Heat conduction in potato cubes: an experiment to estimate thermal diffusivity using arduino

Abstract

This work presents a low-cost experiment to investigate heat conduction inside potato cubes, using thermistors as a thermometric instrument and Arduino as a data acquisition platform. In the theoretical framework, a heuristic argument was proposed to switch the 3D heat equation to an equivalent and simpler 1D equation, which is valid only along points on the cube’s symmetry axis. In the experimental domain, an empirical and realistic time-dependent temperature function was used as boundary conditions instead of assuming simple boundary conditions, i.e., the temperature of the cube surface in contact with the air is kept constant. An approximate solution of the one-dimensional heat equation with nonhomogeneous and time-dependent empirical and actual boundary conditions was provided. Adjusting the experimental data to the theoretical solution predicted by the heat equation provides the thermal diffusivity of the potato. The thermal diffusivity obtained agrees with the reference values found in the literature.

Keywords:
Arduino; heat equation; physics teaching; thermal diffusivity

1. Introduction

Thermal diffusivity is the fundamental quantity in the heat conduction analysis when the temperature inside the conductive material is time-dependent. The homogeneous body’s thermal diffusivity is defined by the ratio between the thermal conductivity and the specific volumetric heat, which is the product of the density and the specific heat of the material [¹]. Thermal diffusivity can be understood as the ratio between the material’s ability to conduct thermal energy and its resistance to temperature variation. It represents an inverse measure of the material’s thermal inertia.

In 2003, a very enlightening article was published on the physical meaning of the physical quantities related to heat propagation by conduction [²]. This work masterfully explains the meaning of thermal conductivity, thermal diffusivity, and thermal effusivity. The last two are generally little known and misunderstood among physics students and instructors. Indeed, it is usually said that a material placed in contact with a hot/cold reservoir gets warm/cold quickly because it is a good thermal conductor, but actually, the reason is that it is a good diffuser. In short, thermal conductivity is the only property required when temperature distribution along the body does not vary over time (steady problems). However, the complete characterization of time-varying temperature distribution involves the knowledge of two properties: thermal diffusivity and thermal effusivity [²]. Inspired by this article, we can understand the following conceptual issues.

Consider two solid and homogeneous cylinders of the same mass and cross-sectional area, wrapped in thermal insulators, except by their ends. The first cylinder is composed of iron, and the second is made of lead. The values of the three thermal parameters associated with heat conduction for iron and lead are in Table 1.

Table 1
Three thermal parameters associated with heat conduction for iron and lead.

	Conductivity (W/m.K)	Diffusivity (m2/s)	Effusivity (J/m2.K.s${}^{-0.5}$)
Iron	71.97	$20.4\times {10}^{-5}$	15924
Lead	34.31	$23.3\times {10}^{-5}$	7107

Naturally, since the density of lead ($11.35\times {10}^3$ kg/m³) is 1.44 times greater than the density of iron ($7.87\times {10}^3$ kg/m³), the length of the iron cylinder is 1.44 times greater than the length of the lead cylinder.

Question 1: If the ends of the two cylinders are kept in contact with two thermal reservoirs of different temperatures, choosing any fixed inner point in each cylinder, in which cylinder the heat flux at that point is greater?

Question 2: If cylinders are taken out of a refrigerator at the same temperature and their ends are placed in thermal contact with the same thermal reservoir (higher air temperature, for example) and, before reaching thermal equilibrium, the temperatures were simultaneously measured at an inner point located at the same distances from the reservoir to each cylinder, in which cylinder the temperature of this point is higher?

Question 3: If cylinders are taken out of a refrigerator at the same temperature, and you immediately touch the ends of both cylinders with your hands, which one will you “feel” colder as it will conduct a greater flow of heat from your hands?

The answer to question 1 is the iron cylinder, whose thermal conductivity is about twice as high. Thermal conductivity measures the flow of heat per unit of time through a unit area due to the temperature difference between two adjacent layers of material. It is important to emphasize that thermal conductivity is defined for a stationary situation when the temperature inside the conducting cylinder has a gradient along its length, but the temperature at each point has already stabilized and does not vary with time [²].

