The Bernoulli's Principle
Hydraulic networks are designed to distribute water from one main source to multiple end users. These end users will be located at different locations within a certain stipulated region. What pushes the water from the source to the end user? As you might be aware, in the water distribution networks, we do not use active devices likes pumps to carry water from the source (ESR) to users; rather we use the energy stored/ available in the water itself for this purpose. Understanding the concept of head and head loss is crucial to understand how this is achieved. For that let us first understand, the classical principle stated by Daniel Bernoulli (1700 - 1782) who published it in his book Hydrodynamica in 1738.
The principle states that, for a perfect in-compressible liquid, flowing in a continuous stream, the total energy of a particle remains the same, while the particle moves from one point to another.
Mathematically, and with reference to Fig. 1, it can be written as:
In hydraulics, energy is converted to energy per unit weight of water and is reported in length units (m) called “head”. In other words, the total energy associated with fluid per unit weight of fluid is called head.
To put it simply, ‘head’ is the total sum of energy possessed by a fluid in all possible forms. Fluids typically possess energy in three forms - fluid movement (kinetic energy), elevation (potential energy) and pressure (pressure energy).
Thus. the equation has three forms of energy:
Pressure - P / Ɣ
Velocity - V^2 / 2g, negligible
Elevation - Z
where,
P = pressure
Ɣ = specific weight of fluid
V = velocity
g = gravitational acceleration
Z = elevation
As V^2 / 2g is very small, it can be considered as 0. The remaining term 'P / Ɣ + Z' is called as the Hydraulic Head or the Hydraulic Grade.
Head Loss
Passing a certain volume of liquid through a pipe requires certain amount of energy. Energy or pressure difference must exist to cause liquid to flow. A portion of energy is lost to overcome the resistance to the flow. At the same time, other factors such as change in velocity and elevation also lead to energy dissipation. This dissipation of energy leads to energy loss, which is technically defined as ‘head loss’. Consider 2 sections on the same pipeline (refer Fig. 1 and the equation below).
The head loss is primarily linked to the friction which in turn depends on the properties of fluid as well as other parameters like velocity, diameter of pipe and internal roughness of pipe etc.
Friction losses occur as fluid passes through pipe fittings, bends and pressure drop due to change in elevation of fluid. While calculating head loss, one must account for total of energy losses due to length of pipe and those due to fittings, valves and other system structures.
The total energy losses are classified as:
major energy losses, and
minor energy losses.
Friction losses in pipe are termed as Major losses while loss of energy due to change of velocity of flowing fluid in magnitude or direction is termed as Minor loss. Major energy losses are calculated by Darcy Weisbach formula, Chezy’s formula, Hazen Williams formula, modified Hazen Williams formula, etc. Minor losses are caused due to sudden expansion or contraction of pipe, bends, fittings like expanders and reducers, and other obstructions in the pipe.
Impact of Flow Type on Friction Losses
As we have seen, friction losses are the primary source of head loss. Friction losses closely depend upon the type of fluid flow. By finding out the Reynolds's number we can find out whether the flow is laminar or turbulent. For closed conduit flow, if Re <2000 the flow is laminar and if Re >4000 then the flow is turbulent. The flow with Reynolds's number in between 2000 & 4000 is called transition flow.
For laminar flows, fluid friction is proportional to velocity of flow, is independent of pressure, proportional to contact surface area and independent of nature of surface.
For turbulent flow, the fluid friction is proportional to velocity of flow, is independent of pressure, proportional to contact surface area (longer the pipe more will be the loss) and density of flowing fluid, depends on nature of the surface.
Calculating Major Losses Using Different Formulas
Darcy Weisbach Formula -
Where,
hf = loss of head due to friction,
f = coefficient of friction which is function of Re number, derived from Moody's Diagram,
L = length of pipe,
V = mean velocity of pipe,
d = diameter of pipe.
Chezy’s Formula -
V=C √mi
Where,
C = Chezy’s constant,
m = hydraulic mean depth (Area / Perimeter),
i = hf / L, which is head loss of head per unit length of pipe.
Hazen-Williams Formula -
Where,
h = head loss (m)
D = diameter (m)
V = velocity (m/s)
C = Hazen-Williams C-factor
L = length (m)
k = 6.79 for V in m/s, D in m or
The C-factor ranges between 0 to 150 depending on the material and age of the pipe.
Modified Hazen-Williams Formula -
where,
V = velocity of flow in m/s,
Cr = pipe roughness coefficient, ( 1 for smooth pipes; < 1 for rough pipes),
r = hydraulic radius in m,
S = friction slope,
h = friction head loss in m.
Difference between the Darcy Weisbach and Hazen-Williams formulas:
As a part of academic curriculum, Darcy Weisbach formula has enjoyed much popularity as compared to any other formula owning to extreme precision in the values of the resulting head loss.
But, in practice, the engineers are challenged with high time consumption in determining the value of f - coefficient of friction for every pipe using Moody's Diagram. Because of this reason, its popularity among practitioners is very less.
Hazen-Williams formula, perhaps, generates reliable and reasonably precise values of head loss. The Hazen Williams C-factor ranges between 0 to 150. Rougher the pipe is, smaller would be the value of C and smoother the pipe is, higher would be the value of C.
Minor Losses
Apart from friction loss, there are also minor losses that occur in the system. These are prominently caused due to:
Sudden expansion of pipe
Sudden contraction of pipe
Bends on pipe
The total minor loss would be the sum of minor losses due to each of the above three losses.
How to minimize head losses in the pipe?
As the best practice, the piping system should be designed to keep head losses at a minimum. Here are some practices to minimize the head losses:
Correct sizing of pipe: Generally, undersized pipes would lead to increased head losses due to increased friction. However, over sizing pipes beyond reasonable limits would increase the Capex and commissioning cost. Hence, the pipes should be sized optimally for minimal head loss.
Optimizing the network length: As the length of the network increases, the head losses increase. Attention should be paid to the network design to ensure that water reaches all the end users with minimum length of the pipe.
Inner surface roughness of the pipe: The inner surface roughness of the pipe plays key role in determining the friction losses. The roughness coefficient for different diameters and materials can be found in the guideline for designing water supply networks for e.g. India's Central Public Health and Environmental Engineering Organization's (CPHEEO) Manual on Water Supply and Treatment, etc.
Streamlined design: As the number of pipes, valves, fittings and other obstructions in the system increase, both major and minor losses pile up. Therefore, a good hydraulic consultant should be engaged for a streamlined pipe design.
Can we neglect the minor losses?
Major losses in the pipe get accumulated over the length. However, major losses can be brought down to minimum by adhering to best design practices. If the pipe is long enough, the minor losses can usually be neglected as they are much smaller than the major losses. However, there are practices in some countries where minor losses are assumed to be 10% of the major losses; instead of calculating minor losses for every pipe. This can be achieved by simply increasing the hydraulic length of pipes by 10%.
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Quick revision of theories for all the design engineers (wet utilities) out there. Thank you for sharing.