Pie Dimensional Analysis

Dimensional Analysis Skill

At a Glance

Dimensional analysis is the mathematical verification layer that ensures physical calculations are dimensionally consistent -- catching unit errors before they become calculation errors.

When to activate:

• Verify multi-step calculations for unit consistency
• Mix SI and Imperial units in the same calculation
• Scale experimental data to new conditions via dimensionless groups
• Identify governing parameters of a physical system
• Validate Calculator agent outputs before committing to CalculationRecord

Key capabilities:

• Unit tracking via exponent maps (PhysicalQuantity pattern)
• Compound unit algebra: multiply, divide, power, dimensional homogeneity
• Dimensional mismatch detection at every arithmetic step
• SI to Imperial conversion for all infrastructure engineering domains
• Buckingham pi theorem for deriving dimensionless groups
• Infrastructure dimensionless numbers: Reynolds, Nusselt, Prandtl, Grashof, Froude, Strouhal

Integration: Cross-cutting skill -- applies to outputs from fluid-systems, power-systems, and thermal-engineering. Acts as verification layer before Calculator agent commits to CalculationRecord.

NOTE: Dimensional analysis verifies mathematical self-consistency only. It does not replace engineering judgment or safety verification. Dimensionally correct equations can still be physically wrong if incorrect constants or assumptions are used.

Quick routing:

• Unit conversions only --> @references/unit-algebra.md for full tables
• Pi theorem derivation --> @references/buckingham-pi.md for worked examples
• Tolerance stack-up --> see Tolerance Stack-Up Analysis section below
• Spatial fit checking --> see Spatial Constraint Verification section below

Unit Tracking Algebra

The Seven SI Base Units

Compound Units as Exponent Maps

Every physical quantity carries its unit as a map of base unit exponents. This representation makes unit algebra mechanical -- multiply means add exponents, divide means subtract.

Examples:

• Velocity: 2.4 m/s --> { value: 2.4, units: { m: 1, s: -1 } }
• Pressure: 101325 Pa --> { value: 101325, units: { kg: 1, m: -1, s: -2 } }
• Power: 1000 W --> { value: 1000, units: { kg: 1, m: 2, s: -3 } }
• Thermal conductivity: 385 W/(m*K) --> { value: 385, units: { kg: 1, m: 1, s: -3, K: -1 } }

Common Infrastructure Units -- Exponent Map Reference

The PhysicalQuantity Interface

The Calculator agent implements unit-safe arithmetic using this TypeScript pattern. The SKILL documents the knowledge; lib/units.ts provides the implementation.

interface PhysicalQuantity {
  value: number;
  units: { [baseUnit: string]: number }; // exponent map
}

function multiply(a: PhysicalQuantity, b: PhysicalQuantity): PhysicalQuantity {
  const result: PhysicalQuantity = { value: a.value * b.value, units: { ...a.units } };
  for (const [unit, exp] of Object.entries(b.units)) {
    result.units[unit] = (result.units[unit] || 0) + exp;
  }
  // Remove zero exponents
  for (const unit of Object.keys(result.units)) {
    if (result.units[unit] === 0) delete result.units[unit];
  }
  return result;
}

function divide(a: PhysicalQuantity, b: PhysicalQuantity): PhysicalQuantity {
  const negated = { value: 1 / b.value, units: Object.fromEntries(
    Object.entries(b.units).map(([k, v]) => [k, -v])
  )};
  return multiply(a, negated);
}

function assertSameUnits(a: PhysicalQuantity, b: PhysicalQuantity): void {
  const aKeys = Object.keys(a.units).sort().join(',');
  const bKeys = Object.keys(b.units).sort().join(',');
  if (aKeys !== bKeys || !Object.entries(a.units).every(([k, v]) => b.units[k] === v)) {
    throw new Error(`Unit mismatch: [${aKeys}] vs [${bKeys}] — cannot add/subtract`);
  }
}

Unit Algebra Rules

The dimensional homogeneity rule is absolute: you cannot add quantities with different units. If assertSameUnits() throws, the calculation has a structural error that must be fixed before proceeding.

