A simple lowcost method was used to measure the thermal diffusivity of nine different types of foods: potato, sweet potato, pumpkin, taro, radish, eggplant, lemon, tomato, and onion. We cut the foods into spherical shapes, inserted thermocouple sensors into their centers, and immersed them in boiling water. Fitting the time dependence of the center temperature to a heatconduction model yielded a value for the thermal diffusivity with good consistency between spheres of different radii. This method can be generalized to determine thermal diffusivity of a wide variety of samples.
I. INTRODUCTION
Thermal diffusivity controls the heat transfer process via the equation $ \u2207 2T= \alpha \u2212 1(\u2202T/\u2202t)$, where T is the temperature and α is the thermal diffusivity. This equation describes one of the most important physical phenomena governing the worldheat transfer and exchange. Thermal diffusivity is expressed mathematically as α = k/ρc, where α is the thermal diffusivity in m^{2}/s, ρ is the density (kg/m^{3}), and c is the specific heat (J/(kg K)).
In 1979, Unsworth and Duarte presented a simple method to determine the thermal diffusivity for rubber spheres by measuring the rate of cooling of their centers.^{1} The process required gluing a spiralshaped thermocouple to the center of a rubber hemisphere and joining two hemispheres to make a whole sphere. Probably due to the difficulty of preparing the samples, only spheres of a single diameter were studied. We show that, by varying the diameter of the spheres, students can discover for themselves the simple scaling relationship between diameter and time, and by making the spheres out of a variety of foods, they can easily prepare interesting samples.
II. THEORETICAL MODEL
III. EXPERIMENTAL METHOD
Our method is similar to Unsworth and Duarte's work,^{1} but we extend their method in the following ways: (1) We insert a straight, rather than spiral shaped, thermocouple into the spherical sample. In Unsworth and Duarte's work, a spiral shaped thermocouple was used, intended to slow any potential thermal leakage through the thermocouple itself. We found that a straight thermocouple works well enough, and the smallsized thermocouple tip can be inserted directly into the spherical sample. Hence, the process of preparing the sample and the measurement setup is simpler than that of Ref. 1. (2) We take the diameter of the sphere as an additional experimental parameter. We measure the time dependence of the center temperature with multiple samples of different diameters, which adds an independent parameter through which the model can be tested. (3) We apply this method to measure some interesting materials: various types of food.
Nine types of food were used in this project: potato, sweet potato, pumpkin, taro, radish, eggplant, lemon, tomato, and onion. All the food samples were shaped into spheres with radii ranging from 15 to 31 mm. Figure 1 shows three of these samples.
The sample preparation process is straightforward. Spheres of potato, sweet potato, pumpkin, taro, radish, and eggplant were shaped manually with a regular kitchen vegetable “Y” peeler such as a Westmark Gallant Vegetable Y Peeler, taking about 5 min. For samples such as onion, lemon, and tomato, we started with foods that were already nearly spherical and with diameters very close to what we want for the final samples; thus, we only needed to remove the skin, since shaping these foods would be difficult.
During the sample shaping process, calipers were used to monitor diameter of the sample. For any spherical sample used in this work except for onion, lemon, and tomato, the maximal deviation between ten measurements of diameters in random orientations was less than 1 mm. In addition, we also measured the weight of each sample and then calculated the density. For the same material, the difference in the densities between different samples was less than 4%. For onion, lemon, and tomato, the maximal deviation in diameter between random orientations was less than 2 mm.
Because we need to measure the temperature at the center of the sample, the size of the temperature sensor head needs to be small. A Proster digital temperature thermometer was used, which included Ktype thermocouple probes, shown in Fig. 2. The thermocouple probe has a diameter of around 1 mm and connects to the meter via 1 m wires. The plastic insulation coating wrapped around the thermocouple's conductive wires is about 0.75 mm in thickness. The small tip, i.e., the junction of the thermocouple, is exposed. The thermocouple is inserted into the center of the spherical food sample to measure the temperature rising as a function of time during the heating process with an uncertainty of ±1 °C.
After the thermocouples are inserted, the samples are immersed but suspended in boiling water (100 °C), as shown schematically in Fig. 3. We then track the center temperature increase as the function of time.
The total cost for the materials, tools, and instruments used in this project is about $100. The thermocouple and its voltmeter come together with the thermocouple cost $60, and all the food materials cost less than $50. It takes about 10–30 min to measure each sample, depending on the sample diameter, and the measured results are consistent and accurate.
IV. EXPERIMENTAL RESULTS
Figure 4 shows the measured center temperature as the function of time of five spherical potato samples with radii from 16 to 30 mm.
In a course in which students are not asked to fully understand the solution to the diffusion equation, the data shown in Fig. 4 may not be very meaningful. We suggest two approaches for these students. The first would be to ask them to consider the time required for the center to reach a particular temperature: We plot the time required to reach 75 °C in Fig. 5. The dashed curve is a bestfit quadratic relationship. Depending on the amount of time available to the class, this plot could be created by individual lab groups, or it could be created by sharing data for a single radius from each lab group among the entire class. If desired, students could then apply Eq. (5) to find the thermal diffusivity, α, which for these data is α = 1.47 × 10^{−7} m^{2}/s.
The second approach further confirms that the heating time has a quadratic relationship with the radius of the sample. In Fig. 6, we normalize the time (dividing by the square of the radius) and replot three heating curves for samples with radii of 16, 23, and 30 mm, respectively, vs the normalized time. This process of collapsing a dataset onto a single curve to verify an equation is an important tool of experimental physics that is seldom illustrated in the introductory labs, and that students may find very convincing.
A careful look at the fit of the heating curves reveals some small but significant discrepancies. Figure 7 shows the measurement result (dots) for a potato sample with a radius of 23 mm. The two theoretical curves use different thermal diffusivities: For the solid curve, α = 1.32 × 10^{−7} m^{2}/s, and for the dashed curve, α = 1.42 × 10^{−7} m^{2}/s. The solid curve fits the measured data almost perfectly for t > 400 s. However, the dashed curve fits the experimental data better during the initial phase (0–250 s). This discrepancy may be due to the temperature dependence of thermal diffusivity, which is not included in our model. We also observe that the potatoes change their physical properties as they are heated (they “cook”), and that, where the literature includes separate thermal diffusivities for cooked and uncooked potatoes,^{5} the diffusivities are lower for cooked potatoes, consistent with our results.
More measurement results and simulations for other samples, including several other types of foods, can be found in the supplementary material.^{15}
V. RESULTS AND ANALYSIS
Table I summarizes all the results of the measured thermal diffusivities for the nine different types of food. For each sample, we provide both a lowend value and a highend value, since, as discussed in Sec. IV, the data are not perfectly fit by the use of a single thermal diffusivity. We summarize these data and compare to values from the literature in Table I.
Food .  Diameter (mm) .  Thermal diffusivity (10^{−7} m^{2}/s) measured in this work .  Thermal diffusivity (10^{−7} m^{2}/s) from the literature .  

