Thermal conductivity of solid paraffins and several n-docosane compounds with graphite

The technical importance of paraffins as phase change materials (PCM) in heat storage systems increases. Knowledge on the thermal conductivity of paraffins is necessary for the design and optimization of heat storage systems. However, for most paraffins solely the thermal conductivity of the liquid state has been sufficiently investigated. For the solid state, precise thermal conductivity data are only known for a few paraffins, while only generalized values are available for the remainder, some of which contradict each other. In this study, a measurement setup based on the modified guarded hot plate method is developed. It is used to investigate the thermal conductivity of several paraffines in the solid state, including pure n-docosane and its compounds with different types and concentrations of graphite. For n-docosane in the solid state, the thermal conductivity is determined to be 0.49 W m−1 K−1. A particle size of 200 μm with a spherical shape turns out to be optimal to increase the thermal conductivity. This allows the thermal conductivity of a compound with 10% graphite to increase by a factor of three compared to the pure paraffin. Furthermore, significant differences to thermal conductivity data from the literature are found.


Introduction
The technical importance of phase change materials (PCM) has increased in recent years. PCM are used for the shortterm and long-term storage of heat in order to save energy in a wide range of applications, such as household appliances, PV module cooling or heating systems [1,2]. In particular, the temperature range around the ambient of about −50 to +50 °C is of great technical importance. In this temperature range, a large number of materials, like salt hydrates, water or paraffins, can be used as PCM [3]. In addition to an adequate melting temperature for the given process and a large enthalpy of fusion, further selection criteria, such as cycle stability, corrosivity, volume expansion, toxicity or flammability, have to be met for the design of PCM storage systems [4]. Despite the comparatively low enthalpy of fusion, paraffins offer great advantages for many processes due to the relatively freely selectable melting temperature, excellent cycle stability and non-corrosive behavior. Hasnain published a comprehensive review on the thermophysical properties of numerous PCM, such as density and isobaric heat capacity. However, the thermal conductivity was not dealt with in depth in this review [5].

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Especially in processes with short-term heat storage, the thermal conductivity is an important property because it determines how quickly the PCM can be charged and discharged [6]. Technically, this can be optimized with additives that increase the thermal conductivity or the structural design of the PCM container [2]. In practical use, the relatively low thermal conductivity of paraffins is often compensated for by applying the paraffin over a large area with a small layer thickness [7][8][9][10]. Furthermore, it should be noted that the thermal conductivity of solid and liquid states differs significantly from one another.

Thermal conductivity of paraffins
In 1969, Bogatov et al. measured the thermal conductivity of n-tetradecane, n-pentadecane and n-hexadecane in the liquid state over a temperature range from 20 to 200 °C and a pressure range from 0.98 to 490 bar [11]. Shortly after that, Naziev et al. investigated the thermal conductivity of n-decane [12]. In 1994, Vargaftik et al. published a comprehensive collection of thermal conductivity data for hydrocarbons in liquid and gaseous states [13].
Investigations of the thermal conductivity of paraffins are listed in Table 1, indicating the melting temperature and enthalpy of fusion. This includes various paraffins from n-decane to n-triacontane, n-tetracontane and n-pentacontane. For most of the paraffins addressed in Table 1, only the thermal conductivity of the liquid state was investigated, or the literature does not indicate which state the values refer to. In 1983, a review article by Abhat assumed that the thermal conductivity of all paraffins from n-tridecane to n-pentacontane is generally = 0.21 W m −1 K −1 , with the exception of octadecane, = 0.15 W m −1 K −1 [14]. However, these values contradict other studies. The thermal conductivity of the solid state is only known for n-hexadecane, n-octadecane and n-eicosane. The results for these four paraffins were published by Vélez et al., Sasaguchi et al. and Sharma et al. for n-octadecane [15,16,17] and are highlighted in boldface in Table 1. Furthermore, Sharma et al. stated that the thermal conductivity of solid n-dodecane, n-tetradecane, n-hexadecane, n-nonadecane and n-hexacosane is throughout 0.21 W m −1 K −1 , which partially conflicts with the results discussed above (in italics in Table 1).
It can be concluded that there is an extensive literature dataset for the thermal conductivity of liquid n-docosane, but the thermal conductivity of this paraffin in its solid state was not addressed yet.

