Rack Liquid Cooling: Cold-Plate Flow, Pressure Drop and CDU Sizing

Reserved liquid heat is $P_d=N_rP_rf_{liq}M_Q$. With $N=N_rN_b$ balanced paths, select $\dot V_s=\max[P_d/(\rho c_p\Delta T_{max}),Nq_{min}]$ and $\dot V_b=\dot V_s/N$. Rack rise is $P_d/(\rho\dot V_sc_p)$; equal plate loads give $\Delta T_{plate}=\Delta T_{rack}/N_s$.

Circular-pipe losses use $\Delta p=(fL/D+\Sigma K)\rho v^2/2$ with Churchill's all-regime Darcy friction factor (ASHRAE Fundamentals 2025, Chapter 3, equations 30–34). One plate uses the local measured fit $\Delta p_{cp}=\Delta p_{ref}(q_b/q_{ref})^n$. Add the series plates, one branch hose path, common pipes and quadratic auxiliary loss; then apply the pressure reserve. Parallel path losses are not summed. Electrical pump duty is $P_e=\Delta p_{pump}\dot V_s/\eta_w$. Exchanger duty is $Q=P_d+P_e$.

For the liquid-to-liquid CDU, $C_s=\dot m_sc_{p,s}$, $C_f=\dot m_fc_{p,f}$, $C_{min}=\min(C_s,C_f)$ and $C_r=C_{min}/C_{max}$. Combined fouling resistance $R_f$ is referred to the stated heat-transfer area $A$:

The counterflow relation is

The implementation uses an equivalent small-argument expansion to remain continuous and accurate around equal capacity rates. NPTEL Heat Exchangers, PDF pp. 11–12 and 27–29, supplies the resistance and ε–NTU relations.

Close the recirculating secondary loop at its imposed heat load:

All pump electrical power is conservatively added to the liquid after the exchanger: $T_s=T_{h,out}+P_e/C_s$. Rack return is $T_s+P_d/C_s$, equal to $T_{h,in}$. Facility return is $T_f+Q/C_f$. Both exchanger terminal differences are calculated. The delivered supply approach $T_s-T_f$ includes pump heat; the separate exchanger cold-end approach $T_{h,out}-T_f$ excludes it. This distinction follows the FWS/TCS supply-temperature convention in OCP's CDU Test Methodology, revision 1.0 (2024), §3.3.2.1.

For equal series plate heat loads, the warmest device inlet is $T_{in,max}=T_s+(N_s-1)\Delta T_{plate}$. Headroom is the entered accelerator inlet limit minus this temperature. Positive headroom checks only coolant inlet temperature, not junction temperature.

OCP OAI Liquid Cooling Guidelines (2023), PDF p. 7, informs topology and the example 10 K rack-rise target. NIST water properties at 30 °C and 1 bar supply the water defaults. APALTEK's eight-GPU example checks thermal flow and topology; ASHRAE Example 2 checks hydraulic loss. CMU's published countercurrent example checks the equal-capacity ε–NTU limit. Sensitivity curves use 101 directly evaluated points.

An independent distributed energy-balance boundary-value integration (SciPy 1.15.3 solve_bvp) verifies the reduced-flow, fouled and equal-capacity closed-loop temperatures without using ε–NTU. These regressions are identified separately from published reference data.

Steady, incompressible, single-phase Newtonian fluids with constant properties at representative mean temperatures. Primary and secondary properties are independent. Identical racks, balanced branches, equal heat per plate, no bypass, leakage or pipe heat exchange. The pump allowance enters immediately after the exchanger and before the rack.

The exchanger is an ideal two-stream counterflow unit with uniform effective UA, no axial conduction and no heat loss to the room. Enter clean UA at the intended facility and selected secondary flows. Fouling from both sides is combined on one common area basis; if the areas differ, convert each resistance to that basis before summing. Enter clean rather than already fouled UA to avoid counting fouling twice.

Clean UA stays fixed when varying facility flow, heat load or rack-rise target. The model captures finite primary/secondary heat-capacity rates and added fouling resistance; it does not infer flow-dependent film coefficients from exchanger geometry. Re-enter a flow-specific clean UA when assessing another final duty point.

The default 18 kW/K UA, 10 m² area, 40 °C accelerator inlet limit, clean condition, 150 L/min facility flow, component losses, geometry and pump efficiency are illustrative design inputs, not a manufacturer's ratings. The 40 °C limit applies at every accelerator inlet in this example and must be replaced with the chosen device's qualified limit.

Full-flow common pipe length excludes the distributing portions of a manifold. Branch hose length includes both legs. The cold-plate power law and quadratic auxiliary loss are local fits for the entered coolant. The plate reference-flow warning is a proximity reminder; the plotted hydraulic sweep is not a wider validation of that fit. Heat and pressure reserves are explicit design choices.

Use for duty-point planning of balanced rack loops. A passed coolant inlet check does not establish acceptable junction temperature or guarantee all branches receive the calculated flow. Validate the hottest downstream accelerator with vendor thermal data. The bulk coolant rise is not a cold-plate thermal resistance.

Counterflow ε–NTU is appropriate to the stated idealized exchanger, not an uncorrected multipass or crossflow unit. Validate effective UA with supplier performance data. Flow sweeps hold clean UA fixed: reduced facility flow can also reduce convective heat transfer and raise the real approach further. Fouling is thermal resistance only; blockage and its hydraulic penalty are not inferred. No CDU control valve, bypass, dew-point setpoint, fouling growth rate or transient thermal storage is solved.

At very low facility flow or UA the required steady temperatures may be outside a fluid's liquid range or equipment ratings. Such results indicate an unsuitable duty point; this constant-property model has no phase-equilibrium or pressure-rating data. Check both exchanger terminal differences, temperature limits, freeze protection, pressure ratings and condensation separately.

Churchill friction does not replace measured losses for corrugated hoses, developing passages or cold-plate microchannels. Check Reynolds-dependent fitting coefficients and transitional flow. A filled closed loop has no net circulation elevation head; fill pressure, absolute pressure, NPSH, expansion, relief and surge are outside this model.

Pump pressure includes auxiliary losses and reserve. Compare the external circuit loss with a supplier's available external differential pressure without subtracting internal losses twice. A pump curve must meet the selected flow and pressure at the actual efficiency and redundancy configuration. CDU heat duty includes reserve once and the conservative full electrical pump allowance. Primary-side pressure losses and facility pump sizing are not calculated. Check against the standard in force where applicable. Installation work belongs to a qualified person.