$$\dot{Q} = \dot{m}\,c_p\,\Delta T = \rho\,\dot{V}\,c_p\,\Delta T$$

Equation 1 · Datacenters & Infrastructure

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

Cooling power equals fluid mass flow × specific heat capacity × temperature rise. Density converts volume flow into mass flow.

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

Meanings in this article

• $\dot{Q}$: The rate at which the cooling stream removes heat from the load. The dot means “per unit time”. Its SI unit is watts, or joules per second.
• $\dot{m}$: How many kilograms of cooling fluid pass through the loop each second. Its SI unit is kilograms per second.
• $c_p$: The heat needed to raise one kilogram of the fluid by one kelvin at approximately constant pressure. Its SI unit is joules per kilogram per kelvin.
• $\Delta T$: The fluid’s temperature increase while crossing the load. A change of one kelvin is the same size as a change of one degree Celsius; the sign follows outlet minus inlet here.
• $\rho$: Mass per unit volume of the cooling fluid. Its SI unit is kilograms per cubic metre.
• $\dot{V}$: The volume of fluid passing through the loop each second. Its SI unit is cubic metres per second. Multiplying it by density gives the mass flow rate.
• $\rho\,\dot{V}$: Density × volume per second = mass per second, so ρV̇ can replace ṁ in the first version of the formula.

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