Thermal conductivity of n-docosane with graphite
Composite materials with a higher thermal conductivity can be created by mixing n-docosane with graphite. Using Eshelby's equivalent inclusion method, Hatta derived a theoretical model for the thermal conductivity of composite materials consisting of different phases [23]. Figure 1 shows the results of investigations on the thermal conductivity of graphite-compounded n-docosane by Sari et al. [20] from 2007 and Li et al. (2013) [21] from 2013. In another study by Li et al. (2014), the addition of graphite resulted in an increase of the thermal conductivity from 0.26 to 0.59 W m −1 K −1 , but the authors did not specify the mass fraction of graphite [22].

Methods for the measurement of thermal conductivity
A large number of methods has been proposed for measuring the thermal conductivity [13].
With the laser point method, a defined amount of radiation energy is introduced into the sample in a temperaturecontrolled vacuum chamber by means of a pulsed laser or a xenon discharge lamp. A photodiode and a temperature sensor attached to the back of the sample are used to determine the ensuing temperature rise in the sample. With the measured temperature profile, the thermal conductivity of the sample can be calculated. This method can be used for solids, regardless of their surface properties, up to temperatures of 2000 °C. However, applying this method to materials with a melting point close to the ambient temperature can result in melting of the sample at certain points so that it is not usable for paraffins [24].
In the hot wire method, a platinum wire is surrounded by the sample material. The heat introduced thermally into the sample can be determined from the electrical current or voltage and the resistance of the wire. The platinum wire serves simultaneously as a heating element and as a temperature sensor. This method can be used when the thermal conductivity is in the range from 0.005 to 5 W m −1 K −1 and the evaluation is carried out using either the transient or the periodic method. In the transient method, the thermal conductivity can be calculated from the heating curve, and in the periodic method, it is determined via a frequency analysis of the alternating periods of heating and cooling of the sample [18,[25][26][27][28].
The measurement setup of the plate conductivity meter is primarily used to determine the thermal conductivity of materials with a higher heat conduction. In this measurement setup, a cylindrical sample is placed between a heat sink and a hot plate in a vacuum chamber. The thermal conductivity can be determined by the imposed heating power and the sample geometry. In order to minimize external influences, a protective heater system is also used in the vacuum chamber.
Due to the low thermal conductivity of paraffins, this measurement method is not suitable for such materials [29,30].
For solids with a low thermal conductivity, like insulation materials or paraffins, the guarded hot plate method is used. The basic measurement setup is shown in Fig. 2. In this symmetrical setup, a temperature gradient ΔT is generated in two identical samples via a centrally located electrical heating element and cooling plates attached to the top and bottom of the stack. By measuring the power P supplied to the heating element, the thickness of the samples s and the surface area of the heater A , the thermal conductivity of the samples can be determined by The heating element is surrounded by a ring-shaped protective heater, the temperature of which is regulated to be identical with that of the heating element so that radial heat flow through the sample is prevented. This measurement method can be used for solid materials with a low thermal conductivity of up to 5 W m −1 K −1 over a wide temperature range [31,32].

