Velocity profile ducts

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

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... 

The CDC-NIH document describes, in detail, the different uses of the different classes and types of BSCs and the type of protection (personnel, product, and environmental) each type provides. The document also provides a detailed comparison of filtration (air cleaning), airflow pattern (into the cabinet from the room or from the supply duct), and necessary performance tests (leak, velocity profile, differential pressure, etc.) for each type of BSC (see also Simons ). 

The volume flow rate is calculated as the arithmetical mean of the measured velocities multiplied by the duct cross-sectional area. The number of diameters along which the traversing occurs is not defined. If a near-symmetrical velocity profile is expected, an even travetse along one diameter may be sufficient. In case of a more disturbed profile, traversing along two or more diameters is recommended. 

Because all measurement methods and instruments are sensitive to the velocity profile, the choice of the measurement cross-section is of vital importance. In most ventilation systems there is seldom enough straight duct to allow a fully developed velocity profile to develop, which is the most favorable for flow measurement. Thus, the principle in selecting the measurement cross-section is to find the place where the velocity profile is as near to the fully developed profile as possible. In practice the distance from the nearest source of disturbance upstream is maximized, ensuring that the distance to the nearest downstream disturbance is at least 3 to 5 duct diameters. 

Air at 323 K and 152 kN/m2 pressure flows through a duct of circular cross-section, diameter 0.5 m. In order to measure the flow rate of air, the velocity profile across a diameter of the duct is measured using a Pitot-static tube connected to a water manometer inclined at an angle of cos-1 0.1 to the vertical. The following... 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

In both building locations, the velocity profile indicated duct floe turbulence. The drilling operation of building 1619 had flow velocities and negative static pressures that were significantly higher than the operations in building 1611. These differences can be attributed to the duct diameters, sizes, and number of dust cleanouts found in the two removal systems. 

The mass transfer coefficient is usually obtained from correlations for flow in non-porous ducts. One case is that of laminar flow in channels of circular cross-section where the parabolic velocity profile is assumed to be developed at the channel entrance. Here the solution of LfivfiQUE(7), discussed by Blatt et al.(H>, is most widely used. This takes the form ... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

The model results show the velocity profile of a FCCU flue gas traveling vertically upward then making a 90° turn leading to the SCR. Notice the uniformity of the stream velocity as it travels upward. As soon as the stream encounters the 90° turn, the velocities stratify with nearly stagnant flow at the comer, and very high velocities at the far wall. This occurs because the denser catalyst particles are carried out further in the duct by their momentum relative to the lighter gas molecules. 

Kintner et al (K7) and Damon et al. (Dl) have discussed photographic techniques applicable to the study of bubbles and drops. Sometimes it is desirable to hold a bubble or drop stationary, to study internal or external flow patterns and transfer processes. To prevent the particle from migrating to the wall, it is desirable to establish a minimum in the velocity profile at the position where the particle is to reside, and various techniques have been devised (D4, FI, Gl, Pll, M15, R15, S20) to do this. Vertical wandering of such particles may occur (W7), and may be reduced by using a duct tapered so that the area decreases towards the top (D4). Acoustic levitation of liquid drops may also be used (A3). 

Fig. 4.8 Illustration of the axial-velocity profile in the entry region of a circular duct.

We have just discussed several variations of the flow in ducts, assuming that there are no axial variations. In fact there well may be axial variations, especially in the entry regions of a duct. Consider the situation illustrated in Fig. 4.8, where a square velocity profile enters a circular duct. After a certain hydrodynamic entry length, the flow must eventually come to the parabolic velocity profile specified by the Hagen-Poiseuille solution. 

Consider a long circular duct in which an incompressible, constant-property fluid is initially at rest. Suddenly a constant pressure gradient is imposed. The axial momentum equation that describes the transient response of the velocity profile for this situation is... 

Fig. 4.9 Transient nondimensional axial velocity profiles in a long circular duct, responding to a suddenly imposed pressure gradient. The fluid is initially at rest.

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

The flow field is symmetric over the period, with velocities in both directions at different times. Because of the symmetry there is no net flow through the duct, and thus the mean velocity profile is exactly zero. The average root-mean-square velocity, however, does have a radial dependence as shown in Fig. 4.11. The root-mean-square velocity is defined as... 

Fig. 4.16 Illustration of the Graetz problem. A fully developed parabolic velocity profile is established in a circular duct and remains unchanged over the length of the duct. There is a sudden jump in the wall temperature, and the fluid temperature is initially uniform at the upstream wall temperature. The thermal-entry problem is to determine the behavior of the temperature profile as it changes to be uniform at the downstream wall temperature. Because the flow is incompressible, the velocity distribution does not depend on the varying temperatures.

Fig. 4.17 Nondimensional temperature profiles (left-hand panel) in the thermal entry of a circular duct with a fully developed velocity profile. The profiles are shown at various nondimensional downstream locations z. Also shown is the nondimensional heat-transfer coefficient, Nu as a function of the nondimensional downstream position.

A numerical solution procedure is reasonably flexible in accommodating variations of problems. For example, the Graetz problem could be solved easily for velocity profiles other than the parabolic one. Also variable properties can be incorporated easily. Either of these alternatives could easily frustrate a purely analytical approach. The Graetz problem can also be worked for noncircular duct cross sections, as long as the velocity distribution can be determined as outlined in Section 4.4. 

Jeffery-Hamel analysis leads to either full or local similarity of the velocity profiles in certain channels and ducts. Since it is based on incompressible flow, there are certainly limitations on its applicability. Nevertheless, given the very significant mathematical reductions, the analysis can be used effectively to provide some important insights about channel flows. 

Consider the flow of an incompressible fluid in the entry region of a circular duct, assuming that the inlet velocity profile is flat. As is often the case, the problem can be generalized by casting into a nondimensional form. A set of nondimensional variables may be chosen as... 

This section describes a spreadsheet to solve for the two-dimensional velocity profile in a rectangular duct. It also determines the factor /Re, given an aspect ratio. The spreadsheet is laid out to correspond to the mesh shown in Fig. D.5. The spreadsheet itself is shown in Fig. D.6. 

Even without chemical interaction the effect of a velocity profile across the section of the duct leads to a similar effect. The result, again, is an attenuation of variance. Lenschow and Raupach (13) investigated this aspect and presented their results in terms of the half-power frequency, /0i5, the frequency at which the variance has been diminished by 50%. This halfpower frequency is given by the expression... 

Henri de Pitot invented the Pitot tube in 1732. It is a small, open-ended tube that is inserted into the process pipe with its open end facing into the flow. The differential between the total pressure on this open impact port and the static pipeline pressure is measured as an indication of the flow. The Pitot tubes provide a low-cost measurement with negligible pressure loss and can also be inserted into the process pipes while the system is under pressure (wet- or hot-tapping). They are also used for temporary measurements and for the determination of velocity profiles by traversing pipes and ducts. 

The velocity profile is assumed fully developed at x = 0. The heating (or cooling) section starts at x = 0 the thermal boundary layer grows in thickness as x increases until it reaches the center line where it meets the boundary layer from the other wall of the duct. 

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

Therefore, as was the case with fully developed pipe flow, the velocity profile in fully developed plane duct flow is parabolic. 

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