Heat Transfer in Granular Material: Experimental Measurements and Parameters Identification of Macroscopic Heat Conduction Model

Abstract

The paper presents experimental results regarding heat transfer in granular materials in the cylindrical domain during heating by the outer surface of the container. Sensors (K-type thermocouples) were used to measure the temperature changes at several points inside granular material (the fine-grained table salt was used in the experiment). Knowledge of measurement data allows the verification of a mathematical model (based on Fourier’s law) to describe the macroscopic heat conduction in granular materials. An iterative algorithm for the inverse heat conduction problem consisting of the estimation of the thermal diffusivity coefficient of granular material, the parameters of initial boundary conditions and the position of the thermocouple tips during the experiment was developed. Several computational simulations were performed. Based on the experimental results and the computational simulation results, one can conclude that the analytical solution of the direct heat conduction problem calculated for the optimal values obtained from the inverse heat conduction problem gave us the confirmation of the validity of Fourier’s model.

1. Introduction

In modern technology, many measurements of physical quantities are performed using various technologies [1]. One can note that measurements have become an indispensable part of virtually every field of technology. The need for measurement arises from the necessity of comparing certain values of physical quantities with certain reference values, for example when controlling the quality of production, monitoring a control system or the course of a specific physical phenomenon. One of the more popular measurements is temperature measurement. Depending on the specifics of the phenomenon under consideration, various devices, such as thermometers, thermocouples, resistance temperature sensors, pyrometers or thermal imaging cameras [2,3,4], can be used for this purpose. In many cases, the selection of a specific metrology technology is determined primarily by the specifics of the phenomenon or process under study, and sometimes the knowledge and applicability of a particular measurement technology [5].

For temperature measurements carried out in industry [2,5], thermocouples or resistance temperature sensors are mainly used. The use of both technologies is determined by several factors, including accuracy, measurement range or ease of application. Thermocouples have numerous applications even in extreme conditions, such as measurements in chemically aggressive environments. Due to the widespread use of thermocouples, in this work, we will deal with measurements carried out precisely with their use. A typical thermocouple has a measuring junction made of various materials (e.g., chrome/alumina for the type K thermocouple). If two different metals are connected and one of the connections is heated, a voltage proportional to the temperature difference is generated. This voltage is measured and converted to a temperature value. Nevertheless, it is important to realize that every measurement, and regardless of the technology used, results in so-called measurement errors [1,2]. These errors, of course, are not intentional; they arise as a result of still imperfect measurement techniques and can be defined as the inconsistency of the test result with the actual value of the measured quantity.

The measurements considered in this work concern temperature changes inside granular materials. Such materials are characterized by certain grain structures and physical–mechanical properties that affect the results of at least temperature measurements. In the case of such materials, there are a number of problems related to the determination of their properties, for example, bulk density, thermal conductivity or heat capacity. The specific values of these parameters depend on many factors, including grain size and structure, moisture content, degree of compaction, temperature or pressure at which tests are performed. Among the recent works on this topic are the following articles: [6,7,8,9,10]. Descriptions of different methods of measuring temperature in granular materials can be found, e.g., in the works [11,12,13].

Heat conduction in the granular material is the process of transferring heat from one part of the material to another and can be described by various mathematical models. The most popular model for describing macroscopic problems is the Fourier heat conduction model [14,15]. Fourier’s law of heat conduction is characterized by an infinite speed of propagation of thermal signals, which are treated by many researchers as non-physical behaviour. The non-Fourier models (for example, the Cattaneo model, the dual phase lag model, different versions of fractional models) [16,17,18,19,20] differ in their behaviour in terms of the finite velocity of heat propagation. These models introduce one or two parameters, known as the phase lags or the delay times. Recently (since 1990, after the publication of the work [21]), there has been an ongoing discussion on the validity of using specific models to describe heat flow in granular materials. Kaminski in [21] reported on the basis of the conducted experiment that, e.g., the relaxation times are equal to about 20 s for sand and 29 s for sodium bicarbonate in the Cattaneo model. However, e.g., in two later independent works [22,23], the authors tried to reproduce the experiment conducted by Kaminski, but finally they stated in the conclusion that there are doubts about the application of the Cattaneo model to describe heat conduction in materials with the non-homogeneous inner structure in the considered parameter ranges. These above-mentioned remarks motivated us to conduct similar experimental studies and make measurements to confirm the observations in the work [23], where the authors have used a cylindrically symmetric system filled with granular material heated by an electrical heat source along the axis. Our experiment differs in that the system is heated by the outer surface; details are presented in the next Section.

