I could use a hand understanding heat loss.  I took this formula on heat loss from a Shelter Institue article that I found on the web.

It appears that the more R's there are the less btu loss/hr and the relationship looks linear to me.....more R's, less btu loss proportionately.  I think I have my flat head around that part.  But as was previously pointed out (by Ed with graphics) (a page that I can no longer access due to a parsing error...what ever that is) the early R's do most of the stopping power (I once read the first 5 R's do 50%).  Seems like we have the formula and the real world here.  What gives?

you are dividing by R-value. that is not a linear progression.

let's say you have a value of 500 for A x dT.

Divide by 1, it's 500. Divide by 2, and you have 250. a linear progression would then drop 250 every time you increment your divisor by 1.

But, divide by 3 and you get about 166. much less loss was avoided with the addition of another R, eh?

The reason that is not linear is that division is multiplication by fractions, not whole numbers. divide by 2 is multiply by one half. divide by three is multiply by 1/3rd... your increment is not linear like it is when you are multiplying by whole numbers.

hope that helps!

Also consider air infiltration loses - double the insulation doesn't 1/2 the heat loss - because infiltration losses (and other losses) can stay the same.

Rob, Thanx for that but could still use some help here for the less cranial.  Let's try this example:

Let's say R = 10,   surface area =1 sq.ft.  and   dT= 100

So formula looks like  Hmax =  1/10 x 1 x 100

Answer: 10 btu's per sq.ft. per hr.


2nd case:

double the R-value so R = 20  same 1 sq.ft. and same dT

Formula-            Hmax = 1/20 x 1 x 100

Answer:  5 btu's per sq.ft. per hr.

So I don't get it!  Double the R-value cuts the btu heat loss in half.  Looks proportional to me.  Granted, inversely proportional.....but proportional.  I know it doesn't work that way but I'm not getting that from the formula. 

Thanx Jon,  You can correct me here but this calc would be against a basement wall (concrete) so I assume there would be no infiltration.

To double is violates the definition of linear.

"Linear" means the change in the result is the same every time you change the variable.

2x=y is linear. Every time you increase X by 1, Y increases by 2.. those are fixed amounts. doesn't matter if x goes from 0 to 1 or 100 to 101, it will still increase Y by 2.

what you just described is not linear. It's a simple and easy to understand relationship, but it is not linear. "Double the R valve" depends on the R value you started with, and "halving the heat loss" depends on the heat loss you started with.

For instance, going from R1 to R2 cuts a massive amount of heat loss in half with only an R1. That is cutting an R1 heat load (huge) in Half (big savings) for only R1 (easy).

Going from R100 to R200 also cuts the heat load in half. But that's cutting an R100 heat load (tiny) in half (tiny savings) for adding R100 (very hard).

A Linear relationship would cut the heat load by a fixed amount every time you added R1. This is not a proportional relationship either, I don't think, though I understand why you called it that. This is an exponential relationship, I think, as all "curved line" relationships are.

O.K. Got this thing hooked up to a battery and inverter so I'm good for a few minutes!

Rob,  Thanx for that explanation.  I have a better handle on that now.

Next question: 

If you bury a cube of some dimension that is filled with sand or earth or gravel and you install a heat source in the center of it will the heat loss be the same on all six sides or will heat loss be greater on the top than the bottom due to heat rising over time.  Question is posed wondering if much heat is lost to a downward direction as compared to an upward direction.

Heat doesn't rise. On the other hand, things that convect heat (like air or water) do rise as they heat up.

Jon, For all intents and purposes would compacted sand, which has some air in it, be considered equal heat diffusion in all directions?

Flathead

My guess would be that very little air would rise through it.

if the air is trapped, it can't convect except within each bubble. I don't know what degree of resolution you are looking for, but technically that would make the upward flow greater. However I think the effect would be so miniscule as to be unnoticeable or at least not very significant.

this also depends on how deeply you are buried. normal residential depths will lose more heat upward because the heat can conduct out faster as it's colder up there. but if you were at some theoretical depth which was the same temp on all sides, heat loss would be more or less equal on all sides.

The formula is a geometric progression
An exponential progression increases with the square, cube,... of the variable
Sorry could not resist

thanks. Math class was a long time ago

Interesting.....thanx for your insight guys!

If X = 1, Y = 2
If X = 2, Y =4
Increase X to 3 and Y = 6, not 8.

The definition of Linear is based on the order of magnitude and direct correlation, not doubling the end result every time. In your linear and non linear examples, the order of magnitude of both elements is 1. No X squared or Y to the third. X to the first power and Y to the first power meant a linear relationship.

So, your idea of non-linear equations isn't quite right.

I also think that we are forgetting that the valuation of R factor is, itself, non linear....it is exponential. Kind of like decibel level. In the audio world, 20 decibels is not equal to 2 times 10 decibels. It is a logarithmic function where 20 decibels is actually a power of 10 greater than 10 decibels. The same is true with regards to seismic activity and the Richter Scale.

The R Value is defined as:

R Value = (Temperature Difference x Area x Time)/heat loss....or to write it in terms of the units (degrees F x sq ft X hrs)/BTU's

Note that the square footage is not a linear element, and therefore any usage of the R Value in an otherwise linear looking equation is not linear, instead, it is exponential.

Posted By Cgallaway on 19 Mar 2010 09:56 AM
I also think that we are forgetting that the valuation of R factor is, itself, non linear....it is exponential. Kind of like decibel level. In the audio world, 20 decibels is not equal to 2 times 10 decibels. It is a logarithmic function where 20 decibels is actually a power of 10 greater than 10 decibels. The same is true with regards to seismic activity and the Richter Scale.

The R Value is defined as:

R Value = (Temperature Difference x Area x Time)/heat loss....or to write it in terms of the units (degrees F x sq ft X hrs)/BTU's

Note that the square footage is not a linear element, and therefore any usage of the R Value in an otherwise linear looking equation is not linear, instead, it is exponential.

To understand  heat transfer you've got to get away from looking at R value.  It's a neat little term that was invented for people to wrap their arms around a nice whole number and makes it easier to envision a complex idea in a simple fashion.   "Bigger number, better insulative quality"  But to understand the mechanics, you look at it as 'U' value (the inverse of R) which is the way heat transfer is normally expressed  and then you can see what happens a little easier.  U value is the transmittance of heat. The higher the U, the higher the heat flow ( but the numbers are usually quite small when you are trying to restrict the flow of heat as in a building, and thus a little tougher to grasp). 

Take R values of 5, 10, and 20.  This converts to U values of   ---- 0.2; 0.1; 0.05.  

However, you'll notice you get a diminishing return  --  doubling the R value from 5 to 10 cuts the U value in half from 0.2 to 0.1 (Heat flow cuts in half)  Doubling the R value from 10 to 20 cuts the heat flow again by 1/2.  But it is 1/2 of a smaller number.  And it took an increase of 10 R points instead of 5.  Thus diminished returns. 

If graphed, it would be assymptotic ( I may know heat transfer, I just can't spell).  Meaning you'll never hit the other axis, kind of like radioactive decay and it's half-life.


Additionally, other structural elements with much higher U values (windows, studs) will eventually become a controlling force.

And no, the ft2 part of the equation doesn't enter into whether this is linear or geometric.  You'll notice I never had to reference it at all in the example.