PIPE FLOW PROBLEMS

There are three typical problems encountered in pipe flows, depending upon what is known and what is to be found. These are the unknown driving 

We note first that the Bernoulli equation can be written 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables  

---

We will illustrate the procedure for solving the three types of pipe flow problems for high-speed gas flows unknown driving force, unknown flow rate, and unknown diameter. 

In some problems it is advantageous to eliminate obvious dependent variables to reduce the number of equations that must be included as constraints. You can eliminate linear constraints via direct substitution, leaving only the nonlinear constraints, but the resulting equations may be too complex for this procedure to have merit. The following example illustrates a pipe flow problem in which substitution leads to one independent variable. 

Two common pipe flow problems are calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given the pressure drop. When flow rate is given, the Reynolds number may be calculated directly to determine the flow regime, so that the appropriate relations between/and Re (or pressure drop and flow rate or velocity) can be selected. When flow rate is specified and the flow is turbulent, Eq. (6-39) or (6-40), being explicit in/ may be preferable to Eq. (6-38), which is implicit in/and pressure drop. 

FIGURE 8 Hydraulic grade line (HGL) method design for pipe flow problem showing placement of pumping stations and change of diameter of pipe to handle excess head downstream of control point. 

Finally, we consider the type (ii) homogeneous boundary condition in physical terms. For the pipe flow problem, if we had stipulated that the tube wall was well insulated, then the heat flux at the wall is nil, so... 

Assume that a fluid of mass density p flows through a pipe of diameter d =2a (a is the radius) as shown in Fig. 5.6 (Hagen-Poiseulle flow). When the velocity field is one-dimensional, the differential equation governing the pipe flow problem along with the boundary conditions (BC) is given in cylindrical polar coordinates (r, z) as follows ... 

In problems of forced convection, it is usually the cooling mass flow that has to be found to determine the temperature difference between the cooling substance and the wall for a given heat flow. In turbulent pipe flow, the iol-low ing equation is valid ... 

Figure 2-31 is useful in sohing the usual steam or any vapor flow problem for turbulent flow based on the modified Darcy relation with fixed friction factors. At low vapor velocities the results may be low then use Figure 2-30. For steel pipe the limitations listed in (A) above apply.

The flow problems considered in previous chapters are concerned with homogeneous fluids, either single phases or suspensions of fine particles whose settling velocities are sufficiently low for the solids to be completely suspended in the fluid. Consideration is now given to the far more complex problem of the flow of multiphase systems in which the composition of the mixture may vary over the cross-section of the pipe or channel furthermore, the components may be moving at different velocities to give rise to the phenomenon of slip between the phases. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

Figure 9-2 provides a convenient way of solving compressible adiabatic flow problems for piping systems. Some iteration is normally required, because the value of K( depends on the Reynolds number, which cannot be determined until G is found. An example of the procedure for solving a typical problem follows. 

Simpler optimization problems exist in which the process models represent flow through a single pipe, flow in parallel pipes, compressors, heat exchangers, and so on. Other flow optimization problems occur in chemical reactors, for which various types of process models have been proposed for the flow behavior, including well-mixed tanks, tanks with dead space and bypassing, plug flow vessels, dispersion models, and so on. This subject is treated in Chapter 14. 

When analysing simple flow problems such as laminar flow in a pipe, where the form of the velocity profile and the directions in which the shear stresses act are already known, no formal sign convention for the stress components is required. In these cases, force balances can be written with the shear forces incorporated according to the directions in which the shear stresses physically act, as was done in Examples 1.7 and 1.8. However, in order to derive general equations for an arbitrary flow field it is necessary to adopt a formal sign convention for the stress components. 

The flow problems considered in Volume 1 are unidirectional, with the fluid flowing along a pipe or channel, and the effect of an obstruction is discussed only in so far as it causes an alteration in the forward velocity of the fluid. In this chapter, the force exerted on a body as a result of the flow of fluid past it is considered and, as the fluid is generally diverted all round it, the resulting three-dimensional flow is more complex. The flow of fluid relative to an infinitely long cylinder, a spherical particle and a non-spherical particle is considered, followed by a discussion of the motion of particles in both gravitational and centrifugal fields. 

A sketch of a pipe section is given in Figure E4.3.1, with the coordinate system and the vector notation. Typically, a length of 50 diameters from the inlet is required before a fully developed (long pipe) flow is assumed to be reached. The following can be assumed for this problem statement ... 

---