How does a heatsink on top of a CPU, which is hot, cool down your CPU?

In general, a heatsink should feel hot, if it's doing its job right. If it feels hot, that means it's transferring energy to your hand, which means it's transferring energy away from the CPU. Conversely if it felt cool, it'd be heating up the CPU.

This also holds for other cooling things. The cooling coils on a fridge or an old A/C should feel hot for the same reason: the heat you're taking away has to be dumped somewhere.

A heat sink really shouldn't be hot (although it is a relative term, of course) at all. At higher temperature than ambient air certainly but not hot.

When a heat sink is in intimate contact with a source of heat, like a CPU, heat is transferred in accordance with Newton's Cooling Law:

$$\frac{\mathrm dQ}{\mathrm dt}=uA(T_\textrm{CPU}-T_\textrm{Sink})$$

Where $u$ is a heat transfer coefficient (CPU to sink) and $A$ is the area of contact between the CPU and the heat sink.

Note that $\frac{\mathrm dQ}{\mathrm dt}$ is the heat carried off from the CPU per unit of time.

Higher values of $T_\textrm{Sink}$, as the basic formula shows, actually decrease $\frac{\mathrm dQ}{\mathrm dt}$, which becomes effectively zero when $T_\textrm{CPU}-T_\textrm{Sink}=0$.

To prevent this from happening, the heat sink itself has to transfer accumulated heat, usually to the surrounding air, in which case another heat transfer equation comes into play:

$$\frac{\mathrm dQ}{\mathrm dt}=hA_\textrm{Sink}(T_\textrm{Sink}-T_\textrm{air})$$

Where $h$ is the convection heat transfer coefficient (sink to air) and $A_\textrm{Sink}$ the sink's surface area exposed to the air. $h$ is very dependent on speed or airflow which explains why forced air circulation (fan assisted ventilation) is often used.

In steady state ($T_\textrm {CPU}\approx \text{Constant}$), we have, with $\dot{Q}_\textrm{CPU}$ power generated by the CPU:

$$\dot{Q}_\textrm{CPU}=(T_\textrm{CPU}-T_\textrm{air})\left[\frac{1}{uA}+\frac{1}{hA_\textrm{sink}}\right]=(T_{CPU}-T_\textrm{air})\frac{1}{K}$$

Or:

$$T_\textrm{CPU}=T_\textrm{air}+K\dot{Q}_\textrm{CPU}$$

With:

$$K=\frac{uhAA_\textrm{Sink}}{hA_\textrm{Sink}+uA}$$

The influence of the various factors on $T_\textrm{CPU}$ can be readily appreciated.

Ingenious ways of increasing both $h$ (apart from forced circulation) and $A_\textrm{sink}$ to lower $T_\textrm{CPU}$ have been used like cooling fins or this design (ST-HT4 CPU Cooler Riser):

The highly heat conductive copper bands mostly release the heat from the U-bends.

That nice image of a cooler riser looks as though the four copper U's are heat pipes; they have effective thermal conductivity far better than copper. Look up "heat pipes" for more info. They're worth knowing about.

Regards,

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