INTERNATIONAL STANDARD ISO 18555 First edition 2016-02-01 Metallic and other inorganic coatings — Determination of thermal conductivity of thermal barrier coatings Revêtements métalliques et autres revêtements inorganiques — Détermination de la conductivité thermique des revêtements barrières thermiques Reference number ISO 18555:2016(E) © ISO 2016 ISO 18555:2016(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 © ISO 2016 – All rights reserved ISO 18555:2016(E) Contents Page Foreword iv Introduction v Scope Normative references Terms and definitions Principle Apparatus for measuring thermal diffusivity Specimen 6.1 Shape and dimensions 6.2 Surface treatment 7 Measuring procedure 7.1 Specimen thickness ff 7.2.1 Measurement of temperature-rise curve f ff f 7.2.3 Calculation of thermal diffusivities of BC and TC 10 10 Thermal conductivities of BC and TC 11 Report 11 9.1 Items to be reported 11 9.2 Additional items to be selected for the report 12 Annex A (informative) Areal heat diffusion time method 13 Annex B (informative) Examples of theoretical temperature-rise curves 16 Bibliography 18 7.2 Thermal di 7.2 us ivity C alculatio n o 7.3 S p ecific heat cap acity 7.4 B ulk dens ity © ISO 2016 – All rights reserved thermal di us ivity o s ub s trate iii ISO 18555:2016(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 WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary in formation The committee responsible for this document is ISO/TC 107, Metallic and other inorganic coatings iv © ISO 2016 – All rights reserved ISO 18555:2 016(E) Introduction Thermal barrier coatings are highly advanced material systems They are generally applied to sur faces o f hot-section components made o f nickel or cobalt-based superalloys, such as combustors, blades, vanes of power-generation gas turbines in thermal power plants and aero-engines operated at elevated temperatures The function of these coatings is to protect metallic components for extended periods at elevated temperatures by employing thermally insulating materials which can sustain an appreciable temperature di fference between load bearing alloys and coating sur faces These coatings permit the high-temperature operation by shielding these components, thereby extending their lives Although thermal conductivity is one o f the most important properties o f thermal barrier coatings, the existing International Standard (ISO 18755:2005) includes only the method for determining the thermal di ffusivity o f monolithic ceramics, regarding the heat conduction in thermal barrier coating © ISO 2016 – All rights reserved v INTERNATIONAL STANDARD ISO 18555:2016(E) Metallic and other inorganic coatings — Determination of thermal conductivity of thermal barrier coatings Scope T h i s I nternationa l Sta ndard s p e ci fie s the me tho d for de term i n i ng the therma l conduc tivitie s o f therma l barrier coatings consisting of metallic bond coats and ceramic top coats, in a direction normal to the coating surface, at room temperature 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 ISO 1463, Metallic and oxide coatings — Measurement of coating thickness — Microscopical method ISO 18755:2005, Fine ceramics (advanced ceramics, advanced technical ceramics) — Determination of thermal diffusivity of monolithic ceramics by laser flash method EN 821-3, Advanced technical ceramics — Monolithic ceramics Determination of specific heat capacity Thermophysical properties — Part 3: ASTM E1269-11, Standard Test Method for Determining Specific Heat Capacity by Differential Scanning Calorimetry Terms and definitions Fo r the purpose fo l l o w i n g o f th i s s ta nd a rd , the te r m s a nd de