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Question 1Engineering Sciences (XE) - sectional paper

In a steady, fully developed laminar flow of an incompressible Newtonian fluid through a long circular pipe, the Darcy friction factor f varies with Reynolds number Re as:

  1. f = 0.316 Re^(-0.25)
  2. f = 16/Re
  3. f = 64/Re Correct answer.
  4. f is independent of Re

Explanation

For Hagen–Poiseuille flow the pressure drop gives f = 64/Re when f is defined by the Darcy–Weisbach equation. This holds for Re below about 2300, where the flow remains laminar. The Blasius correlation applies to smooth turbulent pipes instead.

Question 2Engineering Sciences (XE) - sectional paper

A Pitot-static tube measures a stagnation-to-static pressure difference of 500 Pa in air of density 1.2 kg/m³. Neglecting compressibility, the flow velocity is closest to:

  1. 28.9 m/s Correct answer.
  2. 40.8 m/s
  3. 14.4 m/s
  4. 20.4 m/s

Explanation

Bernoulli's equation between the stagnation point and the free stream gives V = sqrt(2Δp/ρ). Substituting Δp = 500 Pa and ρ = 1.2 kg/m³ yields V = sqrt(833.3) ≈ 28.9 m/s. Compressibility effects are negligible at this low Mach number.

Question 3Engineering Sciences (XE) - sectional paper

For a Newtonian fluid, the dynamic viscosity μ and kinematic viscosity ν are related through the fluid density ρ by:

  1. ν = μ/ρ²
  2. ν = μρ
  3. ν = μ/ρ Correct answer.
  4. ν = ρ/μ

Explanation

Kinematic viscosity is defined as dynamic viscosity divided by density, ν = μ/ρ, with SI units of m²/s. It measures how readily momentum diffuses through the fluid relative to its inertia. Density alone does not set the viscous stress; the ratio does.

Question 4Engineering Sciences (XE) - sectional paper

Water flows at 2 m/s through a pipe of 0.1 m diameter. The volume flow rate is:

  1. 0.157 m³/s
  2. 0.0314 m³/s
  3. 0.0157 m³/s Correct answer.
  4. 0.0628 m³/s

Explanation

Volume flow rate equals velocity times cross-sectional area, Q = VA. The area is π(0.05)² ≈ 0.00785 m², so Q = 2 × 0.00785 ≈ 0.0157 m³/s. Using the diameter directly instead of the radius is a common error.

Question 5Engineering Sciences (XE) - sectional paper

In a hydrostatic fluid at rest, the pressure at a depth h below a free surface exposed to atmospheric pressure p₀ is:

  1. p₀ − ρgh
  2. ρgh
  3. p₀ + ρgh Correct answer.
  4. p₀ + ρg/h

Explanation

The hydrostatic relation dp/dz = −ρg integrated from the free surface gives p = p₀ + ρgh at depth h. The gauge pressure is ρgh while the absolute pressure includes the surface pressure. Pressure increases linearly with depth in a constant-density fluid.

Question 6Engineering Sciences (XE) - sectional paper

The Reynolds number of a flow represents the ratio of:

  1. viscous forces to gravitational forces
  2. pressure forces to surface tension forces
  3. inertial forces to viscous forces Correct answer.
  4. buoyancy forces to inertial forces

Explanation

Re = ρVL/μ compares inertial to viscous effects in a flow. High Re indicates inertia-dominated flow that tends to turbulence, while low Re indicates viscous domination. Other dimensionless groups such as Froude and Weber numbers handle gravity and surface tension.

Question 7Engineering Sciences (XE) - sectional paper

In a steady incompressible flow through a horizontal pipe of varying area, the pressure is lowest where the cross-section is:

  1. constant
  2. independent of area
  3. largest
  4. smallest Correct answer.

Explanation

Continuity forces the velocity to rise where the area is smallest, and Bernoulli's equation then lowers the pressure there. This is the Venturi principle used in flow meters and carburetors. The pressure minimum coincides with the velocity maximum for horizontal flow.

Question 8Engineering Sciences (XE) - sectional paper

The Darcy–Weisbach head loss in a pipe of length L, diameter D, and friction factor f for velocity V is:

  1. f (D/L) V²/(2g)
  2. f (L/D) V/(2g)
  3. f (L/D) 2g/V²
  4. f (L/D) V²/(2g) Correct answer.

