© ISO 2016 Measurement of apparent thermal conductivity of wet porous building materials by a periodic method Détermination de la conductivité thermique apparente des matériaux de construction poreux[.]
INTERNATIONAL STANDARD ISO 16957 First edition 2016-06-15 Measurement of apparent thermal conductivity of wet porous building materials by a periodic method Détermination de la conductivité thermique apparente des matériaux de construction poreux et mouillés par une méthode périodique Reference number ISO 16957:2016(E) I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n © ISO 2016 ISO 16957:2016(E) COPYRIGHT PROTECTED DOCUMENT © ISO 2016, Published in Switzerland All rights reserved Unless otherwise specified, no part o f this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission Permission can be requested from either ISO at the address below or ISO’s member body in the country o f the requester ISO copyright o ffice Ch de Blandonnet • CP 401 CH-1214 Vernier, Geneva, Switzerland Tel +41 22 749 01 11 Fax +41 22 749 09 47 copyright@iso.org www.iso.org ii I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n © ISO 2016 – All rights reserved ISO 16957:2016(E) Contents Page Foreword iv Introduction v Scope Normative references Terms and definitions Symbols and units Determination of thermal conductivity of a wet porous material by non-steadystate method (periodic method) Measurement by periodic method 6.1 Test procedure 6.2 Measuring apparatus 6.2.1 Overall design 6.2.2 Generator of the sinusoidal or stepwise electric wave 6.2.3 Heater 6.2.4 Specimen 6.3 Specimen preparation and preconditioning 6.3.1 Initial uniform moisture content and adiabatic and impermeable boundaries 6.3.2 Embedding and the position of the thermocouples f ff f Annex B) f f f 7 Test report Annex A (informative) Theoretical background Annex B (informative) Derivation of thermal conductivity from measured temperatures 11 Annex C (informative) Example of measurement by periodic method 17 Bibliography 20 6.4 D erivatio n o thermal di ro m meas ured temp eratures (s ee S o lutio n o r heat flo w witho ut mo is ture S o lutio n o r heat flo w with mo is ture 6.5 E s timatio n o 6.6 Thermal co nductivity © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n us ivity meas uring uncertainty due to mo is ture (vap o ur) movement iii ISO 16957:2016(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies) The work o f preparing International Standards is normally carried out through ISO technical committees Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters o f electrotechnical standardization The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part In particular the different approval criteria needed for the di fferent types o f ISO documents should be noted This document was dra fted in accordance with the editorial rules of the ISO/IEC Directives, Part (see www.iso.org/directives) Attention is drawn to the possibility that some o f the elements o f this document may be the subject o f patent rights ISO shall not be held responsible for identi fying any or all such patent rights Details o f any patent rights identified during the development o f the document will be in the Introduction and/or on the ISO list of patent declarations received (see www.iso.org/patents) Any trade name used in this document is in formation given for the convenience o f users and does not constitute an endorsement For an explanation on the meaning o f ISO specific terms and expressions related to formity assessment, as well as information about ISO’s adherence to the World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html The committee responsible for this document is ISO/TC 163, Thermal performance and energy use in the built environment, Subcommittee SC 1, Test and measurement methods iv I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n © ISO 2016 – All rights reserved ISO 16957:2016(E) Introduction Most building materials, with the exception of glass and metals, are porous and, thus, absorb moisture due to conden s ation, rai n a nd water up ta ke materi a l s th rough, e g ro tti ng or fro s t from the ground T he ab s orb