Each mathematical model of heat conduction can be described by the appropriate partial differential equation (or system of such equations) along with geometric, physical, boundary and initial conditions [14,15,19,24]. Solutions to such direct heat conduction problems require determining appropriate boundary conditions at the boundaries of the considered domain, as well as knowledge of the initial condition and the assumption of all thermophysical parameters occurring in the model. These parameters include the specific heat capacity (that determines the quantity of heat that must be added to one unit of mass of the material in order to raise its temperature by one unit), the thermal conductivity (that indicates the ability of the material to conduct heat, so-called an intensive property) and density. All these quantities can be constants or determined by functional relationships. A solution to the direct heat conduction problem is the spatiotemporal temperature field in the considered domain. The inverse heat conduction problems [2,24,25] concern, among others, temperature measurements in order to estimate unknown parameter values appearing in the mathematical formulation of problems. In the inverse problems, the unknown parameters of the adopted mathematical model are determined on the basis of discrete-in-time temperature measurements (both on the surface and inside the considered domain) using sensors. This approach also allows the validation of experimental results with theoretical ones. Due to the fact that inverse problems are treated as ill-posed problems [26], the solutions may be unstable. It should also be taken into account here that the measured temperature data used in the analysis of the inverse problems includes measurement errors. The estimated parameter values in the inverse analysis are not exact and should be treated with the associated uncertainties.

In this research, in order to find the solution to the inverse problem, we use minimization algorithms that are also used in many optimization problems [27,28,29,30]; the details are presented in Section 3. The problem of identification of unknown thermophysical parameters of materials can also be solved, e.g., using the sensitivity analysis methods [31], the evolutionary algorithm [32] or the Monte Carlo simulation [33]. The essence of sensitivity analysis is to show the impact of different values of an independent variable on the dependent variable taking into account a given set of assumptions. Often, the inverse heat conduction problem is solved using the criterion of the least squares by applying the sensitivity coefficients. In recent times, artificial intelligence (AI) algorithms have been increasingly used. Baghbani et al. [34], have considered various mathematical models (including AI algorithms) to predict the thermal conductivity of the quartz sand, while, in the [35], AI is used to reconstruct the heat transfer coefficient during the heat conduction in the solidification process.

2. Description of the Measurement System

We present a system used to measure temperature changes inside granular material during its heating. In the thin-walled metal cylindrical can filled with granular material, several thermocouples were placed at different positions of the radius of the can. Such a system, being at a predetermined ambient temperature (so-called room temperature) at the initial moment, was placed rapidly (i.e., rapidly immersed within about 1 s to about 80% of the can’s height) in boiling water. Water was continuously heated by the heater when the measurements were performed in order to allow the assumption of almost constant temperature (as the boundary condition of the first kind on the outer surface of the considered domain in a mathematical model), i.e., about 100 °C (this temperature depends on atmospheric pressure, or dissolved substances in the water and other factors). The choice of boiling water as the external heating medium ensures, in general, ideal thermal contact at the interface between the metal sheet of the can and water. Moreover, the boiling water ensures very well mixing of the water, providing a uniform temperature in the entire volume. The volume of boiling water in the container (pot) should be much larger (e.g., about 20 times) than the volume of the metal can with granular material so that the temperature of the water does not drop significantly just after the can is immersed. The thermal diffusivity coefficient for sheet metal (e.g., for aluminium is about $9.7\times {10}^{-5}$ m²/s), which is approx. 300 times greater than, for example, for fine-grained table salt. For thin-walled sheet metal (e.g., 0.2 mm), the heat flow through the wall is very fast and, in principle, after about 0.02 s the other side of the sheet metal wall under adiabatic conditions reaches the temperature of the heated side. If the temperature measurements are registered with an accuracy of ±1 s, it is of negligible importance, and in this case, it becomes justifiable to omit the sheet in the thermal analysis. Measuring the temperature inside the granular material using thermocouples seems to be justified in this case.

Figure 1 shows a schematic diagram of the measurement system. In addition to the thermocouples placed inside the granular material, there is one sensor placed outside the can to record the moment of placing the container in boiling water. All thermocouples are connected to the eight-channel thermocouple data logger (the Pico Technology TC-08 data logger: (access date: 20 January 2025) https://www.picotech.com/data-logger/tc-08/thermocouple-data-logger), which has rated accuracy of the sum of ±0.2% and ±0.5 °C. The data logger is connected to the PC using the USB wire. Moreover, approximately 10 mm thick cardboard was placed in the lower and upper parts of the can to provide thermal insulation of heat conduction through the bases of the cylindrical container.

3. Mathematical Models

In this section, two mathematical models of the direct and inverse heat transfer problems are considered.