Mixed Unit Systems

Infrastructure engineering inherently mixes SI and Imperial units. This is not optional -- it is driven by standards bodies, building codes, and equipment manufacturers.

Common mixed-unit scenarios:

• Pipe sizes: NPS in inches (ASME B36.10 standard)
• Flow rates: GPM (US practice) or L/s (SI practice)
• Pressure: PSI (US safety codes, ASME BPVC) or kPa (SI engineering)
• Temperature: degrees F (US weather/safety) or degrees C (SI engineering)
• Power: BTU/hr (US HVAC industry) or kW (SI)
• Conductors: AWG (US NEC) or mm^2 (EU/IEC)

Strategy: Convert Immediately, Work in SI, Convert Output

1. Receive mixed-unit inputs (e.g., pipe diameter in inches, flow in GPM)
2. Convert ALL inputs to SI at the boundary -- lib/units.ts handles this
3. Calculate entirely in SI using PhysicalQuantity arithmetic
4. Convert final outputs to user-preferred units for display

This eliminates mixed-unit errors in the calculation core. Conversion errors are isolated to the boundary layer where they are easy to audit.

Common Conversions -- Quick Reference

Full conversion table for all infrastructure engineering domains --> @references/unit-algebra.md

Buckingham Pi Theorem

The Theorem

For a physical equation relating n dimensional variables that involve k independent base dimensions, the equation can be rewritten using (n - k) independent dimensionless groups.

This means: if you have 6 variables and 3 base dimensions, you need only 3 dimensionless groups to describe the physics -- regardless of what unit system you use. The universe is dimensionally self-consistent.

Procedure (5 Steps)

1. List variables -- identify every physical quantity that could influence the outcome. Include the dependent variable.
2. Write dimensions -- express each variable in base units (m, kg, s, A, K).
3. Count k -- number of independent base dimensions appearing. Usually 2-4 for engineering problems.
4. Choose k repeating variables -- variables that together span all k dimensions. Avoid the dependent variable. Common choices: (rho, v, L) for fluid flows; (k, L, h) for heat transfer.
5. Form pi groups -- multiply each remaining variable with the repeating variables raised to unknown powers; solve exponents so that the result is dimensionless.

Worked Example: Pipe Flow Pressure Drop

Variables: delta_P (pressure drop), L (pipe length), D (pipe diameter), rho (fluid density), mu (dynamic viscosity), v (flow velocity)

n = 6 variables, k = 3 base dimensions {kg, m, s}

Repeating variables: rho, v, D (together they contain all three dimensions: rho has kg and m, v has m and s, D has m)

Form n - k = 3 dimensionless groups:

Physical law recovered: Euler = f(Re, L/D)

This is exactly the structure of the Darcy-Weisbach equation: delta_P = f * (L/D) * (rho * v^2 / 2). Dimensional analysis recovers the equation form without any physics -- only dimensional reasoning.

For full theorem derivation, dimensional matrix method, and five infrastructure worked examples --> @references/buckingham-pi.md

Common Infrastructure Dimensionless Groups

Why these matter for infrastructure:

• Re determines whether pipe flow is laminar or turbulent -- which selects the friction factor equation (Moody chart vs 64/Re)
• Nu determines convective heat transfer coefficient -- drives heat exchanger sizing
• Pr groups fluid thermal properties -- used in every forced convection correlation
• Dimensional analysis reveals which parameters dominate -- enables experimental scale-up (same Re = same physics at any size)

Calculation Verification Workflow

The Calculator agent applies this skill as a verification pass on all numerical outputs. Every multi-step calculation follows this protocol:

Step 1: Convert Inputs

All inputs converted to SI via lib/units.ts. Record each as a PhysicalQuantity with explicit exponent map.