fi n i tio n s g i ve n in ISO 18755:2005 and the ap p l y 3.1 thermal barrier coating TBC f heat transfer from outside of the top coat through the coating to the substrate two -l ayer co ati ng s i s ti ng o N o te to entr y: S e e a me ta l l ic b ond co at (B C ) a nd a ceram ic top co at ( TC ) , i n order to re duce Figure © ISO 2016 – All rights reserved ISO 18555:2016(E) Key top coat (TC) bond coat (BC) substrate thermal barrier coating (TBC) Figure — Diagrammatic view of a section of TBC [S O U RC E : I S O 141 8 : 01 , defi n ition , mo d i fie d] 3.2 apparent thermal diffusivity therma l d i ffu s ivity o f the s p e c i men s [s ub s trate with b ond co at (B C ) a nd s ub s trate with therma l b arrier co ati ng ( T B C ) ] i n a d i re c tion norma l to the co ati ng s u r face 3.3 normalized temperature rise T(t)/ΔT va lue wh ich i s de term i ne d b y d ivid i ng the d i fference b e twe en the temp eratu re o f the s p e c i men re ar surface after the pulse heating and the temperature of the specimen rear surface before the pulse ff f f temperature of the specimen rear surface before the pulse heating he ati ng by the d i erence b e twe en the ma xi mu m temp eratu re o the s p e ci men re ar s ur ace a nd the T(t ) = T (t ) − T ∆T T − T where max 1( ) T t i s temp eratu re o f s p e ci men re ar s ur face a fter pu l s e he ati ng b y a fla sh me tho d; t i s ti me; T max T i s temp erature o f the s p e c i men re ar s u r face b e fore pu l s e he ati ng; is maximum temperature of specimen rear surface N o te to entr y: S e e Figure © ISO 2016 – All rights reserved ISO 18555:2016(E) a) Flash method Key pulse heating specimen substrate TBC front surface rear surface infrared radiometer b) Temperature-rise curve under ideal conditions X time (s) Y normalized temperature rise T(t A areal heat diffusion time (s) ) / Δ T Figure — Flash method and temperature-rise curve under ideal conditions 3.4 temperature-rise curve curve which shows the variation in the normalized temperature rise of the specimen rear surface with time N o te to entr y: S e e the th ick s o l id l i ne i n Figure 2b 3.5 half rise-time 1/2 t time required for the normalized temperature rise to reach 0,5 in the temperature-rise curve N o te to entr y: S e e Figure 2b 3.6 areal heat diffusion time A are a with ti me - d i men s ion wh ich i s b ordere d b y the hori zonta l l i ne at the height o f the ma xi mu m temp eratu re -ri s e and b y the temp eratu re -ri s e c u r ve N o te to entr y: S e e Figure 2b 3.7 heat diffusion time τ0 time period beginning with pulse heating of the specimen front surface until time at which the specimen temperature becomes uniform τ0 =d α © ISO 2016 – All rights reserved ISO 18555:2016(E) where τ0 is heat di ffusion time (s); d is thickness o f specimen (m); is thermal di ffusivity (m /s) α Note to entry: See Figure 2b Principle Thermal conductivities of the substrate, BC, and TC are determined according to calculations using the thermal di ffusivities, specific heat capacities and bulk densities The fundamental procedures are shown in Figure The fundamental procedures for determining the thermal diffusivities of the substrate, BC, and TC consist o f the measurement o f temperature-rise curves o f three types o f specimens (substrate, substrate with BC, and substrate with TBC) by a flash method, and o f calculations The thermal di ffusivities o f the BC and TC are obtained by applying a multi-layer analytical model to the temperature-rise curves The specific heat capacities and bulk densities o f the substrate, BC, and TC are measured separately Figure — Fundamental procedures for determining thermal conductivity