The [5] For measuring the effective thermal conductivity, a uniformity of the five different size fractions was checked commercial probe TP02 thermal conductivity probe from under a microscope for random samples.
For all size the Dutch company Hukseflux was used. The design of this fractions the effective thermal conductivity was first deter- sensor is based on the principle of the hot wire method, a mined under atmospheric conditions and then at different well known non steady state technique. The TP02 probe, gas pressures.
Supplementary, the effective thermal conduc- shown in Figure 2, consists of a heating wire and two tivity of a defined mixture of glass beads, consisting of the thermocouples embedded in a stainless steel tube. One three biggest grain sizes 1. There- and reacts to the applied heating, the other is placed at the fore, equal volumes of each of the three grain sizes were tip and acts as the reference. This configuration allows the mixed. For this case one container was used to scoop equal verification that the heat flow only occurs in radial direction volumes of each of the three types of glass spheres into the in this case the readings of the thermocouple located at the sample container.
Additionally, a was carefully filled up to edge. The different size fractions temperature-dependent resistor is placed at the base of the were added in an alternating order. In all from each grain instrument. The probe method has already been used very size the same amount of scoop containers was added.
The porosity of all samples ; Koemle et al. Data Evaluation and applying equation 1. This was done five times per [6] The data evaluation for probe measurements is based grain size fraction each time the volume was refilled and on the theory of an infinitely long and thin line heat source embedded in an infinitely extended medium. The medium is heated by the line source with a constant power supply. A Table 1a. Chemical Composition of the Used Glass Spheres as solution of the heat conduction equation for this configuration Given by the Producer Sigmund Lindner was given by Carslaw and Jaeger [].
Figure 2. Hukseflux TP02 thermal conductivity measurement probe. The locations of the two thermocouples TC1 and TC2 and the heating wire are indicated.
The operational reliability of the measurement thermal conductivity of the surrounding medium. For a probe was assured by calibrating the sensor in Agar and in known heating power the latter can be calculated easily Teflon. The linear part was determined by sequential values. Sequential fitting 5. Experimental Setup and Measurement means, that a linear fit is applied to a part of the Description measurement curve e.
In the next step the data are fitted in between 80 [8] The thermal conductivity of the glass beads was and s, and again the results are stored. This is repeated investigated in a pressure range from atmospheric pressure until the fit interval oversteps the end of the measurement. A typical measurement curve is shown in Figure 3. For the evaluation of the thermal conductivity the part indicated in red was used. Boundary Effects [7] Equation 10 is an approximation only valid for the temperature development after the initial transient period, which can easily be identified in Figure 3.
It is reasonable to estimate the duration of the transient time period, if relation 10 is ought to be used. This was done using equation 11 which was derived by Vos [] and has been successfully applied to measurements performed with the same type of measurement probe as used in this work by Goodhew and Griffiths []. A second important fact is that the samples have finite dimensions and boundary effects may occur. The dashed For the estimation of ttrans and tmax tabulated thermal lines indicate the part used for the calculation of the thermal diffusivity values for dry and water saturated glass beads conductivity by linear regression.
The thermal conductivity of the glass sphere samples for the corresponding pressure level was deter- mined as the arithmetic mean of these three values. The time interval between the measurements had to be chosen to be long enough that the sample and the probe could reach again a thermal equilibrium state after completion of the heating period. Depending on the particular grain size and pressure level this took quite a long time.
The temperature gradient in the sample was logged continu- ously, not only during measurement. This was used to check if thermal equilibrium was reached. For measure- ments at pressures of approximately hPa the time periods between measurements were about 10 min. Under vacuum it took about a few hours until thermal equilibri- um was reached again.
The longest waiting periods were those for the smallest glass spheres under high vacuum only two measurements per day were possible. Corre- spondingly, the effective thermal conductivity for these samples was the smallest Table 2. Results [18] In the following the results obtained from the per- Figure 4. Setup of probe and sample during measurements. The thermal conduc- tivity values determined for the single grain sizes at twelve pressure levels are given in Table 2.
