Validation

Tier A · Conduction · reference: Fourier's law

One slab, one heat flow: R = L/(kA). The simplest thing that can be wrong, and the one every other conduction result is built on.

Tier A · Conduction · reference: Series resistance addition

Steel, insulation and aluminium in a stack. Resistances add and every layer carries the same heat.

Tier A · Conduction · reference: Parallel conductance addition

A bolted joint and a gap pad side by side. Conductances add, and each takes its share of the heat in proportion.

Tier A · Conduction · reference: Overall U-value chain, Incropera ch. 3

The textbook building-wall chain: inside film, wall, outside film. Tests that convection and conduction links compose.

Tier A · Radiation · reference: Stefan-Boltzmann

Q = eps sigma A (T1^4 - T2^4), exact for this geometry. Checks that the linearised conductance recovers the fourth-power law once converged.

Tier L · Radiation · reference: Two-surface enclosure network

EXPECTED TO DEVIATE. The plain radiation link uses one emissivity and a view factor, so two grey plates joined by it exchange 1.4 times the grey-body heat. The case measures that ratio at fixed temperatures and checks both exact remedies: the effective emissivity and the link's two-surface model (or an enclosure).

Tier A · Transient · reference: Newton cooling, exact exponential

An aluminium block quenched into still air. T(t) = T_inf + (T0 - T_inf) exp(-t/tau), checked at one, two and three time constants.

Tier A · Transient · reference: Truncation error of an implicit first-order scheme

The integrator's order measured rather than assumed: halving the step must halve the error. Catches a scheme that is accidentally explicit.

Tier A · Transient · reference: Eigenvalues of C^-1 G

A two-node chain has two time constants, so the response is not a single exponential. Both the steady state and the slow eigenvalue are checked.

Tier A · Transient · reference: First law over a run

Energy leaving equals the drop in stored energy, integrated over two time constants. An accounting identity, which is why it catches an integrator that loses heat.

Tier B · Board · reference: AtlasPCB thermal via design guide · a correlation check, not a model

A 0.3 mm via with 25 um plating through 1.6 mm: about 190 K/W open and 58 K/W copper filled in the published working.

Tier A · Board · reference: Lumped energy balance

In-plane conductivity set high enough that the board cannot have a gradient, so dT = Q / ((h_top + h_bot) A). Tests the convective area the mesh assembles, which is invisible where conduction dominates.

Tier A · Board · reference: Fin equation with an adiabatic tip

A clamped strip losing heat off its faces is a one-dimensional fin with an exact cosh profile. Constrains the whole curve, not one number.

Tier A · Board · reference: K_0 Bessel solution; Abramowitz and Stegun 9.8.5-6

theta(r) = Q/(2 pi k t) K_0(r/L). The spreading physics the two-dimensional mesh exists to capture, against its exact solution at one, two and three decay lengths.

Tier A · Board · reference: First law; grid convergence

What is dissipated leaves, and the answer does not follow the grid. A board is solved again with every cell halved, components included, and the automatic mesh's error is estimated by Richardson extrapolation.

Tier A · Board · reference: Datasheet resistance network

With both destinations at the same temperature the junction rise is Q / (1/theta_JB + 1/theta_JC), which is the only configuration where the two paths can be checked apart from the board's spreading.

Tier A · Board · reference: Rule of mixtures · a correlation check, not a model

Copper and resin in parallel across the board and in series through it. Exact for the idealised stack; vias and uneven coverage are not modelled.

Tier A · Board · reference: Symmetry of the governing equation

A centred source on a symmetric board must give a symmetric field. Catches a size field that grades asymmetrically about a feature.

Tier A · Board · reference: One-dimensional series resistance

A heater covering the whole board with only the far face cooled: the heat must cross the laminate, and nothing moves sideways, so the junction rise is Q (theta_JB + t/(k_z A) + 1/(h A)) exactly. The case a single-sheet board model could not express.

Tier A · Board · reference: Symmetry through the mid-plane

A chip on the bottom with the face coefficients swapped is the top-mounted problem turned over, so its junction and fields must match exactly. Checks both plates are wired alike.

Tier A · Board · reference: Parallel conduction: barrels, fill and laminate

A25's full-board heater with a 0.3 mm via array at 1.2 mm pitch under it. The array's effective k_z is worked from the geometry and the junction rise follows exactly, since nothing moves sideways.

Tier A · Board · reference: Fin equation with the pour's sheet conductance

A bare, barely conducting strip with one ounce of copper poured over both faces: the profile must follow the fin solution for k t + 2 k_Cu t_Cu.

Tier A · Board · reference: Series film resistance, bare area in parallel

An isothermal board coated on both faces with a keepout on one: the rise is Q over h A for the bare part plus the coated film 1/(1/h + t_c/k_c) for the rest.

