Here we discuss the role of Rayleigh number on the initiation of convection.
Continuing with our convection discussion from the introduction of the phenomenon and the subsequent explanation for the mechanism of free convection , we shall discuss in this post under what conditions free convection is initiated in an enclosed fluid.
At the end of the post on the mechanism of free convection we realized that free convection should 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. The onset of buoyancy driven free convection in an enclosed fluid has to take into consideration two more modes of energy dissipation in the fluid.
The pressure and buoyancy force imbalance equation, which explains the convection motion, needs to be recast to accommodate two more forces. One of our initial assumptions while explaining the mechanism of buoyancy driven convection is that before the temperature gradient prevails the fluid is at rest and is not subjected to any external influence which might induce motion. So when the fluid tries to move, or circulate (i.e., convect), it does so with minimum velocity. When the fluid packet moves, its motion is impeded by the viscous drag between the packet and the surrounding larger fluid bulk.
Viscosity, as we know, is the internal fluid resistance offered to a change in the momentum. For any fluid it can be evaluated from the constitutive relation
The above ‘equation’ means, dynamic viscosity mu is equal to the ratio of the applied shear stress tau on the fluid and the perpendicular direction change in the velocity component of the fluid (du/dy ) . If the fluid could be imagined as a stack of newspaper loosely tied together, when the top paper is ‘pushed’ along its surface, the stack would react to the ‘shear’, which is akin to the resistance offered by the fluid to the shear stress and is attributed to the fluid’s property viscosity.In our fluid packet, this viscous resistance acts against the buoyancy force and tries to impede motion. If the magnitude of the viscous drag force equals that of the buoyancy force, motion will cease.
The second dissipative effect is from the fact that Convection is not the only mode of heat transfer that could happen in the given circumstance – Conduction and Radiation being the other two. Out of these two, Radiative effects are predominant only at very large temperature values but Conductive heat transfer cannot be altogether ignored. Since the hot fluid packet from the bottom is displaced up by the buoyancy force into a colder region of fluid, the hot packet can ‘leak’ energy as heat into the colder surrounding, because of the temperature difference.
To explain in another way, the microscopic definition of heat assumes the molecules in the warm packet (look, who is using ‘molecules here!) to have higher average velocity than that of the surrounding. This makes the molecules in the packet to jiggle more freely thereby exchanging energy with the surrounding (colder) molecules of lesser velocity resulting in the equalization of their velocities. This results ultimately in the premature cooling of the fluid packet that began to raise from the bottom. For the fluid packet coming down from a cooler environment the heat transfer is in the other way leading to similar results.
So if the local temperature difference (say, between the fluid packet and its surrounding fluid) is reduced by heat diffusion (conduction) it results in a reduction in the buoyancy force. It is necessary that the buoyancy force, which is the result of the temperature gradient (because of the dependence of density on temperature, as we saw in the previous post ), must exceed the dissipative forces of viscous drag and heat diffusion to ensure the onset of convective flow.
Hence, for the fluid to convect, the buoyancy force, resulting in the displacement of the fluid packet up and down, must be more than the magnitude of the ‘fluid brake’ and ‘heat diffusion’. These requirement are expressed as a non-dimensional number, called the Rayleigh Number in honor of Lord Rayleigh who came up with the explanation for this convection behavior of an enclosed fluid subjected to a temperature differential.
The Rayleigh number is the buoyant force divided by the product of the viscous drag and the rate of heat diffusion. In equation form using symbols it reads
where 
Convection sets in when the Rayleigh number exceeds a certain critical value. Thus Rayleigh Number (a non-dimensional number, if you notice) is a quantitative measure/ representation of when the ‘switch’ from conductive to convective transport happens for a given fluid plus geometry configuration. Once the Ra exceeds a critical value, henceforth, the dominant energy transport mechanism in the fluid would be convection.
Lord Rayleigh’s analysis of the problem of convective flow was initiated by the 1901 experiments of Henri Benard. I am not going to explain the detailed analysis done by him. For the records, it can be found with all of the minute derivation details and gaps filled in, in Subrahmanyam Chandrashekar’s book Hydrodynamic and Hydromagnetic Stability, Dover Publishers. Convection lovers should take a look at this remarkable book just to see what scientific lineage we hail from in India and what legacy it has left us to cherish.
(There is another reason I am not going into the equations. I don’t know how to type differential equations in its original form in a blog. Suggestions are welcome.)
While trying to explain those experiments of Benard, Rayleigh devised the theory explained above. For instance, for a thin fluid layer (H is very small when compared to the length) confined between two sufficiently long horizontal parallel plates, with the bottom plate hotter than the top one by a \Delta T, it would take the Ra to be greater than 1708 for convection motion to set in the fluid. For any fluid (water or air or mercury or, you get the idea…). This requirement of Ra > 1708 for the above configuration has been shown experimentally to be true over the years by many researchers. For interested readers, more on these experiments can be read from the book by A. V. Getling, Rayleigh Benard Convection: Structures and Dynamics , 1998, World Scientific publishers.
I have also done this experiment.
And so have you, in a slightly modified form, when you made hot water using a stove (not microwave oven – that is a different type of convection).
A little digression: while searching for information on Benard, I realized Wikipedia doesn’t have a page on him. I should do it someday now. And funnily, after a few google clicks, I ended up with a page that explained about the experiments of Benard in a paragraph vaguely familiar to me. I realized later, the paragraph was directly from one of my early papers!
Before proceeding with other issues on Convection Stability; Convection Carnot Engine; Marangoni Convection, I shall end this post with a question that nags me always: Why 1708? What is special about this number? It is neither 1729 nor 1776, nor is it a prime number or 1709 or 1707. I don’t have a simple, convincing answer yet. I invite my readers to comment.
