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Significant hidden temperature gradients in thermogravimetric tests
Polymer Testing 68 (2018) 388–394
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Short Communication: Analysis Method
Significant hidden temperature gradients in thermogravimetric tests a
b
T
b
Carlos Gracia-Fernández , Begoña Álvarez-García , Silvia Gómez-Barreiro , Jorge López-Beceirob, Ramón Artiagab,∗ a b
TA Instruments-Waters Cromatografía, Alcobendas, 20108, Madrid, Spain University of A Coruña, EPS, Avda. Mendizábal s/n, 15403, Ferrol, Spain
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermogravimetry Temperature gradient Error Simulation
In thermal analysis, correct measurement of temperature is usually ensured by means of a calibration procedure. However, in addition to make sure that the right value of temperature is measured, estimation of temperature gradients into the sample is important. That is especially important in thermogravimetric (TG) analysis, where combinations of some of the common sample sizes heating rates could lead to important differences of temperature into the sample. If there is a significant gradient between different parts of the sample, then the temperature of the thermocouple, although correct, does not actually represent the temperature of the whole sample. While the correctness of the temperature is always important, errors in temperature measurement are critical in kinetic studies. Thus, estimations of the temperature gradients that appear into the sample as a result of a given treatment are of highest interest to choose the right operational conditions that minimize that gradient. That is particularly important for kinetic studies. In this work, thermal gradients originated into a sample during a typical TG test are estimated through a simulation study performed on the Comsol ™ software. A typical vertical TG furnace, sample size of about 125 mg, and several heating rates were used. Additionally, samples of different void contents were considered. The results of the simulation show that significant gradients of temperature can be achieved into the sample with experimental conditions like those that are often used. It is also observed that the difference of temperature between the sample and the furnace wall not only depends on the heating rate, which can be easily corrected by calibration at the corresponding heating rate, but also varies with temperature, which makes highly recommended to calibrate in more than one temperature point when broad ranges of temperature are considered.
1. Introduction Thermogravimetry is a technique that measures the mass of a sample as a function of temperature or time while it is subjected to a controlled temperature program in a controlled atmosphere [1]. The origin and first developments of this technique were thoroughly described by different authors [2–5]. Most of the early thermobalances were constructed by individual investigators [6], such as Nernst and Riesenfeld [7], Brill [8], Truchot [9], Urbain and Boulanger [10] and Honda [11] at the beginning of the twenty-first century. It is also remarkable the work of Duval [12], who developed an automated analytical method based on thermogravimetry. His work provided a strong impetus for this technique [13]. The first commercial thermobalance appeared in 1945 and it was based on the work of Chevenard [14]. The evolution has been fast from the beginning up to now, and the sensibility and precision of the thermobalance were increasing continuously. Nowadays, TG is one of the most common thermal analysis techniques
∗
Corresponding author. E-mail address: [email protected] (R. Artiaga).
https://doi.org/10.1016/j.polymertesting.2018.04.039 Received 4 December 2017; Received in revised form 5 April 2018; Accepted 26 April 2018 Available online 30 April 2018 0142-9418/ © 2018 Elsevier Ltd. All rights reserved.
and it is used in many industrial and scientific fields. A clear review of TG and other thermophysical characterization techniques has been provided by K.P. Menard [15]. Temperature calibration is routinely performed in any thermal analysis technique. While there are many works discussing the importance of temperature calibration and several standards indicating the right procedures to calibrate temperature of different instruments [16–26], only a few works paid attention to the possible gradients of temperature originated into the sample while subjected to a typical thermo-analytical temperature program [27,28]. Thermal gradients originated into the sample during a typical thermogravimetric (TG) test are estimated here through a simulation study. For the simulation, a typical vertical TG furnace, samples of different porosity and thermal conductivity, and a few of the most common heating rates were used.
Polymer Testing 68 (2018) 388–394
C. Gracia-Fernández et al.
temperature distribution into the sample, four levels of porosity were considered: 0, 8, 16, and 25%. B-type: In order to evaluate the effect of thermal conductivity four samples with no porosity but with different thermal conductivities were considered: 0.42, 0.21, 0.084, and 0.042 W/(m. K). In all cases the density was 2650 g/m³ and heat capacity 758 kg/J/K. 3. Fundamentals of the simulation The thermal gradients originated into the sample during a typical TG test are estimated through a simulation study performed by means of the Comsol software [29]. In order to perform the simulation, the three-dimensional geometry described above is reduced, for computation, to a two-dimensional geometry. That simplification should not significantly affect the results since the system, including furnace, platform and sample, is cylindrically symmetric except for the entry and exit holes. These three holes were conveniently located on the same plane so that a two-dimensional simulation can take into account the gas flow effect. On the other hand, the Navier-Stokes equations are coupled with the heat transfer model, being the pressure and the velocity field the solution of the Navier-Stokes equations, while the temperature is solved through the heat transfer model. The buoyancy exerted by the fluid, which depends on temperature and density, is introduced in the NavierStokes equations for compressible fluids. Simultaneously, the heat equation accounts for convective heat transfer. The Laminar Flow and the Heat Transfer in Fluids interfaces are coupled through the “Temperature Coupling” and “Flow Coupling” features of the software. The governing equations in the Laminar Flow and the Heat Transfer in Fluids interfaces are:
Fig. 1. Layout of the furnace geometry considered for this work.