Apparatus for measuring thermal diffusivity An example o f the apparatus for measuring the thermal di ffusivity is schematically shown in Figure The apparatus consists of the following 5.1 Pulse heating light source 5.2 Data recorder 5.3 Measurement circuit 5.4 Infrared radiometer © ISO 2016 – All rights reserved ISO 18555:2016(E) for the TBC specimen g) T he d i fference b e twe en ma xi mu m and m i n i mu m th ickne s s sh a l l b e ≤ , 01 d h) T he co ati ng s u r face s hou ld b e p ol i she d me chan ic a l ly i n order to s mo o th the co ati ng s u r face j) For for the BC and TBC specimens i) The selections of the shape, the dimension and the thickness shall be decided according to the agreement between parties involved in the transaction me a s urement o f the s p e c i fic he at of B C c ap ac itie s substrates shall be used as the specimen a) Substrate specimen and TC , the b) BC specimen co ati ngs s tripp e d o ff the c) TBC specimen Key substrate bond coat top coat diameter S thickness of substrate BC thickness of bond coat TC thickness of top coat D d d d F i g u r e — S a) Substrate specimen h a p e o f f l a t d i s k s p e c i m e n s b) BC specimen c) TBC specimen Key substrate bond coat top coat side length S thickness of substrate BC thickness of bond coat TC thickness of top coat l d d d F i g u r e — S h a p e o f f l a t s q u a r e p l a t e s p e c i m e n s © ISO 2016 – All rights reserved ISO 18555:2016(E) T a b l e — T h i c k n e s s e s o Designation thickness of substrate thickness of BC thickness of TC thickness of TBC specimen S ymb ol dS dBC dTC d f s u b s t r a t e , B C , a n d T C Thickness (× 10 m) −3 1,00 ≤ dS ≤ , 0 0,15 dS dBC 0,20 (dS + dBC dTC d = dS + dBC + dTC ≤ ) ≤ ≤ ,00 6.2 Surface treatment B o th s u r face s o f s p e ci men s for me a s uri ng therma l d i ffu s ivity sh a l l b e co ate d with a th i n, op aque, pre ferably black layer accord i ng to I S O 75 : 0 Measuring procedure S p e c i m e n t h i c k n e s s The specimen thickness shall be measured as follows a) The specimen thickness shall be measured according to ISO 18755:2005 b) The thickness of BC and TC shall be measured on the image of the coating cross section according to ISO 1463 7.2 Thermal diffusivity 7.2.1 Measurement of temperature-rise curve According to ISO 18755:2005, the temperature-rise curve (Figure 2b) shall be measured as follows a) The specimen shall be placed in the specimen holder of the chamber The BC and TBC specimens f and TC surfaces shall be detected (See Figure 2a) b) The atmosphere is decided according to the agreement between parties involved in the transaction c) The temperature of the specimen rear surface before pulse heating, T0 , shall be measured with the thermocouple d) With variation of the specimen temperature minimized (0,2 K or less per minute) before pulse f f heating to measure the temperature rise s l l b e s o fi xe d that thei r s ub s trate s u r ace s a re he ate d b y pu l s e l ight a nd the temp eratu re s at B C he ati ng a nd with the output o i n rare d rad iome ter s tabi l i z e d; the s p e c i men i s s ubj e c t to pu l s e 7.2.2 Calculation of thermal diffusivity of substrate T he d i ffu s ivity o f s ub s trate s l l b e c a lc u late d b a s e d on the temp eratu re -ri s e c u r ve o f the s ub s trate specimen according to ISO 18755:2005 7.2.3 Calculation of thermal diffusivities of BC and TC The calculation of thermal diffusivities of BC and TC shall be made based on the temperature-rise curve and with appl ic ation o f a mu lti-layer ana lytic a l mo del T he a re a l he at d i ffu s ion ti me me tho d or the the ore tic a l temp erature -ri