# Thermal Physics - 2016 Fall

## Introduction

In Fall 2016 semester, the Thermal Physic class ran an experiment to measure the performance of a Yeti 20 ounce insulated coffee cup. These Yeti cups had become very popular over the previous year, most people that had one said that it was unbelievable how long they would keep coffee hot, so we wanted to find out just how long that was.

## Procedure

### Materials used

Vernier's original LabQuest (multi-meter), that we ran in the computer with the Logger Pro software Logger Pro 3.5.8.1
2 Insulated cups of the brand YETI
1 Off-brand insulated coffee cup
Standard Styrofoam coffee cups
Insulated coozie of the brand YETI
Empty regular size pop can
Water heater
Marker
Digital Scale

The first step was to put plenty of water in the heater. The heater raised the temperature of the water to about 80 Celsius degrees (which is regular coffee temperature). Next we decided to connect the Vernier Labquest multi-meter to the computer and changed the correspondent settings in the data collection section of the program (Logger Pro 3.5.8.1). The Vernier Logger Pro multi-meter that was used can be seen at image 1. The multi-meter had 3 probes connected to it, two for each YETI cup, and the third one was used to measure the temperature of the off-brand cup. We changed the data collection time to about 240 minutes, which is the time it took the water to cool down to a little further below what we considered "unacceptable" coffee temperatures once it was placed inside the YETI insulated coffee cups. In those 240 minutes the water reached temperatures of about 40 degrees Celsius.

Vernier's LabQuest multi-meter used in the experiment measuring the temperature of hot water (75.1 degrees Celsius)

We grabbed a conventional white Styrofoam coffee cup and drew a mark inside of the cup with a marker. This mark was the limit of the amount of volume of water that was used in all of our measurements. Once the multi-meter was running and the data collection parameters were suitable to our experiment we decided to pour the hot water in the marked Styrofoam cup, and then transferred this water to the YETI cups.

We used the same Styrofoam cup in order to be consistent with the amount of water being used (same volume of water used with YETI cups only; different amount used for off-brand cup) and tried to rush in the process for the water not to lose much heat to the colder environment. But how much colder was it? We decided to make measurements of the room temperature with the probes connected to the multi-meter before and after the data collection process, in order to have more accurate data. The room environment oscillated around 22 degrees Celsius. Once the data collection started, the human labor was done, we just had to come back after those 240 minutes and collect the data.

This exact same process was repeated two more times (which gives us a total of three measurements). This was repeated to reduce human error and get more accurate data by using the average of our values.

We also tested the efficiency of the YETI coozie. Since we wanted to find out about the insulation efficiency and compare it to the YETI coffee cups, we decided to place hot water inside the, previously empty, pop can that was placed in the coozie. This sounds a little unnatural since what most people would do with a can is storage a cold beverage; however, our goal was to compare the insulation properties of both the coozie and the coffee cup, and for this reason we decided to use the same contents for both cases.

Once the data collection was done, it was time to run calculations to actually calculate the insulation properties of these containers. The formula used was the one introduced by Newton in the early 16th century: the famous Newton's Law of cooling.

$\displaystyle dQ = -h(T-T_a)dt = mcdT$

Since the Law of cooling involves the mass of water placed inside of cup, we measured the mass of the water placed in the Styrofoam cup with a digital scale. Mass measurements were effectuated twice, because the amount of water was different for the off-brand cup than for the YETI cups. We had to use a different amount of water for the off-brand cup, because this cup was a little taller. Its height did not allow the probe to be left inside and collect the data, so we did a second mark on the white Styrofoam cup in order to be consistent with the amount of water used for every trial (3 total).

In the Law of Cooling, the h corresponds to the cooling constant. This depends on the mass and the specific heat capacity of the liquid placed inside the cup. This cooling constant is responsible for the "efficiency" of the cup in keeping the coffee warm, or of the coozie in keeping the beverage cold. So the main focus of this project was on finding and comparing the different values of h. We derived a formula for h basing ourselves in the Law of cooling itself, and all the data we collected from the experiment. This cooling constant told us how much better, or worse, the YETI cups were in comparison to the off-brand one. Further information can be found in the Analysis section below.

## Data

All data from the Newton's Law of Cooling experiment was plotted in gnuplot, and () was fit to the data to find the value for the cooling constant for each configuration.

