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International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Determination of thermal diffusivity of fibrous insulating materials at high temperatures by thermal wave analysis H. Brendel ⇑, G. Seifert, F. Raether Fraunhofer Institute for Silicate Research, Center for High Temperature Materials and Design, Bayreuth 95448, Germany a r t i c l e i n f o Article history: Received 15 September 2016 Received in revised form 4 January 2017 Accepted 6 January 2017 Keywords: Thermal diffusivity Fibrous insulation Thermal wave analysis High temperature a b s t r a c t This work investigates the potential of measuring thermal diffusivity of materials in the high temperature range up to 1580 °C by thermal wave analysis of cylindrical samples. In particular, aiming at the typical situation of fiber insulation materials, the method is evaluated for investigations on heterogeneous media with low thermal conductivity, low density and heat capacity, and significant internal radiative heat transport (participating media). Furthermore, a model for combined conductive and radiative heat transfer is introduced. The model is adapted to a commercial insulation material using experimental scattering and absorption spectra evaluated at 1000 °C. The predictions of this numerical model are in accordance with the thermal diffusivity data obtained from the thermal wave analyses. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction THERMAL conductivity and thermal diffusivity are the most important parameters for describing the heat transport properties of a material. Fibrous insulations are a common solution for high temperature processes. They are applied in all kind of furnaces due to their favorable properties such as low thermal conductivity, low heat capacity and high thermal shock resistance. Conductive and radiative heat transfer are the predominant heat transport mechanisms at high temperatures. The established methods for measuring thermal conductivity and thermal diffusivity differ from one another primarily by recommended temperature range, required sample volume and achievable range of thermal conductivity and diffusivity values. The methods can be distinguished by their operation mode: Heat-Flow meters or Guarded-Hot-Plates are measuring in steadystate, whereas the Hot-Wire technique is a transient method [1]. Due to their capability for investigating large samples, Guarded-Hot-Plate [2,3] and Hot-Wire technique [4,5] are the methods primarily used for high temperature measurements on heterogeneous sample materials, but only the Hot-Wire technique is applicable for measuring above 1000 °C. One of the main drawbacks of the Hot-Wire method is that sample preparation requires a great deal of effort to provide good thermal contact between the sample and the heater wire [5,6]. Furthermore, one has to ensure that test material and heater wire show ⇑ Corresponding author. E-mail addresses: [email protected] (H. Brendel), gerhard.seifert@ isc.fraunhofer.de (G. Seifert), [email protected] (F. Raether). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.01.063 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved. no chemical reaction [4,6]. To overcome the disadvantage of thermal contact uncertainty we investigated the less widespread Ångstrom’s method [7], nowadays mostly called thermal wave analysis (TWA) to determine thermal diffusivity of high temperature insulation materials. For TWA, an oscillating temperature boundary condition is applied to the sample. The thermal diffusivity is then obtained from an analysis of phase and amplitude of the temperature oscillation measured inside the sample specimen [8–11]. In previous publications, TWA was realized at temperatures up to 1000 °C by applying the oscillating temperatures to the faces of the sample. This paper presents a high temperature application (T >¼ 1000 C) of TWA, where the temperature oscillation is applied uniformly at the lateral area of a long cylindrical sample of ceramic-fiber insulation material. Furthermore a theoretical model for combined conductive and radiative heat transfer is developed to predict the thermal diffusivity of the material. The model employs radiative transport parameters experimentally derived at 1000 °C. 2. Analytical models This section describes the analytical model for an infinitely long, periodically heated cylinder. The problem has an analytical solution, which is first used to calculate the general correlation between the measuring quantities phase and amplitude (of temperature oscillation inside the sample) and the transfer properties of the material. Thereafter the theory for radiative transfer in a radially symmetric participating medium is presented. Next, a H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 2515 Nomenclature velocity of light specific heat 1 2 Rð pffiffiffix=jÞ , dimensionless parameter 2Rarel , dimensionless parameter connection