Two reservoirs having different water levels are connected by two long parallel pipelines of same length and same material but having diameters of 600 mm and 400 mm. Using Darcy-Weisbach equation, the ratio of flowrate of water in the bigger diameter pipe to that in the smaller diameter pipe is

The correct option is 2.76.

Step-by-Step Explanation:

When two reservoirs are connected by parallel pipelines, the head loss due to friction ($h_f$) remains the same for both pipelines because they share the same inlet and outlet water levels. Thus, we have:
$$h_{f1}=h_{f2}=h_f$$

According to the Darcy-Weisbach equation, the head loss in a pipe of length L, diameter D, friction factor f, and carrying a discharge (flow rate) Q is given by:
$$h_f=\frac{8fL{Q}^2}{{\pi }^{2}g{D}^5}$$

Since the pipes are made of the same material (friction factor $f$ is identical) and have the same length ($L$), the variables $f$, $L$, $g$, and $\pi$ are constants. Therefore, the head loss equation simplifies to a proportionality relation:
$$h_f\propto \frac{Q^2}{D^5}$$

Since the head loss is the same for both parallel pipes, we can write:
$$\frac{{Q_1}^2}{{D_1}^5}=\frac{{Q_2}^2}{{D_2}^5}$$

Rearranging this equation to solve for the ratio of the flow rates gives:
$${\left(\frac{Q_1}{Q_2}\right)}^2={\left(\frac{D_1}{D_2}\right)}^5$$

Taking the square root of both sides, we express the ratio of the flow rates as:
$$\frac{Q_1}{Q_2}={\left(\frac{D_1}{D_2}\right)}^{2.5}$$

Given the diameter of the bigger pipe $D_1=600\text{mm}$ and the diameter of the smaller pipe $D_2=400\text{mm}$:
$$\frac{D_1}{D_2}=\frac{600}{400}=1.5$$

Substituting this ratio back into our flow rate ratio equation:
$$\frac{Q_1}{Q_2}={\left(1.5\right)}^{2.5}$$

Calculating the numerical value:
$$\frac{Q_1}{Q_2}\approx 2.7557\approx 2.76$$

Thus, the ratio of the flow rate in the bigger diameter pipe to that in the smaller diameter pipe is approximately 2.76.