(a) Prove that the power transmission through nozzle is maximum when d/D = √(D/8fl). Neglect the minor losses. (10 marks)

(b) Soil from a particular site yields a maximum dry unit weight of 18 kN/m³ at an optimum moisture content of 16% during a sta

Question

(a) Prove that the power transmission through nozzle is maximum when d/D = √(D/8fl). Neglect the minor losses. (10 marks)

(b) Soil from a particular site yields a maximum dry unit weight of 18 kN/m³ at an optimum moisture content of 16% during a standard Proctor test. If the value of G is 2·65, what is the degree of saturation? What is the maximum dry unit weight, it can be further compacted to? Take the unit weight of water as 9·81 kN/m³. (10 marks)

(c) Draw the possible gradually varied flow profiles for critical slope. Indicate very clearly the boundary conditions. (10 marks)

(d) In a fluid machine, the torque T of the impeller is known to depend on the diameter D and speed N of the impeller, the density ρ and dynamic viscosity μ of the fluid. Obtain the relationship in a dimensionless form using Buckingham method. Specify the use of non-dimension numbers in design problems. (10 marks)

(e) A 2 m × 2 m square footing is founded at a depth of 0·8 m in a homogeneous bed of sand having a unit weight of 19 kN/m³ and an angle of shearing resistance of 38°. Assuming the water table to be at a great depth, compute the safe load that can be carried by the footing. Use Terzaghi's theory and assume a factor of safety of 3. For φ = 38°, take Nq = 65 and Nγ = 80. (10 marks)

UPSC Answer Check · Accepted Answer

This multi-part question requires proving a nozzle efficiency condition in (a), solving soil compaction calculations in (b), drawing GVF profiles in (c), deriving dimensionless parameters in (d), and computing bearing capacity in (e). Allocate approximately 20% time to each part given equal 10-mark weighting. Begin with clear statements of given data, show complete derivations for (a) and (d), present neat diagrams for (c), and demonstrate all calculation steps for (b) and (e) with proper units and FOS application.
- For (a): Derive power transmission equation P = ρgQ(H - hf), express hf = 4fLV²/2gD, substitute V = Q/(πD²/4), differentiate dP/dd = 0 to obtain d/D = √(D/8fL)
- For (b): Calculate void ratio e = (Gγw/γd) - 1 = 0.441, then degree of saturation S = wG/e = 96.1%; for maximum possible γd at S=100%, use γd(max) = Gγw/(1+wG) = 19.35 kN/m³
- For (c): Draw C1, C2, and C3 profiles on critical slope (yc = yn), showing C1 above normal depth, C2 between critical and normal, C3 below critical depth with proper boundary condition annotations
- For (d): Apply Buckingham π-theorem with 6 variables and 3 fundamental dimensions to obtain power coefficient P/ρN³D⁵ = f(ρND²/μ) or torque coefficient T/ρN²D⁵ = φ(Re); explain use of power coefficient for pump similarity and Reynolds number for viscous scale effects
- For (e): Apply Terzaghi's bearing capacity equation for square footing: qu = 1.3cNc + γDfNq + 0.4γBNγ with c=0, compute qu = 0 + 19×0.8×65 + 0.4×19×2×80 = 988 + 1216 = 2204 kPa, then qsafe = qu/3 = 734.7 kPa and safe load = qsafe × 4 = 2938.8 kN

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that (a) requires maximization of power with respect to nozzle diameter, (b) involves phase relationships in soil mechanics, (c) requires understanding that yc = yn on critical slope, (d) needs proper identification of repeating variables for Buckingham π-theorem, and (e) applies Terzaghi's bearing capacity for cohesionless soil with shape factors; no conceptual confusion between sub-critical and super-critical flows in (c)	Minor errors in one sub-part such as confusing standard with modified Proctor in (b), or using wrong shape factors in (e), or misidentifying profile types in (c); core concepts mostly correct but some muddled application	Fundamental misconceptions such as treating (a) as energy minimization instead of power maximization, using wrong bearing capacity equation (general instead of Terzaghi) in (e), or drawing M-profiles instead of C-profiles in (c)
Numerical accuracy	20%	10	All calculations precise: (b) S = 96.1% and γd(max) = 19.35 kN/m³, (e) qu = 2204 kPa and safe load ≈ 2940 kN; proper significant figures, unit conversions correct, no arithmetic errors in algebraic manipulation of (a) and (d)	Minor calculation errors in one sub-part (e.g., wrong decimal place in S calculation, or arithmetic error in bearing capacity), but method and formula selection correct; final answers slightly off but traceable	Major numerical errors such as forgetting factor of safety in (e), calculating degree of saturation >100% without comment in (b), or algebraic mistakes in differentiation leading to wrong exponent in (a)
Diagram quality	20%	10	For (c): Neatly drawn C1, C2, C3 profiles with properly labeled axes (y vs x), normal depth line, critical depth line coinciding, flow direction arrows, and boundary conditions specified (reservoir for C1, free overfall/control section for C2/C3); for (a) and (d), clear schematic diagrams of nozzle setup and impeller geometry if drawn	Profiles drawn but lacking clear boundary condition labels, or yc and yn lines not clearly distinguished; diagrams understandable but missing essential annotations for full credit	Missing diagram for (c), or profiles drawn without understanding that C-profiles have yc = yn; confused with S-profiles or M-profiles; illegible sketches or wrong profile curvature (concave vs convex)
Step-by-step derivation	20%	10	(a) Complete derivation from P = γQ(H - hf) through substitution of V=Q/A, expressing hf in terms of d, differentiation, and simplification to required form; (d) proper Buckingham π steps—listing variables, determining m=3, n=6, forming π terms, solving exponents by dimensional homogeneity, final dimensionless relationship with clear explanation of power coefficient and Reynolds number	Derivations present but skips key steps (e.g., jumps to final derivative without showing chain rule application in (a), or assumes π terms without solving for exponents in (d)); logically follows but examiner must fill gaps	Missing derivations entirely—states final results without proof in (a) and (d), or completely wrong differentiation approach; numerical parts (b) and (e) show only final answers without intermediate steps
Practical interpretation	20%	10	For (a): Explains practical significance of optimal nozzle diameter in penstock design for hydroelectric plants; for (b): Discusses field compaction control and why 100% saturation is theoretical limit; for (d): Explains pump scaling laws and model testing applications; for (e): Comments on applicability of Terzaghi's theory vs Meyerhof or Hansen for high φ soils, and effect of water table lowering	Brief mention of practical relevance without elaboration; standard comments on factor of safety significance or pump similarity without connecting to Indian engineering practice	No practical interpretation provided; treats all parts as purely academic exercises; fails to mention that (e) result is for dry conditions and water table correction would be needed in actual design

Q5

Directive word: Prove

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Civil Engineering 2022 Paper I