3.1. Direct Heat Transfer Problem

On the basis of the assumptions presented in the previous section, the macroscopic mathematical model of the heat transfer problem in granular material, based on Fourier’s law, as the one-dimensional axisymmetric task in the circle (the cross-section of the cylinder) of radius R [m] (ignoring the thin-walled metal sheet of the can) can be formulated in the form [14,15]

$$c\rho {\displaystyle \frac{\partial T\left(r,t\right)}{\partial t}}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(\lambda r{\displaystyle \frac{\partial T\left(r,t\right)}{\partial r}}\right),0<r<R,t>0,$$

(1)

where T denotes temperature [°C], t and r denote time [s] and the radial coordinate [m], c, $\rho$ and $\lambda$ are the specific heat capacity [J/(kg °C)], density [kg/m³] and thermal conductivity [W/(m °C)] of the granular material, respectively. Equation (1) is supplemented by the following initial boundary conditions

$${\left.T\left(r,t\right)\right|}_{t=0}=T_{init},$$

(2)

$${\left.{\displaystyle \frac{\partial T\left(r,t\right)}{\partial r}}\right|}_{r=0}=0,$$

(3)

$${\left.T\left(r,t\right)\right|}_{r=R}=T_b.$$

(4)

Assuming that the thermophysical parameters c, $\rho$ and $\lambda$ are constants, and $a=\lambda /\left(c\rho \right)$ is the thermal diffusivity [m²/s] (that describes the speed of propagation of thermal energy in the material during changes in temperature over time), then Equation (1) takes the form

$${\displaystyle \frac{\partial T\left(r,t\right)}{\partial t}}=a\left({\displaystyle \frac{{\partial }^{2}T\left(r,t\right)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T\left(r,t\right)}{\partial r}}\right),0<r<R,t>0.$$

(5)

The analytical solution of the above direct problem using Green’s function in the general form has been presented in [14,36], but by adopting the constant values of $T_{init}$, $T_b$ and a, then the solution reduces to the simpler form and can be represented as

$$T\left(r,t\right)=T_b+\left(T_{init}-T_b\right)\sum _{k=1}^{\infty }{\displaystyle \frac{2}{{\mu }_k{J}_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{at}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r}{R}}\right),$$

(6)

where $J_0$ and $J_1$ are called the Bessel functions of the first kind of orders zero and one. The values of ${\mu }_k$ in Equation (6) are positive zeros of the Bessel function of order zero, i.e., $J_0\left(\mu \right)=0$. Usually, the roots of this equation are determined numerically (by using one of many known root-finding algorithms [37]) and some of them with a precision of six decimal places are presented in

Table 1. Selected values of positive zeros of the Bessel function of order zero, i.e., $J_0\left(\mu \right)=0$.

k	${\mathit{\mu }}_{\mathit{k}}$	k	${\mathit{\mu }}_{\mathit{k}}$	k	${\mathit{\mu }}_{\mathit{k}}$
1	2.404826	15	46.341188	200	627.533332
2	5.520078	16	49.482610	300	941.692531
3	8.653728	17	52.624052	400	1255.851763
4	11.791534	18	55.765511	500	1570.011008
5	14.930918	19	58.906984	600	1884.170260
6	18.071064	20	62.048469	700	2198.329516
7	21.211637	30	93.463719	800	2512.488774
8	24.352472	40	124.879309	900	2826.648034
9	27.493479	50	156.295034	1000	3140.807295
10	30.634606	60	187.710827	2000	6282.399929
11	33.775820	70	219.126658	3000	9423.992576
12	36.917098	80	250.542513	4000	12,565.585226
13	40.058426	90	281.958384	5000	15,707.177878
14	43.199792	100	313.374266	10,000	31,415.141142

. In the numerical calculations of the values of function T, it is worth taking the values of ${\mu }_k$ with much greater precision.

In practical computations, the infinite sum in Equation (6) can be reduced to about 100–200 terms, giving fairly good approximate results.

3.2. Inverse Heat Transfer Problem

Let us consider the following problem, in which the measured temperature values are known in several places in the considered domain at different moments of time. In the case of the one-dimensional axisymmetric cylindrical domain, the sensors should be placed at different positions of the radial coordinate of the cylinder in order to capture the most diverse experimental measurement results. Let $po{s}_s$, for $s=1,\dots ,M$, denote the locations of M sensors on the radius axis r. It is assumed that the exact positions $po{s}_s$ are not known in advance and they will be estimated as a result of solving the inverse heat transfer problem. Let $t_f$, for $f=0,\dots ,F$, denote $F+1$ moments of time that are known. The values of $t_f$ can be arbitrary (but it is recommended that they are equidistant), while $t_F$ is the final time. The measured temperature values can be tabulated as

$${\widehat{T}}_{s,f}\equiv T_{measurement}\left(po{s}_s,t_f\right),\mathrm{for}s=1,\dots ,M,f=0,1,\dots ,F.$$

