Drainage Systems

• Continuity Equation in Cartesian Coordinates:
• General Form (Mass Conservation):

• Universality of the Equation:
• Steady and Unsteady flow: Directly accounts for temporal density fluctuations ($\frac{\partial \rho}{\partial t} = 0$ for steady flow).
• Uniform and Non-uniform flow: Spatial derivative terms handle velocity and area variations along coordinate axes.
• Compressible and Incompressible fluids: Retains fluid density $\rho$ inside spatial gradients for gas dynamics while simplifying to $\nabla \cdot \vec{v} = 0$ when $\rho = \text{constant}$.
• Conclusion: (A), (B) and (C) (Option 3).

Correct option: (3) (A), (B) and (C).

• Capillary Rise & Minimum Piezometer Tube Diameter:
• Capillary Rise Equation:

For clean water in a glass tube, contact angle $\theta = 0^\circ \implies \cos\theta = 1$.

• Calculation:
• Surface tension $\sigma = 0.073575\text{ N/m}$
• Density $\rho = 1000\text{ kg/m}^3$, $g = 9.81\text{ m/s}^2$
• Maximum allowable capillary rise $h = 3\text{ mm} = 0.003\text{ m}$

• Conclusion: Minimum diameter of glass tube is 1 cm (Option 4).

Correct option: (4) 1cm

• Most Economical (Efficient) Open Channel Section:
• Statement (I): A channel section is most economical when maximum discharge is obtained for a given cost of construction. $\implies$ Correct.
• channel section is hydraulically most efficient/economical when it conveys maximum discharge ($Q$) for a given cross-sectional area and excavation volume (which corresponds to minimum wetted perimeter $P$).
• Statement (II): Discharge is maximum when the flow velocity is minimum. $\implies$ Incorrect.
• By continuity equation $Q = A \cdot V$, for a given cross-sectional area $A$, discharge is maximum when flow velocity $V$ is maximum (not minimum).
• Conclusion: Statement (I) is correct but Statement (II) is incorrect (Option 3).

• Steady Uniform Flow in Open Channels:
• Statement (I): The depth of flow is the same at every cross section of channel. $\implies$ Correct.
• In uniform flow, the depth of flow, water cross-sectional area, velocity, and discharge remain constant along the longitudinal axis of the channel ($\frac{\partial y}{\partial x} = 0$). The channel bed, water surface, and energy grade line are all parallel.
• Statement (II): The depth of the flow does not change with time. $\implies$ Correct.
• In steady flow, hydraulic parameters such as flow depth, velocity, and discharge at any given cross-section do not change with respect to time ($\frac{\partial y}{\partial t} = 0$).
• Conclusion: Both Statement (I) and Statement (II) are correct (Option 1).

Correct option: (1) Both Statement (I) and Statement (I) are correct