Computing R Value

[thechimneysweep]

I'm looking at a new manufactured hearth. The company claims their product is equivalent to 3/8" of millboard with k value of 0.84. So what's the R value of this hearth?

 

[pumiglk]

R=1/K x Thickness

If my math is right its a shade under 3.2

 

[fossil]

Interesting question. As an ME with lots of heat transfer way back in my academic career, I can't answer it. And I can't access whatever old textboooks I still have because they're all in storage while our house is remodeled. Gotta be careful about which system of units is being used...SI or English, and keep in mind that "k" and "r" values are not simple numbers...they have interesting units, like "R=xx.xx square feet degrees F hours per BTU. R is thermal resistivity, while K is thermal conductivity. These properties of a material are all about conduction of heat energy through the material. The higher the R-value (or lower the K-value), the better the material is at acting as what we call a thermal insulator. If the material is simply described as having a k value of 0.84, that begs the question ,"0.84 what?"...is it SI or English? Are we talking square meters, or square feet? Degrees Celsius or Farenheit? I don't think deriving one from the other would be difficult, but more information is needed. Rick

 

[pumiglk]

As a cell biologist who tried most of his academic life to avoid math I knew I shoudn't try to answer the question %-P . My answer is based on the Hearthstone manual I have spent a lot of time looking at recently, where they provide that formula as a way to calculate the hearth requirements.

 

[webbie]

in mathematics, the number 1/x, which multiplied by x gives the product 1, also known as a reciprocal

.84 (K or U) x 1.2 = 1 (appox)

So R=1.2

(I think)

 

[jtp10181]

Web is on the right track.... I love the info in the Quad Manuals, makes sense to me.
You have to factor in the thickness because R is always per inch.

----------------------------------------------


B. Calculating Alternate Floor Protection

Material

Thermal Conductivity: k value
The k value indicates the amount of heat (in BTU’s) that will flow in 1 hour through 1 square foot of a uniform material 1 inch thick for each degree (F) of temperature difference from one side of the material to the other. The LOWER the k factor means less heat is being conducted through the non-combustible material to the combustible material beneath it. The k value of a material must be equal or smaller then the required k value to be acceptable.

(BTU) (inch)
(foot2 (hour) (oF)

Thermal Resistance: R value

The R value is a measure of a material’s resistance to heat transfer. R value is convenient when more than one material is used since you can add the R values together, whereas you can not do this for k value. The HIGHER the R factor means less heat is being conducted through the noncombustible material to the combustible material beneath it.
The R value of a material must be equal or larger then the required R value to be acceptable.

Converting k to R:

Divide 1 by k and multiply the results times the thickness in inches of the material.

R = 1/k x inches of thickness

Converting R to k:

Divide the inches of thickness by R.

k = inches of thickness/R

Calculations:
Example: Floor protection requires k value of 0.84 and 3/4 inch thick. Alternative material has a k value of 0.6 and is 3/4 inch thick. Divide 0.6 by .75 = k value of 0.80. This k value is smaller than 0.84 and therefore is acceptable.

 

[fossil]

K and U are not the same thing. K is thermal conductivity, and is equal to BTU/hr/sq. ft./unit temperature gradient, where the temperature gradient is the temperature change per foot through the material. U is the overall coefficient of heat transfer, and is equal to BTU/hr/sq. ft./temperature differential, where the temperature differential is the difference across the outside surfaces of the material. It may seem subtle, but it's not the same thing.

R is thermal resistance, and can be expressed as R=1/UA, where U is the overall coefficient of heat transfer as defined above, and A is the heat transfer surface area.

The industry has made various efforts to simplify these terms...as in "R-11", or "R-19" labelling of wall insulation. Bigger is better, no doubt, when it comes to "R", if you're looking to discourage the flow of heat. But where are the units of measurement? They drop them because to most people they're meaningless.

I have to assume that the insulation values advertised here in America are English units, but I've no idea how insulation is rated in the rest of the world which long ago shifted to the use of the much more sensible SI system.

I'll keep digging. I'm getting a headache and having flashbacks of my professors. %-P Rick

 

[thechimneysweep]

Just to complicate matters, I understand there's a difference between lower-case k and Upper Case K. This rating was stated in lower-case k.

