Open window in hot room, how fast will temperature equalize? I am currently sitting in a warm room. I just opened a window to the outdoors, where it is cooler.
Middle school science says the warm air should move toward the cool air, which should (in theory) result in equal temperatures inside and outside. How long should I expect that process to take? 
For the sake of getting a formula that will help a lot of people figure out how cool a room will be, assume the room is square, has 8 foot ceilings, there's no fan inside and no wind outside, and the window does not have a pane or a casement - it's simply a hole in the wall. If humidity is a relevant factor let's assume 50%.
How much will the temperature drop inside in one hour? How long will it take to get within a degree Fahrenheit of the outside temperature? (Also, what is the name for this type of problem?)
This probably seems like a simple question to a physicist but it has been surprisingly difficult for me to find even an approximation for, on this site or via searching Google. For the sake of demonstrating completeness and avoiding "this is a homework question" moderation, I have been searching for over an hour now, and the answers I have found assume one of the following: 


*

*the asker is concerned about heat loss through a closed window 

*is willing to (pay for and) read through PDF's of papers laying out fairly complex formulas

*assume you know some heat transfer coefficient, or just assume one (0.6, for example) without showing any work

*want to know how an open window will affect the performance of an A/C unit.


I am not asking about those things (or I don't know how to calculate e.g a heat transfer coefficient) which is why I'm looking for help here.
The best result I have been able to find so far is this paper which seems to be more complex than strictly necessary, is difficult for a layperson to follow, has a key formula with at least one parameter (g) that does not appear to be defined anywhere in the paper, and another formula that relies on solving the integral of the heat flux density as a function of time. I'm trying my best to generalize those results but not having much luck so far.
The second best result I have been able to find is this paper which defines a heat loss coefficient and describes the results in general terms but does not explain how to translate between a heat loss coefficient or ACH (air changes per hour) and the temperature of the room.
 A: Jeffrey's answer has already explained how heat flows via conduction, convection, or radiation. I will explain how one can analyze just the conductive heat flow in the room.
The temperature in the room is a function $T(x,y,z,t)$ that satisfies the heat equation
$$\frac{\partial T}{\partial t}=\alpha\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right).$$
This is a 3D diffusion equation, and the coefficient $\alpha$ is the thermal diffusivity of the medium in which the heat is flowing. According to Wikipedia, the value for air is $\alpha=19\text{ mm}^2/\text{s}$, which in more convenient units is $18\text{ feet}^2/\text{days}$.
There are ways to solve it analytically which are too involved to get into here. Instead, one can simply solve it numerically on a computer. I used Mathematica, in which one call to NDSolve[] suffices.
To solve this equation within a room, one must specify the dimensions of the room, the dimensions and placement of the window, the initial temperature distribution in the room, and the boundary conditions at the walls and the window.
I chose a 10'$\times$10'$\times$10' room with a 5'$\times$5' window centered on one wall. I assumed that the initial temperature was uniform throughout the room. For boundary conditions, I assumed that the walls were perfectly insulating (no heat flow through them) and that at the open window the temperature was a constant, cooler, exterior temperature.
Each point in the room cools differently. The point at the center of the room cools according to the following curve:

The horizontal axis is time in days. The vertical axis is a temperature percentage scale where 100 means the initial room temperature and 0 means the outside temperature.
It takes 3.8 days for the center point to cool 50%, 10.3 days to cool 90%, and 19.6 days to cool 99%.
From this you can see that heat conduction through air is very slow. In fact, we use air all the time as a heat insulator, such as when we layer clothing to stay warm. The long time scale was basically clear early on, from the value of $\alpha$ expressed in feet and days; for a geometry measured in feet, the cooling time scale is going to be in days. Convective effects, and probably also radiative effects, are going to be much more relevant. A nice breeze or a good fan is extremely important!
A: General Analysis
First, let's correct one general misconception.
Warm air does not move toward the cooler air. In a gravitational field, a fluid that is less dense moves upward while a fluid that is more dense moves downward. Warm air is less dense than cold air. So, warm air moves upward while colder air moves downward. Better said, when warm air is placed laterally next to colder air, the warm air will not move sideways into colder air due solely to a temperature difference.
Now, appreciate the types of heat transfer that can occur.
Heat flows in one of three ways: conduction, convection, or radiation. Conduction is energy transport through a system due to a temperature difference across the system. It is transfer by (molecular) collisions, lattice vibrations, or free electron collisions. Convection is energy transport by a moving fluid. It is transfer by the fact that anything with mass also has enthalpy. Radiation is energy transfer directly due to temperature.
Finally, consider your problem in general.
The system is a room. The surroundings are the outside. The boundary is the walls to the room with one closed window. Assume that the air in the room is perfectly mixed (using whatever Maxwell's demon that you happen to have handy to do this particular job).
The transfer between the inside of the room and the outside of the room includes these steps: convection from the inside of the room to the inside surfaces of the walls and window, conduction through the walls and window, and convection from the outside surfaces of the walls and window to the outside air. To a first approximation, the combination of conduction through the walls and window function as a set of parallel thermal resistors.
The above system + surroundings can be solved by first principles or by numerical analysis (which of course requires a proper first principles set of governing equations). The above case is cumbersome but not generally so unwieldy, and basic answers are likely found by reference. So, let's assume that we have a cooling time for this case as $\Delta t^\star$.
Now, open the window.
Assume for a perfect case that the pressures inside and outside the room are equal. In this perfect case, the air on either side experiences no net force to cause it to flow into or out of the room. So, in the perfect case, when you open the window, all you do is take away the thermal resistance of the window in the parallel resistor equation. By intuition, the room should cool faster. For this case, $\Delta t < \Delta t^\star$. It stands to reason that the degree to which the cooling time is smaller will depend on the size (area) of the window.
Finally, let's relax one of the "perfect system" approximations. If we allow the air to flow through the open window by any type of convection, we increase the heat transfer from the inside to the outside and correspondingly decrease the time needed to cool the room. We can put fans in the window. This is called forced convection. Alternatively, to emphasize the opening statement about fluid density, we can tilt the wall at the window so that the warmer air inside the room sees some component of an "up" direction to leave the room and the colder air sees some component of a "down" direction to enter the room. This is equivalent to having the open window allow natural convection between the room and the outside. Again, to emphasize, in the case with a perfectly vertical window with stagnant air on both sides at the same pressure on both sides, no air will flow of its own accord from the room out or from the outside air in.
Specific Questions
The problem addressed for the perfect system arises in unsteady state conduction + convection. It can be solved using calculations of factors such as Biot, Nusselt, and Fourier numbers. It might eventually end up using Heisler charts. A entire chapter or two in this textbook is devoted to such analysis.
How much will the temperature drop $\Delta T$ in a time $\Delta t$ (or how long will it take to drop by a given temperature change)? The most direct answer to these questions in the perfect system starts by calculating the Biot and Fourier numbers. For lumped systems, you would end up using the equation given at this link. For systems where the temperature profiles inside the room vary as a function of both time and distance, the analysis shown by G. Smith gives an example of what will be required to start, and the Heisler charts give the foundations of where the work can lead.
