Free Convection for Dummies

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A fluid layer heated from below and relatively cooled from the top (say, water kept in a vessel heated on a stove) is a simple free convection system, which experiences a local force imbalance in a gravitational field resulting in the convection of the fluid. This convection or displacement of some portion of the fluid (water, in our vessel) from one local position to another (inside the vessel) is the result of the buoyancy of the heated layer and the magnitude of it depends on the temperature difference prevailing between the top and bottom portion of the fluid layer (vessel).The earliest description of (free) convection was written in the 1790s by Benjamin Thompson, Count Rumford, a description he used to account for the transfer of heat in an apple pie. The most significant and systematic experimental work was done by the Frenchman Henri Benard in the early 1900s. The results of these experiments, i.e., why there are convection currents in a heated fluid layer, by a theory put forth by John William Strutt, Lord Rayleigh. Some physicists have had the insight of originating a new field of study thereby making the first word(s) about it. But Lord Rayleigh, according to Subrahmanyam Chandrashekar, had said the last words of many subjects and sealed them once and for all. In one of his last articles, published in 1916, he attempted to explain what is now known as Rayleigh-Benard Convection. Though his explanation was superceded in later years, his work remains as the starting point for most of the modern theories of convection.

In what follows, we shall discuss qualitatively, how and why there is convection motion in a fluid heated differentially. For doing this, a simplified fluid, as against a real one, is considered in the two dimensional model. A thin layer of the fluid is confined fully between two semi-infinite flat plates so that there is no gap (free surface) between the plates and the sandwiched fluid. By semi-infinite we mean plates of definite thickness vertically and long, infinite length horizontally and by a thin layer we mean that the horizontal dimension (length, in Figure 1) of the fluid layer is very large when compared to that of the vertical. These requirements ensure minimal interference of the side boundaries (walls etc.) in the convection.

We can now subject the thin layer of the fluid to a controlled experiment where the fluid is heated from the bottom in such a way that the temperature of the bottom portion of the fluid remains uniform (spatial invariance) and steady (temporal invariance). A similar cooling condition is imposed at the top portion so that the temperature gradient across the height is uniform. This means that the graph between Temperature and Time is a straight line as shown in Figure 1. In the language of calculus this means the change in temperature with respect to height or time is a non-zero constant.

Further, by simplified fluid we mean the fluid is assumed to be incompressible (why?) with its density being the only property that is changing with respect to the change in temperature (what other property could change w.r.t. to T?), while the fluid experiences uniform gravitational force on the entire volume (when can the fluid experience non-uniform gravitational influence?).

Now we shall construct a model in which a packet of the above fluid is considered with a random dimension. The displacement of this packet above or below from its present position is controlled by forces whose nature is yet to be seen but are nevertheless responsible for convection. The packet considered can be of any size and shape but the displacement must be small. The initial displacement of the packet need not be due to any imbalance in the forces we are studying but a random displacement from the mean position – which will eventually occur if we wait long enough – is sufficient. To further understand how convection would result in this simplified model, we need to be familiar with buoyancy, viscosity, surface tension and thermal diffusivity of fluids.

Let us begin with buoyancy and proceed to explain free convection motion and used the other concepts as and when required to refine our understanding.

We all have heard of buoyancy in our school stories and how it made Archimedes run naked shouting Eureka!, to explain his method involving buoyancy force differences to find the silver impurities in the royal crown of the King of Heron supposedly made of pure gold. This kind of buoyancy force in a fluid arising without heating is due to the pressure difference across the fluid packet, which when balanced by the weight force of the packet, ensures static equilibrium.

However when heated, the fluid packet considered in Figure 2 has lesser density at the bottom relative to the surrounding which makes it raise up, when the gravitational force is acting downwards. This is because of the increase in the buoyancy force which disrupts the static equilibrium. Suppose in the fluid packet in Figure 2, if water is filled and allowed to get immersed to a certain depth as shown, it will stay in the position inside the trough maintaining its static equilibrium. This is because the weight of the packet is balanced by the upward reaction force, by the water in the trough, called the buoyancy force.

The static pressure in the water trough increases as we go down because the weight of the water layer above each point also contributes to the net force experienced by that point. So this static pressure is greater on the bottom side of the packet than the upper. This balance in forces can get affected by the actual weight of the fluid in the packet and the pressure difference across the packet. For a heavier packet the weight force increases causing the packet to sink to a different height where the upward buoyancy force equals the weight force to make the packet float. So heavier, more massive, objects of identical volume hence with higher density, sink when compared to lighter objects.

Now consider the same fluid packet at the bottom of the trough in Figure 2 with the heat supplied to it from below, as shown. This packet considered has a higher temperature and so has lesser density (density, in general, decreases with increase in temperature for fluids) when compared to the average density of the entire layer. This means, the fluid packet of certain mass ‘dm’ expands as it is heated. A similar packet at the top side of the domain in Figure 2 will have relatively higher density due to its lesser temperature. Hence this packet, also of mass ‘dm’, is cooler and occupies lesser volume than the hotter one in the bottom. As long as the fluid packets remain in their respective position they are surrounded by fluid of identical average density and so maintain their static equilibrium with the surrounding. Suppose now due to some random fluctuation a very small displacement is given to the bottom fluid packet in the upward direction. This will result in an imbalance in the forces acting on the packet.

The fluid packet which is originally of lesser density than the surrounding average density due to its higher temperature now is pushed up into a region of higher density (and lower temperature). This creates a positive buoyancy as explained earlier, which causes the packet to raise. The raise will be sustained till the density of the fluid packet while raising equals that of the surrounding. At this point it will simply float as the static equilibrium is restored. The upward force is proportional to the density difference and volume of the packet. As the fluid packet raises through regions of relatively colder fluid whose average density progressively increases due to the lack of additional heat, it results in an increased density gradient between the packet and the surrounding, which accelerates the raise.

On similar analysis the downward push of a packet of colder fluid in Figure 2 makes it enter a region of lesser average density resulting in the `heaviness’ of the packet thus propelling it down. It would sink getting its initial disturbance enhanced. Both of these motions are simultaneous and continuous. The ‘hole’ left by the hotter fluid packet as it raises up is (and has to be) filled by fluid from somewhere else in the domain of Figure 2. In principle, this is done by the colder fluid packet that is propelled down from the top. Essentially, the differential heating results in the hotter and colder fluid packets exchanging places in the domain of Figure 2. Thus the whole of the fluid layer is eventually overturned resulting in a continuous circulation of the fluid between the hot and cold ends. This circulation should prevail as long as the temperature gradient could be maintained by the heating, thus maintaining the local buoyancy force imbalance.

A graphic representation of the convection flow in such a simplified experiment is shown in the animation below.

R-B cell

It would seem from the above explanation that (free) convection will be observed in a fluid region whenever there is a temperature gradient, however small it may be. But such sensitive dependence of the initiation of the flow on the temperature gradient is not observed in actual circumstances. There is a critical value of some variable, beyond which only convection flow results. Remember, we are yet to use the other concepts like viscosity and thermal diffusivity.

And this is where Lord Rayleigh showed his wit. And it merits another note.