(7)

If we take a look at the solution (6), we can see that this function depends on many parameters, i.e., r, t, a, $T_{init}$, $T_b$ and R. In general, function $T(r,t)$ can be replaced by the multi-argument function $T(r,t,a,T_{init},T_b,R)$, whose value depends on the above-mentioned arguments. In the inverse heat transfer problem, one can try to estimate each value of these unknown parameters. But, if in solution (6) we notice the occurrence of two independent relations $a/R^2$ and $r/R$, where the individual ratios of parameters may be the same, then it is worth setting one of the parameters as fixed, e.g., R, which in this case means that the radius R of the cylinder is fixed (or was precisely measured). Therefore, the parameter R is excluded from the further considerations. So, in the problem under consideration, we will look for parameters $r_1$, $r_2$, …, $r_M$, a, $T_{init}$, $T_b$, where $0\le r_s\le R$, for $s=1,\dots ,M$, are searched sensor positions $po{s}_s$. In Figure 2, the schematic position of thermocouple tips in the cross-section of the container with granular material and considered domain are shown.

In principle, speaking in colloquial language, we are dealing with temperature measurements using several sensors placed in certain unknown positions inside any granular material at different moments of time, and we do not know the value of its thermal diffusivity coefficient; we also do not know the exact constant initial temperature and the constant temperature assumed at the outer surface of the cylindrical can. In the considered task, we want to estimate these unknown parameters by solving an optimization problem.

Based on the above assumptions, the following functional S is constructed

$$S\equiv S\left(r_1,r_2,\dots ,r_M,a,T_{init},T_b\right)={\displaystyle \frac{1}{F+1}}{\displaystyle \frac{1}{M}}\sum _{f=0}^F\sum _{s=1}^M{\left({\left.T\left(r,t,a,T_{init},T_b\right)\right|}_{r=r_s,t=t_f}-{\widehat{T}}_{s,f}\right)}^2.$$

(8)

The above functional describes the error of the solution in a discrete form and is based on the least squares criterion. The aim of the further considerations is to find the minimum of functional $S\to min$, sometimes called the best solution, which leads to finding the optimal values of the particular $M+3$ parameters: $r_1$, $r_2$, …, $r_M$, a, $T_{init}$, $T_b$. In order to simplify further notations, we rewrite Equation (8) as

$$S\left(\mathbf{P}\right)={\displaystyle \frac{1}{F+1}}{\displaystyle \frac{1}{M}}\sum _{f=0}^F\sum _{s=1}^M{\left(T_{s,f}-{\widehat{T}}_{s,f}\right)}^2,$$

(9)

where

$$\mathbf{P}={\left[r_1{r}_2\dots r_{M}a{T}_{init}{T}_b\right]}^{⊺}$$

(10)

and

$$T_{s,f}\equiv {\left.T\left(r,t,a,T_{init},T_b\right)\right|}_{r=r_s,t=t_f}.$$

(11)

3.2.1. Optimization of Functional S

In order to minimization of functional (9), the modified Newton’s method [27,28,29,30] is applied. In general, Newton’s method is used in numerical methods to find zeros of nonlinear functions. This method belongs to the iterative methods. If this method is used for the optimization of functional then the following necessary condition for local minimizer

$$\nabla S\left({\mathbf{P}}^k\right)=0,\mathrm{for}k=0,1,\dots$$

(12)

is required to be met. Parameter k denotes the step of iteration. ${\mathbf{P}}^0$ is the vector of the initial guess and the determination of a reasonable first iteration can sometimes be problematic. Subsequent iterations should bring the solution parameters closer to the optimal state.

Let us consider the second-order approximation of the functional S using the Taylor series expansion

$$S\left({\mathbf{P}}^{k+1}\right)=S\left({\mathbf{P}}^k+{\mathbf{d}}^k\right)\cong S\left({\mathbf{P}}^k\right)+{\left(\nabla S\left({\mathbf{P}}^k\right)\right)}^{⊺}{\mathbf{d}}^k+{\displaystyle \frac{1}{2}}{\left({\mathbf{d}}^k\right)}^{⊺}{\nabla }^{2}S\left({\mathbf{P}}^k\right){\mathbf{d}}^k,$$