2. Furnace geometry and operating conditions
- The Navier-Stokes equations
∂u 2 + u⋅∇u ⎞ = −∇p + ∇⋅⎛μ (∇u + (∇u)T ) − μ (∇⋅u) I ⎞ + F ρ⎛ 3 ⎝ ⎠ ⎝ ∂t ⎠
For the simulation, a typical vertical TG furnace geometry was considered. Fig. 1 shows a layout of the furnace where the locations of the gas inlet and outlet holes and of the sample holder can be observed. The sample holder consists of a typical open platinum pan. For simplicity, the hang-down system was not included in the simulation. The inner walls of the furnace are supposed to be made of alumina. In order to realistically reproduce the actual operating conditions, a purge of nitrogen is applied through the A and B inlets at 80 and 20 mL/min, respectively. While entering through two different holes, all gas goes out through one single outlet. All the simulated experiments consist of linear heating ramps. It is assumed that the temperature at the inner wall surface is always uniform. The heating ramp is directly applied on the furnace wall. This is an ideal situation since, in real instruments, the temperature is not completely uniform at the furnace wall surface and the wall surface is normally heated from a resistance embedded in the furnace material. Four heating rates, 5, 10 and 20 °C/min, were used for the different cases as it will be described below. These heating rates are very common in thermogravimetry. In order to evaluate how some features of the sample may affect the temperature distribution into the sample, two different types of samples were considered: A-type: this sample is defined as a cylindrical slice of 2.00 mm height and 4.16 mm radius, made of a hypothetical material with thermal conductivity and specific heat capacity values similar to those of a commercial polyamide 66 at 20 °C. For simplicity, it is supposed that the material keeps these features in all the range of temperature considered in the tests. Of course, many polymers would melt and degrade in that range, but including possible transformations of the samples would imply a very high complexity and it would be very difficult to generalize the results to any kind o material. Thus, the values used for simulation were 1140 kg/m³, which results in a sample mass of 124 mg, thermal conductivity of 0.43 W/(m. K), and specific heat capacity of 1670 J/(kg. K). In order to see the effect of voids in the
where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density and μ is the fluid dynamic viscosity. The left-hand side term corresponds to the inertial forces, the first term of right-hand side represents the pressure forces, the second term the viscous forces and the third term the external forces applied to the fluid. This equation is always solved together with the continuity equation:
∂ρ + ∇⋅(ρu) = 0 ∂t Solving these equations, for a particular set of boundary conditions, allows to predict the fluid velocity and its pressure for a given geometry. - Heat equation
q = hf (Tw − Tf ) where q is the heat flux, hf is the heat transfer coefficient of the fluid, Tf is the local fluid temperature and Tw is the wall temperature. The Reynolds number (Re) allows to predict laminar or turbulent patterns for different fluid flow situations. For flow in a cylindrical pipe, the Reynolds number can be defined as:
Re =
ρvs D μ
where ρ is the fluid density, vs is the average velocity of the fluid, μ is the fluid dynamic viscosity and D is the hydraulic diameter of the pipe. According to the Reynolds number prediction, considering the furnace and sample geometry and the gas flow rate, the flow will be always of the laminar type. 389
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Fig. 2. Temperature distribution into the furnace at different times.
390
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4. Results and discussion 4.1. Temperature distribution and gradients within the sample In this case an A-type non porous sample was chosen. Fig. 2 shows the temperature distribution into the furnace at different times, assuming that a 10 °C/min heating rate is applied on the inner furnace wall. It is assumed that the temperature is uniformly distributed in the entire furnace wall at any time. In practice, physical furnaces were designed to approach to that ideal condition but there are always some deviations that depend on the geometry and operation conditions. Those deviations would imply an additional complexity that is not included in this work. It can be observed that although the scales of temperatures are different, the pictures obtained at different times are almost the same: there is no important gradient of temperature except for the sample and the regions around the sample. That is not surprising considering that sample and sample holder have very different values of heat capacity (Cp) and thermal conductivity than the nitrogen atmosphere into the furnace. That differences of Cp lead to different increments of temperature depending on the material of which the different parts are made. On the other hand, the thermal conductivity of the gas originates a delay in the sample temperature with respect to that of the wall surface. Similarly, the thermal conductivity of the sample originates a delay in the temperature at the center of the sample with respect to the surface of the sample. Fig. 3 shows how the temperature difference between the center of the sample and the wall inner surface varies along time. A maximum difference is observed at about 12 min when using a 10 °C/min heating rate. Fig. 4 shows the gas speed distribution and the direction of the flow into the furnace at a given moment. Both the peed distribution and the pattern of flow practically did not change along time. The highest speed values are observed at the entering and exit points. The speed is almost zero for the sample-gas interface, but it increases as we move upward away from the sample. This is important to sweep off the volatiles resulting from the thermal degradation of the sample. Once the volatiles go out from the sample, they go into the gas flow and are swept to the exit hole. On the other hand, no significant variations in the gas speed distribution or the pattern of flow were observed along time with respect to those of Fig. 4. Fig. 5 plots the x-component of the gas speed along a vertical axis passing through the center of the sample. A maximum speed is observed at about 2 mm above the sample top. At 18 mm the speed is practically null. Fig. 6 plots the temperature difference between the center of the sample and the inner wall of the furnace for the three heating rates considered. In all cases a very asymmetric peak is observed: a relatively fast increase of temperature up to a maximum which is followed by a slower decrease. A practical consequence of this observation is that,
Fig. 4. Gas speed distribution into the furnace at t = 98 min.
