Velocity profile axial, flat

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. 

The reactor is turbulent, the velocity profile is flat, radial mixing is complete, and there is some transfer of heat and mass in the axial direction. The tube can be either packed (e.g., with a catalyst) or open. This model provides an estimate of reaction yields in highly turbulent reactors that is more conservative than assuming piston flow. 

We illustrate a sketch of the physical system in Fig. 1.2. It is clear in the sketch that we shall again use the plug flow concept, so the fluid velocity profile is flat. If the stream to be processed is dilute in the adsorbable species (adsorbate), then heat effects are usually ignorable, so isothermal conditions will be taken. Finally, if the particles of solid are small, the axial diffusion effects, which are Fickian-like, can be ignored and the main mode of transport in the mobile fluid phase is by convection. 

A PFR can be visualized as a tubular reactor for which three conditions must be satisfied (i) the axial velocity profile is flat (ii) there is complete mixing across the tube, so that all the reaction variables are a function of the axial dimension of the reactor (named z) and (iii) there is no mixing in the axial direction. PFRs have spatial variations in concentration and temperature. Such systems are caUed distributed, and analysis of their steady state performance requires the solution of differential equations. 

Real tubular reactors approach axial plug flow if the viscosity of the fluid decreases with increasing rate of shearing and the resulting velocity profile is flat across the tube. 

At lower Reynolds numbers, the axial velocity profile will not be flat and it might seem that another correction must be added to Equation (9.14). It turns out, however, that Equation (9.14) remains a good model for real turbulent reactors (and even some laminar ones) given suitable values for D. The model lumps the combined effects of fluctuating velocity components, nonflat velocity profiles, and molecular diffusion into the single parameter D. 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

When the velocity uz varies with radial position, equation 12.7.28 must be solved by a stepwise numerical procedure. Experimental evidence indicates that the axial velocity does indeed vary with radial position in fixed bed reactors. The velocity profile is relatively flat in the center of the tube. As one moves radially outward, the velocity increases gradually until a maximum is reached at a point about one pellet diameter from the tube wall. It then falls rapidly, until it reaches zero at the wall. If the ratio of the tube diameter to the pellet diameter... 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

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. 

Here, ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid... 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective... 

When the velocity u varies with radial location, a stepwise numerical solution of Eq. (13-10) or Eq. (13-2) would be needed. Axial velocities do vary with radial position in fixed beds. The typical profile is flat in the center of the tube, increases slowly until a maximum velocity is reached about one pellet diameter from the wall, and then decreases sharply to zero at the wall. The radial gradients are a function of the ratio of tube to pellet diameter. Excluding the zero value at the wall, the deviation between the actual velocity at any radius and the average value for the whole tube is small when djdp > 30. In single-tube reactors this condition is usually met e.g., for pellets this corresponds to a tube diameter of about 8 in. 

The tube is packed with catalyst pellets. Flow may be either laminar or turbulent. The velocity profile is assumed to be flat. Transfer of heat and mass in the radial direction is modeled using empirical diffusion coefficients that combine the effects of convection and true diffusion in the radial direction. There is no axial diffusion. Details are given in Chapter 9. This model is important only for nonisothermai reactors. It reduces to piston flow if the reaction is isothermal. 

The boundary conditions are zero velocity at the walls and zero slope at any planes of symmetry. Analytical solutions for the velocity profile in square and rectangular ducts are available but cumbersome, and a numerical solution is usually preferred. This is the reason for the transient term in Equation 16.7. A flat velocity profile is usually assumed as the initial condition. As in Chapter 8, is assumed to vary slowly, if at all, in the axial direction. For single-phase flows, u can vary in the axial direction due to changes in mass density and possibly to changes in cross-sectional area. The continuity equation is just AcUp = constant because the cross-channel velocity components are ignored. 

Figure 13 shows the axial velocity profiles computed from equation 124. One can observe the variation of the axial velocity with the normalized bed radius, DF1/2/2. When the normalized bed radius is zero, the axial velocity displays a parabolic profile that corresponds to the Hagen-Poiseuille solution. As the normalized bed radius increases, the axial velocity profile flattens. When the normalized bed radius is infinite, the axial velocity corresponds to a unidirectional flow (flat) profile. 

Plug flow in a tube is an ideal-flow assumption in which the fluid is well mixed in the radial and angular directions. The fluid velocity is assumed to be a function of only the axial position in the tube. Plug flow is often used to approximate fluid flow in tubes at high Reynolds number. The turbulent flow mixes the fluid in the radial and angular directions. Also in turbulent flow, the velocity profile is expected to be reasonably flat in the radial direction except near the tube wall. 

With turbulent flow in a straight pipe, the velocity profile is blunter than with laminar flow but still quite different from the flat profile assumed for plug flow. The ratio of maximum velocity to average velocity is about 1.3 at Re = 10", and this ratio slowly decreases to 1.15 at Re = 10 . A pulse of tracer introduced at the inlet gradually expands, but the distribution of residence times at the exit is fairly narrow. The effect of the axial velocity profile is largely offset by rapid radial mixing due to the turbulent velocity fluctuations. 

The ideal tubular reactor is one in which elements of the homogeneous fluid reactant stream move through a tube as plugs moving parallel to the tube axis. This flow pattern is referred to as plug flow or piston flow. The velocity profile at a given cross section is flat and it is assumed that there is no axial diffusion or backmixing of fluid elements. 

The tracer is injected at the inlet at the reference time 0 = 0. C,(0) and C(0) are the tracer concentrations, at the inlet and the outlet, respectively. In an ideal PFR, there is complete mixing in the radial direction and no mixing in the axial direction. So, the tracer material injected at the inlet, at time 0 = 0, spreads uniformly in the radial direction (due to complete mixing) and all the tracer elements move at the same velocity in the axial direction (no axial mixing and flat velocity profile). Thus, all the fluid eluents have the same residence time, which is equal to the mean residence time Q = V/q. Thus, C(0) is the same as Cj(0) shifted along the time axis by 0. 

PFRs, on the other hand, have an extremely narrow RTD. In fact, in an ideal PFR, the output is exactly the same as the input, as there is no axial dispersion of information along the reactor, due to its flat velocity profile (usually due to achieving high levels of turbulence). Real reactors RTDs lie between these two extremes no PFR has an infinitely narrow RTD, and no CSTR is perfectly mixed. 

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