Download ENGR-45_Lab_03_Specific_Heat_Cp

Engineering-45
Specific Heat of Metals
Lab-04
Lab Data Sheet – ENGR-45 Lab-04
Lab Logistics
Experimenter:
Recorder:
Date:
Equipment Used (maker, model, and serial no. if available)
Executive Summary – Student Lab-Report Work-Product




Apply a Heat Source to Aluminum and Copper Specimens Placed in Simple
“Calorimeters”
Complete Data Tables for the Al and Cu test runs
X-Y Plots using MATLAB or MS Excel
o Specimen Temperature-Change vs Time: Plot of T vs. t
o Specimen Temperature-Change vs Cumulative Heat-Applied: Plot of T vs. Q
Calculate using several Different Techniques
o The SPECIFIC HEAT of the Specimens
o The HEAT LOST to the calorimeter structure
© Bruce Mayer, PE • Chabot College • 875611562 • Page 1
Theoretical Development
In this lab we will measure, using simple techniques, the Specific Heat of Aluminum and
Copper. Recall from Lecture that when a heat source is applied to a material object the
temperature of the object increases. For a solid object in a constant-pressure environment the
temperature increase may be quantified by:
Q  M  c p  T

Equation 1
Where
o ΔQ ≡ The Heat, or Energy, absorbed by the object in Joules
o M ≡ Constant Quantity of matter, either on a mass-basis (kg) or molar-basis
(mols)
o ΔT ≡ The Temperature Increase in Kelvins
o cp ≡ Specific Heat of the material in
 J/kg-K (mass-basis)
 J/mol-K (molar-basis)
Now CHOOSE some baseline-time and call that time, t = 0. Then at t = 0 Let
T t  0  T 0  T0
Qt  0  Q0  Q0

The Baseline Temperture
The Baseline Heat Input
Equation 2
Note that Both T0 and Q0 are CONSTANTS
Now expand ΔT and ΔQ in Equation 1 using Equation 2.
ΔT  T t   T0  T  T0

ΔQ  Qt   Q0  Q  Q0
Equation 3
Note that in most Calorimetry Experiments Q0 = 0.00 Joules
Next Substitute for ΔT and ΔQ in Equation 1 using Equation 3.
Q  Q0  M  c p  T  T0 
Now take the derivative of Equation 4 with respect to TIME:
© Bruce Mayer, PE • Chabot College • 875611562 • Page 2
Equation 4


d
Q  Q0  M  c p T  T0  
dt
d
d
d 
d
Q  Q0  M  c p  T  T0  
dt
dt
dt 
 dt
dQ
 dT

 0  M  c p 
 0 
dt
 dt

dQ
dT
 M  c p 
dt
dt
Equation 5
The quantity dQ/dt represents the HEAT FLOW into to the object. The Heat Flow:
 is given the symbol lower-case q,
 has units of J/s or Watts
 is often called the “Power InPut”
Replacing dQ/dt with q in Equation 5
dQt 
dT t 
 qT   M  c p 
dt
dt
Equation 6
Now if we assume that M, cp and q are all CONSTANT, then using Equation 6
q  M  c p 
dT
dt
or
dT
q

dt M  c p 
Equation 7
Since M, cp and q are all constant, the quantity q/(M• cp) = dT/dt is also CONSTANT. Under
these circumstances a plot T vs t should yield a STRAIGHT LINE with (constant) slope of
dT/dt. Now let dT/dt = mTt, the slope of the T vs t line. Then solving Equation 7 for cp.
q  M  c p mTt
or c p 
q
M  mTt 
We will use Equation 8 to make one calculation for cp.
© Bruce Mayer, PE • Chabot College • 875611562 • Page 3
Equation 8
Return to Equation 4 with Q0 = 0 joules
Q  0  M  c p  T  T0 
Q  M  c p T  T0 
Equation 9
Solving Equation 9 for T
T
Q
 T0
M  c p 
or
Equation 10
 1 
T 
Q  T0
M

c

p

Equation 10 reveals that the plot of T vs Q should be LINEAR in form:
 1 
T 
Q  T0
 M  c p 
 1 
T  
 Q  T0
M

c

p 





y
or

mTQ
x

b
y  mTQ x  b
Thus we can calculate cp by finding the SLOPE of a plot of T vs Q by:
© Bruce Mayer, PE • Chabot College • 875611562 • Page 4
Equation 11
mTQ 