Fig. 5. Plot of the x-component of the gas speed along a vertical axis passing through the center of the sample. Zero is taken at the bottom of the sample.
since the temperature difference along a range of temperatures is not constant, temperature calibration should be performed at several points of temperature. On the other hand, the higher the applied heating rate the higher the resulting difference of temperature for any temperature in the studied range. It is also observed that the maximum difference of temperatures shifts to higher temperatures when increasing the heating rate. Thus, with respect to temperature calibration, it is clear that calibration is strongly convenient not only at several temperature points but also at the heating rates that will be used for experiments. Fig. 7 shows a layout where five points are indicated as a reference for the study of temperature into the sample. Fig. 8 shows the plots of the temperature difference observed between the points B and D versus the temperature at the furnace wall obtained at three heating rates. At the beginning of the experiment the temperature of the sample is the same as that of the furnace wall. As
Fig. 3. Plot of the temperature difference between the inner furnace wall and the geometrical center of the sample along time. 391
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Fig. 9. Plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at the indicated heating rates.
Fig. 6. Plots of the temperature difference between the center of the sample, B, and the inner wall of the furnace, Tfw, for three heating rates (5, 10 and 20 °C/ min).
Fig. 7. Layout of the sample indicating a few reference points.
Fig. 10. Temperature profile along the AC trajectory into the sample for the instant where Tfw = 400 °C at three heating rates.
Fig. 8. Temperature difference between the points B and D versus the temperature at the furnace wall obtained at 5, 10 and 20 °C/min.
soon as the heating ramp starts, a temperature difference appears. That difference increases until reaching a maximum and then decreases slightly to stabilize. The range of temperature preceding the maximum of temperature difference increases when increasing the heating rate. A maximum temperature difference is observed at the maximum heating rate and is about 0.15 °C. In principle, that temperature difference does not seem very important. Fig. 9 shows the plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at three heating rates. Like in Fig. 8, the higher the heating rate, the higher the temperature difference. But in this case the maximum temperature difference is about 1.3 °C. That difference is important and can be significant in many thermogravimetric studies. Fig. 10 shows the temperature profile along the AC trajectory into the sample for the instant where Tfw = 400 °C at three heating rates. It confirms that the lower the heating rate the closer the sample temperature to that of the furnace wall. In order to better see the temperature gradients into the sample for the three heating rates, Fig. 11 plots the difference of temperature for each point along the AC trajectory with respect to that at the B point. It can be observed how the temperature difference increases as getting
Fig. 11. Plots of the difference of temperature for each point along the AC trajectory with respect to that at the B point at three heating rates at the moment when Tfw = 400 °C.
closer to the A and C ends. The effect is more notorious for the higher heating rates, reaching a maximum value of about 1.8 °C with the 20 °C/min heating rate. That increase of the temperature difference is not linear and not perfectly symmetric, but it can be approximated to a straight line. The little asymmetry comes from the geometrical effects of the entry and the exit of the purge gas, which are located at opposite points, as indicated in Fig. 1. 4.2. Effect of voids In order to see the effect of voids in the temperature distribution into the sample, samples of the A-type, with levels of porosity 0, 8, 16, and 25% were simulated. A 5 °C/min heating rate was used. 392
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Fig. 14. Plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity.
between the points B and D versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity. As expected, the higher the thermal conductivity, the lower the temperature difference. A change of 100 times in the thermal conductivity produces a change of the same order in the temperature difference. However, even in the lowest thermal conductivity case, 0.042 W/M/K, the temperature difference is only about 0.6 °C, although that value increases continuously with temperature. Fig. 14 plots the temperature difference between the points E and B versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity. Although the temperature difference stabilizes at about 150 °C, very important values are observed, with a maximum of 2.7 °C for the case of the lowest thermal conductivity.
Fig. 12. Plots of the difference of temperature along the AC trajectory with respect to that at the B point for samples of different void content when Tfw = 400 °C. A 5 °C/min heating ramp is assumed.
