09pipesandpumps.ppt

Pipe Flow and Water Distribution Systems

CEE 311—Hydroscience

BackgroundWill now consider pipe systems

under pressure flowWater distribution systemsSewer interceptors

Problems can be solved using:Continuity equationSteady-state energy equation


Continuity EquationBetween two points in the system,

continuity equation states that flows are equal:

Q1 =A1V1 =Q2 =A2V2


Steady-State Energy EquationTotal head is the sum of elevation,

pressure and velocity headsBetween two points in the system, can

write the steady-state energy equation:

Z1 +P1

γ +α V12

2g =Z2 +P2γ +α V2

2

2g +hf +hm

Elevation Pressur

e Velocity

Friction lossesMinor

losses

(11.1)


Definition of TermsHydraulic grade line (HGL)

Line depicting elevation of pressure head + elevation head along the pipe

Energy grade line (EGL)Line depicting elevation of total

head along the pipeFor uniform pipe, V1=V2 thus EGL is

parallel to HGL


Hydraulic and Energy Grade Lines

Gupta, Fig. 11.1

L

Sf =hf

L


Energy Grade Line ConceptsThe EGL will have discontinuities at

fittings due to minor lossesPumps and turbines also add

discontinuitiesFor pump, add energy term to

LHSFor turbine, subtract energy term

from LHS


Computing Friction LossesIn some cases, can determine hf

from energy balanceCan compute hf directly from

equationsChezy equationDarcy-Weisbach equationHazen-Williams equation


Chezy EquationRecall the form of the Chezy equation:

Substituting S = hf/L and solving for hf:

V =C RS

hf =LV 2

RC2 =4LV 2

dC2

(10.11)


Darcy-Weisbach EquationIn 1845, Darcy and Weisbach found a

corresponding model for pipe flow:

hf =f L

dV2

2g

Where f = friction factor [-], L = pipe length [L], d = pipe diameter [L], V = velocity [LT-1], and g = gravity [LT-2]

(11.2)


Darcy-Weisbach EquationEquating this to the Chezy formula,

see the relationship between Chezy coefficient and the friction factor:

f =8g

C2 C = 8g

for


Darcy-Weisbach EquationThe friction factor depends on the flow

Different relationship for laminar and turbulent flow

Recall that Reynolds number (Re) can be used to define flow conditions:

Re=Vd

ν (11.3)


Darcy-Weisbach EquationFor laminar flow (Re < 2000),

friction factor is a simple function of Re (64/Re)

For turbulent flow (Re > 4000), friction factor is a function of Re and pipe roughness

Pipe roughness determined from equivalent sand roughness ()


Roughness terms for pipes

Gupta, Table 11.1


Darcy-Weisbach EquationBased on experiments in the 1930s

and 1940s, determined relationship between friction factor and Re

Expressed in graphical form as the Moody diagram


Moody Diagram

Gupta, Fig. 11.4

Relative roughness (/d)


Darcy-Weisbach EquationIn 1976, Jain derived an empirical

relationship for the family of curves presented in the Moody diagram:

1f

=−2log ε3.7d +5.72

Re0.9⎛ ⎝ ⎜

⎞ ⎠ ⎟ (11.8)


Minor LossesTo determine head losses due to pipe

fittings (bends, valves, transitions):

The loss coefficient (K) is a function of the type of fitting (Table 11.2)

hm =K V2

2g (11.12)


Minor Loss Coefficients

Gupta, Table 11.2


Darcy-Weisbach EquationThere are three main variables in

the Darcy-Weisbach equation: hf, d and V(or Q)

Thus there are three main classes of problems:Compute hf given d and VCompute V given d and hf

Compute d given hf and Q


Compute hf (given d, V)Determine Re from V, d and

(function of temperature)Determine f using either Moody

diagram or equation (laminar eqn or Jain formula)

Compute hf directly from D-W equation


Compute V (given d, hf)Since V and f are unknown, cannot

determine Re directlySolution is to use a trial-and-error

procedure:Assume a value for fCompute V from D-W eqnDetermine Re and hence fIterate until f converges


Compute d (given hf, Q)Since V and f are unknown, cannot

determine Re directlySolution is to use a trial-and-error

procedure:Assume a value for fCompute d from D-W eqn and

continuity (V=Q/A)Determine Re and hence fIterate until f converges


Hazen-Williams EquationAnother relationship for hf is

commonly used for pipe flow in water-supply systems:

V =1.318CR0.63S0.54

Where V is velocity in ft/s, C is a coefficient, R is hydraulic radius (d/4) in ft, and S is slope of EGL (hf/L)

(11.9)


Hazen-Williams EquationFor circular pipes, can substitute

V=Q/A, A=d2/4, and R=d/4 to get

Q =0.432Cd2.63S0.54

Where Q is flow in ft3/s (m3/s), d is diameter in ft (m), and S is slope of EGL (hf/L), dimensionless

(11.10a)

Q =0.278Cd2.63S0.54 (11.10b)

English

Metric


Pipes in SeriesConsider a compound pipeline,

with the pipes in series:

The general approach is to compute the equivalent length of a single diameter pipe


Pipes in SeriesFor pipes in series, know that:

Q1=Q2=…=Qn

hf=hf1+hf2+…+hf3

If we assume a value of Q, can compute individual losses

Setting this equal to loss for a single pipe (diameter d), can solve for equivalent length (Leq)


Pipes in ParallelConsider a compound pipeline,

with the pipes in parallel:


Pipes in ParallelFor pipes in parallel, know that:

hf=hf1=hf2=…=hf3

Q=Q1+Q2+…+Qn

If we assume a value of hf, can compute individual discharges (Qi) for each pipe

Setting total loss for single pipe (diameter d) equal to hf, can solve for equivalent length (Leq)


Pipe NetworksFlow through pipe networks has

multiple pathsSolution techniques based on

corrections to assumed flowsHardy-Cross methodLinear theoryNewton-Rhapson method


Example Pipe Network

Linsley, Fig. 11.7


Pipe NetworksFor a valid solution, two

conditions must hold:The algebraic sum of the

pressure drops around any closed loop must be zero

The flow entering a junction must equal the flow leaving it


Pipe NetworksRecall the Darcy-Weisbach and

Hazen-Williams formulas:

€

hf =16π2 f L

d5Q2

2g

€

hf = 4.727LC1.85d4.87Q1.85

Both formulas are of the general form hf=KQn, where K is equivalent resistance (see Table 11.5 in Gupta)


Hardy-Cross MethodMethod is based on successive

iterationsA flow is assumed in each pipe to

satisfy continuityA correction to each flow is

computed based on pressure drops around closed loops


Hardy-Cross MethodThe pressure drop condition can

be written as

€

hfloop∑ =0

From general resistance formula, this can be written as

€

K Qa +δ( )loop∑ n =0


Hardy-Cross MethodUsing a binomial expansion and

neglecting 2nd and higher order terms,

€

δ =− hf

loop∑

n hf Qaloop∑

(11.18)

Where hf is the head loss for the assumed Qa and n is the exponent in the resistance formula


Systems with PumpsTo accommodate pumps in the

system, the energy equation needs to be modified:

Z1 +P1

γ +α V12

2g +H p =Z2 +P2γ +α V2

2

2g +hf +hm

Hp is the head added to the system by the pump


System with a Pump

Gupta, Fig. 11.8

=Hp Z

Z1 +P1

γ +α V12

2g +H p =Z2 +P2γ +α V2

2

2g +hf +hm

=Z+hf+hm

H p = Z2 −Z1( )+

P2 −P1( )γ +α V2

2 −V12

2g⎛ ⎝ ⎜

⎞ ⎠ ⎟ +hf +hm

V1=V2

P1=P2


Pump Brake HorsepowerEfficiency of pump () is the ratio of

horsepower out (BHP) and power in, thus BHP can be computed as

BHP =γQHp

550η (11.14)

Where BHP is power (HP), is specific weight (lb/ft3), Q is flow (cfs), Hp is head added (ft), and is efficiency


PumpsThere are many different types of

pumpsClosed radial

Open radialMixed flowPropeller


PumpsFor design purposes, pumps are

selected based on performancePerformance parameters include:

Rotational speedDischarge capacityPumping headPower appliedEfficiency


Affinity LawsFor geometrically similar (homologous)

pumps, dimensional analysis produces

€

Q2Q1

=N2N1

D2D1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

3

€

H2H1

= N2N1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 D2D1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

€

P2P1

= N2N1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

3 D2D1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

5


Pump System TermsStatic suction lift: Vertical

distance from source water level to centerline of pump

Gupta, Fig. 11.17b


Pump System TermsStatic discharge lift: Vertical

distance from centerline of pump to water level at outlet

Gupta, Fig. 11.17b


Pump System TermsTotal static head: Sum of static

suction lift and static discharge lift

Gupta, Fig. 11.17b

=Z


Pump System TermsTotal dynamic head (Hp): Sum of

total static head and head losses

Using Darcy-Weisbach formula for friction losses, in terms of Q:

€

Hp =ΔZ+hloss=ΔZ+hf +hm

€

Hp =ΔZ+0.81g f LQ2

d5 + KQ2∑d4

⎛ ⎝ ⎜

⎞ ⎠ ⎟ (11.31)


Example System-Head Curve

Gupta, Fig. 11.18


Pump CharacteristicsSo far, we’ve assumed the efficiency

() of a pump is constantIn practice, for a given pump running

at a given speed, there are relationships among Q, Hp, and

The relationships are called the pump characteristics or performance curves


Pump CharacteristicsThe pump characteristic curves

are experimentally derivedGenerally shown as a function of Q

Pumping head (Hp)Brake horsepower (P)Efficiency ()


Example Pump Characteristic Curve

Gupta, Fig. 11.19


Pump SystemsIf we superimpose the system-

head curve with the pump-characteristic curve, the intersection will determine the operating point of the pump in the system

If efficiency is too low, then select another pump


Economics ReviewIn many cases, comparison of

different water-resources alternatives is difficult due to differing types of costs, benefits

Economic analysis offers a basis for comparison

There are 2 general bases for comparison: money and time


Economics ReviewThe time to be considered will be one of

the following:Economic life

Point where benefits < costsPhysical lifeAnalysis period

Planning horizon, generally less than economic or physical life

Since projects have different lives and cash flows, need to reduce to a common unit


Time Value of MoneyIn an inflation-free world, there is a

time-value associated with moneyGiven $1 today, I could invest the

money, earn i percent interest, and have $1(1+i) in a year

If we let P = present value, and F1 = future value in 1 year, see that

€

F1 =P 1+i( ) ⇒ P = F11+i( )


Time Value of MoneyThus $1 a year from today is worth

less than $1 in terms of today’s moneyIn general, for a future amount at the

end of N years, see that

€

FN =P 1+i( )N

€

P = FN1+i( )Nand


Time Value of MoneyUsually use the following notation:

€

FP i,N( )= 1+i( )N

€

PF i,N( )= 1

1+i( )N“Present worth of

a future sum”

“Future worth of a present sum”


Time Value of MoneyNow consider a series of payments

which result in $A each yearThis is termed an annual seriesFrom the formula for present worth of a

future sum, see that

€

P = A1+i( )1 + A

1+i( )2 +L + A1+i( )N


Time Value of MoneyIt can be shown that:

€

PA i,N( )= 1+i( )N −1

i 1+i( )N“Present worth of an annual series”

“Equivalent annual series of a

present sum”

€

AP i,N( )= i 1+i( )N

1+i( )N −1


Time Value of MoneyIt is easy to shown that:

€

FA i,N( )= 1+i( )N −1

i“Future worth of an annual series”

“Equivalent annual series of a

future sum”

€

AF i,N( )= i

1+i( )N −1


Economic AnalysisWith the discounting formulas, can

put projects on an equivalent basis in terms of money and time

In practice, there are three approaches:Present worth methodBenefit-cost ratio methodNet annual benefit method


Economic AnalysisConsider the choice between two projects:

Project A

Project B

Initial cost $50,000 $50,000Annual benefits

$12,000 $12,500

Project life 50 yrs 25 yrsDiscount rate 4% 4%Which project should be chosen?


Present Worth MethodThe idea is to compute the net present

worth (B-C) of each projectSelect the project with the largest net

present worthRules of analysis:

Bring benefits, costs back to the presentUse the same discount rateUse the same period of analysis


Benefit-Cost Ratio MethodThe idea is to move to the next alternative

if benefits increase more than costsRules of analysis:

Bring benefits, costs back to the presentUse the same discount rateUse the same period of analysisRank alternatives from least to greatest

costIf B/C >1, move to next alternative


Net Annual Benefit MethodThe idea is to chose the project

with the largest net annual benefit (B-C)

Rules of analysis:Compute annual benefits, costsUse the same discount rateUse the same period of analysis


Choosing a Discount Rate (i)The discount rate represents the

“cost” of moneyFor private industry, use the

interest rate corresponding to the least expensive source of capital

For the public sector, generally use the rate paid by the Treasury on securities with terms to maturity exceeding 15 yrs

09pipesandpumps.ppt

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