Apparatus and methods for measurement of thermal conductivity and thermal diffusivity

Abstract

A method of deriving a thermal conductivity value and a thermal diffusivity value for a sample including the steps of providing a sample container comprising a solid annular cylinder of a material with a known thermal conductivity and a known thermal diffusivity, the internal radius of the annular cylinder defining a cylindrical sample space; providing the sample in the cylindrical sample space; applying heat to the sample along the central axis of the cylindrical sample space; taking temperature readings over time in the sample or in the annular cylinder at a first point of known radial distance from the central axis; and using the temperature readings to derive the thermal conductivity value and the thermal diffusivity value.

Description

APPARATUS AND METHODS FOR MEASUREMENT OF THERMAL CONDUCTIVITY AND THERMAL DIFFUSIVITYFIELD

[0001] In one of its aspects, the present disclosure relates generally to measurement of thermal conductivity (K) and thermal diffusivity (a) of a material of interest.BACKGROUND

[0002] The transient hot wire method is a method for measuring only thermal conductivity. In this method, a constant source of heat per unit length and time "q" is applied to a sample using a heated wire. The temperature of a point on or close to the heated wire is then measured with time. With the assumption that the sample is large and infinite (which is not true), the only parameter defining the log time-temperature slope (S) of temperature versus time is the thermal conductivity. The slope S is used to calculate the thermal conductivity (K) using the equation K=q / (4.iT.S). The hot wire method only measures thermal conductivity, and it is based on heating a point and measuring the temperature at the same point.

[0003] An apparatus and method for measuring both thermal conductivity and thermal diffusivity would be desirable.SUMMARY

[0004] In one aspect, the present disclosure relates to a method of measuring both thermal conductivity and thermal diffusivity for a sample by applying a line source of heat along the center of a cylindrical sample and measuring the temperature at one or two known radial distances with time. The thermal conductivity and diffusivity of the sample is calculated by fitting the measured temperature values with an analytical solution of heat conduction in a finite hollow composite cylinder with a line source.

[0005] In another aspect, the present disclosure relates to a method of measuring both thermal conductivity and thermal diffusivity for a sample including the steps of placing thesample in the well of an annular cylinder, placing a sample in the well, placing a heat source probe in the sample, and placing a temperature sensor either in the sample or the annular cylinder. If the sample is a solid, two to three holes (depending upon whether one or two temperature sensors are used) are made in the sample, one in the center of the sample for the heat source probe which in one aspect is a needle-like heat source probe and one or two for each of the temperature sensors, which in one aspect are needle-like thermocouples.

[0006] The present disclosure in another aspect relates to a method of calculating or deriving a thermal conductivity value and a thermal diffusivity value for a sample including the steps of: providing a sample container comprising a solid annular cylinder of a material with a known thermal conductivity and a known thermal diffusivity, the internal radius of the annular cylinder defining a cylindrical sample space; providing the sample in the cylindrical sample space; applying heat to the sample along the central axis of the cylindrical sample space; taking temperature readings over time in the sample or in the annular cylinder at a first point of known radial distance from the central axis; and using the temperature readings to derive the thermal conductivity value and the thermal diffusivity value.

[0007] In another aspect, the present disclosure relates to an apparatus for measuring both thermal conductivity and thermal diffusivity for a sample including an annular cylinder of a homogeneous material of known thermal conductivity and thermal diffusivity, a lower insulation layer base for the annular cylinder, the inner radius of the annular cylinder and in the lower insulation layer base defining a well for receiving a sample to be tested, an upper insulation layer for capping the top of the annual cylinder and the well, a heat source probe for insertion into the sample along the central axis of the annual cylinder for heating the sample, a power source operably connected to the heat source probe for providingpower to heat the heat source probe, a thermocouple temperature sensor for insertion into the sample or the annual cylinder at a radial distance from the central axis for measuring temperature at the radial distance, and a temperature data acquisition unit operably connected to the thermocouple temperature sensor for acquiring and recording temperature data from the thermocouple temperature sensor.

