EXPERIMENT NO. 4-4 (revised 2011)
UNSTEADY STATE CONDUCTION IN A SOLID
OBJECTIVE:
The objective of this experiment is to study unsteady-state conduction and to determine
the thermal conductivity of the solid. If all the necessary physical properties of the solid
are known, the thermal history within the body may be evaluated for a given set of
energy input conditions. A calculated temperature will be compared with the
experimental results.
NOTATION:
Cp d h k
n
r
s
t
T
λ
ß
θ
ρ
φ
-
specific heat (energy/mass * absolute temperature)
difference between calculate and measured temperatures
convective heat transfer coefficient (energy/(time * length2 * absolute
temperature)
sphere thermal conductivity (energy/[time * length * absolute temperature])
number of terms in the series in Equation 4
sphere radius (length)
sum of the square residuals
time
temperature
thermal diffusivity (length2/time)
arbitrary function in the error function
zenith angle (radians)
density (mass/length3)
azimuthal angle (radians)
Subscripts
i
s
= initial
= surface
THEORY:
In problems involving conduction of heat in the transient state, as in the warming or
cooling of a solid body, the temperature varies with both time and position in the body.
The relationship between temperature and time for various positions in solids is highly
dependent on the thermal conductivity, density, specific heat, and size and shape of the
body. The external conditions such as the surrounding temperature and the heat transfer
coefficient between the surroundings and the surface also affect the heat transfer rate.
In this experiment heat conduction through a solid sphere is being studied. The heat
equation for spherical coordinates is as follows:
1
2
r
  2 T 
1
 kr
 2
2
r 
r 
r sin 


 T 
1
 k
  2



 r sin 
 
T 
T
 k sin 
  q  C p
 
 
t
1
The solutions to this equation can be attained via analytical, numerical or graphical
methods. In this experiment we will investigate the analytical and the graphical methods.
Analytical Solution
For one-dimensional heat transfer in the radial direction, with no internal heat sources,
and constant thermal diffusivity, Equation (1) reduces to:
 2
T
2T
  T
 

2
t
rr
 r






2
where:
for t<0; The sphere is maintained at a constant initial temperature Ti.
for t>0; The surface of the sphere r=ro is maintained at a constant temperature To>Ti.
If an excess temperature (v) is defined as
v = T - Ti .
Then, Equation (2) can be expressed as:
 2
v
  v
 
2
t
 r






3
where the following initial condition and boundary condition, respectively apply:
for t<0 T = Ti and v = Ti-Ti =0 for 0<r<ro, and
for t>0 and r = ro T = To and v = vo = To-Ti.
and:
v
T
Ti
To
=
=
=
=
Excess temperature (T-Ti)
Temperature at a distance (r) from the center at a time (t)
Initial temperature of the sphere
Temperature at radius (ro) from center. Note: This is the surface of the sphere
r = Distance at any point from center of sphere
ro = Radius of sphere
α = Thermal diffusivity (k/ρCp)
t = Time
After applying the initial condition and boundary condition, making several
substitutions, and solving by LaPlace Transforms, the solution takes the form of an
infinite series as shown by Equation (4) (see reference 4).
v
r  
 (2n  1)ro  r

 erf 
vo
ro n 1  
4t




 (2n  1)ro  r

erf


4t









4
where n is the number of terms in the series the error function is defined as,
erf ( z ) 
2


exp



d



2
z
0

5
If r = 0, the center of the sphere, the solution reduces to:
v(0, t )

vo
2ro
t
2 2


(
2
n

1
)
ro

 exp 
n 1
4t








6
If the goal of the experiment is to determine the thermal conductivity of the object
being studied, the solution to Equations (4) or (6) becomes trial and error. Remember:

k
C p
7
Since the thermal conductivity is a variable in the error function term in Equation 4, it
is not possible to solve directly for the thermal conductivity. To determine the thermal
conductivity from the infinite series solutions, Equations 4 or 6, the experimental timetemperature history in the sphere must be obtained. The time-temperature history is
obtained by measuring the temperature at various locations in the sphere along with the
corresponding times. With these data the only unknown in these equations is the thermal
diffusivity (α).
The procedure is then to guess a value of (α), and for a given position and time
calculate the temperature. This calculated value is then compared with the value that was
measured at the same position and time value. A least squares-type method is then
applied to calculate the difference (or square residual) between the two temperatures as
shown in Equation (8)
d2 = (Tcalc-Tmeasured)2
8
The procedure is repeated for each temperature and corresponding time at each
thermocouple position and the sum of the square residuals is computed for all the data
points. The sum of the square of residuals is
n  2
S    di 
i 1 

9
A new guess for (α) is made and the same procedure, listed above, is followed. This
sum of the square residual for (α) the first alpha guess, is compared to the sum of the
square residual for (αi+1), the next alpha guess. The smallest (or least) sum is the (α)
value closest to the true valve since in a perfect fit, the sum of the residuals would be
zero. An iterative procedure is then followed until the minimum sum of squares is found
corresponding to the best (α) value. The thermal conductivity can then be calculated
from Equation (7).
Graphical
The graphical solutions for various shapes are known as the Gurney-Lurie Charts
(1,2,3,5). In these charts the ordinate Y is on log scale.
Y
T T
s
10
Ts  Ti
The abscissa, X, is on a linear scale
X 
kt
2
C p rm
where,
k
ρ
Cp
rm
Ts
Ti
T
t
is thermal conductivity of the solid (energy/time length temperature)
is density of the solid (mass/length3)
is heat capacity of the solid (energy/mass * temperature)
is the distance from center to surface of the body (length)
is surface temperature
is initial uniform temperature throughout the body
is temperature at position r at time t
is time
Various curves are provided for different positions in the body,
11
n = r/rm , and
for different resistance ratios
m = k/rmh
where, r is the distance from center of the body to the point under consideration, and h is
the convective heat transfer coefficient.
PROCEDURE:
Overview
The student will collect time-temperature data using the same computer as in the level
control experiment from ChE 361. Two thermocouples will be connected to the OPTO
22 box and placed inside a rubber ball submersed in a boiling water bath. OPTO 22 will
log data from inside the ball. This data will be fit to the model above by calculating the
diffusivity coefficient using and a program called SPHERE, residing on an SDSM&T
server.
Thermocouples
Two ‘type T’ copper/coppernickel (constantan) thermocouples were inserted in a
polymer sphere. The two thermocouples are positioned such that one is in the center of
the polymer sphere (r = 0) and the other is half way between the center and the surface of
the sphere (r = r0/2). The polymer sphere has a circumference of 28.42 cm. and weighs
993.5 grams. ‘Type T’ thermocouples can measure temperatures within the range of 0 oC
to 400 oC.
1.
2.
3.
4.
5.
6.
7.
Start Up
Turn on SNAP PAC Learning Center
Open ‘PAC Control Basic’.
Go File -> 1 UnsteadystateSphere.
Click the Debug Mode button.
Click Stop Strategy button and then start strategy button.
Open ‘PAC Display Runtime Basic’.
If the display does not automatically load, select:
a. File
b. Open Project
c. Select the file “SphereTempDisplay. UUI” from the Unsteadystate
Conduction folder on the desktop.
Running Experiment
Set up a constant temperature heat source, such as a boiling water bath. An additional
beaker of boiling water is useful to replenish the primary bath of boiling water as it
evaporates. Suspend the polymer ball in the bath, once the water is boiling. Once center
temperature becomes constant data acquisition is complete.
To Save Your Data
1.
2.
3.
4.
5.
6.
7.
Navigate to the ‘Unsteady state conduction v1.0Z’ folder on the desktop.
Locate the file called ‘DataLog’.
