Pie Thermal Engineering

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)

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:

Convection -- Newton's Law of Cooling

Heat transfer between a surface and a moving fluid:

q = h x A x (T_surface - T_fluid)

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

Convective coefficient reference values:

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)

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

Emissivity reference values:

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:

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:

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 LED (was 50 W/m^2 for fluorescent). All lighting converts to heat -- add to cooling load.

UPS losses: Heat generated by power conversion inefficiency:

Fan and pump power: CRAC/CRAH fan power typically 5-15% of served IT load; pump power <3% of IT load. Add to heat balance as these also convert to heat within the space.

Envelope loads: Usually <5% for well-insulated interior data halls. Include solar gain if windows or poorly insulated walls face direct sun. Q_envelope = U x A x CLTD (cooling load temperature difference; typically 5-15C for insulated walls).

Total Cooling Load

Q_cooling = Q_IT x (1 + UPS_loss_fraction) + Q_lighting + Q_fans + Q_envelope

Add 15-20% safety margin for future expansion.

Per-Rack Analysis

Cooling approach by density:

• < 10 kW/rack: air cooling (raised floor CRAC/CRAH sufficient)
• 10-30 kW/rack: rear-door heat exchangers or in-row cooling units

• 30 kW/rack: direct liquid cooling (DTC) per ASHRAE TC 9.9 guidelines

Heat Exchanger Sizing

LMTD Method

Use when both inlet and outlet temperatures are known (rating existing equipment or fixed design conditions).

Q = U x A x F x LMTD

Log Mean Temperature Difference:

LMTD = (DeltaT1 - DeltaT2) / ln(DeltaT1 / DeltaT2)

For counterflow: DeltaT1 = T_h,in - T_c,out; DeltaT2 = T_h,out - T_c,in

For parallel flow: DeltaT1 = T_h,in - T_c,in; DeltaT2 = T_h,out - T_c,out

F = LMTD correction factor: F = 1.0 for pure counterflow (ideal). F = 0.7-0.95 for shell-and-tube and crossflow arrangements. If F < 0.75, consider adding shell passes or switching to counterflow.

Required area: A = Q / (U x F x LMTD)

Typical overall heat transfer coefficient U:

Epsilon-NTU Method

Preferred when only inlet temperatures and desired capacity are known (sizing new equipment).

Effectiveness:

epsilon = Q_actual / Q_max = Q_actual / (C_min x (T_h,in - T_c,in))

Number of Transfer Units:

NTU = U x A / C_min

Heat capacity rates:

C_hot  = m_dot_hot x Cp_hot    (W/K)
C_cold = m_dot_cold x Cp_cold  (W/K)
C_min  = min(C_hot, C_cold)
C_r    = C_min / C_max

Epsilon-NTU relationships:

Design process: Choose target epsilon (typically 0.7-0.85), solve for NTU, then A = NTU x C_min / U.

When to use each method:

• LMTD: rating existing equipment or when both inlet AND outlet temperatures are specified
• Epsilon-NTU: sizing new equipment when only inlet temperatures and desired capacity are known

For LMTD correction factor charts and epsilon-NTU tables for all flow arrangements -- @references/heat-exchangers.md

Efficiency Metrics

PUE -- Power Usage Effectiveness

PUE = Total Facility Power / IT Equipment Power

Total includes: IT, UPS, cooling (chillers, pumps, fans, towers), lighting, power conditioning, security. Excludes non-data-center loads on the same meter.

Measurement: Annual average (preferred for reporting), not instantaneous. Use 15-minute interval samples for trending.

TUE -- Total Usage Effectiveness

TUE = IT Equipment Energy / (Total Energy - Energy Reused Externally)

Accounts for heat recovery: district heating, absorption chillers, aquifer thermal storage.

TUE = PUE x (1 - reuse_fraction)

If heat recovery reuses 25% of total energy and PUE = 1.3: TUE = 1.3 x 0.75 = 0.975. TUE < 1.0 is achievable when heat recovery is substantial. TUE <= PUE always; equality holds when no heat is reused.

WUE -- Water Usage Effectiveness

WUE = Annual Site Water Usage (L) / Annual IT Equipment Energy (kWh)

Sources of water use: cooling tower evaporation (~90%), humidifiers, chiller heat rejection.

Trade-off: air-side economizer uses zero water; evaporative cooling uses more water but delivers lower PUE.

CUE -- Carbon Usage Effectiveness

CUE = Annual Total CO2 Emissions (kg) / Annual IT Energy (kWh)
CUE = PUE x local_grid_carbon_intensity (kg CO2/kWh)

Reduce CUE by: renewable energy procurement (PPAs, RECs), carbon-aware workload scheduling, on-site generation, heat recovery (reduces total energy via TUE).

For PUE measurement methodology (Green Grid Annex A), TUE derivation, and WUE water budget -- @references/dc-efficiency-metrics.md

Airflow Management

Hot/Cold Aisle Layout

Servers face into the cold aisle (front intakes aligned), exhaust into the hot aisle. This is the foundational airflow pattern for all air-cooled data centers.

• Cold aisle supply: 15-27C (ASHRAE A-class), delivered from raised floor tiles or overhead diffusers
• Hot aisle return: 35-45C, returned to CRAC/CRAH inlets -- never allow recirculation back to cold aisle
• Row spacing: cold aisles 4 ft (1.2 m) minimum, hot aisles 3 ft (0.9 m) minimum

Containment Systems

CAC captures supply air; HAC captures exhaust air. HAC is generally preferred for new builds because it prevents hot air mixing with the room and allows higher CRAC supply temperatures.

Raised Floor Airflow

• Plenum height: 12 inches minimum, 18-24 inches recommended for high-density deployments
• Perforated tiles: 150-500 CFM per tile at 0.05 inches water gauge plenum pressure
• Tile placement rules: cold aisle only; no tiles in hot aisles or under power/network equipment
• Blanking panels: fill ALL empty rack U-spaces to prevent hot air recirculation -- highest ROI single action for thermal management
• Cable management: route cables to avoid blocking plenum airflow; use overhead trays where possible

Economizer Modes

Air-side economizer requires air filtration and humidity control. Water-side economizer avoids contamination risk by keeping outdoor air outside. Evaporative saves compressor energy but increases WUE.

Reference Documents

Thermal Engineering Skill v1.0.0 -- Physical Infrastructure Engineering Pack Phase 435-02 | References: ASHRAE 90.1, ASHRAE TC 9.9, Green Grid WP#32, ASHRAE Fundamentals 2021 All outputs require verification by a licensed Professional Engineer.