Assume we know the mass, surface area, starting temperatures etc of each substance, is their a simple way to know the time it will take for them to reach the same temperature.
for example say we put a hot metal ball in a glass of water, could you calculate the time it would take for the ball to cool down and the water to heat up.
Posted Nov 18, 2014, 10:45 PM UTC
Yes, it's commonly done using finite element solvers. But your example problem is simple enough to be solved analytically, using some basic assumptions.
Set the center of your metal sphere to be at the origin. Assume the glass of water goes to infinity (i.e. the ball is surrounded by enough water that we can neglect the container). You'll want to use spherical coordinates to fully exploit the symmetry of this problem.
For a treatment of this problem, see this LINK section 4.4, The Quenching Problem for a Sphere with Convective Heat Losses at its Boundary.
That equation will give you the temperature as a function of position within the sphere. (It's an infinite sum, but you can truncate it once you have the accuracy you need.)
You could say that the two objects are within thermal equilibrium when the temperature difference between the water and the center of the sphere are within 1%. Use that equation and back-solve for the time.
For more complex problems you'd typically just simulate the system with a computer.
If the ball was surrounded by a finite amount of water of known mass, and very well insulated (assume infinite), is there a way to calculated the final temperature and the time it takes to get there?
Yes. The final temperature in that case is easy to calculate. Find the total thermal energy in the system. (Heat capacity of water)(temp of water) + (heat capacity of ball)(temp of ball).
You know that, in equilibrium, the total amount of thermal energy will be the same (insulated system), but everything will have to be the same temperature.
So find the temperature T such that, if both the water and ball have temperature T, the total thermal energy is what you calculated before.
Approximating Cooling Time
Assuming the ball conducts heat very quickly, you could assume it has a uniform temperature at all times. Then, it transfers heat to the surrounding water at a rate proportional to (Twater - Tball). The constant of proportionality is the thermal conductivity between the ball and the water.
You can write what I said above as a 1st order differential equation. It's called Newton's law of cooling, and the solution to this equation will have the form of an exponential (e^x ).
Use that equation to estimate when the system will reach equilibrium (or 1% difference in temperature, remember, in the ideal case the temperatures will never quite be equal.)
The equations for heat transfer are very simple. Like all differential equations it's the boundary conditions that are a pain. Spherical coordinates usually end up using Legendre Polynomials. So there typically isn't a neat analytical solution.
Yes. If it's an extremely simple case you should be able to solve the heat equation analytically. If it's more complex, it's best to solve it either with a computer simulation or by applying experimentally derived non-dimensional equations (if they exist for your problem).
That said, "thermal equilibrium" will not exactly happen. The hot ball with get colder over time and the water will get warmer, but the average temperatures will never be equal (at least theoretically) as the behaviour is asymptotic. You have to specify some tolerance to equilibrium which is close enough (eg: the temperature difference is 1% of the original difference).
How simple are we talking in order to approximate it, within the right order of magnitude?
The heat equation is three dimensional (+time). It would be a bit easier if your problem is one dimensional, classically an insulated bar with a boundary conditions at either end and an initial temperature distribution. The 1-D assumption actually works for 3-D objects as well, as long as they only do something interesting in 1 dimension (eg: large flat plate with heat flowing in the normal direction).
Solving it analytically requires integration of the heat equation, and application of the relevant boundary and initial conditions. It's easier if the geometry is simple (1-D). Solving something like a metal ball in a glass of water is a bit tricky because there are fluid mechanics equations to solve for the water. If you had two solids, it would be a bit easier.
It's probably best to find an experimental description of the behavior. If I had to solve such a problem, that's how I would do it. The method is described in the linked book, below, which is what I learned it from. It involves finding non-dimensional numbers for your problem (similar to Reynold's number for fluid problems), and then basically looking up your problem type and plugging in the right numbers into equations (or reading them off charts). Various assumptions are made based on the relative material properties and geometries. (eg: a very thin object with high conduction can be assumed to be constant temperature throughout).
Relevant book: http://books.google.com.au/books?id=vvyIoXEywMoC&printsec=frontcover&dq=fundamentals+of+heat+and+mass+transfer&hl=en&sa=X&ei=ReRrVNvZGIe1mwWMt4LQAg&redir_esc=y#v=onepage&q=fundamentals%20of%20heat%20and%20mass%20transfer&f=false
http://en.wikipedia.org/wiki/Heat_equation
Uhhhhhh.. It's a pain in the butt.
It's not one of these back of the envelope kind of things. You have to dig up a bunch of thermo equations that are rarely used, create an energy balance, know the heat capacities of the materials, the details of the thermal interface, surface roughness, environmental temperature (unless we're assuming adiabatic)....
It's not simple.
EDIT: Just saw your description. For a metal ball in a glass of water it's a little easier as you don't have to deal with contact resistance, but there's still convection, conduction, plus the leidenfrost effect (which is also a pain in the butt to model), boiling, etc. If you just wanted an equlibrium temperature (that will be asymptotically approached) it's easier, because then it's just heat capacities, masses of the water and ball and initial temperatures. But since you're dealing with time, it's going to involve calculus and a bunch of complicated math.
I am starting to get the idea that this might not be nearly as simple as I predicted.