e d moi s ture may damage the damage a nd, thu s , may c au s e thei r p er forma nce to de teriorate In particular, an increase in the moisture content of insulation material causes a reduction of its thermal resistance, which must be avoided as much as possible to preserve its performance However, i n fi ltration o f rai n water i nto fou ndation (fo o ti ng) a brick wa l l or j oi nts o f ti le s and up ta ke o f grou nd water i nto the a re ver y d i ffic u lt to avoid T here fore, it i s i mp or tant to u nders tand the cha nge s i n the therma l prop er tie s (therma l conduc tivity a nd he at cap acity) o f p orou s materia l s due to change s i n their moisture content I S O 10 51 s p e ci fie s a s te ady- s tate me tho d for me as u ri ng the therma l conduc tivity o f a moi s t bu i ld i ng materi a l I n the s te ady- s tate me tho d , a nonu n i form d i s tribution o f moi s tu re content i n the te s t pie ce i s inevitable, since the imposed temperature gradient causes moisture transfer The nonuniform moisture d i s tribution ma ke s it d i ffic u lt to defi ne wh ich moi s tu re content the me a s u re d therma l conduc tivity corre s p ond s to I S O 10 51 c ategori z e s the moi s tu re d i s tribution i n the te s t pie ce i nto s evera l typ e s a nd e s ti mate s the therma l conduc tivity corre s p ond i ng to e ach typ e Si nce the ore tic a l and exp eri menta l re s e arch s re cently b e en p er forme d concern i ng he at and moi s tu re transfer in porous materials (see References [5], [7], [8], [9] and [10]), along with measurements and f f f the s truc tion o a datab a s e o hygro therma l prop er tie s (s e e Re erence [ ] ) , hygro therma l b ehaviour c an now b e pre d ic te d with re a s onable acc u rac y This I nternationa l Sta nda rd de s c rib e s a tra n s ient me tho d for me a s uri ng the therma l conduc tivity o f a we t p orou s bu i ld i ng materi a l and a me tho d o f eva luati ng the me a s urement uncer ta i nty, on the basis of both theoretical developments for heat and mass transfer and the constructed database of hygro therma l prop er tie s T he eva luation o f the me a s u rement u ncer tai nty ma ke s p o s s ible a s i mple and, thu s , prac tic a l me tho d NO TE for me as u ri ng therma l conduc tivity T her m a l conduc tivity i s one o f the ne ce s s a r y hygro ther m a l pro p er tie s S i nce he at tra n s fer a nd m a s s tra n s fer i n p oro u s m ater i a l i nterac t with e ach o ther, a n e xac t va lue o f the ther m a l co nduc tivity mu s t b e given i n order to e xa m i ne the va l id ity o f the the ore tic a l mo del s T hu s , p re c i s el y s p e a ki n g , the ab ove -mentio ne d the ore tic a l mo del s h ave no t b e en va l idate d , a nd the s tr uc tion o f the mo del a nd the me a s u rements o f the hygro ther m a l properties must be carried out in parallel Nonetheless, it seems reasonable to expect that measurement of the for the present document ther m a l conduc ti vity with a n a l lowab le acc u rac y i s p o s s ib le u s i n g a s u itab le me a s u r i n g me tho d T h i s i s the b a s i s © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n v I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n INTERNATIONAL STANDARD ISO 16957:2016(E) Measurement of apparent thermal conductivity of wet porous building materials by a periodic method Scope T h i s I nternationa l Sta ndard de s crib e s a me tho d o f me as u ri ng the therma l conduc tivity (d i ffu s ivity) o f a we t p orou s bui ld i ng materi a l a nd a me tho d o f eva luati ng the me as u rement uncer ta i nty While ISO 10051 change with a short period as an input Along with the measurement, an evaluation of the measurement is the current I nternational Standard, b as ed on a s teady- s tate metho d, thi s I nternational Standard propo ses a metho d that makes us e o f a non- s te ady- s tate metho d which us es a s mal l temperature uncer tai nty is describ ed, which makes p os s ible a s imple and prac tical meas uring metho d T h i s I nternationa l Standa rd i ntend s to me as u re the app arent (e ffe c tive) therma l conduc tivity, i