[0008] Other advantages of the present teachings may become apparent to those of skill in the art upon reviewing the present specification.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Embodiments of the present invention will be described with reference to the accompanying drawings, wherein like reference numerals denote like parts, and in which:

[0010] FIG. 1 is a perspective view showing an apparatus and set-up according to an embodiment of the present invention;

[0011] FIG. 2 is an enlarged view of the apparatus of FIG. 1 with the upper insulation disk removed and a sample shown in the sample space with an electric heating probe and two thermocouple sensors inserted in the sample;

[0012] FIG. 3 is a perspective view from the top of the apparatus of FIG. 1 with the upper insulation disk removed and a sample shown in the sample space with three holes visible in the sample for receiving the electric heating probe and the two thermocouple sensors of FIG. 2;

[0013] FIG. 4 is a perspective view from the top with only the sample and lower insulation disk of FIG. 1 shown;

[0014] FIG. 5 is a diagram depicting a cross-sectional view the annular cylinder and lower insulation disk of FIG. 1 with the upper insulation disk and sample removed;

[0015] FIG. 6 is a graph of temperature with time for the testing of the sample of FIG. 2 for thermal conductivity K and thermal diffusivity a;

[0016] FIG. 7 is a graph of temperature with time for the testing of the sample of FIG. 2 for thermal conductivity K and thermal diffusivity a before fitting of the data is completed;

[0017] FIG. 8 is a graph of temperature over time where the data of FIG. 7 has been fitted; and

[0018] FIG. 9 is a top schematic diagram of an alternate configuration for the annular cylinder and sample space of the apparatus of FIG. 1 .DETAILED DESCRIPTION

[0019] Referring to Figures 1 to 5, a testing apparatus indicated generally at 1 according to an embodiment of the present invention is shown. The apparatus includes a lower insulation disk 2 of sheet foam insulation, an upper insulation disk 4 of sheet foam insulation, and an annular cylinder 6 composed of a standard material. A standard material as referred to herein is a material that is homogenous and for which the thermal conductivity and thermal diffusivity are known. In the present embodiment, the standard material is acrylic. In other embodiments, other materials such as aluminum or steel can be used as the standard material. In other embodiments, other suitable insulating materials can be used for the disks 2 and 4. The additional foam base 7 is not essential.

[0020] The inner radius of annular ring 6 defines a cylindrical well that serves as a sample space indicated generally at 8 for holding a sample 10 to be tested. Cylindrical sample space 8 is best seen in Figure 5 which depicts a cross-sectional view of annular cylinder 6 sitting on lower insulation disk 2 with upper insulation disk 4 removed.

[0021] In the present example, sample 10 is a clay sample that was prepared at the University of New Brunswick geotechnical laboratory. Sample 10 can be any solid, granular, aggregate, powdered sample or liquid for which testing is desired to determine the thermal conductivity and thermal diffusivity properties of the sample.

[0022] Apparatus 1 further includes a needle-like electric heating probe 12 (see Fig. 2) with a known power per unit length and which is operably connectable with wires 14 to a power source 16 (see Fig. 1 ) for supplying a controlled voltage to electric heating probe12. Wires 14 are depicted as discontinuous but are continuous between probe 12 and source 16. Apparatus 1 also includes a first thermocouple sensor 18 and a second thermocouple sensor 20 which are operably connected by wires 22 and 24, respectively, to data acquisition unit 26 for collecting temperature readings from sensors 18 and 20 (see Fig. 1 ). Wires 22 and 24 are depicted as discontinuous but are continuous between sensors 18 and 20 and unit 26.

[0023] Since sample 10 is a solid clay sample, holes 28, 30 and 32 are drilled into sample 10 (see Fig. 3) to enable probe 12 and sensors 18 and 20, respectively, to be inserted into sample 10 (see Fig. 2). In the case of a granular, aggregate, powdered or liquid sample, probe 12 and sensors 18 and 20 can be inserted into the sample without the need to first drill holes in the sample. Hole 28 corresponds to the central axis of annular cylinder 6.Measurement of Thermal Conductivity and Thermal Diffusivity Using Line Source and Two Radial Thermocouples

[0024] According to one embodiment of the present invention, a method for measuring thermal conductivity and thermal diffusivity of sample 10 includes inserting electric heating probe 12 and thermocouple sensors 18 and 20 into sample 10; measuring: radius (rb) of sample 10, radial distance (r) from the central axis of annular cylinder 6 to thermocouple sensor 18, and radial distance (r2) from the central axis of annular cylinder 6 to thermocouple sensor 20; switching on power source 16 to energize heating probe 12 to apply a line source of heat to sample 10; switching on unit 26 and acquiring temperature readings with time from sensors 18 and 20, representing a radial temperature response along parallel lines at the known radial distances (r) and (r2); and deriving the thermal conductivity (K1) and thermal diffusivity (a) of the sample that best fit the analytical solution temperature by solving equations (1) to (12) below with the measured temperature readings.

