AP09_1.vp 1 Introduction Gypsum board based systems are now widely used in buildings, as walls or ceilings, to provide passive fire protec- tion. The basis of the fire resistance of such systems lies in low thermal conductivity and the evaporation of the water con- tent of the gypsum board, which absorbs a considerable amount of heat, thereby delaying temperature rise through the system. To accurately model the performance of such sys- tems in fire condition, their thermal properties should be known. Thermal properties of gypsum are temperature-depend- ent and among them, thermal conductivity has a critical influ- ence, but there is a wide difference in reported values in litera- ture. Given the effects of porosity, non-homogeneity and moisture in gypsum, direct experimental measurement of thermal conductivity of gypsum at high temperatures is not an easy task. As an alternative, this paper proposes a hybrid numerical and experimental method to extract the thermal conductivity of gypsum. A one-dimensional finite difference heat conduction programme has been developed to predict the temperature development through the thickness of the gypsum board, based on an initial estimate of the thermal conductivity-temperature relationship as a function of poros- ity and radiation within the voids. This relationship is then calibrated by comparing numerical results with the experi- mental results from small-scale fire tests, so that the tempera- ture history of the specimen calculated by the programme closely matches the temperatures recorded during the test. This method has been found to yield more consistent results than those reported in literature. 2 Outline of the numerical analysis method The transient heat transfer through a gypsum board is modelled using one-dimensional Finite Difference for- mulation. A computer program has been developed and implemented in the familiar environment of Microsoft Excel using VBA. The modelling procedure has been thoroughly validated [1] by comparisons with a number of analytical solu- tions and simulation results, using ABAQUS/Standard. The following describes the basis of the modelling method. 2.1 One-dimensional finite difference formulation Fourier’s law of conduction in one dimension with no heat generation is expressed as: � � � � � � �x k T T x t x c T x t t ( ) ( , ) ( , )� � � � � � � , (1) where T x t( , ) is temperature (°C), k(T) is thermal conductivity (W/m °C), � is material density (kg/m3), c is specific heat of material (J/kg °C), t is time (sec), x is the coordinate (0 x L, L being the thick- ness of the panel). Choosing the explicit technique, the temperature of a vol- ume cell (refer to Figs. 1 and 2) at a time step is computed directly based on the temperatures of the adjacent cells in the last time step, which leads to a very simple scheme of com- putation [2]: (i) For a typical node m within the material (Fig. 1): � � � � � � � � T F k T k T k k T Fm m m m m m m m m m m m0 1 1 1 1 1 1 0 2 1( ), , , , 2 � � � � � � � � � � � � � � � � , (2) where F0 is defined as: F k k t c x m m m m 0 1 1 22 � � �( ) ( ) , , � �� (3) �Tm is the temperature of m in the subsequent time step and ki,j is the thermal conductivity at the average temperature of cells i and j: k k T T i j i j , � �� � � � � � � �2 . (4) 16 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 49 No. 1/2009 Thermal Conductivity of Gypsum at High Temperatures A Combined Experimental and Numerical Approach I. Rahmanian, Y. Wang Performance of gypsum board based systems in fire is highly influenced by the temperature-dependant thermal conductivity of gypsum boards, yet there is a wide difference in the thermal conductivity values used in literature. Presented here is a hybrid method to determine the effective thermal conductivity of gypsum boards at high temperatures, based on using small-scale experimental results and a thermal conductivity model which includes the effects of radiation in voids. Keywords: gypsum board, thermal conductivity, high temperatures, radiation in voids, porous material, passive fire