Civil Engineering 2024 Paper II 50 marks Derive

Q6

(a) Explain an aquifer. Deduce the expression for discharge (Q) through a well in an unconfined aquifer taking usual symbols. If the permeability of the aquifer, K = 10⁻⁴ m/s, radius of drawdown curve, R = 500 m, radius of well, r = 5·0 m, total aquifer thickness, H = 30 m and depth of water in well, h = 10 m; find the steady discharge. (20 marks) (b) Determine the size (i.e., diameter and depth) of a circular rapid mixing tank having a mechanical mixer, which is to be designed for treatment of water flow of 10 × 10⁶ litres per day and for mean hydraulic detention time of 45 seconds. Also, calculate the power required to achieve a mixing intensity (G) of 450 s⁻¹. Assume viscosity of water = 0·89 × 10⁻³ N.s/m² and depth of water to diameter of tank ratio of 2 : 1. (15 marks) (c) (i) What is the "5 R's" concept in waste management ? How do they contribute to managing the ill-effects of waste ? Briefly explain. (7 marks) (ii) A dumpsite fire emits 4 g/s of NOₓ. Write an equation of NOₓ concentration at 2·0 km downwind from the dumpsite if the wind speed U₁₀ = 5 m/s and its stability is 'D' type. What would be the maximum NOₓ concentration at 2·0 km from the dumpsite at the ground and also at 50 m above ground ? Assume diffusion coefficients σᵧ = 150 m and σᵤ = 50 m at the downwind distance of 2·0 km from the source for 'D' type stability. (8 marks)

हिंदी में प्रश्न पढ़ें

(a) एक जलवाही स्तर की व्याख्या कीजिए। सामान्य प्रतीकों को लेकर एक अपरिबद्ध जलवाही स्तर में एक कुएँ के निस्सरण (Q) के लिए व्यंजक प्राप्त कीजिए। यदि जलवाही स्तर की पारगम्यता, K = 10⁻⁴ मीटर प्रति सेकंड, अपकर्ष वक्र की त्रिज्या, R = 500 मीटर, कुएँ की त्रिज्या, r = 5·0 मीटर, सकल जलवाही स्तर की मोटाई, H = 30 मीटर और कुएँ में पानी की गहराई, h = 10 मीटर है, तो अपरिवर्ती निस्सरण ज्ञात कीजिए। (20 अंक) (b) एक वृत्ताकर द्रुत मिश्रण टंकी, जिसमें एक यांत्रिक मिश्रक लगा है, के आमाप (यानी व्यास और गहराई) को ज्ञात कीजिए, जिसे 10 × 10⁶ लीटर प्रति दिन के जल प्रवाह प्रशोधन और 45 सेकंड के औसत जलीय अवरोधक अवधि के लिए अभिकल्पित किया जाना है। 450 s⁻¹ की मिश्रण तीव्रता (G) को प्राप्त करने के लिए आवश्यक शक्ति की गणना भी कीजिए। जल की श्यानता = 0·89 × 10⁻³ N.s/m² और जल की गहराई तथा टंकी के व्यास का अनुपात 2 : 1 मान लीजिए। (15 अंक) (c) (i) अपशिष्ट प्रबंधन में "5 आर" की अवधारणा क्या है ? ये अपशिष्ट के दुष्प्रभावों के प्रबंधन में कैसे योगदान करते हैं ? संक्षेप में व्याख्या कीजिए। (7 अंक) (ii) एक क्षेपण स्थल पर लगी आग से 4 ग्राम प्रति सेकंड NOₓ निकलता है। यदि वायु की गति U₁₀ = 5 मीटर प्रति सेकंड है और इसका स्थायित्व 'D' प्रकार का है तो क्षेपण स्थल से 2·0 किमी हवा की दिशा में NOₓ सांद्रता का समीकरण लिखिए। क्षेपण स्थल से 2·0 किमी दूर जमीन पर और जमीन से 50 मीटर ऊपर अधिकतम NOₓ सांद्रता क्या होगी ? 'D' प्रकार के स्थायित्व के लिए स्रोत से 2·0 किमी की दूरी (हवा की दिशा में) पर विसरण गुणांक σᵧ = 150 मीटर और σᵤ = 50 मीटर मान लीजिए। (8 अंक)

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Directive word: Derive

This question asks you to derive. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Begin with the directive 'derive' for part (a), which demands rigorous mathematical derivation of Dupuit's equation for unconfined aquifer flow, followed by systematic numerical substitution. Allocate time proportionally: ~40% for (a) including derivation and calculation, ~30% for (b) on rapid mixing tank design with power calculation, and ~30% for (c) covering 5 R's explanation and Gaussian plume dispersion modeling. Structure as: (a) definition → derivation → substitution → result; (b) tank sizing → power calculation; (c)(i) conceptual explanation → (c)(ii) equation setup → concentration calculations at specified locations.

