Plug flow assumption/condition

In the holding section of a continuous sterilizer, correct exposure time and temperature must be maintained. Because of the distribution of residence times, the actual reduction of microbial contaminants in the holding section is significantly lower than that predicted from plug flow assumption. The difference between actual and predicted reduction in viable microorganisms can be several orders of magnitude therefore, a design based on ideal flow conditions may fail. 

Since Re > 2300, flow will be in the turbulent regime and the plug flow assumption will be a good approximation to conditions in the reactor. With a longer, narrower pipe of the same volume, the Reynolds number would have been higher and plug flow would have been approached even more closely. 

Under conditions where the plug flow assumption is valid, that is, concentration gradients are negligible so that the linear flow velocity of the carrier gas is the same as that of the reactants, the time (t) for A and B to travel a distance d along the flow tube is given by... 

Plug-flow assumption Using die Michell and Furzer correlation (eq. (3.417)), the liquid-phase Peclet value is 0.66 and die minimum value of Z/d, evaluated by using the Mears criterion (eq. (3.421)) 4.81, which is much lower that the experimental one, which is 100. Then, condition (c) is satisfied. 

Under fairly low gas velocity conditions where U is close to t/mf or the bed is in particulate fluidization, the plug flow assumption for the gas phase can be reasonably made [Wen and Fane, 1982]. Considering the sublimation of species A from the solid phase to the gas phase, the mass balance on the concentration for species A in the gas phase, Ca, over the incremental height AH can be expressed as... 

This reactor may follow the plug flow assumptions, or it may be equilibrium limited depending on the operating conditions. 

Dead time can result from measurement lag, analysis, and computation time, communication lag or the transport time required for a fluid to flow through a pipe. Figure 2.27 illustrates the response of a control loop to a step change, showing that the response started after a dead time (td) has passed and reaches a new steady state as a function of its time constant (t), defined in Figure 2.23. When material or energy is physically moved in a process plant, there is a dead time associated with that movement. This dead time equals the residence time of the fluid in the pipe. Note that the dead time is inversely proportional to the flow rate. For liquid flow in a pipe, the plug flow assumption is most accurate when the axial velocity profile is flat, a condition that occurs when Newtonian fluids are transported in turbulent flow. 

If the plug flow assumption holds and the reactor truly behaves in a differential manner, a plot of Xgg Vs. W/Fgg should be linear with the slope equal to the reaction rate. However, as is evident from Figure 1, slight curvature persists in each plot. Typical calculations revealed that intra and interparticle heat and mass transfer problems should not exist at the operating conditions. The reaction rates, therefore, were obtained by evaluating the slope of each curve at the origin and as such can be called initial rates of reaction, Rq. 

For the case with porous particles, the pore fluid can be treated as a mass transfer medium rather than a separate phase thus enabling it to be combined with the bulk fluid in the overall mass balance. Under plug flow transfer conditions, at the end of each time increment, the pore fluid was assumed to remain stagnant, and only the bulk fluid was transferred to the next section. Based on these assumptions and initial conditions, the concentrations of the polypeptide or protein adsorbate in both liquid and solid phase can be calculated. The liquid phase concentration in the last section C , is the outlet concentration. The concentration-time plot, i.e., the breakthrough curve, can then be constructed. Utilizing this approach, the axial concentration profiles can also be produced for any particular time since the concentrations in each section for each complete time cycle are also derived. 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

The solutions below give the conditions when the effects of mass and momentum transfer through the chaimel/GDL interface are small and the plug flow assumption is justified. Note that we ignore the possible presence of liquid droplets in the chaimel and assume that the cathode flow is purely gaseous. 

In actual practice, TF reactors deviate from the plug flow model because of velocity variations in the radial direction (see Figure 10.4(b)-(d)). For any of these conditions, the residence time for annular elements of fluid within the reactor will vary from some minimal value at a point where the velocity is a maximum to a maximum value near the wall where the velocity approaches zero. The concentration and temperature profiles, as well as the velocity profile are therefore also not constant across the reactor. The describing equations based on the plug flow assumption are then not applicable. 

