sign in sign up
<<
Home , Thermal insulation

I Ser

no. i44 )$r National Research Conseil national c. 2 Council Canada de recherches Canada Institute for lnstitut de Research in recherche en Construction construction

Heat Tmnsport Through Thermal Insulation: An Application of the Principles of of Irreversible Processes

by M.K. Kumaran and D.G. Stephenson

Reprinted from ASME Paper No. 86-WA/HT-70, p. 1-4 ASME Winter Annual Meeting Anaheim, California, December 7 - 12, 1986 (IRC Paper No. 1445)

NRC - ClSTl I R C Price $2.00 LIBRARY NRCC 27451 MAY 28 ;ss7

BIBLIOTH~QUE IRC CNRC - IClST Les auteurs utilisent les principes de la thermodynamique des processus irrgversibles pour analyser la transmission de chaleur 3 travers les isolants thermiques secs. 11s considbrent le flux de chaleur comme la combinaison des modes de transfert de chaleur par conduction et par rayonnement, et ils en tirent des Squations ph'enom6nologiques. Les mesures rgalisiks en laboratoire leur permettent de determiner le flux de chaleur traversant un spscimen d'isolant en fibre de verre pour verifier la validit6 de cette mgthode. 11s formulent une expresshon prgcisant la dgpendance de la conductibilitg thermique apparente de l'isolant sec 3 l'bgard de la tempgrature. Les donnges expgrimentales concernant un panneau de cgramique sont bien reprGsent6es par l'expression d'une gamme de de 800 K. Les auteurs proposent une extension de la mgthode th'eorique pour dgcrire le transport simltang de chaleur et dthumidit6 3 travers l'isolant thermique. THE A OF MECH ,ew York,b

a Socrety 9hall not be rest~olsibls far s raremania or oprnlonr sdv,anted In papers or in d~s sston a* meellngs of Ine SI3ciery or of it.! D.v~srons or S=c!luns. or vrinted In Its publicalrons scussjon Ic pr~ntedgnly ~f the paper la publ~shedIn an PSME JUIJrnnl Papers arp ava~laille Irn ASME for ?siteennontl) IS a!ter the mee!llg rntfd ~n USA

Heat Transport Through Thermal Insulation: An Application of the Principles of Thermodynamics of Irreversible Processes

M. K. KUMARAN and 0. G. STEPHENSON Services Section Institute for Research in Construction National Research Council Canada Ottawa, Canada, KIA OR6

ABSTRACT

The principles of irreversible thermodynamics are where (c)is the temperature gradient in the direction used to interpret through dry thermal dx of the heat transfer and (A') and (A") are insulation. The heat flux is treated as coupled conductive and radiative modes of heat transfer and proportionality constants that describe conductive phenomenological equations are accordingly derived. (solid + gas) and radiative modes of heat transfer, Laboratory measurements of heat flux through a specimen respectively. The use of Fourier relation and of fibre insulation are used to check the Rosseland's approximation, together with the theory of validity of the approach. An expression is derived for irreversible processes (T.I.P.) (1-6) can provide a method to look at the interaction between conductive the temperature dependence of the apparent of dry thermal insulation. The and radiative modes of heat transfer in thermal experimental data for a ceramic board are well insulation. This paper describes such an application represented by the expression for a temperature range of T.I.P. to heat transfer through thermal insulation.' of 800 K. An extension of the theoretical method to A three-term expression is derived for the heat flux describe simultaneous heat and moisture transport through the insulation. One term represents the through thermal insulation is suggested. conductive part, the second, the radiative part and the third. the interaction between the two Darts. KEY WORDS The basic postulates of T.I.P. are: 1) The rate (0) of entropy production per unit volume Heat transfer, thermal insulation, thermodynamics is given by the sum of the products of fluxes (Ji) of irreversible processes, rate of entropy production, and associated forces (I$~)i.e. phenomenological equations, moisture. 0 = C Jiei (3) INTRODUCTION i 2) The components of the flux are linear functions of Heat is transferred through dry thermal insulation the components of forces and are given by by a combination of conduction and radiation. Models phenomenological equations as described in the literature interpret the heat transfer as the sum of solid conduction, gas conduction and radiation. In engineering applications, the , interaction between conduction and radiation is usually neglected and the apparent thermal conductivity (Aapparent) of a thermal insulation is written as where the coefficients Lik are called phenomenological coefficients. 3) The cross phenomenological coefficients, subject to certain mathematical restrictions, obey the Onsager where (As) is the contribution from solid conduction, reciprocal relations (g)i.e. (A ) from gas conduction and (Ar) from radiation. g These models use Fourier relation and Rosseland L.. = L. (5) 1J ~i approximation to write expressions for the heat flux (9) as

Presented at the Winter Annual Meeting Anaheim, California - December 7-12, 1986 Thus for an irreversible process, associated with where a, b and c are the phenomenological two simultaneous fluxes, J1 and J2 coefficients. Since qc and qr are in the same direction, they may be added algebraically to obtain the total heat flux where and @2 are the forces corresponding to J1 and J2, respectively and

The first term in the bracket gives the conductive part of the heat transfer, the third term, the radiative part and the second term, the interactive part. This indicates that the apparent fv$rmal conductivity should be a quadratic function of T . The objective of this paper is to check this hypothesis.

