Mathematical verification for physical calculations: unit tracking algebra (exponent maps), PhysicalQuantity pattern for compound units, SI/Imperial mixed-unit handling, Buckingham pi theorem for dimensionless groups, and common engineering dimensionless numbers. Activates for unit verification, dimensional consistency checks, scaling analysis, and calculation validation across all infrastructure domains.
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Mathematical verification for physical calculations: unit tracking algebra (exponent maps), PhysicalQuantity pattern for compound units, SI/Imperial mixed-unit handling, Buckingham pi theorem for dimensionless groups, and common engineering dimensionless numbers. Activates for unit verification, dimensional consistency checks, scaling analysis, and calculation validation across all infrastructure domains.
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
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
Symbol
Quantity
Notes
m
length
meter
kg
mass
kilogram (only SI base unit with a prefix)
s
time
second
A
electric current
ampere
K
temperature
kelvin (absolute; not degrees Celsius)
mol
amount of substance
mole
cd
luminous intensity
candela (rarely used in infrastructure)
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.
Common Infrastructure Units -- Exponent Map Reference
Quantity
SI Unit
Symbol
Exponent Map
Force
Newton
N
{ kg:1, m:1, s:-2 }
Pressure
Pascal
Pa
{ kg:1, m:-1, s:-2 }
Energy
Joule
J
{ kg:1, m:2, s:-2 }
Power
Watt
W
{ kg:1, m:2, s:-3 }
Dynamic viscosity
--
Pa*s
{ kg:1, m:-1, s:-1 }
Heat transfer coeff
--
W/(m^2*K)
{ kg:1, s:-3, K:-1 }
Thermal conductivity
--
W/(m*K)
{ kg:1, m:1, s:-3, K:-1 }
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.
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)
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
Receive mixed-unit inputs (e.g., pipe diameter in inches, flow in GPM)
Convert ALL inputs to SI at the boundary -- lib/units.ts handles this
Calculate entirely in SI using PhysicalQuantity arithmetic
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
From
To
Factor
inches
meters
x 0.0254
feet
meters
x 0.3048
PSI
kPa
x 6.8948
GPM
L/s
x 0.06309
BTU/hr
W
x 0.29307
degrees F
degrees C
(F - 32) / 1.8
degrees C
K
+ 273.15
ft/s
m/s
x 0.3048
lb/ft^3
kg/m^3
x 16.018
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)
List variables -- identify every physical quantity that could influence the outcome. Include the dependent variable.
Write dimensions -- express each variable in base units (m, kg, s, A, K).
Count k -- number of independent base dimensions appearing. Usually 2-4 for engineering problems.
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.
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:
Group
Combination
Result
Name
pi_1
delta_P * D / (rho * v^2)
dimensionless
Euler number (pressure coefficient)
pi_2
rho * v * D / mu
dimensionless
Reynolds number
pi_3
L / D
dimensionless
Length ratio
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
Group
Formula
Physical Meaning
Activation Threshold
Reynolds
Re = rhovL / mu
Inertia / viscous force
Re > 4000 = turbulent flow
Nusselt
Nu = h*L / k
Convection / conduction
Heat transfer coefficient
Prandtl
Pr = mu*Cp / k
Momentum / thermal diffusivity
Nu correlation parameter
Grashof
Gr = gbetadT*L^3 / nu^2
Buoyancy / viscous force
Natural convection indicator
Froude
Fr = v / sqrt(g*L)
Inertia / gravity
Open channel flow regime
Strouhal
St = f*L / v
Vortex shedding frequency
Pipe vibration analysis
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
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.
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
Scenario
Method
Rationale
Safety-critical (pressure containment)
Worst-case
Zero tolerance for interference
Small assembly (<5 components)
Worst-case
RSS benefit is marginal
Large assembly (5+ components)
RSS
Statistical saving is significant
Expensive rework
RSS with verification tests
Balance economy vs risk
Worked Example -- Pipe Assembly in Wall Chase
Setup: One 2" pipe (NPS) with insulation in a field-cut wall chase.
Component
Nominal
Tolerance
Chase width
8.000"
+/- 0.250" (field cut)
Pipe OD
2.375"
+/- 0.010" (manufacturing)
Insulation thickness (each side)
1.000"
+/- 0.125" (installation)
Required clearance (each side)
0.500"
--
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)
Voltage to Ground
Condition 1 (live one side)
Condition 2 (live + grounded both sides)
Condition 3 (live both sides)
0-150V
3 ft
3 ft
3 ft
151-600V
3 ft
3.5 ft
4 ft
601-2500V
4 ft
4 ft
5 ft
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)).
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:
Sum all pipe-plus-insulation widths
Add (n - 1) x minimum separation clearances
Add 2 x wall clearance (typically 1" each side)
Compare required total to chase width W
Apply worst-case tolerance: available = W - T_chase; required = sum + T_pipes
If required > available at worst case, the chase must be widened
Cable Tray Fill (NEC 392.22)
Cable Type
Max Fill
Notes
600V multiconductor, <=2000 kcmil
50% of tray cross-sectional area
Area = cable OD^2 x pi/4
Power cables >2000 kcmil
Single layer only
No stacking; check current derating
Control/instrument (<=1" OD)
50% of area
Mixed power + control
40% for control portion
Separation between power and control recommended
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
Material
Minimum Bend Radius
Steel pipe (Schedule 40)
5-6 x OD
Copper pipe (Type L/K)
4 x OD
PEX tubing
8 x OD minimum
Rigid conduit (>1" trade)
6 x trade size
EMT conduit
5 x trade size
THWN-2 conductor
8 x OD (NEC 300.34)
Armored cable (AC/MC)
7 x smallest OD dimension
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
System
Minimum Slope
Notes
Drain/waste/vent (horizontal)
1/4" per foot (2.08%)
IPC/UPC gravity drain
Storm drainage (horizontal)
1/8" per foot (1.04%)
Minimum; more is better
HVAC condensate drain
1/4" per foot
Away from air handler
Steam condensate return
1/2" per foot
In direction of flow
Cold water supply
None required
Pressurized system
Hot water supply
None required
Pressurized; slope for drainability preferred
Safety Warden Integration
Interference checking triggers Safety Warden findings at these severity levels:
Condition
Severity
Domain
Action
Equipment bounding box overlap
critical
structural
Block -- cannot place overlapping equipment
Clearance < code minimum (electrical panel)
critical
voltage
Block until clearance verified by PE
Pipe chase fill > 100% at worst-case tolerance
blocking
structural
Redesign chase or reroute pipes
Cable tray fill > 50%
warning
structural
Review tray sizing; may need larger tray
Bend radius below minimum
warning
structural
Flag for field review (critical if pressure piping)
Missing required slope on gravity drain
warning
plumbing
Verify routing elevation changes
Reference Documents
Reference
When to Read
Coverage
@references/unit-algebra.md
Full SI/Imperial conversion tables, all infrastructure domains
Complete unit reference
@references/buckingham-pi.md
Full theorem derivation, dimensional matrix, five worked examples
Deep pi theorem guide
@references/tolerance-stack-up.md
Statistical analysis, GD&T basics, Monte Carlo simulation
Tolerance engineering
@references/spatial-constraints.md
OBB algorithm, full NEC 110.26 tables, cable tray details
Spatial verification
Dimensional Analysis Skill v1.0.0 -- Physical Infrastructure Engineering PackPhase 437 | References: BIPM SI Brochure (9th ed.), Buckingham (1914), Bridgman (1922), NEC 2023, ASME Y14.5Dimensional verification is a mathematical check -- not a substitute for engineering judgment.