(13)

where ${\mathbf{d}}^k$ is called the Newton direction at state ${\mathbf{P}}^k$, $\nabla S\left({\mathbf{P}}^k\right)\equiv {\mathbf{g}}_S\left({\mathbf{P}}^k\right)$ is the gradient vector and ${\nabla }^{2}S\left({\mathbf{P}}^k\right)\equiv {\mathbf{H}}_S\left({\mathbf{P}}^k\right)$ is the Hessian matrix of S calculated for values from the vector ${\mathbf{P}}^k$. In this method, it is assumed that ${\nabla }^{2}S\left({\mathbf{P}}^k\right)$ is non-singular in each iteration. If we apply the differentiation of Expression (13) with respect to ${\mathbf{d}}^k$ that meets the Condition (12), then we have

$$\nabla S\left({\mathbf{P}}^k\right)+{\nabla }^{2}S\left({\mathbf{P}}^k\right){\mathbf{d}}^k=0,$$

(14)

and hence

$${\mathbf{d}}^k=-{\left({\nabla }^{2}S\left({\mathbf{P}}^k\right)\right)}^{-1}\nabla S\left({\mathbf{P}}^k\right).$$

(15)

Remarks on the calculation of vector ${\mathbf{d}}^k$ in Equation (15): From the computational point of view, calculating the inverse matrix of the Hessian is more complex and it may even be numerically unstable. It is worth calculating it better from the numerical solution of the system of linear equations ${\nabla }^{2}S\left({\mathbf{P}}^k\right){\mathbf{d}}^k=-\nabla S\left({\mathbf{P}}^k\right)$, using, e.g., iterative methods.

In the modified Newton’s method, it is assumed that

$${\mathbf{P}}^{k+1}={\mathbf{P}}^k+{\alpha }_k{\mathbf{d}}^k,$$

(16)

where ${\alpha }_k\in (0,1]$ is a scalar selected to satisfy the condition: $S\left({\mathbf{P}}^{k+1}\right)<S\left({\mathbf{P}}^k\right)$. The value of ${\alpha }_k=1$ corresponds to the classical Newton’s method. The optimal value of the parameter ${\alpha }_k$ should be determined by the minimizing of $S\left({\mathbf{P}}^k+{\alpha }_k{\mathbf{d}}^k\right)$, to ensure the fulfilment of the above-mentioned condition.

There are many stopping criteria for an iterative algorithm. The final values of $\mathbf{P}$ should be stabilized with respect to errors in the input data. Here, we use the following criterion

$$∥{\mathbf{P}}^{k+1}-{\mathbf{P}}^k∥=\sqrt{{\left({\mathbf{P}}^{k+1}-{\mathbf{P}}^k\right)}^{⊺}\left({\mathbf{P}}^{k+1}-{\mathbf{P}}^k\right)}<\epsilon ,$$

(17)

where $\epsilon$ is user prescribed tolerance.

3.2.2. Determination of the Gradient Vector $\nabla S\left({\mathbf{P}}^k\right)$ and the Hessian Matrix ${\nabla }^{2}S\left({\mathbf{P}}^k\right)$

Calculations of the values of the gradient vector and the Hessian matrix can be very time-consuming. Here, we use analytical methods. The particular elements of the vector and matrix for the considered problem are as follows:

$$\nabla S={\left[\begin{array}{ccccccc}{\displaystyle \frac{\partial S}{\partial r_1}}& {\displaystyle \frac{\partial S}{\partial r_2}}& \cdots & {\displaystyle \frac{\partial S}{\partial r_M}}& {\displaystyle \frac{\partial S}{\partial a}}& {\displaystyle \frac{\partial S}{\partial T_{init}}}& {\displaystyle \frac{\partial S}{\partial T_b}}\end{array}\right]}^{⊺},$$

$${\nabla }^{2}S=\left[\begin{array}{ccccccc}{\displaystyle \frac{{\partial }^{2}S}{\partial r_1^2}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_1\partial r_2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial r_1\partial r_M}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_1\partial a}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_1\partial T_{init}}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_1\partial T_b}}\\ {\displaystyle \frac{{\partial }^{2}S}{\partial r_2\partial r_1}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_2^2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial r_2\partial r_M}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_2\partial a}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_2\partial T_{init}}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_2\partial T_b}}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{{\partial }^{2}S}{\partial r_M\partial r_1}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_M\partial r_2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial r_M^2}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_M\partial a}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_M\partial T_{init}}}& {\displaystyle \frac{{\partial }^{2}S}{\partial r_M\partial T_b}}\\ {\displaystyle \frac{{\partial }^{2}S}{\partial a\partial r_1}}& {\displaystyle \frac{{\partial }^{2}S}{\partial a\partial r_2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial a\partial r_M}}& {\displaystyle \frac{{\partial }^{2}S}{\partial a^2}}& {\displaystyle \frac{{\partial }^{2}S}{\partial a\partial T_{init}}}& {\displaystyle \frac{{\partial }^{2}S}{\partial a\partial T_b}}\\ {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}\partial r_1}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}\partial r_2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}\partial r_M}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}\partial a}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}^2}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_{init}\partial T_b}}\\ {\displaystyle \frac{{\partial }^{2}S}{\partial T_b\partial r_1}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_b\partial r_2}}& \cdots & {\displaystyle \frac{{\partial }^{2}S}{\partial T_b\partial r_M}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_b\partial a}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_b\partial T_{init}}}& {\displaystyle \frac{{\partial }^{2}S}{\partial T_b^2}}\end{array}\right].$$