parameter for fiber-fiber contact ratio between the thermal conductivity of gas and the solid material rext =q, specific extinction oscillation frequency anisotropy parameter Planck p ffiffiffiffiffiffiffi constant 1 Planck-Intensity distribution Bessel-function (order n) of the first kind Boltzmann constant length of fibrous probe index of refraction pressure radiative heat flux outer radius of sample radial coordinate time radial temperature external temperature amplitude mean temperature of oscillation wall temperature thermal conductivity ratio of the volumes of fiber and gas Young’s modulus fiber orientation parameter relative heat transfer coefficient height of a fiber-gas cell in units of the contact area emissivity amplitude loss factor tc=ðcp qÞ, thermal diffusivity wavelength phase shift cosð#0 Þ, cosine of mean fiber orientation angle cl cp c1 c2 C sb Cr e f g h i ipk Jn kB l n p0 qrad R r t TðrÞ T ex Tm Tw tc Vr Y Z arel d g j k m l oþ1 #0 q r rext rsca rabs 1 s sR x0 x area of a fiber-gas cell in units of the contact area mean fiber orientation angle density Stefan-Boltzmann constant extinction coefficient scattering coefficient absorption coefficient Poisson’s number r rext , optical depth Rrext , total optical depth albedo 2pf , angular frequency Superscripts low,high low, high quantity lit literature value exp experimentally derived quantity num numerically derived quantity Subscripts k indicates a spectral quantity m index of spectral-band bulk attribute of the solid bulk-material fs attribute of the sample g attribute of the gas b refers to fiber-fiber contact ext,sca,abs extinction, scattering and absorbtion eff refers to effective quantities w refers to quantities of the wall boundary ^ refers to scaled quantities Abbreviations RT ambient temperature Im imaginary part Re real part TWA thermal-wave analysis description of heat conduction by solid components and the enclosed gas in the material is presented; this combined model also accounts for the heat transfer due to fiber contacts. Finally, the inhomogeneous diffusion equation for combined gaseous, solid and radiative heat transfer is introduced. A brief summary of the numerical implementation of the models concludes this section. 2.1. The periodically heated cylinder An infinite cylinder heate periodically by a harmonic cosinetype temperature profile is considered (Fig. 1). Heat transfer is assumed to be radiative between cylinder lateral surface and its surroundings. In the case of small temperature differences the radiative heat transfer coefficient can be approximated by a convection term; the corresponding relative heat transfer coefficient arel reads: arel ¼ 4 Here eff rT 3 the ð1Þ tc effective emissivity is defined as eff Fig. 1. Schematic diagram of the periodically heated cylinder. ¼ ð1=fs þ ð1=w 1Þ R=Rw Þ1 where fs and w are the spectrally integrated emissivities of sample and the surrounding wall, respectively. R and Rw are the radii of the inner medium (sample) and the surrounding wall. tc is the thermal conductivity and r the Stefan- Boltzmann constant. In [12] an analytical solution of the problem is given for a cylinder at zero temperature with sinusoidal periodic radiation at its surface. For experimental realization a cosine-type modulation is more beneficial because it facilitates a slope zero 2516 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 for the temperature variation at starting time instead of a technically unrealizable kink. The equivalent solution for a cylindrical medium of radius R heated by a cosine-type temperature profile is given by the equation T ðr; tÞ ¼ Re h c2 T ffiffiffi p=2ÞÞ i 2ic1 J 1 ðc1 wÞþic2 J 0 ðc1 wÞ 1 pffiffiffi P 2 c2 T ex þ J ðc wr=RÞ exp ðiðxtþ ex p0 1 n¼1 J 0 ðbn r=RÞ exp ðb 2n xt=c12 Þ ð2Þ ðb4n =c 41 þ1Þðb2n þc22 =2ÞJ0 ðbn Þ the dimensionless parameters c1 ¼ Rðx=jÞ 1 = 2 and pffiffiffiffi 2 R arel . Further parameters are r: radial coordinate; j: therc2 ¼ mal diffusivity; x: angular velocity of the harmonic oscillation; T ex : external amplitude of excitation; Jn: Bessel-function of first kind of order n; w ¼ expð3i=4pÞ. Finally, bn are the roots of the following equation [12]: with pffiffiffi 2bJ 1 ðbÞ ¼ c2 J 0 ðbÞ ð3Þ The first part of Eq. (2) describes the steady-state periodic oscillations, while the second part of Eq. (2) accounts for the transient oscillation whose contribution converges to zero with progressing time. A proof for the validity of Eq. (2) is given in the Appendix. If one is primarily interested in the stationary temperature profile at radial coordinate r, the first part of Eq. (2) can be converted to the following form: Tðr; tÞ ¼ T ex gðrÞ cosðxt mðrÞÞ ð4Þ In Eq. (4), gðrÞ is the amplitude loss factor and mðrÞ refers to the phase-shift at radial coordinate r. By using the abbreviations f1 ðc1 ; c2 Þ ¼ c1 c2 ½ReðJ 1 ðc1 wÞÞ Im ðJ 1 ðc1 wÞÞ ImðJ 0 ðc1 wÞÞ ð5Þ f2 ðc1 ; c2 Þ ¼ c1 c2 ½Re ðJ 1 ðc1 wÞÞ þ Im ðJ 1 ðc1 w ÞÞ þ Re ðJ 0 ðc1 w ÞÞ ð6Þ was