# How the thermal conductivity is measured in a solid?

The non‐steady‐state or transient technique records a measurement during the heating process. The method determines thermal conductivity properties by means of transient sensors. These measurements can be made relatively quickly, which garners an advantage over steady‐state techniques. For this reason, numerous solutions have been derived for the transient heat conduction equation by using one‐, two‐, three‐dimensional geometries. Transient methods generally employ needle probes or wires.

1. How the termal conductivity is measured in a solid?

There are a number of methods to measure thermal conductivity. In general, there are two basic techniques for measuring thermal conductivity: steady‐state methods and transient or non‐steady‐state methods.

The steady‐state technique records a measurement when a tested material’s thermal state reaches complete equilibrium. A steady‐state condition is attained when the temperatureat each point of the specimen is constant and the temperature does not change with time. A disadvantage, however, is that it generally takes a long time to reach the required equilibrium. The method involves expensive method apparatus since a well‐designed experimental installation system is usually needed. Nevertheless, it is the primary and most accurate measurement method.

Figure 1: Common methods for determining thermal conductivity of solids.

Steady‐state methods apply Fourier’s law of heat conduction to measure thermal conductivity. The solution to the problems with the different steady heat‐flow methods is to convert the heat transfer problem to a one‐dimensional problem, thus simplifying the mathematics. The calculations change for the models of an infinite slab, an infinite cylinder, or a sphere. The typical specimen geometry, the configuration of a measurement system, and the magnitude of the thermal conductivity are used to distinguish between different types of thermal conductivity measurements. The thermal magnitude of the measuring object is determined by the following measuring techniques using the direction of the heat flow, the conservation of the heat flow, and an auxiliary layer having a known thermal property.

2.1.Guarded hot plate (GHP)

The guarded hot plate, also known as the Poensgen apparatus, is the most commonly used and most effective method for measuring the thermal conductivity of insulation materials. The geometry of the sample or the chamber where the sample is a plate or a cylinder with axial heat flow. Depending on the thermal conductivity and the homogeneity of the material under investigation, the thickness of the sample varies between a few millimeters and a few decimeters. Its operation is based on the establishment of a stationary temperature gradient over a known thickness of a sample and on controlling the flow of heat from one side to the other.

It is very important to control the heat flow so that it is unidirectional and perpendicular to the sample. In fact, determining the thermal conductivity of the sample uses Fourier’s Law in one dimension. Under this approximation, the amount of energy supplied to the hot plate to create a particular temperature gradient through the sample is proportional to the thermal conductivity of the material

Figure 2. The apparatus of guarded hot‐plate method for thermal conductivity measurement. (a) two specimen with/ without auxiliary heaters and secondary guards, (b) single‐specimen

One disadvantage is that establishing a steady‐state temperature gradient through a specimen is time‐consuming when using the GHP and other steady‐state techniques. Other potential disadvantages are that the temperature gradient must be relatively large, the specimen width must be large, and also that the contact resistance between the thermocouple and the specimen surface poses a major source of error

Hot plate is electrically heated and the cold plates are Peltier coolers or liquid‐cooled heat sinks. The configuration is arranged symmetrically, with guarded hot plates located on the sides while the heater unit is sandwiched between two specimens or a single specimen and an auxiliary layer (Figure 2). The different types of guarded hot‐plate apparatus are shown in Figure 2. In the single‐sided system state, the heat flow passes through one specimen, while the top of the main heater acts as an insulating guard, thus ensuring an adiabatic environment. These heat measurements are recorded by differential thermocouples, which are instruments that control a flat electrically heated metering area that is surrounded on all lateral sides by a guard heater section.

2.2.Guarded heat flow meter method.

This method is similar to the hot plate, except that instead of measuring temperature differences, the heat flux through the sample is measured. This is achieved by means of one or two heat flow sensors permanently installed in the appliance.

In many cases, heat flow sensors consist of a series connection of thermocouples across a thermal resistor, for example a thin ceramic or plastic plate. In that case the signal is a thermal voltage proportional to the temperature drop across the board. In more modern designs, thermoelectric or Peltier effect modules are used as heat flow sensors, which generate an electric current proportional to the flow of heat that passes through them.

Figure 3. Schematic design of heat‐flow meter basically

2.3. Direct heating method

Two disadvantages of steady‐state methods are the lengthy time requirements and the difficulty of determining heat loss, especially at high temperatures. These disadvantages can be overcome by the direct heating method, which can be used for electrically conductive materials such as metals. The specimen, such as a wire, pipe, or rod, is placed in a vacuum chamber, clamped between two heat sinks cooled by liquid, and the specimen is heated up to temperatures in the range of 300–4000 K. Figure 4(a) portrays the schematic of the design of the direct heating method.

