraldi 2005) which considers all the transitions possible between each... 

Equation (1.7) was proposed by Walstra (1996). Walstra s equation shows that an optimal salt concentration in the internal phase exists between the Laplace and osmotic pressures exerted on the inner aqueous droplets. 

Using equation (4.28), which was proposed by Walstra (1996), we can calculate the minimum percentages difference of NaCl required (for various... 

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