On the other hand, the answer to question 2 is lead because its thermal diffusion is slightly higher. The heat flux per unit of time in the iron cylinder is higher than in lead because iron has higher thermal conductivity. Yet, the temperature in iron varies more slowly than in lead. Although the density of lead is 1.44 times that of iron, the specific heat of iron is 3.45 times that of lead. Therefore, the volumetric specific heat of iron is 2.4 times greater, which exceeds its thermal conductivity, which is only two times greater, making its thermal diffusivity smaller than that of lead.

The answer to question 3 is iron because its thermal effusivity is higher than lead. Thermal effusivity measures the ability of the material to exchange heat through an interface. It can be calculated by the square root of the product between the volumetric specific heat and the thermal conductivity [²]. As the volumetric specific heat of solids has close values [²], generally, the material with the highest thermal conductivity will also have the highest thermal effusivity, so using these two concepts as if they were synonyms is common.

The study of the thermophysical properties of materials is of great importance for the research and design of thermal processes in general. Its applicability is widespread in engineering. Specifically, the knowledge of these properties by the food industry, particularly thermal diffusivity, makes it possible to optimize processes such as cooking, drying, and freezing of food.

Experimental methods for measuring the thermal diffusivity of food were reviewed by Dickerson [³] and Singh [⁴] based on the analysis of the behavior of a material with a usually cylindrical geometry when exposed to a temperature gradient. The choice of a cylindrical shape is not a practical convenience but a mathematical one because the solution of the three-dimensional heat equation becomes simpler in an object with cylindrical symmetry.

Several papers are on the thermal diffusivity of vegetables in food science and food engineering. For instance, Magee and Bransburg filled a chrome-plated brass cylinder with boiled mashed potatoes, inserting thermocouples in the center and its edge. Using both the slope method¹ and the log method², they obtained the solution of the one-dimensional heat conduction equation and calculated the thermal diffusivity of the material [⁵]. In turn, Murakami molded potatoes and carrots into cylinders and directly measured their density and thermal conductivity using the line-heat source technique³. The specific heat and thermal diffusivity were calculated using these values without solving the heat conduction equation [⁶]. Vidaurre-Ruiz and Salas-Valerio cut raw potatoes into cubes with different edges and inserted K-type thermocouples in the center and edges. Using the explicit finite difference method⁴, they obtained the thermal diffusivity concerning the temperature increase with a second-degree polynomial function [⁷]. Kumar et al. cut raw potatoes into cylindrical shapes and inserted K-type thermocouples in the center and edges. The thermal conductivity and diffusivity of raw and blanched/par-fried potatoes were measured directly with a handheld conductivity meter using the dual probe method⁵ [⁸].

Recently, in a physics teaching context, Hanisch and Ziese presented a simple experiment on thermal conduction. Cylindrical cross sections of heated carrots were cut, and their temperature profiles were measured with an infrared (IR) camera. They solved the heat conduction equation in cylindrical coordinates and thus calculated the thermal diffusivity of the carrot [⁹].

Cylindrical geometry is predominant because it substantially simplifies the heat equation. Still, a few papers explore heat conduction in foods with cubic geometry, such as potato cubes [⁷] and cheese cubes [¹⁰]. In these articles, it is shown that the temperature at different points located next to the faces of the cube are different. The vertices are the points of the highest temperature because they are exposed to the heat transfer from three sides [¹⁰]. However, the function that provides the temperature distribution on each face and in straight sections of the cube equidistant from the center is symmetric [⁷].

We propose a simple and low-cost experiment to investigate the heat conduction inside potato cubes (Solanum Tuberosum L.) with different dimensions exposed to air. The main original contribution of this work lies in two aspects, one of a theoretical nature and the other experimental. In the theoretical framework, we propose a heuristic argument to switch the three-dimensional heat equation to a one-dimensional one, allowing an approximated solution only along points on the cube’s symmetry axis. This procedure substantially simplifies the resolution of the heat equation for non-trivial boundary conditions.