The measurement setup for this task consisted of ments taken at each pressure level. In Figure 5 the the following: dependence of the determined effective thermal conductiv- [9] 1. For all spheres. A vacuum chamber with a rotary vane pump to observed. A pressure measurement system. The measurement results for the glass beads [12] 4. An adjustable supply of nitrogen gas, which of 0.
The TP02 measurement probe described in section 4 with the corresponding data logger. The setup of probe and sample during a measurement is shown in Figure 4. Table 2. The sample container was filled with the particular Grain Size mm glass spheres and placed in the vacuum chamber. Then the Pressure hPa 0. After making sure that the probe and the 0.
Next, vacuum conditions were established by first 0. This process took up to several hours depend- 5. Constant pressure was 50 0. The different grain sizes are outgoing gas flux via the pumps was reached. The standard deviations mainly ranged [17] 3. Then a thermal conductivity measurement was between 0. For the case of the two smallest grain size fractions and pressure levels below 5. At each pressure level three measurements order of magnitude smaller.
Figure 5. Measured effective thermal conductivity of glass spheres of different grain size as a function of the respective pressure displayed on a semilogarithmic scale. At a higher pressure of []. Pore Sizes 0. Our value for the 0. At the beginning and end of this region the conductivity found for the 0. Below smaller 0. Furthermore, the measured a pressure of approximately 0. This mental results reported by Reichenauer et al. Therefore, the effective conductivity in this pressure mately 0.
Figure 6. Reciprocal thermal conductivity change versus reciprocal pressure plotted for the different grain size fractions. The higher ratio of Woodside and Messmer [] is maybe due to the fact that the used conductivity value was too high, since the lowest pressure level they investigated was above 0. A second possible explanation, mentioned by Presley and Christensen [a], is that Woodside acknowledged that the conductivity measurement probe used by Woodside and Messmer [] sometimes produced higher values.
Furthermore, the pore sizes were also determined using equation 3. Except for the smallest glass beads, these strongly differ from the values derived from the measurement data. For the smallest grain size fraction the pore size derived from the measurements and the pore size determined from equation 3 are in the same range.
In Figure 7 the pore sizes determined from the data and estimated by equation 3 are shown as a function of grain size. Figure 7. Pore sizes determined from the measurement data using relation 9 chain-dotted line , and pore sizes 6. Knudsen Number estimated by equation 3 solid line ; these sizes are plotted [20] The behavior of the gas in the pores under different against the mean grain size.
Therefore, this dimensionless factor was estimated for these measurements. In Figure 8 the radiation. This meets relation 8 for the vacuum conduc- derived Knudsen number as a function of pressure is tivity. From the given measurement data the pore sizes of displayed for various conditions in a double logarithmic the different samples were determined using relation 9.
In each case the black line. First, the pore sizes calculated from equation conductivity value measured at the lowest pressure level 3 were used to estimate the Knudsen number for the given of 0. The before stated pressure limit of 0. Below this pressure the gaseous conduc- beads fraction. Woodside and Messmer [] calculated a tivity becomes negligible.
For a lead shot sample of 1. This causes a shift of the Knudsen grain size they computed a ratio of Second, the Knudsen number was also evaluated 0. Figure 8. Derived Knudsen number versus pressure for different conditions of grain size, temperature, and interstitial gas displayed on a double logarithmic scale. Figure 9.
Thermal conductivity as function of the grain size for two low pressure levels. Both show a linear behavior. These pore sizes are significantly smaller than the was evaluated.
The derived radiative conductivity as a estimated ones in the case of the biggest glass spheres, but function a of temperature for different particle sizes is the values get closer with decreasing grain size. The largest given in Figure It can be observed that the larger the pore size difference occurs for the 4. Fur- This causes a shift of the pressure limit for the largest grain thermore, for larger particles a stronger temperature depen- size fraction toward higher pressures, while hardly any shift dence can also be noted.
The radiative contribution calculated occurs for the 0. Furthermore, it for the biggest glass spheres 0. In the case of the smallest glass beads of 0. Vacuum Conductivity is two magnitudes smaller than the measured vacuum [21] Below a pressure of about 0.