1
M  cp
 cp 
1
M  mTQ
Equation 12
Where “mTQ” is the constant SLOPE of a plot of T vs. Q
Thus in summary:
 If M, cp and q are constant we can calculate cp using Equation 8. Ref. Figure 1
 if M and cp are constant we can calculate cp using Equation 12. Ref. Figure 2
The TWO POINT approximation assumes that the cp(Q) relationship is PERFECTLY Linear. In
this case Equation 1 can be solved for cp as:
t end
 qu du
t PwrOff
 V t   I t dt
Qtot
t start
cp 

 0
M  T M  Tmax  Tstart  M  Tmax  Tstart 
Equation 13
Thus integrating the power over the time during which the heater is turned-on, and noting the
starting and MAXIMUM temperatures gives an easily calculated estimate of the specific heat.
Note that using Tmax, which will occur some time after heater turn-off, reduces the effect of the
TIME LAG that is present in the system. That is, the application of power to the heater is not
instantaneously observed as a temperature rise as measured by the thermometer. The Time
Lag is due to the R•C product (see Figure 7) which is called the “Time Constant” for the
system.
Finally, note that the Specific Heat is an INTRINSIC material property; i.e., Specific Heat is
independent of the size of the object as it is “normalized” to unit-mass or unit-mol. The Specific
Heat is measure of the ability of a material to “store” heat. This heat-storage “capacity” is
directly analogous to charge-storage in electrical capacitors. For this reason Specific Heat is
occasionally referred to synonymously as “Heat Capacity”.
The so-called Thermal Capacitance of a solid specimen with mass M and heat-capacity, cp, is
calculated as:
Ctherm  M  c p
kg J
J
 Units 

1 kg  K K
© Bruce Mayer, PE • Chabot College • 875611562 • Page 5
Equation 14
The THERMAL capacitance units of Joules per Kelvin are analogous to the ELECTRICAL
capacitance units of Coulombs per Volt
HeatUp for Al Block
95
Block Temperature, T (°F)
90
dT
F
mK
 mTt  0.9174
 8.4944
dt
min
s
85
80
75
T(t) = 0.9174t + 60.935
2
R = 0.9986
PARAMETERS
• Date = 06Jan09 • B. Mayer
• 1x1 Kapton Heater, 159.3 ohm
• Material = 6061 Al Block
• Size = 0.5x3.055x2.965 Cu-inches
• Mass Al = 0.444 lbm = 0.201 kg
• Tambient = 61.6 °F
• Power Input Apporx. 2.28W
• Power Off at 33 min
70
65
60
0
Cp_Al_Test_0901.xls
5
10
15
20
25
Time, t (minutes)
Figure 1 - Find mTt by Linear Regression
© Bruce Mayer, PE • Chabot College • 875611562 • Page 6
30
35
T vs. Q for Al Block
34
32
dT
C
 mTQ  3.7257
dQ
kJ
Block Temperature, T (°C)
30
28
26
24
T = 3.7257Q + 16.076
2
R = 0.9986
22
PARAMETERS
• Date = 06Jan09 • B. Mayer
• 1x1 Kapton Heater, 159.3 ohm
• Material = 6061 Al Block
• Size = 0.5x2.965x3.055 Cu-inches
• Mass Al = 0.444 lbm = 0.201 kg
• Tambient = 61.6 °F
• Power Input Approx. 2.28W
20
18
16
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Heat Stored, Q (kJ)
Cp_Al_Test_0901.xls
Figure 2 - Find mTQ by Linear Regression
Experiment Description
Recall from lecture the schematic diagram for
a solid-material calorimeter shown in Figure 3.
In this configuration Heat is added to the
block of material in the form of electrical
power.
INSULATION
The diagram in Figure 3 uses an Ammeter
(“A” inside a circle) and a Voltmeter (“V”
inside a circle). In this configuration the
Ammeter measures the Electrical Current, I,
and the Voltmeter measures the Electrical
Potential, V, delivered to the electricresistance heater.
Figure 3 - Schematic of a Sold-Material
Calorimeter
© Bruce Mayer, PE • Chabot College • 875611562 • Page 7
The concepts of Engineering 43 describe the relationship between I, V, and power dissipated
by the heater:
P V  I