The gradient vector and the Hessian matrix contain the first and second derivatives of S, respectively. One can express these derivatives of function (8) in the following forms

$${\displaystyle \frac{\partial S}{\partial p_i}}=2{\displaystyle \frac{1}{F+1}}{\displaystyle \frac{1}{M}}\sum _{f=0}^F\sum _{s=1}^M\left(T_{s,f}-{\widehat{T}}_{s,f}\right){\displaystyle \frac{\partial T_{s,f}}{\partial p_i}},$$

$${\displaystyle \frac{{\partial }^{2}S}{\partial p_i\partial p_j}}=2{\displaystyle \frac{1}{F+1}}{\displaystyle \frac{1}{M}}\sum _{f=0}^F\sum _{s=1}^M\left({\displaystyle \frac{\partial T_{s,f}}{\partial p_i}}{\displaystyle \frac{\partial T_{s,f}}{\partial p_j}}+\left(T_{s,f}-{\widehat{T}}_{s,f}\right){\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial p_i\partial p_j}}\right),$$

where $p_i,p_j\in \left\{r_1,r_2,\dots ,r_M,a,T_{init},T_b\right\}$. As can be seen, there are the partial derivatives of function T with respect to each considered differentiation variable that can be determined in the analytical way. The first-order partial derivatives of T are as follows

$${\displaystyle \frac{\partial T_{s,f}}{\partial r_q}}=\left\{\begin{array}{cc}{\displaystyle -\left(T_{init}-T_b\right)\frac{1}{R}\sum _{k=1}^{\infty }\frac{2}{J_1\left({\mu }_k\right)}exp\left(-{\mu }_k^2\frac{a{t}_f}{R^2}\right)J_1\left({\mu }_k\frac{r_s}{R}\right),}\hfill & \mathrm{if}q=s,\\ 0,\hfill & \mathrm{if}q\ne s,\end{array}\right.$$

for $q=1,2,\dots ,M$,

$${\displaystyle \frac{\partial T_{s,f}}{\partial a}}=-\left(T_{init}-T_b\right){\displaystyle \frac{t_f}{R^2}}\sum _{k=1}^{\infty }{\displaystyle \frac{2{\mu }_k}{J_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right),$$

$${\displaystyle \frac{\partial T_{s,f}}{\partial T_{init}}}=\sum _{k=1}^{\infty }{\displaystyle \frac{2}{{\mu }_k{J}_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right),$$

$${\displaystyle \frac{\partial T_{s,f}}{\partial T_b}}=1-\sum _{k=1}^{\infty }{\displaystyle \frac{2}{{\mu }_k{J}_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right).$$

While the second-order derivatives of T take the forms

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_q^2}}=\left\{\begin{array}{cc}{\displaystyle \left(T_{init}-T_b\right)\frac{1}{R^2}\sum _{k=1}^{\infty }\frac{{\mu }_k}{J_1\left({\mu }_k\right)}exp\left(-{\mu }_k^2\frac{a{t}_f}{R^2}\right)\left(J_2\left({\mu }_k\frac{r_s}{R}\right)-J_0\left({\mu }_k\frac{r_s}{R}\right)\right),}\hfill & \mathrm{if}q=s,\\ 0,\hfill & \mathrm{if}q\ne s,\end{array}\right.$$

for $q=1,2,\dots ,M$,

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial a^2}}=\left(T_{init}-T_b\right){\displaystyle \frac{t_f^2}{R^4}}\sum _{k=1}^{\infty }{\displaystyle \frac{2{\mu }_k^3}{J_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right),$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_{init}^2}}=0,$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_b^2}}=0,$$

and the second-order mixed partial derivatives (they are symmetric) can be expressed as

