PART 7

Multiple-Pipe Systems

7.1. Head losses in a simple pipelineThe simple pipeline is a pipeline with constant diameter without

branches.The portion of simple pipeline is described in Fig. 7.1.

Lt’s write the Bernoulli equation for cross-sections 1 and 2, assumed that kinetic energy factors and reducing the velocity heads in left an right sides of it:

or

where – summary head losses along stream from section to section

.

The difference between the pressure heads in left side of an equation is the required head - . If this value is given, we will call it as available head -

. We can express the head loss as a function of discharge:

, (7.1)

where the coefficients and have different values depending on the flow mode.

For laminar flow using the portions of pipe with equivalent lengths instead of the local losses ( ) we will have:

Fig 7.1. Simple pipeline

and . (7.2)

For turbulent flow the values for and change, depending on whether the Darcy-Weisbach, Hazen-Williams equation is used.

Darcy-Weisbach equation.Let’s write the expression for head losses considering booth the friction

losses and local losses:

.

Then factor out the common factor and using the dependence

between velocity and discharge in round pipes : , we can rewrite

the exp. (7.2) as:

, (7.3)

where is the pipeline characteristic factor. The

coefficient in this case is equal 2.For the Hazen-Williams equation the exponent is and the coefficient is:

, (7.4)

where and CHW is the Hazen-Williams roughness

coefficient. Table (7.1) gives values for for some common pipe

materials.Pipe Material Pipe Material

PVC 150 Wood, Concrete 120Very Smooth 140 Clay, New Riveted Steel 110Cement-lined Ductile Iron 140 Old Cast Iron, Brick 100New Cast Iron, Welded Steel 130 Badly corroded Cast Iron 80

Table 7.1. Hazen-Williams Roughness

In summary, the best equation for computing the frictional head loss in a given pipe for a given discharge, or the best equation for the discharge if the head loss is known, regardless of the fluid, is the Darcy-Weisbach equation. The range of applicability for the empirical equations is much more restricted. Consequently, all engineers should consider using the Darcy-Weisbach equation in professional practice even if it is sometimes more difficult to use than the empirical equations.

7.2. Pipeline characteristicPipeline characteristic is the diagram of dependence of required head

from discharge . Where the value of is equal to head losses in pipeline. According to expression (7.1) we can introduce the head losses in

pipeline as: . The values of power index and coefficient are

different for the laminar and turbulent mode of flow and are defined according Exp. (7.2) and (7.3) correspondingly.

The example of this dependence for laminar and turbulent flow is described in Fig. 7.2.

Fig.7.2. Pipeline characteristic for laminar and turbulent flow

7.3. Three Types of Pipe-Flow Problems

The Moody chart or Murin chart can be used to solve almost any problem involving friction losses in long pipe flows. However, many such problems involve considerable iteration and repeated calculations using the chart because the standard Moody chart is essentially a head-loss chart. One is supposed to know all other variables, compute , enter the chart, find , and hence compute . This is one of three fundamental problems which are commonly encountered in pipe-flow calculations:1.Given , , and or , , and , compute the head loss (head-loss problem).2.Given , , , , , and , compute the velocity or flow rate Q (flow-rate problem).3.Given , , , , , and , compute the diameter of the pipe (sizing problem).

For the solution the first type of problem it’s necessary to calculate the Reynolds number and define the flow mode in pipeline or in its sections. Then, using the Moody chart or corresponding formula to define the value of .

After obtaining you should calculate the head loss according the follow expression:

(7.5).

The second type of problem we solve with help of iteration method.Let’s consider that flow is turbulent and the value of in first

approximation : . From exp. (7.5) we can obtain the value of and Reynolds number . Using the Moody chart taking into consideration

the given value of pipe relative roughness , we obtain the next

approximation of . Then we repeat this algorithm of calculations while the difference between the values of velocities in adjacent iterations will not be less then given value .

The algorithm of solution of the third type of problem will be follows:Let’s write the expression for head losses (7.3) and express the :

. Then designate as: and

, we can obtain the equation which gives the solution of

problem: .

It’s convenient to solve this equation graphically. For this purpose take into consideration some values of diameter and

calculate the friction losses coefficient for each one using the corresponding formulas depended on the flow mode. The required value of

we can obtain as intersection point of two curves :

and .

7.4 Multiple-Pipe SystemsIf you can solve the equations for one-pipe systems, you can solve

them all; but when systems contain two or more pipes, certain basic rules make the calculations very smooth. Any resemblance between these rules and the rules for handling electric circuits is not coincidental.

Fig 7.3. Pipes in series

Fig. (7.1)-(7.3) shows three examples of multiple-pipe systems. The first is a set of three (or more) pipes in series. Rule 1 is that the flow rate is the same in all pipes

or

(7.6)

Rule 2 is that the total head loss through the system equals the sum of the head loss in each pipe

(7.7)In terms of the friction and minor losses in each pipe, we could rewrite

this as

(7.7)

and so on for any number of pipes in the series. Since and are proportional to from Eq. (7.5), Eq. (7.7) is of the form

(7.8)

where the a, are dimensionless constants. If the flow rate is given, we can evaluate the right-hand side and hence the total head loss. If the head loss is given, a little iteration is needed, since , , and all depend upon through the Reynolds number. Begin by calculating , , and , assuming fully rough flow, and the solution for will converge with one or two iterations. EES is ideal for this purpose.

