Technical Design Portfolio
Whole System Design Suite
Taking a whole system approach to achieving
sustainable design outcomes
Worked Example 1 – Appendix 6A
July 2007
This Course was developed under a grant from the Australian
Government Department of the Environment and Water Resources as
part of the 2005/06 Environmental Education Grants Program.
(The views expressed herein are not necessarily the views of the
Commonwealth, and the Commonwealth does not accept the
responsibility for any information or advice contained herein.)
TDP
ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Appendix 6A
Calculating the total energy balance
Symbol nomenclature
Symbol Description
Unit
A
Pipe cross sectional area
m2
D
Pipe diameter
m
f
Friction factor
g
Acceleration due to gravity 9.81 m/s2
h
Head loss
KL
Loss coefficient
L
Pipe length
m
p
Pressure
Pa
P
Power
W
Q
Volumetric flow rate
m3/s
R
Universal gas constant
8.314 kJ/kmolK
Re
Reynolds number
T
Temperature
K
V
Average velocity
m/s
z
Height
m
α
Kinetic energy coefficient
γ
Specific weight
kN/m3
ε
Equivalent roughness
mm
μ
Dynamic viscosity
Ns/m2
ν
Kinematic viscosity
m2/s
ρ
Density
kg/m3
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m
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Calculating the total energy balance
There are 4 kinds of energy changes associated with fluid flow through a pipe and pump
system:
1. Pressure, kinetic energy and potential energy changes
2. Friction losses
3. Component losses
4. Pumping gains
1. Energy balance of a steady, inviscid (zero viscosity), incompressible flow in a frictionless
pipe system (an ideal system) is governed by the Bernoulli equation, which indicates that the
sum of the pressure, kinetic energy, and potential energy changes is constant along a
streamline. The equation is given in terms of heads:
p/ρg + αV2/2g + z = constant along a streamline
For uniform velocity profiles α = 1. For non-uniform velocity profiles α > 1.
2. Friction head loss for a fully developed, steady, incompressible flow in a single pipe is given
by the Darcy-Weisbach equation:
hF = f (L/D)(V2/2g)
The total friction head loss through all pipes in a pipe system is the sum of the individual friction
losses.
Calculating the friction factor, f, depends on the type of flow. The Reynolds number is used to
distinguish between laminar and turbulent flow:
Re = ρVD/μ
Reynolds number Type of fluid flow
Re < 2100
Laminar
2100 < Re < 4000
Transitional
Re > 4000
Turbulent
If the flow is laminar the friction factor is given by:
f = 64/Re
If the flow is turbulent the friction factor is a function of Re and the ratio ε/D, where ε is the
equivalent roughness. Table 6A.1 gives values of ε for various types of pipe.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Table 6A.1: Equivalent Roughness for New Pipes
Pipe
Equivalent roughness, ε
Feet
Millimetres
Rivited steel
0.003 – 0.03
0.9 - 9.0
Concrete
0.001 – 0.01
0.3 – 3.0
Wood stave
0.0006 – 0.003 0.18 – 0.9
Cast iron
0.00085
0.26
Galvanised iron
0.0005
0.15
Commercial steel or wrought iron 0.00015
0.045
Drawn tubing
0.000005
0.0015
Plastic, glass
0.0 (smooth)
0.0 (smooth)
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998), p4921
1
Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) Fundamental of Fluid Mechanics, 3rd edn, Wiley & Sons, New York, p492.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Figure 6A.1. The Moody Chart: Friction Factor as a function of Reynolds number and relative
roughness for round pipes – the Moody Chart shows the relationship between f, Re and ε/D
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p4932
2
Ibid, p493.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Alternatively, the Colebrook formula is valid for the non-laminar range of the Moody chart:
1/√f = –2log10 [(ε/D)/3.7 + 2.51/(Re√f)]
3. Component head losses refer to losses associated with flow though components such as
pipe contractions, expansions, bends, joins and valves. The component head loss through a
single component is given by:
hC = KLV2/2g
The total head loss through all components in the system is the sum of the individual component
losses. The loss coefficient, KL, depends on the type of component.