Even when the plug flow assumption is not valid, transportation processes usually can be modeled approximately by the transfer function for a time delay given in Eq. 6-28. For liquid flow in a pipe, the plug flow assumption is most nearly satisfied when the radial velocity profile is nearly flat, a condition that occurs for Newtonian fluids in turbulent flow. For non-Newtonian fluids and/or laminar flow, the fluid transport process still might be modeled by a time delay based on the average fluid... 

Ejfects of Gas and Liquid Mixing As noted previously, it is necessary in most instances to convert point efficiency E g to Murphree plate efficiency E, ,. This is true because of incomplete mixing only in small laboratoiy or pilot-plant columns, under special conditions, is the assumption E g = E, , likely to be valid. For a crossflow plate with no hquid mixing there is plug flow of hquid. For this condition of liquid flow, Lewis [Ind. Eng. Chem., 28, 399 (1936)] analyzed effects of gas mixing on efficiency. He considered three cases ... 

Since all of the tube diameters of interest are less than 1 ft, it is evident that in all three cases the flow will be highly turbulent and assumption of plug flow conditions will be quite appropriate. 

An approximate design procedure for packed tubular reactors entails the assumption of plug flow conditions through the reactor. Discuss critically those effects which would ... 

The part of this process that is described by a force balance is the liquid flowing through the pipe. It will have a mass equal to the volume of the pipe (j4j,L) times the density of the liquid p. This mass of liquid will have a velocity v (ft/s) equal to the volumetric flow divided by the cross-sectional area of the pipe. Remember we have assumed plug-flow conditions and incompressible liquid, and therefore all the liquid is moving at the same velocity, more or less like a solid rod. If the flow is turbulent, this is not a bad assumption. 

Here, issues in relation to the trickle flow regime—isothermal operation and plug flow for the gas phase—will be dealt with. Also, it is assumed that the flowing liquid completely covers the outer surface particles (/w = 1 or aLS = au) so that the reaction can take place solely by the mass transfer of the reactant through the liquid-particle interface. Generally, the assumption of isothermal conditions and complete liquid coverage in trickle-bed processes is fully justified with the exception of very low liquid rates. Capillary forces normally draw the liquid into the pores of the particles. Therefore, the use of liquid-phase diffusivities is adequate in the evaluation of intraparticle mass transfer effects (effectiveness factors) (Smith, 1981). 

In general, the material balances and the corresponding solutions for trickle and bubble bed reactors are the same, under the assumption that the plug-flow condition holds for both phases. Of course, the appropriate correlations should be used for the estimation of mass transfer coefficients. However, in packed bubble bed reactors, the liquid-phase is frequently found in a complete mixed state, and thus some adjustments have to be made to the aforementioned models. Two special cases will be presented here. 

Emulsion phase gas in plug flow Solutions for bubble phase free of solids In the following, a simplified solution is presented under the following assumptions first-order reactions, gas flow only through the bubble phase (fh = 1), and absence of solids in the bubble phase (yb = 0). Under these conditions, the material balances (3.519) and (3.520) become the following. 

If all the dimensionless parameters in a reaction model are kept constant with scale change, a similarity in the reactor performance is expected, provided that the basic assumptions of the model remain unchanged in both scales, e.g. in our example the plug flow condition of gas in the bubble phase. 

Plug Flow Reactor. A PFR is a continuous flow reactor. It is an ideal tubular type reactor. The assumption we make is that the reaction mixture stream has the same velocity across the reactor cross-sectional area. In other words, the velocity profile across the reactor is a flat one. In a PFR there is no axial mixing along the reactor. The condition of plug flow is met in highly turbulent flows, as is usually the case in chemical reactors. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

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