[0025] According to an embodiment of the method of the present invention, the acquired temperature readings are processed wherein thermal conductivity and thermal diffusivity of sample 10 are calculated by fitting the results with an analytical solution of heat conduction in an annular cylinder with a line source, by solving equations (1 ) to (12):where equation (7) is solved first, followed by solving equation (2) with all variables defined in equations (3), (4), (5), (6) and (8), (9), (10), (11 ), and (12).

[0026] Figure 6 is a graph of temperature over time for the testing of sample 10 for thermal conductivity (K) and thermal diffusivity (a). The circles 34 represent the temperature response at a point with radial distance r = 10mm over time and line 36 represents the best fit (K and a) of equations (1) to (12). The best fit results are then used to predict the temperature response at an outer radius with r = 15.5 (line 40) using equations (1) to (12) and the best fit parameters and the results are compared with the measured temperature response represented by circles 38 over time at the point r = 15.5. The results show a very good agreement.Measurement of Thermal Conductivity and Thermal Diffusivity Using Line Source and One Radial ThermocoupleIn another embodiment of the present invention, the method can be carried out using apparatus 1 with a set-up that uses only one of the thermocouple sensors 18 and 20. Thethermocouple sensor used can be inserted either in sample 10 or in annular cylinder 6. In such an embodiment, the steps outlined above are the same except that temperature readings are only taken from one thermocouple sensor (either sensor 18 or 20) and the equations to solve for the best fit solution are the same except that equation (2b) is substituted for equation (2):where equation (7) is solved first, followed by solving equation (2b) with all variables defined in equations (3), (4), (5), (6) and (8), (9), (10), (11 ), and (12).Example using One Thermocouple Sensor

[0027] The following parameters were used in a test with a set-up with only one thermocouple sensor:Power: 10 wattSample thickness: 37.56 mmSample radius (rb) = 25 mmAnnular cylinder material is acrylic with known thermal conductivity and thermal diffusivityThermal diffusivity (the ratio of the thermal conductivity to the specific heat) a2 = 1.095E-07 mm2 / sThermal conductivity k2 = 0.19 W / mKThermocouple radial distance = 10 mm

[0028] Table 1 sets out lab data for a time from 0 to 270.08 seconds.Table 1

[0029] Unknowns are a1 , K1 (thermal diffusivity, thermal conductivity) of the sample 10.

[0030] A first try with a1 and K1 values from a possible range:K1 = W / mK 0.3 a1 = m2 / s 0.1 E-6

[0031] Steps of calculation of temperature with time at the same location of the measured temperature include the steps of:Step 1 , Equation 7: F0 = a1 * time I (rbA2)Step 2, Equation 2:where: v: integration function range from 0 to ooJo, Bessel Function of the First Kind and zero order,]n, Bessel Function of the First Kind and n order]'n, derivative Bessel Function of the First Kind and n orderYn, Bessel Function of the Second Kind and n orderY'n, derivative Bessel Function of the Second Kind and n order<p, ip, g, and f as defined in equations 3, 4, 5, and 6, respectavely

[0032] Step 1 and step 2 are run for each time step, from the same time range of the lab data (in this example for a time from 0 to 270.08 second) to obtain “model” temperature shown in dashes 44 in Figure 7, and to be compared with the lab data shown in dots 42 in Figure 7.

[0033] The total error between “model” and “lab”:

[0034] Keep iteration of K1 and a1 to the best results (minimum error < ) to match the model temperature with the lab temperature data. The result of the fitting method is shown graphically in Figure 8. The method in this example yielded the following values for Thermal diffusivity (a1 ) and Thermal conductivity (K1 ) of sample 10:K1 = W / mK 0.56 a1 = m2 / s 0.22E-6

[0035] Apparatus 1 can be modified according to another embodiment of the present invention. Referring to Figure 9, instead of using a full annular cylinder 6, a partial annular cylinder 46 can be used. A solid insulating material 48 completes the rest of the cylinder. The inner radius of partial annular cylinder 46 and walls 48 and 50 of material 48 define a sample space 52 for holding a sample of material to be tested. The embodiments for the set-up of apparatus 1 as discussed above are otherwise the same, as are the embodiments of the methods for deriving thermal conductivity and thermal diffusivity values for the sample material being tested.