24 responses so far ↓
Dr.Katte // August 3, 2006 at 8:54 am |
you can type the equations in MSWord or Powerpoint, using equation editor (presently, if equation editor is not installed, reinstall MSOffice by customising and selecting these options). Then save the equation as a picture (by right clicking on the equation) in any format you wish. Upload the picture, so simple.
you are welcome!
btw, you are yet to register your comments on a couple of articles in my blogsite.
Arunn // August 3, 2006 at 9:13 am |
Yes, I use equation editor with Word regularly. Converting into an image renders the equation text quality a bit smudgy, which I don’t like. I may still do this, if this proves out to be the only option with a blog.
Arunn // August 3, 2006 at 10:18 pm |
And I am unable to access Blogspot blogs from inside IITM for the past two days, for me to comment!
mdmohiddin // August 11, 2006 at 11:31 am |
sir i want an experimental set up for free convection in vertical cylinder…….can u help me in this ………….
Arunn // August 14, 2006 at 1:48 pm |
mdmohiddin:
If by your comments you mean you want to buy one, you should search the web for manufacturers.
If it is for some specific laboratory-level research purpose, you would be better of taking a look at some of the research papers published in the ASME Journal of Heat Transfer and the Int. Journal of Heat and Mass Transfer. Go to http://www.sciencedirect.com and use appropriate keywords and browse through the paper abstracts that might give you a good lead.
Alternately, you could start with the research references given at the back of natural convection chapters in Heat Transfer text books such as that by J. P. Holman or F. Incropera and DeWitt or A. Bejan.
Sai // January 13, 2007 at 10:05 am |
Hi
Just came across your blog and was fascinating to read it. It is nice to find a blog (and through yours, a whole bunch of them) with good scientific content. Thanks.
Just to provide a quick solution to the issue of printing equations in blogs – you can try tex2gif. It’s a small python app where you type the equation in latex and it converts it to a gif image.
Arunn // January 13, 2007 at 10:51 am |
Sai: Thanks for the visit and comments. Keep conversing here. Will check the tex2gif suggestion. (seems you are an IITM alumni. Have we met?)
Sai // January 13, 2007 at 11:29 am |
I think I graduated just before you came to IITM (~May-June 2003). So, I don’t think we have met. I was in a 5-year dual degree program in Chemical Engg. I am currently working on my Ph.D at Johns Hopkins.
Incidentally, I am also working on convection in porous media using a little bit of theory (stability, Rayleigh number effects etc) and a lot of computational work (spectral simulations).
Arunn // January 13, 2007 at 11:41 am |
Sai: Interesting to note that you are working in stability of flows in PM. I am working in this are of late. Should share research notes in due course (have sent an email to you). Good luck with your Ph. D.
Sai // January 24, 2007 at 11:27 pm |
This is small note concerning the critical value of the Rayleigh number (1708, as you mentioned). From the definition, Rayleigh number is small when either the viscosity/thermal diffusivity is large, or, when the temperature difference is very small. Under these conditions, the dominant physical mechanism is diffusion of heat, and any deviations from the solution of the diffusion equation (the linear temperature profile) get damped out. When the Rayleigh number is large enough, we will have convection.
A simple analogy is trying to push a block of wood on a table. If we apply a very weak force, the block does not move because we have to overcome friction (diffusion, for the convection problem). There is a minimum force that needs to the applied to cause the block to move. This minimum force depends on the material properties, amount of friction etc. For the convection problem, this minimum force (Rayleigh number) was found to be 1708. This value can be derived analytically, and was done by Reid and Harris in 1958 (Phys. of Fluids, vol 1).
Arunn // January 25, 2007 at 4:56 pm |
Sai: Thanks for the proper “value addition” to the post (one of the reasons for the existence of the “comments” section in a blog)!
Pradeep // March 30, 2007 at 9:02 pm |
Sir
While studying Solidification of ALLOY in mushy layer in one of papers it is given that temprature gradient induces stable density fields so thermal buoyancy and compostoinal variations also induces density variations which are conectively unstable. does it(thermally stable) mean that “thermal convection( because of density variations) is balanced by thermal diffusivity and visocity variations. so it is stable” i.e. its reyleigh number is less than 1708? is there any counter acting forces to balance compostional variations like in the case of thermal convection?
Yogesh // April 2, 2007 at 1:09 pm |
We are studying the heat transfer from vertical & horizontol surfaces by natural convection. Please help me hoe to calculate this mathematically.
parseval // April 15, 2007 at 12:36 pm |
Professor,
This is a nice blog, with fantastic scientific content!
About displaying equations, it’s possible to use LaTeX in WordPress.
Arunn // April 16, 2007 at 4:46 pm |
Pradeep and Yogesh: sorry I am unable to answer your questions.
parseval: thanks for the suggestion. The plugin has been implemented for sometime now at this blog. Do check the recent posts. I should update the content of this post.
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OAnttila // September 2, 2007 at 6:35 am |
Dear Arunn,
You wrote concerning experiment that you have performed to demonstrate R-B convection
“I have also done this experiment.
And so have you, in a slightly modified form, when you made hot water using a stove (not microwave oven – that is a different type of convection).”
If you have ever heated regular vegetable oil on a stove, e.g., in order to deep fry something, you have probably noticed that distinctly separated areas develop into the fluid. The boundaries between the areas move slowly, some disappear and new ones are born. That is, if you heat vegetable oil, you may actually see the phenomenom and even take video for educational purposes. The optical properties of oil change significantly with its temperature, contrary to water. Oil is also much more viscous, and that is why everything takes place slowly and the convective cells are relatively large in size.
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Arunn // September 7, 2007 at 8:59 am |
OAntilla: thanks for the suggestions.
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