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_s\partial r_q}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_q\partial r_s}}=0,\mathrm{if}q\ne s,$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_q\partial a}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial a\partial r_q}}=\left\{\begin{array}{cc}{\displaystyle \left(T_{init}-T_b\right)\frac{t_f}{R^3}\sum _{k=1}^{\infty }\frac{2{\mu }_k^2}{J_1\left({\mu }_k\right)}exp\left(-{\mu }_k^2\frac{a{t}_f}{R^2}\right)J_1\left({\mu }_k\frac{r_s}{R}\right),}\hfill & \mathrm{if}q=s,\\ 0,\hfill & \mathrm{if}q\ne s,\end{array}\right.$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_q\partial T_{init}}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_{init}\partial r_q}}=\left\{\begin{array}{cc}{\displaystyle -\frac{1}{R}\sum _{k=1}^{\infty }\frac{2}{J_1\left({\mu }_k\right)}exp\left(-{\mu }_k^2\frac{a{t}_f}{R^2}\right)J_1\left({\mu }_k\frac{r_s}{R}\right),}\hfill & \mathrm{if}q=s,\\ 0,\hfill & \mathrm{if}q\ne s,\end{array}\right.$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial r_q\partial T_b}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_b\partial r_q}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{R}\sum _{k=1}^{\infty }\frac{2}{J_1\left({\mu }_k\right)}exp\left(-{\mu }_k^2\frac{a{t}_f}{R^2}\right)J_1\left({\mu }_k\frac{r_s}{R}\right),}\hfill & \mathrm{if}q=s,\\ 0,\hfill & \mathrm{if}q\ne s,\end{array}\right.$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial a\partial T_{init}}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_{init}\partial a}}=-{\displaystyle \frac{t_f}{R^2}}\sum _{k=1}^{\infty }{\displaystyle \frac{2{\mu }_k}{J_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right),$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial a\partial T_b}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_b\partial a}}={\displaystyle \frac{t_f}{R^2}}\sum _{k=1}^{\infty }{\displaystyle \frac{2{\mu }_k}{J_1\left({\mu }_k\right)}}exp\left(-{\mu }_k^2{\displaystyle \frac{a{t}_f}{R^2}}\right)J_0\left({\mu }_k{\displaystyle \frac{r_s}{R}}\right),$$

$${\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_{init}\partial T_b}}={\displaystyle \frac{{\partial }^2{T}_{s,f}}{\partial T_b\partial T_{init}}}=0.$$

Finally, when the analytical forms of the gradient vector and the Hessian matrix are known, then one can substitute the values of vector ${\mathbf{P}}^k$ estimated in the k-th iteration into them in order to determine the numerical values of $\nabla S\left({\mathbf{P}}^k\right)$ and ${\nabla }^{2}S\left({\mathbf{P}}^k\right)$.

4. Illustrative Example

We present one of the many experiments performed. Fine-grained evaporated table salt was poured into the cylindrical container made of aluminium (the beverage can with the radius R = 33 mm, a height of about 150 mm, and a sheet thickness of 0.2 mm). Inside the granular material (about the middle height of the can), 4 sensors (thermocouples) were placed at different positions on the radius. Temperature measurements were recorded at 1-s time intervals (time measurement with an accuracy of ±1 s). After immersing the can in boiling water, the temperature measurements were recorded for a period of 800 s, wherein the water was heated continuously to ensure the boiling state and the constant temperature (approximately) on the outer surface of the can. It is common knowledge that the boiling point of water is 100 °C, but depending on various external factors (e.g., pressure), this temperature may be different.

The recorded temperature measurements using thermocouples at four different radius positions inside the granular material placed in the can with the time step of 1 s are shown in Figure 3. In the zoomed subplot, the fluctuations in measurement data can be observed.

The second part of this task was the identification of seven unknown parameters: $r_1$, $r_2$, $r_3$, $r_4$, a, $T_{init}$ and $T_b$ using the mathematical model described in the previous section. For this purpose, three different sets of initial values of these parameters were assumed and their values are presented in

Table 2. Initial sets of parameter values.

Variant	${\mathit{r}}_1$ [mm]	${\mathit{r}}_2$ [mm]	${\mathit{r}}_3$ [mm]	${\mathit{r}}_4$ [mm]	a [m2/s]	${\mathit{T}}_{\mathbf{init}}$ [°C]	${\mathit{T}}_{\mathit{b}}$ [°C]
1	10	15	20	25	$3.2\times {10}^{-7}$	30	100
2	1	7	12	30	$3.3\times {10}^{-7}$	25	95
3	2	10	18	24	$3.5\times {10}^{-7}$	20	90

. If we used these three sets of values directly to calculate the values of function (6), the approximations of measurement data would look like in Figure 4. As can be seen, these preliminary estimates are difficult to accept. The values of functional S for these data are 211.129, 149.124 and 14.768, respectively.

