Simplified Explicit Model to Measure Transient Heat Transfer in Foamcrete Panel System

Transcription

1 Australian Journal of Basic and Applied Sciences, 77: -, ISSN Siplified Eplicit Model to Measure ransient Heat ransfer in Foacrete Panel Syste Md Azree Othuan Mydin School of Housing, Building and Planning, Universiti Sains Malaysia, 8, Penang, Malaysia Abstract: he copleity of calculating transient heat transfer in foacrete panel is cople due to its porous and ulti-phase aterial nature which can require the nuerical solution of two or threediensional transient conduction equations. his paper reports the basis of one-diensional Finite Difference ethod to attain theral properties of foacrete. In addition, this paper also integrates the ipleentation of the ethod and the justification of theral properties odel of foacrete. A onediensional finite difference heat conduction prograe has been developed to predict the teperature growth through the thicness of the foacrete panel, based on preliinary estiation of the theral conductivity-teperature relationship as a function of porosity and radiation within the voids. he precision of the odel was then assessed by coparing predicted and eperiental teperature profiles acquired fro sall scale heat transfer test on foacrete panel, so that the teperature history of the specien calculated by the prograe closely atches those recorded during the eperient. Key words: lightweight concrete, foacrete, foaed concrete, theral analysis, theral properties, heat transfer, fire INODUCION With rapid developent in coputer and software engineering and technology, nuerous designers and building product developers loo towards coputer aided odelling as an econoic eans to suppleent fire resistance testing as a ethod of perforance verification. According to heat flow anaylsis studies and structural verification analysis, and on the nowledge of the behaviour of aterials at high teperatures and fire condition, the coputer-generated approach ay offer the engineer and designer with a solution of substantial practical value. he epansion of coputational techniques in fire safety engineering has forerly been rather ignored, but the general level of consciousness in the area of nuerical odelling is ounting. ising deand has eerged for enhanced predictive ethods based on refined analysis to establish the fire resistance of the building coponents. At this tie, non-linear structural analysis techniques are progressively allowing designers to envisage the structural perforance under a given set of tie-varying teperature. his effort has enabled the ost responsive ebers to be identified. Nevertheless, if the design is not to be over conservative, it is essential to be able to predict the teperature distribution in all structural ebers with a realistic precision. he ajor part of the current inforation on the perforance of passive fire protection aterials and systes has been derived fro standard fire tests. Faced with this situation, there has been ongoing research with the ai of analyzing the behaviour of eleents of building aterial in fire by studying their fire resistance in the standard furnace tests and developing sufficient techniques for interpolation and etrapolation. As understanding increases, a natural progression is the epansion of analytical procedures for the best possible design of building aterial eleents to offer a specified fire resistance. At present, researches on building aterial heat transfer and fire analysis have becoe progressively ore significant. According to the current design codes, the fire resistance of building aterials are traditionally deterined either by assessing the capability of load-bearing capacity or by evaluating fire resistance rating ties through epirical relations. Practical and precise teperature forulations for aterial and structural ebers subjected to fire conditions are of paraount significance for researches in structural fire engineering. At this tie, the proble of a porous ediu lie foacrete eposed to elevated teperature is of great attention in civil engineering. Foacrete can be defined as a ceentitious aterial having a iniu of per cent by volue of echanically entrained foa in the ortar slurry in which air-pores are entrapped in the atri by eans of a suitable foaing agent. he air-pores are initiated by agitating air with a foaing agent diluted with water; the foa then carefully ies together with the ceent slurry to for foacrete. Integrating the air-pores into the base atri gives a low self-weight, high worability, ecellent insulating values, but lower strength in contrast to noral strength concrete. During disclosure to elevated teperatures, any nonlinear phenoenons concerning the different phases constituting the porous edia in foacrete should be taen into account It should be pointed out that not only heat conduction and vapour diffusion ust be considered, but also liquid water flow due to pressure gradients and capillary effects caused by interface curvature inside pores Corresponding Author: Md Azree Othuan Mydin, School of Housing, Building and Planning, Universiti Sains Malaysia, 8, Penang, Malaysia