[0036] In further embodiments, the present invention can be used in large scale field testing of materials such as soil, landfill, gravel and course material. In Canada, and cold regions, thermal properties in the permafrost area is an essential aspect in the design of infrastructure. Also, the design of nuclear spent fuel deep geological repositories depends on reliable values of the thermal properties of the backfill materials used within the system. Researchers in heat transfer, geotechnical engineers, and designers are all potential users.

[0037] Various apparatuses or processes were described above to provide an example of an embodiment of each claimed invention. No embodiment described above limits any claimed invention and any claimed invention may cover processes or apparatuses that differ from those described above. The claimed inventions are not limited to apparatuses or processes having all of the features of any one apparatus or process described above ioor to features common to multiple or all of the apparatuses described above. It is possible that an apparatus or process described above is not an embodiment of any claimed invention. Any invention disclosed in an apparatus or process described above that is not claimed in this document may be the subject matter of another protective instrument, for example, a continuing patent application, and the applicants, inventors or owners do not intend to abandon, disclaim, or dedicate to the public any such invention by its disclosure in this document.

[0038] While the teaching herein includes illustrative embodiments and examples of some aspects of an invention, the description is not intended to be construed in a limiting sense. Thus, various modifications of the illustrative embodiments, as well as other embodiments of the invention, may be apparent to persons skilled in the art upon reference to this description. It is therefore contemplated that the appended claims will cover any such modifications or embodiments.All publications, patents, and patent applications referred to herein are incorporated by reference in their entirety to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference in its entirety.

Claims

CLAIMSWhat is claimed is:

1. A method of deriving a thermal conductivity value and a thermal diffusivity value for a sample comprising the steps of: providing a sample container comprising a solid annular cylinder of a material with a known thermal conductivity and a known thermal diffusivity, the internal radius of the annular cylinder defining a cylindrical sample space; providing the sample in the cylindrical sample space; applying heat to the sample along the central axis of the cylindrical sample space; taking temperature readings over time in the sample or in the annular cylinder at a first point of known radial distance from the central axis; and using the temperature readings to derive the thermal conductivity value and the thermal diffusivity value.

2. The method of claim 1 , wherein the material of the annual cylinder is substantially homogeneous.

3. The method of claim 2, comprising carrying out a best fit method to derive the thermal conductivity value and the thermal diffusivity value.

4. The method of claim 3, wherein the best fit method comprises fitting the temperature readings with an analytical solution of heat conduction in an annular cylinder with a line heat source.

5. The method of claim 4, wherein the best fit method comprises solving equations (1 ) to (12):wherein equation (7) is solved first, followed by solving equation (2b) with all variables defined in equations (3), (4), (5), (6) and (8), (9), (10), (11 ), and (12).

6. The method of claim 4, further comprising taking temperature readings over time in the sample or in the annular cylinder at a second point of known radial distance from the central axis, and wherein the best fit solution comprises solving equations (1 ) to (12):wherein equation (7) is solved first, followed by solving equation (2) with all variables defined in equations (3), (4), (5), (6) and (8), (9), (10), (11 ), and (12).

7. An apparatus for measuring both thermal conductivity and thermal diffusivity for a sample comprising: an annular cylinder of a homogeneous material of known thermal conductivity and thermal diffusivity; a lower insulation layer base for the annular cylinder, the inner radius of the annular cylinder and in the lower insulation layer base defining a well for receiving a sample to be tested; an upper insulation layer for capping the top of the annual cylinder and the well; a heat source probe for insertion into the sample along the central axis of the annual cylinder for heating the sample; a power source operably connected to the heat source probe for providing power to heat the heat source probe;a thermocouple temperature sensor for insertion into the sample or the annual cylinder at a radial distance from the central axis for measuring temperature at the radial distance; and a temperature data acquisition unit operably connected to the thermocouple temperature sensor for acquiring and recording temperature data from the thermocouple temperature sensor.