In the experimental domain, instead of assuming that the surface temperature instantaneously corresponds to the temperature of the thermal bath in which the body is immersed [⁵, ⁶, ⁷], it was used an empirical and realistic function of temperature-dependent on time as boundary conditions. Although the air in the environment is a thermal reservoir, the air in contact with an object has no fixed temperature. Then, thermistors were used to measure the temperature of the geometric center of the potato and an external point very close to the body’s surface, at the center of one of the surfaces. This procedure provided actual boundary conditions to the heat equation for the potato cubes.

Another original technical aspect was using Arduino as a data acquisition platform due to the accessibility and versatility of its programming language [¹¹]. Fitting the experimental data to the approximate solution predicted by the heat equation provides the thermal diffusivity of the cubic potato with nonhomogeneous and time-dependent boundary conditions.

2. Theoretical Model

The heat equation in three dimensions, $T(x,y,z,t)$, is:

(1) $$\frac{\partial T}{\partial t}=\alpha \left[\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right]$$

being $\alpha$ the thermal diffusivity of the solid.

If the initial temperature distribution of the whole body is a constant value T₀, immersed in an ideal thermal reservoir (very high thermal conductivity) of temperature equal to zero, then the solution to the heat equation with simple homogeneous Dirichlet boundary conditions is [¹²]:

(1a) $$\begin{array}{l}T\left(x,y,z,t\right)={\displaystyle \sum _{m,n,p=1}^{\infty }{b}_{m,n,p}{e}^{-\alpha \left[{\left(\frac{m\pi }{L}\right)}^2+{\left(\frac{n\pi }{L}\right)}^2+{\left(\frac{p\pi }{L}\right)}^2\right]t}}\\ .\sin \left(\frac{m\pi x}{L}\right)\sin \left(\frac{n\pi y}{L}\right)\sin \left(\frac{p\pi z}{L}\right),\end{array}$$

where

(1b) $$b_{m,n,p}=\frac{8T_0}{L^3}{\displaystyle \int {\displaystyle \int {\displaystyle {\int }_0^L\sin \left(\frac{m\pi x}{L}\right)}\sin \left(\frac{n\pi y}{L}\right)}\sin \left(\frac{p\pi z}{L}\right)}dxdydz$$

Note that for points that belong to the symmetry axes of the cube, for example, $y=z=L/2$, the terms of the Fourier series in which m, n, or p are even, are null. Therefore, the second non-zero term of the specific solution on the symmetry axes corresponds to the value n = 3, m = 1, and p = 1. It entails that the second non-zero term of the series is significantly smaller than the first term due to the large increase in the negative exponent of the exponential decay in time.

If we consider only the first term of the Fourier series and calculate its value only at points on one of the symmetry axes (x axis), taking $y=z=L/2$, the approximate specific solution is:

(1c) $$T(x,\frac{L}{2},\frac{L}{2},t)=\frac{64T_0}{{\pi }^3}{e}^{-3\alpha {\left(\frac{\pi }{L}\right)}^{2}t}\sin \left(\frac{\pi x}{L}\right)$$

Note that, short of a multiplicative constant, this last specific approximate solution to the 3D heat equation has the same functional form as the first-term solution to a one-dimensional heat equation, but with three times the thermal diffusivity value. We can interpret the multiplying factor “3” on the thermal diffusivity due to the cubic symmetry. The heat flow in one dimension corresponds to “1/3” of the total flux in three dimensions.

Therefore, considering the following assumptions: (i) the actual boundary conditions are time-dependents and symmetric on all faces of the cube [⁷], and (ii) we are looking for an approximate solution to the three-dimensional heat equation, that is, only the first term of the Fourier series, along one of the symmetry axes of the cube, it is enough to solve the one-dimensional heat equation, but with three times the value of the thermal conductivity.