Equation 15
Where
o V ≡ The Electrical Potential across the Heater in Volts
o I ≡ The Electrical Current thru across the Heater in Amps
o P ≡ The Power Dissipated in the form of HEAT by the electrical-resistance
Heater
In the case of the solid-material calorimeter all the dissipated power is assumed to be
absorbed by the solid specimen. With reference to the heat-flow, q, defined earlier it is
assumed that q = P.
ThermoMeter
Low Density PolyStyrene
Thermal Insulation
Material Specimen
Kapton-Enclosed
Metal-Film Heater
Figure 5 - Calorimeter Cross Section
Figure 4 - Solid-Material Calorimeter
Figure 4 contains a photograph of one design for a simple solid-material calorimeter. Figure 5
shows a schematic cross section of this calorimeter. A comparison of in Figure 3 and Figure 5
indicates that in the ENGR45 design a metal-film heater replaces the “cartridge” heater used in
in Figure 3. Figure 6 displays a close-up view of the film heater applied to the test specimen.
The operation of the calorimeter is described by the lumped-parameter thermal-circuit
schematic shown in Figure 7. The quantities depicted in the schematic include:
 Tspec ≡ The Temperature of the specimen in Kelvins (or °C or °F)
 Tambient ≡ The Temperature of the ambient air surrounding the calorimeter in Kelvins (or
°C or °F)
© Bruce Mayer, PE • Chabot College • 875611562 • Page 8








