A rather wonderful piece of software called Therm can answer the question of how much heat is going to be lost from an actual wall structure, so we can see how close the actual U factor is of a wall with a wooden pillar running down it, compared to the prediction from the U values of 10% wood and 90% insulation.
You start by drawing the structure and setting each polygon to the appropriate material from the library of data the system has. In this picture, you can see the three layers of glass wool insulation in blue, a wooden pillar in the middle in orange, and a couple of layers of structural board in the other colour. Is that puce?
Next you set the boundary conditions. You can tell the software whether each surface is inside or outside, or whether to ignore it. You can consider a surface it adiabatic, in other words that heat is not going to flow through it at all. It would be very time consuming, and not particularly helpful, to model the whole house, and you usually want to find out about a particular bit of wall, or a boundary between roof and wall, or some kind of junction.
To model a wall, you can slice it in two places and put in an adiabatic surface in each, so you can get some meaningful estimation of what's going on. The main concern is heat flowing from inside the house out, so once you get far enough away from the part you're interested in, you can ignore any heat flowing along the walls.
In this case, the left side is outside, the right side is inside, and the top and bottom are adiabatic, so we're just looking at heat flowing from inside (where the temperature is assumed to be 20 degrees C) to outside (where it's assumed to be very cold - 18 degrees below zero). Of course the temperature will be changing all the time, as will the humiditiy, but this is just looking at a steady state in the worst case. Another piece of software called Wufi http://www.wufi.de/index_e.html will simulate the humidity conditions over a year or two, and show where moisture could build up in a wall or roof structure. That's not avaiable as a free download though!
To find out the thermal bridge effect of the wooden pillar running through an insulated wall, I compared three different structures. First, I made an ideal wall with a 50mm insulation on the inside, 120 mm in the middle, 12 mm of structural board, then 100 mm of insulation on the outside (1). Ideal, but of course it would not hold up very well! This has a U value of about 0.131 W/m2K.
Next I made a wall with the 120 mm middle layer completely made of wood (2). This has a U value of 0.187 W/m2K. In both cases 1 and 2, the U factor can be calculated directly from the U values of each component part. To do this you have to add up the R values (the reciprocals of the U values) which measure thermal resistance. There are also surface effect factors to account for convection, inside and outside, and factors to account for radiation. When you get to the surface, convection is the biggest cause of heat loss, but across the wall the heat is conducting.
Next, I made a wall with a 120x240 wooden beam in the middle. This is close to the real situation. As there is 240 mm of wood and 760mm of insulation, we would assume that the U factor of this bit of wall is 0.24 x U1 + 0.76 x U2, or 0.145 W/m2K. The Passive house spreadsheet also assumes this. In fact, the wall is conducting 0.147 W/m2K. This represents a difference of 0.002W/m2K. This corresponds to 0.002 W/mK along the length of the beam, and is the thermal bridge effect. This is small enough that we need not worry about it. Larger thermal bridge effects need to be added to the passive house spreadsheet. You can see the isotherms on this picture, showing how the temperature is distributed.
This picture, much more pretty, shows the temperature by colour, as you'd see from an infrared camera.
The next picture, perhaps even prettier still, shows the heat flux, with white representing the highest flux. So we can see which parts of the wall the heat is rushing through.
This software uses what's called a finite element grid, which I can remember hearing about in my lectures at university. I think I nodded off shortly after them, only to wake up just before my finals, but along with the thermodynamics, I realise now that at least something stuck from those days, and at least in some tiny way I can call myself an engineer. I'm not sure whether it was the result of my university study, or whether it was instilled in me from earlier by my father. Perhaps the essence of engineer goes further back, and courses through my veins from generations living in the harsh and unyielding environment of the North of England, with nothing but their ingenuity, which the word engineer probably came from before they ever got around to making engines. I digress.
Calculating heat flow over complex shapes with different materials is tricky, but if we imagine a small rectangle or triangle with a constant temperature along each side, we can easily work out how temperature is going to flow through it. Therm breaks any structure up into such polygons, then goes from one end to the other working out how much heat is going through each part until it reaches some kind of equilibrium.
Therm can be downloaded for free from here. http://windows.lbl.gov/software/therm/6/index.html