chosen as control parameter, because it is proportional to the thermal diffusivity j. Experimentally, phase shift and amplitude decay are the measured quantities from which thermal diffusivity is deduced by the help of Eqs. (5)–(14). Insofar Fig. 2 may be used as a guide to design a TWA experiment, e.g., for large g and m (to increase signal-to-noise ratio). However, besides practical limits of an existing setup (maximum sample size, heating rate and precision, etc.) the inverse relation of amplitude and phase effects (as clearly seen in Fig. 2) requires always a compromise. The same statement holds for the parameter c2 . If it is increased (meaning a higher heat transfer coefficient and/or a higher cylinder radius R), the observed phase shift decreases while the observed amplitude decay increases. 2.2. Scaled diffusion approximation For optically dense media the diffusion equation for radiative transfer is an appropriate approximation for calculating the radiative flux qrad [14,15]. A medium can be considered as optically dense when the optical depth s has at least a value of 1. The physical meaning of this value is that the mean free path of the photons is much lower than the size of the medium. Using the scaling rules given in [16] and the definition of the diffusion equation for an absorbing and anisotropically scattering cylindrical enclosure given in [17], the scaled diffusion equation transforms to d2 db s 2 k qrad b sk þ 1 d bs k dbs k b 0k þ qrad b sk 3 1 x b 0k ¼ 4p 1 x d db sk p ik T b sk 1 bs 2 k qrad b sk ð15Þ b 0k are defined as [16]: bk ¼ r r b extk and x The scaled quantities s b 0k ¼ x0k b s k ¼ sk ð1 x0k g k =3Þ; x 1 g k =3 1 x0k g k =3 ð16Þ f3 ðr; c1 Þ ¼ ReðJ 0 ðc1 r=RwÞÞ ð7Þ f4 ðr; c1 Þ ¼ ImðJ 0 ðc1 r=RwÞÞ ð8Þ In Eq. (16), g k is the anisotropy parameter; g k =3 can be interp preted as the factor of anisotropy [16]. Furthermore ik represents the Planck-function ð9Þ ik ðT Þ ¼ and CA ¼ CB ¼ f3 f1 þ f4 f2 f21 þ f22 f1 f4 f3 f2 f21 þ f22 2 ð10Þ the phase shift at r reads tan mðrÞ ¼ CA CB p n2eff;k 2hcl k 5 exp 1 hcl kkB T 1 ð17Þ According to [18] it is necessary to consider the refractive index n of the medium for calculating the radiative flux. For the case of fibrous media an effective index of refraction neff has to be defined, ð11Þ and the expression for the amplitude decay g yields gðrÞ2 ¼ C2A þ C2B ð12Þ For r ¼ 0, Eqs. (11) and (12) reduce to tan m ¼ f1 ; r¼0 f2 1=g2 ¼ f21 þ f22 ; r ¼ 0: ð13Þ ð14Þ This means that the harmonic oscillation from the surroundings remains harmonic inside the cylinder with a reduction in amplitude given by g and a lag in time corresponding to m. Eqs. (5)– (14) in connection with Eq. (4) were derived already by [13], but with a different definition of the parameters c1 and c2 . The definition in the current work was made such that, for the case of r ¼ 0, both g and m can be described solely by the two parameters c1 and c2 . Thus they can be displayed in one line chart as done in Fig. 2. c2 1 Fig. 2. Amplitude decay g and phase shift parameter c2 at r ¼ 0. m as a function of c2 for different 1 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 which typically shows a strong dependence on the porosity or the fiber volume fraction, respectively [19,20]. For our numerical model, neff of fibrous media is obtained from the index of refraction of bulk-alumina according to the model introduced in [19]. The temperature dependence of the bulk materials index of refraction is implemented by usage of the data introduced in [21]. The boundary condition at r ¼ R is the one given in [17] and is transformed by application of Eq. (16) to: 1 x 0k 1 b 1 d bs k dbs k b s k qrad ^ R;k þ2 2w;k w;k s b R;k qrad s ð18Þ Here, b s R;k is the scaled total optical depth. For symmetry reasons qrad ¼ 0 at r ¼ 0. A proof for the validity of Eqs. (15) and (18) is given in the Appendix. The temperature dependent spectral emissivity w;k of the wall is taken from [22] for Al2 O3 . 2.3. Solid and gas conductivity For a dense fibrous insulation the thermal conductivity tcsgb of solid fibers, gas and their coupling can be approximated according to [23]: 1 d oþ1 1 tcsgb ¼ ðd þ 1Þ þ tcsg o tcg þ tcbulk C sb ð19Þ C sb is a dimensionless connection parameter representing fiber to fiber contacts. It has to be derived experimentally. The parameters d and o are calculated using the relations given in [23] (see Appendix). The thermal conductivity of the gas is denoted as tcg and the thermal conductivity of the fiber bulk material as tcbulk . The expression tcsg is defined as Cr 1 1 þ V r ð1 þ ZðC r 1Þ=ðC r þ 1ÞÞ ð20Þ where C r ¼ tcg =tcbulk is the ratio between the thermal conductivities of the gas and the solid material. V r qfs =ðqbulk qfs Þ represents the volume ratio of fiber material and gas. Z accounts for the fiber fraction oriented perpendicular to the macroscopic heat flow. For randomly oriented fibers, Z has a value of 0.66; this value is used throughout the paper. 