ME 315 - Heat Transfer Laboratory
Experiment No. 2
TIME DEPENDENT HEAT CONDUCTION
Nomenclature
Cp
specific heat, J/(kg K)
Di
inner diameter of cylinder, m
Do
outer diameter of cylinder, m
Fo
Fourier number, g
gravitational constant, m/sec2
overall heat transfer coefficient, W/(m2 K)
h
k
thermal conductivity, W/(m K)
overall Nusselt number, Nu D

qrad
net radiative heat flux, W/m2
Pr
Prandtl number, RaD
Rayleigh number, t
time, sec
T
temperature, K
Th
temperature of heating band, K
Ti
initial temperature, K
T
ambient temperature, K
Greek Symbols
thermal diffusivity, m2/s


volumetric thermal expansion coefficient, 1/K
surface emissivitiy, 
Stefan-Boltzmann constant, W/(m2 K4)

kinematic viscosity, m2/s


density, kg/m3
Objectives
The purposes of this experiment are to measure the propagation of heat through a solid, to
deduce the thermal diffusivity, and to compare the results with data available in tables.
Concepts Emphasized
1.
One dimensional, unsteady heat conduction.
2.
Experimental techniques for measuring and analyzing transient temperatures.
3.
Energy balances for control volumes in an unsteady state.
4.
Modeling of boundary conditions.
2.1
Pre-Lab Section: Theoretical Analysis
The purpose of this exercise is to formulate a mathematical model which can be used to calculate
the temperatures in a thin walled metal cylinder, heated at one end, as functions of time and
position. The model calculations can then be used to compare with those actually measured.
1. To analyze the heat flow through the solid, consider the experimental apparatus shown in
Fig. 1. The heating unit, which is simply a resistor, is confined within a metal band for safety
purposes. As the electrical power is turned on, electrical energy is converted into heat which
is then conducted through the metal band into the hollow cylinder. Although the contact
between the metal band and the hollow cylinder is quite good, it is not perfect, i.e., the effect
of contact resistance could be significant. To reduce heat losses by convection to the
surrounding air, the cylinder is enclosed within a transparent enclosure.
T ransparent Enclosure
Heater
T hermocouples
Metal Tube
Signal
Conditioning
Card
70 V
1 A
PC
Power
Supply
Data Acquisition
System
Figure 1. Schematic diagram of overall experimental setup for measurements of timedependent heat conduction in a hollow metal cylinder.
2. Due to the effects of contact resistance, it is necessary to separately analyze the temperature
history of the heating unit and the temperature history of the hollow cylinder. To perform
such analysis, you should specify one control volume for the heating element and another for
the hollow cylinder utilizing the provided sketches in Figs. 2 and 3.
3.81cm
Heating Band
Specifications
mh = 0.077kg
Cp,h=402.6 J/(kg K)
r
Node 0
r=0
x
Dhi
x=0
Figure 2. Cross-section of the heating band.
2.2
Heating Band
Dimensions
Dho=3.14 cm
Dhi=2.54 cm
Thickness=0.3 cm
Lh=3.81 cm
3. Consider the sketch of the heating unit shown in Fig. 2. Specify the control volume,
including losses/gains of energy. Complete the list of assumptions that are required for
performing the energy balance:
(a) the heater temperature does not vary in either the circumferential or the axial direction;
(b)
(c)
(d)
(e)
4. Perform the energy balance for the control volume treating the heater as Node 0. Your
equation should be a simplification of Equation 5.15 in the text according to the above
assumptions.
5. The surface heat flux term qs must be considered in this case as conduction heat transfer
from the heater into the metal cylinder. To simplify calculating qs , we will make the
approximation that the radial heat conduction in the band can be calculated as if it were plane
conduction, as we will do in the x direction. This requires that the wall thickness of the band
be much smaller than its diameter. This approximation will suffice for our calculations.
Write qs in terms of the thermal conductivity and a radial temperature gradient and
substitute it into your equation:
6. Now approximate the terms in this equation using finite differences between the Nodes 0 and
1, thus giving a finite difference equation for Node 0:
7. Consider the sketch of the hollow cylinder shown in Fig. 3. In a similar manner as in item 3
above, proceed by specifying the control volume for each of the eight nodes. Prior to
performing the energy balance for each node, complete the following list of assumptions:
(a) temperature is dependent on time (t) and the axial (x) direction only,
(b)
(c)
(d)
(e)
2.3
x=0
x
Node 0
Metal Tube
Node 1
2
3
4
5
6
7
8
Di
Heater
Figure 3. Cross-section of Metal Cylinder
8. Perform an energy balance according to the explicit method for each of the eight nodes of the
hollow cylinder, starting with node 1. Follow the notation used in Sect. 5.9.1, of the text by
Incropera and DeWitt; it may be useful for you to review this section as well. Here, the
temperature is denoted Tmp , where the subscript m denotes the m-th node and the superscript p
the time-step (or level). Rearrange your equations so that the Fo number appears in them.