Low end value .  High end value .  
Potato  45  1.32  1.42  1.30^{3} (The log method), 1.44^{3} (the slope method), 1.70^{4} (average value), 1.23^{5} (cooked, mashed), 1.70^{5} (whole), 1.3^{6–8} 
Potato  45  1.32  1.42  
Potato  50  1.32  1.42  
Potato  40  1.32  1.42  
Potato  32  1.32  1.48  
Potato  60  1.32  1.50  
Potato  42  1.32  1.48  
Potato  46  1.32  1.50  
Potato  53  1.32  1.50  
Potato (measured the second time)  51  1.52  1.60  
Pumpkin  50  1.50  1.66  ⋯ 
Pumpkin  35  1.50  1.72  
Sweet potato  51  1.66  1.84  1.20^{9} 
Sweet potato  46  1.66  1.75  
Taro  52  1.50  1.60  ⋯ 
Taro  53  1.40  1.50  
Radish  40  1.30  1.40  1.869^{10} 
Radish  41  1.55  1.65  
Onion  63  1.60  1.78  1.1–1.5^{11} 
Eggplant  47  2.20  5.00  
Lemon  52  1.50  1.70  1.16–1.785^{12} (Lemon juice) 
Tomato  50  1.40  1.60  1.42^{13} (Tomato paste) 
Food .  Diameter (mm) .  Thermal diffusivity (10^{−7} m^{2}/s) measured in this work .  Thermal diffusivity (10^{−7} m^{2}/s) from the literature .  

