Frictional Losses in One-Dimensiuonal Flow
Frictional Losses in One-Dimensional Flow
1. Conservation of Energy – Steady Flow Energy Equation
The Bernoulli equation was found useful in relatively simple applications in which: the flow
is steady, the fluid is incompressible and frictionless, there is no heat/work transfer to or from
the fluid, and the flow process is thermodynamically reversible.
It is obvious that these conditions are quite limiting since most, if not all, practical flows
include friction, energy or work transfer. Because work and energy are interchangeable, and
because friction acts to degrade mechanical energy into heat (thermal energy), such effects can
be accounted for in terms of energy conservation.
The 1st law of Thermodynamics expresses the energy conservation principle and states that
the energy, E, of a system can be increased or decreased only by work, W, or heat, Q,
interactions, otherwise it remains constant, although conversions from one kind of energy to
another can occur within the system:
Q  W  E
or in specific terms, i.e., per unit mass:
Q W E



 q  w  e
m m
m
where small letters are used for specific terms.
If the 1st law of thermodynamics is to be applied to an open system in which there is a
continuous flow of fluid mass, then one has to recognise that the term E, representing the total
energy content of the fluid, can be made up of different terms. Each term indicates a different
form of energy. The four forms of energy that are of primary interest in open systems are
discussed below.
The Different Forms of Energy a Fluid can possess:
- Kinetic Energy
- Potential Energy
- Internal Energy
- Flow or Displacement Energy
a) Kinetic Energy:
This is given by:
1 2
V (per unit mass of the fluid).
2
Kinetic energy is the energy, which a fluid possesses by virtue of its speed. Since viscous
friction acts to retard the flow, especially near solid surfaces, the kinetic energy term must
1
Frictional Losses in One-Dimensiuonal Flow
account for this by using a suitably averaged value of the velocity V across the cross-sectional
flow area.
b) Potential Energy: This is given by: gz (per unit mass of the fluid).
Potential energy is the energy, which a fluid possesses by virtue of its vertical position with
respect to an arbitrary datum. The choice of the datum is irrelevant because only changes in
potential energy are significant.
c) Internal Energy:
This is usually denoted by u (per unit mass of the fluid).
It is the amount of energy stored within a fluid by virtue of its temperature and pressure. In the
general case:
u  f ( p, T )
whilst for a perfect gas:
u  cv T
where cv is the specific heat capacity at constant volume.
d) Flow or Displacement Energy:
This is the amount of energy required to push a fluid across the boundary of a system.
Consider the flow system shown in Fig. 1 where a packet of fluid of length l and area A1 is
just about to cross the system boundary at section 1.
If the pressure at section 1 is p1, then the force required to push the fluid packet across A1 is:
F  p1 A1
2
A2
p2
V2
1
l
A1
p1
V1
Figure 1
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Frictional Losses in One-Dimensiuonal Flow
Hence, by the time the fluid packet is fully imbedded in the system, just downstream of
section 1, the amount of work expended by the fluid pushing it in will be:
W  F  Distance  Fl  p1 A1l  p1  Volume
This energy must have come from the flowing fluid, and if it is calculated per unit mass it
becomes:
Displacement Energy 
W
Volume
 p1 
m
m
DisplacementEnergy 


At section 2, a displacement energy equivalent to
p2

p1

is required to push a fluid packet out of
the system because the pressure there is p2.
Assembling the various energy terms back into the 1st law of thermodynamics, and ensuring
all quantities are specific (i.e., taken per unit mass):
q12  w12
 V22 V12 
p 
p
  gz 2  gz1   u 2  u1    2  1 
 

2 

 
 2
The above equation is the Steady Flow Energy Equation (S.F.E.E.); a representation of the
1st Law of Thermodynamics suitable for handling an open flow system. Note that the sum

p
u    h , is the specific enthalpy.


2. The Effects of Friction
The S.F.E.E., unlike Bernoulli’s, applies to both reversible and irreversible processes. It may
therefore be used in calculations where viscous friction is a significant factor.
Friction always acts to convert mechanical energy into thermal energy (the energy required to
overcome friction is transformed into thermal energy). The temperature of the fluid rises
above that of frictionless flow and, in general, the heat transferred from the fluid to its
surroundings per unit mass is increased. The increase of temperature and consequently of heat
transfer is generally of no worth and thus corresponds to a loss of energy from the system.
Hence, the S.F.E.E. may be rearranged with the heat transfer and internal energy terms lumped
together, to give a single term which accounts for frictional dissipation:
w12
 p 2 V22
  p1 V12

 
 gz 2    
 gz1   L
2
2

 