o The ambient temperature is
effectively the Thermal “Ground”
for the circuit; that is, all the heat
supplied by the heater is
eventually dissipated to the
room thru the PolyStyrene
Resistance
qH ≡ The Heat Flow supplied by the
Metal-Film Heater in Watts
Cspec ≡ The Thermal Capacitance of the
in J/K (or J/°C or J/°F)
qspec ≡ The “Charging” Heat Flow into
the Material Specimen in Watts
CPS ≡ The Thermal Capacitance of the
PolyStyrene Insulation J/K
Figure 6 - Adhesive Backed,
qC,PS ≡ The “Charging” Heat Flow into
Serpentine, Metal-Film Resistance
the PolyStyrene Watts
Heater Applied to the Specific Heat Test
RPS ≡ The Thermal Resistance of the
Specimen
PolyStyrene Insulation K/W (or °C/W or
°F/W)
qR.PS ≡ The Heat Flow thru PolyStyrene
Insulation Watts
qleak ≡ The “Leakage” Heat Flow drawn by the PolyStyrene Capacitance and Resistance
in Watts
o By Conservation of Energy: qleak = qC,PS + qR,PS
In conducting this experiment it is ASSUMED that the Leakage Heat Flow is “small” compared
Tspec
qleak
qspec
qH
Cspec
qC,PS
qR,PS
CPS
RPS
Tambient
Figure 7 - Thermal Circuit Schematic for the Solid Material Calorimeter. See text
for symbol definitions.
© Bruce Mayer, PE • Chabot College • 875611562 • Page 9
to the specimen-charging heat flow. A Mathematical statement of this assumption:
qleak << qspec
Note that any leakage heat flow will erroneously INCREASE the calculated Specific Heat for
the specimen. We will use this fact to estimate the heat flow leakage.
The total heat, Q, absorbed by the test specimen is simply the time-integral of the “charging”
heat-flow into the specimen:
Qt    qu du  Pu du   V u  I u du
u t
u t
u 0
u t
u 0
u 0
Equation 16
In this experiment V & I will be measured periodically; not continuously. That is, the V & I data
will be collected in TABULAR form. In this case the incremental added heat, ΔQ, is simply the
P•I product multiplied by the time-period over which P and I are assumed to be constant.
If the time increment between V & I measurements, Δt, is itself constant, then for n
measurements the integration of Equation 16 becomes a summation:
k n
k n
k n
k n
Qn   Qk   qk  t   Pk  t  Vk  I k   t
k 1
k 1
k 1
Equation 17
k 1
The TOTAL Energy applied to the specimen is simply Equation 17 taken up the time at which
power is no longer applied to the specimen. That is, the summation concludes at the Power
OFF time.
Qtot 
k @ PwrOff
 V
k 1
k
 I k   t
Equation 18
Performance of this experiment entails that measurement and collection of these quantities:
 Once-Only measurements
o Specimen Mass, M (done earlier by the instructor)
o Ambient air temperature, Tamb
o Heater Electrical Resistance, RH
 The Time-Incremented Measurements
o Elapsed Time, t
o Heater Electrical Potential, V
o Heater Electrical Current, I
o The Specimen Temperature, T
© Bruce Mayer, PE • Chabot College • 875611562 • Page 10
Notes on PolyStrene Mechanical and Thermal properties
 Mass Density, ρM = 25.6 kg/m3 (1.6 lb/ft3)
 Thermal Conductivity, kth = 0.0288 W/m•K (0.2 BTU•in/[hr•ft2•°F])
 Thermal Resistivity, , ρth =1/kth = 34.7 m•K/W
Exercise Directions
1. Equipment, Instruments & Supplies
 Electrical Power Supply with Constant DC Voltage output
 Bench-Style Digital Electrical-Quantity MultiMeter (DMM) to Measure Current
 Hand-Held Digital Electrical-Quantity MultiMeter (DMM) to Measure Voltage
 Red & Black Probe-Leads for use with the Power Supply and DMM
 Aligator Clip-Lead; 1 each, any color
 SetUp Electrical Resistor, 390Ω nominal
 ALUMINUM-specimen Calorimeter Assembly
 COPPER-specimen Calorimeter Assembly
 Digital ThermoMeter, 0.1 °F, or better, resolution
 StopWatch
For the First Specimen Make Once-Only PreRun Measurements to complete
© Bruce Mayer, PE • Chabot College • 875611562 • Page 11
2. Table I (or Table III)
 Read the Specimen Mass from the label on the calorimeter
 Measure the Heater Electrical Resistance with the DMM
 Measure the SetUp Resistor with the DMM
 Measure the Ambient Air Temperature in °F using the digital thermometer
o All measurements will be done in Degrees Fahrenheit as this temperature scale
provides higher resolution than does the Celcius scale
3. Construct the SetUp Electrical Circuit and Verify Electrical Current (milliamps) & Potential
(volts) measurements
 Construct the SetUp circuit per the electrical schematic contained in Figure 8.
o Use the BENCH DMM as the
AMMETER
o Use the HANDHELD DMM as
the VOLTMETER
 Turn on the Power Supply and use the
DMM (NOT the Power Supply Display)
to set the Voltage Supply Level to
o 20.50 Vdc for the COPPER
Specimen
o 11.00 Vdc for the ALUMINUM
Specimen.
 Verify that the electrical current is in the
range of
o 40-60 mA for the COPPER
voltage of 20.50 Vdc
o 20-40 mA for the ALUMINUM
voltage of 11.00 Vdc
 Turn Off the Power Supply
Figure 8 - SetUp circuit used to adjust
4. Construct the Test System
the Voltage supply, and to verify the
 Remove the SetUp Resistor from the
Current and Voltage measurements.
circuit of Figure 8.
VS = 11.00 V. RSU = 100-200 Ω.
 Connect in place of the SetUp resistor
the leads to the Calorimeter metal-film
heater to arrive at the electrical circuit configuration shown in Figure 9.
o Use the BENCH DMM as the AMMETER
o Use the HANDHELD DMM as the VOLTMETER
 Be sure to connect the VoltMeter right at the SPECIMEN connections to
avoid measuring the voltage drop associated with Ammeteri.
 Install the ThermoMeter probe in the Calorimeter. See Figure 4.
o Be sure that the probe is properly seated in the specimen “ThermoWell”; a
shallow, blind hole drilled into the top surface of the specimen. See Figure 10
 If so directed by the instructor, apply a small amount of Heat Transfer
Compoundii to the tip of the probe prior to inserting into the probe into the
calorimeter
i
The Series Resistance of the Ammeter is 3-4 Ω.
© Bruce Mayer, PE • Chabot College • 875611562 • Page 12