2 Aust. J. Basic & Appl. Sci., 77: -, produced by surface tension that alters the equilibriu between liquid water and gas. In order to consider fire resistance of any aterial, it is crucial to distinguish teperature history of the aterial in fire conditions. Although siilar principles of heat transfer govern the teperature developent throughout aterial, different analytical or nuerical ethods ay be realistic to attain the solution. In sequence to get theral properties of foacrete, a one-diensional heat transfer odel is adequate and Finite Difference Method FDM eerges to be the siplest ethod to be eecuted. his paper focuses on the developent of one-diensional Finite Difference odel which can be eployed to resolve transient heat conduction probles in ulti-layer panels. he convection and radiation boundary conditions on both fire eposed side and uneposed side of the aterial are also considered. In addition, theral property odels and their validation using the eperiental results will also be presented. he net section will depict the basis of the odelling ethod which includes the developent of one-diensional Finite Difference forulation which can be eployed to solve transient heat conduction probles for porous aterial lie foacrete. Epirical and heoretical Model: he atheatical odel used in the analysis considers foacrete as a ultiphase porous aterial. he transient heat transfer through foacrete is odelled using one-diensional Finite Difference forulation. A coputer progra to odel the transient heat transfer through foacrete has been developed and ipleented in the failiar environent of Microsoft Ecel using VBA based on one-diensional Finite Difference forulations. he odelling procedure has been systeatically validated by coparisons with a nuber of analytical solutions and siulation results using ABAQUS/Standard. he general three-diensional transient heat-conduction equation based on Fourier s law of conduction in Cartesian coordinates is Holan, : c t y y z z where,y,z,t is teperature C; is teperature dependent theral conductivity W/K; is aterial density g/ ; c is specific heat of aterial J/g C; t is tie sec;, y, z are Cartesian coordinates. he right hand side of Eq. stand for the net heat conduction in a solid aterial, whereas the left hand side represents the accuulated internal energy. Calculation of heat flow in solids is based on the solution of this differential equation. Due to copleity of any geoetric shapes and boundary conditions of practical interest, analytical solutions to Eq. are not always possible. Finite Difference Method is a relatively siple technique which can provide approiate nuerical solutions to any practical cases. FDM replaces derivative epressions in Eq. with approiately equivalent partial difference quotients. wo approaches can be taen to transfor the partial differential equation into finite difference equation; atheatical replaceent approach and physical heat balance approach. he two techniques often result in the sae finite difference equations. However, the heat balance technique ight suit better to irregular boundaries with convective heat loss Croft and Lilley, 977 and therefore is adopted in this study. he consequential finite difference equations can be either eplicit or iplicit. In the iplicit approach a set of siultaneous equations has to be solved at each step in tie. Although the solution process is relatively cople, it allows for the use of larger tie intervals. In the eplicit approach, on the other hand, the teperature of a volue cell at a tie step is coputed directly based on the teperatures of the adjacent cells in the last tie step, leading to a very siple schee of coputation Croft and Lilley, 977. Given the siplicity and effectiveness the eplicit approach in finite difference ethod provides, it was found suitable for the purpose of this study. Although tie intervals required in this ethod are sall, it does not cause any further coplications to the coputations, since sall tie steps are already necessary in fire resistance analysis due to rapid changes of teperature in fire conditions. If the thicness of the foacrete panel is sall in coparison to the other diensions, the proble will reduce to a one diensional heat transfer analysis, i.e. the heat flow is perpendicular to the face ecept near the edges. Hence, the governing Eq. with no heat generation reduces to:

3 Aust. J. Basic & Appl. Sci., 77: -, t t t c,, where L, for t >, L is the thicness of the panel Fig. : Finite Difference discretization for node within the aterial Assuing a hoogeneous aterial and choosing the eplicit ethod, the teperature of a volue cell Fig. and Fig. at a tie step is coputed directly based on the teperatures of the adjacent cells in the last tie step which leads to a very straightforward schee of coputation Wang, 995: i For a typical node within the aterial Fig. : ] [,,,, F F where F is defined as:,, c t F 4 is the teperature of in the subsequent tie step and j i, is the theral conductivity at the average teperature of cells i and j:, j i j i 5 Nuerical stability under the eplicit schee requires:,, c t 6 Fig. : Finite Difference discretization for boundary node ii For a boundary node, when subjected to convective and radiative boundary conditions Fig. :

4 Aust. J. Basic & Appl. Sci., 77: -, ' h F [ where F is F F t c h e[ t ] c 7 h is convection heat transfer coefficient W/ K; is the abient teperature C; is a geoetric view factor e is the effective eissivity is Stefan-Boltzann constant W/.K 4. Nuerical stability liits the tie step to: t c 4.5 h E [ 7 ] 8 It should be pointed out that foacrete panel is recognized to have a unifor initial teperature equal to the abient teperature. On the uneposed boundary, the convective heat transfer coefficient h is assued to be constant and the value is taen as W/ K Othuan Mydin and Wang,. he eissivity of the surface depends on the aterial. For concrete, the surface eissivity will be taen as.9 Ozisi, 985. he eposed boundary conditions are the recorded teperatures on the eposed surface of fire test speciens. heral Properties Of Foacrete At Elevated eperatures: Foacrete is a coposite aterial ade fro a cobination of sand, ceent binder and water. After iing, the ceent hydrates and hardens into a stone lie aterial. In theory, the cobination of ass and heat transfer should be carried out to obtain teperatures in foacrete construction. However, odelling ass transfer is cople. A coon approiation is to conduct heat transfer only, but odifying the aterial theral properties to reflect the effects of water oveent. In order to utilize the proposed nuerical ethod for heat transfer analysis, data on density, specific heat and theral conductivity should be provided. his section will only present suary of these data which were based on coprehensive eperiental investigation on theral properties of foacrete density of 65g/ conducted by the author Othuan Mydin and Wang,. Foacrete Dry Density: Moisture flow in foacrete can be epressed in two different ways; either in ters of the evaporable or free water. Evaporation of free and of soe of the cheically bonded water will cause dehydration in foacrete, which will influence all the aboveentioned three ites of theral properties. he dehydration process starts as early as 9 C Othuan Mydin and Wang,. Fig. shows recorded density of foacrete at different teperatures, as ratio of the original density for initial density values of 65 g/. 5 Original Mass % eperature C Fig. : Density of foacrete as a function of teperature % of the original density 4

5 Aust. J. Basic & Appl. Sci., 77: -, Foacrete Specific Heat: In ters of specific heat of foacrete, it can be divided into two parts; the base value corresponding to a iture of the dry coponents and the effect of water evaporation. he teperature-dependent specific heat of foacrete eperience one pea corresponding to the dehydration reaction of foacrete between 9 C to 7 C as shown in Fig. 4. his pea represents the energy consued to dissociate and evaporate water and include the effect of water oveent and recondensation of water in cooler regions of foacrete. 5 Specific heat J/g. C 5 5 Fig. 4: Specific heat of foacrete as a function of teperature eperature C he calculated base value of specific heat at abient teperature for 65 g/ density was J/g C Othuan Mydin and Wang, and the additional specific heat at the dehydration reaction can be epressed by: 6.6 c ew f J / gc 9 in which the value of.6 6 J/g is the latent heat of evaporation of water, Δc is the average additional specific heat, e is dehydration water content percentage by total weight, Δ is the agnitude of the teperature interval during which water is evaporated and f is a odification factor accounting for water oveent. A value of f =.4 was used for standard fire conditions. Fig. 4 shows the teperature-dependent specific heat of 65 g/ density. Foacrete heral Conductivity: he theral conductivity is the ey paraeter for evaluating the perforance of foacrete based syste. ather than using the siplest parallel and serial odels, a odel eploying a theory closer to reality has been introduced into this study. Given that foacrete is a porous aterial, heat transfer through this aterial is a cobination of all three odes: conduction through the solid and convection and radiation through the pores. herefore the effective theral conductivity of foacrete should include these effects. If the acroscopic scale is uch larger than the randoly distributed individual pores and the pores are of the sae size and uniforly distributed, it is possible to develop an analytical ethod to calculate the overall theral conductivity of the coating. his effective theral conductivity can be affected by any factors such as teperature, density, oisture content and porosity of the aterial. Assuing foacrete is ade of solid substrate and uniforly distributed spherical pores, the effective theral conductivity of foacrete ay be calculated using the following equation Yuan, 9: s g g s s where * is the effective theral conductivity of foacrete, g is effective theral conductivity of gas to account for heat transfer in the pores, s is the theral conductivity of the solid and ε is the porosity of the aterial the ratio of the volue of void pore to the overall volue. In this study, the theral conductivity of 5

