Study on designing measurement system of thermal conductivity based on one-dimensional steady-state method

Thermal conductivity has wide application in various fields such as in agricultural and food sectors. In a food industry, measuring the thermal conductivity is designed to optimize the thermal process such as drying, sterilization, and freezing. In this study, a system to measure thermal conductivity is designed according to one-dimensional steady state method through Fourier formula. The experiment uses samples consisting of rice-based product (rice grain, rice powder, rice flour, cooked rice) and whole milk powder. The analysis will be performed in the range of 15-30 minutes on its steady condition. The results are verified with the literature data and evaluated through degree equivalence calculation. As a result, the system can distinguish the samples which has different texture and composition. Furthermore, the system can figure the increasing trend of thermal conductivity with the raise of moisture content. However, in most case, thermal conductivities of the measured samples have higher value compared to the literature data (2-4 times higher). The disparity due to the variation of heating temperature, sample condition, measurement technique, and analysis. Several finding to gain more reliable system can be done by improving the insulation or evaluating the mathematical calculation into a two-dimension approximation. In general, the design of the system is plausible for measuring the thermal conductivity.


Introduction
Thermal conductivity has wide application in various fields study.As for instant, in agricultural sector, measurement the thermal conductivity of soil is to know the soil characteristic for conducting heat [1].Furthermore, the thermal conductivity is a key parameter in a thermal process such as drying, sterilization, and freezing in a food industry [2].These processes will determine the success of yielding the food product.For example, giving high temperature in the sterilization process can destroy microorganism faster, but can degrade the food nutrition.By measuring the thermal conductivity, the amount of heat that must be given into a system can be predicted and adjusted.Thus, an optimum thermal process can be achieved efficiently.
To date, commercial equipment based on transient needle probe method is the most common tool to measure conductivity of food because of its fast measurement (10-15 minutes) and moderate accuracy (up to 10%) [3].However, several international standards do not specify that the transient method is 1230 (2023) 012136 IOP Publishing doi:10.1088/1755-1315/1230/1/012136 2 purposed for food application only [4,5].This method can be utilized for particulate samples as for instance soil, foam, flour [6][7][8][9], or steel [10] .Meanwhile, a steady state method has advantages on its accuracy which is approximately 2% [3].Furthermore, constructing the one-dimensional system can simplify the mathematical model and analysis.Thus, designing other measurement system is expected to give a reliable result with simple analysis.
In this study, designing a system to measure the thermal conductivity based on the steady state method in one-dimensional direction is performed.The aim of this study is to design a construction system that can measure the thermal conductivity with simplified mathematical model and higher accuracy.The constructed system is intended to give the easiness for measuring the thermal conductivity with an inexpensive budget.

Materials and Methods
Before constructing a complete system with automatization, the experiment is tested in a simple set up as presented in figure 1.

Figure 1. Set up of the measurement system
The system consists of a heater, a sample cube, four temperature sensors, and power meter.A hotplate magnetic stirrer from Thermo Scientific acts as heater and it is set at constant temperature of 70±5 o C. The samples are put in the sample cube which has dimension of 90 mm length, 90mm wide, and 30mm height.The cube is created from a glass and isolated with plywood to prevent a heat loss in the lateral direction.The upper and bottom side of the cube is made from aluminium to transfer the heat from the heater.The temperature sensors are Thermocouple Lutron TP-01 (Type K Temp Probe).Two sensors are placed inside the cube, whilst the other two are outside (at the heater and outer wall of the cube).In the inner cube, the sensors are positioned in the middle of both plates.A Power meter from Taffware is utilized to measure the input power, voltage, and current.
The samples consist of rice-based product (cooked rice, rice grain, rice powder, rice flour) and whole milk powder.The rice is cooked with the ratio of the rice grain and water of 1:1.The rice powder is produced by grinding the rice grain.The moisture content of the samples is performed using Ohaus MB120.In addition, the sample density is obtained by calculating the mass to volume ratio of the sample.The thermal conductivity measurement was carried out in 3 times repetition.whole milk powder A one-dimensional heat transfer in the axial direction is shown in figure 3. The calculation of the thermal conductivity is based on Fourier Formula [11][12][13][14].The mathematical models are stated in the Equation ( 1) and ( 2) Heat per area unit is equivalent to the temperature gradient.
≈   Inserting a constant and minus sign as the gradient temperature experience a temperature drop, thus the formula for absolute one-dimensional heat transfer on its steady condition.
q : The amount of heat given to the system (W) dL : the axial length of the sample (m) A : Cross-sectional area toward the heat direction of the sample (m 2 ) dT : Temperature difference on both end of the sample ( o C) The measurement result is verified through degree equivalence analysis as described in Equation (3) [15,16].The equation required the calculation of the uncertainty measurement.In this case, the uncertainty approximation is taken from the standard deviation of the measurement result.

