FE Other Disciplines (NCEES Fundamentals of Engineering) — Study Guide

Fluid mechanics is where fluency pays off most. Almost every FE fluids problem collapses into one of a few tools: fluid statics (pressure with depth), continuity (conservation of mass), the Bernoulli energy equation, or a friction-loss model. The exam skill is recognizing which tool a problem calls for. Once you name the right tool, the arithmetic is usually short.

1.1 Fluid properties and specific weight

A fluid is described first by its density ρ (mass per unit volume, kg/m³) and its specific weight γ = ρ·g (weight per unit volume, N/m³).

• For water at standard conditions, ρ = 1000 kg/m³, so γ = 1000 × 9.81 = 9,810 N/m³ ≈ 9.81 kN/m³.
• Specific gravity SG is the ratio of a fluid's density to that of water: SG = ρfluid / ρwater. An oil with SG = 0.8 has ρ = 800 kg/m³.
• Viscosity μ (dynamic, Pa·s) measures resistance to shear. Kinematic viscosity is ν = μ/ρ (m²/s).

The FE almost always hands you ρ, g, and SG directly. The skill being tested is converting cleanly among density, specific weight, and specific gravity.

1.2 Fluid statics: pressure with depth

In a static fluid, gauge pressure grows linearly with vertical depth:

p = ρ·g·h = γ·h

where h is the vertical distance below the free surface. Pressure at a point acts equally in all directions and does not depend on container shape or total fluid volume.

Absolute pressure adds atmospheric pressure: pabs = pgauge + patm (patm ≈ 101.3 kPa).

Manometers apply the same rule column by column. A mercury column (SG 13.6) of height 0.2 m represents p = 13,600 × 9.81 × 0.2 ≈ 26.7 kPa.

The classic trap: mixing gauge and absolute pressure, or using the slant length of a tube instead of the true vertical height h.

1.3 Buoyancy and submerged forces

Archimedes' principle: the buoyant force on a submerged or floating body equals the weight of the fluid it displaces:

Fb = ρfluid · g · V_displaced

A body floats when its average density is less than the fluid's, sinking until displaced weight equals body weight.

For a flat submerged surface, the resultant hydrostatic force is F = pcentroid · A = γ · hc · A, where h_c is the depth of the centroid. This force acts at the center of pressure, which lies below the centroid because pressure increases with depth. Recognizing that the force uses the pressure at the centroid — not the maximum pressure at the bottom — is a common point-earner.

1.4 Continuity and volumetric flow

Conservation of mass for steady incompressible flow gives the continuity equation:

A₁·V₁ = A₂·V₂

Velocity rises where area shrinks — halving the pipe area doubles the velocity.

• Volumetric flow rate: Q = A·V (m³/s). For a round pipe, A = π·D²/4.
• Mass flow rate: ṁ = ρ·Q (kg/s).

Worked example. Water flows at V = 3 m/s in a pipe of diameter D = 0.1 m. Find Q.

A = π/4 × (0.1)² = 7.854 × 10⁻³ m² Q = A·V = 7.854 × 10⁻³ × 3 = 0.0236 m³/s

The frequent mistake: forgetting that area scales with diameter squared, so a small change in diameter produces a large change in velocity.

1.5 The Bernoulli energy equation

Along a streamline with no losses, total head is conserved:

p/γ + V²/(2g) + z = constant

Each term is a "head" in meters — pressure head, velocity head, and elevation head. Energy simply trades among the three.

Worked example (Torricelli). A tank drains through a small opening a depth h = 5 m below the free surface. Find the jet velocity.

At the surface, pressure and elevation head convert entirely to velocity head at the jet: V = √(2·g·h) = √(2 × 9.81 × 5) = √98.1 ≈ 9.9 m/s

Real systems append a head-loss term h_L to the downstream side to account for friction:

p₁/γ + V₁²/(2g) + z₁ = p₂/γ + V₂²/(2g) + z₂ + h_L

1.6 Flow regime, friction, and open channels

The dimensionless Reynolds number sets the flow regime:

Re = ρ·V·D/μ = V·D/ν

• Re < ~2,100 → laminar (smooth, ordered)
• Re > ~4,000 → turbulent
• In between → transitional

Pipe friction loss uses Darcy–Weisbach:

h_L = f · (L/D) · V²/(2g)

where the friction factor f depends on Re and relative roughness (read from a Moody chart). For laminar flow, f = 64/Re directly.

Open-channel flow uses Manning's equation for velocity as a function of slope, roughness, and hydraulic radius. On the FE, the first move is almost always to compute Re, because it tells you which loss model and friction correlation are valid.

Common traps (Fluid Mechanics)

• Using slant height instead of vertical depth in p = γ·h.
• Mixing gauge and absolute pressure.
• Forgetting Q depends on D² (area), so velocity changes dramatically with diameter.
• Applying Bernoulli where friction is significant (long pipes) without a head-loss term.
• Using ρ where γ is needed (off by a factor of g).

What the FE tests here

Expect several statics/manometer problems, at least one continuity problem, one or two Bernoulli/Torricelli problems, and a Reynolds-number or Darcy–Weisbach friction problem. The Handbook has all the equations — your job is to identify the regime and pick the tool.