Fig. 12 plots the difference of temperature for each point along the AC trajectory with respect to that at the B point for samples with different void content. In this case only the lowest heating rate was used and the reference temperature was Tfw = 400 °C. As described for Fig. 11, the temperature difference increases as getting closer to the A and C ends. The effect is stronger for the higher void content, with a maximum value of about 0.55 °C which is significant compared to 0.42 °C, which corresponds to the non-porous sample.
5. Conclusions Temperature calibration at several heating rates and in several temperature points is highly recommended because the temperature difference between the sample and the furnace wall is not constant in a range of temperatures like those normally used in TG operation. The heating rate has always an important effect in the temperature gradient into the sample. Low heating rates are recommended to minimize that gradient. Void content of the sample clearly influences the temperature gradients developed into the sample, but that effect is not serious compared to the other effects. Thermal conductivity of the sample has a clear effect on the temperature gradients observed within the sample. Since all above mentioned factors have a clear effect on the temperature gradient developed into the sample and on the difference of sample temperature with respect to that of furnace wall, the heating rate and sample size should be carefully adjusted depending on the sample or furnace features that sometimes cannot be adjusted (thermal conductivity, porosity). Additionally, the results can be carefully scrutinized for possible deviations from the theoretical sample temperature depending on the aim of the studies.
4.3. Effect of thermal conductivity In order to evaluate the effect of thermal conductivity four B-type samples with thermal conductivities 0.42, 0.21, 0.084, and 0.042 W/(m · K) were considered. In all cases the density was 2650 g/m³ and the heat capacity 758 kg/J/K. A heating rate of 5 °C/min was used. Fig. 13 shows the plots of the temperature difference observed
Acknowledgement The authors acknowledge the financial support of the Spanish Ministerio de Economía, Industria y Competitividad for partially funding this work through the MTM2014-52876-R project. Fig. 13. Temperature difference between the points D and B versus the temperature at the furnace wall obtained at 5 °C/min for four thermal conductivity values.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. 393
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doi.org/10.1016/j.polymertesting.2018.04.039.
[16] P. Gabbott (Ed.), Principles and Applications of Thermal Analysis, Blackwell Pub, Oxford , Ames, Iowa, 2008. [17] J.D. Menczel, R.B. Prime (Eds.), Thermal analysis of Polymers: Fundamentals and Applications, John Wiley, Hoboken, N.J, 2009. [18] R. Artiaga Díaz, Thermal analysis, fundamentals and applications to material characterization, proceedings of the international seminar : thermal analysis and rheology, Ferrol, Spain, 30 Juny-4 July 2003, Universidade da Coruña, La Coruña, Spain, 2005, http://search.ebscohost.com/login.aspx?direct=true&scope=site& db=nlebk&db=nlabk&AN=331434 (accessed January 31, 2017). [19] G.W.H. Höhne, H.K. Cammenga, W. Eysel, E. Gmelin, W. Hemminger, The temperature calibration of scanning calorimeters, Thermochim, Acta 160 (1990) 1–12, http://dx.doi.org/10.1016/0040-6031(90)80235-Q. [20] J.V. Nicholas, D.R. White, Traceable Temperatures: an introduction to temperature measurement and calibration, 2nd edn, Meas. Sci. Technol. 13 (2002), http://dx. doi.org/10.1088/0957-0233/13/10/701 1651–1651. [21] ASTM E1582-17, Standard Test Method for Temperature Calibration of Thermogravimetric Analyzers, (2017) www.astm.org. [22] ASTM E2069-06(2012), Standard Test Method for Temperature Calibration on Cooling of Differential Scanning Calorimeters, (2012) www.astm.org. [23] ASTM E2509-14, Standard Test Method for Temperature Calibration of Rheometers in Isothermal Mode, (2014) www.astm.org. [24] ASTM E1867-16, Standard Test Methods for Temperature Calibration of Dynamic Mechanical Analyzers, (2016) www.astm.org. [25] ASTM E967-08(2014), Standard Test Method for Temperature Calibration of Differential Scanning Calorimeters and Differential Thermal Analyzers, (2014) www.astm.org. [26] ASTM E1363-16, Standard Test Method for Temperature Calibration of Thermomechanical Analyzers, (2016) www.astm.org. [27] E. Donoghue, T.S. Ellis, F.E. Karasz, The effect of sample temperature gradients on DSC thermograms at the glass transition temperature, in: J.F. Johnson, P.S. Gill (Eds.), Anal. Calorim, Springer US, Boston, MA, 1984, pp. 325–341, , http://dx.doi. org/10.1007/978-1-4613-2699-1_27. [28] R. Comesaña, M.A. Gómez, M.A. Álvarez, P. Eguía, Thermal lag analysis on a simulated TGA-DSC device, Thermochim. Acta 547 (2012) 13–21, http://dx.doi.org/ 10.1016/j.tca.2012.08.008. [29] COMSOL Multiphysics, Multiphysics Simulation Software - Platform for PhysicsBased Modeling, (n.d.). http://www.comsol.com/comsol-multiphysics (accessed April 1, 2015).