2.4. Coupling conduction and radiation A comprehensive description of heat transfer in the considered case requires including both radiative and conductive heat transfer. Therefore the appropriate description of the transient temperature evolution in the cylindrical sample is given by the following inhomogeneous diffusion equation qfs cp @t@ T ðr; tÞ tcsgb where the 1 d ðrqrad ðr; tÞÞ r dr h @2 @r 2 þ 1r @ @r i T ðr; t Þ ¼ div qrad ðr; t Þ divergence of the radiative appears as source term. flux ð21Þ divqrad ðr; tÞ ¼ 2.5. Numerical implementation Eqs. (21) and (15) are solved in a finite difference scheme. The boundary condition at r ¼ R is updated after every time step by the cosine-type wall intensity. 2jMt ðMrÞ2 <1 The temperature profile is considered as converged if the maximal difference in temperature between the preceding and the present iteration is less than 106 K. The nonlinear equations describing the temperature distribution in the periodically heated cylinder are solved by the Scilab built-in nonlinear-solver which is based on the Powell hybrid method [25]. 3. Sample preparation and characterization h i p s R;k ipw;k ¼ 4p ik b tcsg ¼ tcbulk 1 þ 2517 ð22Þ Equidistant time steps are chosen in accordance with the Neumann stability condition Eq. (22). The nonlinearity in Eq. (21) is solved iteratively by the method introduced in [24] (Appendix). The sample material is the commercial fiber insulation UltraBoard A99 of the M.E. SCHUPP Industriekeramik GmbH & Co. KG [26]. According to manufacturer information the fiber-insulation is made of Al2 O3 with 99% purity and average fiber diameters of 3–5 lm. SEM images recordings suggest a random orientation of the fibers. Before sample preparation the insulation material was tempered at its maximum application temperature of 1600 °C to avoid excessive shrinkage during the actual measurements. The cylindrically shaped sample specimens are manufactured using a conventional lathe to provide geometrical accuracy. The cylindrical samples for thermal diffusivity measurements had a diameter of 26:0 0:2 mm and a length of 95:0 0:5 mm. This corresponds to a radius to length ratio of 0.14 which is sufficient to consider the sample as being of infinite length (see Appendix). Furthermore two drillings were set for the placement of thermocouples, one on the cylinder axis and a parallel one off-axis. The on-axis drilling was realized on the lathe; the off-axis one was drilled by help of a gadget on a conventional drilling machine to provide parallelism of the two drilling axes. The average sample density was determined geometrically yielding a value of 616 43 kg=m3 . In order to evaluate density variations within a sample, one cylindrical specimen was cut in small disks (1–3 mm); their average density was measured to be 636 kg=m3 with a standard deviation of 77 kg=m3 . 3.1. Determination of albedo and extinction Disk-shaped samples with diameter of 20:00 0:15 mm and thickness of 1:00 0:15 mm have been prepared for the determination of the optical transport parameters extinction and albedo b extk =qfs defined in Eq. (16). The effective specific extinction b ek ¼ r b 0k were measured at 1000 °C by a black body boundand albedo x ary conditions method in the wavelength interval of 2–8 lm. The same quantities were determined at ambient temperature by the integrating sphere method in the wavelength interval between 0.5 lm and 2 lm. The integrating sphere method as well as the black body boundary conditions method are described in detail in [27–29]. Both measurement techniques exhibit an experimental uncertainty of about 10%. This explains the difference at the intersecting wavelength of 2 lm in Fig. 3 where the effective specific extinction determined at ambient temperature is 8% above the corresponding high temperature quantity. For numerical reasons the continuous variable k is replaced by discrete spectral bands m ¼ ½km ; kmþ1 where the interval limits are selected such that every spectral band contains identical black body radiative power (dotted-line in Figs. 3 and 4). The Planck-averaged albedo and extinction for every spectral band is defined by: 1 R kmþ1 dkX k k5 ðexp ðhcl =kkB T Þ 1Þ X m ¼ kRmk 1 mþ1 dk k5 ðexp ðhcl =kkB T Þ 1Þ km ð23Þ X represents either the albedo or the extinction. For simplification it is assumed that the experimentally derived extinction and albedo remain constant for temperatures above 1000 °C. 2518 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 the unknown connection Parameter C sb . The average value obtained is C sb ¼ 0:00391 0:00030. 