Thermal conductivity and resistance

In summary, my teacher wants to know how much heat can be blocked from flowing through a wall when the temperatures are not allowed to change. If the two rooms are insulated from everything else except between each other, then eventually the temperatures would equalize and the flow of heat from one room to the other would stop.
  • #1
justinlj
13
0

Homework Statement


i need to find the thickness of the insulating material that is needed to block out the heat that is coming into the room.

the area of the affected wall is 3.3m2 the temp in first room is 18 degree celsius while the 2nd room is 45 degree celsius.

the thermal conductivity of the material is 0.0698 W/mK

Homework Equations



thermal conductivity, lambda= W/mK

thermal transmittance, U= W/m2K

thermal resistance, R=1/U=thickness/lambda

The Attempt at a Solution


1.find K, the temp difference between the 2 rooms
2.square both sides of equation 1, since I am only given area. Find the value of W.
3. subsitute the value of W into equation 2 to find U.
4. substitute the value of U into the third equation to obtain the thickness.

my teacher said the way i dealt with the area was incorrect. Is there any other way to obtain the amount of heat transfer W when I am only given the temp difference and the area of the wall?
 
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  • #2
I am an electrical kind of guy and typically solve thermal problems like this as an analogy to electrical network...I use Ohm's Law V=IR and turn it into T=QR

I typically refer to conductivity with the greek letter kappa, I will use ' k ', here.

I typically refer to resistivity with the greek letter rho, I will use ' r ' here. And so r = 1/k

I refer to a total resistance value of a given part with dimensions with the letter R, and so

R = r L/ A = L / kA

L is the length (thickness of the wall)
A is the cross-sectional area (of the wall between rooms)

In your case, these condcutivities and resistivities and resistance are thermal ones.

If Q is the amount of watts flowing through the wall...we can simply apply ' Ohm's Law ' (remember, I am electrical)

T2 - T1 = Q R
T2 - T1 = Q ( L / kA )

so

Q = (T2 - T1) k A / L

And so, the larger the L the less the heat...but I doubt very much that you will ever block all the heat...

so, when
L = (T2 - T1) k A
you have Q=1 watts going through

when L is 10 times larger, you will have Q = 0.1 watts

how small a value for Q does your teacher desire to consider " blocking " the heat?
 
  • #3
gsal said:
I am an electrical kind of guy and typically solve thermal problems like this as an analogy to electrical network...I use Ohm's Law V=IR and turn it into T=QR

I typically refer to conductivity with the greek letter kappa, I will use ' k ', here.

I typically refer to resistivity with the greek letter rho, I will use ' r ' here. And so r = 1/k

I refer to a total resistance value of a given part with dimensions with the letter R, and so

R = r L/ A = L / kA

L is the length (thickness of the wall)
A is the cross-sectional area (of the wall between rooms)

In your case, these condcutivities and resistivities and resistance are thermal ones.

If Q is the amount of watts flowing through the wall...we can simply apply ' Ohm's Law ' (remember, I am electrical)

T2 - T1 = Q R
T2 - T1 = Q ( L / kA )

so

Q = (T2 - T1) k A / L

And so, the larger the L the less the heat...but I doubt very much that you will ever block all the heat...

so, when
L = (T2 - T1) k A
you have Q=1 watts going through

when L is 10 times larger, you will have Q = 0.1 watts

how small a value for Q does your teacher desire to consider " blocking " the heat?

hmm.. so i would actually need to know the maximum amount of heat allowed to flow through? i don't really have that data though. without that i can't solve the question rite?

what if i assume an ideal case where all the heat is blocked and Q=0? can i still solve this question?
 
  • #4
You mean, like L=infinity?

I think there is something missing in the statement of the problem.

You see, if the temperatures of 45C and 18C are not allowed to change for the purposes of this problem, but heat is allowed to flow from one room to the other one, then, clearly, the 45C room is being continuously heated somehow for it to remain at 45C...and the one at 18C is being cooled, somehow, for it to no heat up when the heat comes through the wall


On the other hand, if both rooms where totally insulated from everything else except between each other...then, eventually, the temperatures would equalized and only then would the flow of heat from one room to the other would stop.
 
  • #5
gsal said:
You mean, like L=infinity?

I think there is something missing in the statement of the problem.

You see, if the temperatures of 45C and 18C are not allowed to change for the purposes of this problem, but heat is allowed to flow from one room to the other one, then, clearly, the 45C room is being continuously heated somehow for it to remain at 45C...and the one at 18C is being cooled, somehow, for it to no heat up when the heat comes through the wall


On the other hand, if both rooms where totally insulated from everything else except between each other...then, eventually, the temperatures would equalized and only then would the flow of heat from one room to the other would stop.

in that case i would be juggling with too many unknown variables right?

how do u suggest i make the question solvable? will it be ok if i set these conditions?
1. the max Q allowed is 20W
2. there is a constant heat source in the room at 45C
3. the max allowed temp rise in 18C is around 2-5C
 
  • #6
I say Option 1

Meaning, suppose the 45C second " room " is actually the outdoors being heated by the sun and the 18C room is an actual room with an A/C in it that keeps cooling it down...in this case, the maximum heat (Q) allowed would be whatever the A/C can handle...anymore and the room would start warming up

For example, let's supposed that the 18C room is a room that is being cooled down by a small window A/C equipment that consumes 500 watts...assuming that the A/C has 100% efficiency, it means that all the energy the A/C uses is used to suck 500 watts of heat out of the air in the room...and, hence, the room can keep taking in 500 watts through the wall ...

so
L = (T2-T1) k A / Q
L = (45 - 18) x 0.0698 x 3.3 / 500
L = 0.012
and so, a 1.2 centimeter insulation in the wall would do...
 
Last edited:

1. What is thermal conductivity and how is it measured?

Thermal conductivity is a measure of a material's ability to conduct heat. It is typically measured in units of watts per meter-kelvin (W/mK). It can be measured experimentally by applying a temperature difference across a material and measuring the rate of heat flow.

2. What factors affect thermal conductivity?

Thermal conductivity is affected by factors such as the material's composition, density, and temperature. Materials with higher thermal conductivities typically have higher densities and are made of materials that are good conductors of heat, such as metals.

3. What is thermal resistance and how is it related to thermal conductivity?

Thermal resistance is the measure of how much a material impedes the flow of heat. It is the inverse of thermal conductivity and is measured in units of meters-kelvin per watt (mK/W). Materials with higher thermal resistances are poor conductors of heat.

4. How does thermal conductivity and resistance impact heat transfer in buildings?

Thermal conductivity and resistance play a crucial role in determining the efficiency of heat transfer in buildings. Materials with high thermal conductivity, such as insulation, can help to keep buildings warm in cold weather by preventing heat from escaping. On the other hand, materials with low thermal resistance, such as windows, allow heat to transfer more easily, making them less efficient for insulation.

5. How do engineers and scientists use thermal conductivity and resistance in their work?

Engineers and scientists use thermal conductivity and resistance in various fields, including building and construction, materials science, and energy systems. They use this knowledge to design efficient insulation, heating and cooling systems, and to understand how heat flows through different materials. These concepts are also important in fields such as thermodynamics and heat transfer engineering.

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