Wolfgang POESSNECKER and Ulrich GROSS*

Transcription

1 Proceedings of the Asian Thermohysical Proerties onference -4 August, 007, Fukuoka, Jaan Paer No. 0 A QUASI-STEADY YLINDER METHOD FOR THE SIMULTANEOUS DETERMINATION OF HEAT APAITY, THERMAL ONDUTIVITY AND THERMAL DIFFUSIVITY WITH THE USE OF TEMPERATURE MEASUREMENT IN ONE POINT ONLY Wolfgang POESSNEER and Ulrich GROSS* Institute of Thermal Engineering, Technische Universität Bergakademie Freiberg D Freiberg, Germany, *corresonding author: gross@iwtt.tu-freiberg.de Abstract Heat caacity, thermal conductivity and thermal diffusivity are measured simultaneously using a hollow cylindrical samle with adiabatic boundary conditions at its outer surface in axial and radial direction. Starting from thermal equilibrium a constant heat flow is sulied to the inner surface of the samle. The temeratures throughout the samle are raised asymtotically aroaching a certain constant radial rofil which is time-linear shifting uwards. Measurements of the temerature at only one osition allow the determination of all three roerties and no temerature differences are needed. eywords: Simultaneous method, thermal conductivity, thermal diffusivity, secific heat, measuring technique. Introduction For simultaneous determination of the thermohysical roerties various transient, ulse, ste-wise and eriodic heat flow methods (wire and lane sources) are widely alied (for an overview see [], []). The eculiarity of these methods mostly consists in considerable deviations between mathematical model and ractical realisation. These deviations tyically cause comlicated formulas for the evaluation on the one hand and not seldom a strong limitation of exactness on the other hand. Because of it, a new simultaneous method with different than usual new features has been develoed. A similar rocedure for the

2 simultaneous measurement of the three roerties has been ublished earlier by the resent authors [] where continous internal heating of a hollow adiabatic cylinder has been alied. The long-term behaviour has been evaluated based on temerature measurements at two radial ositions and all three roerties are obtained for bad conducting materials. In contrast to this the resent method is focused on the short-time transient temerature at only one location.. Princile of the Method A hollow cylindical samle is used ( R = m, R = m as the inner and outer radius) with an axial borehole (at radius r ) containing one single well contacted temerature sensor. Starting with uniform temerature, a constant heat flow is sulied to the inner surface of the samle. The temerature field develos deending on the thermohysical roerties and also on axial and radial heat losses. The latter ones increase with time distorting the measurements. The resulting error can be reduced and finally avoided by alication of controlled rotection shields and/or alication of long cylinders and short measuring times.. Mathematical Solution Neglecting axial losses the temerature field is one dimensional and it can be calculated from the solution of the following initial boundary condition roblem with thermal diffusivity a, thermal conductivity λ and the excess temerature T starting from time zero: T t T = a + r r T r where T = q& T λ and λ = α( TR ) () r r R R The exact analytical solution has been obtained by Lalace transformation. It is not included in this aer due to its sace consuming comlexity, however, it has been evaluated numerically for some examlary cases ( a = m s, λ = Wm, inside heat & flow rate q = Wm and outside heat transfer coeffcients 0 α 000 Wm ) and the resulting excess temerature is lotted in Figure for one location close to the inner surface (at r = m ). In case of erfectly adiabatic conditions ( α = 0 Wm ) the sloe of the temerature increase (bold line) is found to aroach a straight asymtote as exected which is reached

3 after a settling time of about t set = 4 s. This line is exactly reresented by the quasi-stationary art of the extended solution (for α = 0 Wm and t ) with the transient T & = AB 4 4 R R R R R ln + R qr & R a r R r ( ) R 4 RR T r, t = t R ln + () λ(r R ) R { R R R R A B In the exeriment, the samle will be ositioned inside a with or without secial recautions to avoid radial heat losses. Evacuation will hel to reduce heat conduction and convection effects to be negligible. Remaining radiation losses can be minimized by surrounding the samle by controlled radiation shields (see, e.g., []) to guarantee adiabatic conditions in a best ossible way. Remaining radial losses will affect the temerature increase to be weaker exhibiting a saturation characteristic (horizontal asymtote) where heat inut is balanced by radial heat losses (see the dash-dotted curve in Figure, as calculated from the detailed solution with α = 000 Wm just for demonstration of a high heat loss case). Figure : Temerature increase starting with time zero for various outside heat transfer condictions with t * as the intersection time, see eq. (3)

