Flat velocity profiles

This section considers three special cases. The first is a flat velocity profile that can result from an extreme form of fluid rheology. The second is a linear profile that results from relative motion between adjacent solid surfaces. The third special case is for motionless mixers where the velocity profile is very complex, but its net effects can sometimes be approximated for reaction engineering purposes. 

Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste flow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although Vflr) = u and z(7) = 1, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is 

Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution. Calculations that assume a parabolic profile will be conservative in the sense that they will predict a lower conversion than would be predicted for the actual profile. The changes in velocity profile due to variations in temperature and composition are normally much more important than the fairly subtle effects due to non-Newtonian behavior. 

Flow in a Slit. Turning to a slit geometry, a flat velocity profile gives the simplest possible solution using Euler s method. The stability limit is independent of y  

Note that Equation (8.49) applies for every point except for y=Y where the wall boundary condition is used, e.g.. Equation (8.27). When i = 0, Ogidi— 1) = ow(+ )  

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Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

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. 

This system produces a steady laminar flow with a flat velocity profile at the burner exit for mean flow velocities up to 5m/s. Velocity fluctuations at the burner outlet are reduced to low levels as v /v< 0.01 on the central axis for free jet injection conditions. The burner is fed with a mixture of methane and air. Experiments-described in what follows are carried out at fixed equivalence ratios. Flow perturbations are produced by the loudspeaker driven by an amplifier, which is fed by a sinusoidal signal s)mthesizer. Velocity perturbations measured by laser doppler velocimetry (LDV) on the burner symmetry axis above the nozzle exit plane are also purely sinusoidal and their spectral... 

The plug flow reactor has a flat velocity profile and no longitudinal mixing. These idealizations imply that all fluid elements leaving the reactor have the same age (T). The F(t) function for this system must then be... 

This is identical to equation 2.24 for turbulent flow of a Newtonian fluid because n = 0 corresponds to a flat velocity profile and this is a good approximation for turbulent flow. 

The possible existence of an interface resistance in mass transfer has been examined by Raimondi and Toor(12) who absorbed carbon dioxide into a laminar jet of water with a flat velocity profile, using contact times down to 1 ms. They found that the rate of absorption was not more than 4 per cent less than that predicted on the assumption of instantaneous saturation of the surface layers of liquid. Thus, the effects of interfacial resistance could not have been significant. When the jet was formed at the outlet of a long capillary tube so that a parabolic velocity profile was established, absorption rates were lower than predicted because of the reduced surface velocity. The presence of surface-active agents appeared to cause an interfacial resistance, although this effect is probably attributable to a modification of the hydrodynamic pattern. 

When velocity profiles are present across the cross-sections of the tracer injection and sampling points, then great care must be taken over the appropriate tracer injection and collection methods if the true system RTD is to be recovered (see, for instance, ref. 4 and also refs. 12 and 26). Our attention is, for the moment, restricted to situations where flat velocity profiles are present. 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

Even with constant dispersion coefficients, accounting for the velocity profile still creates difficulties in the solution of the partial differential equation. Therefore it is common to take the velocity to be constant at its mean value u. With all the coefficients constant, analytical solution of the partial differential equation is readily obtainable for various situations. This model with flat velocity profile and constant values for the dispersion coefficients is called the dispersed plug-flow model, and is characterized mathematically by Eq. (1-4). The parameters of this model are Dr, Dl and u. 

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... 

The simplest analytical solution assumes that the contact time is very short and the following assumptions are reasonable (a) the film moves with a flat velocity profile v0 (see Fig. 4a) (b) the film may be taken to be infinitely thick with respect to the penetration of the absorbed material and (c) the concentration at the interface x = 0 is cA0. The analytical... 

Pig. 4. Diffusion into a falling film (a) with flat velocity profile, and (b) with parabolic velocity profile. 

Another solution to Eq. (159) for the diffusion into a falling film with a flat velocity profile is obtained by taking into account the finite thickness of the liquid film 8 and its effect on the concentration profiles that is, the boundary conditions are taken to be... 

Clearly the assumption of a flat velocity profile is not correct. For a film in steady, laminar motion one may obtain an expression for the velocity distribution from the Navier-Stokes equations of motion [Eq. (9)]. For this case the Navier-Stokes equations simplify to... 

Although at first glance the solution in Eq. (184) would seem to be more realistic than that in Eq. (182), it has been found that the solution for the flat velocity profile agrees better with experiment than does that for the parabolic. This discrepancy has been explained by Boelter (B14,... 

Of interest in applied kinetics is the study of chemical reactions taking place in flow systems which are hydrodynamically simple, so that the kinetics effects may be properly calculated. A simple example is the flow (with flat velocity profile v0 in the z direction) of a fluid through a circular tube the fluid is an inert material S containing a small quantity of substance A. The inside of the cylindrical tube is coated with a catalyst which converts A into B according to a first-order reaction, with k as reaction-rate constant. Let it then be desired to obtain the percentage of conversion after the fluid has flowed through the reactor tube of length L and radius R. 

Assuming an initially flat velocity profile, calculate and plot the Nusselt number as a function of the inverse Graetz number. Compare the soultions for a range of Prandtl numbers. Explain why the Graetz number may not be an appropriate scaling for the combined entry-length problem. 

Take the case of flow and diffusion in a long tube with a flat velocity profile but no reaction. The concentration of solute is governed by... 

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