Modified guarded hot plate
A modified guarded hot plate measurement setup was developed for this study. The solid sample was placed between a cooling plate and an inner main heater. By specifying the temperature gradient between the main heater and the cooling plate, a heat flow was generated through the sample. A protective heater cover and a protective heater ring prevented heat loss from the main heater to the ambient. For this purpose, the protective heaters were regulated to the same temperature as the main heater with a closed loop controller. The basic measurement setup is shown in Fig. 3. Compared to other methods, this new measurement setup is characterized by a simple structure and straightforward use.   The heater had a diameter of 116 mm, the cooling plate had a diameter of 158 mm and the gap between the main heater and the protective heater was 1 mm. The protective heaters had a thickness of 20 mm both in vertical and radial directions.
The solid PCM sample was placed between the heaters and the cooling plate. Since surface roughness of the heater and the sample leads to air pockets between the sample and the heater and also between the sample and the cooling plate, a smooth, soft heat conducting foil was placed on both surfaces of the sample. The foil compensated for the remaining unevenness of the sample surface, thus minimizing the trapped layer of air. The heat conducting foil (SB-HIS-5 from DETAKTA Isolier-und Messtechnik GmbH & Co KG) was made of a silicone glass fabric and had a thermal conductivity of 5 W m −1 K −1 and a thickness of 0.2 mm.
The heat flow Q through the cross-sectional area A of the sample with a thickness s is given by Fourier's law [33] The thickness of the air s a summarily represents thin layers of air on top and bottom of each foil. With measurements of samples with a known thermal conductivity and a varying thickness s , the average layer thickness of the air inclusions s a was found to be 0.8 µm. To adequately deal with the rather uncertain air layer thickness, a large error bar was assigned to it, i.e., s a = 0.8 ± 4 µm. By measuring the electrical heating power P of the main heater, the sample thickness s and the temperature gradient ΔT between the main heater and cooling plate, together with thickness and thermal conductivity of foil and air inclusions, the thermal conductivity of the sample can be determined by The electrical heating power was applied and measured over the range from 0 to 200 W with a power supply Agilent E3633A with an accuracy of ± 1% of the measured value.
To determine the sample thickness, a caliper gauge with an accuracy of 0.02 mm was used.
With the uncertainties of the individually measured values listed in Table 3, the total uncertainty of the measured thermal conductivity can be determined via the error propagation law by Gauß.
All measurements were carried out at an ambient temperature of 293.15 K and a pressure of 0.1 MPa.

Sample materials
For the measurement of the pure paraffins and compounds of n-docosane with various types and quantities of graphite, cylindrical samples were produced. Once the samples were made, they initially had uneven and curved surfaces. Consequently, these were smoothed in two steps by sandpaper grinding with a grain size of P120 and P320. The n-docosane samples contained an additive to prevent the graphite particles from sinking to the ground of the sample so that they remained homogeneously distributed. Its influence on the thermal conductivity was investigated by a comparative measurement with a pure sample without additive. The difference was 0.06 W m −1 K −1 , which is negligible compared to the increase of the thermal conductivity due to the presence of graphite. The compositions of the samples are listed in Table 4. The paraffins (pure n-octadecane,  n-eicosane, n-docosane and n-docosane with additive) were provided by Axiotherm GmbH. Several expanded graphite powders from Sigratherm® GFG of types GFG5, GFG200, GFG600 and GFG1000HD were used. The technical characteristics of these graphite powders are listed in Table 4.