The first trial using the Yeti cup. This trial yielded a value of 4.3 ± 0.2 (1/min) for the cooling constant.
The second trial using the Yeti cup. This trial yielded a value of 4.50 ± 0.09 (1/min) for the cooling constant.
The third trial using the Yeti cup. This trial yielded a value of 4.61 ± 0.03 (1/min) for the cooling constant.
The first trial using the MyBevi cup. This trial yielded a value of 5.51 ± 0.02 (1/min) for the cooling constant.
The second trial using the MyBevi cup. This trial yielded a value of 5.16 ± 0.02 (1/min) for the cooling constant.
The third trial using the MyBevi cup. This trial yielded a value of 5.34 ± 0.02 (1/min) for the cooling constant.
The first trial using the styrofoam cup. This trial yielded a value of 12.8 ± 0.2 (1/min) for the cooling constant.
The second trial using the styrofoam cup. This trial yielded a value of 12.2 ± 0.3 (1/min) for the cooling constant.
The third trial using the styrofoam cup. This trial yielded a value of 11.5 ± 0.3 (1/min) for the cooling constant.
The first trial using the Yeti coozie. This trial yielded a value of 4.89 ± 0.02 (1/min) for the cooling constant.
The second trial using the Yeti coozie. This trial yielded a value of 5.20 ± 0.07 (1/min) for the cooling constant.
The third trial using the Yeti coozie. This trial yielded a value of 5.97 ± 0.09 (1/min) for the cooling constant.

After all of the data sets were collected, the average for each data set was found and fit to determine the cooling constant. Since the initial and ambient temperatures were different for each trial, the data sets were trimmed so that they would all start at the same temperature and the ambient temperature was subtracted from each set. The graphs for the average sets are of the temperature difference versus the time.

The average data set for the Yeti coozie. This trial yielded a value of 5.48 ± 0.06 (1/min) for the cooling constant.
The average data set for the styrofoam cup. This trial yielded a value of 12.40 ± 0.07 (1/min) for the cooling constant.
The average data set for the Yeti cup. This trial yielded a value of 4.43 ± 0.05 (1/min) for the cooling constant.
The average data set for the MyBevi cup. This trial yielded a value of 5.65 ± 0.07 (1/min) for the cooling constant.

All of the values for the cooling constant can be found in the table below:

Table of the cooling constants for all of the configurations in the experiment

All of the data collected in the experiment can be found in the spread sheets below:

The spreadsheet for all of the data collected: File:Thermal Project Data.xlsx

The spreadsheet for the averaged data with the ambient temperature subtracted from the data: File:Thermal Project Averages.xlsx

## Analysis

### Newton's Law of Cooling

$\displaystyle dQ = -h(T-T_a)dt = mcdT$

$\displaystyle Q$ is thermal energy

$\displaystyle h$ is the heat transfer coefficient

$\displaystyle T$ is the temperature of the water

$\displaystyle T_a$ is the temperature of the air (environment)

$\displaystyle t$ is time

$\displaystyle m$ is mass of water

$\displaystyle c$ is specific heat capacity of water

The heat transfer coefficient h is depended on the material. $\displaystyle \alpha$ is dependent on the surface area

$\displaystyle \alpha = h/A$

### Derivation

Solve equation for temperature as a function of time

$\displaystyle - \frac {h}{mc} \int\limits_{0}^{t}\, dt = \int\limits_{T_a}^{T(t)}\frac {dT}{T-T_a}$

$\displaystyle -\frac {ht}{mc} + C = ln(T(t)-T_a))$

$\displaystyle e^{-\frac {ht}{mc}} + C = T(t)-Ta$

$\displaystyle Ce^{-\frac {ht}{mc}} + T_a = T(t)$

Solving for C using initial conditions

$\displaystyle Ce^{0} + T_a = T(0)$

$\displaystyle C = T(0) - T_a$

Substituting C

$\displaystyle (T_o - T_a)e^{-\frac {ht}{mc}} + T_a = T(t)$

Solving for h

$\displaystyle (T_o - T_a)e^{-\frac {ht}{mc}} = T(t) - Ta$

$\displaystyle e^{-\frac {ht}{mc}} = \frac {T(t) - T_a}{T_o - T_a}$

$\displaystyle -\frac {ht}{mc} = \ln \left(\frac {T(t) - T_a}{T_o - T_a}\right)$

$\displaystyle -\frac {ht}{mc} = \ln \left(\frac {T(t) - T_a}{T_o - T_a}\right)$

$\displaystyle h =- \frac{mc}{t} \ln \left(\frac {T(t) - T_a}{T_o - T_a}\right)$