Voltage drops and temperatures are measured: in the middle of the rod and on each end of the rod. From these three measurements obtained in the direct heating method, the thermal conductivity and the specific electric resistivity k can be calculated

2.4. Pipe method

The pipe method takes advantage of a radial heat flow in a cylindrical specimen. A core heater, which is a tube, rod, or wire, is inserted into the central axis of the pipe‐shaped specimen. There are heaters at both ends of the specimen. The combination of the specimen and heaters is surrounded by thermal insulation and then a water jacket or a liquid‐cooled heat sink. Figure 4(b) shows the schematic and components of the pipe method. End guard heaters can be used to minimize axial heat loss, and also increasing the specimen’s ratio of length to diameter can achieve the same purpose

Figure 4. (a) The schematic design of the direct heating method, (b) the schematic design of the pipe method.

• simple mathematical expression,
• absolute and primary method for low conductivity specimens
• acceptable time consumption
• partially suitable for powdered, granular, or solid forms
• uncertainties of 1–2% for insulations near room temperatures
• acceptable small test specimens (except for concentric sphere)

• complexity of the apparatus giving high accuracy
• uncertainties of 10% or higher to the conditions
• time consuming
• immeasurable error due to contact resistance
• difficulty of measuring geometrically shaped specimens (concentric cylinder or concentric
sphere)
•  heat losses from especially in parallel plate and concentric cylinder methods
• difficulty of measurement of heat‐flow value for two specimens
• use error of specimens containing moisture

3. Transient methods

The advantages of transient methods are mainly distinguished by the short amount time needed, so that various thermal values can be determined in the measurement process. Therefore, this method is based on a signal measurement and an acceptably small temperature differential. The transient technique is measured by evaluating the feedback response after a signal is transmitted to the specimen for heat generation in the specimen. Therefore, test time is obtained in a few minutes or a subsecond time intervals for transient methods. This method is also more appropriate for high moisture content materials because of the signal and response in the specimen. In many cases, it is possible to replace the temperature measurements at two opposite surfaces with a measurement as a function of time at only one position on the specimen

Among transient methods, the hot‐wire and the laser flash methods are commonly used for measuring the thermal conductivity of different materials. A modification of the hot‐wire method is the hot‐strip or disk technique, which can be applied to solid non electrically conducting materials in order to measure the thermal diffusivity and conductivity

3.1.Hot‐wire method (THW)

The hot wire technique consists of keeping a conductive wire immersed (eg platinum (Pt) or tantalum) in the material under study and pass a constant electric current through it, so that it is heated by the Joule effect.

A schematic representation of the hot wire model is shown in the Figure. The speed with which the temperature of the conductive wire increases depends on the heat dissipated by conduction towards the surrounding material. Therefore, by measuring said speed, the thermal conductivity of the material can be obtained. In fact, in the case of liquids, it has become the standard reference. Some authors use mercury capillaries to measure refrigerant mixtures

Figure 5. The schematic principle and the design of the hot wire.

3.1.1.Heated-Needle Probe.

It is a variant of the THW, in which the fine conductive wire is replaced by a slightly more thick and much more resistant. The greater robustness, compared to THW, has allowed the development of commercial devices. This technique or device is especially suitable for measurement of the thermal conductivity of granular materials such as powders and soils, natural materials such as stone and concrete, and even food.

Figure 5A. The schematic principle and the design of the needle probe.

The Figure schematically shows the probe used in this technique. It consists of a fine hollow metal needle (3mm diameter) that contains a separate heating resistor and thermal resistor. The needle acts simultaneously as a heat source and a temperature probe afterwards. The history of the probe temperature is usually interpreted with the help of the same equations of a THW but in a relative way, that is, by calibrating its response against known standards. This rather simplistic approach to the analysis of a somewhat complex cell inevitably restricts the accuracy that can be achieved. However, it provides a measurement capability where perhaps no other technique is viable. It is often used for measurements on inhomogeneous samples such as rocks or soils where thermal conductivity is simply required.

3.1.2.Transient hot- strip method.

The heat foil method consists of three parallel nickel bands connected to an electrical circuit that provides a constant flow of heat, the outer bands act as thermal protectors, forcing the heat flow vector to be perpendicular to the surface of the the probe.

The circuit is designed to monitor the temperature of the central strip, and the data acquisition allows obtaining the increase in the temperature of said central strip, and the time elapsed since the start-up of the heat output. When this device is in contact with the flat surface of a sample, the temperature of the center strip depends on the thermal properties of the sample.