nclud i ng latent he at tran s fer c au s e d b y vap ou r movement T he s ituation i n wh ich moi s tu re and/or r movement o cc u r due to conve c tion or gravity i s e xclude d T he appl ic ation o f th i s I nternationa l Stand ard to h igh moi s ture content i s exclude d s o that the gravity e ffe c t c a n b e ne gle c te d T h i s I nternationa l Standard can be applied to a porous material heavier than about 100 kg/m3 , in which radiative heat transfer can be neglected T h i s I nternationa l Standard s p e ci fie s the fol lowi ng: a) a non- s te ady- s tate me tho d o f me a s uri ng therma l conduc tivity; b) a n approxi mation formu la for the me a s u rement u ncer tai nty c au s e d b y moi s tu re movement a nd nonuniform moisture distribution (and, thus, a determination of the measuring conditions that s ati s fy the upp er l i m it o f me a s u rement u ncer tai nty) ; c) a n e s ti mate o f the he at tran s fer c au s e d b y moi s tu re (vap our) movement Normative references T he fol lowi ng i nd i s p en s able c u ments , i n whole or i n p ar t, are normatively re ference d i n th i s c u ment a nd are for its appl ic ation For date d re ference s , on ly the e d ition cite d appl ie s For u ndate d re ference s , the late s t e d ition o f the re ference d c u ment (i nclud i ng any amend ments) appl ie s There are no normative references in this document Terms and definitions For the pu r p o s e s o f th i s c u ment, the term s and defi n ition s given i n I S O , I S O 10 51 a nd the fol lowi ng apply ISO and IEC maintain terminological databases for use in standardization at the following addresses: — IEC Electropedia: available at http://www.electropedia.org/ — ISO Online browsing platform: available at http://www.iso.org/obp © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n ISO 16957:2016(E) 3.1 apparent thermal conductivity of a wet material λ* i ntri n s ic materia l prop er ty dep endent up on moi s tu re content and temp erature, but no t on te s ti ng conditions N o te to entr y: Si nce th i s va lue i nclude s the i n fluence o f he at tra n s fer due to p h a s e ch a nge (co nden s ation a nd evap o ration) , it i s c a l le d app a rent ther m a l conduc ti vity Symbols and units Symbol a A , B, C, D c c cs DT ′ D θ D Tl D D Tv D E, E1 , E2 I0 kv R S θl θv t T T0 T1 , T2 , θ1 , θ2 Quantity Units m 2/s — m3/m J/(kg·K) J/(kg·K) kg/(m·s·K) ther m a l d i ffu s ivity o f the m ater i a l de fi ne d i n Formula (B.14) p o ro s ity s p e c i fic he at o f the m ater i a l s p e c i fic he at o f the d r y m ater i a l moi s tu re (vap o u r a nd l iqu id water) conduc ti vity rel ate d to temperature gradient M o i s tu re (vap ou r a nd l iqu id water) co nduc tivity rel ate d to water content gradient Formula (B.12) amplitude of the input surface temperature kg/(m·s·K) m 2/s kg/(m·s·K) m 2/s — K vap o u r d i ffu s i vity kg/[m· s· ( kg/ kg′ ) ] l iqu id water co nduc tivity rel ate d to temp eratu re grad ient l iqu id water co nduc tivity rel ate d to water content grad ient vap o u r co nduc tivity rel ate d to temp eratu re grad ient vap o u r co nduc tivity rel ate d to water co ntent grad ient de fi ne d i n latent heat of vaporization s p e c i fic s u r face a re a i n s ide the m ater i a l , i e ratio o f p ore s u r face area to the material volume time temperature initial temperature the fi rs t a nd s e cond ter m s o f the p er tu rb ation s olution o f the temp eratu re a nd water content, re s p e c ti vel y x X Xi U3 , U7, Q1 – Q6 , R1 – R6 , V18 α1 , α2 α′ γ γ′ γs η , ζ1 , β1 , ξ1 , κ1 , γ1 η , ζ2 , β2 , ξ2 , γ2 λ I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n m 2/s J/kg m 2/m s K K K, kg/m coordinate m hu m id ity ratio o f mo i s t a i r i n the p o re s o f the s p e c i men kg/ kg′ e qu i l ib riu m hu m id ity ratio with l iqu id or c ap i l l a r y water at the kg/ kg′ interface in the material co e ffic ients ap p e a ri n g s olution s de fi ne d i n θ2 and T2 — Formula (B.13) — e ffe c ti ve vap o u r tra n s fer co e ffic ient at the i nter face s p e c i fic weight o f d r y a i r den s ity o f the m ater ia l den