pie-thermal-engineering

Thermal Engineering Skill

At a Glance

Calculate heat transfer rates, size heat exchangers, and analyze data center thermal performance from component-level junction temperatures to facility-wide efficiency metrics.

Activation: InfrastructureRequest type='thermal', any heat exchanger sizing request, PUE/WUE calculation, airflow management design, or cooling load analysis.

Key capabilities:

• Three-mode heat transfer: conduction (Fourier), convection (Newton), radiation (Stefan-Boltzmann)
• Thermal resistance networks (series/parallel, analogous to electrical circuits)
• Data center cooling load breakdown (IT, UPS, lighting, fans, envelope)
• Heat exchanger sizing via LMTD and epsilon-NTU methods
• PUE/TUE/WUE/CUE efficiency metric calculations with target benchmarks
• Airflow management: hot/cold aisle containment, raised floor, economizer modes

Integration: Works in tandem with pie-fluid-systems: the fluid skill sizes cooling loop pipes and pumps; this skill quantifies the heat and determines exchanger performance.

ENGINEERING DISCLAIMER: All calculations must be verified by a licensed Professional Engineer before use in construction or installation. HVAC and mechanical codes (ASHRAE 90.1, IMC) impose requirements not captured here. User assumes all responsibility for verification.

Quick routing: Heat transfer modes -- see Heat Transfer Fundamentals. Resistance networks -- see Thermal Resistance Networks. Data center loads -- see Data Center Cooling Load. Exchanger sizing -- see Heat Exchanger Sizing. PUE/WUE metrics -- see Efficiency Metrics. Airflow layout -- see Airflow Management.

Heat Transfer Fundamentals

Conduction -- Fourier's Law

Heat flow through a solid material by molecular vibration:

q = -k x A x (dT/dx)

| Variable | Definition | Units | |----------|-----------|-------| | q | Heat flow rate | W | | k | Thermal conductivity | W/(m K) | | A | Cross-sectional area perpendicular to heat flow | m^2 | | dT/dx | Temperature gradient | K/m |

The negative sign indicates heat flows from hot to cold (opposite to the temperature gradient).

Thermal resistance (conduction): R_cond = L / (k x A), analogous to electrical resistance R = rho L / A.

Thermal conductivity reference values:

| Material | k (W/(m K)) | Application | |----------|------------|-------------| | Copper | 385 | Heat sinks, cold plates | | Aluminum | 205 | Enclosures, fins, cold plates | | Carbon steel | 50 | Structural, pressure piping | | Stainless steel 304 | 16 | Corrosion-resistant piping | | Concrete | 1.4 | Building structure | | Gypsum board | 0.16 | Wall construction | | Mineral wool | 0.04 | Pipe insulation | | Polyurethane foam | 0.025 | Refrigeration insulation | | Thermal paste (TIM) | 1-8 | CPU/GPU mounting |

Convection -- Newton's Law of Cooling

Heat transfer between a surface and a moving fluid:

q = h x A x (T_surface - T_fluid)

| Variable | Definition | Units | |----------|-----------|-------| | q | Heat flow rate | W | | h | Convective heat transfer coefficient | W/(m^2 K) | | A | Surface area exposed to fluid | m^2 |

Thermal resistance (convection): R_conv = 1 / (h x A)

Convective coefficient reference values:

| Flow Type | Medium | h (W/(m^2 K)) | |-----------|--------|---------------| | Natural convection | Air | 5-25 | | Forced convection | Air | 25-250 | | Forced convection | Water (low velocity) | 500-2,000 | | Forced convection | Water (high velocity) | 2,000-10,000 | | Boiling | Water | 3,000-60,000 | | Condensing | Steam | 5,000-100,000 |

For convection correlations (Nusselt number, Reynolds, Prandtl relationships) -- @references/heat-transfer.md

Radiation -- Stefan-Boltzmann Law

Heat transfer by electromagnetic emission between surfaces:

q = epsilon x sigma x A x (T1^4 - T2^4)

| Variable | Definition | Units | |----------|-----------|-------| | epsilon | Surface emissivity (0 = perfect reflector, 1 = blackbody) | dimensionless | | sigma | Stefan-Boltzmann constant = 5.67e-8 | W/(m^2 K^4) | | T1, T2 | Surface temperatures | K (Kelvin only) |

CRITICAL: Temperatures MUST be in Kelvin for radiation calculations. K = C + 273.15.