Step 2: Track Through Calculation

Each arithmetic step propagates units via multiply/divide rules. Intermediate results carry their units.

Step 3: Check Dimensional Homogeneity

At each addition or subtraction, verify units match via assertSameUnits(). If mismatch: stop, flag error, do not proceed.

Step 4: Verify Result Units

Final result should have expected unit exponents:

• Pipe sizing result should have units { m: 1 } (a length/diameter)
• Pressure drop should have units { kg: 1, m: -1, s: -2 } (Pascal)
• Flow rate should have units { m: 3, s: -1 } (cubic meters per second)

If result has wrong units, something was inverted or an exponent was applied incorrectly. Flag the error.

Step 5: Check Reasonableness

Dimensionless numbers should be in expected ranges:

• Re < 2300: laminar (unusual for data center cooling loops -- flag if unexpected)
• Re > 4000: turbulent (normal for infrastructure piping)
• Nu > 100: high convective transfer (expected for turbulent water flow; flag if < 10)
• Pr ~ 7 for water at room temperature; flag if wildly different

Worked Multi-Step Example: Pipe Sizing Verification

Input: Heat load Q = 40 kW, temperature difference dT = 10 K, max velocity v_max = 2.4 m/s

Step 1 -- Flow rate:

m_dot = Q / (rho * Cp * dT)
Units: W / (kg/m^3 * J/(kg*K) * K)
     = kg*m^2/s^3 / (kg/m^3 * kg*m^2/(s^2*kg*K) * K)
     = kg*m^2/s^3 / (m^3/s^2 * 1/m^3)
     ... simplifies to m^3/s  [PASS]

Step 2 -- Minimum cross-sectional area:

A_min = V_dot / v_max
Units: (m^3/s) / (m/s) = m^2  [PASS]

Step 3 -- Minimum diameter:

D_min = sqrt(4 * A_min / pi)
Units: sqrt(m^2) = m  [PASS]

Step 4 -- Pressure drop (Darcy-Weisbach):

dP = f * (L/D) * (rho * v^2 / 2)
Units: dimensionless * (m/m) * (kg/m^3 * m^2/s^2) = kg/(m*s^2) = Pa  [PASS]

Each step is dimensionally verified before proceeding. If any step produces wrong units, the error is caught before it propagates through the rest of the calculation chain.

Tolerance Stack-Up Analysis

Real manufactured components have dimensional tolerances. When multiple components assemble in series, tolerances accumulate. The critical question: does the worst-case assembly actually fit in the available space?

Worst-Case Method (Conservative)

Formula: T_total = sum of |t_i| (sum of all individual tolerance magnitudes)

This assumes all tolerances are simultaneously at their worst values. It is always conservative -- the result is the absolute maximum possible deviation.

When to use:

• Safety-critical assemblies (pressure containment, seismic bracing)
• Small assemblies with 4 or fewer components (RSS benefit is marginal)
• Zero tolerance for field rework

Decision rule: If nominal_gap >= nominal_assembly + T_total, the assembly always fits regardless of manufacturing variation.

RSS Method (Root Sum of Squares, Statistical)

Formula: T_total = sqrt(sum of t_i^2)

This assumes tolerances are independent, normally distributed, and centered on nominal values. The probability that all tolerances simultaneously reach their worst values is vanishingly small -- for n=5 components, P(all worst) = (0.0027)^5 = 1.4 x 10^-13.

Result: RSS total is typically 40-70% of worst-case total for 5 or more components. This represents significant material and space savings.

When to use:

• Large assemblies with 5 or more independent components
• Moderate cost of occasional interference (rework is feasible)
• Tolerances are truly independent (no shared manufacturing process)

Choosing the Right Method

Worked Example -- Pipe Assembly in Wall Chase

Setup: One 2" pipe (NPS) with insulation in a field-cut wall chase.

