How much heat from a fire actually warms your home? A fire in a hearth disperses heat to, I guess, three places:


*

*the bricks of the chimney

*out the hearth (where the person tending the fire is standing)

*out the chimney, above the house


How would you go about figuring out how much of a fire's heat goes to each of these three places? I assume the specific heat(†) of the bricks plays a role in both capturing and retaining heat from the flames.  But what do you need to measure, and what calculations would you do with those measurements, to predict how much heat would go to each of those places?
Just a pointer to the right approach would be helpful.
† (BTW, this seems like a good example of temperature vs heat: if the volume of air in the house is much larger than the volume of flames, then even a hot fire cannot warm a cold house very much.)

Bonus question: I want to contrast colonial homes with one fireplace in the centre, versus homes with several small fireplaces around the edges. I'm thinking of modelling this as a plate with one large central heat source, versus a plate with several small heat sources around the edges. Any reasons why this is wrong would be appreciated.
 A: This is such a complicated question! The worst part is that as heat leaves the chimney, it draws air from the room with it - air that needs to be replaced from outside. This actually makes fires quite good as ventilation systems.
Whether a fire heats a building depends in very large part on the degree to which cool air can flow past parts heated by the fire. This makes a wood stove (metal pot "in the room" with a small pipe leading out) much more efficient than a hearth. But in many cases a fire place might have other mechanisms to improve heat transfer - for example with ducts that run alongside the chimney and that draw in and heat cool air from the room.
Measuring this is quite tricky. I suppose you start with measuring the temperature of the effluent at the point in the chimney still just inside the house (air might cool more when you get further out but that only heats the external chimney and boer the house). You also need the flow rate of the air - this will allow you to estimate "heat going out the chimney" (assume outside temperature for the initial temperature because you are drawing outside air in). Next, for a fireplace that is on the outside wall, measure the temperature of the outside wall (IR non contact thermometer) and make reasonable assumptions about the h factor of air flowing past the wall and the heat transport. I would not worry about bricks of the chimney and their heat capacity - in the steady state that neither gains not loses energy.
Finally measure how much wood you are burning per hour and use this to estimate the net heat going in- this allows you to estimate how much of the heat is making it into the house. You could also measure how warm the house gets with the fire (again use steady state) and see what kind of (electrical) heater you would need for the same effect.
Note that a fireplace is most effective as a radiant heater - you sit/ stand nearby and feel warm, even though the room might be quite chilly.
As I said - this is a really hard problem.
A: This is an excellent question, but not readily answered, as discussed in Floris's Answer. 
Here is a stab at how to get an estimate of efficiency. It is the method which I believe is reasonably accurate: the actual values will need to be refined by experimental measurement: I am not too confident of the actual numbers that fall out owing to the calculation's high sensitivity to the variables used (in particular the fourth power influence of the flame temperature).
For an open hearth, it would seem reasonable to assume that most of the heat transfer to the room is through radiation; this will make you feel warm and will also warm things in the room up which then raise the temperature through conduction and convection. This is because, as discussed in Floris's Answer, the fire draws in a great deal of air. The chimney and hearth shape are chosen so that the fire's heat sets up a considerable draught through the fire and up the chimney to keep the fire burning lustily and this would limit convection / conduction severely.
From here, I get a flame temperature for burning wood of about $1300{\rm K}$ (i.e. $1027{\rm ^o C}$). From here I get a $\Delta H$ for burning wood of about $15{\rm MJ \,kg^{-1}}$. Assume that we can burn $50{\rm kg}$ of wood for one hour to give a flame area of $\frac{1}{2}{\rm m^2}$.
Then, the total heat output ($\Delta H$) per second is $50\times 15\times 10^6/3600\approx 208{\rm kW}$.
The total radiant power, from the Stefan Boltzmann law is of the order of $\frac{1}{2}\times\sigma\,T^4$ (with a $\frac{1}{2}{\rm m^2}$ flame area as seen from the room), i.e. $\frac{1}{2}\times 5.7\times 10^{-8}\times 1300^4\approx 81{\rm kW}$.
So our efficiency here is estimated to be in the tens of percent range. I don't believe it is quite as high as the $30\%$ implied here. I expected the answer to be minuscule - of the order of $1\%$. I am skeptical of this exact answer, but I believe you could get a good estimate from its method using experiments to measure wood burning rate (you'd measure this over many hours to get a good estimate of the rate), a pyrometer to probe the flame's temperature and temperature uniformity and perhaps some ad hoc image processing of a video of the fire to estimate the radiant area. You could even do some crude pyrometry with a colour video, using a laboratory pyrometer to help you calibrate your measurement. Grounded on this calculation, I would estimate between $10\%$ and $30\%$ efficiency. As I said, this is a great deal better than I thought. However, you would need to experiment to refine this method: note the very high sensitivity of the calculation to the variables used - particularly the flame temperature (which has a fourth power influence).
For a closed wood burning stove, you can see the situation would be much more complicated. The stove's housing is raised to some temperature by the reverberating furnace within: several hundreds of degrees celsius, which then heats the room by convection, conduction as well as radiation. I daresay there are heating furnace efficiency specifications and standards which will let you estimate the temperature of the stove and its power output.
A: The answer is, it depends on the type of fireplace you have.  An ordinary brick chimney consisting of a fireplace directly venting up and to the outside has a very low efficiency.  The draft generated by the fire pulls warm air from the house, and most of the heat travels directly up.
There are various types of fireplaces which specifically attempt to address this problem.  These are really less of fireplaces and more of wood stoves.  However, they sometimes retain some of the features of a regular fireplace such as a brick chimney, bricks to retain heat, and a hearth.  There are typically several features to increase efficiency, such as baffles to redirect the air flow in the unit to allow for more heat transfer, and various material such as soapstone to absorb and retain heat.
I have seen a figure of 5-10% quoted as the efficiency of a classical fireplace, with a maximum of about 20% if glass fireplace doors are used.  I don't know if these figures are calculated or measured.  This excludes heat lost to the need to pull warm air out the room to vent exhaust gases.  The EPA provides figures which suggest that wood stoves are generally in the 60-70% range.
Given the above, there are a lot of things which affect efficiency.  For instance, incomplete combustion reduces efficiency - the presence of visible smoke indicates incomplete combustion, and hence, some of the fuel is going up the chimney.  The type of wood burned affects efficiency, because different woods have different flame temperatures and this has non-linear effects.  What is needed is to maximize the temperature differential between the exhaust gases above the flame and the exhaust temperature when the insulated portion of the chimney is reached.  However, since fireplaces rely on a natural draft induced by temperature differential to pull air into the flu, this temperature difference cannot be too low, or the smoke won't go up the chimney.
