"Archimedes" by Domenico Fetti. Public Domain from Wiki Commons

If you have done any construction or home improvement you have probably had to buy insulation. Here in Duluth, at Mission Creek, we’ve probably installed a boat load of the pink stuff, of the itchy and non-itchy variety.

When you went to buy insulation, you probably also noticed numbers like R-9. If you didn't know already, I'm sure you quickly discovered the higher the R-value, the cheaper it will be to heat your home in the winter and cool in the summer. But, you may have wondered "what exactly does that number mean?"

It turns out the number after the "R" isn't some number like in "describe how you are feeling on a scale of 1 to 10." The R-value is a precise number that can be put in equations and used to calculate stuff. Here is R-value in a nutshell:

**R-value is the amount of square feet of insulation needed to transfer 1 btu/hr, when the temperature difference is 1 ℉.** Since, "btu" is a unit of heat energy and each of these units costs a little to produce, it is obviously better to need a lot of square feet to transfer each btu, or a higher R-value.

That's the nutshell definition. But, if you'd like to go a little further down the rabbit hole, let’s look at what happens when we have a warm inside and a cold outside separated by a wall.

To help us picture heat flow, I'm bringing back the same heat tank analogy I used in a different blog. Here I am showing that heat flow from hot to cold across some insulation is kind of like a liquid flowing from a tank at a high level to one with a low level. Like before, the height represents temperature and the volume of the liquid is the heat energy. **A btu is like a drop of liquid. If a lot of those drops flow in a given time (high btu/hr) you are losing a lot of heat.**

The valve is like insulation. It does its best to slow down the flow of these drops. But the best valve (insulation) in the world can't stop it all.

What about in the summer?

Even here in Duluth, a lot of people have at least some air conditioning. When in use, you are trying to keep your side of the insulation (your tank) low instead of high. The summer animation is just a mirror image of the winter one.

One important thing to note, ** heat flows at the same rate both ways through the same insulation, if the temperature difference is the same.**

This means if it is 70 degrees inside and 50 degrees outside or 90 degrees outside and 70 degrees inside, heat will flow at the same rate, if the same amount of insulation is present, **just in opposite directions.**

Let's look at temperature differences.

Here we have kept the insulation the same, but increased the differences in temperatures. Just like larger drops in height will make liquid flow faster, larger temperature differences will cause heat to flow faster and the furnace to work harder.

This is where some extra insulation would help.

Now that's better. The increased insulation has the same effect as tightening the valve, it slows the flow.

The R-value is a measure of how good insulation is, at holding back this flow. **A higher R-value is like a tighter valve.**

Here is where the math comes in. We can figure out the heat flow through an insulated wall, with the following equation:

Equation 1: Heat_Flow=(Area / R_value)(Temp_{inside}-Temp_{outside}).

We want to note a couple things about this equation

**1) This equation can apply to a single layer of insulation or many layers.** If your whole wall was somehow nothing but a single layer of polystyrene, then your R-value is just the number printed on the sheets. For the total R-value of the wall, you would have to add up, all the individual R-values of other layers of insulation, wood, ect.

**2) This equation just gives the heat loss through a single wall.** You would have to use this equation on each of the walls to find the total heat loss, at a given time.

This formula says there will be more heat flow when the temperature difference is big and the R-value is low.

Enough for these fuzzy statements, lets talk some hard cold numbers.

You may remember, from even high school chemistry, that you have to plug the units into an equation, cancel out the appropriate units, and get some sort of units that describe what you are solving for.

R-values have the rather strange units of [(ft^{2})(hr)℉] / btu. So, the last time you saw a sheet of polystyrene insulation that said R-8 it really means 8[(ft^{2})(hr)℉] / btu.

Since that would scare the heck out of most people and make them think you have to be some kind of rocket surgeon to install the stuff, they just label it R-8. Maybe that's the same reason hardware stores call hydrochloric acid "muriatic acid." "Hydrochloric" just sounds like too much work.

Now let's plug some real life numbers into this equation, and see how it works.

Let's say we have a wall that is 1000 square feet, just to make the math easy. To make the math even easier let's say we have R-10 insulation and it is 70 degrees inside and 30 degrees outside. We also are going to assume all the heat-blocking ability comes from this R-10 insulation.

Since, I rigged the problem to have all these nice numbers that end with zero, I don't even need a calculator to find out, this wall's heat loss is:

[ 1000ft^{2} / { (10ft^{2}hr℉) / (btu) } ](70F-30F)=4000 btu/hr.