3.2. Hot‐disk method

The transient plane source (TPS) technique is a recent development of the hot‐strip method. It is also known as the Gustafsson probe or the hot‐disk method. The technique is designed to measure both thermal conductivity and thermal diffusivity. The advantage of transient technique to steady‐state technique is that the effect of the contact resistance is eliminated in the analysis. This method ensures accurate measurements over a thermal conductivity range from 0.005 to 500 W/(m K) over temperatures from 30 to 1200 K .The TPS technique is used for measuring the thermal conductivity of insulation materials and electrically conducting materials. The main advantages of the hot‐disk measurement are that it produces results quickly (usually in under 10 min), and that different sensor sizes can be used to accommodate different specimen types. Furthermore, the hot disk requires using specimen sizes that are usually much smaller than those used in other techniques

The hot‐disk method utilizes a sensor in the shape of a double spiral of nickel covered material. The TPS sensor consists of a number of concentric circles that are made into a double spiral so that the current will travel from one end to the other. A thin polymer coating material is used as electrical insulation and sensor protection on the spiral. The coating materials are most commonly Kapton for measuring temperature ranges between 30 and 450 K, Mica for higher temperatures of up to 1200 K, and Teflon. The sensor acts as both a heat source and a thermometer. The source and the thermometer are used to determine the changes in the temperature of the specimen and the increase in the time‐dependent temperature, respectively.

The sensor is sandwiched between two pieces of the specimen, as shown in Figure 6. During testing, a current travels through the nickel spiral and causes an increase in temperature. The generated heat dissipates throughout the specimen on either side. By comparing the temperature versus the time response in the sensor, thermal conductivity or diffusivity can be
calculated accurately.

Figure 6. The schematic and principle of the hot disk

3.3. Laser flash method

The laser flash method is the most commonly used method for ascertaining the thermal properties of solids. The method can investigate to properties of glasses, metals, and ceramics without significant limitations due to uncertainties of the achievable measurement. The property can be measured in a temperature range between −100 and about 3000°C.

In the method, a laser pulse is send to the front side of a specimen, and the temperature change on the back side is measured. The method is conducted through heating a specimen with a short laser pulse of 1 ms width on the front side of the specimen. The temperature increase at its rear side is measured and determined.

Figure 7. The schematic and principle of the laser flash method

3.4-Photothermic method.

The principle of this family of methods is based on a light-induced change in the thermal state of a material in solid, liquid or gaseous state. When light is absorbed by a sample, changes in temperature, pressure or density occur, which can be detected. There are methods in which the sample is in contact with the detection system and others that involve non-contact remote sensing systems. A disadvantage of these methods is the poor availability of optical properties of the material that are necessary

Photothermic methods for the determination of optical absorption and thermal properties of materials can be classified according to the detection technique used. These are based on measuring changes in:

• Temperature: Temperature changes are usually investigated by means of contact thermometry (for example, the photopyroelectric technique), radiation thermometry or calorimetric methods
• Pressure: The pressure changes are obtained by acoustic methods
•  Density: Density changes include the detection of variations in the index of refraction or deformations of the surface. The most important techniques are the thermal lens method, the thermal wave technique, beam deflection, refraction or diffraction methods

3.5.Thermal comparator method

The method is based on the well-known observation that when two materials at different temperatures are brought into contact in a small area, heat transfer takes place from the warmer to the colder body, which is a function of thermal conductivity. of the materials. As a result, an intermediate temperature is reached very early at the point of contact. The contact temperature depends on the thermal conductivity of the two materials. This technique is used to measure the thermal conductivity of organic liquids and liquid mixtures.

3.6.Temperature swing method.

One of the techniques of transient methods is the temperature swing method. The basic principle of this method is in the application of a periodic heat source in the contours. This produces the temperature swing for a sample location along the along its length with the same frequency as the applied heat source. The measure of the amplitude and phase shift of the temperature wave propagation can give the estimation of thermophysical properties.

3.7. The 3ω method

The method called the 3ω method is commonly used for measuring the thermal conductivity of thin films and solid materials. An AC current with frequency ω of angular modulation is passed through the wire. The wire is used simultaneously as a heater and a thermometer. The heat generated by this process diffuses into the specimen. Since the electrical resistance of the metal heater is proportional (linear) to the temperature, the temperature oscillation can be measured indirectly by measuring the associated 3ω voltage.

Because the current is driven at a frequency ω and the resistance changes at a frequency 2ω, a voltage at 3ω results. In this method, a thin electrically conductive wire is patterned on the specimen, as shown in Figure 8.

Figure 8: The schematic of the 3‐ω method for thin film

3.8. Fitch method

The Fitch method developed by Fitch is used to measure the materials of low thermal conductivity by using a plane source of heat. This method consists of two components: a heat source and a heat receiver. The heat source is a vessel filled with a constant temperature liquid that functions as a sink. The heat receiver is a sink in the form of a copper plug insulated all sides but the one facing the vessel. The roles of the heat source and the heat receiver can be changed if the vessel is at a temperature lower than that of the copper block. The specimen is interposed between the vessel and the open face of the plug. The sample is firstly in thermal equilibrium with the copper block as shown in Figure 9. The vessel is brought into contact with the specimen under a temperature differential. The temperature history of the copper block and the temperature of the bottom of the vessel are measured by thermocouples. It is assumed to have a uniform temperature distribution.

Figure 9. The schematic of the Fitch method

Javier Echave

PhD. Chemical Engineer

Innovation I+D Manager NTC (Nuevas Técnicas de Coloración)

Temperature and Heat management in Power Electronics for EVs

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