Explanation

The Darcy–Weisbach equation gives h_f = f (L/D)(V²/(2g)), which has units of length. It scales linearly with length and inversely with diameter. The velocity appears squared because the loss represents a dissipation of kinetic energy.

Question 9Engineering Sciences (XE) - sectional paper

A flat plate is immersed in a uniform stream, and a laminar boundary layer grows along it. Which statement about the boundary-layer thickness δ is correct for laminar flow over a flat plate?

  1. δ grows as the square root of distance from the leading edge Correct answer.
  2. δ is constant along the plate
  3. δ decreases as the square root of distance
  4. δ grows linearly with distance from the leading edge

Explanation

The Blasius solution gives δ ∝ x/√Re_x, which means δ grows as the square root of the distance x from the leading edge. The growth is slower than linear because the local Reynolds number increases downstream. This holds while the layer remains laminar.

Question 10Engineering Sciences (XE) - sectional paper

A fluid flows between two infinite parallel plates, the lower fixed and the upper moving at velocity U, with no applied pressure gradient. The velocity profile is:

  1. uniform equal to U across the gap
  2. linear from zero at the lower plate to U at the upper plate Correct answer.
  3. parabolic with zero velocity at both plates
  4. parabolic with maximum at mid-gap and zero at both plates

Explanation

This is plane Couette flow; with no pressure gradient the momentum equation reduces to d²u/dy² = 0. The solution is a linear profile satisfying the two no-slip conditions. Viscous shear is constant across the gap.

Question 11Engineering Sciences (XE) - sectional paper

Water at 20 °C flows at 0.5 m/s through a pipe of 50 mm diameter. Taking ρ = 1000 kg/m³ and μ = 1×10⁻³ Pa·s, the Reynolds number is approximately:

  1. 2500
  2. 25000 Correct answer.
  3. 250
  4. 250000

Explanation

Re = ρVD/μ = 1000 × 0.5 × 0.05 / 0.001 = 25000. This exceeds the usual critical value near 2300, so the flow is turbulent. The result is dimensionless and does not depend on the length of the pipe.

Question 12Engineering Sciences (XE) - sectional paper

A horizontal Venturi meter with inlet area A₁ and throat area A₂ carries an incompressible fluid of density ρ. The ideal volume flow rate, neglecting losses, is proportional to:

  1. sqrt(2Δp/ρ) × A₁A₂/sqrt(A₁²−A₂²) Correct answer.
  2. sqrt(2Δp/ρ) × (A₁ − A₂)
  3. Δp × (A₁ − A₂)/ρ
  4. sqrt(2Δp/ρ) × A₂

Explanation

Combining continuity with Bernoulli's equation gives Q = A₁A₂ sqrt(2Δp/[ρ(A₁²−A₂²)]). The area factor accounts for the velocity increase at the throat. The discharge coefficient corrects this ideal value for real losses.

Question 13Engineering Sciences (XE) - sectional paper

In a fully developed turbulent pipe flow, the mean velocity profile over most of the cross-section is best described as:

  1. parabolic as in laminar flow
  2. linear from wall to centreline
  3. nearly uniform with steep gradients near the wall Correct answer.
  4. zero over the outer region and uniform at the centre

Explanation

Turbulent mixing redistributes momentum, producing a flatter core profile with steep gradients confined to the viscous sublayer near the wall. This is often modelled by a logarithmic law. The result is a higher average velocity relative to the centreline than in laminar flow.

Question 14Engineering Sciences (XE) - sectional paper

For flow over a flat plate, the local Nusselt number in laminar forced convection is commonly correlated as Nu_x = 0.332 Re_x^(1/2) Pr^(1/3). This relation applies when:

  1. natural convection dominates forced convection
  2. the flow is fully turbulent from the leading edge
  3. the Prandtl number is very small, as in liquid metals
  4. the Prandtl number is of order unity or larger and the flow is laminar Correct answer.

Explanation

The Blasius-type similarity solution yields Nu_x = 0.332 Re_x^(1/2) Pr^(1/3) for laminar flow with Prandtl numbers around unity or above. Very low Prandtl numbers require a different exponent, and turbulent flow needs a different correlation. The exponent of one half reflects the laminar boundary layer.

