# Thermal Analysis of Espresso Cup Materials

In our blog this week focuses on espresso coffee and the material of the cup in which it is served. The three standard materials that will be tested are paper, Styrofoam and ceramic. Paper cups are designed for cold drinks, so it is likely that they will result in the lowest amount of thermal energy retention in the espresso coffee. Styrofoam cups are designed as a cheaper and single-use alternative to standard ceramic coffee cups. Since they have the ability to handle a large range of temperatures, they will be able to manage the thermal energy in the espresso coffee. Ceramic cups are designed to handle extremely hot temperatures and then be washed by cool water without cracking as would typical glassware. Most likely the ceramic espresso cup will prove to be the best alternative when attempting to keep espresso coffee hot for any duration of time.

One important physical property in this experiment is specific heat. Specific heat is the amount of heat energy per unit mass required to raise the temperature of the substance by one degree Celsius. The values for each materials specific heat are seen in Figure 1 below.

#### Fig. 1: Specific Heats of Paper, Styrofoam and Ceramic Espresso Cups

The second important property is thermal conductivity. Heat transfer will occur at a high rate across materials of high thermal conductivity. Therefore a good thermal insulator is a material with a low thermal conductivity. The thermal conductivities for the materials tested in this experiment are seen in Figure 2 below.

#### Fig. 2: Thermal Conductivities of Paper, Styrofoam and Ceramic Espresso Cups

The third important physical property is the Biot number (Bi). The Biot number is a dimensionless value that is used in heat transfer calculations. The value of a Biot number gives a direct indication of the impact of convection and conduction on a body being heat or cooled. Unlike thermal conductivity and specific heat, Bi is not a material property but instead is the ratio of the resistances to heat transfer of the conduction and convection processes. A high Biot number (>>1) indicates that convective heat transfer is much faster than the conductive heat transfer, while a low Biot number (<<1) indicates the opposite.

**Thermal Management Experiment**

**Purpose**

The purpose of this experiment is to understand the material properties of a variation of standard coffee cups and how well they are able to help the espresso coffee retain its thermal energy. Simply, how much hotter is coffee in a cup of a certain material compared to cups of other materials after time has passed.

**Scope**

The scope of this experiment is to create an espresso coffee with maximum heat retention by testing varying cup materials. Each temperature will be measured using an Omega ^{TM}HH806AU J-type thermocouple. We will preheat each of the three espresso cup materials (paper, Styrofoam and ceramic) with hot water and then fill each with three ounces of espresso to compare the temperature of the espresso over time.

**Procedure**

The basic procedure for the experiment will be as follows:

- Assemble the paper espresso cup, the Styrofoam espresso cup and the ceramic espresso cup.
- Measure the height at which each espresso cup holds 3 ounces and mark it on the side of each cup to maintain consistency.
- Fill the paper espresso cup with hot water and record temperatures for 2 ½ minutes as it cools.
- Compress espresso grinds into basket, filling basket about ¾ full.
- Run machine for 5 seconds, in order to wet the grinds which will help avoid “jetting”.
- Run machine until espresso coffee reaches the 3 ounce line, this time collecting temperature of the cup and of the coffee as the espresso comes out.
- Continually measure temperature for the first three minutes that the espresso coffee is in each cup.
- Repeat the previous 4 steps for the remaining cup materials.
- Collect temperature readings at 2 minute intervals for the next twenty minutes.

**Experimental Results of Paper Espresso Cup Preheating**

- The measured starting temperature for the paper espresso cup before adding hot water to it was 26.2
^{°C}(79.1^{°F}). - The temperature of the paper espresso cup jumped to 38.7
^{°C}(101.6^{°F}) in just ten seconds, as seen in Figure 3 below.

#### Figure 3: Preheating of Paper Espresso Cup with Hot Water

- At 45 seconds, when the hot water would normally be poured out, the measured temperature of the paper espresso cup was 46
^{°C}(114.8^{°F}). - From the 45 second mark to the final time at 150 seconds (2.5 minutes) the paper espresso cup rose another 2.5
^{°C}(4.5^{°F}) to 48.5^{°C}(119.3^{°F}). - The paper espresso cup reached a peak temperature of 48.5
^{°C}(119.9^{°F}) at 116 seconds and remained within a degree of that temperature for the remainder of the measurement. - To find the time constant, the time it takes to reach 63.2% of the maximum value, the equation used is:

T_{o}*(e^{-t/τ}) Eqn. 1

where T_{o} is the initial temperature, e is Euler’s constant, t is the time and τ is the time constant, and was calculated to be 75.8.

- The change in the hot water temperature from its maximum (78.6
^{°C},173.5^{°F})at time zero and its minimum (75^{°C}, 167^{°F}) at 150 seconds was only 3.6^{°C}(6.5^{°F}).

**Experimental Results for Styrofoam Espresso Cup**

- The measured starting temperature for the Styrofoam espresso cup before adding hot water to it was 26.17
^{°C}(79.1^{°F}). - The temperature of the Styrofoam espresso cup rose very slowly and, as can be seen in Figure 4 below, did not reach 37.8
^{°C}(100^{°F}) until 136 seconds of being filled with hot water.

