Date: Wed, 23 Nov 94 01:06:45 CST From: rusin (Dave Rusin) To: ab49@ab49.cityscape.co.uk Subject: Re: COOLING CURVES Newsgroups: sci.math In article <3atdqe$ghv@ns.cityscape.co.uk> you write: >I need some help. I'm doing an A level applied maths project the aim >of which is to *Model the cooling of a glass of boiling water*. I really Hey, cool! (sorry, couldn't resist) >need to know what the actual equation for this is, and was hoping you Sigh. I spend most of my educating life trying to convice students that math is _not_ equations or computation, but rather a language. There's no point in asking for _the equation_ until you've defined the variables. So T is the temperature, perhaps -- at time t I suppose. But do you intend to take an _average_ temperature for T? Or is T also a function of the three spatial variables? (And do you intend to write it as a function defined oon points in R^3, or on Cartesian triples (x,y,z), or perhaps cylindrical triples (r, theta, z)? Not to nag unnecessarily, but it would seem appropriate to clarify first the question before asking if anyone knows the answer. Now to be more helpful: It does sound like you've thought about the expected cooling of a hot point in an infinite cool medium, in which dT/dt = c . T for some constant c, where T is the difference in temperature between the hot point and the cool medium. As a next step, perhaps you can replace the point with a cylinder (the glass) consisting of several nestd cylinders, each assumed to be of homogeneous temperature at each time t. This will give you a system of DE's to solve, coming from: (1) temperature of each shell is proportional to heat energy in the molecules in the cell, and (2) heat flow between consecutive shells is proportional to the difference in temperature between those shells. Now you get to think about those constants of proportionality. At least one of them depends on the surface area of one of the shells -- do you see why? This model could be refined to reflect heat loss out the top (and bottom) of the cylinder(s) too. Then you might consider going whole hog: why assume the temperature distribution admits any kind of symmetry? At this point you begin to assume T depends not just on t but on spatial coordinates x y z as well. Here you need to look up the heat equation in a differential equations book. At this stage you can begin to ask whether you think the initial distribution of temperature is uniform or not -- the initial conditions will of course affect the later values of T. Is the air around the glass likely to be an infinite heat sink? Unlikely. You can adapt your model to consider also the temperature of points in space outside the glass, too. This may seem like something beyond the requirements, but if you think for a moment you'll see you do need it to study temperatures in the glass -- heat lost to the air starts to slow down after the air warms up. Now, typical treatments of heat flow as I just recommended you consult will assume the media are solids. This is clearly not the case, and this difference will affect temperature distributions greatly. Even absent any deliberate mixing you know that warmer fluids, being less dense, will rise in their mixture (nearly-freezing water being the most common exception) Thus you need to consider the effects of convective heat loss as well as conduction. Ordinarily a difficult topic, this becomes terribly complex in your case since you are considering boiling water, which is highly turbulent (among other things, the solubility of gasses in water decreases as the water heats, so that dissolved air in the water collects and bubbles up to the top -- hmm, now is that air carrying out any heat?). Oh, while we're at it, let me point out the third main method of heat loss: radiation. You know hot metals glow; that glow you see represents the loss of energetic photons of certain wavelengths. The collective enrgy of those photons, across all wavelengths, is proportional to the fourth power of the (Kelvin) temperature. OK, so hot water doesn't glow very much in the visible spectrum, but you ought to borrow an infra-red camera and monitor the hot glass; you'll see quit a bit of energy being lost at those wavelengths. (The water tends to absorb very little energy from radiation since it is clear -- I'm not sure myself whether that's true at all wavelengths or only the visible ones). Look up 'blackbody radiation' in a quantum mechanics book. Lest you think this too much effort to expend on a cup of hot water, allow me to remind you of a study conducted some 15 years ago by the Royal Society attempting to determine the best way to brew tea. Good luck on your project. Let me know what you discover. dave ============================================================================== From: c2a192@ugrad.cs.ubc.ca (Kazimir Kylheku) Newsgroups: sci.math Subject: Re: COOLING CURVES Date: 26 Nov 1994 03:08:18 -0800 In article <3atdqe$ghv@ns.cityscape.co.uk>, Williams wrote: >I need some help. I'm doing an A level applied maths project the aim >of which is to *Model the cooling of a glass of boiling water*. I really >need to know what the actual equation for this is, and was hoping you >folks could help me. > >From my experiments it does not just seem to be a basic exponential >relationship between time and the difference between the water temp. and >room temp. When I plotted the log graph, a straight line wasn't produced >and I wouldn't expect this to be much due to experimental error. > >I would much appreciate any helpful comments > >TIA > >John Did you consider the possible effects of convection of both the water inside the glass, and the air around the glass? Did you allow vapours to escape? Did you assume that the heat cpacity of H2O is a constant over the temperature range you are looking at? How did you measure the temperature of the water? Did you sample it at the bottom of the glass, near the surface? If at only one place, did you stir the cooling mixture? Hope you arrive at a more realistic model!