6 Aust. J. Basic & Appl. Sci., 77: -, solid dried foacrete s is.5 W/. C and the porosity of 65 g/ density is 75% Othuan Mydin and Wang,. Foacrete heral Conductivity Of he Gas Inclusion: he heat transfer through the pores of a porous aterial should norally include all echaniss of heat transfer: conduction, radiation and convection. However, since the pores in an intuescent coating norally range fro several hundred icrons to less than 5, natural convection within the pores is sall and can be neglected. In studies of porous aterials, the radiative coponent of heat transfer has often been neglected because these studies deal with heat transfer at abient teperature or low teperatures. However, for fire applications where the teperatures are high, accurate calculation of heat transfer within the pores should also incorporate radiative heat transfer. hus for this study, the equivalent theral conductivity of the gas will consist of contributions fro pure conduction conductance through gas and radiation. Foacrete Conductance hrough Gas Phase: he available eperiental data shows that the conductance related theral conductivity of gas typically has strong teperature and pressure dependencies Sith, 98. Under atospheric pressure, the gas theral conductivity as a function of teperature: cond g.8 where λ g =.46W/K, is theral conductivity of gas at teperature =7K C. A siilar forulation has been given ore recently by Di Blasi. he theral conductivity of gas, regardless its nature, is epressed as: 4.77 cond 4.85 W / K Figure 5 copares these two odels of calculating conductance related theral conductivity of gas. he results show that the two odels produce very close values over a wide range of teperatures fro 7K to 7K. In this study, the Di Blasi s odel has been selected for calculation Di Blasi,. Fig. 5: wo different odels of relationship between conductance related gas theral conductivity and teperature adiation hrough Gas: As teperature increases, theral radiation becoes iportant and cannot be neglected in porous aterials. he effect of theral radiation across pores was discussed by Loeb 954, who has been accredited for the following equation to calculate the contribution of theral radiation to the overall theral conductivity of a pore: rad 4Gde where G value is the average width of the pore divided by the aiu width of the pore and both quantities should be deterined in the direction of the theral gradient. herefore, G is for lainar pores and 6

7 Aust. J. Basic & Appl. Sci., 77: -, cylindrical pores with aes parallel to the heat flow direction, G is π/4 for cylindrical pores with aes perpendicular to the heat flow direction. For spherical pores, G = /. Due to the difficulty in accessing the original article by Loeb 954, the following will present a derivation of Eq.. efer to Figure 6 which shows a spherical pore. Fig. 6: Illustration of a general spherical pore for calculation of radiation For a surface eleent on the solid-pore boundary at a, the net value of noral coponent of radiation Q a is equal to the difference between the heat flu leaving and the energy coing to the surface: r 4 4 Qa ea e ' ' df ' A a dada 4 where a is any other eitting eleent on the surface of this spherical pore; A and A represent allocated area for a and a, respectively; geoetric consideration as: ' da df ' dada 4 df ' da da is view factor, and for a spherical enclosure it is obtained fro siple 5 hus, in this case, the second ter on the right hand side of Eq. 4 becoes constant: r 4 4 ' Qa e a ' ' da a 4 A Fro siple consideration, the ean teperature is: r a ' ' a 4 A ' da herefore the noral coponent of radiation becoes: Q e 4 4 a Assuing a, then: Q 4e r a a where 9 7

