Plug laminar pipe flow

Equations for the design of laminar pipe flow can be derived by integrating Eqn. (5) over the circular pipe geometry (Govier Aziz, 1972). Because of the yield stress, a central solid plug is formed where the point shear stress is less than the yield stress, as shown in Figure 10. 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

Fig. 10. Axial dispersion in laminar flow in pipes, dispersed plug flow model. Adapted from (B13).

Example 2-6 Consider the situation where the reactants at constant density are fed continuously into a pipe of length L instead of a tank of volume V as in the batch reactor. The reactants react as they flow down the tube with a speed u, and we assume that they flow as a plug without mixing or developing the laminar flow profile. Show that the conversion of the reactants is exactly the same in these very different reactor configurations. 

In laminar flow of Bingham-plastic types of materials the kinetic energy of the stream would be expected to vary from V2/2gc at very low flow rates (when the fluid over the entire cross section of the pipe moves as a solid plug) to V2/gc at high flow rates when the plug-flow zone is of negligible breadth and the velocity profile parabolic as for the flow of Newtonian fluids. McMillen (M5) has solved the problem for intermediate flow rates, and for practical purposes one may conclude... 

Once again Fit) can be calculated from Eq. 9.2-36 in conjunction with Eq. 9.2-28. Figure 9.11 plots the RTD function F(t) versus reduced time t/t and compares it to the RTD function of Newtonian laminar flow in a pipe and that in a well-stirred vessel. The RTD function in the melt extruder is quite narrow, approaching plug-type flow. Only about 5% of the flow rate stays more than twice the mean residence time in the extruder. 

To facilitate measurement and control of larger flow rates, the heat-transfer-type flowmeters can also be placed in a small bypass around a capillary element in the process pipe. The laminar flow element serves to ensure laminar flow and also acts as a restriction forcing a portion of the flow into the sensor tube. The small-size bypass serves to minimize the electric power requirement and to increase the speed of response, but it requires upstream filters to protect it against plugging. 

In addition to the CSTR and batch reactors, another type of reactor commonly used in industry is the tubular reactor. It consists of a cylindrical pipe and is normally operated at steady state, as is the CSTR. For the purposes of the material presented here, we consider systems in which the flow is highly turbulent and the flow field may be modeled by that of plug flow. That is, there is no radial variation in concentration and the reactor is referred to as a plug-flow reactor (PFR). (The laminar flow reactor is discussed in Chapter 13.)... 

Figure 23. Schematic of laminar and plug flow in a pipe. Radius of central plug is Rp.

As discussdd in Sec. 6.3, the velocity profile for laminar flow in a tube is parabolic. For turbulent flow it is much closer to plug flow, i.e., to a uniform velocity over the entire pipe cross section. Furthermore, as seen from Fig. 11.7, as the Reynolds number is increased, the velocity profile approaches closer and closer to plug flow. At the wall the turbulent eddies disappear so the shear stress at the wall for both laminar and turbulent flow of newtonian fluids is given byj dVJdy. Although it is ve difficult experimentally to... 

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. 

Performance problems related to maldistribution also exist in cooling applications, especially where viscosity increases as temperature is lowered. At worst, case equipment could become inoperable, due to plugging of all but the center of the flow channel. This condition can be eliminated by the use of static mixer internals, discussed later. Heat transfer coefficients for laminar flow in empty pipe are correlated by... 

This section of the report compares laminar and plug flow along pipes of internal diameter 12 mm and 18 mm. Simulations were performed on a 30 m length of pipe which consisted of a 10 m length of lead pipe between two equivalent lengths of non-lead pipes. A series of one litre samples were taken and the proportion of each sample that started in the lead section... 

COMPARISON OF PLUG AND LAMINAR FLOW ALONG A STRAIGHT PIPE... 

The value of this variable is set to zero at the outflow boundary and its values represent the time it will take fluid at any location to travel to the outflow boundary given the velocity field u. Multiplying this variable by 4-q/(jzD ), the flow rate divided by the pipe cross sectional area, produces a variable that can be used to indicate where each drawn litre starts from. The starting point of the water that contributes to the nth litre drawn from the pipes is identified by the positions where the value of P = AqContour plots of P for the four combinations of 12 mm or 18 mm diameter pipes and plug or laminar flow are contained in Figure 3.1 to 3.4. Each shade in those images represents a different litre. 

The results from the single pipe model under laminar flow conditions matched fairly well the results from actual surveys at two sites. At a third site, the match was better under plug flow conditions. It should be borne in mind that the flow conditions at an individual property will be somewhere in between the idealised plug and laminar conditions that have been reported. 

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