Emissivity reference values:

| Surface | epsilon | Notes | |---------|--------|-------| | Blackbody (ideal) | 1.0 | Theoretical maximum | | Painted steel | 0.9 | Most painted surfaces | | Oxidized copper | 0.7 | Aged copper surfaces | | Glass | 0.9 | Window and enclosure glass | | Polished aluminum | 0.04 | Reflective radiation shield | | Anodized aluminum | 0.8 | Common enclosure finish |

When radiation matters: High temperatures (>200C), large temperature differentials, or vacuum/low-pressure environments. Usually negligible for data center operating temperatures (15-55C); important for outdoor equipment and industrial processes.

Thermal Resistance Networks

The thermal-electrical analogy maps heat transfer directly onto circuit analysis:

| Thermal Domain | Electrical Domain | |---------------|-------------------| | Heat flow q (W) | Current I (A) | | Temperature difference DeltaT (K) | Voltage V (V) | | Thermal resistance R (K/W) | Electrical resistance R (ohm) |

Series Network (Heat Path Stack)

When heat flows through sequential layers, resistances add:

R_total = R1 + R2 + R3 + ...
q = (T_hot - T_cold) / R_total

Example: server die to coolant path:

R_total = R_die_spreading + R_TIM + R_cold_plate_wall + R_cold_plate_convection

R_die_spreading   = t_die / (k_silicon x A_die)
R_TIM             = t_TIM / (k_paste x A_contact)
R_cold_plate_wall = t_wall / (k_copper x A_plate)
R_cold_plate_conv = 1 / (h_coolant x A_internal)

Temperature rise check: DeltaT_junction = q x R_total. Verify T_junction < TjMax (typically 95-105C for modern CPUs/GPUs).

Parallel Network (Multiple Heat Paths)

When heat has multiple simultaneous paths, resistances combine as parallels:

1/R_total = 1/R1 + 1/R2 + 1/R3

Use when heat splits between paths: fins parallel to base, combined air cooling + liquid cooling, multiple heat sinks on a shared substrate. The path with lowest resistance carries the most heat.

Contact and Interface Resistance

Real surfaces have microscopic asperities -- actual contact area is only 1-2% of nominal area.

R_contact = 1 / (h_c x A)

Thermal interface material (TIM) specific resistance:

| TIM Type | R (K cm^2/W) | Application | |----------|-------------|-------------| | Bare metal contact | 0.5-5.0 | Avoid -- poor performance | | Thermal grease | 0.1-0.5 | Standard CPU mounting | | Phase-change pad | 0.05-0.2 | Factory-applied, consistent | | Solder/braze | 0.02-0.1 | Permanent, best performance |

Reduce contact resistance by: smoother surfaces, higher clamping force, higher-conductivity TIM.

Mathematical Connection

Heat flow against a temperature gradient follows the same mathematical structure as gradient descent in machine learning:

Heat:  q = -k nabla(T)                         (heat flows opposite to temperature gradient)
ML:    theta_n+1 = theta_n - alpha nabla(L)     (parameters move opposite to loss gradient)

Both describe movement opposing the direction of increasing potential. The temperature field T(x,y,z) in a data center is a scalar field whose gradient points from cold to hot. Heat flows against it. The student who has trained neural networks already understands the physics of heat conduction.

For thermal network examples and contact resistance data -- @references/heat-transfer.md

Data Center Cooling Load

Load Components

IT equipment load (dominant term, typically 60-80% of total):

Q_IT = sum(server_nameplate_TDP) x diversity_factor
     = rack_count x avg_density_kW x utilization

Diversity factor: 0.7-0.9 (servers rarely sustain nameplate power continuously). Maximum demand method (NEC 220.87 adapted): use measured 15-minute peak power x 1.25.

Lighting: 10-20 W/m^2 for