s e c u r ve me tho d s l l b e u s e d as the mu lti-layer ana lytic a l mo del T he mo del shall be chosen according to the agreement between parties involved in the transaction © ISO 2016 – All rights reserved ISO 18555:2016(E) 7.2.3.1 The areal heat diffusion time method (See Annex A) a) Calculation of the areal heat diffusion time across the specimen The areal heat diffusion time shall be calculated as follows Either o f the following 1) or 2) may be used 1) Calculation using numerical data of temperature-rise curve The areal heat diffusion time, A (hatched portion of Figure 2b) shall be determined directly from the data of temperature-rise curve 2) Calculation using apparent thermal di ffusivity Using the apparent thermal di ffusivity, α , the areal heat diffusion time, A shall be calculated according to Formula (1) A = d 6α app (1) app ( ) where is thickness o f specimen (m); d /s) The apparent thermal di ffusivity o f the specimen, αapp , shall be calculated according to equiareal method or half-time method (see ISO 18755:2005) 3) The correction factors for heat loss and non-uniform heating should be calculated in reference to αapp is apparent thermal di ffusivity o f specimen (m Annexes B and D respectively in ISO 18755:2005 b) Calculation of thermal diffusivities of substrate, BC, and TC 1) The thermal di ffusivity o f BC shall be calculated according to Formulae (2) and (3) on the basis o f the measurement made using the BC specimen, and the thermal di ffusivity o f substrate obtained in 7.2.2 The thermal di ffusivity o f BC can be expressed as follows: =d α BC BC (2) / τ BC where αBC is thermal di ffusivity o f BC (m /s); dBC is thickness o f BC (m); is heat diffusion time of BC (s) (c ρ d + c ρ d ) A − ( c ρ d + 3c = 3c ρ d + c ρ d τBC τ BC where S S BC BC BC BC-S S S S S BC S S BC BC ) ρ BC dBC ( dS / α S ) (3) BC ρs is specific heat capacity o f substrate [J/(kg·K)]; is bulk density o f substrate (kg/m ); αs is thermal di ffusivity o f substrate (m /s); cs S © ISO 2016 – All rights reserved ISO 18555:2016(E) ds i s th ickne s s o f s ub s trate (m) ; cBC i s s p e ci fic he at c ap ac ity o f B C [ J/( kg· K) ] ; ρ BC i s bu l k den s ity o f B C ( kg/m A BC-S ); is areal heat diffusion time of BC specimen (s) ) T he therma l di ffus ivity o f TC sha l l b e calcu lated accordi ng to Formulae (4) and (5 ) on the b as i s o f the measurement made using the TBC specimen, αs obtained in 7.2.2 and αBC obtained in Formula (2) T he therma l d i ffu s ivity o f TC ca n b e e xpre s s e d a s α TC = d TC fol lows: (4) τ TC where αTC i s therma l d i ffu s ivity o f TC (m dTC i s th ickne s s o f TC (m) ; τTC /s) ; is heat diffusion time of TC (s) ( c S ρ SdS + c BC ρ BCdBC + c TC ρ TCd TC ) A TBC-S − 3c ρ d + c ρ d + 3c ρ d + 6c S ρ SdSc TC ρ TCd TC dBC τ TC = BC BC BC TC TC TC α BC c B C ρ BC d BC 3c S ρ S d S + c BC ρ BC d BC + c TC ρ TC d TC S S S dS c ρ d 3c ρ d − ( S S S + BC BC BC + 3c TC ρ TC d TC ) α S (5) where cTC i s s p e c i fic he at c ap acity o f TC [ J/( kg· K ) ] ; ρ TC i s bu l k den s ity o f TC ( kg/m ATBC-S 7.2.3.2 ); is areal heat diffusion time of TBC specimen (s) Calculation using theoretical temperature-rise curve method T he therma l di ffu s ivitie s o f B C a nd TC s l l b e c a lc u l ate d b y applyi ng the the ore tic a l temp eratu re -ri s e c u r ve b a s e d on a mu lti-l ayer mo del ana lys i s to the me a s u re d temp eratu re -ri s e c ur ve The theoretical temperature-rise curve shall be chosen according to the agreement between parties involved in the transaction Examples of the theoretical temperature-rise curve are given in Annex B The measured temperature-rise curve should