Note: you should be able to perform the energy balance without consulting the text.
Node 1:
Node 2:
Nodes 3-7:
Node 8:
2.4
Do
The above nine energy balances (including node 0) should yield nine coupled equations for
the nine unknown temperatures. Please have your instructor come by and check your work. If
have time and experience, then you may wish to program these equations on the PC using
Matlab. Also note that this finite difference problem can be solved using Excel by assigning
a column to each node and writing the proper equation for each cell in each column.
9. The above system of equations has been programmed on the PC using C. The program can
be opened by clicking on the desktop icon labeled Finite Diff Aluminum or Finite Diff
StSteel. Once the graphical interface opens, run the computational model by clicking on the
DO CALCULATION button. The program will plot the resulting temperature as a function
of time for node 1 and node 8 and will automatically store the results for all the nodes in a
file. Note approximately how long it takes for Node 1 to reach 70C. Exit the program by
clicking on the QUIT button. Knowing that the Fourier number must be equal to or less
than 0.5 for the computational model to remain stable, determine the time step (t) and
the number of time steps that are necessary to simulate ten minutes of real time. The
thermodynamic properties are specified in Table 1 below.
Table 1 Properties of the Metal Cylinders.
Size and Properties
T-6061 Aluminum
T-304 Stainless Steel
Inner Diameter
0.0210 m
0.0238 m
Outer Diameter
0.0254 m
0.0254 m
Thermal Conductivity
177 W/(m K)
14.9 W/(m K)
Thermal Diffusivity
73  10-6 m2/s
3.95  10-6 m2/s
Density
2770 kg/m3
7900 kg/m3
Specific Heat
875 J/(kg K)
477 J/(kg K)
10. The results from these finite difference calculations are stored in the files
C:\Temp\AluminumFD.txt and C:\Temp\StSteelFD.txt. Using Excel, import the data
from the calculations for the aluminum and stainless steel cylinders. Plot the temperature as
a function of time for nodes 1-8. Your graphs should resemble the one shown in Fig. 3
below.
Calculated Aluminum Tube Temperatures
350
Temperature (K)
Node 1
340
Node 2
330
320
310
300
290
0
20
40
60
80
100
Time (sec)
Figure 3. Typical Finite Difference Results for the Aluminum Cylinder.
2.5
DAQView Setup Parameters
Thermocouple Type:
Units:
Number of Thermocouples:
Start Condition:
Stop Condition:
Scan Rate:
Averaging:
Suggested Monitoring Method:
T
°C
8 ( +CJC )
Manual Start
Manual Stop
0.4 scan/sec (1 scan/2.5 secs)
Enabled: 100
Digital Meters/Chart
Experimental Procedures
1. Obtain the equipment needed for the experiment. This includes the power supply unit,
stainless steel and aluminum heat conduction system, and thermocouple attachment unit.
2. Complete the connections for the stainless steel cylinder system as shown in Fig. 1. The
thermocouple wires snap very easily. BE CAREFUL while handling them. DO NOT
turn on the power supply.
3. Configure the data acquisition software.
4. Finally, click on the Digital Meters… icon. Using the drop down menu, select 8 meters to
be displayed. The digital meter will be used to monitor the temperature of node 1, so that the
power supply can be turned off and data acquisition can be stopped once the temperature of
node 1 has reached 70°C. If any thermocouples behave strangely, seek help from the
instructor.
5. With the heat conduction system unplugged from the power supply, turn on the power supply
and adjust the voltage to 50 V for the steel cylinder or 70 V for the aluminum cylinder.
Turn off the power supply and plug in the heat conduction system.
6. Turn on the power supply to the steel cylinder and make any final adjustmenst so that the
voltage is set to about 50 V (if you started with the aluminum cylinder, then use 70 V). Do
not rely on the dial setting; consider the meter reading as being accurate. Simultaneously,
start the data acquisition. Also, click the Start button in the digital meters window so that
you can monitor the temperature of node 1.
7. Once the temperature of node 1 has reached 70°C, turn off the power supply immediately
(there is a danger of overheating) and stop the data acquisition and the digital meter.
8. Once the data acquisition has been stopped, the data will automatically be imported into
Excel. Column A corresponds to the cold-junction circuit temperature (which can be
deleted), while Columns B through I correspond to the temperatures of nodes 1 through 8.
The rows correspond with time; knowing that the scan rate is 0.4 scans per second or 1 scan
every 2.5 seconds, insert a column on the far left and enter the time.
9. Create a new spreadsheet and repeat steps 4 through 8 for aluminum cylinder with voltage
set at 70V.
2.6
Data Analysis
1. Refer to Chapters 2 (The Heat Diffusion Equation) and 5 (Transient Conduction) of the text.
Under assumptions of one-dimensional conduction heat flow, negligible convection and
radiation, constant properties, and no internal heat generation the general heat diffusion
equation can be reduced to:
  2T
 2
 x