#### Figure 4: Preheating of the Styrofoam Espresso Cup with Hot Water

- At 45 seconds, when the hot water would normally be poured out, the measured temperature of the Styrofoam espresso cup was 32.9
^{°C}(91.2^{°F}). - From the 45 second mark to the final time at 150 seconds (2.5 minutes) the Styrofoam espresso cup rose another 5.6
^{°C}(10.1^{°F}) to 38.5^{°C}(101.3^{°F}). - The Styrofoam espresso cup reached its peak temperature of 38.5
^{°C}(101.3^{°F}) at the final time of150 seconds with the trend that indicates it would still increase if allowed to continue. - The calculated time constant (τ), again using Eqn. 1 seen above, is 64.1.
- The change in the hot water temperature from its maximum (84.3
^{°C},183.7^{°F})at time zero and its minimum (79.7^{°C}, 175.5^{°F}) at 150 seconds was only 4.6^{°C}(8.2^{°F}).

**Experimental Results of Ceramic Espresso Cup Preheating**

- The measured starting temperature for the ceramic espresso cup before adding hot water to it was 26.5
^{°C}(79.7^{°F}). - The temperature of the ceramic espresso cup rose steadily to 45.2
^{°C}(113.3^{°F}) in 96 seconds, as seen in Figure 5 below, before flattening out for the remainder of the measurement.

#### Figure 5: Preheating of Ceramic Espresso Cup with Hot Water

- At 45 seconds, when the hot water would normally be poured out, the measured temperature of the ceramic espresso cup was 40.1
^{°C}(104.1^{°F}). - From the 45 second mark to the final time at 150 seconds (2.5 minutes) the ceramic espresso cup rose another 8.4
^{°C}(10.4^{°F}) to 48.5^{°C}(114.5^{°F}). - The ceramic espresso cup reached a peak temperature of 46
^{°C}(114.8^{°F}) at 132 seconds and essentially remained constant for the remainder of the measurement. - The calculated time constant (τ), again using Eqn. 1 seen above, is 72.6.
- The change in the hot water temperature from its maximum (80.2
^{°C},176.4^{°F})at time zero and its minimum (69.1^{°C}, 156.4^{°F}) at 150 seconds was 11.1^{°C}(20^{°F}).

**Experimental Results of All Cups with Espresso Coffee**

- When the espresso coffee was poured into the paper espresso cup the espresso coffee was at a temperature of 81.6
^{°C}(178.8^{°F}) and the paper espresso cup was at a temperature of 29.8^{°C}(85.6^{°F}). - As can be seen in Figure 6 below, the paper espresso cup temperature rose steadily for the duration of the measurement and reached a maximum temperature of 47.8
^{°C}(118^{°F}).

#### Figure 6: Temperature Readings from Paper Espresso Cup with 3 ounces of Espresso Coffee

- It can also be seen in Figure 6 above that the espresso coffee steadily declined in temperature reaching a minimum of 68.3
^{°C}(155^{°F}) at the end of the measurement. - When the espresso coffee was poured into the Styrofoam espresso cup, the espresso coffee was at a temperature of 80.5
^{°C}(176.9^{°F}) and the Styrofoam espresso cup was at a temperature of 32.4^{°C}(90.4^{°F}). - It is seen in Figure 7 below, the Styrofoam espresso cup reached a peak temperature of 53
^{°C}(127.4^{°F}) at a time of 154 seconds.

#### Figure 7: Temperature Readings from Styrofoam Espresso Cup with 3 ounces of Espresso Coffee

- Also in Figure 7 it is seen that the espresso coffee declined to a minimum temperature of 64.9
^{°C}(148.9^{°F}). - When the espresso coffee was poured into the ceramic espresso cup the temperature of the espresso coffee was 78.6
^{°C}(170.3^{°F}) and the temperature of the ceramic espresso cup was 32.8^{°C}(91^{°F}). - In Figure 8 below it is seen that the ceramic espresso cup reached a peak temperature of 52.3
^{°C}(126.1^{°F}) at 154 seconds.

#### Figure 8: Temperature Readings from Ceramic Espresso Cup with 3 ounces of Espresso Coffee

- It can be seen in Figure 9 below, that in the following 20 minutes after the first measurement (Figs.4,5,6) the temperature exponentially dropped over time.

#### Figure 9: Temperature Readings over an Extended Period of Time for Each Cup

- The temperature in the paper espresso cup was measured to be 60.7
^{°C}(141.2^{°F}) at 5 minutes after being poured and 38.9^{°C}(102^{°F}) at 23 minutes after being poured. This is a difference of 21.8^{°C}(39.2^{°F}). - The temperature in the Styrofoam espresso cup was measured to be 63.4
^{°C}(146.1^{°F}) at 5 minutes after being poured and 46.6^{°C}(115.8^{°F}) at 23 minutes after being poured. This is a difference of 16.8^{°C}(30.3^{°F}). - The temperature in the ceramic espresso cup was measured to be 59.4
^{°C}(139^{°F}) at 5 minutes after being poured and 41.7^{°C}(107.1^{°F}) after being poured. This is a difference of 17.7^{°C}(31.8^{°F}).