Results and discussion
The results of this work are compared to literature values for the solid state of the paraffins n-octadecane, n-eicosane and n-docosane in Fig. 4 and are listed in Table 4. The thermal conductivity data for n-octadecane and n-eicosane measured in this study agree well with the results of Vélez et al. [15] and Sasaguchi et al. [16]. All of these data are significantly higher than the results from Sharma et al. [17], Li et al. (2013) [21], Li et al. (2014) [22], Sari et al. [20] and Abhat [14]. From measurements of the thermal conductivity in the liquid state, it is known that it increases with rising carbon chain length [34]. With the thermal conductivity data determined in this study, this can now also be confirmed for the paraffins n-octadecane, n-eicosane and n-docosane in the solid state. Figures 5 and 6 show the results for the thermal conductivity of n-docosane as a function of the particle size and varying mass fraction of graphite. In the present work, a thermal conductivity of pure n-docosane of 0.49 W m −1 K −1 was measured, which is indicated by the horizontal line in Fig. 5. The addition of graphite increases the thermal conductivity, while particle size and shape of the graphite were found to have a significant influence. With a mass fraction  [22] Sari et al. [20] Thermal conductivity/W m -1 K -1 Carbon chain length Thermal conductivity of n-docosane-graphite compounds as a function of particle size and varying graphite content of 10% for the graphite types GFG200 and GFG600, the thermal conductivity increased almost threefold to a value of 1.41 W m −1 K −1 . The finest graphite GFG5 considered in this study only showed an increase by a factor of ~ 1.4 to a value of 0.67 W m −1 K −1 . With the largest particle size, the GFG1000HD graphite led to the smallest rise of the thermal conductivity. Graphite incorporated into the compound creates local thermal bridges that conduct heat within the paraffin. Mathematically, this can be described by a combination of serial and parallel connections of more thermally conductive graphite within the paraffin. Models of Lewis and Nielsen from 1970 that rest on this idea have prevailed in the literature [35].
The results of the paraffin samples compounded with graphite powder with a particle size of 1000 μm show a significantly lower thermal conductivity compared to the smaller particle sizes. This is due to the more rod-shaped form of the GFG1000HD particles. The thermal conductivity deteriorates significantly due to the smaller extent of the rods in transverse direction. With a theoretical arrangement of the rods in longitudinal direction aligned with the heat flow, the thermal conductivity should increase significantly. But such an application is practically impossible, since the graphite particles rearrange themselves with each phase change cycle of the paraffins, resulting in a disordered particle arrangement.
In comparison with the results of Sari et al. [20] and Li et al. (2013) [21], the thermal conductivity measured in this study shows higher values and a larger increase of the thermal conductivity upon the addition of graphite, cf. Figure 6. It should be noted that the temperature of the measurements of Sari et al. [20] and Li et al. (2013) [21] was not clearly defined so that their values may refer to the liquid state.

Conclusions
In addition to the enthalpy of fusion and the melting temperature, the thermal conductivity is an important design parameter for PCM. With the knowledge of these properties, thermal energy storage systems can be adequately designed and the time period of a phase change can be determined. Despite the extensive studies on the thermal conductivity of paraffins in the liquid state from in the 1970s to the early 1990s, the thermal conductivity of paraffins in the solid state has been poorly investigated so far. A thermal conductivity of solid n-docosane between 0.205 and 0.26 W m −1 K −1 specified in older investigations seems to be too low. The thermal conductivity of solid n-docosane was determined in the present work to be 0.49 W m −1 K −1 . The results of Vélez et al. and Sasaguchi et al. for the thermal conductivity of n-octadecane and n-eicosane were confirmed in this study.
By adding graphite powder, the thermal conductivity of n-docosane can be significantly increased by more than a factor of three with a graphite mass fraction of 10%. The selection in terms of quantity and shape of the graphite powder has to be optimized so that the desired thermal conductivity is achieved, while minimizing costs. Spherical particles were found to be adequate.
The presented measuring setup is a suitable tool for determining the thermal conductivity of solid paraffins. With this setup, a reliable determination of the thermal conductivity in a range from below 0.2 to 2 W m −1 K −1 was possible. Since the thermal conductivity of only a few paraffins in the solid state is known, these materials should be further investigated because of their technical relevance.
Author Contributions AP contributed to conceptualization, methodology, validation, formal analysis, investigation, resources, data curation, writing-original draft, writing-review and editing, visualization, supervision, project administration. EB contributed to conceptualization, methodology, resources, writing-review and editing. MOD contributed to conceptualization, methodology, investigation, resources, writing-review and editing. IHD contributed to conceptualization, methodology, resources, writing-review and editing. GS contributed to writing-review and editing. JV contributed to conceptualization, methodology, resources, writing-review and editing, project administration, funding acquisition.
Funding Open Access funding enabled and organized by Projekt DEAL.
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