Fan Curves, System Impedance, and Where a Cooler Actually Operates

A heat sink’s thermal resistance is not a constant. It is a function of the air velocity passing through it, and the velocity is not chosen by the heat sink designer. It is the result of an intersection between two curves: the fan’s pressure-flow characteristic and the chassis’s resistance to flow. Reading that intersection is the whole of fan selection, and it explains why a higher-rated fan sometimes cools worse than a lower-rated one.

The energy balance that sets the required flow

Air carries heat with its sensible enthalpy, so the flow required to remove a given power at a given temperature rise is

$$\dot{V} = \frac{P}{\rho , c_p , \Delta T}$$

with $\rho$ the air density, $c_p$ the specific heat, and $\Delta T$ the allowed rise of the air above inlet. At sea level and room temperature, air has a density near 1.2 kg/m³ and a specific heat near 1005 J/(kg·K), so the product is about 1200 J/(m³·K). A 300 W load with a 10 K air rise therefore needs about 0.025 m³/s, which is roughly 53 cubic feet per minute.

That is the total flow through the chassis, not the flow through the fins. The distinction matters because a heat sink sits in a duct of its own making, and only the flow that passes between the fins removes heat from them. Air that goes around the sink, over the top, or through a bypass gap cools the chassis but not the die. The ratio of the flow through the fin field to the total is the ducting efficiency, and in a chassis with an open layout it can be well below half. Where the fins themselves sit, and how far heat must travel sideways to reach them, is the subject of heat pipes and vapor chambers.

The temperature rise allowed is a design choice with a direct cost. Allowing 20 K instead of 10 K halves the required flow, which halves the fan’s speed and reduces acoustic noise substantially, at the price of a warmer inlet to the heat sink and therefore a warmer junction. Since the junction limit is fixed, the entire system budget is a chain of such allocations.

Two curves, one operating point

A fan is a pressure source with an internal resistance. Its published characteristic plots static pressure against volumetric flow at a fixed speed: maximum pressure at zero flow, maximum flow at zero pressure, and a curve between them that is roughly parabolic for a well-designed axial fan. The curve is measured on a standard test plenum, so it describes the fan alone, not the fan installed in anything.

The chassis and heat sink present a resistance that rises with the square of flow: $\Delta P = k \dot{V}^2$, where $k$ lumps every restriction in the path, from the intake grille through the fin gaps to the exhaust. Plotting that parabola on the same axes as the fan curve gives an intersection, and that intersection is the operating point. It is the only point where the fan’s pressure rise equals the system’s pressure drop, so it is the only point the system can actually run at.

The consequence is immediate and counterintuitive. A fan’s free-air delivery, the number quoted at zero pressure, is never achieved in a system. A fan with a steep pressure curve holds its flow well as restrictions are added; a fan with a flat curve collapses toward zero flow as soon as the system needs pressure. Two fans with the same free-air rating can differ by a factor of two in delivered flow through a dense fin stack, and the steep one wins.

Adding a dust filter, a front bezel, or a drive cage raises $k$, pivoting the system parabola upward and moving the intersection down and to the left. The airflow falls, and because the sink’s resistance rises as velocity falls, the junction temperature rises more than proportionally. That is the mechanism behind the loss of cooling margin as dust accumulates over a product’s life.

Fan laws and why speed is the wrong knob to turn casually

For a given fan geometry, changing the rotational speed scales the entire characteristic:

$$\dot{V} \propto N, \qquad \Delta P \propto N^2, \qquad P_{\text{shaft}} \propto N^3$$

so doubling the speed doubles the flow, quadruples the pressure, and multiplies the power by eight. The acoustic consequence is worse than the power one: sound power radiated by a fan rises roughly with the fifth power of tip speed for the broadband component, and the tonal component at the blade-pass frequency rises with the sixth or more. A fan run at 60 percent speed is not 40 percent quieter than one at full speed; it is dramatically quieter, which is why fan curves that plateau at low duty are common in quiet systems.