be corrected for heat loss and non-uniform heating in reference to ISO 18755:2005 (Annexes B and D re s p e c tively) a) C a lc u l ation o f therma l d i ffu s ivity o f B C 1) The theoretical temperature-rise curve of the BC specimen is given in Formula (6) © ISO 2016 – All rights reserved ISO 85 5 : 01 6(E) (6) T t ∆T) th = F(dS , d BC , α S , α BC , c S , c BC , ρ S , ρ BC , t) ( ( )/ where (T(t ) /Δ T ) th i s the ore tic a l temp eratu re -ri s e c u r ve; is a function F ) I nput the d ata e xcep t 3) T he therma l for the therma l d i ffu s ivity o f B C i nto th i s e quation d i ffu s ivity o f B C s l l be temperature-rise curve of the BC specimen b) c a lc u late d b y fitti ng th i s e quation to the me as u re d C a lc u lation o f therma l d i ffu s ivity o f TC 1) The theoretical temperature-rise curve of the TBC specimen is given in Formula (7) Tt (7) T = G(d , d , d , α , α , α , c , c , c , ρ , ρ , ρ , t ) ( ( ) / ∆ ) th S BC TC S BC TC S BC TC S BC TC where G is a function ) I nput the data excep t 3) T he therma l for the therma l d i ffu s ivity o f TC i nto th i s e quation d i ffu s ivity o f TC s l l be c a lc u l ate d temperature-rise curve of the TBC specimen b y fitti ng th i s e quation to the me as u re d 7.3 Specific heat capacity T he s p e ci fic he at c ap ac ity s l l b e de term i ne d a s me as u rement of a) T he b) T he l i teratu re va lue s ASTM E1269 11 the fo r s p e ci fic he at fol lows c ap ac ity sh a l l be accord i ng to EN 82 1-3 : 0 or the co ati ng p ro duce d b y the s a me co ati n g p ro ce s s wi th the p o wder o f the same composition should be used, if the preparation of the coating stripped off the substrate i s d i ffic u lt NOTE The value measured for the raw material powders of the respective coating can be used, when the literature values are not available 7.4 Bulk density The substrate, BC, and TBC specimens shall be used as follows for determining the bulk densities of BC and TC a) Measure the mass of the substrate, BC and TBC specimens b) M e a s u re the d i men s ion s o f the s ub s trate, B C and T B C s p e c i men s b y u s i ng the m ic rome ter and c) T he d) T he bu l k den s ity o f B C sha l l b e de term i ne d accord i ng to Formu la (8) calculate their volumes bu l k den s ity o f s ub s trate substrate specimen ρ BC = (ρ ( where ρBC-S e) 10 BC-S d S + dBC ) − ρ S dS ) s l l d be de term i ne d u s i ng the ma s s d i men s ion s o f the (8) BC i s bu l k den s ity o f the B C s p e ci men ( kg/m and the ) T he bu l k den s ity o f TC sha l l b e de term i ne d accord i ng to Formu la (9) © ISO 2016 – All rights reserved ISO 18555:2016(E) ρ TC = ( ρ (d + d where ρTBC-S TBC-S s BC + d TC ) − ρ s d s − ρ BC dBC )d i s bu l k den s ity o f the T B C s p e ci men ( kg/m (9) TC ) Thermal conductivities of BC and TC Thermal conductivities of BC, λBC, , and TC, λTC, (11): , s l l b e de term i ne d re s p e c tively b y Formu lae (10) a nd λ BC = α BC c BC ρ BC (10) λTC = α TCcTC ρ TC (11) Report 9.1 Items to be reported T he rep or t sh a l l contai n the described in the report a) Specimen: fol lowi ng item s When the rep or te d va lue s are c ite d , they s hou ld b e 1) materi a l o f s ub s trate; 2) materi a l s and s prayi ng cond ition s o f B C and TC , i nclud i ng s u r face prep a ration o f the s ub s trate; 3) s hap e o f the s p e ci men (d i s k or s quare plate) ; 4) d i ame ter or s ide leng th o f the s p e ci men; 5) thickness of substrate, BC and TC b) Measurement conditions: 1) typ e o f the app a ratu s 2) atmo s phere for for me as u ri ng the therma l d i ffu s i vity (mo del o f the i n s tru ment) ; me a s u ri ng the temp