OBJECTIVES
- To investigate Fourier’s Law of linear conduction.
RELEVANT THEORY AND WITH RELEVANT EQUATIONS
Heat transfer has direction as well as magnitude. The rate of heat conduction in a specified direction is proportional to the temperature gradient, which is the change in temperature per unit length in that direction. Heat conduction in a medium, in general, is three-dimensional and time dependent. That is, T _ T(x, y, z, t) and the temperature in a medium varies with position as well as time. Heat conduction in a medium is said to be steady when the temperature does not vary with time, and unsteady or transient when it does. Heat conduction in a medium is said to be one-dimensional when conduction is significant in one dimension only and negligible in the other two dimensions, two-dimensional when conduction in the third dimension is negligible, and three-dimensional when conduction in all dimensions is significant.
Conduction is a mode of heat transfer in which energy transfer takes place from high temperature region to low temperature region when a temperature gradient exists in a body. The basic law of conduction was established by Fourier. According to Fourier’s law, heat flow by conduction in a certain direction is proportional to the area normal to that direction and to the temperature gradient in that direction.
Q=-kA
Where Q=transferred heat
k = thermal conductivity
A = area
= temperature gradient
The minus sign in the equation above shows that heat flows in the direction of decreasing temperature.
Thermal conductivity is the property of materials which shows heat conduction per unit length of material per degree of temperature difference.
Heat is conducted in solids in two ways: transport of energy by free electrons and lattice vibration. In good conductors a large number of free electrons move about in lattice structure of the material which transports heat from high temperature region to the low temperature region. The portion of the energy transported by the free electrons is larger than that by the lattice vibrations. An increase in temperature causes increase in both the lattice vibration and the speed of free electrons, but increased vibration of lattice disturbs the movement of free electrons causing reduction in the transport of energy by free electrons which means the overall conduction is reduced. In insulators and alloys, the transport of energy is mainly due to the lattice vibration and an increase in temperature increases conduction.
Conduction of heat along a simple bar
Let us consider Fourier’s law of conduction for the case of a simple bar with lateral surface insulated as shown in Fig. 1.1.
This is approximation of one dimensional conduction for a plane wall as shown in Fig 1.2. For steady state condition, it is assumed that the power generated by an electrical heater enters at one end and leaves from the other end uniformly. Then the thermal conductivity of the specimen can be determined as:
k(T) = = w/m.K
where, Q = heat power
k (T) = mean value of thermal conductivity between and .
T=mean value of and .
APPARATUS USED
Arm field Thermal Conduction apparatus.
- ON switch
- Wattmeter reading
- Wattmeter reading
- Cooler
- Temperature measurement points at 10mm each
- Heater
- Temperature selector switch
- Wattmeter selector switch
- Temperature reading
OBSERVATIONS
Specimen material: Brass
Thermal conductivity of the specimen from tables:
Diameter of the specimen: 25 mm
Length of specimen: 30 mm
Distance between temperature probes: 10 mm
Observation table:
Test no. | Wattmeter Watts, Q | T_{1} ^{o}C | T_{2} ^{o}C | T_{3} ^{o}C | T_{4} ^{o}C | T_{5} ^{o}C | T_{6} ^{o}C | T_{7} ^{o}C | T_{8} ^{o}C | T_{9} ^{o}C |
1 | 10 | 40.7 | 40.1 | 36.7 | 29.4 | 26.6 | 25.8 | 19.5 | 18.7 | 18 |
2 | 15 | 45.4 | 44.4 | 40.3 | 31.6 | 27.5 | 20.7 | 19.5 | 18.5 | 17.6 |
3 | 20 | 52.3 | 51.2 | 46.3 | 34.5 | 29.5 | 28.5 | 20.7 | 18.7 | 18 |
CALCULATIONS
Area of cross section (A) =4.91×10^{-4} m^2
For Q= 10
SN. | ∆x(m) | ∆T(K) | k=Q*∆x/(∆T*A) |
1 | 0.1 | 1.2 | 107.06 |
2 | 0.1 | 4.7 | 43.42 |
3 | 0.1 | 11.8 | 17.14 |
4 | 0.1 | 5.0 | 40.8 |
5 | 0.1 | 1.1 | 185.52 |
6 | 0.1 | 7.8 | 26.16 |
7 | 0.1 | 1.9 | 107.41 |
8 | 0.1 | 0.8 | 255.1 |
k(avg)= 105.70
For Q= 15
SN. | ∆x(m) | ∆T(K) | k= Q*∆x/(∆T*A) |
1 | 0.1 | 1.0 | 306.12 |
2 | 0.1 | 3.9 | 78.5 |
3 | 0.1 | 9 | 34.01 |
4 | 0.1 | 4 | 76.53 |
5 | 0.1 | 0.8 | 382.65 |
6 | 0.1 | 7.3 | 41.5 |
7 | 0.1 | 0.8 | 382.65 |
8 | 0.1 | 0.9 | 340.1 |
k(avg)= 205.3
For Q=20
SN. | ∆x(m) | ∆T(K) | k= Q*∆x/(∆T*A) |
1 | 0.1 | 0.9 | 453.5 |
2 | 0.1 | 3.0 | 136.0 |
3 | 0.1 | 7.3 | 55.9 |
4 | 0.1 | 2.6 | 156.98 |
5 | 0.1 | 1.3 | 313.98 |
6 | 0.1 | 5.6 | 72.88 |
7 | 0.1 | 1.1 | 317.15 |
8 | 0.1 | 0.7 | 583.09 |
k(avg)= 267.93
therefore,
Thermal conductivity of material (k)= (105.7+205.3+267.92)/3 =192.97 W/mK
RESULTS AND ANALYSIS^{}
Arm field Thermal Conduction Apparatus was used during this experiment. In the apparatus the specimen material used was brass with 9 temperature probes having each temperature probes at a distance of 10mm. On different power input, we found out the temperature of the brass rod on different point and calculate the temperature gradient of the rod in order to find the thermal conductivity of the rod. From the calculation we found two different value of thermal conductivity at different power input as per 192.97 Wm^{-1}K^{-1 .}
But we know that the thermal conductivity of the substance depends only on its material and for Brass it should be constant for a given power input. The standard value of thermal conductivity as published is 109 Wm^{-1}K^{-1.} The fluctuation in the conductivity might be because of the insufficient cooling system. It might be because of the malfunctioning of the temperature probes or due to errors in the procedure. However, the mean thermal conductivity from the three calculated value was found to be 192.97 Wm^{-1}K^{-1}. We also found that heat that flows in certain direction by conduction is directly proportional to the cross sectional area normal to that direction and the temperature gradient in that direction. The Temperature vs Length graph is shown below.^{}
CONCLUSION
From the experiment, it was found that the heat that flow through the conducting rod is directly proportional to the cross sectional area normal to the direction of flow of heat and temperature gradient. Hence, Fourier’s law of linear conduction was verified. And the value of thermal conductivity calculated is 192.97Wm^{-1}K^{-1}.
SAFETY CONSIDERATIONS
- Temperature should not exceed 100^{o}C in linear conduction apparatus.
- After finishing the experiment, the specimen should be removed from the linear conduction apparatus, which may be hot, so should be careful to handle.
- A continuous flow of cooling water is necessary for the experiment, otherwise the apparatus may get damaged.