The fan laws also explain why the operating point moves along the system curve rather than staying at a fixed flow fraction. If the speed falls by half, the fan curve scales down, and the new intersection with $\Delta P = k\dot{V}^2$ is at a lower flow, but not necessarily half: the shape of the intersection depends on where the two curves are steep. In the steep part of the fan curve, near zero flow, speed changes move the operating point a lot; in the flat part, near free delivery, they move it little.

Series and parallel, and what each buys

Fans in parallel add flow at a given pressure. Fans in series add pressure at a given flow. The system curve decides which is useful.

If the restriction is low, the operating point sits in the flat, high-flow part of the fan curve, and two fans side by side deliver nearly twice the flow. If the restriction is high, the operating point sits in the steep part near zero flow, and parallel fans deliver almost no additional flow because each is already near its pressure ceiling; series fans, which double the available pressure, move the intersection substantially. This is why a dense server with a long, narrow airflow path uses fans in a row, while a large open desktop case uses several fans spread across the panels.

Series operation has a failure mode worth knowing. Two identical fans in series do not simply double the pressure curve; each sees the other’s exit swirl, and if the second fan’s blades are not matched to the first’s outflow the combined curve is worse than the sum. In practice series fans are separated by a duct or a gap precisely to let the swirl decay.

Heat sink resistance is a function of velocity, not a number

Forced-convection fin arrays have a thermal resistance that falls with velocity roughly as $v^{-0.5}$ in laminar flow and closer to $v^{-0.8}$ in turbulent flow, because the heat transfer coefficient follows a Nusselt correlation of the form $Nu \propto Re^{m} Pr^{1/3}$ with $m$ near 0.5 for laminar flat-plate flow. The exact exponent depends on the geometry, but the shape is consistent: doubling the airflow through the fins reduces the sink’s resistance by about 30 percent, not by half.

That diminishing return is the economic core of air cooling. The last doubling of fan speed buys a third of the resistance and costs eight times the fan power and far more noise. It is why a bigger sink with a slower fan usually beats a small sink with a fast one, and why the design point is chosen where the marginal kelvin per decibel is still acceptable.

The resistance this velocity produces is the sink-to-ambient term in the junction temperature stack, and it is usually the largest of the four.

Fin spacing is the other half of the trade. Densely packed thin fins have more surface area per volume, but the boundary layers in the gaps merge and the flow becomes fully developed, which lowers the heat transfer coefficient; and the narrow gaps raise $k$, reducing the flow the fan can deliver. There is an optimum spacing that depends on the available pressure, typically a few millimetres for the pressures a case fan can produce, and pushing below it makes both the sink and the fan worse at once.

Altitude, temperature, and the derating nobody plans for

Air density falls with altitude, roughly 1 percent per 100 m in the lower atmosphere, and the fan’s pressure capability falls with it because pressure rise scales with density. A system rated at sea level loses roughly 10 percent of its cooling capacity at 1000 m and substantially more at 3000 m, where density is about 70 percent of sea level. Fan curves published at standard conditions therefore overstate performance in a high-altitude installation, and the derating must come out of the thermal margin rather than being discovered later.

Air temperature has a double effect. It raises the inlet temperature directly, which raises the junction temperature one-for-one, and it lowers density, which reduces the mass flow for the same volumetric flow. Both effects push the same direction, and the standard rating conditions (typically a specified ambient and altitude) exist so that a quoted thermal resistance means something.

What the intersection leaves out

The operating point gives flow, and flow gives sink resistance through a correlation, and resistance gives junction temperature. Each step in that chain has uncertainty, and they compound. The fan curve is measured on a plenum that does not resemble the chassis; the system impedance is estimated from a model that is usually calibrated only after a prototype exists; the sink correlation assumes uniform approach velocity that the fan’s outflow does not provide; and the power number is a workload assumption.

The practical consequence is that airflow design is iterative, and the iteration is measurement rather than calculation. The useful output of the analysis is not a predicted junction temperature but a sensitivity: how many kelvin per percent of flow, and therefore how much margin is needed for the uncertainties. A design with 20 K of margin can absorb a badly estimated impedance; one with 3 K cannot, regardless of how careful the model was.