A TEST OF THE HYPOTHESIS

Heat Transport and Entropy Production The form of the temperature dependence of the The entropy flux through any material is: apparent thermal conductivity that is indicated by Eq. (18) can be checked by measuring the total rate of heat flow through semi-transparent materials. It is convenient to define a quantity Q as The direction of the S vector is the same as the direction of the heat flux (q), since the temperature (T) is a scalar. If we make the x axis parallel to the entropy and heat flux vectors, the derivative is dx Then dx = (a + 2be~l.' + c.T3) dTdx (20) -dS=l.*-q .dT. (9) dx T dx T2 dx Hence from (18) and (20). The second term of Eq. (9) is the rate at which then q=--.dQ (21) entropy is being produced in unit volume of the dx material; i.e., If one measures the,total heat flux (q1-2) through a sample of thickness t, when the surfaces are kept at T1 and T2, When heat is transferred by conduction,

Thus for heat conduction, from Eqs. (3), (4), (10) and (11) Similar equations can be written when the surfaces are maintained at T3 and T,,, T5 and T6, etc. These equations can be combined into a matrix form

If the heat is transferred by radiation

qr - T3 dT (14) dx (this is the Rosseland approximation for ). In this case

When heat conduction and heat transfer by radiation occur together, the flux and force vectors have two components, and the phenomenological equation that interrelates these components becomes:

b and solved for 5 , - and 2 . tt t Test Specimens laboratories in Canada and the United States (10). Two test specimens were prepared from a sheet of These data are in the form of the apparent thermal medium glass board produced by Johns conductivity at four mean ranging from Manville Co. for the National Bureau of Standards, 533 K to 1363 K. These data were fitted by a quadratic Washington, D.C. The average density of the sample was in ~1.~.Table 2 gives these data and the values of 51 kg*m-j. Specimens were cut and placed in wooden a, b and c that were derived from them, and shows the frames. The dimensions of the specimens were 58.42 x calculated values of the apparent thermal conductivity. 58.57 x 2.64 cm. The two specimens weighed 461.2 g and The quadratic in T~*~represents the data very well 461.3 g, and were identical in all other respects. over this relatively wide range of temperatures. The These specimens were dried in an oven at 323 K for one values of the coefficients a, b, and c depend on the week and then they were mounted on a 60 cm guarded hot nature of the material, and may also depend on the plate apparatus built according to ASTM specifications of the bounding surfaces and the thickness (2). Steady state measurements were made at ten of the sample. Measurements on similar samples of different sets of boundary temperatures. different thicknesses are needed to check on these possibilities. These results show, however, that these EXPERIMENTAL RESULTS characteristic coefficients are not functions of temperature or temperature gradient, and that the The test results and the derived values of a, b analysis based on T.I.P. is valid for this type of and c are given in Table 1, along with values of heat insulation material. The interaction between the heat flux calculated using these values of a, b and c and conduction and radiation is taken into account without the boundary temperatures for the various tests. The having to establish the temperature distribution standard deviation between the measured and calculated through the material. values of heat flux is 0.04 W/m2 when the flux ranged from 23.4 to 52.6 w/m2. Table 2

Table 1 The apparent thermal conductivity (happarent) of a ceramic board at different temperatures. Heat flow through a dry sample of medium density insulation (51 kg*m-3); T1 and T2 are surface 'apparent (W/meK) tempertures and q1-2 is the heat flux T (K) Measured Calculated* 41-2 533 0.073 0.073 Measured Calculated* 2) 2) w/m2 w/m2 813 0.114 0.113 293.92 273.97 23.47 23.43 1088 0.189 0.189

303.11 283.16 24.24 24.29 1363 0.312 0.311

312.63 293.48 24.21 24.24 a = 5.40 x W/m-K b = 1.92 x w/~*K~.~ 322.08 303.25 24.75 24.78 c = 9.40 x 10-I W/m*K4