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Frictional Losses in One-Dimensiuonal Flow
Here, L  u2  u1   q12 is the loss term, which describes the amount of mechanical energy
that is converted into thermal energy by friction. The velocity terms represent mean velocities
over the corresponding sections.
Often when performing calculations that involve friction, it is most convenient to work in
terms of heads. Dividing the S.F.E.E. throughout by g gives:
  p V2

w12  p 2 V22
 
 z 2    1  1  z1   hL
g
 g 2 g
  g 2 g

In the above:
 p V2


 z2   H

 g 2 g

hL 
L

g
u
2
 u1   q12 
g
is the Total Head
is the Head Loss
The S.F.E.E. can therefore be written in a more concise form:
w12
 H 2  H 1  hL
g
In either form, it is a mathematical statement of how the mechanical energy or head within the
flow are affected by work and friction as the fluid moves from a point 1 to a point 2. If there is
no work done by or on the fluid and there is no variation in the pipe cross sectional area or
elevations, then the frictional losses will result in the reduction of the fluid static pressure:
hL 
1
P1  P2 
g
3. Major Causes of Head Loss
1. Friction between fluid and solid surfaces, degrading mechanical energy into thermal energy
(heat).
2. Sudden changes of section causing friction between fluid particles.
3. Junctions at inlet or exit to pipes/reservoirs.
4. Pipe fittings (e.g., valves, taps, couplings, etc.) where the sudden changes in flow direction
cause large velocity gradients, and hence, friction.
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Frictional Losses in One-Dimensiuonal Flow
4. Head Loss due to Friction in a Pipe
4.1. Darcy’s Equation
The dissipation of energy by fluid friction results in a drop of the static pressure head in the
direction of flow. In the nineteenth century, many experiments were conducted to study the
head losses in pipes. The results suggested that the head losses are directly proportional to the
pipe length, l, inversely proportional to its diameter, d, (in the case of circular pipes), and
proportional to the so-called friction factor. Hence:
hL, pipe  4 f
l V2
d 2g
The equation above is known as Darcy’s Equation.
The friction coefficient f depends on the surface finish of the pipe interior wall: the rougher
the pipe, the greater the friction factors and the greater the loss. It also depends on the nature
of the fluid flow as described by a dimensionless quantity called Reynolds number (Re).
4.2. Reynolds Experiments – Reynolds Number
Reynolds carried out a series of experiments, which established the existence of two distinct
types of flow. The experiments involved injecting a dye into the flow under different
conditions and observing what happened to it. The dye either travelled in the fluid in orderly
straight lines or became totally mixed. The first type of flow is called laminar flow, Fig. 2, and
the second turbulent flow, Fig. 3.
Figure 2. Laminar flow
Figure 3. Turbulent flow
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Frictional Losses in One-Dimensiuonal Flow
The Reynolds number represents the ratio of the inertia forces to the viscous forces within the
flow. Low values of Re indicate that viscosity effects are dominant and hence laminar flow is
obtained. High Re values indicate that the effects of inertia forces are dominant and hence
turbulent flow occurs.
For flow in circular pipes or ducts, the Reynolds number is given by:
Re 
Vd Vd



