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Equation 1 · Part 7 · The Physical Plant: Power, Cooling, and Networks in an AI Datacenter

Mass flow from volume flow

Q˙=m˙ cp ΔT=ρ V˙ cp ΔT,\dot{Q} = \dot{m}\,c_p\,\Delta T = \rho\,\dot{V}\,c_p\,\Delta T,
ρ V˙\rho\,\dot{V}

What this part means

Density × volume per second = mass per second, so ρV̇ can replace ṁ in the first version of the formula.

Its job in the formula

rhoV̇ has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

The passage around this formula

The rate at which a fluid stream removes heat is Q˙=m˙ cp ΔT=ρ V˙ cp ΔT\dot{Q} = \dot{m}\,c_p\,\Delta T = \rho\,\dot{V}\,c_p\,\Delta T. with mass flow m˙\dot{m} , specific heat capacity cpc_p , and the temperature rise Δ\Delta T across the load. What matters for a datacenter is not cpc_p per kilogram but the volumetric heat capacity ρ\rho cpc_p , because the room has to physically contain the flow.

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Learn the underlying idea

Multiplication scales one quantity by another. A dot, a cross, or adjacent symbols can indicate a product.

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Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.

Further reading for this equation