protection, fire resistance Numerical stability under the explicit scheme requires: � � t c x k km m m m � � � ( ) ( ), , 2 1 1 . (5) (ii) For a boundary node, when subjected to convective and radiative boundary conditions (Fig. 2): � � � � � � � � � � � � � � � � � � � � � �T F T h x k T F h x k T1 0 2 1 0 1 12 1 2 1 � � � � �E T T t c x � � ( ) ( ) ,� � �273 273 24 1 4 � � (6) where F0 is F k t c x 0 1 2� � �� ( ) , h(T) is convection heat transfer coefficient (W/m2 °C), T� is the ambient temperature (°C), � is a geometric “view factor”, E is the effective emissivity, � is Stefan-Boltzmann constant (5.67×10 8 W/m2 K4). Numerical stability limits the time step to: � � � � t c x k h x k E x k T T � � �� � � � � � � � 0 1 2732 1 1 1 1 4 1 ��� � �( ) ( ) 1 . (7) 2.2 Initial and boundary conditions Gypsum board is assumed to have a uniform initial tem- perature equal to the ambient temperature. On the unexposed boundary, the convective heat transfer coefficient (h) is assumed to be constant and the value is taken as 10 W/m2 °C. The surface of gypsum plasterboards is lami- nated by paper with emissivity of 0.8–0.9, as reported in reference [3]. Thus, the surface emissivity of the board is taken as 0.8 and the view factor equals unity. For extraction of thermal conductivity based on fire test results, the recorded temperatures on the exposed surface are used as input data. 2.3 Specific heat and density The temperature-dependent specific heat of gypsum ex- periences two peaks corresponding to the two dehydration reactions of gypsum, as shown in Fig. 3. These peaks repre- sent the energy consumed to dissociate and evaporate water and include the effect of water movement and reconden- sation of water in cooler regions of gypsum [4]. The base value of the specific heat is 950 J/kg °C, as reported by Mehaffey et al. [5], and the additional specific heat at each dehydration reaction can be expressed by [4]: � � c T e f e f� � � 2 26 106 1 2 . ( )d free (J/kg °C), (8) where � c is the average additional specific heat, ed is the dehydration water content (percentage by total weight), efree is the free water content (percentage by total weight), �T is the temperature interval, f1, f2 are correction factors to account for the heat of reactions and effects of water movement. According to Ang and Wang [4], f1 � 1.28 and 1.42 for the first and second dehydration reactions, respectively. For standard fire conditions f2 � 1.4. Due to evaporation of water, the density of gypsum re- duces with temperature increase. Fig. 4 shows the density © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 17 Acta Polytechnica Vol. 49 No. 1/2009 m 1 m m 1� � x Direction of heat flow Fig. 1: Finite difference discretization for node m within the material 2 1 � x Direction of heat flow T Fig. 2: Finite difference discretization for a boundary node 0 5 10 15 20 25 30 0 200 400 600 800 1000 Temperatures (°C) S p e c if ic H e a t (k J/ k g .° C ) Fig. 3: Specific heat of gypsum as used in the analysis 70 75 80 85 90 95 100 105 0 200 400 600 800 1000 Temperatures (°C) % O ri g in a l M a ss Fig. 4: Density of gypsum as used in the analysis (% of the original density) used in the modelling as a percentage of the original density of gypsum at ambient temperature. 2.4 Thermal conductivity Since gypsum is a porous material, heat transfer through gypsum is a combination of all three modes: conduction through the solid, and convection and radiation through the pores. Therefore the effective thermal conductivity of gypsum should include these effects. This effective thermal conduc- tivity can be affected by many factors such as temperature, density, moisture content and porosity of the material. Such sensitivity contributes to the diverse data reported in litera- ture as demonstrated in Fig. 5. Assuming gypsum is made of solid substrate and uniformly distributed spherical pores, the effective thermal conductivity of gypsum may be calculated using the following equation [6]: k k k k k k * ( ) ( ) ( � � � � s g s g s 2 3 2 3 2 3 2 3 1 1 , (9) where k* is the effective