4. Experimental setup and measurement procedure 4.1. Setup and temperature control Fig. 3. Effective specific extinction ^ e (solid-line) measured at room temperature in the short wavelength interval and at 1000 °C in the long wavelength interval and their band approximation (dotted-line). The cylindrical sample is fixed concentrically inside a ceramic muffle by means of a gadget made of the same fiber insulation material. The ceramic muffle is made of alumina with a purity of 99.7%. The setup was placed in the middle of the self-constructed furnace TOM_air which is controlled by an Eurotherm Nanotac unit. The sample temperature is analyzed by butt-welded type B thermocouples of 0.75 mm diameter inside the on-axis and offaxis excentric drilling (positions A and B in Fig. 5). The distance between these thermocouples is 8:0 0:5 mm. The muffle temperature is also analyzed by a thermocouple of type B which is located midway between the outer and inner muffle wall. For fixation of the thermocouples at the muffle a high temperature adhesive (Ceramobond 552) is used. The thermovoltage is logged by the Eurotherm Nanotac controller unit and by an external data acquisition device. To provide time synchronicity the temperature of the guidance thermocouple is logged simultaneously by the controller and the external data acquisition device. Data acquisition is done with a sampling-rate of 1 Hz; the signal is filtered by a time constant of 1.6 s. To realize the cosine-type muffle wall temperature a script for calculating the time dependent target values with an update rate of 2 Hz was written. To enable accurate temperature control the controller parameter setup has to be adapted for every frequency and furnace-temperature before the actual thermal diffusivity measurements. 4.2. Temperature homogenity and limitation of operating parameters ^ (solid-line) measured at room temperature in the short Fig. 4. Effective albedo x wavelength interval and at 1000 °C in the long wavelength interval and their band approximation (dotted-line). 3.2. Determination of the connection parameter C sb The connection parameter C sb used in Eq. (19) to describe the conductive heat transfer due to fiber-to-fiber contacts has to be derived experimentally. The idea is to measure the thermal conductivity tcsgb of the material at low temperatures, where radiative contributions are negligible, and then extract C sb by help of Eq. (19) using known values for the parameters of the components (gas and fibres). In practice, thermal diffusivity of the samples was measured by laser-flash measurements (Netzsch LFA 457) at a temperature of 200 °C to avoid detrimental effects of air moisture. The thermal conductivity tcsgb was calculated using the measured density of the fibrous samples and the heat capacity taken from [30]. The other material parameters required were obtained as follows: due to the observed random orientation of the fibers, a value of 0.66 for the orientation parameter Z was used. The thermal conductivity of bulk-alumina was assumed to be similar to the values given in [31]. A value of 3950 kg=m3 is used for the bulk-density of alumina [32]. The gaseous properties for dry air are taken from [33]. The literature data used for calculation at 200 °C are summarized in Table 3. With these input data Eq. (19) could be solved for The temperature homogeneity over the inner muffle wall during periodic heating was evaluated at six positions of the inner muffle wall. For this, the muffle after each complete measurement cycle was rotated such that the position of thermocouple C takes the previous position of the auxiliary thermocouple D. Comparing the phase shifts obtained for a frequency of f ¼ 0:025/min in the six different positions, a maximum difference of approximately 0.15 rad was observed at 1000 °C, whereas a value of only 0.012 rad was measured at 1580 °C. The obvious improvement of temperature homogeneity towards higher temperature is attribu- Fig. 5. Experimental setup for measurement of the thermal diffusivity via thermal wave analysis (TWA). H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 ted to some cracks in the furnace insulation, which are closed by thermal expansion at higher temperature. The maximal frequency being achievable for a given amplitude (T ex ¼ 10 K) is limited for two reasons: first, the furnace must have a self-cooling rate high enough to follow the specified temperature profile. Second, the homogenity of the temperature at the inner muffle wall is deteriorated at higher frequencies. We found that the maximum achievable frequency in our setup is f ¼ 0:05/min at 1000 °C. Above 1200 °C the limit is f ¼ 0:15/min. 4.3. Determination of phase shift, amplitude and thermal diffusivity Fig. 6 shows the temperature response signal in a fibrous sample for an excitation frequency of 0:05= min at a temperature of 1000 °C. To obtain phase shift and amplitude of the sample signal, only the stationary part of the temperature curve is fitted by the equation TðtÞ ¼ A1 sin xt þ A2 cos xt þ A3 þ A4 t 2519 Convergence is attained when the variation of the evaluated thermal diffusivities is lower than 0:75% around the final value in each interval. In addition the deviation of the averages and the starting values of both intervals must be lower than 0.5%. Since the described evaluation was done during the experiment