 

[fossil]

In engineering terminology, typically a lower case indicates a "specific" measurement, meaning it's "per" something...per cubic foot, per inch, per pound-mass, or whatever, whereas an uppercase indicates the parameter figured with all variables included (unless otherwise defined). I'm sure that clears things up a lot. Rick

 

[pumiglk]

I read the equation such that the whole term K X T was the reciprocal which jives somewhat with the R=1/UA equation Fossil explained.

At least if you make the assumption that K is being used as a surrogate for U then: 1/UA=

1/KA which is equal to 1/(K x Thickness ) = 1/(0.84)(0.375)= 1/0.315 = 3.17

But the quad manual clearly states to do it the way webmaster does it -which I think my Algebra teacher would have written:

R=(1/K) x T to separate those terms and the Value is 1.2

seems like a a pretty big difference to make up to not be absolutely clear on.

 

[pumiglk]

Well pretty much ignore that last post - I reread fossils explanation and the A of UA is an area or two-dimensional term - thickness probably isn't a good surrogate. Still think it should be written more clearly (but that is mine and any other hearthstone customer who tries to do this conversions tough luck).

However I came across another equation that is clearer R=1/C. is C = UA? Do they provide you with a C value Cimmneysweep?

 

[webbie]

I'll bet ya that the manufacturer of the board doesn't even know!

R value is USUALLY per inch and may be that way scientifically, but when we discuss R it is for an assembly, not per inch. That confuses the situation.

I still think it is the simply inverse without thickness, being as we are not specifying per inch:
https://www.hearth.com/econtent/index.php/articles/k_values_what_does_it_all_mean

 

[fossil]

I've always found that particular article a bit confusing, as the author mixes his systems of units. In the table, he says these values are "per inch", whereas in his conversion equations, he refers to the thickness in meters, and he never defines "U". I'll try to find one of my old books or snother reference and see if I can come up with some semi-coherent discussion. Part of the confusion arises from the fact that we're not careful with our use of notation, and we don't specify the units. If you're looking for an overall rate of heat transfer through a structure, then the surface area is definitely a factor. If you're looking simply to define the specific resistance to heat flow through a structure, then only the thermal resistances and thicknesses of the material making up the structure are important. Gotta be careful about the terms...are they "per inch", or "per square inch" (thickness, or surface area)? Rick

 

[thechimneysweep]

Thanks all, I was little muddy on this issue, and now I'm deeply mired. In the product literature, the manufacturer goes on to state as an example that this hearth would not provide sufficient floor protection if the appliance requires a "greater amount of protection than 3/8" of k = .84"

Since the lower-case k seems to imply that the .84 rating of the millboard in question would be per inch, it seems that 3/8" of that millboard would have a greater k rating by a bit more than double. Else, why mention the 3/8" thickness at all?

In the first statement, the mfr says his hearth is, "equivalent to 3/8” of millboard with k value of 0.84."
Then, he says his hearth would not work if the stove requires a "greater amount of protection than 3/8" of k = .84"

If the 3/8" thickness wasn't a factor, wouldn't he just say his hearth has a k value =0.84?

Unless I'm way off in the wilderness on this one, I think the question I'm looking to have answered is,

If 1" of a given material has a k value of .84, what is the R value of the same material at 3/8" thickness?

 

[savageactor7]

.38 or around there. Divide .84 by 8 x 3 = R value

 

[fossil]

For a thermal insulating material, the r-value has to be greater than the k-value for the same thickness. The r-value is a measure of the resistance to heat flow, the k-value is a measure of thermal conductivity. Until I can find and dust off a couple of my books (not to mention my brain), I don't think that for our purposes we're doing anything unsafe by simply taking the reciprocal. Thus, if a 3/8" thickness of some material has a k-value of 0.84, then assume the r-value to be around 1/0.84 = 1.19.

Until the folks start putting the units of measure consistently with their numbers, it's going to be tough to make a lot of sense of them (I'm not holding my breathe). But one thing we do know without a doubt...Low k = good; high r = good. I'll try to put together a more definitive and scholarly article when I get a chance. Rick