eratu re -ri s e c u r ve; 3) surface treatment of the specimen for measuring the temperature-rise curve (coating material, co ati ng pro ce du re) ; 4) temperature of specimen rear surface before pulse heating, T0 5) c a lc u lation me tho d o f the therma l d i ffu s ivity; 6) me tho d for ; me a s u ri ng the s p e c i fic he at c ap acity c) Results of measurement and calculation: 1) bu l k den s itie s o f s ub s trate, B C and TC; 2) s p e ci fic he at c ap ac itie s o f s ub s trate, B C a nd TC; 3) therma l d i ffu s i vitie s o f s ub s trate, B C and TC; 4) thermal conductivities of substrate, BC and TC © ISO 2016 – All rights reserved 11 ISO 18555:2016(E) 9.2 T he Additional items to be selected for the report rep or t may contai n add itiona l item s T he add itiona l item s a re according to the agreement between parties involved in the transaction a) ye ar/month/day o f me as u rement and the me as u rement l ab orator y; b) manu fac turer o f the me a s uri ng s ys tem; c) de tai l s o f therma l d i ffu s ivity me a s urement; 1) correction factors used in 7.2.3.1a) and 7.2.3.2 s ele c te d from the fol lowi ng , ; 2) typ e o f thermo couple and p o s itiona l relation sh ip b e twe en the thermo couple and s p e ci men; 3) typ e o f the i n fra re d rad iome ter; 4) pu l s e he ati ng l ight s ou rce (typ e, i nten s ity, pu l s e width, pu l s e b e am centre o f gravity a nd the me tho d to de term i ne the centre o f gravity) ; 5) nu mb er o f the me as u rement; 6) data re gard i ng the temp eratu re -ri s e c ur ve (the repre s entative data i n c as e o f ne ce s s ity) ; 7) half-rise time or areal heat diffusion time 12 © ISO 2016 – All rights reserved ISO 18555:2016(E) Annex A (informative) Areal heat diffusion time method A.1 General A multi-layer model is shown in Figure A.1 According to the multi-layer model, based on the response function method,[2] the areal heat diffusion time A [s] across the multi-layer specimen is expressed as: A= ∫ ∞ − b τ ⋅ T (t ) dt = r where lim ξ →0 − b τ ⋅ T? (ξ ) ξ r b is thermal e ffusivity across the multi-layer specimen; τ is heat di ffusion time across the multi-layer specimen; Tr (t) = T(t) / ∆Τ is normalized temperature rise (see 3.3 ); T?r ( ζ )= ∞ ∫ () Tr t exp (−ζ t) dt ξ (A.1) is Laplace transform of Tr(t); is a Laplace parameter The heat diffusion across the n -layered specimen is systematically analysed using Formula (A.1) © ISO 2016 – All rights reserved 13 ISO 18555:2016(E) Key pulse heating infrared radiometer di thickness of the i th layer (i = 1,2,…, n ) αi ci ρi bi τi thermal di ffusivity o f the i th layer (i = 1,2,…, n ) specific heat capacity o f the i th layer (i = 1,2,…, n ) bulk density o f the i th layer (i = 1,2,…, n ) thermal e ffusivity o f the i th layer (i = 1,2,…, n ) heat diffusion time of the i th layer (i = 1,2,…, n ) Figure A.1 — Multi-layer model A.2 Single-layer model The areal heat di ffusion time across a single-layer specimen A1-S is expressed as: A1 -S =τ (A.2) /6 From Formula (A.2), the heat di ffusion time o f the first layer is determined by Formula (A.3): τ1 (A.3) = A1 -S A.3 Two-layer model The areal heat di ffusion time across a two layer specimen A 2-S is expressed as: A2 -S = b1τ 13/ + b2τ 1τ 21 /2 + b1τ 11/2τ + b2τ 23/2 b1 τ + b τ ( (A.4) ) From Formula (A.4), the heat di ffusion time o f the 2nd layer is determined by Formula (A.5): τ2 = ( A2 -S c1 ρ d1 + c ρ d ) − ( c ρ d + 3c ρ d ) τ 3c ρ d + c ρ d 14 2 1 1 2 (A.5) 2 © ISO 2016 – All rights reserved ISO 18555:2016(E) A.4 Three-layer model The areal heat di ffusion time across a three layer specimen A 3-S is expressed as: b τ (τ + 3τ + 3τ ) + b τ = A -S 1 2 (A.6) + τ + 