Axial flow correction

If the constant speed characteristic of a pump is superimposed on a system curve, there is usually one intersection point, shown in Figure 32.39. If a flat system curve is being matched with a mixed or axial flow machine there can be flow instability, as illustrated in Figure 32.40, which is only corrected by changing pump speed or the static lift, or selecting a different pump. 

In viscometry, it is usually necessary to correct for end effects (e.g., entrance and exit effects in axial flow instruments) and for slip between sample and viscometer surfaces. 

Thus the presence of a small amount of curvature of the tube axis is seen to produce a significant departure in the nature of the flow even at the leading-order correction, 0(a/R). The motion is no longer simply in the axial direction, but there is also motion in the cross-sectional plane. This motion, superposed upon the primary axial flow, is known as a secondary flow. A sketch of the streamlines for the secondary flow, given by contours of constant f, is shown in Fig. 4-5. The motion at the center of the tube is from the inside to the outside, driven essentially by the centrifugal force exerted on the fluid as it traverses the curved path followed by the tube axis. The return flow nearest the tube walls at the top and bottom is necessary to satisfy continuity. 

Figure 3-56. Viscosity performance correction chart for centrifugal pumps. Note do not extrapolate. For centrifugal pumps only, not for axial or mixed flow. NPSH must be adequate. For Newtonian fluids only. For multistage pumps, use head per stage. (By permission. Hydraulic Institute Standards for Centrifugal, Rotary, and Reciprocating Pumps, 13th ed.. Hydraulic Institute, 1975.)...

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

Correlations for E are not widely available. The more accurate model given in Section 9.1 is preferred for nonisothermal reactions in packed-beds. However, as discussed previously, this model degenerates to piston flow for an adiabatic reaction. The nonisothermal axial dispersion model is a conservative design methodology available for adiabatic reactions in packed beds and for nonisothermal reactions in turbulent pipeline flows. The fact that E >D provides some basis for estimating E. Recognize that the axial dispersion model is a correction to what would otherwise be treated as piston flow. Thus, even setting E=D should improve the accuracy of the predictions. 

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

Most biochemical reactors operate with dilute reactants so that they are nearly isothermal. This means that the packed-bed model of Section 9.1 is equivalent to piston flow. The axial dispersion model of Section 9.3 can be applied, but the correction to piston flow is usually small and requires a numerical solution if Michaehs-Menten kinetics are assumed. 

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