s ity o f the d r y m ater i a l D , D , D Tl , D Tv, c γ λ f D , D , D Tl , D Tv, λ water co ntent co e ffic ients o f temp eratu re co e ffic ients o θl θl θv ′ ′, θv ther m a l conduc ti vity witho ut mo i s tu re mo vement kg/(m· s· kg) / kg′ kg/m kg/m kg/m — — W/(m·K) © ISO 2016 – All rights reserved ISO 16957:2016(E) Symbol Quantity λ* θ θ0 Θ0 ω ther m a l conduc tivity de fi ne d a s water content of material initial water content of material Units W/(m·K) kg/m3 kg/m3 % rad/s λ* = λ + RD Tv water content b y weight a ngu l a r velo c ity o f the i np ut s u r face temp eratu re Determination of thermal conductivity of a wet porous material by nonsteady-state method (periodic method) When me a s uri ng the therma l conduc tivity of a p orou s materia l in wh ich moi s tu re tra n s fer may o cc u r, the nonun i form ity o f the water content d i s tribution mu s t b e kep t a s low a s p o s s ib le under the temp erature grad ient I n order to m i ni m i z e the nonu n i form ity, a p erio d ic me tho d i s adop te d a s Annex C) Since positive and negative temperature gradients are generated in turn in this method, the (time-averaged) water content distribution can be expected to remain uniform a tra n s ient me tho d o f me as u ri ng therma l conduc tivity (s e e example i n B y me a s uri ng the temp eratu re s at two p oi nts i n the s ample (u s ua l ly one o f them i s at the s a mple s ur face) , the therma l d i ffu s ivity (no t conduc tivity) c an b e de term i ne d b as e d on the ampl itude ratio or phase difference of these two temperatures If moisture movement does occur, a similar method can b e u s e d to de term i ne the therma l d i ffu s ivity i f the i nput c ycl ic temp eratu re fluc tuation i s kep t s ma l l enough that the ch ange i n tran s p or t prop er tie s i s a l s o s ma l l enough that the s ys tem c a n b e regarde d a s linear (see Annex B) Measurement by periodic method 6.1 Test procedure A schematic diagram of the apparatus for the periodic method is given in Figure installed in a climate chamber whose temperature is kept at the mean temperature of the sample under measurement The sample is preconditioned at a certain water content, and then the whole surface is made impermeable to moisture movement A periodic temperature variation is imposed on the sample, T he whole s ys tem i s and the temp eratu re s at (at le as t) two p oi nts i n the s ample are me as u re d b y thermo couple s Figure — Schematic diagram for measuring thermal diffusivity © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n ISO 16957:2016(E) 6.2 Measuring apparatus 6.2.1 Overall design A detailed schematic of the measuring apparatus is shown in Figure Refrigerant kept at a low constant temperature is circulated in a metal refrigerant bath [200 mm (length) × 200 mm (wide) × 50mm (height)] in order to avoid a temperature increase due to heating by the heater A heater, a damping layer and a sample are placed on the re frigerant bath in order, and another damping layer is placed on the sample to reduce the influence o f the temperature fluctuations o f the climate chamber surrounding the measuring apparatus Either a sinusoidal or stepwise electric current is generated in the heater The the sample surface Thermocouples are inserted into the sample and connected to the recorders The output from the thermocouples is recorded by both an analogue recorder and a digital recorder The digital recorder is used for a long-term record for temperatures at multiple points, while the analogue recorder is used for recording the temperature wave in a short-term measurement stepwise wave becomes almost sinusoidal as it flows through the damping layers be fore arriving at Dimensions in millimetres Key thermal insulation material specimen thermocouple rubber film heater copper plates refrigerant bath refrigerant in refrigerant out Figure — Vertical cross section of apparatus 6.2.2 Generator of the sinusoidal or stepwise electric wave A sinusoidal electric wave is generated by an arbitrary wave generator and is sent to the heater When such an apparatus is not available, a cyclic stepwise electric