**Analysis of Preheating Paper Espresso Cup**

For the purposes of this analysis the specific heat of the paper espresso cup is assumed to be 1400 J/KgK, which is the specific heat of paper. The mass of the cup was calculated to be 17.4 g with a volume of 266.2 ml (9oz).

In order to find the maximum theoretical temperature the equation:

c_{paper}*m_{paper}*ΔT_{paper}= c_{water}*m_{water}*ΔT_{water }= Q_{ }Eqn. 2

where Q is the thermal energy, c is the specific heat, m is the mass and ΔT is the change in temperature. By using this relationship the maximum temperature calculated to be possible was 77.4^{°C} (171.32^{°F}). This temperature is 98.5% of the maximum hot water temperature. However this is not physically possible because in order to reach 98.5% of the hot water’s temperature then there would need to be almost perfect energy transfer. This is made unachievable due to energy losses in the water because of convection and, in a small part, evaporation.

**Analysis of Preheating Styrofoam Espresso Cup**

For the purposes of this analysis the specific heat of the paper espresso cup is assumed to be 1300 J/KgK, which is the specific heat of polystyrene. The mass of the cup was calculated to be 190 g with a volume of 354.9 ml (12oz).

Equation 2 was again used to calculate the maximum possible temperature that the cup could reach, yielding a value of 75.7^{°C} (168.7^{°F}). You may have noticed that even though the Styrofoam espresso cup was preheated with slightly hotter water than the paper espresso cup, it is unable to achieve the same theoretical maximum temperature. This is due to the slight differences in the specific heats of polystyrene and paper.

**Analysis of Preheating Ceramic Espresso Cup**

For the purposes of this analysis the specific heat of the paper espresso cup is assumed to be 1090 J/KgK, which is the specific heat of standard ceramic. The mass of the cup was calculated to be 553 g with a volume of 207.2 ml (7oz).

Once again, using equation 2, the maximum theoretical temperature that the ceramic espresso cup could reach is 57.4^{°C} (135.3^{°F}). This is significant because the paper and Styrofoam espresso cups’ maximum temperatures were 77.4^{°C} and 75.7^{°C}, leaving a gap of about 20^{°C} (about 35^{°C}) between the three cups. As proved in the last blog, the preheating phase can play a major role in the overall temperature of the espresso coffee and if the Styrofoam and paper espresso cups can be preheated more it is likely that they would be the more beneficial choices.

**Analysis of Cups with Espresso Coffee**

As was seen in Figure 7, over time each of the cups allowed the espresso coffee to cool at different rates, with each following a fairly parabolic path. The largest drop was seen in the paper espresso cup, which had an espresso coffee temperature change of 39.7^{°F} (76^{°F}) after 23 minutes of being in the paper espresso cup. The ceramic espresso cup had the second largest drop in espresso coffee temperature from 78.6^{°C} (170.3^{°F}) at the time it was poured to 41.7^{°C} (107.1^{°F}) after 23 minutes in the cup, a difference of 36.9^{°C} (63.2^{°F}). The styrofoam espresso cup has the smallest temperature drop with a difference of 33.9^{°C} (61.1^{°F}) between its starting temperature of 80.5^{°C} (176.9^{°F}) and its final temperature after 23 minutes, 46.6^{°C} (115.8^{°F}).

**Conclusion**

After examining the results from the materials of espresso cups experiment it can be seen that as predicted the paper espresso cup performed the worst in terms of keeping the most amount of thermal energy in the espresso coffee. The surprise, however, is that the Styrofoam espresso cup just slightly outperformed the ceramic espresso cup by retaining just 3^{°C} more in the espresso coffee than the ceramic espresso cup. The likely reason for this is that the Styrofoam espresso cup worked as a slightly better insulator and did not draw as much energy from the espresso coffee as the ceramic espresso cup did.

Based on all of these results it may seem logical to recommend the Styrofoam espresso cup for all of your espresso coffee needs, however there is a glaring issue with relying on the Styrofoam espresso cup which was touched on earlier. The Styrofoam cup in general is designed as a single use cup, it is not something that can be put in the dishwasher and reused. One of the biggest problems with Styrofoam is that it is very slow to biodegrades and therefore is an abundant form of litter in the environment. Using a new Styrofoam espresso cup every day will only increase this, as well as put a heavy dent into your personal expenses. Because the temperature difference between the ceramic and Styrofoam espresso cup was minimal it is in your best interest to utilize a reusable ceramic espresso cup for all of your coffee needs.

Christine Dapoc2020-03-13 at 12:15 AMMay I ask how you calculated the maximum theoretical temperature? Cant seem to figure out how it was calculated from eqn 2.

Glew Engineering2020-03-31 at 5:49 PMEquation 2 does not refer to the maximum theoretical temperature. As a general rule, the maximum theoretical temperature arises from the conservation of energy with no losses. The cup and liquid would experience zero energy loss, but eventually reach equilibrium and be at the same temperature. Using the thermal mass of the liquid and cup, along with the starting temperature of each, one can calculate the change in temperature for each that conserves energy but yields the same temperature for both.