3τ ) + b τ (3τ + 3τ + τ ) + b b τ τ τ b b τ +b τ +b τ (τ ( 1 3 3 ) 2 2 From Formula (A.6), the heat di ffusion time o f the 3rd layer is determined by Formula (A.7): τ = + 3c ρ d + c ρ d 3c1 ρ d1 A − S ( c ρ d + c ρ d + c ρ d ) − ( c ρ d + 3c ρ d + 3c ρ d ) τ − 3c ρ d + c ρ d + 3c ρ d + 6c ρ d c ρ d τ cρd 2 3 1 2 3 1 1 2 3 1 3 2 3 3 (A.7) 1 2 A.5 Thermal diffusivity The thermal di ffusivity o f each layer α is determined by Formula (A.8): i αi = d i2 / τ i , ( i = 1, 2, 3) (A.8) A.6 Convenient calculation method of areal heat diffusion time [3] [4] By considering the temperature-rise curve o f a two or a three-layer specimen as that o f a single-layer specimen, the areal heat diffusion time is assumed as follows: A = ( d1 +d 6α -S ) (A.9) app where αapp is apparent thermal di ffusivity of a two-layer specimen A3 -S = ( d1 + d + d3 ) α app (A.10) where αapp is apparent thermal di ffusivity of a three-layer specimen This is a simple technique and has advantages practically, although it is an approximation © ISO 2016 – All rights reserved 15 ISO 18555:2016(E) Annex B (informative) Examples of theoretical temperature-rise curves B.1 General Repre s entative e xample s o f the the ore tic a l temp eratu re -ri s e c u r ve s b as e d on a mu lti-layer mo del (s e e Figure A.1 The theoretical temperature-rise curve is chosen according to the agreement between parties involved in the transaction ) ana lys i s a re given b elow [5 ] B.2 Two-layer model γ ⋅ t (ω1 X1 + ω X2 ) exp − dk α T (t ) ∞ = + ∆T k=1 ω X1 cos (ω 1γ k ) + ω X cos (ω 2γ k ) th where ∑ X1 = ρρ cc1 dd1 ⋅ d22α + , ω = X2 = 2 d1 α ρ c d1 ρ 2c 2d2 d22α d12α ⋅ −1 , ω = d12α d22α +1 d12α d22α −1 (B.1) γk i s defi ne d b y the k-th positive root of following equation: ( ) + X (ω γ ) = X1 sin ω1γ sin B.3 Three-layer model γ ⋅ t (ω1 X1 + ω X2 + ω3 X3 + ω X4 ) exp − dk α T (t ) ∞ = + (B.2) ∆T ω X1 cos (ω 1γ k ) + ω X2 cos (ω 2γ k ) + ω X3 cos (ω 3γ k ) + ω X 4cos (ω 4γ k ) k = th where ∑ 2 X = ρ c d d22α + ρ c d d32α + … ω ρ c d d1α ρ c d d 2α 1 2 2 2 3 16 = d12α d22α + +1 d32α d32α © ISO 2016 – All rights reserved ISO 85 5 : 01 6(E) ρ c d d 22α ρ c d d32α X = + − … ω ρ c d d1α ρ c d d 2α ρ c d d 22α ρ c d d32α X = −1 − … ω ρ c d d1α ρ c d d 2α ρ c d d 22α ρ c d d 32α X = −1 + … ω ρ c d d1α ρ c d d 2α 1 2 2 2 3 1 2 2 2 3 1 2 2 2 3 3 4 γk i s defi ne d b y the k-th positive root of the following: = d12α d22α + −1 d32α d32α = d12α d22α + +1 d32α d3 α = d12α d22α + −1 d32α d3 α X1 sin ( ω1γ ) + X2 sin ( ω2γ ) + X3 sin ( ω3γ ) + X4 sin ( ω 4γ ) = © ISO 2016 – All rights reserved 17 ISO 18555:2016(E) Bibliography [1] ISO 14188:2012, Metallic and other inorganic coatings — Test methods for measuring thermal cycle [2] B aba T Analysis o f One-dimensional Heat Di ffusion a fter Light Pulse Heating by the Response Function Method Jpn J Appl Phys 2009, 4805EB04 [3] A koshim a M., Tanaka T., E ndo S., B aba T., H arada Y., Kojim a Y et al Thermal Di ffusivity Measurement for Thermal Spray Coating Attached to Substrate Using Laser Flash Method Jpn J Appl Phys 2011, 5011RE01 [4] resistance and thermal shock resistance for thermal barrier coatings Takah ashi S., A koshim a M., Tanaka T., E ndo S., O gawa M., Kojim a Y et al Determination of Thermal Conductivity o f Thermal Barrier Coatings, Proc The 5th Asian Thermal Spray Conference, 2012, pp 11-12 [5] A raki , N., M akino , A., I shiguro , T., M ih ara , J An Analytical Solution o f Temperature Response in Multilayered Materials, for Transient Methods, International Journal of Thermophysics, 13-3 (1992) pp 515-538 18 © ISO 2016 – All rights reserved ISO 85 5 : 01 6(E) ICS 25.220.20 Price based on 18 pages © ISO 2016 – All rights reserved