where ρ, μ and ν are the fluid density, dynamic viscosity and kinematic viscosity, respectively,
V the velocity and d the pipe diameter.
The critical Reynolds number is the value when the flow changes from laminar to turbulent. It
is usually expressed in a range because the change does not happen at a fixed point. There is a
transition from one type of flow to the other. For flow in pipes or ducts:
The flow is laminar for Re < 2000.
The flow is turbulent for Re > approx. 3500 - 4000.
Transition region: 2000 < Re < approx. 3500 - 4000.
4.3. Friction Factor
For laminar flow (Re< 2000), the friction factor can be found by theoretical analysis and it is
equal to: f = 16/Re. However, in the case of turbulent flow, theoretical analysis is complicated
and almost impossible. Thus, experimental results are often used to determine the friction
coefficient. The Moody diagram as prepared by Lewis F. Moody, gives the friction coefficient
for various commercial pipes. The diagram is shown in Fig. 4 (from Mechanics of Fluids by
Massey). The ratio k/d or often ε/d is the relative roughness of the pipe; k or ε is the wall
roughness (‘mean height of roughness’).
From the Moody diagram, the following can be observed:
 For Re < 2000, (Laminar Flow Region), the friction coefficient is represented by a straight
line corresponding to f=16/Re,
 For the 2000 <Re < 4000 region, where the flow is transitional, the friction factor is not
well defined and this region is known as the critical zone,
 For Re > 4000, for each value of the surface relative roughness (ratio of mean height of
surface roughness to the diameter of the pipe), there is a curve representing the
relationship between the friction coefficient and the Re number. As the Re number
increases, the friction coefficient decreases until the curve flattens out and becomes almost
horizontal. The region where the curves flatten out represents the region of Rough Pipe
Behaviour. In this case, the friction coefficient depends only on the surface relative
roughness.
For relative roughness of less than 0.001, the curves approach the Smooth Pipe Line with
decreasing Re number. The smooth pipe line represents that flow behaviour where the friction
coefficient depends on the Re number and does not depend on the relative roughness.
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Frictional Losses in One-Dimensiuonal Flow
Figure 4. Moody diagram
The smooth and rough pipe concepts can be explained in view of the viscous sub-layer. In any
flow, even for the highly turbulent one, there is inevitably a very thin layer immediately
adjacent to the wall in which random motions are negligible and the effect of viscosity is
dominant. This layer is known as the viscous sub-layer. The higher the Re number, the smaller
the thickness of the viscous sub-layer. In the smooth pipe zone, the viscous sub-layer is thick
enough to cover completely the irregularities of the surface. Thus the size of the irregularities
has no effect on the main flow and all the curves for different roughness coincide. With
increasing Re number, the thickness of the viscous sub-layer decreases and the bumps can
protrude through it and act as centres for eddies generation. The rougher the pipe, the lower
Re number at which the rough pipe behaviour occur. In the complete turbulence of the rough
zone of flow, simple viscous effects are negligible and f is constant.
Equations for the friction coefficient were developed from the Moody diagram such as:
For the smooth pipe curve: f = 0.079 Re-0.25 for 3000<Re<105.
For the entire range of  /d and Re, the best yet produced correlation of f is probably that by
S.E. Haaland:
 6.9   1.11 
1
 
 3.6 log10 

f
 Re  3.71d  
For many practical problems which involve long runs of pipe, the head loss due to friction
dominates all other losses and it is simply known as the pipe loss.
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Frictional Losses in One-Dimensiuonal Flow
Note: Darcy’s equation is sometimes written (usually in American textbooks) as:
hL , pipe  f
l V2
d 2g
This form will not be used in the present course. The friction factor in this form is 4 times the
factor f as determined in this section. Care should be taken which friction factor and which
Moody diagram is used when applying Darcy’s equation.
5. Head Loss due to Sudden Changes of Pipe Sections - Loss Coefficients
a) Sudden Enlargement
Consider the flow within the control volume bound by the dashed lines in Fig. 5. Experiment
shows that just after the enlargement, the streamlines diverge, and there is a dead zone of
eddying flow near the pipe corners. These eddies cause considerable friction between fluid
particles and are responsible for significant frictional dissipation.
2
1
A1
p1
V1
Eddies
A2
p2
V2
Figure 5
Using the S.F.E.E. and the conservation of momentum principle, it can be shown that the head
loss due to sudden expansion is given by:
2

V2  A
V2 
A
hL  2  2  1  1 1  1 
2 g  A1 
2g 
A2 
2
This agrees with physical intuition. It is reasonable to expect that the dissipation of
mechanical energy is dependent on the speed with which the motion occurs - i.e., the loss is
proportional to the kinetic energy. This type of head loss / K.E. relationship applies to all
other cases where frictional effects are of importance.
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Frictional Losses in One-Dimensiuonal Flow
b) Sudden Contraction
Experiment shows that the flow pattern at the contraction is similar to that shown in Fig. 6.
1
Eddies
2
V ena
C ontracta
A1
p1
V1
A2
p2
V2
Figure 6
A vena-contracta forms downstream of the contraction and a region of eddying flow is set up
between the vena and the pipe walls. Because of the high velocities found there, this region is
primarily responsible for the energy dissipation and hence the head loss. Beyond the vena, the
flow settles and expands to a parallel motion at section 2.
The problem can be treated as a sudden enlargement if one is to consider the flow between the
vena and section 2. Thus, based on the findings of the previous section:
2


