The continuity equation in cartesian co-ordinates form is... | Agrigyan

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

$$h = \frac{4\sigma \cos\theta}{\rho g d}$$

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 $h = 3\text{ mm} = 0.003\text{ m}$

$$d = \frac{4 \times 0.073575}{1000 \times 9.81 \times 0.003} = \frac{0.2943}{29.43} = 0.01\text{ m} = 1\text{ cm}$$

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

Correct option: (4) 1cm

• Hydraulic Jump Sequent Depth Calculation:
• Bélanger's Momentum Equation:

$$y_2 = \frac{y_1}{2} \left(\sqrt{1 + 8 Fr_1^2} - 1\right)$$

• Parameters:
• Channel width $B = 1\text{ m}$, Discharge $Q = 3.132\text{ m}^3\text{/s} \implies q = Q/B = 3.132\text{ m}^2\text{/s}$.
• Initial depth $y_1 = 0.8\text{ m}$.
• Initial velocity $v_1 = \frac{q}{y_1} = \frac{3.132}{0.8} = 3.915\text{ m/s}$.
• Initial Froude number:

$$Fr_1 = \frac{v_1}{\sqrt{g y_1}} = \frac{3.915}{\sqrt{9.81 \times 0.8}} = \frac{3.915}{\sqrt{7.848}} = \frac{3.915}{2.8014} = 1.3975$$ $$Fr_1^2 = (1.3975)^2 \approx 1.953$$

• Post-Jump Depth ($y_2$):

$$y_2 = \frac{0.8}{2}\left(\sqrt{1 + 8(1.953)} - 1\right) = 0.4 \times (\sqrt{16.624} - 1) = 0.4 \times (4.077 - 1) = 0.4 \times 3.077 \approx 1.23\text{ m}$$

• Conclusion: Depth of water after jump is 1.23 m (Option 1).

Correct option: (1) 23m

• Operational Characteristics of Air-Lift Pumps:
• Compressed air operation: Operates by injecting compressed air into the footpiece at the bottom of a submerged discharge pipe, producing an air-water mixture of lower bulk density that rises by hydrostatic buoyancy. (Correct)
• No moving parts: Contains no submerged pistons, impellers, or valves, eliminating mechanical wear from gritty abrasive sands. (Correct)
• High borehole discharge capacity: In a well of given casing diameter, an air-lift pipe occupies minimal cross-sectional area compared to submersible pump bowls, yielding maximum discharge rate. (Correct)
• Low energy efficiency: Overall thermal-pneumatic pumping efficiency is low ($20 - 40\%$) due to air compressor thermodynamic losses and slip between air bubbles and water. (Correct)
• Conclusion: (A), (B), (C) and (D) (Option 3).

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

• Confined (Artesian) Aquifers Hydrogeology:
• Artesian designation: An aquifer bounded above and below by impermeable confining strata (aquicludes), maintaining pore water under hydrostatic pressure. (Correct)
• Recharge localization: Recharge occurs exclusively in exposed outcrop recharge areas where permeable strata contact ground surface. (Correct)
• Piezometric surface: The imaginary hydrostatic head surface (piezometric level) can coincide with or rise substantially above the upper confining layer. (Correct)
• Semi-confined / Leaky aquifer: A confined aquifer bounded by semi-pervious strata (aquitards) through which vertical cross-leakage occurs is classified as a leaky aquifer. (Correct)
• Conclusion: (A), (B), (C) and (D) (Option 3).

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