Thus, using this heuristic argument, Eq. (1) can be simplified as the following heat equation of one dimension:

(2) $$\frac{\partial T}{\partial t}=3\alpha \frac{{\partial }^{2}T}{\partial x^2},$$

and the initial condition is

(3) $$T(x,t_0)=T_{c0},$$

where T_c0 is the temperature of the center of the potato at t₀.

Although the air in the potato’s environment can be considered a thermal reservoir, experience reveals that the air in contact with the potato does not have a fixed temperature. This result can be explained by the fact that air, even a poor conductor of heat, is a good diffuser, and its thermal diffusivity is similar to that of various metals. Firstly, the air layer close to the surface of the potato undergoes rapid heating from the heat exchange with the potato and reaches a higher temperature than the ambient air. As time passes, the heated air layer surrounding the potato loses thermal energy by conduction, convection, and radiation. Although these three processes of heat transfer are pretty different from each other, it was shown that Newton’s law of cooling is mostly valid if temperature differences are below a certain threshold, which varies from 50K to 200K [¹³].

In order to obtain empirical and more realistic boundary conditions for the heat equation, thermistors were used to measure the temperature of the geometric center of the potato and an external point very close to the body’s surface, at the center of one of the surfaces. This procedure provided experimental data to fit a time-dependent temperature function as boundary conditions. Indeed, the sensor placed in the air close to the potato’s surface revealed that its temperature pattern resembles an exponential decay.

Therefore, to describe the temperature decay of the surface of a cubic potato of side L, we assume the following time-dependent boundary conditions:

(4) $$T_s(t)=T_{air}+T_{s0}{e}^{-a(t-t_0)}$$

being t₀ the instant of time which the measure began, $T_{air}+T_{s0}$ the surface temperature in t₀, T_air the air temperature in the environment, and a the potato surface temperature decay rate.

In Appendix A, the following solution of Eq. (2) with the initial condition (3) and time-dependent boundaries conditions imposed by Eq. (4) is calculated:

(5) $$\begin{array}{l}T(x,t)=T_{air}+T_{s0}{e}^{-a(t-t_0)}\\ +{\displaystyle \sum _{n=1,3,\mathrm{5...}}^{\infty }\{\frac{\left(\frac{4}{n\pi }\right)a{T}_{s0}}{\left[3\alpha {\left(\frac{n\pi }{L}\right)}^2-a\right]}{e}^{-a(t-t_0)}}\\ +\left(\frac{4}{n\pi }\right)\left[T_{c0}-T_{s0}-T_{air}-\frac{a{T}_{s0}}{\left[3\alpha {\left(\frac{n\pi }{L}\right)}^2-a\right]}\right]\\ \cdot e^{-3\alpha {\left(\frac{n\pi }{L}\right)}^2(t-t_0)}\}\sin \left(\frac{n\pi x}{L}\right)\end{array}$$

The position x varies from 0 to L, which corresponds to the length of the side of the potato cube and T_c0 corresponds to the temperature of the center of the potato at t₀.

Considering only two terms of the sum, n = 1 and n = 3, measuring the temperature decay at the center of the potato ($x=L/2$, considering the coordinate system of the developed theoretical model) as a function of time, $T_c(t)$, Eq. (5) becomes:

(6) $$\begin{array}{l}{T}_c(t)=T_{air}+T_{s0}{e}^{-\alpha (t-t_0)}+\frac{\left(\frac{4}{\pi }\right)a{T}_{s0}}{\left[3\alpha {\left(\frac{\pi }{L}\right)}^2-a\right]}{e}^{-a(t-t_0)}\\ -\frac{\left(\frac{4}{3\pi }\right)a{T}_{s0}}{\left[3\alpha {\left(\frac{3\pi }{L}\right)}^2-a\right]}{e}^{-a(t-t_0)}\\ +\left(\frac{4}{\pi }\right)\left[T_{c0}-T_{s0}-T_{air}-\frac{a{T}_{s0}}{\left[3\alpha {\left(\frac{\pi }{L}\right)}^2-a\right]}\right]\\ \cdot e^{-3\alpha {\left(\frac{\pi }{L}\right)}^2(t-t_0)}\\ -\left(\frac{4}{3\pi }\right)\left[T_{c0}-T_{s0}-T_{air}-\frac{a{T}_{s0}}{\left[3\alpha {\left(\frac{3\pi }{L}\right)}^2-a\right]}\right]\\ \mathrm{}\cdot e^{-3\alpha {\left(\frac{\pi }{L}\right)}^2(t-t_0)}\end{array}$$

By fitting the temperature surface decay given by Eq. (4), we can obtain the parameters t₀, T_s0, T_air, and a (decay rate of the potato surface temperature). These values are assumed constants and inserted into Eq. (6). Only the temperature of the center of the potato at t₀ (T_c0) and the factor $\alpha /L^2$ are free parameters to be adjusted in the graphs of temperature drop. This procedure provides a reasonable estimate of the thermal diffusivity of the potato.

3. Experimental Setup

The potato cubes were prepared in the following dimensions: cubes of 55, 50, 45, 40, and 35 mm, and the final adjustments were made using a mandolin with a precision of 1 mm.

An axial hole slightly smaller than the thermistor diameter was made in the geometric center of one of the surfaces to create an internal pressure of the potato under the thermistor, thus decreasing its relative movement concerning the geometric center of the cube during the heating and cooling processes. The thermistor was insulated (see the first section of Appendix B) and positioned according to the mark made on the surface of the potato (Fig. 1 and Fig. 2). Detailed information regarding thermistor insulation, characterization of the acquisition system and its associated uncertainty can be found in Appendix B and references [¹⁴, ¹⁵, ¹⁶].

Figure 1
2D representation of the position of the thermistors (orange spheres) in the potato cube.

The cube was placed on a ceramic dish isolated from the bottom of the pan by pegs to avoid overheating the bottom surface of the cube. The cube was entirely immersed in a pan filled with water and slowly heated (Fig. 2). The temperature of the thermal bath was monitored and maintained at around 70°C by manual control of the stove.

Figure 2
Heating of the 40 mm side potato cube. The cube was entirely immersed in water and slowly heated.

The potato cube was removed from the bath when the temperature of its geometric center reached 70°C. Soon after, the cube was rested on a suspended wooden stick base to minimize heat loss from the potato to the support and allow air circulation along the supported surface (Fig. 3). The second thermistor was positioned so that its geometric center aligned with the potato’s surface (Fig. 1).

Figure 3
Cooling the potato cube on the 40 mm side.

Temperature data were collected until the difference between the core and surface temperature of the potato reached zero. This process was done for potato cubes of 35-, 40-, 45-, 50- and 55-mm sides (Fig. 4).

Figure 4
Potatoes cubes with 55-, 50-, 45-, 40-, and 35-mm sides are used in this work.

4. Analysis of Results

All analyses started when the center of the cube’s temperature reached 65°C to avoid the initial turbulent air movement due to the water evaporation on the potato’s surface.

Figure 5
Cooling curves and their fits according to the theoretical models on the surface (left column) and in the center (right column) of the potato cubes of 30-, 35-, 45-, 50- and 55-mm sides.

Figure 5 (graphs on the left column) shows the fitting of the temperature surface decay according to Eq. (3) and the obtained parameters t₀, T_s0, T_air, and a (decay rate of the potato surface temperature). Using those values as constants to be inserted in the theoretical model presented in Eq. (6), leaving only T_c0 (potato center temperature at t₀) and the factor $\alpha /L^2$ as free parameters to adjust the temperature decay at the center of the potato. Fig. 5 (graphs on the right column) presents the theoretical model fitting and the obtained parameters.