8 Aust. J. Basic & Appl. Sci., 77: -, At the top point of this spherical pore, it has: Q 4 e Fig. 7: Calculation of radiation along heat flow direction As deonstrated in Figure 7, heat flow is along the z ais, therefore only the z direction coponent of heat flu fro every point a is considered to ae contribution to radiation through the whole pore. Because theral gradient is linear along z direction, then: a Z Z Q a Where Z is coordinate of point a along z ais central point of sphere is Q Q a, z Q a, z he z coponent of radiation at point a is: Z Q a By siple calculation, the z direction coponent of heat flu fro point a is described as: Z Q Fig. 8: Calculation of total aount of radiation for a heisphere efer to Figure 8, a ring strip is on the surface of heisphere, with an infinitesial height dz. For a spherical surface, it can be proved that every ring strip that has the sae height provides the sae surface area. herefore, for every single infinitesial ring in Figure 8, the surface area can be epressed as: 8

9 Aust. J. Basic & Appl. Sci., 77: -, Sring dz 4 Heat responsible for z direction radiation is therefore calculated, for this ring surface: Z H z Q dz 5 he total aount of z direction radiation of this heisphere is obtained by integration: Z H half Q dz Q Z dz Q 6 In a siple way, the lower half of sphere provides the sae effect as the upper half, so the total aount of heat transferred by radiation is: 4 Hwhole Hhalf Q 7 he equivalent heat flu along the z direction over the projected area is: H whole 4 4 Qeq Q Qeq 4 e 8 he theral gradient in this case is d, where d is the diaeter of sphere. herefore, the effective theral conductivity for radiation, within a sphere, is described as: rad 4 de 9 Foacrete Effective heral Conductivity: Since the size of the pores is very sall never larger than 5, natural convection in the pores can be neglected Yuan, 9. Cobining the contributions to heat transfer in a pore by pure conduction and radiation, the total theral conductivity of the gas within the pore can be given as: 4.77 g d e where is absolute teperature and d e is the effective diaeter of the pores. In this study d e =.7 for 65 g/ density Othuan Mydin and Wang,. he first ter is the gas theral conductivity without the effect of theral radiation and the second ter represents the effect of radiation within the air pores. Hence, the effective theral conductivity-teperature relationship of 65 g/ density foacrete consists of three parts as deonstrated in Fig. 9: i Constant theral conductivity up to 9 C before water evaporation, equal to that at abient teperature reported by the anufacturer; ii Linear reduction of conductivity to.4 W/. C at 7 C; iii Non-linear increase in theral conductivity based on Eqs...5 heral Conductivity W/ C eperature C Fig. 9: Effective theral conductivity of foacrete as a function of teperature 9