Drag this file to the desktop.
Rename file according to group (DataLogGroup1, etc.)
Move file to flash drive or email it to yourself.
Open original DataLog file in ‘Unsteady state conduction V1.0Z’ folder to notepad.
Delete all data off of the DataLog file (excluding header) and save it.
To Access Your Data (on other PC)
1. Double click your data file (DataLogGroup1, etc.)
2. Select Microsoft EXCEL, Notepad, or a similar program from the list.
3. Open.
NOTE: If a previous group did not rename their DataLog prior to exporting the
data, it is possible your data file will contain data from their experiment. Be sure
you check the time stamps in the EXCEL file so you use the right data for your
calculations.
Shutting Down
1.
2.
3.
4.
Close out of PAC Display Runtime Basic.
Click ‘Stop Strategy’ in PAC Control Basic, and close out of PAC Control Basic 9.0.
Log off of Windows XP.
Turn off the OPTO 22 box.
After the student has created the file, TEMP, to collect the temperature-time data of
the two thermocouples, he or she will use a program called SPHERE to calculate the
thermal diffusivity.
5. Running sphere.exe
Open the data file (temp.txt) in excel and delete the headings and the first two
columns containing the time and date. Place the times in the third column and one space
in the fourth column. The sylvan workstation will not handle over 2000 data points.
Resave this file as temp.csv. After the file has been saved, rename it to temp.txt.
Use WS FTP to transfer temp.txt into sylvan. Download WS FTP by going to
F:\instal\internet\ws_ftp32\wsftple.exe. After installing WS FTP, open the program and
type sylvan.che.sdsmt.edu in the Profile Name and Host Name, check Automatic Detect
in Host Type and then enter rwinter as the user id and r13w as the password. Open the
sphere folder in the right hand column and locate your temp.txt file in the left hand
column. Use the > to add temp.txt to the sphere folder. After transferring the file,
rename it TEMP.
The FORTRAN program, sphere.exe, resides on the Sylvan workstation that is
maintained by ITS. To access the Sylvan workstation click on the Start button and then
click on Run. Insert the command “telnet sylvan.che.sdsmt.edu.” Log in as rwinter with
password “r13w.” Once you have successfully log onto Sylvan the program enter cd
sphere and press enter. To run (execute) the sphere program type in “sphere.exe” in the
command line and press enter. You will now be asked a series of questions. When you
are asked to assign numbers to the thermocouple positions:
Position No. 1 r/ro=0
Position No. 2 r/ro=.5
After the minimum sum of the square residuals has been found record the result
and select 2 to end running the program. To log off of Sylvan type “exit” in the
command line.
6. Other options to obtain the thermal conductivity.
If you wish to use another numerical method to solve for the thermal conductivity
you will need to document in detail what technique you applied and how you applied it.
For example, finite difference could be used or evaluation of the series presented in the
text.
REPORTING:
1. Plot the temperature - time profiles of each thermocouple using at least 15 data
points taken during the experiment. The last time should correspond to the end of the
experiment. From the thermal data, dimensions and physical properties, calculate by
means of the Gurney-Lurie Charts the value of the thermal conductivity of the solid
object for at least 4 different times.
2. Using Sylvan, the program sphere.exe and the data obtained from the
experimental section, determine the best value of the thermal conductivity of the solid
sphere. Use all data points which were recorded.
3. Compare the values obtained in parts 1 and 2. Additionally, compare these with
values from the literature. Discuss any differences.
REFERENCES:
1. Bennett and Myers, "Momentum, Heat, and Mass Transfer", 3rd edition, pp 252259, 274-309. McGraw-Hill Book Co., 1982.
2. McCabe, Smith and Harriott, "Unit Operations of Chemical Engineering", 4th
edition, pp 266-273, McGraw-Hill Book Co., 1985.
3. Perry, "Perry's Chemical Engineers Handbook", 6th edition, pp 10-10 to 10-11,
McGraw-Hill Book Co., 1984.
4. Eckert and Drake, "Analysis of Heat and Mass Transfer", pp. 177-180.
(McGraw-Hill Book Co., 1972.
5. McAdams, W. H., "Heat Transmission," 3rd edition, pp. 31-43, McGraw-Hill,
1954.
6. Holman, “Heat Transfer,” 10th edition, McGraw-Hill, 2010.