the measurements are abandoned thereafter. In general, the analysis between two and three stationary periods is sufficient. Note that identical values have been obtained when determining phase and amplitude by fast-Fourier analysis of the entire temperature profile by subtracting the characteristic background spectrum of the transient part. The verification by fast-Fourier analysis was done by using the Scilab built-in algorithm. The reproducibility was analyzed by replication of the measurements at distinct temperatures and frequencies after a holding time of 30 min. The deviation between the obtained thermal diffusivities was below 2% in all cases. Investigations on the evaluation of four to five stationary periods lead to a non-essential improvement of measurement accuracy. ð24Þ Here, x ¼ 2pf is the angular velocity of excitation, A3 represents the mean temperature of oscillation and the expression A4 t considers a drift in furnace temperature. Phase and amplitude are determined by the fit parameters A1 and A2 through the relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T fs ¼ A21 þ A22 and tan mfs ¼ A2 =A1 . As seen in the magnified section on the right-hand side of Fig. 6, the actual value of the muffle temperature is slightly delayed in comparison to the target temperature. To obtain the true phase shift of the sample with respect to the actual muffle temperature one has to subtract the muffle temperature phase shift from the sample temperature phase shift. Therefore phase mex and amplitude T ex of the present excitation temperature curve are also derived via fit-approximation by Eq. (24). Finally, the amplitude decay factor g ¼ T fs =T ex is calculated. The stationary part is separated by comparing the analyzed thermal diffusivity of individual periods of the sample temperature. The onset of the stationary part is indicated when two separately evaluated immediately consecutive periods show a difference in the analyzed thermal diffusivity which is lower than 1.5% and subsequent addition of 0.25 periods (rightarrow in Fig. 6). This criterion is more stringent like that used in [11]. The asymptotic behavior of the thermal diffusivity is investigated by stepwise increasement (1/60 period) of the analysed stationary temperature data. This is done in the direction of increasing time and in the reversed direction whereas the corresponding second halves are the intervals taken for further analysis. Fig. 6. Experimental data derived at 1000 °C with an excitation frequency of 0.05/ min. 4.4. Experimental procedure Initially the setup was heated linearly at a rate of 5 °C/min to the set temperature, following a holding time of about 30 min to enable thermal equilibrium of the furnace. Thereafter the measurements for different excitation frequencies were done over three to four oscillation periods until the mentioned convergence criterion for the evaluated thermal diffusivity is fulfilled. The amplitude of temperature excitation was always 10 K. Measurements were done at temperatures of 1000 °C, 1200 °C, 1400 °C and 1580 °C. The frequencies at 1000 °C are f ¼ 0:025/min and f ¼ 0:05/min. At all other temperatures measurements were done at the frequencies f ¼ 0:05/min, f ¼ 0:1/min and f ¼ 0:15/min. 5. Results and discussion In Fig. 7 experimental and numerical results obtained for a representative measurement at 1580 °C and an oscillation frequency of f ¼ 0:05/min are compared. The measured temperature is represented by a thick solid line; the thin solid curve is a representation of the analytical solution Eq. (4) using the parameters c1 and c2 as obtained from the solution of Eqs. (13) and (14) with the experimentally derived phase delay and amplitude decay as input parameter. The dotted line is the result of the numerical solution of (Eq. (21)). Apparently experimental and numerical results are very close to each other, but not identical: phase shift and Fig. 7. Calculated and measured temperature profile at 1580 °C. 2520 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 amplitude decay obtained from the measurement are mexp ¼ 0:82 rad and gexp ¼ 0:83, leading to a thermal diffusivity of 0:39 106 m2 =s. From the numerical calculation one derives a phase shift of 0:86 rad and an amplitude decay of 0.82, which yield a slightly smaller thermal diffusivity (jnum ¼ 0:35 106 m2 =s). The temperature curves shown in Fig. 7 as well as the thermal diffusivity values derived from it via Eqs. (13) and (14) refer to a position at the symmetry axis of the sample (thermocouple A in Fig. 5). Using the temperature profiles measured by the off-axis thermocouple, a second value for the thermal diffusivity jB can be deduced by help of the general Eqs. (11) and (12). All thermal diffusivities derived from the on-axis and off-axis temperature measurements are shown in Table 1. The error is the standard deviation of the average values. The values in Table 1 suggest that jB is lower than jA by 10– 15% in all