ESE ME Prelims flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Dimensional homogeneity

Every term added or equated in a physical equation must have the same base dimensions. This rule lets you eliminate wrong options and check derived formulas without solving them.

Error propagation formula

For y = f(x1, x2, ...), the relative uncertainty satisfies (σy/y)² = Σ(∂ln y/∂xi)²σxi² for independent random errors. This combines contributions in quadrature rather than by simple addition.

Newton-Raphson convergence

The iteration x_{n+1} = x_n − f(x_n)/f′(x_n) converges quadratically near a simple root, roughly doubling correct digits each step, but can diverge if the derivative is near zero.

Reynolds number

Re = ρVD/µ = VD/ν compares inertia to viscous forces. Pipe flow is laminar below about 2300, transitional to 4000, and turbulent above, which selects the friction-factor correlation.

Bernoulli equation

For steady, incompressible, inviscid flow along a streamline, p/ρ + V²/2 + gz is constant. Real systems add head loss and pump head terms to this ideal balance.

Centre of pressure

On a submerged plane, the centre of pressure lies below the centroid at depth h_cp = h_c + I_c/(A·h_c). The second term grows as the plate becomes shallower or wider.

Pump similarity laws

For geometrically similar pumps, Q ∝ ND³, H ∝ N²D², and P ∝ N³D⁵. These scaling relations allow predicting performance at a new speed or impeller diameter.

First law steady flow

For a steady-flow control volume, q − w = Δh + ΔKE + ΔPE per unit mass. Shaft work and heat transfer are signed quantities, so convention must be fixed before solving.

Isentropic ideal-gas relations

For reversible adiabatic processes, T₂/T₁ = (p₂/p₁)^((γ−1)/γ) and pV^γ is constant. These relations underpin compressor and turbine ideal-work calculations.

Carnot efficiency

The maximum efficiency of any heat engine between reservoirs at absolute temperatures T_H and T_L is 1 − T_L/T_H. No real cycle between the same reservoirs can exceed it.

Entropy generation

For any process, S_gen = ΔS_system + ΔS_surroundings ≥ 0. The lost work equals T₀S_gen, so minimising entropy generation maximises useful output.

Lumped capacitance criterion

A body can be treated as uniform in temperature when Bi = hL_c/k < 0.1. Then T(t) − T_∞ = (T_i − T_∞)e^(−t/τ), with time constant τ = ρVc/(hA).

Nusselt number

Nu = hL/k is the dimensionless convection heat-transfer coefficient, equal to the ratio of convective to conductive heat transfer across the same fluid layer. It comes from correlations in Re and Pr.

Log mean temperature difference

For a heat exchanger, ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), multiplied by a correction factor F for multipass arrangements. It drives the duty q = UAΔT_lm.

Fin efficiency

For a rectangular fin, η_f = tanh(mL)/(mL), where m = √(hP/(kA_c)). It is the ratio of actual heat transfer to the ideal case of the whole fin at base temperature.

Grashof condition

A four-bar linkage has a continuously rotating crank when s + l ≤ p + q, with s and l the shortest and longest links. Equality gives a change-point mechanism.

Kutzbach mobility equation

For planar mechanisms, F = 3(n − 1) − 2j − h, where n is the number of links, j lower pairs, and h higher pairs. It predicts the number of independent inputs needed.

Coefficient of speed fluctuation

C_s = (ω_max − ω_min)/ω_mean describes flywheel speed variation. The energy fluctuation satisfies ΔE = Iω²C_s, which sizes the flywheel for a given load cycle.

Secondary unbalanced force

In a reciprocating engine, the secondary force oscillates at twice crank speed and has magnitude about one-quarter of the primary force at typical connecting-rod ratios. It is not removed by a single counterweight.

Von Mises yield criterion

A ductile material yields when σ′ = √(σ₁² − σ₁σ₂ + σ₂²) reaches the yield strength. It matches distortion-energy theory and is less conservative than the Tresca criterion.

Goodman and Soderberg lines

Fatigue design uses 1/n = σ_m/S_ut + σ_a/S_e (Goodman) or σ_m/S_y + σ_a/S_e (Soderberg). Soderberg is more conservative; Gerber is a parabolic curve between them.

Bearing L10 life

The rated life L10 = (C/P)^p million revolutions, with p = 3 for ball and 10/3 for roller bearings. C is the basic dynamic load rating and P the equivalent dynamic load.

Fillet weld throat

For a 45° fillet weld, throat thickness equals 0.707 times the leg size. Parallel fillet welds carry load in shear, with τ = P/(0.707 h L) for weld length L.

Torsion of circular shaft

Shear stress τ = Tr/J and angle of twist θ = TL/(GJ), with polar moment J = πd⁴/32 for a solid shaft. Maximum stress occurs at the outer surface.

Euler buckling load

P_cr = π²EI/(KL)², where K depends on end conditions: 1 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, and 2 fixed-free. Long slender columns fail by elastic buckling.

Principal stress formula

σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]. The square-root term equals the maximum in-plane shear stress and the radius of Mohr's circle.

Transverse shear stress

τ = VQ/(Ib) gives shear stress at a distance from the neutral axis. For a rectangle the maximum is 1.5V/A at the neutral axis; for a solid circle it is 4V/(3A).

Taylor tool life equation

VT^n = C relates cutting speed V to tool life T. Exponent n is about 0.1–0.15 for HSS, 0.2–0.3 for carbide, and 0.4–0.5 for ceramic tools.

Chvorinov solidification rule

Solidification time t = B(V/A)², where V/A is the volume-to-surface-area ratio. A riser must have a larger modulus than the casting so it freezes last.

Merchant shear angle relation

The shear-plane angle satisfies φ = 45° + α/2 − β/2, with rake angle α and friction angle β. It determines chip thickness, shear area, and cutting-force components.

Economic order quantity

EOQ = √(2DS/H) minimises the sum of ordering and holding costs, where D is annual demand, S ordering cost, and H holding cost per unit per year.

PERT expected time

The three-point estimate t_e = (a + 4m + b)/6 weights the most likely duration m four times. Its variance is ((b − a)/6)², and the critical path has zero total float.