Heat Transfer Examples

## Question:

Hello:
I need your help once again. This time I need to know how to do heat transfer. My question is:

1) A wall is made up of equal areas of wood and glass. The glass is 0.50 cm thick and the wood is 4.0 cm thick. For the same temperature difference, which conducts more heat, and how many times more?

2) A 4.0 ft x 2.5 ft window in a building has a single pane of glass 1/8 in. thick If it is 70 degrees Fahrenheit on the inside and 20 degree Fahrenheit on the outside, how much heat is conducted through the window in 24 hours?

3) The bottom of a round copper teakettle has a radius of 10 cm and a thickness of 3.0 mm. The kettle sits on a stove burner at at temperature of 150 degree Celsius and is full of boiling water.
a) What is the rate of heat conduction?

b) Ideally, how much water is "boiled away" in 5.0 min? Is the answer reasonable on a practical basis? Explain.

Thermal Expansion of Materials:
1) What temperature increase in required to increase the length of a 1.0 ft iron bar by 0.021 in?

2) A copper plate at 10 degrees Celsius is heated to 70 degrees Celsius. What is the percent increase in the area of the plate?

Hi Gloria,

The heat transfer rate through any material is H=-k*A*(dT/dx) where H is the heat transfer rate or heat flux, k is the thermal conductivity of the material, A is the area through which the heat is transferred and dT/dx is the temperature gradient. In steady state heat transfer problems where every part of the system is at a steady temperature, the thermal gradient is just the difference in temperature between the hot and cold side divided by the thickness of the material. The minus sign simply means that the heat flows the high temperature side to the low temperature side.

1) Let's work your first problem using some reasonable values for thermal conductivity.
For glass k=0.8 watts/meter degree centigrade.
For wood k= 0.08 watts per meter degree centigrade
For glass then Hg=0.8*A*(T hot - T cold)/.005
For wood Hw=0.08*A(T hot -T cold)/.04.
The A's and T's are the same so Hg/Hw is (0.8*.04)/(0.08*.005) = 80. Clearly the glass conducts 80 times more heat than the wood.

2) This problem is stated in terms of English units so you will need to convert to metric or get k in terms of English units from a book somewhere. You have the basic ingredients to solve the problem though.
A = 4.0 ft * 2.5 ft
T hot = 70 degrees F
T cold = 20 degrees F
x = 1/8 inch
Just plug the appropriate numbers into the K=-k*A*dT/dx formula.
Remember that the total heat transferred is the heat flux H times the time period.

For the third problem
3a) You have A, x and T hot.
T cold will be the boiling temperature of water of 100 degrees centigrade (assuming standard atmospheric pressure) Just plug into the formula to calculate H.

3b) Once you get H you can calculate the total energy transferred to the water in 300 seconds (5 minutes). Assuming that all the water is at 100 degrees centigrade before any of it starts to boil and that all the water vapor produced by the boiling is removed from the kettle then the amount of water boiled away will just be the amount of energy transferred into the water divided by the energy required to convert a certain mass of water to vapor. This is called the latent heat of vaporization. If I remember right it is 931 British thermal units per pound mass of water. You need to look it up in units consistent with this problem. The thing that might make this ideal amount unrealistic is that some of the water vapor will condense when it hits the cooler air above the water surface or the walls of the kettle and fall back into the kettle.

The linear thermal expansion of a solid is given by DL=a*L*DT where DL is the change in length, L is the original length and DT is the change in temperature and a is the average coefficient of linear expansion. For iron and steel a=1.1e-5 approximately.
1) DL=0.021 inches. L=12 inches. a=1.1e-5. So DT=0.021/(1.1e-5*12).
2) Assuming well behaved material which iron is, the average area coefficient of expansion is just 2*a so DA=2*a*A*DT or DA/A=2*a*DT.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know.

JDJ