Fluid mechanics problem about selecting pump

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Southern Methodist University

Bobby B. Lyle School of Engineering

CEE 2342/ME 2342 Fluid Mechanics

Roger O. Dickey, Ph.D., P.E.

V. STEADY PIPE FLOW

Multiple Pipe Systems – Series, Parallel, and Branching Pipes and Pipe Networks

Reading Assignment:

Chapter 8 Viscous Flow in Pipes

Section 8.5.2 – Multiple Pipe Systems, pp. 456-460

C. Multiple Pipe Systems

Pipes in Series –

Consider pipes in series as illustrated in Figure 8.34 (a), p. 456 where every fluid particle that passes through the CV, entering at Section A and exiting at Section B, passes through each of the pipes in sequence:

Figure 8.34 (a) Series Pipe System, p. 456 – Modified

In this scenario Q is the same in each pipe, but V varies from one pipe to the next because pipe sizes differ. Total head loss, hLA-B , from Section A to Section B is simply the sum of the head losses through each of the pipes of differing size.

Governing equations for any arbitrary number of pipes in series, numbered 1 through n, located between Sections A and B are:

The head loss for any given pipe size i, hLi , is comprised of both the pipe friction loss and minor losses. Thus, the total head loss for a series piping system between any two Sections A and B, hLA-B , is obtained by summing both the pipe friction losses and the minor losses for all n pipe sizes:

where,

hLA-B = total head loss across the n pipe sizes between A and B [L]

hfi = friction loss for pipe size i [L]

sum of the energy losses for all individual minor loss components j, for pipe size i [L]

Expanding the summation over all pipe sizes:

Minor Losses

Pipe Size 1

Friction Loss

Pipe Size 1

Minor Losses

Pipe Size 2

Friction Loss

Pipe Size 2

Friction factors, fi , will generally differ for the various pipes because the Reynolds number, Rei , and relative roughness, εi /Di , tend to vary from one pipe size to the next.

Series pipe problems of Types I, II, and III are solved in exactly the same manner as the corresponding type of simple pipe problem, but with multiple fi and Vi values.

Refer to handout V.C.1. Series Pipes Example for piping systems having multiple pipes in series.

Pipes in Parallel –

Consider parallel pipes as illustrated in Figure 8.34 (b), p. 456 where fluid particles passing through the CV, beginning at Section A along the free surface of the left tank and ending at Section B along the free surface of the right tank, may take any of the available parallel paths, with the total flow rate equaling the sum of the flow rates through the individual pipes:

Figure 8.34 (b) Parallel Pipe System, p. 456 – Modified

In this scenario Qi may differ in each pipe i, but the head loss across each pipe is the same, hLA-B , as can be seen by writing the energy equation along a path through any pipe between Sections A and B.

Governing equations for any arbitrary number of pipes in parallel, numbered 1 through n, located between Sections A and B are:

Head loss across a given pipe i, hLi , is comprised of both the pipe friction loss and minor losses and it must equal the total head loss across the overall parallel piping system between the two Sections A and B, hLA-B :

where,

hLA-B = total head loss across all parallel pipes between Sections A and B [L]

hfi = friction loss for pipe i [L]

sum of the energy losses for all individual minor loss components j, for pipe i [L]

Refer to handout V.C.2. Parallel Pipes Example for piping systems having multiple pipes in parallel.

Branching Pipes and Pipe Networks –

Multiple pipe systems may also involve branching pipes as illustrated in Figure 8.35, p. 457:

In this scenario, continuity at Node N requires that:

Application of the energy equation reveals that the head loss across parallel Pipes (2) and (3) between Node N and Section B are equal, although the pipe sizes and flow rates may differ. Similarly, the head losses across series Pipes (1) and (2) equals the head loss across series Pipes (1) and (3) because these two sets of series pipes operate in parallel.

Branching pipe systems may be quite complex like the common 3-reservoir problem, illustrated in Figure 8.36, p. 458, where the direction of flow in Pipe (2) may not be known a priori:

In other words, water flowing out of the highest Reservoir A may flow into both of the lower Reservoirs B and C. However, it is entirely possible that water flows out of both higher Reservoirs A and B into the lowest Reservoir C.

Refer to handout V.C.3. Three Reservoir Problem Example.

Branching pipe systems may also involve loops forming complex distribution networks as illustrated in Figure 8.37, p. 460:

Pipe network problems are solved by using node and loop equations analogous to those used for electrical circuits—specific variable analogs are pressure-voltage, flow rate-current, and pipe friction-resistance. Net flow rate into a node must be zero (Continuity Principle), and the net pressure difference must be zero when following a path around a given loop that returns to the starting point.

Combining these concepts with head loss equations allows determination of flows and pressures throughout the network.

Computer models are usually used to solve the resulting set of simultaneous equations to determine the direction and magnitude of the flow rate through each pipe, and the pressure at each node in the network.

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