Project 1: Calorimeter

INSTRUCTIONS

This project is an individual assignment. Please watch a detailed review of the Calorimeter methods here:

From this information, you should write your “Methods” section of the report. The data files required to carry out

your analysis will be posted on the Canvas website after the laboratory session. We will assign you a specific

sample (1 of 4) to solve for in this lab. Use the posted data to determine the sample’s material type, selecting from

the posted candidate materials properties. You will do this using the methodology outlined below and presented in

class incorporating error propagation and least squares linear fitting. Compile your findings into a report (outlined

below). Your MATLAB code must be submitted in a zipped folder with all required files for the code to compile.

Submit the report and associated MATLAB code to Canvas before the due date.

REPORT DELIVERABLE

• Report must have the following sections: 1) Abstract 2) Introduction 3) Experimental method 4) Results 5)

Discussion 6) Conclusion 7) References and 8) Appendix containing your code flow chart and an analytical

derivation of error propagated for Eq. (1). Note that your flow chart may be written by hand or using graphical

software, but it must be in the proper symbol format and detail. Your analytical derivation of the error

associated with Eq. (1) may be typed or handwritten.

• The report must be written in the AIAA format. You should use the technical writing principles discussed in

class. You will be assessed on your writing skills in this project.

• Your report may be no longer than 4 pages in length, excluding the References and Appendix (i.e. your code

flow chart and analytical solution of error propagation).

Calorimetry analysis to be included in your report:

a) Include a derivation of Eq. (1) below in your report using the First Law of Thermodynamics.

b) Estimate the specific heat of your sample from Eq. (1) below and identify the material from which the

sample may have been made. Information on candidate materials will be provided.

c) Providing an error estimate for your computed values and relate whether the accepted value of specific heat

of the material you propose the sample may be made from falls within the error estimate. If the deviation

was significant, where the accepted value lies outside the error bar, how would parameters used in your

calculation from Eq. (1) below have to change to account for the observed bias? Provide physical reasoning

behind your assessment.

Calorimetry background

Calorimetry is a basic technique for measuring thermodynamic quantities, especially those involving heat transfer,

i.e., specific heats, heat of fusion, heat of vaporization, and heat of chemical reactions. It is one of the oldest

experimental methods in thermodynamics. Benjamin Thompson used a calorimeter in 1798 to show that work could

be converted to heat (thermal energy), and James Joule conducted elegant experiments in the 1840s to measure the

mechanical equivalent of heat (thermal energy). People who count “calories” can look up the energy value of

ASEN 2012 Experimental and Computational Methods Fall 2020

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various foods in their diet, e.g., three Chips Ahoy cookies (~32 grams) supply 169 Calories (kcal) of energy. We

know this because someone, in this case Nabisco, Inc., conducted calorimetry experiments.

Heat transfer between the sample of a given temperature and the calorimeter of a different given temperature takes

place. The basic set up of a calorimeter is shown in Fig. 1. Assuming the sample combined with the calorimeter are

an adiabatic system, one can show that

(1)

where Cs,av = specific heat of the sample, Cc,av = specific heat of the calorimeter, T0 = initial temperature of the

calorimeter, T1 = initial temperature of the sample, T2 = final temperature of the calorimeter and sample at

equilibrium, mc = mass of the calorimeter, and ms = mass of the sample.

However, adiabatic conditions are difficult to maintain. Fig. 2 is copied from the kit manual for the calorimeter and

shows a typical temperature profile from this calorimeter. As can be seen in the data from 1:00 pm to 1:20 pm, the

temperature of the empty calorimeter is rising slowly in the room. The calorimeter temperature rises rapidly after a

sample at 99.5°C is added at 1:21 pm. By 1:23 pm, however, the temperature rise has slowed. During this time, the

calorimeter loses thermal energy to the surroundings, in spite of the insulation. That temperature loss is clear after

1:25 pm. The description following Fig. 1 describes the procedure to extrapolate values for T0 and T2 from a linear

fit of temperature measurements before the sample was added and after heat transfer to the surroundings began to be

a major factor respectively. The purpose of the extrapolation method is to account for heat loss before, during and

after the experiment.

Figure 1 Schematic of the calorimeter, and adiabatic system.

As one can see from the data in Fig. 2, the calorimeter temperature begins to rise immediately after the sample is

introduced. The rise is at first rapid (time point A to point B), then slower. Then the temperature starts to decrease

(point A to point C), as the calorimeter temperature is observed to fall from 25.5 °C to 25.2 °C in 12 minutes. This

represents a change of 0.025 °C per minute. We must employ a method to compensate for this unaccounted heat

loss. This method is outlined below for the data shown in Figure 2 and employs the following steps:

1. Determine the approximate temperature of the calorimeter before addition of the sample by fitting a line to

the pre-sample temperature data from 1:10pm to 1:20pm.

2. Use the line to determine the temperature for the time the sample was added and its uncertainty, in this case

1:21 pm (time of sample addition). Note the extrapolated temperature at TLow(1:21).

, 2 0 ,

1 2

( )

( )

c c av s av s

mC T T C mT T

– = –

insulation

sample

temperature

probe

calorimeter

material

ASEN 2012 Experimental and Computational Methods Fall 2020

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3. Fit a second line from the maximum temperature reading (i.e. 1:27pm) to the last temperature reading at

1:50pm.

4. Extrapolate this line back to the time when the sample was added (i.e. 1:21pm) and note the extrapolated

temperature and its uncertainty, THi(1:21). In this case TLow(1:21) and THi(1:21) temperatures are 20.3 °C

and 25.7 °C respectively. These are approximately correct.

5. Next, determine the average of these two temperatures (23.0 °C); and from the graph determine the time

associated with this temperature (1:22 pm approximately).

6. Extrapolate the line from the later readings back to 1:22 pm, THi(1:22). TLow(1:21) represents the initial

temperature of the calorimeter, T0, and THi(1:22) represents the final temperature of the calorimeter and

sample at equilibrium, T2, with a compensation for heat loss to enable the adiabatic assumption to be

applied and Eq. (1) to be appropriate. Also calculate the associated uncertainty with this value.

Figure 2. Typical temperature profile from calorimeter

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