Medium quiescent

Vanishing gravitational velocity and therefore a medium, quiescent relative to the gravitational velocity, can be presumed for the diffusion of gases into solids or diffusion in very dilute solutions. 

Industrial separations are conducted in gravity or bath separators for a coarse feed, and in centrifugal separators for a fine feed (2,6,10). In gravity-type separators the feed and medium are introduced to the surface of a large quiescent pool of the medium. The float material overflows or is scraped from the pool surface. The heavy particles sink to the bottom of the separator and are removed using a pump or compressed air. The dmm separator (Fig. 13), up to 4.6 m dia and 7 m long, processes approximately 800 t/h, and treats feed of size up to 30 cm dia, operates in the gravity or the... 

Once the piston-driven flow field is known, the flame-driven flow field is found by fitting in a steady flame front, with the condition that the medium behind it is quiescent. This may be accomplished by employing the jump conditions which relate the gas-dynamic states on either side of a flame front. The condition that the reaction products behind the flame are at rest enables the derivation of expressions for the density ratio, pressure ratio, and heat addition... 

These two conditions (Eqs. (4.97) and (4.98)) are usually sufficient for assuming the medium as quiescent in dilute systems in which both cua.s and cda,oo are small. However, in nondilute or concentrated systems the mass transfer process can give rise to a convection normal to the surface, which is known as the Stefan flow [Taylor and Krishna, 1993]. Consider a chemical species A which is transferred from the solid surface to the bulk with a mass concentration cua.oo- When the surface concentration coa,s is high, and the carrier gas B does not penetrate the surface, then there must be a diffusion-induced Stefan convective outflux, which counterbalances the Fickian influx of species B. In such situations the additional condition for neglecting convection in mass transport systems is [Rosner, 1986]... 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

The lag phase occurs after cell inoculation. In this phase, there is no cell division or division takes place at low specific rates. It is an adaptation period in which adherent cells may resynthesize the glycocalyx elements lost during trypsinization, bind, and spread on the substratum. During spreading, the cytoskeleton reappears and new structural proteins are synthesized (Freshney, 2005). The duration of the lag phase is dependent on at least two factors the point in the growth phase from which cells were taken in the previous culture and the inoculum concentration. Cells originating from an actively growing culture have a shorter lag phase than those from a quiescent culture. Cultures initiated at low cell densities condition the culture medium more slowly and hence increase the duration of the lag phase, which is not desirable. 

Burk (1970) showed that Syrian hamster cells (BHK21/C13) failed to grow when transferred to medium containing 0.25% serum and could be maintained in a quiescent state for 8 days or more. On readdition of serum no DNA synthesis occurred for 9 h and mitotic peaks were observed at about 23 and 33 h. It appeared the cells had come to rest in G1 and on stimulation showed a lag of about 9 h before entering into exponential growth with a generation time of about10-12h. 

Figure 6.2 In vitro recovery versus flow rate for a typical microdialysis probe (CMA/ 10, 4 mm polycarbonate membrane, cut off 20,000 Da) in a quiescent medium and at ambient temperature. Typical flow rates used from brain microdialysis applications are in the range of 0.1 to 5 /ttL/min.

Note that the Biot number is tlic ratio of the convection at the surface to conduction within the body, and this number should be as small as possible for lumped system analysis to be applicable. Therefore, small bodies with high tlieniial conductivity are good candidates for lumped system analysis, especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless. Thus, Ihe hot small copper ball placed in quiescent air, discussed eailier, is most likely to satisfy the criterion for lumped system analysis (Fig. 4-6). 

New cell culture techniques, which may improve the applicability of renal epithelial cultures, are also required. Currenfly there exist two commercially available cell culture perfusion systems, which allow the continuous perfusion of culture media and optimized oxygenation [243]. These systems allow stable longterm culture of quiescent adherent cells [244]. Continuous medium perfusion furthermore may lead to the re-expression of lost functions in continuous cell hues and the maintenance of differentiated properties in primary cells. Recently our laboratory has demonstrated that LLC-PKj cells maintained in a newly developed perfusion system (EpiFlow ) changed from a glycolytic to a more oxidative phenotype [72]. Evidence is also available from experiments in our laboratory that this mode of cultivation helps to prolong the lifetime of primary cultures of proximal tubular cells. Combining perfusion culture with co-culture of a cell type that is an anatomical neighbour in vivo (e.g. epithelial with endothelial, interstitial or immune cells) may improve the state of differentiation of both partner cells and increase the complexity of autocrine and paracrine interaction [73]. 

This is the classic equation relating distance and time for species in a totally quiescent medium where transport is controlled by molecular diffusion. The simple formula, sometimes referred to as Einstein s equation, is veiy useful for order of magnitude calculations determining the distance a substance will travel by molecular diffusion. Eor example, if one wanted to know if a chemical species with rate constant k would react significantly during transport across a molecular diffusion boundary layer of thickness Ax one would compare the time calculated from the above equation with the reciprocal of the reaction rate constant, k (see below). 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

We will deal with a quiescent medium relative to the gravitational velocity, w = 0, so that the diffusional flux is... 

The density at a given position does not change with time. This is stipulated because we have presumed a quiescent medium w = 0. However the density is only constant at a fixed position. It can change locally due to a different composition, g = g(x). 

Drops and bubbles are indeed the same mathematical object. However, in marine water studies, the profile analysis of captive (or emerging) bubbles is preferable in respect to the analysis of drops. Actually, from the physical point of view, bubbles exhibit some differences in respect to drops a) diffusion to the air-water interface from a semi-infinite medium (rather than from the small volume confined by the drop) b) limited evaporation c) possibility of observing bubble properties both in quiescent hydrodynamic conditions or in laminar flow regime. Moreover, a captive bubble can be expanded to very large dimensions. 

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