Test the StopWatch to become familiar
with its operation.
o ReSet the StopWatch to Zero
5. Run the Test
 SIMULTANEOUSLY Turn On the
PowerSupply and the StopWatch.
 As indicated in Table II (or Table IV),
every 30 seconds record the
o Heater Voltage in Volts
 Should be very nearly
20.50V
or
11.00V
depending on whether the
specimen is Cu or Al.
 Do NOT adjust the
Voltage during the test.
o Heater Current in mA
o Specimen Temperature in °F
Figure 9 - Experiment circuit used to
 After 30 min Turn Off the Power Supply
generate the data needed to calculate
o The test duration may, at the
specimen Specific Heat
discretion of the student(s), be
extended beyond 30 minutes
subject to the constraint that the specimen temperature NOT EXCEED 125 °F.
 Due the Time Lag the specimen temperature will continue to RISE for a period of time
AFTER power TurnOff. Continue to record the specimen temperature until the
temperature starts to FALL.
o During this Post-Power Period
the temperature may recorded
every 60 seconds if desired by
the experimenters
6. Repeat steps 0 thru 5 for the second
specimen
7. Return all lab hardware to the “as-found”
condition
ThermoWell
Figure 10 - Detail View of Specimen
ThermoWell. If needed the well may
filled with a small amount of Heat
Transfer Compound
ii
This may have been previously done by the instructor
© Bruce Mayer, PE • Chabot College • 875611562 • Page 13
Table I - UNS C10100 COPPER Specimen Pre & Post Run Measurements
Specimen Mass, M
=
Heater Electrical Resistance, RH
=
SetUp Resistor Resistance, RSU
=
Ambient Air Temperature, Tamb
=
Maximum Specimen Temp, Tmax
=
Total Specimen Energy, Qtot
=
Table II - C10100 COPPER Specimen Data Table (Nominal 20.5 Vdc Power Supply)
Time (h:mm:ss)
V
I
0:00:00
0:00:30
0:01:00
0:01:30
0:02:00
0:02:30
0:03:00
0:03:30
© Bruce Mayer, PE • Chabot College • 875611562 • Page 14
Temperature
Time (h:mm:ss)
V
I
0:04:00
0:04:30
0:05:00
0:05:30
0:06:00
0:06:30
0:07:00
0:07:30
0:08:00
0:08:30
0:09:00
0:09:30
0:10:00
0:10:30
0:11:00
0:11:30
0:12:00
0:12:30
0:13:00
0:13:30
0:14:00
0:14:30
0:15:00
0:15:30
0:16:00
0:16:30
0:17:00
0:17:30
0:18:00
0:18:30
0:19:00
0:19:30
0:20:00
0:20:30
0:21:00
0:21:30
0:22:00
0:22:30
0:23:00
© Bruce Mayer, PE • Chabot College • 875611562 • Page 15
Temperature
Time (h:mm:ss)
V
I
0:23:30
0:24:00
0:24:30
0:25:00
0:25:30
0:26:00
0:26:30
0:27:00
0:27:30
0:28:00
0:28:30
0:29:00
0:29:30
0:30:00
© Bruce Mayer, PE • Chabot College • 875611562 • Page 16
Temperature
Time (h:mm:ss)
V
I
Temperature
Table III - UNS A96061 ALUMINUM (6061-T6 Al) Specimen PreRun Measurements
Specimen Mass, M
=
Heater Electrical Resistance, RH
=
SetUp Resistor Resistance, RSU
=
Ambient Air Temperature, Tamb
=
Maximum Specimen Temp, Tmax
=
Total Specimen Energy, Qtot
=
Table IV – 6061-T6 ALUMINUM Specimen Data Table (Nominal 11 Vdc Power Supply)
Time (h:mm:ss)
V
I
0:00:00
0:00:30
0:01:00
0:01:30
0:02:00
0:02:30
0:03:00
© Bruce Mayer, PE • Chabot College • 875611562 • Page 17
Temperature
Time (h:mm:ss)
V
I
0:03:30
0:04:00
0:04:30
0:05:00
0:05:30
0:06:00
0:06:30