Using the presented iterative algorithm, we found the optimal values of particular unknown parameter values in the subsequent steps. The results of approximations of these parameters in subsequent iterations are shown in Figure 5. Moreover, the change in values of functional S can be traced in the last plot in this figure. As can be seen, these estimated values of particular parameters have stabilized after about 7–10 iterations. For all considered variants of the sets of initial values, almost identical optimal values were found, which, with the accuracy of three decimal places, are as follows: $r_1=0.000$ mm, $r_2=5.148$ mm, $r_3=13.259$ mm, $r_4=22.127$ mm, $a=3.213\times {10}^{-7}$ m²/s, $T_{init}=19.968$ °C and $T_b=99.603$ °C, while the values of functional S are about 0.057. The calculations were carried out for the measured temperatures with the assumed step length $\Delta t=10$ s (in order to reduce the computation time, 81 measurements were taken into account (i.e., $F=80$), instead of all 801 measurements for each sensor). In Figure 6, in addition to the measurement data, the determined heating curves according to the mathematical model for the above-mentioned optimal parameters have been presented.

5. Conclusions

The work presented the measurement system used to record temperature changes inside granular material during its heating. Heating a container with granular material (in the example experiment, fine-grained table salt was used) is carried out by placing it in boiling water. Measuring temperature changes using thermocouples located inside such material seems to be justified in this case.

As is well known, any measurement value (including temperature measurement) is not accurate. Moreover, in the experiment under consideration, it is very difficult to align the exact positions of the measuring tips of the temperature sensors with respect to the rotation axis of the cylindrical can, or it is also difficult to accurately measure the constant positions of the sensors in the can after filling it with granular material (the sensor tips may relocate slightly). When measuring temperature with thermocouples, one can notice fluctuations in the measurement data over time, and one can even read different temperatures for the material that appears to be in an identical steady state. Hence, it became necessary to estimate the constant initial temperature of the granular material, as well as to estimate the constant boundary temperature on the outer surface of the can, during the heating process.

The construction of the measurement system and the selection of such an approach to measuring temperature changes have simplified the next stage of the research problem, the aim of which was to solve the inverse problem, consisting of optimal determination of the position of sensors in granular material, the estimation of the thermal diffusivity coefficient of the considered material and the initial temperature and temperature on the outer surface of the can. The assumed mathematical model of the direct heat conduction problem (based on Fourier’s law) in the axisymmetric domain, as well as the model of the inverse heat conduction problem, which were used for this purpose, seem to be reasonable. In addition, we also considered the Cattaneo heat conduction model containing the lag time (we omitted the details and analysis of its solution in this paper) for the same purpose. However, in this case, the characteristics of the heating curves obtained from the Cattaneo model are different and the estimation error of parameters is larger than for Fourier’s model.

The procedure of searching for optimal values of the unknown parameters with the assumed specified error (based on the modified Newton’s optimization method) gave quite satisfactory results after just a few iterations. One of the disadvantages of this optimization method is filling the gradient vector and the Hessian matrix with values in each iteration, which is computationally time-consuming. Here, analytical methods were used for this purpose, but appropriate numerical methods can also be used for the approximate determination of the first and second mixed partial derivatives. The Newton’s method may also lead to incorrect parameter estimates, mainly due to arbitrarily adopted initial guess values. Therefore, it is recommended that the initial guess should be assumed reasonably, as close as possible to the desired parameter values. In the presented experimental-computational example, the obtained estimated parameter values seem to be consistent with the assumptions (i.e., they correspond to the approximate positions of the sensor tips inside the granular material) and the assumed temperatures in the initial boundary conditions. One can conclude that using Fourier’s model for the mathematical description of the heat conduction in fine-grained table salt in the temperature range from room temperature (about 20 °C) to 100 °C (assuming the constant values of thermophysical coefficients in this range) is justified.

The thermal diffusivity coefficient of granular material may depend on many factors, including the grain size fraction, the humidity of the air filling the space between the grains, atmospheric pressure, packing of the granular material, temperature range of the material during the experiment, etc. The study of the influence of all the above-mentioned factors on the estimation of the thermal diffusivity coefficient of granular material was not analysed. In the example experiment presented in this paper, fine-grained table salt (also known as food salt, which is widely available in grocery stores) that usually contains 99.9% pure sodium chloride was used. The remaining part (about 0.1%) consists of moisture and an anti-caking agent. The work mainly focused on the method of determining the thermal diffusion coefficient of granular material that was heated during the experiment. It can be stated that if more temperature sensors are placed inside granular material at different radius positions, then the estimation of searched unknown parameters in the mathematical model can be more precise.

In future research, we would also like to focus on determining the thermal effusivity coefficient $e=\sqrt{\lambda c\rho }$ [W s^1/2/(m² K)], which is a very important thermophysical feature of each granular material. If two quantities, the thermal diffusivity a and thermal effusivity e are known for any material, then the coefficients $\lambda$ and $c·\rho$ can be determined in a simple mathematical way. In this work, we did not investigate the effect of water content in salt on changes in its thermal diffusivity coefficient. We plan to conduct such studies in the future, as well as present the results obtained for other granular materials with different grain sizes and packing degrees.