Table 2 shows the estimated values of thermal diffusivity and its respective uncertainties inside the brackets, obtained from the fitting of the theoretical model over the experimental data for each potato cube.

It is essential to briefly explain why the factor $\alpha /L^2$ is chosen instead of the parameter $\alpha$. If only the parameter $\alpha$ remained free in the data fit, this would imply that the length L should be input without any uncertainty, which is far from reality. The length L is the most significant source of uncertainty in the experiment, and its uncertainty estimated is around 2 mm, which considers the size of the thermocouples, as they cannot be regarded as a single point and cutting process of potatoes into cubes.

Table 2
The factor $\alpha /L^2$ and its estimated uncertainty (inside the brackets) obtained from the theoretical model fit. The thermal diffusivity and its estimated uncertainty are in the last column.

$\alpha /L^2(\times {10}^{-5}{s}^{-1})$	$L(\times {10}^{-3}$ m)	$\alpha (\times {10}^{-7}{m}^2/s)$
11.05(2)	35(2)	1.4(2)
8.30(1)	40(2)	1.3(1)
5.878(6)	45(2)	1.2(1)
4.918(5)	50(2)	1.2(1)
4.503(4)	55(2)	1.4(1)

The mean value obtained for the thermal diffusivity is $1.30(6)\times {10}^{-7}{m}^2/s$, which is comparable to the values presented in previous works as $1.39(9)\times {10}^{-7}{m}^2/s$ [⁶], $1.43(3)\times {10}^{-7}{m}^2/s$ [⁷] and $1.5(1)\times {10}^{-7}{m}^2/s$ [⁸].

5. Conclusions

The approach to analyzing the cooling of potato cubes through an approximate solution of the heat equation, with empirical and realistic time-dependent boundary conditions, yielded a slightly lower value for thermal diffusivity, although in good agreement with previous results from the literature in food science and technology.

However, an evident limitation of this model comes from the approximation that considered only the first term of the Fourier series. Since all even terms in the expansion are zero, the next non-zero term that was neglected corresponds to values n = 3, m = 1, and p = 1. The contribution of this term to the solution is much smaller, but it has a positive value. Then, the actual theoretical curve will be above the approximate theoretical curve obtained using only the first term of the Fourier series. Therefore, in the temperature-time graph, to raise the approximate theoretical curve so that its fitting coincides with the experimental points, it is necessary to slightly decrease the value of thermal diffusivity provided by the model in comparison with its actual value, since thermal diffusivity it is part of the argument of the temporal negative exponential.

In the context of teaching physics, this experiment can be implemented in laboratory classes, as it uses cheap and easily acquired temperature sensors controlled by an Arduino interface. Discussion with students about the meaning of both thermal conductivity and thermal diffusivity provides a better understanding of these concepts. It contributes to clarifying misconceptions about the process of heat conduction.

Supplementary Material

The following online material is available for this article:

Appendix A Heat equation solution.

Appendix B Acquisition system, calibration, and uncertainties.

Appendix C Arduino Code.

References

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• 1
  The slope method consists on analysing the centre temperature of the cylinder while heated on a constant temperature water bath [⁵].
• 2
  The log method consists on analysing the centre temperature of the cylinder while heated on linearly increased temperature water bath [⁵].
• 3
  The line-heat source technique is used to measure thermal conductivity by externaly heating a cylindrical sample on a constant rate while observing the rise of its centre temperature by a cylindrical sensor [⁶].
• 4
  The explicit finite difference method is a mathematical model based on Fourier’s law and Fick’s second law to calculate numerically the 3D temperature inside the body, in rectangular coordinates [⁷].
• 5
  The dual probe method consists of dual needles at a finite distance from each other. One applies heat from a certain period of time, while the other measures temperature during heating and cooling periods [⁸].

Publication Dates

• Publication in this collection
  16 Aug 2024
• Date of issue
  2024

History

• Received
  20 May 2024
• Reviewed
  09 July 2024
• Accepted
  10 July 2024