Models open with mass transfer

One of the simplest models for convective mass transfer is the stirred tank model, also called the continuously stirred tank reactor (CSTR) or the mixing tank. The model is shown schematically in Figure 2. As shown in the figure, a fluid stream enters a filled vessel that is stirred with an impeller, then exits the vessel through an outlet port. The stirred tank represents an idealization of mixing behavior in convective systems, in which incoming fluid streams are instantly and completely mixed with the system contents. To illustrate this, consider the case in which the inlet stream contains a water-miscible blue dye and the tank is initially filled with pure water. At time zero, the inlet valve is opened, allowing the dye to enter the... 

Open models (mass transport) time of mass transport At > 0, scales t, x,y, z and events Models of mass transport Scales x,y,z,t and geological events Models of mass transport with mass-transfer Scales x,y,z, t, geological and physicochemical events... 

Open models with mass transfer assume that the media in the object of study are not balanced either dynamically or chemically. In these models is necessary simultaneous consideration of both mass transport and mass transfer velocities because the relaxation time At >0 and flow time At>... 

There is no restriction against applying polythermal models in open systems. In this case, the modeler defines mass transfer as well as the heating or cooling rate in terms of . Realistic models of this type can be hard to construct (e.g., Bowers and Taylor, 1985), however, because the heating or cooling rates need to be balanced somehow with the rates of mass transfer. 

A mass-transfer model was proposed to account for the function of the [BMIM]PF6/water biphasic system in the reaction (Scheme 19). According to this model, a small quantity of [BMIM]PF6 dissolved in water exchanged its anion with H2O2 to form Q OOH , which was transferred into the [BMIM]PF6 phase to initiate the epoxidation reaction. In this reaction system, the ring-opening... 

Turner (T14) has proposed two detailed models of packed beds which try to closely approximate the true physical picture. The first model considers channels of equal diameter and length but with stagnant pockets of various lengths opening into the channels. There is no flow into or out of these pockets, and all mass transfer occurs only by molecular diffusion. The second model considers a collection of channels of various lengths and diameters. We will briefly discuss each of these models, which are probably more representative of consolidated porous materials than packed unconsolidated beds. 

Fig. 7a, b Average mass transfer coefficient as a function of a aquifer anisotropy ratio for several variances of the log-transformed hydraulic conductivity distribution b variance of the log-transformed hydraulic conductivity distribution where open circles represent numerically generated data and solid lines represent linear fits. All model parameter values are identical with those used in Fig. 6... 

Suppose you want to solve the heat conduction equation along with a diffusion equation. Choose the Multiphysics menu, and en Model Navigator. The same window opens that you used to select e equation in e first place, as shown in Figure D.23. Navigate to Mass Transfer/Dilfusion/Steady State Analysis and choose Add. The dilfnsion equation is added to your problem and the dependent variable is called c, as shown in Figure D.24. 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

A compilation of available kinetic models shows that, in most cases, the calculated reactive surface areas are one to three orders of magnitude less than the estimated physical surface areas. Commonly, geometric and BET surface areas are used interchangeably in kinetic studies to measure physical surface areas. The models that did produce closer fits were for open systems with short residence times. Comparisons assumed experimentally correct reaction rates and dependent reactive surface areas. In reality, the reaction rate and the reactive surface area are explicitly linked on the basis of surface controlled reactions. The product of these two terms determines the mass transfer for a specific system. 

The boiloff rate was measured to be approx. 1,3 %/d (about 0.7 m /d) for the filled open tank, while it was decreasing to 0.6 %/d at filling levels below 40 %. The decrease of the boiloff rate is attributed to the stronger heat uptake of the steel at higher temperatures which occur when in contact with the gas phase. In the self-pressurization tests, mass transfer from liquid to gas was found to have a dominant influence. The tests with the model tank on the BAM test site near Berlin have been terminated in the meantime, the tank is now being offered to third parties for further experimentation. 

From figure 8 it can be concluded that forced desorption of CO2 can easily be realized under practical conditions and can be also predicted by the models. Measured H2S mole fluxes fall between penetration and film theory calculations. The forced desorption of CO2 agrees better with the film theory than with the penetration theory. It should be kept in mind however that the calculations are extremely sensitive to mass-transfer coefficients, diffusion and equilibrium constants which were obtained from separate experiments and open literature. 

Kashid et al. studied the fiow patterns within the slugs and mass transfer between two consecutive slugs in liquid-liquid slug flow using a finite element-based computational fluid dynamics (CFD) model [51]. The model equations are implemented in the open-source software FEATFLOW (www.featflow.de). Figure 12.18 shows snapshots of the concentration profiles of the extract (acetic acid). These results are compared with experimental results and are consistent with them. 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

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