Convective Carnot Engine

About this note

A plain-language retelling of my Physics Education article on convection as a heat engine. The full citation is under Reference.

Why an analogy

Convection, the transport of heat by a moving fluid, is easy to see and hard to explain. Students meet it as a set of correlations and a dimensionless number with a critical value, and the physics behind them often stays hidden. This article came out of several years of teaching undergraduate thermodynamics and heat transfer, and it rests on a single analogy that students already have the tools to follow: a convecting fluid is, thermodynamically, a heat engine, a device that takes heat from a hot reservoir, rejects heat to a cold one, and delivers work in between (Narasimhan, 2000).

How convection starts

Picture a shallow trough of water, much wider than it is deep so that the side walls hardly matter, heated uniformly and steadily from below and cooled from above. Temperature then falls steadily from the bottom of the layer to the top.

Consider a small packet of fluid at the bottom. Being warmer than the average, it is lighter than the layer as a whole, but as long as it stays put it is surrounded by fluid just like itself and nothing happens. Now let a chance disturbance nudge it upward. It finds itself among cooler, denser fluid, and buoyancy, which is proportional to the density difference and to the packet’s volume, pushes it further up. The higher it goes, the colder and denser its surroundings, so the push grows and the packet accelerates. In the same way a cool, heavy packet at the top, nudged downward, finds itself among lighter fluid and sinks faster and faster. The whole layer eventually overturns into a steady circulation between the hot bottom and the cold top. That is the onset of convection. In a fluid heated from the side the same reasoning holds, only the flow starts more readily because gravity already acts along the direction the fluid wants to move.

The four strokes of the engine

Follow one packet around this circulation and it executes the four steps of a Carnot cycle:

1. Isothermal expansion. At the bottom, the packet takes in heat from the hot surface at a nearly constant temperature and expands.
2. Adiabatic expansion. Lighter than its surroundings, it rises, expanding further as it goes, and in rising it can do work.
3. Isothermal compression. At the top, it gives up heat to the cold surface and contracts.
4. Adiabatic compression. Cooled and heavier, it sinks back to the bottom, contracting further on the way, replacing the fluid that rose and closing the cycle.

Heating, expansion, cooling and compression: the convecting layer is a Carnot heat engine. Its ability to do work is real and can be shown in the laboratory. A light propeller placed in the path of the flow will turn, and aluminium powder strewn in heated water makes the circulation visible. In practice, though, almost all of the work goes into pushing the packet against viscous drag, the fluid’s brake.

Why the engine is so feeble

An order-of-magnitude estimate, drawn from the momentum and energy equations of the convecting fluid, shows how small the output is. The work gained from buoyancy (ignoring viscosity and heat diffusion) scales as

$W\sim \rho U\Delta Tg\alpha$

and the heat taken in scales as

$Q_1\sim \frac{\rho C_{p}U\Delta T}{L}$

where $\rho$ is the density, $U$ the flow speed, $\Delta T$ the temperature difference across the layer, $g$ the acceleration due to gravity, $\alpha$ the thermal expansion coefficient, $C_p$ the specific heat and $L$ the depth of the layer. Their ratio is the efficiency of the convective engine:

$\frac{W}{Q_1}\sim \frac{gL\alpha }{C_p}$

For a layer of water 1 cm deep, with $\alpha \approx 2\times 10^{-4}$ per K and $C_p\approx 4200$ J/kgK, this comes to about five parts in a billion (a figure worked out here for illustration, not in the article). The work is minute and the heat rejected at the top is almost equal to the heat taken in at the bottom. How much heat each packet keeps on the way up, and so how much work it can do, depends on its specific heat.

Where the losses go: the Rayleigh number

Energy is conserved in this engine, as the first law requires: the heat that is not carried through to the top becomes the small kinetic energy of the packet, most of which is spent on overcoming drag. An ideal Carnot engine would also be fully reversible, generating no entropy. The real convective engine is not. Viscous friction and heat conduction generate entropy, and that irreversibility is what limits its work.

Both brakes, together with the buoyancy that drives the flow, are gathered in one dimensionless group, the Rayleigh number, named after Lord Rayleigh, who studied this flow in the early 1900s:

$Ra=\frac{g\alpha \Delta T{L}^3}{\chi \nu }$

Here $\chi$ is the thermal diffusivity, a measure of how quickly heat leaks out of a packet by conduction, and $\nu$ is the kinematic viscosity, a measure of how quickly its momentum is smeared out by friction. The Rayleigh number is thus the ratio of buoyancy, which drives the packet up, to the two diffusive brakes, which dissipate its energy. When it is small the brakes win and heat crosses the layer by conduction alone. For a layer like the one described here, once it exceeds a critical value close to 1708 the engine runs, and convection becomes the main way energy is carried across the layer (Narasimhan, 2000).

Related

• Free Convection and the Rayleigh Number
• Free Convection for Dummies
• Natural Convection, our research on buoyancy-driven flows
• Solar Convection and Sun Spots

Reference

Narasimhan, A., (2000), “Convective Carnot Engine: An Easy Approach to Understanding Convection”, Physics Education, 35(3), 178-181. [DOI 10.1088/0031-9120/35/3/307 ]