Nominal assembly width: 2.375 + 2 x 1.000 = 4.375" Nominal with clearances: 4.375 + 2 x 0.500 = 5.375"

Worst-case tolerance: T = 0.010 + 0.125 + 0.125 = 0.260" (pipe + both insulation layers) Available space (worst case): 8.000 - 0.250 = 7.750" Required space (worst case): 5.375 + 0.260 = 5.635" Margin: 7.750 - 5.635 = 2.115" --> PASSES

Second scenario -- add a second 2" pipe: Total nominal assembly: 2.375 + 2.067 + 2 x 1.000 + 2 x 1.000 = 8.442" This already exceeds the 8.000" nominal chase width --> FAILS at nominal before tolerances are even considered. The chase must be widened or the routing redesigned.

For statistical tolerance analysis with non-normal distributions and Monte Carlo simulation --> @references/tolerance-stack-up.md

Spatial Constraint Verification

Bounding Box Intersection Test

AABB (Axis-Aligned Bounding Box) -- the standard approach for infrastructure equipment placement:

Define each item by its min/max coordinates: {x_min, x_max, y_min, y_max, z_min, z_max}

Overlap test:

overlap = NOT (A.x_max < B.x_min OR A.x_min > B.x_max)
      AND NOT (A.y_max < B.y_min OR A.y_min > B.y_max)
      AND NOT (A.z_max < B.z_min OR A.z_min > B.z_max)

If no overlap on any axis, no collision -- items fit.

OBB (Oriented Bounding Box) -- for rotated equipment: more complex, uses the separating axis theorem. Only needed when equipment is not aligned to a 90-degree grid.

For infrastructure placement, most equipment is axis-aligned. AABB is sufficient and fast (O(1) per pair).

Clearance Verification

After confirming no AABB overlap, verify that the gap between items meets or exceeds code-mandated minimums. Expand one bounding box by the required clearance in all directions and re-test -- if the expanded box overlaps, clearance is insufficient.

NEC 110.26 Working Space (Electrical Panels)

Width: Minimum 30" or width of equipment, whichever is greater. Height: Minimum 6.5 ft or height of equipment above floor. Illumination: Required for all working spaces (NEC 110.26(D)). Dedicated space: Working clearance area must not be used for storage; overhead piping and ducts are prohibited in dedicated electrical space (NEC 110.26(E)).

Mechanical Access Clearances (General Rules)

Interference Checking

Pipe Chase Fill

Total assembly width in a pipe chase:

W_required = sum(pipe_OD_i + 2 * insulation_i) + (n-1) * separation + 2 * wall_clearance

Minimum pipe-to-pipe separation: 0.5 x OD of the larger pipe (thermal expansion and installation access).

Algorithm for n pipes in a chase of width W:

1. Sum all pipe-plus-insulation widths
2. Add (n - 1) x minimum separation clearances
3. Add 2 x wall clearance (typically 1" each side)
4. Compare required total to chase width W
5. Apply worst-case tolerance: available = W - T_chase; required = sum + T_pipes
6. If required > available at worst case, the chase must be widened

Cable Tray Fill (NEC 392.22)

Tray fill calculation: sum(cable cross-sectional areas) <= tray_width x tray_depth x fill_fraction

Common trap: Using jacket OD area instead of individual conductor area. Using jacket OD is conservative but acceptable; using conductor area is more accurate but requires knowing the cable construction.

Minimum Bend Radius

All routing changes of direction must have adequate bend radius. Flag tight bends for field review -- undersized bends cause flow restriction in pipes and conductor damage in cables.

Slope Requirements

Safety Warden Integration

Interference checking triggers Safety Warden findings at these severity levels:

Reference Documents

Dimensional Analysis Skill v1.0.0 -- Physical Infrastructure Engineering Pack Phase 437 | References: BIPM SI Brochure (9th ed.), Buckingham (1914), Bridgman (1922), NEC 2023, ASME Y14.5 Dimensional verification is a mathematical check -- not a substitute for engineering judgment.