The formula for Discharge in Rectangular Notch is - (Where B = width of notch, and H = height of liquid above the sill of the notch)
Understanding Discharge in a Rectangular Notch
A rectangular notch, also known as a rectangular weir, is a structure built across a channel to measure the rate of flow, or discharge, of liquid. It is essentially an opening or gap in the side of a tank or channel through which liquid flows. The top edge of the notch over which the liquid flows is called the sill or crest.
The rate of discharge through a rectangular notch depends on several factors, including the width of the notch, the height of the liquid above the sill, the acceleration due to gravity, and the properties of the liquid flow captured by a coefficient.
Deriving the Formula for Discharge
To determine the discharge through a rectangular notch, we consider a small horizontal strip of the liquid flow at a certain depth below the liquid surface. Assuming ideal flow conditions and neglecting velocity of approach, the theoretical velocity of the liquid flowing through this strip can be determined using Torricelli's theorem, which states that the velocity of efflux is equal to the velocity that would be gained by a body falling freely from the free surface level to the level of the opening.
The theoretical velocity \(v\) at a depth \(h\) below the free surface is given by:
\[ v = \sqrt{2gh} \]
where \(g\) is the acceleration due to gravity and \(h\) is the depth below the free surface.
Now, consider a small horizontal strip of thickness \(dh\) at a depth \(h\) below the free surface and across the width \(L\) (or \(B\)) of the notch. The area of this strip is:
\[ dA = L \cdot dh \]
The theoretical discharge \(dQ_{theoretical}\) through this small strip is the product of the velocity and the area:
\[ dQ_{theoretical} = v \cdot dA = \sqrt{2gh} \cdot (L \cdot dh) = L\sqrt{2g} \cdot h^{1/2} dh \]
To find the total theoretical discharge \(Q_{theoretical}\) through the entire notch, we integrate \(dQ_{theoretical}\) from the sill level (\(h=0\)) up to the height of the liquid surface above the sill (\(h=H\)).
\[ Q_{theoretical} = \int_0^H dQ_{theoretical} = \int_0^H L\sqrt{2g} \cdot h^{1/2} dh \]
Since \(L\) and \(\sqrt{2g}\) are constants, we can take them outside the integral:
\[ Q_{theoretical} = L\sqrt{2g} \int_0^H h^{1/2} dh \]
Integrating \(h^{1/2}\) with respect to \(h\) gives \(\frac{h^{3/2}}{3/2}\):
\[ Q_{theoretical} = L\sqrt{2g} \left[ \frac{h^{3/2}}{3/2} \right]_0^H \]
Evaluating the expression at the limits \(H\) and \(0\):
\[ Q_{theoretical} = L\sqrt{2g} \left( \frac{H^{3/2}}{3/2} - \frac{0^{3/2}}{3/2} \right) = L\sqrt{2g} \cdot \frac{H^{3/2}}{3/2} \]
Rearranging the terms:
\[ Q_{theoretical} = \frac{2}{3} L\sqrt{2g} H^{3/2} \]
Introducing the Coefficient of Discharge
The theoretical discharge calculated above assumes ideal conditions. In reality, factors like viscosity, surface tension, and contraction of the nappe (the stream of liquid flowing over the notch) affect the actual discharge. To account for these real-world effects, a coefficient of discharge, \(C_d\), is introduced. The actual discharge \(Q_{actual}\) is given by:
\[ Q_{actual} = C_d \cdot Q_{theoretical} \]
Substituting the theoretical discharge formula:
\[ Q_{actual} = C_d \cdot \frac{2}{3} L\sqrt{2g} H^{3/2} \]
This is the formula for the actual discharge through a rectangular notch.
Comparing with Options
The question asks for the formula for discharge in a rectangular notch, using \(B\) or \(L\) for width and \(H\) for height. The options use \(L\) for width and \(H\) for height above the sill, and include the coefficient of discharge \(C_d\). Comparing the derived formula with the given options:
- Option 1: \(\frac{3}{2}{C_d}L\;\sqrt {2g} \;{H^{3/2}}\) - This has the fraction \(\frac{3}{2}\), which is incorrect.