V2  A
V2  1
hL  2  2  1  2 
 1
2 g  Av
2 g  CC 

2
Note that although A1 does not appear in the above equation, it does have an effect on the
value of Av and hence on the value of the contraction coefficient Cc. In practice, it is
customary to replace the term in brackets by a head loss coefficient, k, which is found from
experiments. Hence:
V22
hL  k
2g
For circular pipes at normal flow rates, the head loss coefficient has values given below (for
different ratios of diameters d2/d1):
Diameter ratio, d2/d1
Head Loss Coefficient, k
0.0
0.5
0.2
0.45
0.4
0.38
0.6
0.28
0.8
0.14
1.0
0.0
Two points should be noted:
1) The losses are referred to the K.E. at the downstream section 2 where the velocity is
greatest and hence most of the dissipation occurs.
9
Frictional Losses in One-Dimensiuonal Flow
2) The diameter ratio, d2/d1 = 0, in the above table should be read as “all cases where the
downstream diameter is much smaller than that upstream”.
c) Exit Losses
The head loss for a sudden enlargement in terms of the velocity in the upstream pipe has been
found as:
V12 
A
hL 
1  1 
2g 
A2 
2
V1
Figure 7
The flow from a pipe discharging into a reservoir, Fig. 7, could be treated as a special case of
A
a sudden enlargement in which A2   . If this is the case, then 1  0 , resulting in:
A2
V12
hL 
2g
i.e., all of the kinetic energy in the flow is dissipated by friction!
d) Entry Losses
For example, consider the flow from a reservoir into a pipe as shown in Fig. 8.
d2
 0 . Thus from the
d1
Table in Section b) (sudden contraction): k = 0.5, if the entry is sharp edged and does not
protrude into the reservoir.
This is a special case of a sudden contraction where the diameter ratio
10
Frictional Losses in One-Dimensiuonal Flow
Figure 8
A number of other entry configurations have been investigated experimentally. They have all
been characterised by the head loss coefficient, k, which is used in the equation:
hL  k
V2
2g
where V is the velocity in the downstream pipe.
Values of k for some configurations are given in Fig. 9.
>d/2
d
k = 0.5
k = 1.0
r>0.14d

d
k=0
k~0.18 for
30 o <<60 o
Figure 9
e) Losses in Fittings (valves, taps, unions, etc.)
Fittings present a restriction to the flow and force it to follow a more complicated path. The
losses again are accounted for by means of a loss coefficient, k, in the usual manner.
hL  k
V2
2g
where V is some measure of the velocity within the fitting. Loss coefficients for individual
fittings are usually quoted by manufacturers.
11
Frictional Losses in One-Dimensiuonal Flow
Problem Sheet
Frictional Losses in One-Dimensional Flow
1.
Water flows through 200 m of a galvanised steel pipe of 150 mm diameter. The water
flow rate is 50 litres/s. Determine the head loss due to friction.
Take the kinematic viscosity  of water as 1.14 mm2/s.
(Ans.: 11.32 m).
2.
Water has to flow through a 180 m long galvanised steel pipe with a head loss due to
friction of 9 m. The water flow rate is 85 litres/s. Determine the required size (diameter)
of the galvanised steel pipe.
Take the kinematic viscosity  of water as 1.14 mm2/s.
(Ans.: 187 mm).
3.
Points A and B are 152.4 m apart in a 203 mm diameter horizontal pipe. They are
connected to a differential manometer by means of tubing. When water flows along the
pipe at a rate of 0.1786 m3/sec, the difference in mercury levels in the manometer is 1.96
m. Find the friction factor of the pipe. The specific gravity of mercury is 13.56.
(Ans.: f = 0.0053).
4.
A medium grade fuel oil, relative density 0.861, is to be pumped up to a storage tank.
The level of oil in the tank is 24.38 m above the pump. The gauge pressure at the pump
inlet is 13.79 kPa and the flow rate 0.198 m3/sec. If the pipe is 1829 m long, 406 mm in
diameter and has a friction factor of 0.0075, find, the power required by the pump. In
your calculation, include the exit loss at the pipe/tank junction.
(Ans.: 65 kW).
5.
A storage tank contains oil of relative density 0.84. It is proposed to move oil from the
tank through a pipeline 153 m long by pressurising the air above the oil. If the surface of
the oil in the tank is 93.9 m above sea level and the outlet from the pipeline is 100 m
above sea level, find the air pressure required to maintain a flow rate of 0.0127 m3/sec.
The inside diameter of the pipe is 153 mm and the average friction factor is 0.005875.
The entry loss coefficient is 0.5.
(Ans.: 55.3 kN/m2 gauge).
6.
Two reservoirs, the surface levels of which differ by 1.5 m, are connected by a pipe
system comprising a 7.5 m long and 75 mm in diameter sloping pipe at each end joined
by a 60 m long 300 mm in diameter horizontal pipe. Taking the head loss coefficient at
the contraction as 0.45 and the friction factor to be f = 0.005(1 + 25/d) where d is the
pipe diameter in mm, find the volumetric flow rate. Note that all other loss coefficients
can be found in the lecture notes.
(Ans.: 8.38 l/s).
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