Fig 7.4. Pipes in parallel;

Fig 7.5. The three-reservoir junction problem

EXAMPLE 1

Given is a three-pipe series system, as in Fig. (7.3). The total pressure drop is , and the elevation drop is . The pipe data are

Pipe1 100 8 0.24 0.002 150 6 0.12 0.003 80 4 0.20 0.00

5The fluid is water, and . Calculate the flow rate Q in m3/h through the system.

Solution

The total head loss across the system is

From the continuity relation (6.105) the velocities are

,

and ,

Neglecting minor losses and substituting into Eq. (7.7), we obtain

or

(*)

This is the form which was hinted at in Eq. (7.8). It seems to be dominated by the third pipe loss . Begin by estimating , , and from the Moody-chart fully rough regime: , ,

.

Substitution into Eq. (*) to find . The first

estimate thus is , from which

, Hence, from the Moody chart,

, , Substitution into Eq. (*) gives the better estimate

, or

A second iteration gives , a negligible change.

The second multiple-pipe system is the parallel-flow case shown in Fig. (7.4). Here the loss is the same in each pipe, and the total flow is the sum of the individual flows

(7.9a) (7.9b)

If the total head loss is known, it is straightforward to solve for in each pipe and sum them, as will be seen in Example 2. The reverse problem,

of determining when is known, requires iteration. Each pipe is related

to by the Moody relation

, where .

Thus each pipe has nearly quadratic nonlinear parallel resistance, and head loss is related to total flow rate by

, where (7.9c)

Since the vary with Reynolds number and roughness ratio, one begins Eq. (7.9c) by guessing values of (fully rough values are

recommended) and calculating a first estimate of . Then each pipe yields a

flow-rate estimate and hence a new Reynolds number and a

better estimate of . Then repeat Eq. (7.9c) to convergence.It should be noted that both of these parallel-pipe cases—finding either

or are easily solved by EES if reasonable initial guesses are given.

EXAMPLE 2

Assume that the same three pipes in Example 1 are now in parallel with the same total head loss of 20.3 m. (Fig. 7.4). Compute the total flow rate , neglecting minor losses.

SolutionFrom Eq. (7.9a) we can solve for each separately

(**)

Guess fully rough flow in pipe , hence . From the Moody chart read ; recompute

, .]

Next guess for pipe 2: , ; then ,

and hence , , .

Finally guess for pipe 3: , ; then ,

and hence , ,

This is satisfactory convergence. The total flow rate is

Ans.

These three pipes carry 10 times more flow in parallel than they do in series.

From the given equations and rules for calculation series and parallel pipelines (7.6), (7.7), (7.9a), (7.9b) it is possible to make follows conclusion:

- at the series connection the discharge is constant in any cross-section and head loss of the system is the sum of head losses of elements;

- at the parallel connection the resulting discharge is the sum а elements discharges and the head losses of all elements are equal to each other and in turn equal to head loss of all system. (Fig.7.6).

Fig.7.6. Pipeline characteristic for series and parallel pipelines

Consider the third example of a three-reservoir pipe junction, as in Fig. (7.5). If all flows are considered positive toward the junction, then

(7.10)which obviously implies that one or two of the flows must be away from the junction. The pressure must change through each pipe so as to give the same static pressure at the junction. In other words, let the HGL at the junction have the elevation

where is in gage pressure for simplicity. Then the head loss through each,

assuming (gage) at each reservoir surface, must be such that

(7.11)

We guess the position and solve Eqs. (7.11) for , , and and

hence , , and , iterating until the flow rates balance at the junction

according to Eq. (7.10). If we guess too high, the sum will be

negative and the remedy is to reduce , and vice versa.

EXAMPLE 3 Take the same three pipes as in Example 1, and assume that they

connect three reservoirs at these surface elevations, ,

Find the resulting flow rates in each pipe, neglecting minor losses.

SolutionAs a first guess, take equal to the middle reservoir height,

. This saves one calculation ( ) and enables us to get the lay of the land:

Reservoir 1 40 -20 0.0267 -3.43 -62.1 1250

2 40 60 0.0241 4.42 45.0 25003 40 0 0 0 2000

Since the sum of the flow rates toward the junction is negative, we guessed too high. Reduce to 30 m and repeat:

Reservoir1 30 -10 0.0269 -2.42 -43.72 30 70 0.0241 4.78 48.63 30 10 0.0317 1.76 8.0

This is positive , and so we can linearly interpolate to get an

accurate guess: . Make one final list:

Reservoir 1 34.3 -14.3 0.0268 -2.90 -52.4

2 34.3 65.7 0.0241 4.63 47.1

3 34.3 5.7 0.0321 1.32 6.0

This is close enough; hence we calculate that the flow rate is

toward reservoir 3, balanced by away from reservoir 1 and

away from reservoir 3.

One further iteration with this problem would give , resulting

in , , and , so that to three-place

accuracy. Pedagogically speaking, we would then be exhausted.

7.5. Multiple pipe system

The ultimate case of a multipipe system is the piping network illustrated in Fig. 6.25. This might represent a water supply system for an apartment or subdivision or even a city. This network is quit e complex algebraically but follows the same basic rules:

Fig.7.7. Multiple pipe system

1. The net flow into any junction must be zero.2. The net head loss around any closed loop must be zero. In other words,

the HGL at each junction must have one and only one elevation.3. 3.All head losses must satisfy the Moody and minor-loss friction

correlations.By supplying these rules to each junction and independent loop in the

network, one obtains a set of simultaneous equations for the flow rates in each pipe leg and the HGL (or pressure) at each junction.