Concentrated suspensions cell models

Several interesting parameter-control models or systems have been reported in recent years. These are a five-state mathematical model for temperature control by Bailey and Nicholson [106], a mathematical-model description of the phenomenon of light absorption of Coffea arabica suspension cell cultures in a photo-culture vessel by Kurata and Furusaki [107], and a bioreactor control system for controlling dissolved concentrations of both 02 and C02 simultaneously by Smith et al. [108]. 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

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In this chapter, we extend the electrokinetic theory of soft particles (Chapter 21), which is applicable for dilute suspensions, to cover the case of concentrated suspensions [1-3] on the basis of Kuwabara s cell model [4], which has been applied to theoretical studies of various electrokinetic phenomena in concentrated suspensions of hard colloidal particles [5-23]. 

Consider a concentrated suspension of charged spherical soft particles moving with a velocity 17 in a liquid containing a general electrolyte in an applied electric field E. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d. We employ a cell model [4] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction 4> throughout the entire dispersion (Fig. 22.1), namely. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

For concentrated suspensions, hydrodynamic interactions among particles must be considered. The hydrodynamic interactions between spherical particles can be taken into account by means of a cell model, which assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume... 

Consider a concentrated suspension of porous spheres of radius a in a liquid of viscosity rj [27]. We adopt a cell model that assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume fraction 4> is given by Eq. (27.2) (Eig. 27.3). The origin of the spherical polar coordinate system (r, 6, cp) is held fixed at the center of one sphere. According to Simha [2], we the following additional boundary condition to be satisfied at the cell surface r = b ... 

For more concentrated suspensions, one can estimate p by using the cell model. 

Cell models have also been applied to the approximate determination of the properties of concentrated suspensions. This is touched upon in our discussion of suspension viscosity at high shear rate in Section 9.3. 

Cell models akin to those discussed in Section 8.5 have also been applied to the determination of the properties of concentrated suspensions (Happel Brenner 1983, van de Ven 1989). Although it is another method which has been used to obtaining approximate expressions for the high shear relative viscosity, we choose not to expand upon it here, instead referring the reader to the references cited. One of the difficulties is that the determination of the boundary conditions at the cell surface is somewhat arbitrary. Furthermore, expressions obtained by this approach indicate that the cell model is inappropriate for highly concentrated suspensions and is most satisfactory only at low to moderate concentrations. 

TA Strout. Attenuation of Sound in High Concentration Suspensions Development and Application of an Oscillatory Cell Model. Thesis. The University of Maine, 1991. 

Dukhin et al. [83-85] have performed the direct calculation of the CVI in the situation of concentrated systems. In fact, it must be mentioned here that one of the most promising potential applicabilities of these methods is their usefiilness with concentrated systems (high volume fractions of solids, 4>) because the effect to be measured is also in this case a collective one. The first generalizations of the dynamic mobility theory to concentrated suspensions made use of the Levine and Neale cell model [86,87] to account for particle-particle interactions. An alternative method estimated the first-order volume fraction corrections to the mobility by detailed consideration of pair interactions between particles at all possible different orientations [88-90]. A comparison between these approaches and calculations based on the cell model of Zharkikh and Shilov [91] has been carried out in Refs. [92,93],... 

This energy-based method of calculation of viscosity rj is due to Einstein [87], who considered hydrod)mamic dissipation in a very dilute suspension of non-interacting spheres. Tanaka and White [86] base their calculations on the Frankel and Acrivos [88] cell model of a concentrated suspension, but use a non-Newtonian (power law) matrix. The interaction energy is considered to consist of both van der Waals-London attractive forces and Coulombic interaction, i.e. 

Studies of the viscosity of suspensions at higher concentrations have usually used cell models. Here, one essentially considers the spheres are arranged in a lattice-like array, and one analyzes the flow in a representative cell. Cell models are critically discussed in the monograph of Happel and Brenner [32]. Such an approach was initiated by Simha [33]. He analyzed the flow around a sphere located in a cell. He predicted... 

Tanaka and White [36] using the Frankel-Acrivos cell theory formulation have modeled the shear flow of a concentrated suspension of spheres in a power law non-Newtonian fluid (Eq. 2.5b). 

In this work, a suspension culture of Taxus chinensis, which produces a bioactive taxoid, taxuyunnanine C (Tc), was taken as a model plant cell system. Experiments on the timing of jasmonates addition and dose response indicated that day 7 and 100 pM was the optimal elicitation time and concentration for both cell growth and Tc accumulation [8]. ITie Tc accumulation was increased more in the presence of novel hydroxyl-containing jasmonates compared to that with methyl jasmonate (MJA) addition. For example, addition of 100 pM... 

Azacitidine, a cytidine analog, causes hypomethylation of DNA, which normalizes the function of genes that control cell differentiation to promote normal cell maturation. The suspension is administered as a subcutaneous injection daily for 7 days for the treatment of myelodysplastic syndrome, a preleukemia disease. The pharmacokinetics of azacitidine are best described by a two-compartment model, with a terminal half life of 3.4 to 6.2 hours, whereas peak concentrations are achieved 30 minutes after a subcutaneous injection.7 Azacitidine has been shown to be clinically active in the treatment of myelodysplastic syndromes. The side effects include myelosuppression, renal tubular acidosis, renal dysfunction, and injection-site reactions. 

Conditioning of the manganese oxide suspension with each cation was conducted in a thermostatted cell (25° 0.05°C.) described previously (13). Analyses of residual lithium, potassium, sodium, calcium, and barium were obtained by standard flame photometry techniques on a Beckman DU-2 spectrophotometer with flame attachment. Analyses of copper, nickel, and cobalt were conducted on a Sargent Model XR recording polarograph. Samples for analysis were removed upon equilibration of the system, the solid centrifuged off and analytical concentrations determined from calibration curves. In contrast to Morgan and Stumm (10) who report fairly rapid equilibration, final attainment of equilibrium at constant pH, for example, upon addition of metal ions was often very slow, in some cases of the order of several hours. 

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