331.61 313.62 24.72 24.66 *'apparent = a + 2b~l.~+ c T~

343.13 323.3,, 28.47 28.43 An Extension to Heat Transfer Through Moist Materials The principal advantage of the TIP approach is 354.02 333.88 30.21 30.24 that it can be extended to include additional components in the flux vector. An enclosed porous 363.46 343.21 31.65 31.66 system that has moist material adjacent to a warm boundary will have vapour flowing from the warm region 314.22 286.33 34.90 34.93 to the region. The heat flux (qv) associated with this vapour flow can be represented as, 333.61 293.86 52.61 52.60 qv=-k(h*-)-dPV dT a = 1.896 x W/m-K dT dx b = 1.764 x 10-7 ~/rn*K~.~ c = 4.520 x 10-lo w/~*K~ where k = vapour permeability of the material, t = 26.4 mm h = of saturated vapour at T, Pv = saturation at T. 2b (T2.5- T$.5) *ql-2 = 5 (Ti-T2) + - t 2.5t dPv Around 300 K the product h canbe approximated by dT + 5 (T: - T;) A.~15*4. 1 4t As the NRC guarded hot plate apparatus is limited lThis approximation has been derived from tabulated to temperatures between 273 K and 365 K, the validity values of enthalpy and vapour pressure of water at of Eq. (16) could only be established over this limited saturation vs temperature. These data are published range of temperature. However, some high temperature in ASHRAE Handbook of Fundamentals, 1985, Ch. 6, data were available from a set of round-robin Table 2: p. 6.6. measurements made on a ceramic material by seven This means that the flux (Jv) should be ACKNOWLEDGEMENT

The authors gratefully acknowledge the technical assistance of Mr. J.G. Thsriault. This paper is a contribution of the Institute for Research in Construction, National Research Council of Canada.

REFERENCES The phenomenological equation for heat transport by conduction, radiation and moisture migration becomes Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, Interscience Publishers, abd T-l dT New York, 134 p. (1967). dx de Groot, S.R., Thermodynamics of Irreversible Processes, North Holland Publishing Company, e- b c e dT dx Amsterdam, 242 p. (1951). Haase, R., Thermodynamics of Irreversible Processes, Addison-Wesley Publishing Company, def T6.7 dx London, Ch. 4, p. 215-470 (1969). Denbigh, K.G., The Thermodynamics of the Steady The coefficients a, b and c can be determined from State, Methuen 6 Company Ltd., London, 97 p. tests on a dry sample and the coefficients d, e and f (1951). from additional tests on a moist sample of the same Fitts, D.D., Non-equilibrium Thermodynamics, material. Of course, the coqonent qv of heat flux McGraw-Hill, New York, 173 p. (1962). will only occur when there is moisture in the warm de Groot, S.R. and Mazur, P., Non-equilibrium parts,of the material. As soon as all the moisture has Thermodynamics, Dover Publications, Inc., New migrated to the coldest region, the moisture flux and York, Ch. I-VI, p. 1-77 (1984). associated heat flux will stop; the conduction and Onsager, L., Reciprocal Relations in Irreversible radiation will continue as long as a temperature Processes, I. Physical Review, Vol. 37, p. 405 gradient exists. (1931). Onsager, L., Reciprocal Relations in Irreversible CONCLUSION Processes, 11. Physical Review, Vol. 38, p. 2265 (1931). The postulates of the Thermodynamics of ASTM C177: Standard Test Method for Steady-State Irreversible Processes simplify the analysis of heat Thermal.Transmission Properties by Means of the transfer in situations where more than one mode of heat Guarded Hot Plate, Annual Book of ASTM Standards, transfer occurs. This approach takes account of energy Part 18, p. 20 (1981). shifting from one mode of transport to another within Bomberg, M., Institute for Research in the material without having to determine the Construction, Private communication. temperature distribution through the material. The approach, which has been shown to be applicable to heat transfer through thermal insulation, may also be applicable to heat transport through semi-transparent gases in the combustion chambers of and internal combustion engines. It appears that the interaction between conductive and radiative modes of heat transfer amounts to approximately 6% of the heat flux through the two specimens, for the range of temperature reported in this paper. Further testing is needed to check the validity of Eq. (27) when moisture migration contributes to the heat transport through porous materials. The current study has demonstrated that heat transport through a semi-transparent medium like glass fiber insulation can be calculated by using three coefficients that characterize the material. The values of these characteristic coefficients can be determined from measurements of the total heat flow through a sample of the material with various combinations of the boundary temperatures. These characteristic coefficients appear to be independent of temperature over a wide range. In this respect, they are preferable to the value of the apparent thermal conductivity, which is a function of temperature. The apparent thermal conductivity can be calculated for any value of mean temperature when the values of the characteristic coefficients are known. This paper is being distributed in reprint form by the Institute for Research in Construction. A list of building practice and research publications available from the Institute may be obtained by writing to the ~ublicatibns Section, Institute for Research in Construction, National Research Council of Canada, Ottawa, Ontario, KlA 0R6.

Ce document est distribud sous forme de tir6-8-part par 1'Institut de recherche en construction. On peut obtenir une liste des publications de 1'Institut pottant sur les techniques ou les recherches en matisre de bttiment en dcrivant B la Section des publications, Institut de recherche en construction, Conseil national de recherches du Canada, Ottawa (Ontario), KIA 0R6.