In this region equation 8 is valid. A grain size — dependent splitting up 6. Effective Thermal Conductivity of a Mixture of the vacuum conductivity can be observed. The highest of Glass Spheres values at hand are those for the biggest glass beads.
More- [22] Additionally, the thermal conductivity of a mixture over, nearly identical thermal conductivity values were of glass spheres was investigated at different pressure found for the two smallest fractions of glass spheres the levels. The mixture was composed of glass spheres with 0.
In Figure 9 the variation of the conductivity as a 1. From each grain size equal volume fractions function of grain size is plotted for two low pressure levels, were added see section 3.
The thermal conductivity values namely, 0. Here the observed decrease of determined for the mixture are listed in Table 4. A plot of conductivity with decreasing particle size is linear. This is in the effective thermal conductivity as a function of pressure agreement with model calculations performed by Tavman is shown in Figure 11 in a semilogarithmic scale.
For [], who also found a decrease of the effective thermal comparison the data are plotted along with the values conductivity with decreasing grain size. The linear relation determined for the single grain sizes. It is striking that between grain size and effective thermal conductivity for the thermal conductivity of the mixture shows a stronger the determined set of conductivity data is valid for pressures rise with pressure than the conductivity determined for the below approximately 0.
Above this pressure con- ductivity versus grain size is linear in the double logarithmic scale. This is in agreement with the results of Presley and Table 3. The vacuum conductivity is 0. For the radiative 0. For those the linear relation fits best.
Figure Radiative conductivity calculated by relation 5 as a function of temperature. Three different particle sizes are displayed. At hPa the thermal conductivity of diameter is 0.
In the analysis of a mixture performed by Woodside and summary, the behavior of the glass sphere mixture signif- Messmer [], who proposed that the effective diameter icantly deviates from the behavior of the single grain size is given as the harmonic mean of the particle diameters of samples.
A possible explanation for this different behavior the mixture components. The derived effective diameter was may be the different texture of the mixture, since the voids used to calculate the mean mixture pore size by equation 3. This point is also indicated by the porosity of 0. Summary and Conclusions mixture. This interpretation would explain the matching conductivities at low pressures, where the mixture acts like [23] The effective thermal conductivity of glass spheres a granular material with a grain size smaller than the grain of different grain sizes was investigated under varying sizes of which the mixture is composed of, while at high pressure conditions.
The determined values are in good pressures an increase of the thermal conductivity over that agreement with data obtained from previous studies.
In of the single grain sizes occurs owing to increased con- addition, a well defined mixture of different grain sizes duction via the solid phase. A similar behavior in the case was examined. This specimen showed a different behavior of a grain size mixture was found by Woodside and compared to the tests with single grain sizes.
The results Messmer []. Natural samples investigated by Presley indicate that at lower pressures the mixture can be described and Craddock [] did not display such an enhancement as an effective granular material with a grain size smaller of the thermal conductivity of a mixture of grain sizes at high pressure, compared to the thermal conductivity of the Table 4.
Effective Thermal Conductivity Values Found for the single grain sizes at high pressure. The conductivity data Mixture of Different Grain Sizesa obtained for the mixture were also analyzed using relation 9. A representative pore size of 0. The evaluation 0. A polynomial fit was performed on 1. From this graph, pore size and grain 5. This 50 0. The corresponding particle a The standard deviation for these values is in the same range as for the conductivity values obtained for the single sizes.
Thermal conductivity values determined for an equivolume mixture of glass spheres of three size fractions compared to the conductivity values determined for all the single-sized samples.
The data are given on a semilogarithmic scale. The mixture components are indicated by chain-dotted lines, while the two smallest grain size fractions are indicated by dashed lines. This case is only slightly from the laboratory conditions.
From the most important, since granular material on planetary surfa- measurements and the Knudsen number can be seen that ces is always a compound of several different grain sizes. For very using the concept proposed by Woodside and Messmer small grain sizes Knudsen diffusion might be reached, for [] and relation 3 given by Kaviany []. For the larger ones not. This implies a significant thermal conduc- single-sized samples pore sizes about two magnitudes tivity difference depending on the grain size and a high smaller than the grain sizes were determined from the sensitivity to pressure changes.
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