Key points expected

  • Part (a): Correct definition of aquifer with geological formation characteristics; rigorous derivation of Thiem/Dupuit equation Q = πK(H²-h²)/ln(R/r) using Darcy's law and radial flow assumptions; accurate substitution yielding Q ≈ 0.091 m³/s or 91 L/s
  • Part (b): Correct tank diameter D = 2.88 m and depth H = 5.76 m from detention time equation; power calculation P = G²μV ≈ 4.23 kW using given mixing intensity and viscosity
  • Part (c)(i): Identification of 5 R's (Refuse, Reduce, Reuse, Recycle, Recover) with specific contribution to waste hierarchy and pollution prevention in Indian context like Swachh Bharat
  • Part (c)(ii): Gaussian plume equation C(x,y,z) = Q/(2πUσyσz)exp[-y²/(2σy²)]{exp[-(z-H)²/(2σz²)]+exp[-(z+H)²/(2σz²)]}; ground level concentration ≈ 33.9 μg/m³ and at 50m ≈ 27.4 μg/m³ using reflection term
  • Recognition that parts (a) and (b) involve hydraulic design while (c) combines environmental management with atmospheric dispersion modeling
  • Proper unit conversions throughout: litres/day to m³/s, detention time consistency, and emission rate to concentration units
  • Physical interpretation of results: well discharge adequacy for irrigation, mixer power reasonableness, and NOx concentration comparison with NAAQS standards

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Precise definition of aquifer as water-bearing geological formation with yield capability; accurate statement of Dupuit assumptions (radial flow, horizontal equipotentials, steady state); correct rapid mixing tank purpose for coagulation; proper identification of 5 R's hierarchy; valid Gaussian plume model selection for D-stability conditionsBasic aquifer definition without geological context; states formula without proper assumptions; generic waste management description without 5 R's specificity; attempts plume equation with minor stability class errorsConfuses aquifer with aquitard/aquiclude; uses confined aquifer formula for unconfined case; omits 5 R's entirely; applies wrong dispersion model or ignores stability class
Numerical accuracy25%12.5Part (a): Q = π×10⁻⁴×(900-100)/ln(100) = 0.091 m³/s (or 91.2 L/s); Part (b): D = 2.88 m, H = 5.76 m, P = 4.23 kW; Part (c)(ii): C(ground) = 33.9 μg/m³, C(50m) = 27.4 μg/m³ with correct exponential decay calculationsCorrect method with arithmetic errors (e.g., ln(100) = 4.6 used correctly but final multiplication error); tank dimensions within 10% tolerance; power order of magnitude correct; concentration calculations with minor σ value misapplicationOrder of magnitude errors in discharge (e.g., missing π or ln term); fundamentally wrong tank geometry (using area instead of volume); power calculation without volume term; concentration units confused (g/m³ vs μg/m³)
Diagram quality15%7.5Clear cross-section of unconfined aquifer showing water table, cone of depression, H, h, R, r with flow lines; rapid mixing tank elevation with impeller, baffles, dimension labels; Gaussian plume schematic with coordinate system, source height, wind direction, and concentration profilesSketch present for aquifer but missing flow net or labels; tank diagram without mixer details; textual description of plume without visual representationNo diagrams despite visual nature of problems; or completely misleading sketches (e.g., vertical well shown as horizontal pipe); missing critical dimension labels
Step-by-step derivation25%12.5Part (a): Full derivation from Darcy's law Q = KA(dh/dr), A = 2πrh, integration from r to R with proper limits, separation of variables, and final rearrangement; Part (b): Clear detention time t = V/Q → V = Qt → cylindrical volume → D and H; Part (c)(ii): Systematic Gaussian equation development with coordinate substitutionsStates starting equation and jumps to final result with 'hence' or 'thus'; shows some integration steps but skips algebraic manipulation; correct final formula but unclear logical flowFormula presented without any derivation; or completely incorrect derivation path (e.g., using Bernoulli instead of Darcy); no intermediate steps shown for tank sizing
Practical interpretation15%7.5Comments on aquifer yield sustainability for Indian agriculture (e.g., comparison with typical tube well discharges of 10-50 L/s); discusses rapid mixer design implications for Delhi/Jaipur water treatment plants; relates 5 R's to Swachh Bharat Mission and circular economy; compares NOx concentrations with NAAQS (80 μg/m³ annual average) and health implicationsGeneric statement about 'adequate discharge' or 'sufficient mixing' without contextual numbers; mentions 5 R's importance without policy linkage; notes concentration values without standard comparisonNo interpretation of numerical results; or absurd conclusions (e.g., 91 L/s well suitable for city water supply); ignores environmental significance of NOx emissions

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