0:07:00
0:07:30
0:08:00
0:08:30
0:09:00
0:09:30
0:10:00
0:10:30
0:11:00
0:11:30
0:12:00
0:12:30
0:13:00
0:13:30
0:14:00
0:14:30
0:15:00
0:15:30
0:16:00
0:16:30
0:17:00
0:17:30
0:18:00
0:18:30
0:19:00
0:19:30
0:20:00
0:20:30
0:21:00
0:21:30
0:22:00
0:22:30
© Bruce Mayer, PE • Chabot College • 875611562 • Page 18
Temperature
Time (h:mm:ss)
V
I
0:23:00
0:23:30
0:24:00
0:24:30
0:25:00
0:25:30
0:26:00
0:26:30
0:27:00
0:27:30
0:28:00
0:28:30
0:29:00
0:29:30
0:30:00
© Bruce Mayer, PE • Chabot College • 875611562 • Page 19
Temperature
Time (h:mm:ss)
V
I
Temperature
Data Reduction
The t, V, I, and T data collected in Table II and Table IV must be reduced to a form that permits
construction of the MATLAB/EXCEL plots shown in Figure 1 and Figure 2.
First a summary of primary ASSUMPTIONS made in the following analysis
 Both V & I are very nearly constant over the 30 second time increment, Δt
 That the internal Thermal resistance of the specimen is very small relative to the
thermal resistance of the of the PolyStyrene
o This “Biot Analysis” implies that the specimen is very nearly ISOthermal; i.e.,
temperature at the top and bottom of the block are very nearly equal
 That the leakage heat flow is very small relative to the specimen heat flow. See Figure
7.
o This implies that virtually ALL the heat generated by the electrical resistance
heater is absorbed by the specimen block
Data reduction entails the performance of these calculations on the raw data:
 Elapsed time converted from h:mm:ss to one of:
o Decimal SECONDS
o Decimal MINUTES (recommended)
o Decimal HOURS
 Temperature converted from °F to a temperature scale that has an “SI Increment”:
o °C (recommended)
o Kelvins
 Use Equation 15 to Calculate the Heater heat-flow from V & I: qH = V•I
 Calculate the Heat Stored in the specimen, Q, as described by Equation 17
© Bruce Mayer, PE • Chabot College • 875611562 • Page 20
The Data Reduction can be done using and EXCEL spreadsheet as outlined in Table V.
Alternatively, the data may be reduced using MATLAB code similar to that shown below:
% Bruce Mayer, PE
% ENGR45 * 07Jan09
% P10.2.18,
% file Cp_DataReduction_0901.m
%
% hand enter data vectors:
%% t for time (min)
%% V for vols (V)
%% I for current (mA)
%% T for temperature (°F)
%
% Alternatively import data from EXCEL Table such as
Cp_Al_Test_MatLab_import_0901.xls
%
Tc = (T-32)/1.8 % in °C
q = V.*I/1000 % in W
for k =1:length(q)
qk = q(1:k)
Q(k)= sum(qk)*30 % in Joules
end
%
plot(t,T) % min vs. °F
display('Showing t vs T plot, hit any key to continue')
pause
plot(Q/1000,Tc) % kJ vs °C
Next create with MATLAB or EXCEL Linear-Regression plots that reveal the slopes of the t vs
T and Q vs T scatter-data. See Figure 1 and Figure 2. With the slopes mTt and mTQ from the
Regression Analysis, calculate two values for cp as described in Equation 8 and Equation 12:
c p ,t 
qavg
M  mTt
c p ,Q 
1
M  mTQ
Where qavg is the average power applied to the specimen over the course of the entire test-run
as calculated by (be sure to use appropriate units):
qavg 
Qtot
Qtot
Qtot