- Option 2: \(\frac{3}{2}{C_v}L\;\sqrt {2g} \;{H^{3/2}}\) - This has the fraction \(\frac{3}{2}\) and uses \(C_v\) (coefficient of velocity), not \(C_d\).
- Option 3: \(\frac{3}{2}{C_c}L\;\sqrt {2g} \;{H^{3/2}}\) - This has the fraction \(\frac{3}{2}\) and uses \(C_c\) (coefficient of contraction), not \(C_d\).
- Option 4: \(\frac{2}{3}{C_d}L\;\sqrt {2g} \;{H^{3/2}}\) - This matches the derived formula with the correct fraction \(\frac{2}{3}\) and the coefficient of discharge \(C_d\). Note that the variable \(L\) is used for the width/length of the notch in the options, consistent with our derivation where \(L\) represents the width \(B\).
Therefore, the correct formula for discharge in a rectangular notch is \(Q = \frac{2}{3}{C_d}L\;\sqrt {2g} \;{H^{3/2}}\).
| Symbol | Meaning | Units (SI) |
|---|---|---|
| \(Q\) | Discharge (Volume flow rate) | m³/s |
| \(C_d\) | Coefficient of discharge | Dimensionless |
| \(L\) (or \(B\)) | Width of the notch | m |
| \(g\) | Acceleration due to gravity | m/s² |
| \(H\) | Height of liquid above the sill | m |
Revision Table: Notch Formulas
| Type of Notch/Weir | Discharge Formula (\(Q\)) | Notes |
|---|---|---|
| Rectangular Notch/Weir | \(Q = \frac{2}{3}C_d L \sqrt{2g} H^{3/2}\) | \(L\) is the width, \(H\) is the head above sill. |
| Triangular (V-Notch) | \(Q = \frac{8}{15}C_d \tan(\frac{\theta}{2}) \sqrt{2g} H^{5/2}\) | \(\theta\) is the angle of the notch, \(H\) is the head. Useful for low discharges. |
| Trapezoidal (Cipoletti Weir) | \(Q = \frac{2}{3}C_d L \sqrt{2g} H^{3/2} + C_{d_{side}} \cdot \frac{8}{15} \tan(\frac{\theta}{2}) \sqrt{2g} H^{5/2}\) For Cipoletti, side slopes are 1 horizontal to 4 vertical (\(\tan(\frac{\theta}{2}) = \frac{1}{4}\)), \(C_{d} \approx C_{d_{side}}\) |
Combines rectangular and triangular sections. Cipoletti weir has specific side slopes. |
Additional Information on Rectangular Notch Discharge
Several factors can influence the actual discharge through a rectangular notch and the value of the coefficient of discharge (\(C_d\)):
- Velocity of Approach: The formula assumes the liquid in the channel approaching the notch has negligible velocity. If the velocity is significant, it adds to the effective head over the notch. A term accounting for the velocity head (\(\frac{V_a^2}{2g}\), where \(V_a\) is the velocity of approach) is sometimes added to \(H\), making the effective head \(H_{eff} = H + \frac{V_a^2}{2g}\).
- Contraction of the Nappe: As the liquid flows over the sill and through the sides, the stream tends to contract. This reduces the effective width and height of the flow stream compared to the notch dimensions. \(C_d\) accounts for these contractions (velocity effects and contraction effects, where \(C_d = C_v \times C_c\)).
- Suppressed vs. Contracted Notch: A rectangular notch is called 'suppressed' if its sides are flush with the sides of the channel, preventing side contractions. It is 'contracted' if the notch width is less than the channel width, allowing side contractions. The formula \(Q = \frac{2}{3}{C_d}L\;\sqrt {2g} \;{H^{3/2}}\) is typically for a contracted rectangular notch.
- Empirical Formulas: While the basic formula provides a theoretical basis, empirical formulas derived from experiments (like the Francis formula or the Bazin formula) are often used in practice to get more accurate discharge values for specific conditions, often modifying the basic formula or providing specific \(C_d\) values.
Understanding the concept of discharge measurement using notches and weirs is crucial in fluid mechanics and hydraulic engineering applications.
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 velocity with which the water approaches a notch is called
The discharge through a V-notch varies as (where, H is the head)
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