At a pipe contraction - where the upstream pipe cross sectional area, A1, is larger than the
downstream pipe cross sectional area, A2 - KL is given by Figure 6A.2 for a rounded inlet edge
and Figure 6A.3 for a sudden contraction.
Figure 6A.2. Entrance loss coefficient as a function of rounding the inlet edge
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p4993
Figure 6A.3: Loss coefficient for a sudden contraction
3
Ibid, p499.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p5004
Figure 6A.4 gives KL for various entrance conditions, where A1 is assumed to be infinite.
Figure 6A.4. Entrance flow conditions and loss coefficient. (a) Re-entrant, KL = 0.8, (b) sharpedged, KL = 0.5, (c) slightly rounded, KL = 0.2, (d) well-rounded, KL = 0.04.
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p4985
At a sudden pipe expansion, where the upstream pipe cross sectional area, A1, is smaller than
the downstream pipe cross sectional area, A2, KL is given by Figure 6A5.
Figure 6A.5: Loss coefficient for a sudden expansion
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p5006
Alternatively, KL for a sudden expansion can be calculated using:
KL = (1- A1/A2) 2
Figure 6A.6 gives KL for various exit conditions, where A2 is assumed to be infinite.
4
Ibid, p500.
Ibid, p498.
6
Ibid, p500.
5
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Figure 6A.6: Exit flow conditions and loss coefficient. (a) Re-entrant, KL = 1.0, (b) sharp-edged,
KL = 1.0, (c) slightly rounded, KL = 1.0 (d) well-rounded, KL = 1.0.
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p4997
Table 6A.2 gives KL for flows through other types of components.
7
Ibid, p499.
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Worked Example 1 – Appendix 6A
Table 6A.2: Loss Coefficients for Pipe Components
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p5058
8
Ibid, p505.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
4. Pumping gains refer to energy from a pump. The pumping head gain for a single pump of
power P pumping a fluid at average velocity V through a pipe of cross sectional area A is:
hP = P/ρgAV
The total pumping gain over all pumps in the system is the sum of the individual gains.
Total energy balance in terms of heads for a pipes and pumps system is given by combining
the 4 sources of energy changes between point 1 and point 2 on a streamline:
p1/ρg + α1V12/2g + z1 + Σ hPi = p2/ρg + α2V22/2g + z2 + Σ hFi + Σ hCi
or
p1/ρg + α1V12/2g + z1 + Σ Pi/ρgAiVi
= p2/ρg + α2V22/2g + z2 + Σ fi (Li/Di)(Vi2/2g) + Σ KLiVi2/2g
Other useful equations
γ = ρg
ν = μ/ρ
p = ρgh
Q = AV
For a circular pipe:
A = ΠD2/4
Comparing pipe 1 with diameter D1 and pipe 2 with diameter D2:
hF1/hF2 = (D2/D1)5
For a perfect gas:
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p = ρRT
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Useful resources
Table 6A.3: Physical Properties of Water (SI Units).
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p. 8539
9
Ibid, p853.
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Worked Example 1 – Appendix 6A
Table 6A.4: Physical Properties of Air at Standard Atmospheric Pressure (SI Units)
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998) p. 85510
10
Ibid, p855.
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Worked Example 1 – Appendix 6A
Table 6A.5: Conversion Factors from BG and EE Units to SI Units
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998)11
11
Ibid.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
Table 6A.6: Conversion Factors from SI Units to BG and EE Units
Source: Munson, B.R., Young, D.F. and Okiishi, T.H. (1998)12
12
Ibid.
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ESSP Technical Design Portfolio: Whole System Design Suite
Worked Example 1 – Appendix 6A
References
Munson, B., Young, D. and Okiishi, T. (1998) Fundamental of Fluid Mechanics, 3rd edn, Wiley &
Sons, New York.
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