cases, but the difference lies within experimental accuracy. It therefore appears highly desirable to improve the precision of the measurements. Guidelines to achieve this can be found by reconsideration of Fig. 2 which shows the general dependencies of phase and amplitude decay as a function of the control parameter c2 1 for fixed parameters c 2 , respectively. In the current situation, the parameter c2 takes values between 3 and 5 for all temperatures and frequencies, the corresponding c2 1 lies approximately between 0:15 and 0:45. In this parameter range the curves for constant c2 are relatively flat. Therefore, already small errors in phase and amplitude lead to comparably large errors for the deduced thermal diffusivity. An obvious way to improve the signal-to-noise ratio would be to move into ranges of steeper variation of the curves. Practically, this would mean to increase the sample diameter or the frequency f of the excitation temperature profile. For the current study it was not possible to implement these ideas due to technical limitations. All values of thermal diffusivities obtained in this work by either TWA measurements or the theoretical model are compared in Fig. 8. The values from the TWA measurements shown there are mean values of the results in Table 1. First we compare the thermal diffusivity from TWA measurements with the theoretical predictions. At every temperature point considered the experimentally derived values are above the theoretically derived values. Towards higher temperatures, the experimental values for the thermal diffusivity show a steeper increase than the theoretical prediction, though mostly still being in accordance within experimental error. A possible explanation for the difference is that the absorption coefficient measured at 1000 °C was used for the calculation also at higher temperatures. According to literature, however, an increase of the spectral absorption coefficient is expected with increasing temperature for pure alumina [21]. Another possible reason is sample imperfection like density fluctuation in the volume or slight surface damage during cylinder preparation. A third point is the model for combined gaseous and conductive heat transfer, which also depends on literature data. The model is calibrated at 200 °C and extrapolated to higher temperatures by assuming that the temperature dependence of the solid and gaseous properties cp and tc is identical to the literature values of Table 1 Measured thermal diffusivity through different thermocouple positions A and B. T in °C 1000 1200 1400 1580 jB jA 106 m2 =s 0:27 0:03 0:32 0:04 0:34 0:04 0:38 0:04 106 m2 =s 0:31 0:05 0:35 0:02 0:39 0:02 0:43 0:05 Fig. 8. Experimental and theoretical thermal diffusivities. Table 2 Comparison of thermal conductivities. T in °C 1000 1200 Thermal conductivity tc in W/(m K) Manufacturer TWA Theory 0:24 0:26 0:22 0:26 0:21 0:23 [30,31,33]. Finally, our measurement results are compared to the ones given by the manufacturer of the fibrous insulation material. Latter were obtained by the hot-wire method at 1000 °C and 1200 °C. The manufacturer specified the thermal conductivity, so we calculated the thermal conductivity from the measured and calculated thermal diffusivities by the help of the measured density and the literature values for the specific heat (Table 3). The results are shown in Table 2. We see that the experimental values obtained by the TWA method and the theoretical results are in accordance with the hot-wire measurements. 6. Conclusion In this work, the feasibility of measuring thermal diffusivity of fibrous insulation materials at high temperatures up to 1580 °C by application of the thermal wave analysis (TWA) technique (periodic temperature excitation) was demonstrated. Furthermore, a model has been developed which allows quantitative numerical prediction of the experimental results, when high temperature experimental input data, in particular spectral extinction and albedo, are available. The thermal diffusivities experimentally obtained are in reasonable agreement with the results from our numerical calculations; as well, the values determined in this work are in good agreement with the manufacturers specification for the thermal conductivities at 1000 °C and 1200 °C. For an optimal implementation of the technique a particular designed furnace should be developed which enables large sample dimensions and high temperature oscillation frequencies. Also, the dynamic temperature homogeneity of the sample environment (muffle) should be improved. A properly sized vertical tube furnace with densely located heating elements in form of a coil with an inner lining, made of a material with a high thermal conductivity (for instance dense alumina), is favorable. The outer thermal insulation has to be rather thin so that the furnace could follow the target heating and cooling curve also at temperatures below 1000 °C (which was the lower limit achieved in this work). 