Need some basic mathematics help, yo

[Zylka]

So I'm sure the equation we've been given to work with in this assignment is wrong. Basically we're to divide a cross section of a wall at home into its' basic elements (ie gypsum, fibreglass, cedar panelling, etc.)

Each element has its own R value, and per each elements' thickness it arrives at a level of insulation that's proportional to the whole.

Element 1 of my wall is about 4.7% of the entire insulation value

Element 2 - 3.9%

Element 3 - 45.5%

Element 4 - 36.4%

Element 5 - 5.3%

Element 6 - 4.3%

Which, (obviously enough) adds up to ~100%

From there, we're to find out what amount of temperature is lost in each element from inside temperature to outside temp. Inside temp is 22c, outside -30c. Simple enough, right?

Though the formula we're being given to do just that is as follows:

ChangeTemp = [To + Ti] x [element R value divided by total R value, ie.) 5% would be equal to 5/100]

Plugging in the numbers that way doesn't work. This hand out is likely a bad photocopy, or somethingI know there's some sort of square or inverse action I have to be applying here, but am not mathematically inclined enough to know! Anyone?

[Zylka]

Therefore for the temp decrease in the %4.7 value element:

ChangeTemp = [52c] x [.047] = 2.44degree loss

while temp decrease in the (better) %5.3 value element

=[52c] x [.053] = 2.756degree loss

Doesn't make sense. As the proportional insulative value goes up so does the heat loss? There's some little square or something missing in that formula and it seems like it should be so easily noted. AHHHHHHH!

[KrazyHorse]

Should be Ti - To

[KrazyHorse]

Other than that, the formula is correct. Your second post applies the formula properly.

[Zylka]

Well in the initial formula it's To + Ti with a little arrow over the plus sign pointing towards Ti. I assume that just means to find the real difference in temperature between the two.

Now changing it to Ti - To does nothing, doesn't it?

[KrazyHorse]

Reasoning is as follows: say you have a pane of glass 10 cm thick. Inside T = 30C outside T = 0C

T(x) = (30 - 3x)C where x is the distance from the inner wall measured in cm. IOW the temperature goes linearly from Ti to To

Now imagine that instead of 10 cm glass you have 9cm glass then 1 cm glass. T at interface is 3C, so delta T through thick piece is 27C and delta T through thin piece is 3C. If total R value is Rt then R of 9 cm piece is 0.9Rt and R of 1 cm piece is 0.1Rt. Formula thus works.

[Urban Ranger]

Originally posted by Zylka
Now changing it to Ti - To does nothing, doesn't it?

When you subtract a negative number it becomes adding a positive number.

[Zylka]

Originally posted by KrazyHorse
Other than that, the formula is correct. Your second post applies the formula properly.

but it isn't correct. By my second post, one layer with a 4.7% proportion of insulative value loses only 2.9 degrees in temperature - when the layer with 45.5% proportion of insulative value loses 23.7 degrees.

You understand how that can't make sense, right?

There are all sorts of little indistinct scribbles in/around the formula.. is there an inverse or something like that that I'm missing?

[Zylka]

Originally posted by Urban Ranger

When you subtract a negative number it becomes adding a positive number.

Uh, yeah. Which comes up with the exact same value as if I had added the two temps by real scale. 52 degrees, either way you cut it

[KrazyHorse]

Basically, delta T does not represent a heat loss. In actuality the heat flow through each of the segments is exactly equal. This is a conservation principle. If more heat flowed through layer one than through layer two then there would be a buildup of heat somewhere. Now, since heat flow through each segment is equal:

H1 = deltaT1/R1
H2 = deltaT2/R2
...
Hn = deltaTn/Rn
Htotal = deltaTtotal/Rtotal

H1 = H2 =...= Hn = Htotal = deltaTtotal/Rtotal
deltaTi/Ri = deltaTtotal/Rtotal
deltaTi = deltaTtotal * (Ri/Rtotal)

which is your formula

[KrazyHorse]

Originally posted by Zylka


but it isn't correct. By my second post, one layer with a 4.7% proportion of insulative value loses only 2.9 degrees in temperature - when the layer with 45.5% proportion of insulative value loses 23.7 degrees.

You understand how that can't make sense, right?

There are all sorts of little indistinct scribbles in/around the formula.. is there an inverse or something like that that I'm missing?

NO. THE FORMULA IS CORRECT. THICKER LAYERS OF INSULATION HAVE A GREATER CHANGE IN TEMPERATURE ACROSS THEM.

[Asher]

LOUD NOISES

[Zylka]

Originally posted by KrazyHorse
Reasoning is as follows: say you have a pane of glass 10 cm thick. Inside T = 30C outside T = 0C

T(x) = (30 - 3x)C where x is the distance from the inner wall measured in cm. IOW the temperature goes linearly from Ti to To

Now imagine that instead of 10 cm glass you have 9cm glass then 1 cm glass. T at interface is 3C, so delta T through thick piece is 27C and delta T through thin piece is 3C. If total R value is Rt then R of 9 cm piece is 0.9Rt and R of 1 cm piece is 0.1Rt. Formula thus works.

I've already accounted for thicknesses in arriving at each elements' R.

As they have been arrived - element 1 has a 4.7% worth of the entire walls' insulative valie. Element 2 has a 3.9% worth of the entire walls' insulative value.

Using the forumla you find correct - the first element (thickness and Btu already accounted for), with a HIGHER proportion of the walls' overall insulation - LOSES more temperature than the lesser element of 3.9%. That does not make sense.

[Koyaanisqatsi]

Originally posted by Zylka
but it isn't correct. By my second post, one layer with a 4.7% proportion of insulative value loses only 2.9 degrees in temperature - when the layer with 45.5% proportion of insulative value loses 23.7 degrees.

You understand how that can't make sense, right?

You mean because 52*.047 != 2.9?

Other than that...

[KrazyHorse]

An analogy:

I have a garden hose which is comprised of two segments: segment one has radius 2 cm and segment two has radius 1 cm. I turn on the hose and measure the velocity of the water flowing through each segment. To my great surprise, the water flows faster through segment 2 than through segment 1.

The total volume flow through each segment is the same (since otherwise the water would build up somewhere), but to maintain that flow through a tight section of hose (i.e. to maintain equivalent heat flow across a good insulator) the water needs to flow faster, velocitywise (i.e. the temperature difference needs to be greater).