References [1] C. Earnest (Ed.), Compositional Analysis by Thermogravimetry, ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 1988, , http://dx.doi.org/10.1520/STP997-EB 19428–2959. [2] A.M.G. Macdonald, Clement duval, inorganic thermogravimetric Analysis,2nd ed, Anal. Chim. Acta 32 (1965) 298–299, http://dx.doi.org/10.1016/S0003-2670(00) 88944-8. [3] J.S. Crighton, An introduction to thermogravimetry, Polymer 11 (1970) 682–683, http://dx.doi.org/10.1016/0032-3861(70)90022-4. [4] W.W. Wendlandt, Thermal analysis techniques - Part I. Thermobalances, J. Chem. Educ. 49 (1972) A571, http://dx.doi.org/10.1021/ed049pA571. [5] C.J. Keattch, D. Dollimore, Studies in the history and development of thermogravimetry: III. The influence of Kotaro Honda and the Japanese school, J. Therm. Anal. 39 (1993) 97–118, http://dx.doi.org/10.1007/BF02235450. [6] W.W. Wenlandt, P.K. Gallager, Chapter 1. Instrumentation, in: Therm. Charact. Polym. Mater., 1981st ed., Academic Press, Inc., n.d. [7] W. Nernst, E.H. Riesenfeld, Über quantitative Gewichtsanalyse mit sehr kleinen Substanzmengen, Berichte Dtsch. Chem. Ges 36 (1903) 2086–2093, http://dx.doi. org/10.1002/cber.190303602126. [8] O. Brill, Über die Dissoziation der Karbonate der Erdalkalien und des Magnesiumkarbonats, Z. Anorg. Chem. 45 (1905) 275–292, http://dx.doi.org/10. 1002/zaac.19050450128. [9] P. Truchot, Rev Quim Pura Apl 10 (1907) 2–12. [10] G. Urbain, C. Boulanger, Sur une balance-laboratoire à compensation électromagnetique à l’étude des systèmes qui dégagent des gaz avec une vitesse sensible, Compt Rend 154 (1912) 347–349. [11] K. Honda, On a Thermobalance, Sci. Rep. Tohoku Imp. Univ. 4, (1915), pp. 1–4. [12] C. Duval, Inorganic Thermogravimetric Analysis, second ed., Elsiever Amst, 1963, pp. 4–8. [13] E.A. Turi (Ed.), Thermal characterization of Polymeric Materials, second ed., Academic Press, San Diego, 1997. [14] P. Chevenard, X. Waché, R. De la Tullaye, Etude de la corrosion seche des metaux au moyen d’une thermobalance, Bull. Soc. Chim. Fr. Mem. (1944) 41–48. [15] K.P. Menard, Thermal transitions and their measurement, Ch. 8, in: W. Brostow (Ed.), Perform. Plast, Hanser, Cincinnati, 2000.
394
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Short Communication: Analysis Method
Significant hidden temperature gradients in thermogravimetric tests a
b
T
b
Carlos Gracia-Fernández , Begoña Álvarez-García , Silvia Gómez-Barreiro , Jorge López-Beceirob, Ramón Artiagab,∗ a b
TA Instruments-Waters Cromatografía, Alcobendas, 20108, Madrid, Spain University of A Coruña, EPS, Avda. Mendizábal s/n, 15403, Ferrol, Spain
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermogravimetry Temperature gradient Error Simulation
In thermal analysis, correct measurement of temperature is usually ensured by means of a calibration procedure. However, in addition to make sure that the right value of temperature is measured, estimation of temperature gradients into the sample is important. That is especially important in thermogravimetric (TG) analysis, where combinations of some of the common sample sizes heating rates could lead to important differences of temperature into the sample. If there is a significant gradient between different parts of the sample, then the temperature of the thermocouple, although correct, does not actually represent the temperature of the whole sample. While the correctness of the temperature is always important, errors in temperature measurement are critical in kinetic studies. Thus, estimations of the temperature gradients that appear into the sample as a result of a given treatment are of highest interest to choose the right operational conditions that minimize that gradient. That is particularly important for kinetic studies. In this work, thermal gradients originated into a sample during a typical TG test are estimated through a simulation study performed on the Comsol ™ software. A typical vertical TG furnace, sample size of about 125 mg, and several heating rates were used. Additionally, samples of different void contents were considered. The results of the simulation show that significant gradients of temperature can be achieved into the sample with experimental conditions like those that are often used. It is also observed that the difference of temperature between the sample and the furnace wall not only depends on the heating rate, which can be easily corrected by calibration at the corresponding heating rate, but also varies with temperature, which makes highly recommended to calibrate in more than one temperature point when broad ranges of temperature are considered.