1  T 
  

t   t  x
(1)
This equation must hold for any interior point where there is no internal heat generation. It
can be used to evaluate the thermal diffusivity, α by calculating the two derivatives:
 T 
  
 t  x
  2T 
 2 
 x t
(2)
2. Using approximations for the above partial derivatives, an Excel worksheet will be used to
obtain values of  from the experimental data. Find this worksheet on the ME 315 website
and open it.
3. The Excel worksheet takes the output data from the data acquisition software and calculates
the expected values of  at the combinations of location and time given in Table 2. To use
the worksheet, paste your thermocouple temperature readouts over the filler data given in the
worksheet. Delete the excess rows of filler data and verify that each selected plot data series
lines up with the appropriate data. Then, copy the coefficients of the polynomial curve fits
into the charts on the right of the plots. The calculated value of  will be reported above the
polynomial tables. DO NOT change any data other than described here. If you believe you
have accidentally adjusted a value in the worksheet, download the original file from the
website and begin the calculation again.
4. Construct a table similar to Table 2 and enter the value of the thermal diffusivity determined
from the experimental data for each of the four combinations of location and time for each
material. For at least one of the combinations, record the coefficients of the polynomial fits
to T  T (t ) and T  T ( x) so that you can calculate α for this case in a sample calculation.
Determine an average value of the thermal diffusivity for each material from these values.
Data Reduction.
1. Plot the experimental data for temperature as a function of time for TC locations 1-8 using
Excel.
2.
Tables 3 and 4 define the locations of the nodal points used in the finite difference
calculations and the locations of the thermocouples used in the experiment. In a separate
graph, plot the temperatures determined by the finite difference method and the
experimentally determined temperatures as a function of time, for the finite difference
nodes that are close to the TC locations. Show these comparisons for the pairs of points
listed in Table 5.
2.7
Table 2. Locations and Times for Determining α from Experimental Data
Material
Positions
Actual Locations (m)
4
0.080
6
0.140
3
0.055
4
0.076
Aluminum
Stainless
Steel
Times
(sec)
40
60
40
60
60
110
60
110
α (m2/s)
3. Calculate the thermal diffusivity for aluminum and steel from your experimental data using
the previously recorded polynomial coefficients for the temperature curve fits. Present one
complete sample calculation. Compare the experimentally determined average values of α
with the values given in the Table 1. Calculate a percentage error.
4. The difference between the experimental and theoretical results may be significant. To
estimate the role of radiation energy transfer, consider the formula for radiation given by
   A (T 4  T4 )
qrad
If the emissivity   0.2 , estimate the rate of energy loss by radiation from the node 1 region
at the beginning of the experiment and at the end of the experiment.
5. Calculate the predicted uncertainty of Fo and α for one for the cases given in Table 2.