Tier B · Contact · reference: CMY 1969; Antonetti and Yovanovich asperity slope

The published correlation recomputed from its own equation, units and all. This is where the 258x asperity-slope unit error lived.

Tier B · Contact · reference: CMY exponents, measured as ratios

h goes as P^0.95 and as sigma^-0.598. Ratios test the exponents independently of every leading constant.

Tier B · Contact · reference: Yovanovich gap model; kinetic theory

The gas is a parallel path and can only help. Helium beats air, but in a micron-scale gap by less than their conductivity ratio, because its gas parameter is larger; thinning the gas makes the joint worse, not better.

Tier C · Contact · reference: Hegazy 1985; Madhusudana, Thermal Contact Conductance; Fletcher reviews

Ground SS304 at 1 MPa in vacuum is reported at roughly 500-5000 W/m2.K. The engine uses bulk hardness where the correlation wants microhardness, so the default reads high; both are shown.

Tier B · Contact · reference: CMY against Mikic elastic

Real joints deform between the two extremes, so the two models must differ. A build returning the same number from both would mean one is not being computed.

Tier A · Contact · reference: Consistency between the two entry points

The same joint solved standalone and as a link. A user comparing the two must not get two answers.

Tier A · Fluid · reference: Exact solution of Navier-Stokes for a round tube

Q = pi D^4 dp / (128 mu L), and f = 64/Re. No fitting constant anywhere: it falls out of the equations.

Tier B · Fluid · reference: Haaland 1983; Colebrook-White 1939 · a correlation check, not a model

The explicit fit against the implicit equation it fits, solved here by iteration. 48 points over five decades of Reynolds number and six roughnesses.

Tier A · Fluid · reference: Mass conservation and resistance addition

Two identical pipes carry the same flow, split the drop evenly, and together equal one pipe of twice the length.

Tier C · Fluid · reference: Crane Technical Paper 410; Idelchik · a correlation check, not a model

The tabulated K values this engine ships, against the standard references, plus a check that the loss really is K rho v^2 / 2.

Tier A · Fluid · reference: Continuity

A header splitting into two branches of different bore. What arrives leaves, both branches see the same drop, and the larger bore takes more.

Tier A · Fluid · reference: Intersection of two quadratics, solved by hand

Where the pump curve crosses the loss curve, worked out analytically and compared against the Newton solve.

Tier B · Fluid · reference: Dittus and Boelter 1930; Gnielinski 1976 · a correlation check, not a model

Two fits to the same tube-flow data. Gnielinski is the better one, and the deviation is reported rather than hidden - it runs in the direction that underestimates wall temperature.

Tier A · Fluid · reference: Effectiveness-NTU, C_min/C_max -> 0

eps = 1 - exp(-NTU), exact. Guarantees the stream cannot leave hotter than the wall however long the passage - which an unbounded UA dT model does not.

Tier A · Fluid · reference: Quadratic pump against a quadratic system

A two-number pump curve against a single K factor meets at Q = sqrt(s / (a + s/Qmax^2)); both are exactly quadratic, so the solver's operating point must be this one.

Tier A · Fluid · reference: Affinity laws; fan law for density · a correlation check, not a model

Speed scales flow by s and head by s^2, parallel units share the flow, and a fan's pressure follows the gas density - checked between a curve's points as well as on them.

Tier A · Fluid · reference: Isenthalpic expansion, h = cp T + p/rho

A valve holds enthalpy constant, so the pressure it destroys appears as dT = dp/(rho cp). The term a temperature-only model loses.

Tier A · Fluid · reference: Effectiveness-NTU, counterflow

Two circuits meeting only in the exchanger: effectiveness from the closed form, duty given equals duty received, each side's drop from its rated point.

Tier A · Fluid · reference: First law of the refrigerant circuit; Carnot fraction

Q_c = Q_e + W, COP at the stated fraction of Carnot between the refrigerant temperatures, chilled water at setpoint, and the plant balanced as one control volume.

Tier A · Fluid · reference: Steady-flow energy equation

No loop: a known supply heated and run to a drain. m (h_out - h_in) = Q for the whole system, and the supply's flow reaches the drain.

Tier A · Fluid · reference: Radial conduction through a cylinder, film in series

Heat gain through pipe insulation and the outside film; the inside film from the flow is the one term the closed form leaves out.

Tier A · Coupled · reference: First law at the coupling node

Everything arriving at the wall leaves it, through the air or into the stream. The two figures are computed on opposite sides of the coupling and nothing forces them to agree.

Tier A · Coupled · reference: m cp eps -> UA as the flow grows

At high flow the coupled model must collapse onto a plain convection link of UA to the inlet temperature - the one place a coupled and an ordinary RC model have to agree exactly.