4 In reality much smaller heat transfer coefficients have to be exected yielding negligible deviations from the adiabatic case at short times. Effective heat transfer coefficients u to α = 3 Wm (corresonding to radiation losses at the outer surface temerature 800 ) have been alied to evaluate the actual temerature increase which roofs to kee below the accuracy limit of temerature measurements within the first 0 s. By this the alication of the zero heat loss solution, eq. (), for the thermohysical roerty calculations is enabled.. Evaluation of the Thermohysical Proerties As the first ste the thermal diffusivity a will be evaluated from the intersection oint of the straight asymtote, eq. (), with the time axis ( T = 0, see Fig. ) t * Bt * + = 0 (3) where t* is the resective negative(!) time obtained from the exeriment. ombining the coefficient B from eq.() with eq.(3) the gained thermal diffusivity is easily obtained as R = (4) t * a with the arameter, see eq.(), only deending on the samle geometry. A ractical way for finding t* consists in linear regression of the measured temerature increase with time for t t set, i.e. a lot analogeous to Fig.. As the second ste the thermal conductivity λ is obtained from eq. () with T & = AB qr & a λ = T(R & R ) (5) As the third ste and in contrast to the evaluation of a and λ, the secific heat c can be calculated directly from its definition with Q & as the heat inut and m as the mass of samle and resectively c = m Q& samle (mc T& m ) samle (6)

5 3. onsideration of Uncertainties by Heat Losses Uncertainties by heat losses should nearly disaear for short times as discussed above for small values of α. By means of eq.(4), diffusivities and their deviations from the adiabatic case have been calculated for selected heat transfer conditions (see Table ). The resulting error by heat losses is clearly shown to kee below about % in any ractical cases. Table : Effect of outside heat transfer conditions on the resulting thermal diffusivity, α (radiation losses) Wm Thermal diffusivity m s deviations from the adiabatic case % Eqs. (6) and (5) for the determination of secific heat and thermal conductivity resectively are only valid for adiabatic conditions. In resence of heat losses the heating rocess (subscrit H) has to be sulemented by considering the dynamic resonse of the system after switching off the heating elements, i.e. for the subsequent cooling rocess (subscrit ) with a resective negative(!!) temerature transient. For heating, subscrit H is valid: After switching off the inut ower is zero: Substraction brings for the samle s secific heat Q & = ((mc ) + (mc ) )T& + Q& samle samle 0 = ((mc ) + (mc ) )T& + Q& mc = (mc ) T& H Q& + T& H heat loss heat loss and analogeous for the thermal conductivity: λ = ( T& + T& )( R R ) H qr & a 4. First Measurements for Testing the Simultaneous Method First measurements with Titanium for testing the method were carried out by very quick ushing of a recedently heated coil ( q& = Wm ) into the samle with the resence of air at room temerature. This coil is made from an oxidized antal wire housed in a quartz tube for revention of short circuit ( m = 60 g and c = 0.48 Jg for the ). A directly welded coated thermocoule was used as temerature sensor ositioned in a narrow borehole near the inner surface. From the measured temerature history the following data

6 have been evaluated: T& = s (sloe) and t* = 4. s (intersection time) and subsequently the roerties from eqs. (4) to (6) - thermal diffusivity: a = m s (Touloukian [3]: a = m s ) - themal conductivity: λ =.97 Wm (d Ans-Lax [4]:.9 Wm λ = ) - secific heat: c = Jg (d Ans-Lax [4]: c = 0.53 Jg ) which is in excellent agreement with the reorted literature data. 5. onclusions The results received for Titanium at room temerature show very good agreement with the resective recommended values by Touloukian and D Ans-Lax. The next aim is the further develoment of the method for higher temeratures. The heat losses which will increase at higher temeratures are to be avoided by comensation and additionally by commonly used constructive measures like automatically controlled rotection shields. Acknowledgements The authors like to thank Andreas Wahl for his tireless hel in assembling the measurement and in carrying out the exeriments. REFERENES [] Poessnecker, W., Gross, U., A Quasi-steady ylinder Method for Simultaneous Thermohysical Proerty Measurements u to 000, J. Thermal Analysis and alorimetry, Vol. 86, , 006 [] ubicar, L., Bohac, V., Review of several dynamic methods of measuring thermohysical arameters, Proc. 4 th Int. Thermal onductivity/ th Int. Thermal Exansion Symosium, eds. P.S. Gaal und D.E. Aostolescu, Technomic Publ. (Lancester/USA), 35-49, 999 [3] Touloukian, Y.S., Thermohysical Proerties of High Temerature Solids, IFI (New York, Washington) Vol. 4, 970 [4] D Ans, J., Lax, E., Taschenbuch für hemiker und Physiker (in german language), ed.:. Schäfer und. Synowietz, Sringer-Verlag (Berlin) Vol. 3, 006