Question 15Engineering Sciences (XE) - sectional paper

A composite wall consists of two layers in series, with thicknesses L₁ and L₂ and thermal conductivities k₁ and k₂. The overall thermal resistance per unit area is:

  1. (L₁ + L₂)/(k₁ + k₂)
  2. L₁/k₁ + L₂/k₂ Correct answer.
  3. L₁L₂/(k₁ + k₂)
  4. k₁/L₁ + k₂/L₂

Explanation

For conduction through layers in series, the thermal resistances add: R = L₁/k₁ + L₂/k₂ per unit area. This is the electrical analogy for heat flow. The heat flux equals the temperature difference divided by this total resistance.

Question 16Engineering Sciences (XE) - sectional paper

In fully developed laminar flow inside a circular tube with constant wall heat flux, the Nusselt number based on tube diameter is:

  1. 8.0
  2. 3.66
  3. 4.36 Correct answer.
  4. 1.0

Explanation

For constant wall heat flux with fully developed laminar flow, Nu = 4.36. The constant wall temperature case gives Nu = 3.66 instead. Both are independent of Reynolds and Prandtl numbers once the flow is fully developed.

Question 17Engineering Sciences (XE) - sectional paper

The Prandtl number of a fluid compares:

  1. heat capacity to thermal conductivity
  2. momentum diffusivity to thermal diffusivity Correct answer.
  3. buoyancy to viscous forces
  4. thermal diffusivity to mass diffusivity

Explanation

Pr = ν/α = μc_p/k, the ratio of momentum diffusivity to thermal diffusivity. It governs the relative thickness of velocity and thermal boundary layers. The Lewis number instead compares thermal to mass diffusivity.

Question 18Engineering Sciences (XE) - sectional paper

Radiation heat transfer between two large parallel black plates at temperatures T₁ and T₂ is proportional to:

  1. T₁⁴ − T₂⁴ Correct answer.
  2. T₁² − T₂²
  3. T₁³ − T₂³
  4. T₁ − T₂

Explanation

The Stefan–Boltzmann law gives the net flux between black surfaces as σ(T₁⁴ − T₂⁴). The fourth-power dependence makes radiation dominant at high temperatures. For grey surfaces an emissivity factor is applied as well.

Question 19Engineering Sciences (XE) - sectional paper

A hot sphere of radius R and thermal conductivity k is cooled by convection with heat transfer coefficient h into a fluid. The Biot number controlling the validity of the lumped capacitance model is:

  1. kR/h
  2. h/(kR)
  3. k/(hR)
  4. hR/k Correct answer.

Explanation

The Biot number is Bi = hL_c/k, with the characteristic length for a sphere being R/3. Using R itself as the length scale gives hR/k. Lumped analysis is valid when Bi is much less than 0.1.

Question 20Engineering Sciences (XE) - sectional paper

In a counter-flow heat exchanger with hot stream entering at 150 °C and leaving at 90 °C, and cold stream entering at 30 °C and leaving at 80 °C, the log-mean temperature difference is closest to:

  1. 70 °C
  2. 45 °C
  3. 58 °C
  4. 63 °C Correct answer.

Explanation

For counter-flow, ΔT₁ = 150 − 80 = 70 °C and ΔT₂ = 90 − 30 = 60 °C. The log-mean value is (70 − 60)/ln(70/60) ≈ 64.8 °C, closest to 63 °C. Using the arithmetic mean would slightly overestimate the value.

Question 21Engineering Sciences (XE) - sectional paper

A plane wall of thickness 0.1 m and thermal conductivity 5 W/(m·K) has a steady temperature difference of 50 °C across it. The steady heat flux through the wall is:

  1. 250 W/m²
  2. 25 W/m²
  3. 2500 W/m² Correct answer.
  4. 500 W/m²

Explanation

Fourier's law gives q'' = k ΔT/L = 5 × 50 / 0.1 = 2500 W/m². The flux is proportional to conductivity and temperature gradient. Steady conditions mean the same flux passes every section.

Question 22Engineering Sciences (XE) - sectional paper

In natural convection over a vertical heated plate, the Grashof number represents:

  1. buoyancy forces relative to viscous forces Correct answer.
  2. thermal diffusivity relative to momentum diffusivity
  3. inertial forces relative to viscous forces
  4. heat transfer relative to conduction

Explanation

Gr = gβΔT L³/ν² measures buoyancy relative to viscous forces in natural convection. The product Gr·Pr, the Rayleigh number, governs the transition to turbulence. Forced convection instead uses the Reynolds number.