Results and Discussions
The temperature profile against time is shown in figure 4. The profile of the power input during measurement of 35 minutes is given in figure 5.As can be seen in the Figure 4, the temperature of T1 (heater), T2 (plate), and T3 (sample downside) rise sharply in the first 5 minutes which means that the heating process of the system is started.The condition in this time is referred as transient.In other hand, there is little change in temperature of T4 which shows that the heat has not been transferred to the end of the sample yet in this stage.Afterward, the temperature change is being stable after 10 minutes.In the meantime, the T4 also gradually increases.Thus, the temperature difference will be analysed in the range of 15-30 minutes on its steady condition.The analysis is not continued after 30 minutes since the T4 is raised very slowly.Transient method measures the power to raise the heat, whereas the steady state method takes the power to stabilize the temperature.Hence, the power analysis will be focused on the time range 1000-2000 s.In addition to prevent an overshoot of the temperature set, the heater is controlled by a PID logic.Initially, the system grabs high amount of power, then gradually feeds the small portion of it.As can be viewed in the small figure 4, the system takes the power of 100-160 W then feed it with only 1.8-2.6W. Thus, the power input for the conductivity measurement will be the average of the feed power.
The dimension of the sample (L, A) is calculated through the sample cube because the food sample is placed in the cube with a fixed dimension.The axial length of the sample cube is known to be 0.03 m and the square area toward the heat direction is 0.09 m x 0.09 m.The sample measurement data and calculation of the thermal conductivity according to Equation (1) are described in the table 1.In general, the system can distinguish the thermal conductivity of food samples with different texture and composition.In the Table 1, the thermal conductivity of rice-based products (cooked rice, rice grain, rice powder, and rice flour) are varied.Those samples are from the same original material but have distinct texture as can be seen in Figure 2. The rice powder in this experiment were grinded from the rice grain and has rougher texture than rice flour.Indeed, the measured thermal conductivity of both samples can be differentiated by this construction system.Meanwhile, the rice flour and whole milk have more similar texture but differ in the composition.Both samples also give slightly difference value of the thermal conductivity.
In the experiment, the samples no 4 and 5 have different weight compared sample no 1-3.The weight of both samples to fit in the cube are less than other samples (only 160 g), This is due to the soft texture of both samples.The fine texture gives larger area to fill the volume of the cube.As result, it can fill the area with less weight, To verify the result, the thermal conductivity from several samples in the Table 1 are compared with the literature data [6] [17].The comparison between the measurement data and literature data are compiled in the Table 2.The calculations of En number based on Equation ( 2) are also presented.The samples in the literature are measured using transient needle probe method.In the table 2, the thermal conductivity of the measured data in most case have higher value compared to the literature data.The magnitude difference is known to be 2-4 times higher.Mostly, the En value of the samples are above 1.This means that the measured values are mostly outlier compared to the published data.The outranging can be due to the discrepancy in the heating temperature, sample conditions (moisture content and density), measurement technique, and analysis [2].As for instant, the whole milk sample is measured at stabilized temperature of 70 0 C, while in the literature the sample is measured at 10-50 o C. The same heating temperature of measured sample and literature of the cooked rice give inline data.The thermal conductivity is a temperature-dependent, the higher temperature will supply more energy in the sample and cause higher atom activity.Moreover, the sample in this study also has different moisture content and bulk density.The variation of the sample condition in the published articles can be the fundamental factor of the disparity.In addition, the effects of the moisture content to the thermal conductivity are presented in the figure 5.
(a) (b) Figure 6.The thermal conductivity variation to the moisture content of sample (a) rice flour (b) whole milk powder In the Figure 6, the rice flour and whole milk have increasing trend of thermal conductivity with the raise of the moisture content.This result is corresponded to the literature data.This because the heat conduction of water is greater than air [9].The increasing trend of sample with lower moisture content (4-20%) is less than (20-40%).Back to Table 1, it is plausible that the cooked rice has the highest thermal conductivity among the samples.The increased quantity of water gives higher thermal conductivity of moisture rich in the sample composition [17] [18].Since the R square is not so close to 1, more detail measurement data are needed to obtain linear relation between the thermal conductivity and moisture content.
Aside from the several factors mentioned, further investigation regarding the system is required to find the cause of the deviation.One-dimensional approximation in the axial direction required a perfect insulation in the lateral side.However, there can be a possible leakage during measurement.Regarding its dimension, the heat has tendency to flow in the shorter distance.The cube design has long lateral size compared to the axial.Based on this, the dimension of the sample cube has no problem, then the insulation of the sample cube should be improved when constructing the complete system for the future.
Another alternative when the heat isolation cannot be fixed, the mathematical model should be corrected.It is suspected there is heat transfer on the lateral or radial side, thus two-dimensional approximation can be taken into consideration [19,20].In the absolute calculation, the amount of heat is a critical parameter.For further improvement, detailed evaluation regarding the estimation of heat input should be done.

Conclusions
A simple construction of thermal conductivity measurement system based on one-dimensional steady state method has been performed to several samples in this study.The system can distinguish those samples according to its texture and composition.Combining with a tool to measure moisture content, the system can figure the increasing trend of thermal conductivity with the raise of moisture content which correspond to the literature data.In most case, the thermal conductivities of measured samples have higher value compared to the literature data (2-4 times higher).The evaluation of the En number also give values above 1 which means out of the range.The disparity due to the variation of heating temperature, sample conditions (moisture content and bulk density), measurement technique, and analysis.Several finding to improve the system can be done by give a perfect insulation or evaluated the mathematical calculation into a two-dimension approximation.In general, the design of the system is plausible for measuring the thermal conductivity.

Figure 4 .
Figure 4. Temperature profile per minute

Figure 5 .
Figure 5.The profile of the power input

Table 1 .
Sample measurement data and calculation (on average)

Table 2 .
Comparison of the measurement and literature data (on average)