Test: Darcy Weisbach Equation - Civil Engineering (CE) MCQ

The layout of the Hydropower plant is shown below: Components are:

1. Reservoir: Reservoirs can be natural (Lake) or Artificial (Dam).
2. Penstock: It is a large-diameter pipe that carries water from the water storage system to the turbine.
3. Surge tanks: It is a reservoir of water placed near the turbine and is used to avoid the water hammer in the penstock.
4. Turbine
5. Tailrace
6. Generator

​Types of head:

• Gross head (H_G): It is defined as the head under which a hydropower plant is working or it is the difference between the head race level and tail race level.
• Net head (H): It is the head available with water at the entry to the turbine or it is the head under which the turbine is working.

H_G - H_F = H; Where H_F = Head loss due to friction
According to Darcy's Weisbach equation for head loss in pipes:
; where F = Darcy's friction factor = 4f ; where f = friction coefficient
From the above equation, we can see that
HF ∝ L       ∝ V²       ∝1D
So, From the above, we can conclude that Head loss due to friction in water flow through the penstock can be minimized by increasing the diameter of the Penstock.

Darcy Weisbach Equation for friction losses in circular pipe:

where,
L = length of the pipe, D = diameter of the circular pipe, V = mean velocity of the flow, f = Darcy’s friction factor = 4 × F’, F’ = coefficient of friction, h_f = head loss due to friction
For Laminar Flow
Friction Fcator

For Turbulent flow

Concept:
The loss of head due to friction in the pipe is given as:

Where f = friction coefficient, L = length of the pipe, V = mean velocity of the fluid, g = acceleration due to gravity, and D = diameter of the pipe.
Head loss in terms of friction factor is given as:

Where F = friction factor of the pipe = 4f
Calculation:
Given:
D = 100 mm, L = 981 m, V = 1 m/s, f = 0.0008
The loss of head due to friction in the pipe when friction coefficient is given:

= 1.6 m

Concept:
Head loss due to friction in a pipe:

h_f = head loss due to friction, f’ = friction factor, L = length of the pipe, V = average velocity of the fluid in the pipe, g = acceleration due to gravity, d = diameter of the pipe.
Hence keeping all other parameters to be same, h_f ∝ V²
Calculations: 
Given:
Final velocity (V₂) = 1.10 × Initial velocity (V₁)

Final head loss (h_f2) = 1.21 × Initial head loss (h_f1)
Hence head loss will increase by 21 %.

Concept:
Darcy friction factor is define as,

where, ρ = density of fluid, V = velocity of fluid, D = Diameter of pipe,
v = kinematic viscosity
If Re > 4000 then the flow become turbulent flow
If Re < 2000 then the flow become laminar flow
Calculation:
Given: D = 10 cm = 0.1 m, v = 0.2 m/s, v = 10-5 m²/s 

Therefore, it is laminar flow

Concept:
When two pipelines are connected in series, the total discharge in each pipe will be the same as individual discharges at each pipe to be connected in series
The discharge through the equivalent pipe,
Q_total = Q₁ = Q₂ 
The total head is equal to the sum of head losses at each individual pipe htotal = h₁ + h₂ 
The darcy wisback factor is given as, 
Here, nothing is mentioned the friction factor for both the pipes which are connected in series so here we need to assume that f is the same for each pipe.
⇒ h_f ∝ 1 / d⁵

Calculation:
Given:
d₁ = d, d₂ = 2d
Therefore,

Frictional Head Loss in Pipe

• Frictional energy loss per length of the pipe depends on the flow velocity, pipe length, pipe diameter, and a friction factor based on the roughness of the pipe, and whether the flow is laminar or turbulent (i.e. the Reynolds number of the flow).
• The total energy of the fluid conserves as a consequence of the law of conservation of energy.
• In reality, the head loss due to friction results in an equivalent increase in the internal energy (increase in temperature) of the fluid.
• The most common equation used to calculate frictional head losses in a tube or pipet is the Darcy–Weisbach equation (head loss form).

where Δp = pressure loss, fD = darcy friction factor, L = length of pipe, D = hydraulic diameter, V = fluid flow average velocity, ρ = fluid density.
As head loss or pressure loss due to friction in the pipe is

1. Directly proportional to the length of the pipe
2. Inversely proportional to the diameter of the pipe.

Hence the reduction in length and increase in diameter of the pipe will reduce frictional head or pressure loss.

Concept:
Head loss due to friction is given by:

d = diameter of pipe, f = friction factor, L = length of pipe, and v = velocity of flow
Calculation:
Given:
d = 1 m, L = 1.5 km = 1500 m, v = 1 m/s, f = 0.02, g = 10 m/s²
Head loss due to friction is

h = 1.5 m

Concept:
Shear stress at any distance ‘r’ from the center of the pipe is given by.

At r = R, i.e. at the, pipe wall, shear stress is maximum and is given by

Where, R = Radius of the pipe
∂p / x = Pressure gradient over the length of the pipe.
So, from the above, τ ∝ R
Calculation:
Given:
δp = 70 kPa, r = 300 mm, x = 30 m
Shear stress 

Concept:
The pressure drop in the duct is given as, 

Where D = 4A/P
Calculation:
Given:
μ = 2 × 10^-5 kg/m-s, L = 1 m, Vavg = 1 m/s