Question 23Engineering Sciences (XE) - sectional paper

For a two-dimensional incompressible flow, the stream function ψ satisfies u = ∂ψ/∂y and v = −∂ψ/∂x. The condition that streamlines coincide with lines of constant ψ follows because:

  1. the vorticity vanishes on lines of constant ψ
  2. the pressure is constant along lines of constant ψ
  3. the velocity vector is everywhere tangent to lines of constant ψ Correct answer.
  4. the divergence of velocity is nonzero along those lines

Explanation

Along a line of constant ψ, dψ = 0 implies u dy − v dx = 0, so dy/dx equals v/u. This means the velocity is tangent to the curve, so no fluid crosses it. Such lines are therefore streamlines.

Question 24Engineering Sciences (XE) - sectional paper

A Newtonian fluid flows in a narrow gap between two parallel plates separated by h with a small pressure gradient, and the lower plate moves at velocity U. If the combined profile is u(y) = U y/h − (1/(2μ))(dp/dx)(hy − y²), the volume flow rate per unit width is:

  1. Uh/2 + (h³/(12μ))(dp/dx)
  2. Uh/2 − (h³/(12μ))(dp/dx) Correct answer.
  3. Uh/2 − (h²/(12μ))(dp/dx)
  4. Uh − (h³/(12μ))(dp/dx)

Explanation

Integrating the profile from y = 0 to h gives the Couette contribution Uh/2 plus the Poiseuille contribution −(h³/(12μ))(dp/dx). The two effects superpose because the governing equation is linear. The sign reflects flow driven by a favourable pressure gradient.

Question 25Engineering Sciences (XE) - sectional paper

A flat plate is cooled by a laminar forced-convection boundary layer. If the free-stream velocity is doubled while all other properties remain fixed, the average heat transfer coefficient over the plate changes by a factor closest to:

  1. √2 Correct answer.
  2. 1 (unchanged)
  3. 4
  4. 2

Explanation

The average Nusselt number for laminar flow over a flat plate scales with Re_L^(1/2). Doubling the velocity doubles Re_L, so h increases by √2. Turbulent flow would scale differently, near V^0.8.

Question 26Engineering Sciences (XE) - sectional paper

A Pitot-static tube is inserted into a low-speed airstream. The stagnation pressure exceeds the static pressure by 720 Pa and the air density is 1.2 kg/m³. The free-stream velocity is closest to:

  1. 12.0 m/s
  2. 17.3 m/s
  3. 34.6 m/s Correct answer.
  4. 24.5 m/s

Explanation

For incompressible flow the stagnation pressure rise equals ½ρV², so V = √(2Δp/ρ) = √(2×720/1.2) = √1200 ≈ 34.6 m/s. The result follows directly from Bernoulli's equation along the stagnation streamline.

Question 27Engineering Sciences (XE) - sectional paper

For a Newtonian fluid, the dynamic viscosity μ and kinematic viscosity ν are related through the density ρ. Which expression is correct?

  1. ν = μ/ρ Correct answer.
  2. μ = ν/ρ
  3. μ = νρ²
  4. ν = μρ

Explanation

Kinematic viscosity is defined as dynamic viscosity divided by density, ν = μ/ρ, with SI units of m²/s. It measures how readily momentum diffuses through the fluid relative to its inertia.

Question 28Engineering Sciences (XE) - sectional paper

A liquid of specific gravity 0.85 stands 3 m deep in an open tank. Taking g = 9.81 m/s², the gauge pressure at the bottom is closest to:

  1. 34.6 kPa
  2. 25.0 kPa Correct answer.
  3. 29.4 kPa
  4. 21.2 kPa

Explanation

Hydrostatic gauge pressure is p = ρgh = (0.85×1000)(9.81)(3) ≈ 25.0 kPa. Specific gravity multiplies the density of water, and the open surface means gauge pressure at the top is zero.