Discuss the meaning of the uncertainty and the effect this result has on the apparent accuracy
of the experiment.
6. Use the following points to organize the discussion section of your report.
(a) Compare the shapes of the curves obtained for aluminum and stainless steel.
(b) Based on the nature of the curves, make qualitative comparisons of the properties of
(c)
(d)
(e)
(f)
aluminum and steel. What can you say about materials that have high thermal
diffusivities compared to those with lower diffusivities?
Do your experimental curves match the theoretical ones at the selected nodal and
thermocouple locations?
Compare the average thermal diffusivity with the value tabulated in the text. Are the
results reasonable? What could account for any discrepancy?
Are other modes of heat transfer significant?
Any other aspect of the experiment.
2.8
Table 3. Specification of thermocouple locations and nodal points used in finite-difference
method for the ALUMINUM cylinder.
Distance to Nodal points
in Finite-difference
Method
1.90 cm
5.60 cm
9.20 cm
12.8 cm
16.4 cm
20.0 cm
23.6 cm
27.2 cm
Location Number
1
2
3
4
5
6
7
8
x= 0
Distance to
Thermocouple Number
Locations
4.0 cm
4.5 cm
6.0 cm
8.0 cm
10.0 cm
14.0 cm
21.0 cm
29.0 cm
x
27.2
20.0
23.6
16.4
9.2
1.9
12.8
5.6
Node 0
Node 2
Node 1
TC1
Node 3
2.0
2.0
TC2 TC3
4.0
4.5
6.0
TC4
Node 4
2.0
TC5
Node 5
3.0
3.0
TC6
Node 6
4.0
Thermocouple
Locations (cm)
8.0
10.0
14.0
21.0
29.0
Units are cm
2.9
Node 7
4.0
TC7
Node 8
4.0
TC8
Table 4. Specification of thermocouple locations and nodal points used in finite-difference
method for the STAINLESS STEEL cylinder.
Distance to Nodal points
in Finite-difference
Method
1.90
5.64
9.30
12.96
16.66
20.32
23.98
27.64
Location Number
1
2
3
4
5
6
7
8
x=0
Distance to
Thermocouple Number
Locations
4.08 cm
4.50 cm
5.51 cm
7.63 cm
9.17 cm
13.25 cm
20.57 cm
25.75 cm
x
27.64
20.32
23.98
16.66
13.00
9.30
1.9
5.64
Node 0
2
Node 1
TC1
0.80 0.80 1.32 0.770.77
4
3
2.04
TC4 TC5
TC2 TC3
4.08
4.5
5.51
7.63
9.17
13.25
20.57
2.04
6
5
3.66
TC6
3.66
Thermocouple
Locations (cm)
8
7
2.59
2.59
TC7
3.68
TC8
25.75
Units are cm
Table 5. Nodal and Thermocouple Points for Temperature Comparisons
ALUMINUM
Nodal Point
Thermocouple
Number
Number
2
3
3
5
6
7
STAINLESS STEEL
Nodal Point
Thermocouple
Number
Number
2
3
3
5
4
6
2.10
M.E. 315 - Heat Transfer Lab
Evaluation Form
EXPERIMENT #2 -- Time Dependent Heat Conduction
This form is to be filled out by each student at the end of each lab experiment, and turned in with
the lab report. The purpose is to help the instructor(s) make changes or modifications for the
future. Your comments will in no way affect your grade--please be honest in your evaluation.
1.
Approximately how much time did you spend on this experiment?
________hours in-class time,
2.
3.
________hours outside of class time
What was your overall impression of the experiment? (You can elaborate on any of these
in 3. below.)
(a)
Terrible! I hated it! Why?
(b)
Didn't like it much, needs these improvements:
(c)
Satisfactory-some minor changes needed:
(d)
I enjoyed it and learned a lot. Comments?
(e)
Fantastic! I loved it! Why?
Please give any other specific comments below which will help us improve the
experiment for next semester.
2.11