Assumption perfect mixing

The dynamic model in Eqs. 2-17 and 2-18 is quite general and is based on only two assumptions perfect mixing and constant density. For special situations, the liquid volume V is constant (that is, dVIdt = 0), and the exit flow rate equals the sum of the inlet flow rates, w = w - - W2. For example, these conditions occur when... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

The name continuous flow-stirred tank reactor is nicely descriptive of a type of reactor that frequently for both production and fundamental kinetic studies. Unfortunately, this name, abbreviated as CSTR, misses the essence of the idealization completely. The ideality arises from the assumption in the analysis that the reactor is perfectly mixed, and that it is homogeneous. A better name for this model might be continuous perfectly mixed reactor (CPMR). 

For the models described, the usual assumption for air nodes in regard to the room air distribution is still valid. This means that each air node represents a volume of perfectly mixed air. Thus, the same limitations as for thermal and airflow models apply Local air temperatures and air velocities as well as local contaminant concentrations can he neither considered nor determined. This also means that thermal comfort evaluations in terms of draft risk cannot be performed. 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

Liquid residence-time distributions in mechanically stirred gas-liquid-solid operations have apparently not been studied as such. It seems a safe assumption that these systems under normal operating conditions may be considered as perfectly mixed vessels. Van de Vusse (V3) have discussed some aspects of liquid flow in stirred slurry reactors. 

Although the above treatment is based on the assumption of perfect mixing, it was found experimentally that deviations from this ideal situation can be taken into account by introducing the additional parameters of Wolf and Resnick (W5) into the kernel (253). 

When this relation is satisfied, the conversion will be limited by the reaction kinetics, not by the mixing rate. As a practical matter, the assumption of perfect mixing is probably reasonable when is eight times larger than... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

Chapter 2 treated multiple and complex reactions in an ideal batch reactor. The reactor was ideal in the sense that mixing was assumed to be instantaneous and complete throughout the vessel. Real batch reactors will approximate ideal behavior when the characteristic time for mixing is short compared with the reaction half-life. Industrial batch reactors have inlet and outlet ports and an agitation system. The same hardware can be converted to continuous operation. To do this, just feed and discharge continuously. If the reactor is well mixed in the batch mode, it is likely to remain so in the continuous mode, as least for the same reaction. The assumption of instantaneous and perfect mixing remains a reasonable approximation, but the batch reactor has become a continuous-flow stirred tank. 

A relatively simple example of a confounded reactor is a nonisothermal batch reactor where the assumption of perfect mixing is reasonable but the temperature varies with time or axial position. The experimental data are fit to a model using Equation (7.8), but the model now requires a heat balance to be solved simultaneously with the component balances. For a batch reactor. 

Copolymerizations. The uniform chemical environment of a CSTR makes it ideally suited for the production of copolymers. If the assumption of perfect mixing is justified, there will be no macroscopic composition distribution due to monomer drift, but the mixing time must remain short upon scaleup. See Sections 1.5 and 4.4. A real stirred tank or loop reactor will more closely... 

In a properly operated bubble-column reactor, the liquid phase can be considered to be perfectly mixed, i.e. concentrations in the liquid are the same everywhere and correspond to those in the effluent. The gas is supposed to flow like a piston, i.e. the reactor is a plug-flow reactor with respect to the gas. These two assumptions are not entirely true, but within a certain flow regime they are not far from the reality. 

The next case to be considered is the ideal continuous stirred tank reactor. The key to the derivation of the F(t) curve for this type of reactor is the realization that the assumption of perfect mixing implies that upon entry in the reactor an element of volume can instantaneously appear in any portion of the reactor. Therefore its past or its future history cannot be derived from its position. Furthermore, the prob-... 

The column contains a total of N-p theoretical trays. The liquid holdup on each tray including the downcomer is M . The liquid on each tray is assumed to be perfectly mixed with composition x,. The holdup of the vapor is assumed to be negligible throughout the system. Although the vapor volume is large, the number of moles is usually small because the vapor density is so much smaller than the liquid density. This assumption breaks down, of course, in high-pressure columns. 

An ice cube is dropped into a hot, perfectly mixed, insulated cup of coffee. Develop the equations describing the dynamics of the system. List aU assumptions and define all terms. 

The values of k a for CO, desorption in a stirred-tank fermentor, calculated from the experimental data on physically dissolved CO, concentration (obtained by the above-mentioned method) and the CO2 partial pressure in the gas phase, agreed well with the k a values estimated from the k a for O, absorption in the same fermentor, but corrected for any differences in the liquid-phase diffusivities [11]. Perfect mixing in the liquid phase can be assumed when calculating the mean driving potential. In the case of large industrial fermentors, it can practically be assumed that the CO, partial pressure in the exit gas is in equilibrium with the concentration of CO, that is physically dissolved in the broth. The assumption of either a plug flow or perfect mixing in the gas phase does not have any major effect... 

To analyze data using these two methods one must make two assumptions (1) that a sorptive entering the chamber can either be sorbed or remain in solution, and (2) the sample is perfectly mixed i.e., the concentration in the mixing chamber equals the effluent concentrations. With these assumptions, one can then develop an equation for mass balance which can be used to analyze time-dependent data using a continuous flow method (Skopp and McAllister, 1986) ... 

It may be preferable to use the directly measurable quantity C rather than indirect ones such as q which are calculated from C, m,J, and t with an assumption of perfect mixing (Skopp and McAllister, 1986). If so, Eq. (3.2) must be solved. This necessitates an expression for dq/dt,... 

From the assumption of perfect mixing, the corresponding residence time distribution probability density function is well known as... 

Another problem arising from the results of the investigation on residence time distribution is the strong mixing of the materials in dispersed phase in the impingement zone. The fact that the model of RTD derived above fits well the concentration of the tracer in the out stream of the device indicates that the assumption of perfect mixing of... 

Assumption 5.2 Both the reactor vessel and the jacket are considered perfectly mixed therefore, the energy balances in the reactor and in the jacket can be written as in (2.30) and (2.31), respectively. 

The above remark is of the utmost importance for evaluating the potential of the proposed observer in a real setup. In fact, exponential stability would ensure robustness of the state estimation against bounded and/or vanishing model uncertainties and disturbances [35], due to inaccurate and/or incomplete knowledge of reaction kinetics and to usual simplifying assumptions adopted for the model derivation (e.g., perfect mixing). 

As discussed in Sect. 2.1, physical and mathematical models of ideal chemical reactors are based on two very simplified fluid dynamic assumptions, namely perfect mixing (BR and CSTR) and perfect immiscibility (PFR). On the contrary, in real tank reactors the stirring system produces a complex motion field made out of vortices of different dimensions interacting with the reactor walls and the internal baffles, as schematically shown in Fig. 7.2(a). As a consequence, a complex field of composition and temperature is established inside the reactor. 

Example 1.7 Suppose a pilot-scale reactor behaves as a perfectly mixed CSTR so that Equation (1.49) governs the conversion. Will the assumption of perfect mixing remain valid upon scaleup ... 

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