Low end value .  High end value .  
Potato  45  1.32  1.42  1.30^{3} (The log method), 1.44^{3} (the slope method), 1.70^{4} (average value), 1.23^{5} (cooked, mashed), 1.70^{5} (whole), 1.3^{6–8} 
Potato  45  1.32  1.42  
Potato  50  1.32  1.42  
Potato  40  1.32  1.42  
Potato  32  1.32  1.48  
Potato  60  1.32  1.50  
Potato  42  1.32  1.48  
Potato  46  1.32  1.50  
Potato  53  1.32  1.50  
Potato (measured the second time)  51  1.52  1.60  
Pumpkin  50  1.50  1.66  ⋯ 
Pumpkin  35  1.50  1.72  
Sweet potato  51  1.66  1.84  1.20^{9} 
Sweet potato  46  1.66  1.75  
Taro  52  1.50  1.60  ⋯ 
Taro  53  1.40  1.50  
Radish  40  1.30  1.40  1.869^{10} 
Radish  41  1.55  1.65  
Onion  63  1.60  1.78  1.1–1.5^{11} 
Eggplant  47  2.20  5.00  
Lemon  52  1.50  1.70  1.16–1.785^{12} (Lemon juice) 
Tomato  50  1.40  1.60  1.42^{13} (Tomato paste) 
We observe that our values are in good agreement with those in the literature,^{3–14} especially given the range of published values, which may reflect both variations within the food categories and also measurement uncertainties.
Some additional sources of measurement uncertainty include:

Due to the sample's nonideal spherical shape and material's nonrigid nature, there is an error in determining the sample's diameter. The deviation from a perfect spherical shape introduces an additional error. We try to reduce the error by measuring the diameter with randomly picked orientations and then using the average diameter value from the ten measurements. Furthermore, we also measure the weight of each sample and then calculate the density using the diameter and the weight. For the same material, the calculated densities between different samples vary by less than 4%, so we suspect that the variations from spherical shape are not the primary source of variations between measurements of thermal diffusivity.

Our method requires measuring the temperature in the center of the spherical sample. There is an error in positioning the thermocouple sensing tip. Furthermore, the thermocouple sensor tip has a size of about 1 mm. The temperature it measures is an average in a small volume, which also introduces an additional error.

Since the sample is immersed in boiling water during the measurement, the thermocouple sensor tip might move in position during the heating process. Such a problem might cause some temperature data irregularity, particularly towards the later part of the heating process, as indicated in Figs. S2 and S3 of the supplementary material.^{15}
VI. SUMMARY
Nine different types of foods, including potato, sweet potato, pumpkin, taro, radish, eggplant, lemon, tomato, and onion, were measured, and their thermal diffusivities were experimentally determined. We cut the foods into spherical shapes, inserted small and thin thermocouple sensors into their centers, and then immersed the samples in the boiling water. The center temperature was recorded through the heating process, and then we compared the heating curve as the function of time with the simulation results, where the thermal diffusivity is used as the fitting parameter. This method allows us to intentionally vary the diameter of the spheres, i.e., adding another variable in order to validate the results. We are able to determine all the thermal diffusivity data with a good matching between the measurement data and the simulation results. This method can be generalized to determine thermal diffusivity of a wide variety of samples.
ACKNOWLEDGMENTS
The authors (L. R. Wang and Y. Jin) would like to thank both The Pennington School and Phillips Academy Andover for their support throughout this study. The authors thank the reviewers for their valuable comments and suggestions throughout the review process.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.