Thermal Conduction
Introduction
The purpose of this lab is to test the formula for thermal conduction and the accepted values of thermal conductivity. The rate of heat conduction from a higher temperature system is given by the following formula:
[1] Q/t = kA(Tc – Th)/L
Q = heat
t = time
k = thermal conductivity of barrier
A = cross-sectional area of barrier
Th = temperature of the higher temperature system
Tc = temperature of the lower temperature system
L = length of barrier
Since conduction of heat can change the temperature of an object, the more precise formula is
[2] ∂Q/∂t = kA(Tc – Th)/L.
The formula relating heat to temperature change for the higher temperature system is the following:
[3] Q = mhcΔT
Q = heat
mh = mass of the higher temperature system
c = specific heat capacity of higher temperature system
ΔT = temperature change for the higher temperature system
Equation [3] can be written in differential form as follows:
[4] ∂Q = mhc∂Th
Equation [4] can be substituted into equation [2]
[5] mhc(∂Th/∂t) = kA(Tc – Th)/L.
Solving for ∂T/∂t in equation [5] gives a differential equation.
[6] ∂Th/∂t = -[kA/(mhcL)](Th – Tc)
If the lower temperature system does not change its temperature because it undergoes a phase change, then equation [6] can be solved for the temperature difference as a function of time.
[7] Th – Tc = Toe-t/τ
[8] τ = mhcL/(kA)
τ = time constant
To = initial temperature difference
Experimental Procedures
Thermal Conduction
1)Measure the width and thickness of the conducting metal. Calculate the cross-sectional area, A.
2)Fill one insulated cup with ice and add cold water so that the water reaches the top of the ice. Cover the cup (skip if you only have one lid).
3)Measure the mass of the second cup with a lid.
4)Fill with hot tap water and cover the second cup.
5)Measure the mass of the cup and lid containing hot tap water.
6)Connect the systems with the metal rod and measure the relevant length of the rod, L, which is the distance from water surface to water surface along the rod, not necessarily the entire length of the rod.
7)Gently shake both cups a few times a minute for a few minutes.
8)Measure the elapsed time in seconds and the temperature of the hot water, Th, every minute or so for half an hour. Gently shake both cups several times per minute. Do not open the cups. You can measure the temperature and shake the cups without opening them.
9)For each measured temperature of the hot water, calculate the temperature difference between the hot and cold water. You may assume that Tc = 2°C ± 2°C throughout the experiment or you may choose to measure the temperature of the ice water.
10)Graph the temperature difference on the vertical axis and the time in seconds on the x axis. Use the approved format for graphs.
11)Create a trendline of type “exponential”. You absolutely must use an exponential trendline. Do not use a trendline of a different type. Include the equation and R2. Does your best fit line go through most of the error bars? If so, then the theory of thermal conductivity made a useful prediction regarding the temperature as a function of time.
Thermal Conductivity
12)You should have a formula from the graph of the form y = Ae-Bx where A and B are constants. Calculate the time constant, τ, by inverting the magnitude of the constant in the exponent (τ = 1/B). You will not be able to estimate the error in this quantity. You may need to increase the number of digits that appear in the exponent to obtain more than one significant digit. Right-click on the equation and select “format trendline label” to do so.
13)Calculate the mass of hot water by taking the difference between full and empty measurements.
14)Solve equation [8] for the experimental thermal conductivity, k, in terms of known quantities.
15)Calculate the experimental thermal conductivity and compare to the accepted value for the substance (k = 217 W/(m*K) for aluminum). You may use the accepted value of c = 4190 ± 15 J/(kg*K) for water. Be sure to use SI units for all your data.
Repeat the entire experiment with a different initial hot temperature, a different mass of hot water, or a different value of L.
Here are some important things to double-check so that you can earn full credit on this report:
1)Use lots of ice. If you don’t have enough ice, the temperature of the cold side will fluctuate and the formulas used will be incorrect.
2)Use SI units at all times. You will have area in m2, length in meters, and time in seconds. You may use K or °C.
3)Gently shake the cups several times per minute.
4)Use temperature difference between hot and cold in the graph, not temperature.
5)Use time in seconds in the graph.
6)Use an exponential trendline.
7)Obtain more than one significant digit in the exponent of your graph.
8)Use the magnitude of the constant in the exponent to calculate τ. See Wikipedia for the definition.
9)Use only equation [8] to calculate conductivity. The rest of the equations are there to show you how equation [8] is derived.