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Control Engineering is the main course of mechanical engineering. This is a major subject in this field of Engineering.
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Experiment 1Measurement of Thermal Conductivity of a Metal (Brass) Bar
Introduction:
Thermal conductivity is a measure of the ability of a substance to conduct heat, determined by
the rate of heat flow normally through an area in the substance divided by the area and by minus
the component of the temperature gradient in the direction of flow: measured in watts per meter
per Kelvin
Symbol K is used for denoting the thermal conductivity
According to the Fourier Law of thermal conductivity of place wall
𝑑𝑇
𝑄∞𝐴
𝑑𝑥
𝑑𝑇
Or 𝑄 = −𝐾𝐴 𝑑𝑥
Where
Q = heat flow (by conduction rate) through the material
A = The section through which heat flows by conduction
𝑑𝑇
= the temperature gradient at the section
𝑑𝑥
The proportionality constant K is a transport property known as thermal conductivity (W/mk)
and is a characteristics of the wall material. It provided an indication of the rate at which energy
is transferred by diffusion process. It depends on the physical structure of matter, atomic and
molecular , which is related to the state of matter. The minus sign is consequence of the fact that
heat is transferred in the direction of decreasing temperature.
The generalized heat conduction equation for constant thermal conductivity in Cartesian co-
ordinate is:
𝑑2 𝑇 𝑑 2 𝑇 𝑑 2 𝑇 𝑞 1 𝜕𝑇
+ + + =
𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2 𝑘 𝛼 𝜕𝑡
,T = temperature distribution at the location x,y,z (ºC)
x,y,z = co-ordinates
q = internal heat generation rate per unit volume (W/m^3)
k = thermal conductivity of the material (W/mK)
𝛼 = Thermal diffusivity (=k/ρc) of the material (m^2/s)
t = time , s
Some assumptions that are given can be followed to simplify the generalized equation:
1. Heat flow is one-dimensional i.e. temperature, varies along x-direction only. This is
achieved by putting insulation on the circumferential surface of the specimen.
2. End effect is negligible
3. The specimen material is isotropic
4. There is no internal heat generation in specimen
5. Steady state is achieved before final data recorded
So, the simplified form of the generalized equation is,
𝑑2 𝑇
= 0
𝑑𝑥 2
When the steady state is attained the following boundary conditions are considered:
(i) At x = 0; T = T0
(ii) At x = L; T = TL
Using these boundary conditions we get the solution of the differential equation as :
𝑇 − 𝑇0 𝑥
=
𝑇𝐿 − 𝑇0 𝐿
Where,
T = temperature of the section at distance x (ºC)
T0 = temperature at section where x = 0 (ºC)
TL = temperature at section where x = L (ºC)
X = Distance of the section of measurement from the section at x = 0, (m)
L = Distance between sections at x = 0 and x = L , (m)
,In this experiment a Brass rod is heated by nicrome wire surrounding the brass bar at one side.
The brass bar was properly insulated is such a way that heat flow remain one dimensional to the
other end of the rod for heat conduction study with a view to fulfilling the following objectives:
(i) To plot temperature vs. distance curve from experimental measurements.
(ii) To plot temperature vs. distance curve from theoretical analysis.
(iii) To determine thermal conductivity of the metal specimen.
Experimental Set up:
Operation procedure
1. Check the room temperature by an analog thermometer and then calibrate the digital
thermocouples.
2. Start the experiment by switching on the Veriac and make suitable heating at the end of
the brass bar by nicrome wire.
3. Carefully measure the distance from one thermocouple to another thermocouple or the
positions of the thermocouples.
4. After every 10 minutes take the reading of every thermocouples along with the reading of
water inlet and outlet.
5. Continue this until the steady state has come.
6. It will take too long time to come steady state. So, take the reading of every thermocouple
after ten minutes and draw the curves.
, 7. If two or more than two consecutive curves show that slopes are similar or equal
(carefully follow the shape of the curve); then we can consider the heat flux through the
brass bar is constant at that time.
8. Take the reading of the water inlet and outlet.
9. Draw the curves of Temperature vs. distance for both experimental case and theoretical
case
10. Find the thermal conductivity of the metal Bar.
11. Find the mass flow rate of the water.
Data Table:
Time
(minutes)
10 20 30 40 50 60 70 80 90 100 110 120
Positions
0 inch
1.8 inch
3.6 inch
5.4 inch
7.2 inch
Time
(minutes)
130 140 150 160 170 180 190 200 210 220 230 240
Positions
0 inch
1.8 inch
3.6 inch
5.4 inch
7.2 inch
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