2521 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 Summarizing our findings, we conclude that the TWA technique appears to be very promising as an accurate method for high temperature measurements of heterogeneous samples, complementing the established, but temperature-restricted methods like the guarded-hot-plate- or the hot-wire technique into the hightemperature regime. Acknowledgments The authors are indebted to J. Manara and M.C. ArduiniSchuster from the ZAE Bayern, Magdalene-Schoch-Straße 3, D-97074 Würzburg for performing the extinction and albedo measurements and for helpful discussions concerning radiative heat transfer. Appendix A In this Appendix, some important mathematical details of the model described in this paper are given. Furthermore, the validity of the model is checked against finite element (FE) calculation of an exemplary situation. We start with a 1-dimensional FE model to prove the validity of the Eqs. (2)–(14). The FE calculation was done using ANSYS 15 with the following parameters: T m ¼ 1190 C, T ex ¼ 10 C, eff ¼ 0:12; f ¼ 0:15=min; q ¼ 593kg=m3 ; j ¼ 0:52 106 m2 =s; r a ¼ 0:013 m. The results and a brief description of the FE model are shown in Fig. 9. It is evident that the prediction of Eq. (2) (solid line) is in perfect agreement with the FE calculation. The same statement holds for the stationary case expressed by Eq. (2) (dashed line) after typically one full cycle of the cosine-type excitation. Next, a 2-dimensional FE model was used to test the validity of assuming an infinite cylinder length (which is the basis of Eqs. (4)–(14)) instead of the real radius to length ratio ð0:14Þ of the fibrous samples used in our thermal diffusivity measurements. The FE model for the finite cylinder features an additional blackboundary condition on the cross-sectional-areas: w ¼ 1. All other parameters are identical to the previous calculations shown in Fig. 9. The FEM results for the time dependent temperature profile for an infinite cylinder and a finite cylinder with radius to length ratio of 0:14 are compared in Fig. 10. The two curves are nearly identical; very small differences, far below experimental accuracy, are only seen in strongly magnified scaling (see insets in Fig. 10). Apparently the additional heat flux through the cross-sectionalareas has almost no influence on the temperature profile at the center of the cylinder. Now we demonstrate by comparison to previous work (Ref. [17]) that our formulation and implementation of the scaled diffu- Fig. 9. Validation of Eqs. (2)–(14) by comparison with FEM-calculations. Fig. 10. FEM-comparison of an infinite and a finite cylinder. sion equation, in particular Eqs. (15) and (18), is correct. Results are collected in Fig. 11. For the parameters g ¼ ð1; 0; 1Þ; x0 ¼ 0:95; sR ¼ 1=ð1 x0 Þ; ¼ 1 and T fs ðrÞ ¼ T w = const the results derived via Eqs. (15) and (18) by application of Eq. (16) must be identical to the results given in Fig. 10 of [17]; there, the unscaled form of Eq. (15) was used for calculation. Another comparison is possible for the radiative flux given in Fig. 6 (curve C) of [17]; in our implementation, this corresponds to the parameters and T fs ðrÞ ¼ T w þ ðT fs ð0Þ T w Þ g ¼ 0; x0 ¼ 0:5; sR ¼ 5; ¼ 1 h i 2 1 ðr=RÞ with the relation T fs ð0Þ=T w ¼ 5. In all cases, the results of our implementation are identical to the literature values within the precision limits of graph digitalization, proving the validity of Eqs. (15), (18) and (16). At this point we give further information about the dimensionless parameters introduced in Eq. (19). According to [23], d is the height of a fibre gas cell in units of the fiber diameter, and o þ 1 is the area of a fibre gas cell in units of the contact area. dþ1¼ 1=2 plqbulk 2qfs ð1 l2 Þ qbulk =qfs oþ1¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l 1 l 1:5p 1 12bulk p0 =Y bulk 2 ð25Þ !1=3 ð26Þ Fig. 11. Comparison of the radiative flux calculated using Eqs. (15), (16) and (18) with results given in (Fig. 6) and (Fig. 10) of [17]. 2522 H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522 Table 3 Literature data according to [30,31,33,34] used for the theoretical model and further calculations. T in °C 200 1000 1200 1400 1580 Air Al2 O3 cp J/(kg K) tcbulk W/(m K) tcg W/(m K) 1022 1261 1295 1317 1341 22:38 6:04 5:70 5:72 6:13 0.038 0.082 0.092 0.101 0.110 l ¼ cos #0 is the mean fiber orientation angle with respect to the external heat flow. p0 is an external pressure, 1bulk represents Poisson’s number and Y bulk is Young’s modulus. Assuming a random fiber orientation (Z ¼ 0:66) and qfs ¼ 616 kg=m3 one finds d ¼ 1:77. Using 1bulk ¼ 0:25; Y bulk ¼ 360 109 N=m2 from [32] and 5 p0 ¼ 10 Pa; o is calculated to be 2:11 103 . Therefore the expression ðo þ 1Þ=o in Eq. (19) is approximately 1. Finally, the algorithm for solving the nonlinearity in the inhomogeneous diffusion Eq. (21) is introduced [24]. 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