wave is generated and input to the heater by switching a constant electric current on and o ff using a relay A cyclic on-o ff switching o f the relay can be realized with a combination of two timers I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n © ISO 2016 – All rights reserved ISO 695 7: 01 6(E) e) f) te s t pro ce dure; — date o f the s tar t a nd duration o f the te s t; — the me tho d o f s a mpl i ng; — he at i nput (me a n temp eratu re, a mp l itude, e tc ) ; — moi s ture content o f s ample; — any fac tors wh ich may have i n fluence d the re s u lts; re s u lts; — table of the measured values (mean and amplitude of temperatures at two depths from sample s ur face , pha s e d i fference b e twe en temp eratu re s at two dep th s , average moi s tu re content) ; — graph showi ng ti me pro fi le s o f temp eratu re s at two dep th s I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n from s a mple s u r face © ISO 2016 – All rights reserved ISO 16957:2016(E) Annex A (informative) Theoretical background A.1 Fundamental formulae of vapour, liquid water and heat transfer Re cently, the me ch an i s m o f the s i mu lta ne ou s he at and moi s tu re tran s fer i n p orou s materi a l h as b e en i nve s tigate d and clari fie d (s e e Re ference s [ properties have been made [6] 5], [7], [8], [9] and [10]), and measurements of the transport Formu lae that de s crib e the s i mu ltane ou s flow o f he at and moi s ture i n a moi s t p orou s materia l are as follows: [9] vapour balance, ∂X cγ = ∂t ∂ ∂X kv +α′ ∂x ∂x i S ( Xi − (A.1) X) liquid water balance, ∂θ = ∂t ∂ ∂θ ∂ ∂T + D Τl +α′ D θl ∂x ∂x ∂x ∂x i (A.2) S ( X − Xi ) heat balance, and ∂T c ′γ ′ ∂t = ∂ ∂T λ + ∂x ∂x (A.3) Rα ′ i S ( X − X i ) absorption isotherm Xi g(θ,T) (A.4) = where the s en s ible he at tran s p or te d b y vap ou r and l iqu id i s negle c te d Assuming i transformed as follows: moisture balance, α′ + cγ is i n fi n ite ( lo c a l e qu i l ibriu m) , u s i ng Formula (A.4), Formulae (A.1) to (A.3) are (A.5) ∂g ∂θ ∂g ∂T ∂ ∂θ ∂ ∂T + cγ = DΤ Dθ + ∂θ ∂t ∂T ∂t ∂x ∂x ∂x ∂x heat balance ∂g ∂θ ∂g + Rcγ + Rcγ ∂θ ∂t ∂T where Dθ = D θv + D θl = kv ∂g ∂θ © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n + ∂T c ′γ ′ ∂t D θl , D T = = ∂ λ + ∂x ( D Tv + D Tl RD Tv ) = kv ∂θ + ∂x ∂g ∂T + R D Tl ∂ ∂θ D θv ∂x ∂x (A.6) (A.7) ISO 16957:2016(E) A.2 Implication of the thermal conductivity of a wet porous material In Formula (A.3), λ represents the thermal conductivity, which is mainly determined by conduction through the constituents (solid skeleton, water and air) in the case of no moisture movement, and of course it varies with the water content On the other hand, as seen in Formula (A.6), λ and RDTv always appear in the conjunct form λ* = λ + RD Tv, and λ* is the coe fficient to the temperature gradient As mentioned above, λ and RDTv never appear separately in the fundamental Formulae (A.5) and (A.6) or in the boundary conditions Consequently, D Θ , D T, λ* = λ + RD Tv and D Θv are necessary to calculate the heat and mass transfer, while the values of λ and DTv are not necessary Because of these properties of the fundamental formulae, only λ* (not λ) is given by any method o f measuring thermal di ffusivity Thus, methods that minimize the e ffects o f moisture movement can mostly minimize the e ffect o f the term R ∂ Dθv ∂θ and measure λ* more accurately The periodic method, for example, satisfies this ∂x ∂x criterion, which is made clear in Annex B It is concluded from the above discussion that in order to solve the formulae of heat and mass transfer, one must measure and use the values of λ*, not λ On the other hand, as λ is defined under the assumption that there is no moisture movement under the temperature gradient, it is a hypothetical quantity As mentioned above, only λ* can be measured in this system, and λ cannot be measured, except in special cases, but may be estimated by assuming an appropriate model 10 I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n © ISO 2016 – All rights reserved ISO 16957:2016(E) Annex B (informative) Derivation of thermal conductivity from measured temperatures B.1 Measurement of