1. Introduction Thermogravimetry is a technique that measures the mass of a sample as a function of temperature or time while it is subjected to a controlled temperature program in a controlled atmosphere [1]. The origin and first developments of this technique were thoroughly described by different authors [2–5]. Most of the early thermobalances were constructed by individual investigators [6], such as Nernst and Riesenfeld [7], Brill [8], Truchot [9], Urbain and Boulanger [10] and Honda [11] at the beginning of the twenty-first century. It is also remarkable the work of Duval [12], who developed an automated analytical method based on thermogravimetry. His work provided a strong impetus for this technique [13]. The first commercial thermobalance appeared in 1945 and it was based on the work of Chevenard [14]. The evolution has been fast from the beginning up to now, and the sensibility and precision of the thermobalance were increasing continuously. Nowadays, TG is one of the most common thermal analysis techniques
∗
Corresponding author. E-mail address: [email protected] (R. Artiaga).
https://doi.org/10.1016/j.polymertesting.2018.04.039 Received 4 December 2017; Received in revised form 5 April 2018; Accepted 26 April 2018 Available online 30 April 2018 0142-9418/ © 2018 Elsevier Ltd. All rights reserved.
and it is used in many industrial and scientific fields. A clear review of TG and other thermophysical characterization techniques has been provided by K.P. Menard [15]. Temperature calibration is routinely performed in any thermal analysis technique. While there are many works discussing the importance of temperature calibration and several standards indicating the right procedures to calibrate temperature of different instruments [16–26], only a few works paid attention to the possible gradients of temperature originated into the sample while subjected to a typical thermo-analytical temperature program [27,28]. Thermal gradients originated into the sample during a typical thermogravimetric (TG) test are estimated here through a simulation study. For the simulation, a typical vertical TG furnace, samples of different porosity and thermal conductivity, and a few of the most common heating rates were used.
Polymer Testing 68 (2018) 388–394
C. Gracia-Fernández et al.
temperature distribution into the sample, four levels of porosity were considered: 0, 8, 16, and 25%. B-type: In order to evaluate the effect of thermal conductivity four samples with no porosity but with different thermal conductivities were considered: 0.42, 0.21, 0.084, and 0.042 W/(m. K). In all cases the density was 2650 g/m³ and heat capacity 758 kg/J/K. 3. Fundamentals of the simulation The thermal gradients originated into the sample during a typical TG test are estimated through a simulation study performed by means of the Comsol software [29]. In order to perform the simulation, the three-dimensional geometry described above is reduced, for computation, to a two-dimensional geometry. That simplification should not significantly affect the results since the system, including furnace, platform and sample, is cylindrically symmetric except for the entry and exit holes. These three holes were conveniently located on the same plane so that a two-dimensional simulation can take into account the gas flow effect. On the other hand, the Navier-Stokes equations are coupled with the heat transfer model, being the pressure and the velocity field the solution of the Navier-Stokes equations, while the temperature is solved through the heat transfer model. The buoyancy exerted by the fluid, which depends on temperature and density, is introduced in the NavierStokes equations for compressible fluids. Simultaneously, the heat equation accounts for convective heat transfer. The Laminar Flow and the Heat Transfer in Fluids interfaces are coupled through the “Temperature Coupling” and “Flow Coupling” features of the software. The governing equations in the Laminar Flow and the Heat Transfer in Fluids interfaces are:
Fig. 1. Layout of the furnace geometry considered for this work.
2. Furnace geometry and operating conditions
- The Navier-Stokes equations
∂u 2 + u⋅∇u ⎞ = −∇p + ∇⋅⎛μ (∇u + (∇u)T ) − μ (∇⋅u) I ⎞ + F ρ⎛ 3 ⎝ ⎠ ⎝ ∂t ⎠
For the simulation, a typical vertical TG furnace geometry was considered. Fig. 1 shows a layout of the furnace where the locations of the gas inlet and outlet holes and of the sample holder can be observed. The sample holder consists of a typical open platinum pan. For simplicity, the hang-down system was not included in the simulation. The inner walls of the furnace are supposed to be made of alumina. In order to realistically reproduce the actual operating conditions, a purge of nitrogen is applied through the A and B inlets at 80 and 20 mL/min, respectively. While entering through two different holes, all gas goes out through one single outlet. All the simulated experiments consist of linear heating ramps. It is assumed that the temperature at the inner wall surface is always uniform. The heating ramp is directly applied on the furnace wall. This is an ideal situation since, in real instruments, the temperature is not completely uniform at the furnace wall surface and the wall surface is normally heated from a resistance embedded in the furnace material. Four heating rates, 5, 10 and 20 °C/min, were used for the different cases as it will be described below. These heating rates are very common in thermogravimetry. In order to evaluate how some features of the sample may affect the temperature distribution into the sample, two different types of samples were considered: A-type: this sample is defined as a cylindrical slice of 2.00 mm height and 4.16 mm radius, made of a hypothetical material with thermal conductivity and specific heat capacity values similar to those of a commercial polyamide 66 at 20 °C. For simplicity, it is supposed that the material keeps these features in all the range of temperature considered in the tests. Of course, many polymers would melt and degrade in that range, but including possible transformations of the samples would imply a very high complexity and it would be very difficult to generalize the results to any kind o material. Thus, the values used for simulation were 1140 kg/m³, which results in a sample mass of 124 mg, thermal conductivity of 0.43 W/(m. K), and specific heat capacity of 1670 J/(kg. K). In order to see the effect of voids in the
where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density and μ is the fluid dynamic viscosity. The left-hand side term corresponds to the inertial forces, the first term of right-hand side represents the pressure forces, the second term the viscous forces and the third term the external forces applied to the fluid. This equation is always solved together with the continuity equation:
∂ρ + ∇⋅(ρu) = 0 ∂t Solving these equations, for a particular set of boundary conditions, allows to predict the fluid velocity and its pressure for a given geometry. - Heat equation
q = hf (Tw − Tf ) where q is the heat flux, hf is the heat transfer coefficient of the fluid, Tf is the local fluid temperature and Tw is the wall temperature. The Reynolds number (Re) allows to predict laminar or turbulent patterns for different fluid flow situations. For flow in a cylindrical pipe, the Reynolds number can be defined as:
Re =
ρvs D μ
where ρ is the fluid density, vs is the average velocity of the fluid, μ is the fluid dynamic viscosity and D is the hydraulic diameter of the pipe. According to the Reynolds number prediction, considering the furnace and sample geometry and the gas flow rate, the flow will be always of the laminar type. 389
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Fig. 2. Temperature distribution into the furnace at different times.