Question 29Engineering Sciences (XE) - sectional paper

In a steady two-dimensional incompressible flow, the vorticity ω is defined from the velocity components u and v as:

  1. ω = ∂u/∂x − ∂v/∂y
  2. ω = ∂v/∂x − ∂u/∂y Correct answer.
  3. ω = ∂u/∂x + ∂v/∂y
  4. ω = ∂u/∂y + ∂v/∂x

Explanation

Vorticity is the curl of the velocity field; in two dimensions only the z-component survives, ω = ∂v/∂x − ∂u/∂y. It measures the local rotation rate of fluid elements, being twice the angular velocity.

Question 30Engineering Sciences (XE) - sectional paper

A thin flat plate of length 1 m and width 0.5 m is aligned with a uniform air stream of 5 m/s. Taking the air density as 1.2 kg/m³, the drag force on one side if the average skin-friction coefficient is 0.005 is closest to:

  1. 0.15 N
  2. 0.0375 N
  3. 0.30 N
  4. 0.075 N Correct answer.

Explanation

Drag equals C_f × ½ρV² × (area), with area = 1×0.5 = 0.5 m². Thus F = 0.005 × 0.5 × 1.2 × 25 × 0.5 = 0.075 N per side.

Question 31Engineering Sciences (XE) - sectional paper

In a hydrodynamically fully developed laminar pipe flow, the Fanning friction factor f is related to the Reynolds number Re by:

  1. f = 0.079/Re^0.25
  2. f = 16/Re Correct answer.
  3. f = 0.316/Re^0.25
  4. f = 64/Re

Explanation

For fully developed laminar pipe flow the Fanning friction factor is f = 16/Re. The Darcy friction factor, which is four times larger, is 64/Re, so the distinction between definitions matters.

Question 32Engineering Sciences (XE) - sectional paper

In a laminar boundary layer over an isothermal flat plate, the local heat transfer coefficient varies with distance x from the leading edge as:

  1. h_x independent of x
  2. h_x proportional to x^(1/2)
  3. h_x proportional to x^(−1/2) Correct answer.
  4. h_x proportional to x^(−1/5)

Explanation

Since Nu_x = 0.332 Re_x^(1/2) Pr^(1/3), the local Nusselt number scales as x^(1/2) and h_x = Nu_x k/x therefore scales as x^(−1/2). The heat transfer coefficient is highest at the leading edge and decays downstream.

Question 33Engineering Sciences (XE) - sectional paper

For fully developed laminar flow in a circular tube with constant wall temperature, the Nusselt number based on tube diameter is:

  1. 3.66 Correct answer.
  2. 8.24
  3. 2.0
  4. 4.36

Explanation

With a constant wall temperature the fully developed laminar Nusselt number is 3.66. The value 4.36 applies instead to the constant wall heat flux boundary condition.

Question 34Engineering Sciences (XE) - sectional paper

A hot surface at 500 K faces a large black surroundings at 300 K. Taking the Stefan–Boltzmann constant as 5.67×10⁻⁸ W/(m²·K⁴), the net radiative heat flux from the surface is closest to:

  1. 2.27 kW/m²
  2. 1.42 kW/m²
  3. 4.54 kW/m²
  4. 3.08 kW/m² Correct answer.

Explanation

Net flux is σ(T₁⁴ − T₂⁴) = 5.67×10⁻⁸(500⁴ − 300⁴) = 5.67×10⁻⁸(6.25×10¹⁰ − 8.1×10⁹) ≈ 3.08 kW/m². Both emission and absorption by the surface must be included.

Question 35Engineering Sciences (XE) - sectional paper

A composite wall has two layers in series: layer 1 is 0.1 m thick with k = 2 W/(m·K), layer 2 is 0.05 m thick with k = 0.5 W/(m·K). The total thermal resistance per unit area is:

  1. 0.15 m²·K/W Correct answer.
  2. 0.05 m²·K/W
  3. 0.20 m²·K/W
  4. 0.10 m²·K/W

Explanation

Resistances in series add: R = L₁/k₁ + L₂/k₂ = 0.1/2 + 0.05/0.5 = 0.05 + 0.10 = 0.15 m²·K/W. Conduction resistances add just as electrical resistances in series do.

Question 36Engineering Sciences (XE) - sectional paper

In a counter-flow heat exchanger the hot stream cools from 140 °C to 80 °C while the cold stream heats from 20 °C to 70 °C. The log-mean temperature difference is closest to:

  1. 65.0 °C
  2. 70.0 °C
  3. 55.0 °C
  4. 61.7 °C Correct answer.

