Frictional Flow in Conduits

Previously it was noted that the Bernoulli Equation with frictional heating was the starting point for many engineering calculations relating to analysis and design. However, in order to use this equation we must be able to satisfactorily handle Fh, frictional heating. 

It will be our purpose in this chapter to develop those techniques that are needed to determine F/,. We will consider both laminar and turbulent flows, conduits with circular and noncircular cross sections, complex piping situations, expansions, and contractions. The overall result will be the ability to handle effectively many of the situations that can and do confront the engineering practitioner. 

For a fluid in laminar flow in a horizontal circular conduit we have 

Now if the Bernoulli Equation is applied to a length of the tube, L, then we obtain 

No z No kinetic No shaft change energy change work 

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In the preceding section, we considered ways and means of describing isothermal flow in conduits. We next direct our attention to the methodologies used to calculate pressure drops or volumetric flow rates. Three approaches will be presented calculation from flow curves, use of friction factors, and computation using a flow equation. 

Culter, J.D. and McClaflin, G.G. "Method of Friction Loss Reduction in Oleaginous Fluids Flowing Through Conduits," US Patent 3,692,676(1972). 

Example 5-6 Friction Loss in a Sudden Expansion. Figure 5-7 shows the flow in a sudden expansion from a small conduit to a larger one. We assume that the conditions upstream of the expansion (point 1) are known, as well as the areas A and A2. We desire to find the velocity and pressure downstream of the expansion (V2 and P2) and the loss coefficient, Kt. As before, V2 is determined from the mass balance (continuity equation) applied to the system (the fluid in the shaded area). Assuming constant density,... 

All the relationships presented in Chapter 6 apply directly to circular pipe. However, many of these results can also, with appropriate modification, be applied to conduits with noncircular cross sections. It should be recalled that the derivation of the momentum equation for uniform flow in a tube [e.g., Eq. (5-44)] involved no assumption about the shape of the tube cross section. The result is that the friction loss is a function of a geometric parameter called the hydraulic diameter ... 

The pitot tube is a relatively complex device and requires considerable effort and time to obtain an adequate number of velocity data points and to integrate these over the cross section to determine the total flow rate. On the other hand the probe offers minimal resistance to the flow and hence is very efficient from the standpoint that it results in negligible friction loss in the conduit. It is also the only practical means for determining the flow rate in very large conduits such as smokestacks. There are standardized methods for applying this method to determine the total amount of material emitted through a stack, for example. 

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

The presence of small amounts of certain polymers can produce spectacular reduction in the frictional losses of fluids in turbulent flow through conduits. Drag reduction has an immense field of applications, both currently and potentially. The list of exploitable situations as described in Sect. 2 could be extended, but a big snag exists drag reduction decreases with flow time. This is believed to be due to mechanical degradation of added polymer (Brostow 1983). In Fig. 32 and Fig. 33 the influence of Mw on drag reduction is displayed. 

A general equation for frictional resistance in a pressure conduit was developed. The same reasoning may now be applied to uniform flow with a free surface. Consider a sloped channel of water flowing over a constant slope of angle 0, which shows the short reach of length L between stations 1 and 2 of a channel in uniform flow with area A of the water section. As the flow is neither accelerating nor decelerating,... 

Expressions for evaluating frictional losses in the flow of fluids through conduits... 

For turbulent flow in a conduit of noncircular cross section, an equivalent diameter can be substituted for the circular-section diameter, and the equations for circular pipes can then be applied without introducing a large error. This equivalent diameter is defined as four times the hydraulic radius RH, where the hydraulic radius is the ratio of the cross-sectional flow area to the wetted perimeter. When the flow is viscous, substitution of 4RH for D does not give accurate results, and exact expressions relating frictional pressure drop and velocity can be obtained only for certain conduit shapes. 

For flow in cylindrical conduits, the viscous work per unit mass of the fluid is expressed in terms of a friction factor and a specific kinetic energy by... 

The vortices formed between the wall and the separated fluid stream, beyond the separation point, cause excessive form-friction losses. Separation occurs in both laminar and turbulent flow. In turbulent flow, the separation point is farther along the conduit than in laminar flow. Separation can be prevented if the angle between the wall of the conduit and the axis is made small. The maximum angle that can be tolerated in a conical expander without separation is 7°. 

Adiabatic frictional flow through a conduit of constant cross section. This process is irreversible, and the entropy of the gas increases, but as shown by Eq. (6.22), since Q 0, the stagnation temperature is constant throughout the conduit. This process is shown in Fig. 6.1b. 

In adiabatic frictional flow, the temperature of the gas changes. The viscosity also varies, and the Reynolds number and friction factor are not actually constant. In gas flow, however, the effect of temperature on viscosity is small, and the effect of Reynolds number on the friction factor / is still less. Also, unless the Mach number is nearly unity, the temperature change is small. It is satisfactory to use an average value for /as a constant in calculations. If necessary,/ can be evaluated at the two ends of the conduit and an arithmetic average used as a constant. 

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow... 

Turbulent flow in many kinds of noncircular conduits can be estimated by substituting 4j times the hydraulic radius for the diameter in the friction factor plot and the friction factor equation. 

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