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4. Results and discussion 4.1. Temperature distribution and gradients within the sample In this case an A-type non porous sample was chosen. Fig. 2 shows the temperature distribution into the furnace at different times, assuming that a 10 °C/min heating rate is applied on the inner furnace wall. It is assumed that the temperature is uniformly distributed in the entire furnace wall at any time. In practice, physical furnaces were designed to approach to that ideal condition but there are always some deviations that depend on the geometry and operation conditions. Those deviations would imply an additional complexity that is not included in this work. It can be observed that although the scales of temperatures are different, the pictures obtained at different times are almost the same: there is no important gradient of temperature except for the sample and the regions around the sample. That is not surprising considering that sample and sample holder have very different values of heat capacity (Cp) and thermal conductivity than the nitrogen atmosphere into the furnace. That differences of Cp lead to different increments of temperature depending on the material of which the different parts are made. On the other hand, the thermal conductivity of the gas originates a delay in the sample temperature with respect to that of the wall surface. Similarly, the thermal conductivity of the sample originates a delay in the temperature at the center of the sample with respect to the surface of the sample. Fig. 3 shows how the temperature difference between the center of the sample and the wall inner surface varies along time. A maximum difference is observed at about 12 min when using a 10 °C/min heating rate. Fig. 4 shows the gas speed distribution and the direction of the flow into the furnace at a given moment. Both the peed distribution and the pattern of flow practically did not change along time. The highest speed values are observed at the entering and exit points. The speed is almost zero for the sample-gas interface, but it increases as we move upward away from the sample. This is important to sweep off the volatiles resulting from the thermal degradation of the sample. Once the volatiles go out from the sample, they go into the gas flow and are swept to the exit hole. On the other hand, no significant variations in the gas speed distribution or the pattern of flow were observed along time with respect to those of Fig. 4. Fig. 5 plots the x-component of the gas speed along a vertical axis passing through the center of the sample. A maximum speed is observed at about 2 mm above the sample top. At 18 mm the speed is practically null. Fig. 6 plots the temperature difference between the center of the sample and the inner wall of the furnace for the three heating rates considered. In all cases a very asymmetric peak is observed: a relatively fast increase of temperature up to a maximum which is followed by a slower decrease. A practical consequence of this observation is that,
Fig. 4. Gas speed distribution into the furnace at t = 98 min.
Fig. 5. Plot of the x-component of the gas speed along a vertical axis passing through the center of the sample. Zero is taken at the bottom of the sample.
since the temperature difference along a range of temperatures is not constant, temperature calibration should be performed at several points of temperature. On the other hand, the higher the applied heating rate the higher the resulting difference of temperature for any temperature in the studied range. It is also observed that the maximum difference of temperatures shifts to higher temperatures when increasing the heating rate. Thus, with respect to temperature calibration, it is clear that calibration is strongly convenient not only at several temperature points but also at the heating rates that will be used for experiments. Fig. 7 shows a layout where five points are indicated as a reference for the study of temperature into the sample. Fig. 8 shows the plots of the temperature difference observed between the points B and D versus the temperature at the furnace wall obtained at three heating rates. At the beginning of the experiment the temperature of the sample is the same as that of the furnace wall. As
Fig. 3. Plot of the temperature difference between the inner furnace wall and the geometrical center of the sample along time. 391
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Fig. 9. Plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at the indicated heating rates.
Fig. 6. Plots of the temperature difference between the center of the sample, B, and the inner wall of the furnace, Tfw, for three heating rates (5, 10 and 20 °C/ min).