tt
30 min 1800 sec
© Bruce Mayer, PE • Chabot College • 875611562 • Page 21
Equation 19
Next, calculate the average, cp,avg, of the above two quantities.
c p ,avg 
c p ,t  c p ,Q
Equation 20
2
Finally, calculate the 2-point specific heat (reference Equation 13 and Equation 18 based on
the assumption of perfect linearity as:
c p,L 
Q Tot
M T max Tstart 
Compare the average value to the room temperature cp values for Al and Cu found in the
technical Literature. Calculate the %-Error using the literature values as the baseline:
% reg
c p ,avg  c p ,lit 100%


c p ,lit
1
Equation 21
% Lin
c p , L  c p ,lit 100%


c p ,lit
1
Equation 22
Now use the LITERATURE Values to estimate for the regression case the total heat missing
due to STORAGE and LEAKAGE, Qmiss, that occurred during the course of the experiment.
Noting that Tfinal is the temperature at TurnOFF, first Calculate the literature-based heat stored
in the specimen by:
Qlit,reg  M  c p ,lit T final  T0 
Use Appriate Units 
Equation 23
Now use the Total Heat supplied by the electrical resistance heater, Qtot, from the data
reduction to calculate the %-Missed as
Qlit,reg  Qtot 100%
Missed % 

Qtot
1
© Bruce Mayer, PE • Chabot College • 875611562 • Page 22
Equation 24
Now use the LITERATURE Values to estimate for the 2-Point case the total heat leakage,
Qleak, that occurred during the course of the experiment. For this assumed-linear situation
Calculate the literature-based heat stored in the specimen by:
Qlit, Lin  M  c p ,lit Tmax  T0 
Use Appriate Units 
Equation 25
Again use the Total Heat Supplied to calculate the %-Leakage as
Leak % 
Qlit,Lin  Qtot 100%

Qtot
1
Equation 26
Complete the Calculation-Summary, and Literature research tables: Table VI, Table VIII, and
Table IX
Work Product Summary
To receive full credit for the laboratory exercise a student must submit this report form with
Completed data tables,
© Bruce Mayer, PE • Chabot College • 875611562 • Page 23



Table I → Table IV
o All data should be listed to at 3 significant figures
o All data must include the units of measure
MATLAB or EXCEL X-Y plots for t vs T and Q vs T attached
o Plots must be properly constructed and labeled as described in Engineering-25
o The Plots must show the Regression-Line along with the Regression-Equations
along with the Coefficient of Determination (or alternatively the Correlation
Coefficient)
Completed Calculation and Literature Research tables: Table VI, Table VII Table VIII,
and Table IX
© Bruce Mayer, PE • Chabot College • 875611562 • Page 24
Table V – Suggested Data Reduction SpreadSheet Table
No.
Raw Data
t
V
I
T
0
0
0
0
T0
0
1
t1
V1
I1
T1
2
•
•
•
59
t2 V2 I2 T2
•
•
•
•
•
•
•
•
•
•
•
•
t59 V59 I59 T59
 t1 60
 t 2 60
T (°C)
 T 0  32 1.8
 T1  32 1.8
 T 2  32 1.8
•
•
•
•
•
•
60
max
t (min)
 t 59 60
t60 V60 I60 T60  t 60 60
tbd
0
0
0
n/a
Reduced Data
qH (W)
Q (J)
0
0
 V1 I1 1000
 V 2  I 2 1000
 q0  V1 I130
 q1  V 2  I 230
•
•
•
•
•
•
 T 59  32 1.8  V 59  I 59 1000
 T 60  32 1.8  V 60  I 60 1000
0
 T max  32 1.8
 q58  V 59  I 5930
 q59  V 60  I 6030
 q 60
Table VI – Specific Heat Calculations for Regression Case (Use SI Units)
Specimen
cp,t
cp,Q
cp,avg
UNS A96061 Al
UNS C10100 Cu
© Bruce Mayer, PE • Chabot College • 875611562 • Page 25
cp,lit
%reg
Table VII – Specific Heat Calculations for 2-Point, Assumed Linear, Case (Use SI Units)
Specimen
cp,Lin
%Lin
cp,lit
UNS A96061 Al
UNS C10100 Cu
Table VIII – Heat-Missed and Heat-Leakage Heat Calculations (Use SI Units)
Specimen
Qtot
Qlit,reg
Qlit,Lin
Missed%
Leak%
UNS A96061 Al
UNS C10100 Cu
Table IX – Specific Heat References
Specimen
Reference
UNS A96061 Al
UNS C10100 Cu
Print Date/Time = 19-Jun-17/19:14
© Bruce Mayer, PE • Chabot College • 875611562 • Page 26