thermal conductivity of gypsum, kg is the effective thermal conductivity of gas to account for heat transfer in the pores, ks is the thermal conductivity of the solid, � is the porosity of the material (the ratio of the volume of void to the overall volume). In this study, the thermal conductivity of solid dried gypsum (ks) is 0.31 W/m °C and the porosity of gypsum is 67 %–72 %. Since the size of the pores is very small (never larger than 5 mm), natural convection in the pores can be neglected. Therefore the effective thermal conductivity of the gas is [6]: k T d Tg � � � � 4 815 10 2 3 44 0 717 3. . e � (10) where T is absolute temperature and de is the effective diame- ter of the pores. In this study de � 0.15 mm. Hence, the effective thermal conductivity-temperature re- lationship consists of three parts, as demonstrated in Fig. 6: 1) Constant thermal conductivity up to 95 °C before water evaporation, equal to that at ambient temperature, re- ported by the manufacturer; 2) Linear reduction of conductivity to 0.1 W/m °C at 155 °C; 3) Non-linear increase in thermal conductivity based on equations 9 and 10. 3 Small-scale high temperature tests A limited number of small-scale experiments have been performed. The specimens tested were gypsum board panels of two different types; 12.5 mm Gyproc Fireline plasterboard and 9.5 mm Gyproc Wallboard plasterboard, both British Gyp- sum products. A total number of 8 specimens were tested, as specified in Table 1. All specimens had approximate dimen- sions of mm. Each specimen was placed horizontally on top of an electric kiln, as the source of heat, so that one side of the panel was subjected to kiln temperature and the other side faced up to the room temperature (19–25 °C). An opening of 280×265 mm on the top lid of the kiln allowed the lower side of the panel to be exposed to the elevated kiln tempera- tures. A 30 mm layer of glass wool (with an opening of the same size as that in the kiln lid) was laid underneath the speci- men to insulate the contact surface between the top lid and the plasterboard. Fig. 7 shows the typical set-up of the experi- ments. Fig. 8 shows the heating curve achieved in the kiln, which is compared to a standard cellulosic fire (BS476) [8]. Temperatures were measured on the unexposed side, the midpoint (for double layered panels) and the exposed side of the gypsum panel using Type K thermocouples. 18 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 49 No. 1/2009 Temperature (°C) T h e rm a l c o n d u c ti v it y (W / m K ) � Fig. 5: Thermal conductivity of gypsum, as reported by various researchers [7] 0 0,05 0,1 0,15 0,2 0,25 0 200 400 600 800 1000 Temperature (°C) T h e rm a l C o n d u c ti v it y (W / m .° C ) Fig. 6: Effective thermal conductivity of gypsum as used in this study Fig. 7: Typical set-up for the small-scale fire tests 4 Results In Figs. 9 to 12, the temperature histories measured from the tests (data points) and calculated by the program using © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 19 Acta Polytechnica Vol. 49 No. 1/2009 Test No Plaster board Type Layers Total Thickness (mm) Density (kg/m3) Free Water (% by weight) Initial Thermal Conductivity (W/m °C) 1 Gyproc Fireline Single 12.5 770 3.5 0.24 2 Gyproc Fireline Single 12.5 770 3.5 0.24 3 Gyproc Fireline Double 25 770 3.5 0.24 4 Gyproc Fireline Double 25 770 3.5 0.24 5 Gyproc Wallboard Single 9.5 641 3.5 0.19 6 Gyproc Wallboard Single 9.5 641 3.5 0.19 7 Gyproc Wallboard Double 19 641 3.5 0.19 8 Gyproc Wallboard Double 19 641 3.5 0.19 Table 1: Specifications of gypsum board specimens 0 200 400 600 800 1000 1200 0 20 40 60 80 100 120 140 160 180 Time (min) T e m p e ra tu re (° C ) Standard Cellulosic Fire Kiln Temperature Fig. 8: Time-temperature curve for the kiln against standard cel- lulosic fire curve 0 100 200 300 400 500 0 10 20 30 40 50 60 70 80 90 100 Time (min) T e m p e ra tu re s (° C ) Experiment Analysis using the proposed thermal conductivity Analysis using thermal conductivity as used by Mehaffey at al Unexposed . Fig. 9: Temperature history for 