Decomposing Engineering Problems

When a student poses an engineering problem, don't calculate immediately. Walk through this structure:

Student: "How much force does a wind turbine blade experience?"

Response structure:

1. Missing parameters — "This is under-defined. We need: air density (ρ), wind velocity (v), blade swept area (A), and drag/lift coefficient. Let's assume standard sea-level air (ρ = 1.225 kg/m³) and v = 10 m/s to start."
2. System/diagram — Sketch the control volume around the blade; identify inflow/outflow of momentum.
3. Governing principle — "This is a momentum flux problem — we'll use the dynamic pressure form of Bernoulli/drag equation: F = ½ρv²A·Cd, because we're treating the blade as a bluff body in a free stream."
4. Unit-tracked calculation:

   F = ½ × (1.225 kg/m³) × (10 m/s)² × A [m²] × Cd
     = ½ × 1.225 × 100 × A × Cd  [kg/m³ · m²/s² · m²]
     = 61.25 × A × Cd            [kg·m/s²] = [N]  ✓ units confirm Newtons
   

5. Sanity check — Compare magnitude to expectation; flag if Cd or A wasn't specified.

Progress:

• Step 1: Extract and state all missing/assumed parameters explicitly
• Step 2: Define the system boundary (open/closed, control volume vs. free body)
• Step 3: Translate the physical scenario into a diagram (FBD, CV, circuit schematic)
• Step 4: Name the governing equation(s) and justify why they apply here
• Step 5: Set up the equation symbolically before plugging numbers
• Step 6: Carry units through every algebraic step; flag mismatches immediately
• Step 7: Compute the numeric result and sanity-check order of magnitude
• Step 8: State remaining assumptions/limitations of the result

Step details:

• Assumption extraction: Never silently assume. List each assumption as a bullet with the reasoning ("steady-state because no time-dependent terms given", "adiabatic because no insulation/heat data provided"). Ask the student to confirm or override defaults.
• System boundary: For mechanics → Free Body Diagram (all external forces/moments, action-reaction pairs). For thermo/fluids → Control Volume (mass/energy in and out, boundary work). For circuits → node/loop identification for KCL/KVL.
• Governing equation selection: State the principle by name (Bernoulli, Navier-Stokes, First Law of Thermodynamics, Newton's Second Law, KVL/KCL, Fourier's Law) and explicitly connect it to the boundary conditions that justify it (e.g., "steady, incompressible, inviscid along a streamline → Bernoulli applies").
• Dimensional analysis: Write units next to every quantity, cancel them algebraically inline, and confirm the final unit matches the expected physical quantity. If units don't cancel to the expected form, stop and flag the error before continuing — don't let it propagate.

Example 1: Steam Turbine (Thermodynamics) Input: "A steam turbine takes in steam at 6 MPa/450°C and exhausts at 10 kPa. Mass flow is 15 kg/s. Find power output." Output:

1. System Boundary: Open system / control volume around the turbine (mass crosses boundary).
2. Assumptions: Steady-state, steady-flow (SSSF); adiabatic (Q̇ = 0); negligible ΔKE, ΔPE.
3. State Properties: Prompt student to find h₁ (6 MPa, 450°C) and h₂ (10 kPa, assume saturated vapor unless quality given) from steam tables.
4. Governing equation: First Law for open system → Ẇ = ṁ(h₁ − h₂), justified because Q=0 and ΔKE/ΔPE neglected.
5. Unit check: ṁ [kg/s] × h [kJ/kg] = [kJ/s] = [kW] ✓
6. Numeric: if h₁ = 3301.8 kJ/kg, h₂ = 2584.7 kJ/kg → Ẇ = 15 × (3301.8−2584.7) = 10,756.5 kW ≈ 10.76 MW.

Example 2: RC Circuit Input: "A 12V battery charges a 100 µF capacitor through a 2 kΩ resistor. How long until it reaches 95% charge?" Output:

1. System boundary: Single-loop circuit, define node between R and C.
2. Governing equation: KVL → V = IR + Q/C, leading to first-order ODE, solution V_c(t) = V(1 − e^(−t/RC)).
3. Assumptions: Ideal battery (no internal resistance), ideal capacitor, wires have zero resistance.
4. Unit check: RC = (2000 Ω)(100×10⁻⁶ F) = 0.2 [Ω·F] = 0.2 [s] ✓ (Ω·F reduces to seconds).
5. Solve: 0.95 = 1 − e^(−t/0.2) → t = −0.2·ln(0.05) ≈ 0.6 s.