Rectangular Notch Discharge Formula

The discharge \(Q\) over a rectangular notch or weir is given by the formula:

$$ Q = C_d \cdot \frac{2}{3} \sqrt{2g} \cdot L \cdot H^{3/2} $$

Where:

• \(Q\) is the discharge
• \(C_d\) is the coefficient of discharge
• \(L\) is the length of the notch
• \(H\) is the height of flow (head) over the notch crest
• \(g\) is the acceleration due to gravity

For a specific rectangular notch, \(C_d\), \(L\), and \(g\) are considered constant. Therefore, the discharge \(Q\) is directly proportional to the height of flow \(H\) raised to the power of 3/2.

$$ Q \propto H^{3/2} $$

Error Calculation

We are given that there is a 1 percent error in the measurement of the height of flow \(H\). We need to find the resulting percentage error in the discharge \(Q\).

Let \(Q = k H^{3/2}\), where \(k\) is a constant comprising \(C_d, \frac{2}{3} \sqrt{2g}, L\).

To find the relationship between the relative errors, we can use differentiation. Taking the natural logarithm of both sides:

$$ \ln(Q) = \ln(k) + \ln(H^{3/2}) $$

$$ \ln(Q) = \ln(k) + \frac{3}{2} \ln(H) $$

Now, differentiate both sides with respect to \(H\):

$$ \frac{1}{Q} \frac{dQ}{dH} = 0 + \frac{3}{2} \frac{1}{H} $$

Rearranging gives the relationship between the differential change in \(Q\) (\(dQ\)) and the differential change in \(H\) (\(dH\)):

$$ \frac{dQ}{Q} = \frac{3}{2} \frac{dH}{H} $$

In terms of small errors or percentage errors, we can write:

$$ \frac{\Delta Q}{Q} \approx \frac{3}{2} \frac{\Delta H}{H} $$

To express this in percentage error, we multiply by 100%:

$$ \left( \frac{\Delta Q}{Q} \times 100\% \right) \approx \frac{3}{2} \left( \frac{\Delta H}{H} \times 100\% \right) $$

We are given that the percentage error in the height of flow (\(H\)) is 1 percent. So, \( \left( \frac{\Delta H}{H} \times 100\% \right) = 1\% \).

Substituting this value into the equation:

$$ \text{Percentage error in } Q \approx \frac{3}{2} \times 1\% $$

$$ \text{Percentage error in } Q \approx 1.5 \times 1\% $$

$$ \text{Percentage error in } Q \approx 1.5\% $$

Therefore, a 1 percent error in the height of flow over a rectangular notch produces approximately a 1.5 percent error in the discharge.

Summary of Error Propagation

For a quantity \(Y\) that depends on another quantity \(X\) as \(Y \propto X^n\), the percentage error in \(Y\) due to a percentage error in \(X\) is given by \(n\) times the percentage error in \(X\). In this case, \(Q \propto H^{3/2}\), so \(n = 3/2\).

Percentage error in \(Q\) = \(\frac{3}{2} \times\) Percentage error in \(H\)

Percentage error in \(Q\) = \(\frac{3}{2} \times 1\% = 1.5\%\)