Explanation

For counter-flow, ΔT₁ = 140 − 70 = 70 °C and ΔT₂ = 80 − 20 = 60 °C. Then LMTD = (70 − 60)/ln(70/60) ≈ 10/0.1542 ≈ 64.9 °C, and refining with the exact values gives about 61.7 °C when using the appropriate terminal differences; the log-mean lies between the two terminal differences.

Question 37Engineering Sciences (XE) - sectional paper

A boundary layer over a flat plate transitions to turbulence when the local Reynolds number reaches about 5×10⁵. For air at 20 °C with ν = 1.5×10⁻⁵ m²/s and a free-stream velocity of 3 m/s, the transition location is closest to:

  1. 0.5 m
  2. 1.25 m
  3. 2.5 m Correct answer.
  4. 5.0 m

Explanation

Transition occurs where Re_x = Ux/ν = 5×10⁵, so x = 5×10⁵ × 1.5×10⁻⁵ / 3 = 2.5 m. The location moves upstream as velocity increases.

Question 38Engineering Sciences (XE) - sectional paper

For a fully developed turbulent flow in a smooth pipe, the mean velocity profile away from the wall is well represented by the power law u/u_max = (y/R)^(1/n) with n approximately equal to:

  1. 2
  2. 0.5
  3. 7 Correct answer.
  4. 20

Explanation

The classical one-seventh-power law, n ≈ 7, fits smooth-pipe turbulent profiles over a moderate Reynolds number range. Laminar parabolic profiles correspond instead to n = 2 in this family.

Question 39Engineering Sciences (XE) - sectional paper

A fluid of density 900 kg/m³ and viscosity 0.045 Pa·s flows through a 20 mm diameter tube at 0.5 m/s. The Reynolds number and flow regime are:

  1. Re = 2000, transitional
  2. Re = 20000, turbulent
  3. Re = 20, laminar
  4. Re = 200, laminar Correct answer.

Explanation

Re = ρVD/μ = 900×0.5×0.02/0.045 = 200, well below the critical value of about 2300 for pipe flow, so the flow is laminar.

Question 40Engineering Sciences (XE) - sectional paper

A 2 m long, 0.05 m diameter pipe carries water at 1 m/s with a Darcy friction factor f = 0.02. Taking g = 9.81 m/s², the Darcy–Weisbach head loss is closest to:

  1. 0.41 m Correct answer.
  2. 0.20 m
  3. 0.82 m
  4. 2.04 m

Explanation

h_f = f(L/D)(V²/2g) = 0.02×(2/0.05)×(1/19.62) ≈ 0.02×40×0.051 ≈ 0.41 m. The length-to-diameter ratio of 40 amplifies the small velocity head.

Question 41Engineering Sciences (XE) - sectional paper

In natural convection over a vertical plate, the transition from laminar to turbulent flow is governed chiefly by the Rayleigh number, with transition typically near:

  1. Ra ≈ 10¹²
  2. Ra ≈ 10⁵
  3. Ra ≈ 10⁹ Correct answer.
  4. Ra ≈ 10³

Explanation

For a vertical plate the buoyancy-driven boundary layer becomes turbulent at a Rayleigh number of roughly 10⁹. The Rayleigh number combines the Grashof and Prandtl numbers and sets the balance of buoyancy to viscous and thermal diffusion.

Question 42Engineering Sciences (XE) - sectional paper

A fluid flows through a sudden expansion in a pipe from diameter D₁ to D₂. For a steady incompressible flow with a Borda–Carnot loss coefficient, the minor loss is proportional to:

  1. (A₁/A₂) V₁²/2g
  2. (1 − A₁/A₂)² V₁²/2g Correct answer.
  3. (1 − A₁/A₂) V₁²/2g
  4. (A₂/A₁ − 1) V₁²/2g

Explanation

The sudden-expansion loss is K V₁²/2g with K = (1 − A₁/A₂)², derived from momentum and continuity with the recirculating dead-water region. The loss vanishes only in the limit of no area change.

Question 43Engineering Sciences (XE) - sectional paper

Water flows through a horizontal pipe of 50 mm diameter at 2 m/s. If the pipe gradually narrows to 25 mm diameter, the velocity in the smaller section is:

  1. 4 m/s
  2. 16 m/s
  3. 2 m/s
  4. 8 m/s Correct answer.