So, we are losing 4000 btus per hour, just through this wall. So if we really had a wall as contrived as this, we would be losing about 4 cents an hour worth of heat (if using natural gas), through this wall, at these temperatures.

But remember, this is just one wall. To get the total heat loss you just add up all the heat loss from all the walls in your house. Then our formula becomes something like:

Equation 2: Heat_Flow_{total}=(Area_{Wall1}/R_{wall1} +Area_{Wall2}/R_{wall2} +...)(T_{inside}-T_{outside})

Here we are also assuming T_{inside} is constant throughout the house and T_{outside} is the same all around the house.

It might be easier just to remember to add up all the individual heat losses through each wall.

Many times, to make things easier, scientist and engineers will just lump together everything in the first parenthesis of equation 2 (including "...") and just call it some constant "K". Then we get a much nicer equation like this:

Equation 3: Heat_Flow_{total}=K(T_{inside}-T_{outside})

Now, this is a really nice equation that can be applied to just about anything, and is mostly correct. Scientists will apply this equation to any object or system that has a temperature difference between it and its surroundings. This equation can apply to a house, camper, heated dog house, or a can of pop.

This equation can be manipulated to produce another equation (which I'm not going to do here) that can predict what the temperature of an object will be at a certain time, if we know the starting temperature and starting time.

A crime scene investigator might then use that equation and determine, based on the temperature of the water, approximately how long ago a person took a bath. The same investigator might take the temperature of a body, to find approximately how long ago the person die.

Sorry, the best examples I came up with are morbid.

The other thing about this equation is as long as you don't change anything else, "K" stays the same! This means if you figure out "K" once, you can use the results as many times as you want.

"K" can be determined experimentally, as well as by adding up all the areas divided by R-factors.

To illustrate the mighty power of "K," let's come up with another amazingly easy to calculate situation.

Let's say we have a camper and to figure out how many 700 watt space heaters you will need, when it is really cold. First, we can rearrange the "K" equation and get:

K=(Heat_Flow_{total})/(T_{inside}-T_{outside})

Figuring out Heat_Flow_{total} is easy, if you have a heat source with a constant output, like an electric space heater.

You just turn off all sources of heating and cooling and turn on your 700 watt space heater. If you wait long enough, you will reach a point where the temperature is not changing. Then, you take temperature readings inside and outside. You find that it is 70 degrees inside and 60 degrees outside. Plugging in the numbers we get:

K= (700 watts)/(70 ℉ - 60 ℉) or:

K=70 watts/℉

You may have noticed I am now using watts instead of BTUs/HRs. They are both units of power (energy per time). There are many different types of units to use for "K." We can convert between them if needed. Watts was just more convenient, in this case, using an electric heater.

Now we can use this equation to figure out many things. For example, armed with this magical "K" number we can ask: how much energy will it take to keep the camper at 70 degrees when it is -20 out.

For this we just insert numbers into the original equation 3, realizing that the heat flow is what our heater will need to put out to maintain the temperature difference.

Heat_Flow_{total}=K(T_{inside}-T_{outside})

Heat_Flow_{total}=70 watts/℉(70F-(-20F))T_{outside})

=6300 watts.
So, you would need 9 of those space heaters, to achieve 70 degrees, on a cold day in January. If these numbers were real, you would face a big challenge, if you wanted to go camping in Duluth, in the winter. You would need to seriously insulate the camper, or give up the idea. It's kind of cool that you could figure this out even in the summer.

I should let you know there is another way to find "K," that I'm not going to cover here. You would use the same equation, the crime scene investigator used, that shows how temperature drops, as a function of time. You would rearrange that equation, turn off all heating and cooling, and measure the temperature rise or drop over time, also plugging in the hopefully constant outside temperature. Actually, the first way sounds much easier.

By the way, this crime-scene equations and equation 3 are different forms of Newton's Law of Cooling. There are very similar equations that describe a number of natural processes from population growth to radioactive decay.

Our old friend R-value can also be really useful in solving the camper dilemma. It provides another way to find "K". You can calculate "K" both ways and see if the numbers match up well. Also, R-value can do what experimentation can't, it can give you an idea of what "K" will be, before you buy the material. Then you can figure out how long it might take to pay for your insulation investment, by spending less on heating and cooling.