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Equation 11 · The Physical Plant: Power, Cooling, and Networks in an AI Datacenter

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ΔT\Delta T

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the temperature rise. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ΔT\Delta T

Symbol Δ T

the temperature rise.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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The following is my own worked calculation, using standard property values rather than any cited source. At around 27 °C and atmospheric pressure, air has ρ\rho cpc_p ≈\approx 1.17\ kJ/(m3⋅K)\mathrm{kJ/(m^3{\cdot}K)} ; liquid water has ρ\rho cpc_p ≈\approx 4180\ kJ/(m3⋅K)\mathrm{kJ/(m^3{\cdot}K)} . The ratio is roughly 3,600 to one. Rejecting 100 kW at a 15 K rise therefore requires about 5.7\ m3/s\mathrm{m^3/s} of air — near 12,000 cubic feet per minute, through a single rack aperture — or about 1.6\ L/s\mathrm{L/s} of water, which is a garden hose. There is no engineering cleverness that closes a factor of 3,600; the only levers are a larger Δ\Delta T or a larger duct, and both run out.

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