Fig. 7. Layout of the sample indicating a few reference points.
Fig. 10. Temperature profile along the AC trajectory into the sample for the instant where Tfw = 400 °C at three heating rates.
Fig. 8. Temperature difference between the points B and D versus the temperature at the furnace wall obtained at 5, 10 and 20 °C/min.
soon as the heating ramp starts, a temperature difference appears. That difference increases until reaching a maximum and then decreases slightly to stabilize. The range of temperature preceding the maximum of temperature difference increases when increasing the heating rate. A maximum temperature difference is observed at the maximum heating rate and is about 0.15 °C. In principle, that temperature difference does not seem very important. Fig. 9 shows the plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at three heating rates. Like in Fig. 8, the higher the heating rate, the higher the temperature difference. But in this case the maximum temperature difference is about 1.3 °C. That difference is important and can be significant in many thermogravimetric studies. Fig. 10 shows the temperature profile along the AC trajectory into the sample for the instant where Tfw = 400 °C at three heating rates. It confirms that the lower the heating rate the closer the sample temperature to that of the furnace wall. In order to better see the temperature gradients into the sample for the three heating rates, Fig. 11 plots the difference of temperature for each point along the AC trajectory with respect to that at the B point. It can be observed how the temperature difference increases as getting
Fig. 11. Plots of the difference of temperature for each point along the AC trajectory with respect to that at the B point at three heating rates at the moment when Tfw = 400 °C.
closer to the A and C ends. The effect is more notorious for the higher heating rates, reaching a maximum value of about 1.8 °C with the 20 °C/min heating rate. That increase of the temperature difference is not linear and not perfectly symmetric, but it can be approximated to a straight line. The little asymmetry comes from the geometrical effects of the entry and the exit of the purge gas, which are located at opposite points, as indicated in Fig. 1. 4.2. Effect of voids In order to see the effect of voids in the temperature distribution into the sample, samples of the A-type, with levels of porosity 0, 8, 16, and 25% were simulated. A 5 °C/min heating rate was used. 392
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Fig. 14. Plots of the temperature difference observed between the points E and B versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity.
between the points B and D versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity. As expected, the higher the thermal conductivity, the lower the temperature difference. A change of 100 times in the thermal conductivity produces a change of the same order in the temperature difference. However, even in the lowest thermal conductivity case, 0.042 W/M/K, the temperature difference is only about 0.6 °C, although that value increases continuously with temperature. Fig. 14 plots the temperature difference between the points E and B versus the temperature at the furnace wall obtained at 5 °C/min with samples of different thermal conductivity. Although the temperature difference stabilizes at about 150 °C, very important values are observed, with a maximum of 2.7 °C for the case of the lowest thermal conductivity.
Fig. 12. Plots of the difference of temperature along the AC trajectory with respect to that at the B point for samples of different void content when Tfw = 400 °C. A 5 °C/min heating ramp is assumed.
Fig. 12 plots the difference of temperature for each point along the AC trajectory with respect to that at the B point for samples with different void content. In this case only the lowest heating rate was used and the reference temperature was Tfw = 400 °C. As described for Fig. 11, the temperature difference increases as getting closer to the A and C ends. The effect is stronger for the higher void content, with a maximum value of about 0.55 °C which is significant compared to 0.42 °C, which corresponds to the non-porous sample.
5. Conclusions Temperature calibration at several heating rates and in several temperature points is highly recommended because the temperature difference between the sample and the furnace wall is not constant in a range of temperatures like those normally used in TG operation. The heating rate has always an important effect in the temperature gradient into the sample. Low heating rates are recommended to minimize that gradient. Void content of the sample clearly influences the temperature gradients developed into the sample, but that effect is not serious compared to the other effects. Thermal conductivity of the sample has a clear effect on the temperature gradients observed within the sample. Since all above mentioned factors have a clear effect on the temperature gradient developed into the sample and on the difference of sample temperature with respect to that of furnace wall, the heating rate and sample size should be carefully adjusted depending on the sample or furnace features that sometimes cannot be adjusted (thermal conductivity, porosity). Additionally, the results can be carefully scrutinized for possible deviations from the theoretical sample temperature depending on the aim of the studies.
4.3. Effect of thermal conductivity In order to evaluate the effect of thermal conductivity four B-type samples with thermal conductivities 0.42, 0.21, 0.084, and 0.042 W/(m · K) were considered. In all cases the density was 2650 g/m³ and the heat capacity 758 kg/J/K. A heating rate of 5 °C/min was used. Fig. 13 shows the plots of the temperature difference observed
Acknowledgement The authors acknowledge the financial support of the Spanish Ministerio de Economía, Industria y Competitividad for partially funding this work through the MTM2014-52876-R project. Fig. 13. Temperature difference between the points D and B versus the temperature at the furnace wall obtained at 5 °C/min for four thermal conductivity values.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. 393
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doi.org/10.1016/j.polymertesting.2018.04.039.
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