Distribution in a pipe

Fig. 2.11. Actual velocity distribution in a pipe, (a) Laminar flow (b) Turbulent flow...

It is possible to find more complex correlations for the velocity distribution in a pipe which do not have the limitations of Prandtl s power rule. In Fig. 11.7 the Reynolds number appears as a parameter in the velocity distribution plot. In trying to produce a universal velocity distribution rule, it seems logical to change the coordinates in Fig. 11.7 so that the Reynolds number enters either explicitly or implicitly in one of the coordinates, in the hope of getting all the data onto one curve. 

Before concluding this section, it is useful to link the apparent power-law index n and consistency coefficient m (equation 3.26) to the true power-law constants n and m, and to the Bingham plastic model constants rf and iib- This is accomphshed by noting that Xy, = (D/4)(—Ap/L) always gives the wall shear stress and the corresponding value of the wall shear rate Yw(= dV /dr) can be evaluated using the expressions for velocity distribution in a pipe presented in Sections 3.2.1 and 3.2.2. 

FIGURE 4-4 Simplified concept of particle distribution in a pipe as a function of volumetric concentration and speed. 

Figure 2-8. Velocity distributions in a pipe. (Adapted from references 1 and 9.)...

Flow distribution in a packed bed received attention after Schwartz and Smith (1953) published their paper on the subject. Their main conclusion was that the velocity profile for gases flowing through a packed bed is not flat, but has a maximum value approximately one pellet diameter from the pipe wall. This maximum velocity can be 100 % higher than the velocity at the center. To even out the velocity profile to less than 20 % deviation, more than 30 particles must fit across the pipe diameter. 

The concentration of gas ions significantly influences the particle-charging process. The high ion concentration is essential for the effective charging of fine particles. The distribution of ion concentration in a pipe-type electrostatic precipitator can be approximated by using the equations presented in the previous section. 

The velocity over the cross-section of a fluid flowing in a pipe is not uniform. Whilst this distribution in velocity over a diameter can be calculated for streamline flow this is not possible in the same basic manner for turbulent flow. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

To investigate a vertical distribution of a chemical, a sediment column is divided into sections with appropriate thickness. The sediment column taken in a pipe should be refrigerated in an ice-cooled container, transported to the laboratory, and removed carefully on to a clean tray so that there is as little disturbance as possible to the soil core structure. In the case of a column in which there is little soil moisture and it tends to collapse, the soil should be pushed out to each required thickness and carved off. It is also possible to take a sediment column up to a 30-cm depth using a pipe that is connected to cylinders (5-cm height) with sealing tape. In this case, the sample in each 5-cm fraction can be obtained as it is, after removing the tape. 

Let us see how to represent changes in properties for a system volume to property changes for a control volume. Select a control volume (CV) to be identical to volume V t) at time t, but to have a different velocity on its surface. Call this velocity, w. Hence, the volume will move to a different location from the system volume at a later time. For example, for fluid flow in a pipe, the control volume can be selected as stationary (w = 0) between locations 1 and 2 (shown in Figure 3.4, but the system moves to a new location later in time. Let us apply the Reynolds transport theorem, Equation (3.9), twice once to a system volume, V(t), and second to a control volume, CV, where CV and V are identical at time t. Since Equation (3.9) holds for any well-defined volume and surface velocity distribution, we can write for the system... 

Universal velocity distribution for turbulent flow in a pipe... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

For steady flow in a pipe or tube the kinetic energy term can be written as m2/(2 a) where u is the volumetric average velocity in the pipe or tube and a is a dimensionless correction factor which accounts for the velocity distribution across the pipe or tube. Fluids that are treated as compressible are almost always in turbulent flow and a is approximately 1 for turbulent flow. Thus for a compressible fluid flowing in a pipe or tube, equation 6.4 can be written as... 

Elastic Behavior. Stresses may be considered proportional to the total displacement strains in a piping system in which the strains are well-distributed and not excessive at any point (a balanced system). Layout of systems should aim for such a condition, which is assumed in flexibility analysis methods provided in this Code. 

Figure 11.13. Velocity and density distributions in vertical pipe flows at negligible gravity effect (from Soo, 1990) (a) NDF = 0.25 (b) NDF = 0.025.

Consider a dilute gas-solid flow in a pipe in which the solid particles carry significant electrostatic charges. It is assumed that (a) the flow is fully developed (b) the gravitational effect is negligible and (c) the flow and the electrostatic field are axisymmetric. Derive an expression to describe the radial volume fraction distribution of the particles and identify the radial locations where the particle volume fractions are maximum and minimum in the distribution. Also, if the electrostatic charge effects are negligible, derive an expression to describe the radial volume fraction distribution of the particles. 

Numerical calculations using MATHEMATICA software were made based on a theoretical model which assumes flow distribution in circular pipes under laminar conditions as described by the Bernoulli equation and applies an electrical circuit model based on Ohm s law [164],... 

Example 7.8 Residence Time Distribution Functions in Fully Developed Laminar Flow of a Newtonian Fluid in a Pipe The velocity distribution... 

A pipe is considered an arrangement to transfer fluids from one equipment to another and not a proper equipment by itself. Recently, much progress has been achieved in mixing or reaction in a pipe line, along with material. However, the position where the second substance should be fed for the establishment of expected mixedness over the shortest axial distance has not yet been clarified. Therefore, it is very important to clarify the spatial distribution of the concentration of tracer when it is injected into an arbitrary radial position. This is because if the spatial distribution of the concentration of the tracer is obtained, it becomes possible to calculate the mixedness at an arbitrary distance in the axial direction in other words, the change in mixedness with distance along the axial direction can be obtained. 

Schleicher, C.A. and Tribus, M., "Heat Transfer in a Pipe with Turbulent Row and Arbitrary Wall-Temperature Distribution , Trans. ASME, vol. 79, pp. 789-797, 1957. 

Therefore let us instead consider the more practical case of the tertiary current distribution. Based on the dependency of the Wagner number on polarization slope, we would predict that a pipe cathodically protected to a current density near its mass transport limited cathodic current density would have a more uniform current distribution than a pipe operating under charge transfer control. Of course the cathodic current density cannot exceed the mass transport limited value at any location on the pipe, as said in Chapter 4. Consider a tube that is cathodically protected at its entrance with a zinc anode in neutral seawater (4). Since the oxygen reduction reaction is mass transport limited, the Wagner number is large for the cathodically protected pipe (Fig. 12a), and a relatively uniform current distribution is predicted. However, if the solution conductivity is lowered, the current distribution will become less uniform. Finite element calculations and experimental confirmations (Fig. 12b) confirm the qualitative results of the Wagner number (4). 

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