thermal diffusivity by periodic method B.1.1 Objective It is estimated from the result of Annex A that the thermal di ffusivity corresponding to λ* can be measured in a non-steady-state method minimizing the liquid movement and, thus, non-uni form moisture distribution As a representative of many transient methods, the periodic method is examined here The objective o f the following analysis is to estimate quantitatively the measurement uncertainty caused by moisture movement, and to derive a formula to give the measurement uncertainty that is composed o f the material properties and is applicable to any material This problem will be solved by a perturbation method assuming that the trans fer coe fficients are linear functions of water content and temperature and that the solution is a power series of the input surface temperature variation amplitude The influences o f moisture are those caused by a) the existence of the moisture movement, and b) variation o f the trans fer coe fficients with water content and temperature Approximately, the first term o f the solution can be considered to represent a), and the second term, b) B.2 Formulation B.2.1 Perturbation from initial conditions Suppose that the material is semi-infinite and that its sur face temperature variations are sinusoidal The basic formulae are Formulae (A.5) and (A.6) (neglecting smaller order terms on the left-hand side) D θ0l 1 + η θ − θ ∂θ ∂ = ∂t ∂x D 1 + ς θ −θ θv ( ) 1 + η (T − T ) ( ) D Τl + β θ − θ ∂ ∂x D 1 + ξ θ −θ Τv 1 + ς (T − T ) ) 1 + β (T − T ) ( ) 1 + ξ (T − T ) 2 0 ∂θ ∂x ∂θ ∂x ( © ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n ∂T ∂x ∂T ∂x + + (B.1) + 11 ISO 16957:2016(E) c ′γ ′ 1 +κ (θ −θ ∂T ) ∂t R with initial conditions T T0 , θ θ0 = λ 1 + γ θ − θ + γ T − T0 ∂ = ∂x RD + ξ Tv θ − θ 1 + ξ T − T0 ( ) ( ∂ D θv + ς ∂x (θ ( ) −θ ( ) 1 ) +ς (T −T ) ∂T + ∂x + ∂T ∂x ) (B.2) ∂θ ∂x (B.3) = and b ou nda r y cond ition s D θl + η θ − θ D θv + ς θ − θ ( ) 1 + η (T − T ) + ( ) 1 + ς (T − T ) D Tl + β θ − θ 1 + ξ D θ −θ θv ( ( θ = Finite ( x → +∞ 0 ) 1 + β (T − T0 ) ) 1 + ξ (T −T ) ∂θ x ∂ + + ∂T = ∂x (B.4) (x = ) (B.5) ) T = T0 + I0 sinω t ( x = ) (B.6) T (B.7) = Finite ( x → +∞ ) where zero superscripts represent the values at Θ0 and T0 Transforming Ө and T to Ө′ and T′ b y the relation T ′ = T − T0 ,θ ′ = θ −θ (B.8) The transformed formulae have the same form as Formulae (B.1) to (B.7), except that Θ0 and T0 are Hereafter, we rewrite Ө′ and T′ as Ө and T B.2.2 Moisture content and temperature as functions of small input We assume a power series solution as = I 0θ + I θ + I 3θ + ⋅ ⋅ ⋅ T = I0T1 + I0 2T2 + I 3T3 + ⋅ ⋅ ⋅ θ Substituting these formulae into Formulae (B.1) to (B.7) of I0 , we obtain the formulae for Ө1 , T1 , Ө2 , T2 , etc 12 I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n (B.9) a nd e quati ng the co e ffic ients o f the s a me order © ISO 2016 – All rights reserved ISO 16957:2016(E) B.3 Solution B.3.1 Solution: First order (periodic solution) θ ( t, x ) −α E exp = α α E exp −α 2 ω T1 ( t, x ) = α EE 1 α B where E e xp −α ω B sin ω t − α ω x × sin x × 2 ωt − α x − (B.10) x ω sin ω t − α ω x − (B.11) x B = (α E1 ω ω x × ωt − α −α x × sin exp EE 2 ω = −α 2) C ( CA − BD ) (α − C − ( CA − BD ) α , E +α α +α 1 C − ( CA − BD ) α 2 = (B.12) ) α1 and α2 are positive roots of the quadratic equation (α1 < α2 ) ( AC − BD ) α A = Dθ , B − ( A + C)α DΤ , C = = λ + + = RD Tv ( c ′γ ′ ) , (B.13) D RD θ0v = ( c ′γ ′ ) (B.14) B.3.2 Solution: Second order The periodic solutions are θ ( t, x ) = 2U3 exp ( −α ω Q exp −2 ω Q exp Q exp −2 Q 6′′ exp − α (α ω ω +α (α ω 2 ) x × cos ω t 2x − α cos ( ω t − α x × cos ω t + Q 6′ exp +α ω ) x ì â ISO 2016 – All rights reserved I n tern ati o n al Org an i z ati o n fo r S tan d ard i z ati o n ) ωx × sin α ω (α ω ωx ) + 2U7 exp ( −α 1x + (α Q exp (α +α α ) x × cos ω cos ( ω t − α x × cos ω t ) x + Q exp −2 +α ω −2 ) ωx × (α ω − ω ωx α 2x + )+ (B.15) α 1x + −α ) x + −α ) x 13 ISO 695 7: 01 6(E) T2 ( t, x ) 2 = − Aα 22 R exp ω −2 R exp ω −2 ω − R exp ω −2 R 6′ exp ω − R 6′′ exp ω − B B 4 A p C p o r o e ) f x f i i c A B y a s s u m i ng C e x × cos ω t α x × cos ω t +α (α +α n t s (α i