The discharge through a V-notch varies as (where, H is the head)
Understanding Discharge Through a V-Notch
A V-notch, also known as a triangular weir, is a device used in open channels to measure the flow rate or discharge of liquid. It is a triangular opening cut into a plate or wall, with the apex pointing downwards. The flow passes through this triangular opening, and the discharge is determined by measuring the head of the liquid above the apex of the V-notch.
Formula for V-Notch Discharge
The theoretical formula for the discharge \(Q\) through a V-notch is derived based on principles of fluid mechanics, considering the flow through infinitesimally small horizontal strips across the notch opening and integrating over the entire height up to the liquid surface.
The standard formula for the actual discharge \(Q\) through a V-notch is given by:
\[ Q = C_d \cdot \frac{8}{15} \cdot \sqrt{2g} \cdot \tan\left(\frac{\theta}{2}\right) \cdot H^{5/2} \]Where:
- \(Q\) is the discharge or flow rate through the V-notch.
- \(C_d\) is the coefficient of discharge, which accounts for the non-ideal flow conditions (like viscosity and surface tension) and is determined experimentally.
- \(g\) is the acceleration due to gravity.
- \(\theta\) is the angle of the V-notch.
- \(H\) is the head of the liquid above the apex of the V-notch.
How Discharge Varies with Head (H)
In the formula for V-notch discharge, several terms are constant for a given setup:
- The coefficient of discharge (\(C_d\)) is usually considered constant for a specific notch and flow conditions.
- The constant factor \(\frac{8}{15}\) is part of the theoretical derivation.
- The acceleration due to gravity (\(g\)) is a constant.
- The angle of the V-notch (\(\theta\)) is fixed for a particular notch, making \(\tan\left(\frac{\theta}{2}\right)\) a constant.
Therefore, the discharge \(Q\) is directly proportional to the head \(H\) raised to the power of \(\frac{5}{2}\). We can write this relationship as:
\[ Q \propto H^{5/2} \]This means that as the head \(H\) increases, the discharge \(Q\) increases significantly, proportional to the \(\frac{5}{2}\) power of the head.
Looking at the options provided, we need to find the power to which \(H\) is raised in the discharge formula for a V-notch.
- Option 1: \(H^{1/2}\) - This corresponds to flow through a small orifice under constant head.
- Option 2: \(H^{5/4}\) - This power is not typically associated with standard notch types.
- Option 3: \(H^{5/2}\) - This matches the power of \(H\) in the V-notch discharge formula.
- Option 4: \(H^{3/2}\) - This corresponds to the discharge through a rectangular weir.
Based on the formula, the discharge through a V-notch varies as \(H^{5/2}\).
Summary of V-Notch Discharge Variation
The discharge \(Q\) through a V-notch is directly proportional to the head \(H\) above the apex raised to the power of 5/2. This relationship is fundamental in fluid mechanics when dealing with flow measurements using triangular weirs.
| Type of Notch/Weir | How Discharge \(Q\) Varies with Head \(H\) | Formula for Discharge \(Q\) |
|---|---|---|
| V-Notch (Triangular Weir) | \(Q \propto H^{5/2}\) | \(Q = C_d \cdot \frac{8}{15} \cdot \sqrt{2g} \cdot \tan\left(\frac{\theta}{2}\right) \cdot H^{5/2}\) |
| Rectangular Notch (Rectangular Weir) | \(Q \propto H^{3/2}\) | \(Q = C_d \cdot \frac{2}{3} \cdot L \cdot \sqrt{2g} \cdot H^{3/2}\) (where L is the length of the weir) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| V-Notch (Triangular Weir) | Used for measuring discharge, especially small flows. |
| Head (H) | Vertical distance from the liquid surface to the apex of the notch. |
| Discharge (Q) | Volume flow rate of liquid passing through the notch. |
| Relationship \(Q\) vs \(H\) | For V-notch, \(Q\) varies as \(H^{5/2}\). |
Additional Information: Weir Types and Discharge
Weirs are overflow structures used to measure or control the flow rate of water in open channels. Different shapes of weirs lead to different relationships between the head and the discharge.
- Rectangular Weir: Has a rectangular opening. The discharge is proportional to \(H^{3/2}\). They are suitable for larger flows.
- Triangular Weir (V-Notch): Has a triangular opening. The discharge is proportional to \(H^{5/2}\). They are more sensitive to changes in head at low flows compared to rectangular weirs, making them better suited for measuring small discharges accurately. The angle of the V-notch affects the constant of proportionality. Common angles are 90 degrees.
- Trapezoidal Weir (Cipoletti Weir): A combination of a rectangular weir and two triangular sections. Designed so that the discharge through the triangular ends compensates for the reduction in the effective length of the rectangular part due to side contractions, simplifying the discharge calculation.
Understanding the relationship between head and discharge for different weir types is crucial for flow measurement applications in hydraulics and fluid mechanics.
Important Questions from Weirs and Notches
The discharge over a rectangular notch is
The horizontal to vertical side slope in case of Cipoletti weir is-
The formula for Discharge in Rectangular Notch is -
(Where B = width of notch, and H = height of liquid above the sill of the notch)
The velocity with which the water approaches a notch is called
While conducting flow measurement using a rectangular notch, an error of 2% in head over the notch and error of 3% in the length was observed. The percentage error in the computed discharge would be