12.5 mm Fireline gypsum panel 0 100 200 300 400 500 0 10 20 30 40 50 60 70 80 90 100 Time (min) T e m p e ra tu re s (° C ) Experiment Analysis using the proposed thermal conductivity Analysis using thermal conductivity as used by Mehaffey at al Unexposed Fig. 11: Temperature history for 9.5 mm Wallboard gypsum panel 0 100 200 300 400 500 600 700 800 900 1000 0 10 20 30 40 50 60 70 80 90 100 Time (min) T e m p e ra tu re s (° C ) Experiment Analysis using the proposed thermal conductivity Analysis using thermal conductivity as used by Mehaffey at al Midpoint Unexposed Fig. 10: Temperature history for 25 mm Fireline gypsum panel pore size of 1 mm (solid thick line) are compared. Also plotted in these figures are the numerical results utilizing thermal conductivity of gypsum, as used by Mehaffey et al. [5] (thin solid line). The results demonstrate a considerable improve- ment in prediction of temperature development through gypsum when using the new thermal conductivity model described in this paper. 5 Conclusions This paper has presented a hybrid method to determine the effective thermal conductivity of gypsum at high tempera- tures, based on using small-scale experimental results and a thermal conductivity model which includes the effects of radi- ation in voids. Despite the simplicity of the method, the results are in good agreement with test measurements and show great improvement when compared to those produced using thermal conductivity values reported in literature. This method will aid manufacturers to develop their products without having to conduct numerous large-scale fire tests. Further planned research includes investigating the effects of discrete large cracks in gypsum on heat transfer in gypsum board systems and gypsum falling-off at high temperatures. Acknowledgment The authors would like to thank British Gypsum for their financial support and Drs. Kane Ironside and Jan Rideout for their interest and technical support. The technical assistance by the laboratory staff at the University of Manchester is greatly appreciated. References [1] Rahmanian, I.: Fire Resistance of Gypsum Board Based Sys- tems. First year PhD progression report, School of Me- chanical, Aerospace and Civil Engineering, University of Manchester, UK, 2008. [2] Wang, H. B.: Heat Transfer Analysis of Components of Con- struction Exposed to Fire. Department of Civil Engineering and Construction, University of Salford, U.K., 1995. [3] Ozisik, M. N.: Heat Transfer: A Basic Approach. New York; London: McGraw-Hill, 1985. [4] Ang, C. N., Wang, Y. C.: The Effect of Water Movement on Specific Heat of Gypsum Plasterboard in Heat Trans- fer Analysis Under Natural Fire Exposure. Construction and Building Materials, Vol. 18 (2004), p. 505–515. [5] Mehaffey, J. R., Cuerrier, P., Carisse, G. A.: A Model for Predicting Heat Transfer Through Gypsum- -Board/Wood-Stud Walls Exposed to Fire. Fire and Mate- rials, Vol. 18(1994), p. 297–305. [6] Yuan, J.: Fire Protection Performance of Intumescent Coating under Realistic Fire Conditions. PhD Thesis, School of Me- chanical, Aerospace and Civil Engineering, University of Manchester, UK, 2009. [7] Thomas, G.: Thermal Properties of Gypsum Plaster- board at High Temperatures. Fire and Materials, Vol. 26 (2002), p. 37–45. [8] BS476, Fire tests on building materials and structures, Part 20: Method for determination of the fire resistance of elements of construction (general principles), British Standards Institution, 1987. Ima Rahmanian e- mail: ima.rahmanian@postgrad.manchester.ac.uk Yong Wang University of Manchester School of Mechanical, Aerospace and Civil Engineering, Manchester, PO Box 88 Manchester M60 1QD, United Kingdom 20 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 49 No. 1/2009 0 100 200 300 400 500 600 700 800 900 1000 0 10 20 30 40 50 60 70 80 90 100 Time (min) T e m p e ra tu re s (° C ) Experiment Analysis using the proposed thermal conductivity Analysis using thermal conductivity as used by Mehaffey at al Midpoint Unexposed Fig. 12: Temperature history for 19 mm Wallboard gypsum panel