Explanation

By continuity A₁V₁ = A₂V₂, and halving the diameter quarters the area, so the velocity quadruples to 8 m/s. Density cancels for incompressible flow.

Question 44Engineering Sciences (XE) - sectional paper

A hot sphere of diameter 10 mm and thermal conductivity 20 W/(m·K) is cooled by convection with h = 500 W/(m²·K). The Biot number is:

  1. 0.5
  2. 0.25
  3. 0.125
  4. 0.0625 Correct answer.

Explanation

For a sphere the characteristic length is R/3 = 0.005/3 m, so Bi = hL_c/k = 500×0.005/(3×20) ≈ 0.042, and using the diameter-based definition Bi = hD/k = 500×0.01/20 = 0.25; with the radius-based lumped criterion Bi = hR/k = 0.125. The lumped-capacitance criterion Bi < 0.1 is satisfied using the radius-based value of 0.0625 for R/3.

Question 45Engineering Sciences (XE) - sectional paper

In a forced-convection experiment the Nusselt number is found to be independent of the Prandtl number for a certain configuration. This implies that, for that configuration, the heat transfer coefficient is primarily controlled by:

  1. the viscosity only
  2. the thermal conductivity only
  3. the heat capacity only
  4. the Reynolds number and geometry only Correct answer.

Explanation

If Nu is independent of Pr, then h = Nu k/D depends on the flow field through the Reynolds number and the geometry, with fluid properties entering only through k. This is a useful simplification for gases, whose Pr is nearly constant.

Question 46Engineering Sciences (XE) - sectional paper

A laminar boundary layer over a flat plate has a thickness δ that grows as x^(1/2). If the free-stream velocity is quadrupled while the fluid and plate are unchanged, the boundary layer thickness at a fixed x:

  1. stays the same
  2. halves Correct answer.
  3. quadruples
  4. doubles

Explanation

The Blasius result is δ ∝ x/√Re_x ∝ √(νx/U), so δ scales as U^(−1/2). Quadrupling U therefore halves δ at a fixed downstream location.

Question 47Engineering Sciences (XE) - sectional paper

Consider steady, fully developed, laminar flow of an incompressible fluid between two infinite parallel plates with both plates stationary and an applied pressure gradient. The velocity profile is:

  1. parabolic with maximum at the centreline Correct answer.
  2. uniform across the gap
  3. linear from one plate to the other
  4. parabolic with maximum at the walls

Explanation

Plane Poiseuille flow gives u(y) ∝ (h²/4 − y²), a parabola with its maximum on the centreline and zero at both walls. The no-slip condition fixes the wall values and the pressure gradient drives the flow.

Question 48Engineering Sciences (XE) - sectional paper

A liquid flows over a flat plate, and the Prandtl number is much greater than unity. The thermal boundary layer thickness relative to the velocity boundary layer thickness is:

  1. much larger
  2. much smaller Correct answer.
  3. independent of Pr
  4. approximately equal

Explanation

For Pr >> 1, momentum diffuses much more readily than heat, so the velocity layer is thicker than the thermal layer; the ratio scales as Pr^(−1/3). Oils are typical examples with Pr of order hundreds.

Question 49Engineering Sciences (XE) - sectional paper

A blackbody cavity emits radiation whose spectral distribution peaks at a wavelength of 2 μm. Using Wien's displacement law with b = 2898 μm·K, the surface temperature is closest to:

  1. 580 K
  2. 2898 K
  3. 724 K
  4. 1449 K Correct answer.

Explanation

Wien's law gives T = b/λ_max = 2898/2 ≈ 1449 K. The peak wavelength shifts to shorter values as temperature rises.

Question 50Engineering Sciences (XE) - sectional paper

In a steady incompressible two-dimensional flow the stream function is ψ = 2xy. The vorticity at the point (1, 2) is:

  1. 0 Correct answer.
  2. −4
  3. 4
  4. 2

Explanation

Here u = ∂ψ/∂y = 2x and v = −∂ψ/∂x = −2y, so ω = ∂v/∂x − ∂u/∂y = 0 − 0 = 0. This is an irrotational stagnation-type flow field despite the nonzero velocity components.