10 Aust. J. Basic & Appl. Sci., 77: -, Sall Scale est on Foacrete Panel: Fig. : Preparation of 65 g/ density foacrete prototype panel with diensions of 4 45 in plan and 5 in thicness for sall scale fore test. Sall-scale eperients have been perfored on 65 g/ density foacrete slab. All foacrete slab speciens had diensions of 4 45 in plan and 5 in thicness Figure. Each specien was placed horizontally on top of an electric iln as the source of heat, so that one side of the panel was subjected to iln teperature and the other side faced up to the roo teperature 9-5 C. he heating chaber has an internal diaeter of 648 and 54 height. here was a 8 65 opening on the top lid of the iln, which allowed eposure of the lower side of the panel to elevated teperatures. A thic layer of glass wool with the sae opening size was laid under the specien to insulate the contact surface of the top lid. he iln teperature was increased to about C. ype K therocouples were placed throughout the thicness of the foacrete specien at the centre of the slab to investigate teperature developents through each foacrete panel. Five therocouples were installed: on the eposed side, on the uneposed side and at quarter, half and three-quarter thicness, being 7.5, 75 and.5 fro the heated surface. One therocouple was place inside the iln, at an approiate distance of 5 fro the eposed surface of the panel, to record the iln teperature. Fig. 6 shows typical setup of the eperients. Figures visualizes the condition of the tested prototype foacrete panels after eposure at elevated teperatures on top of an electric iln as the source of heat Fig. : Condition of prototype foacrete panels after long hour s eposure at elevated teperatures on top of an electric iln Validation Of heral Property Models: It is acceptable to assue that heat transfer in the test saples is one-diensional in the thicness direction of the foacrete slab Othuan Mydin and Wang,. All the easured eperiental teperatures at all recording locations of the test speciens were copared with nuerical analysis results. heral property values theoretical theral property odel results are considered and their prediction results copared. As

11 Aust. J. Basic & Appl. Sci., 77: -, entioned previously, the eposed surface teperatures are used as input data in the heat transfer analysis to eliinate uncertainty in the theral boundary condition on the eposed side. Figures -5 copare the easured and nuerical analysis results for the 65 g/ density speciens. he results shown in Figures - 5 clearly indicate close agreeent between prediction and easured results of teperature throughout the thicness of the foacrete saples. 6 5 Proposed odel est est eperature C ie in Fig. : Coparison between test results and nuerical analysis at 7.5 fro eposed side 6 Proposed odel est est eperature C ie in Fig. : Coparison between test results and nuerical analysis at 75. fro eposed side

12 Aust. J. Basic & Appl. Sci., 77: -, 8 Proposed odel est est eperature C ie in Fig. 4: Coparison between test results and nuerical analysis at.5 fro eposed side 5 4 Proposed odel est est eperature C ie in Fig. 5: Coparison between test results and nuerical analysis at uneposed side Conclusions: his paper has presented the basis of the one-diensional heat transfer odelling, the ipleentation of the ethod and the validation of theral properties odel of foacrete panel. he coparison of test results with the nuerical heat transfer analysis results using the proposed theral property odels is close, confiring the validity of the theral conductivity odels. Despite siplicity, the aforeentioned analytical odels for specific heat and theral conductivity of foacrete of different densities give accurate results. he proposed odel is straightforward yet proficient and can be eploited to assist anufacturers to develop their products without having to carry out nuerous large-scale fire tests in the future. ACKNOWLEDGEMEN he author would lie to show gratitude to Universiti Sains Malaysia for their financial support under USM Short er Grant 4/PPBGN/655.

13 Aust. J. Basic & Appl. Sci., 77: -, EFEENCES Croft, D. and D.G. Lilley, 977. Heat ransfer Calculations Using Finite Difference Equations, Applied Science Publishers, London. Di Blasi, C.,. he State of he Art of ransport Models For Charring Solid Degradation. Poly. Int., 49:-8 Holan, J.P.,. Heat transfer, 9 th ed., McGraw-Hill, London. Loeb, A.L., 954. A heory of heral Conductivity of Porous Materials. J. of A Cera. Soc., 7:96-. Othuan Mydin, M.A and Y.C. Wang.,. Elevated-eperature heral Properties of Lightweight Foaed Concrete. Journal of Construction and Building Materials, 5: Ozisi, M.N., 985. Heat ransfer: A Basic Approach, McGraw-Hill, London. Sith, J.M., 98. Cheical Engineering Kinetics, Mcgraw-Hill Boo Copany. Wang, H.B., 995. Heat ransfer Analysis of Coponents of Construction Eposed o Fire, Departent of Civil Engineering and Construction, University of Salford, Manchester. Yuan, J., 9. Fire Protection Perforance of Intuescent Coating Under ealistic Fire Conditions, Ph.D. hesis, School of Mechanical, Aerospace and Civil Engineering, University of Manchester.