multi-dimensional transient temperature simulation and back-calculation for thermal properties of...

16
Multi-dimensional transient temperature simulation and back-calculation for thermal properties of building materials Hui Li * , John Harvey, David Jones Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA article info Article history: Received 3 August 2012 Received in revised form 28 September 2012 Accepted 30 September 2012 Keywords: Thermal environment Thermal properties Heat transfer Modeling and simulation abstract Thermal properties (i.e. thermal conductivity and heat capacity) are important parameters that inuence the temperature of building materials and thermal performance of built environment. These properties are required as fundamental inputs for modeling and simulating thermal behavior of built environment. Most existing methods for measuring thermal properties of building materials are based on 1-D steady- state heat transfer theory. The critical challenge for these methods has been the difculty in achieving a 1-D heat ow condition for the testing specimen. A multi-dimensional transient method is needed to reduce the challenge and requirement on the testing specimen size and shape, and make it possible to accurately measure all the thermal properties from one single test. This paper rst developed a multi-dimensional transient model and a practical tool to simulate the transient temperature at any location on a beam or cylinder specimen of any size subject to convection heat transfer. Case studies veried that this model can be used to, if the thermal properties are known, simulate the transient temperature at any location for a specimen of various shapes and sizes, and predict the time to reach a specied target temperature for mechanical and other testing. Secondly, this paper developed and validated partly by case studies on both asphalt and concrete materials a procedure for back-calculating the thermal properties of a specimen of various shapes and sizes. Thermal properties of novel building materials (various initiative cool materials such as porous concrete and high thermal resistance materials) can be easily measured with that procedure. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. Background Cool roofs and cool pavements, which have lower temperature than conventional roofs and pavements, have been identied by researchers, US EPA and some state EPAs as a potential strategy to mitigate the heat island effect and improve thermal environment [1e 7]. Cool roof or pavement strategies include increasing the thermal reectivity (also called albedo) [1,8e11], modifying the thermal properties (i.e. thermal conductivity and heat capacity) [3,12e14], enhancing the evaporative cooling [12,15e17], increasing green land covers [18], etc. Many researchers are conducting studies on cool roof and/or cool pavement technolo- gies, to evaluate the effects of these technologies on mitigating the heat island effect, improving air quality, improving thermal environment and human thermal comfort, and reducing building cooling energy use [1e6,8e20]. Thermal properties are funda- mental parameters that inuence the distribution and variation of pavement and other building material temperatures and conse- quently affect the thermal performance of built environment [2,3,5,6,11,13,16,18]. These properties are required as inputs for understanding, evaluating and modeling the thermal behavior and consequent environmental impacts of cool roofs and cool pavements [2,3,8,9,11,15,16,18,21e26]. Moreover, temperature is a critical factor affecting building materialsdeterioration speed and durability, especially for pave- ment materials. High temperature will increase the risk of rutting (permanent deformation) of asphalt pavement [27e29]. On the other hand, low temperature and the adverse thermal gradient at low temperature make both asphalt and concrete pavements more susceptible to thermal cracking [30,31]. From this point of view, it is also of great signicance to measure the thermal properties for predicting and optimally designing the thermal behavior of pave- ment structures and materials. Conventionally, the laboratory testing procedure for thermal properties of building materials is based on ASTM C-177, which employs a one-dimensional (1-D) steady-state method to measure * Corresponding author. Tel.: þ1 530 574 5812; fax: þ1 530 752 1228. E-mail addresses: [email protected] (H. Li), [email protected] (J. Harvey), [email protected] (D. Jones). Contents lists available at SciVerse ScienceDirect Building and Environment journal homepage: www.elsevier.com/locate/buildenv 0360-1323/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.buildenv.2012.09.019 Building and Environment 59 (2013) 501e516

Upload: hui-li

Post on 25-Nov-2016

215 views

Category:

Documents


3 download

TRANSCRIPT

at SciVerse ScienceDirect

Building and Environment 59 (2013) 501e516

Contents lists available

Building and Environment

journal homepage: www.elsevier .com/locate/bui ldenv

Multi-dimensional transient temperature simulation and back-calculationfor thermal properties of building materials

Hui Li*, John Harvey, David JonesDepartment of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA

a r t i c l e i n f o

Article history:Received 3 August 2012Received in revised form28 September 2012Accepted 30 September 2012

Keywords:Thermal environmentThermal propertiesHeat transferModeling and simulation

* Corresponding author. Tel.: þ1 530 574 5812; faxE-mail addresses: [email protected] (H. Li), jthar

[email protected] (D. Jones).

0360-1323/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.buildenv.2012.09.019

a b s t r a c t

Thermal properties (i.e. thermal conductivity and heat capacity) are important parameters that influencethe temperature of building materials and thermal performance of built environment. These propertiesare required as fundamental inputs for modeling and simulating thermal behavior of built environment.Most existing methods for measuring thermal properties of building materials are based on 1-D steady-state heat transfer theory. The critical challenge for these methods has been the difficulty in achievinga 1-D heat flow condition for the testing specimen. A multi-dimensional transient method is needed toreduce the challenge and requirement on the testing specimen size and shape, and make it possible toaccurately measure all the thermal properties from one single test.

This paper first developed a multi-dimensional transient model and a practical tool to simulate thetransient temperature at any location on a beam or cylinder specimen of any size subject to convectionheat transfer. Case studies verified that this model can be used to, if the thermal properties are known,simulate the transient temperature at any location for a specimen of various shapes and sizes, andpredict the time to reach a specified target temperature for mechanical and other testing. Secondly, thispaper developed and validated partly by case studies on both asphalt and concrete materials a procedurefor back-calculating the thermal properties of a specimen of various shapes and sizes. Thermal propertiesof novel building materials (various initiative cool materials such as porous concrete and high thermalresistance materials) can be easily measured with that procedure.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background

Cool roofs and cool pavements, which have lower temperaturethan conventional roofs and pavements, have been identified byresearchers, US EPA and some state EPAs as a potential strategy tomitigate the heat island effect and improve thermal environment[1e7]. Cool roof or pavement strategies include increasing thethermal reflectivity (also called albedo) [1,8e11], modifying thethermal properties (i.e. thermal conductivity and heat capacity)[3,12e14], enhancing the evaporative cooling [12,15e17],increasing green land covers [18], etc. Many researchers areconducting studies on cool roof and/or cool pavement technolo-gies, to evaluate the effects of these technologies on mitigatingthe heat island effect, improving air quality, improving thermalenvironment and human thermal comfort, and reducing building

: þ1 530 752 [email protected] (J. Harvey),

All rights reserved.

cooling energy use [1e6,8e20]. Thermal properties are funda-mental parameters that influence the distribution and variation ofpavement and other building material temperatures and conse-quently affect the thermal performance of built environment[2,3,5,6,11,13,16,18]. These properties are required as inputs forunderstanding, evaluating and modeling the thermal behaviorand consequent environmental impacts of cool roofs and coolpavements [2,3,8,9,11,15,16,18,21e26].

Moreover, temperature is a critical factor affecting buildingmaterials’ deterioration speed and durability, especially for pave-ment materials. High temperature will increase the risk of rutting(permanent deformation) of asphalt pavement [27e29]. On theother hand, low temperature and the adverse thermal gradient atlow temperature make both asphalt and concrete pavements moresusceptible to thermal cracking [30,31]. From this point of view, it isalso of great significance to measure the thermal properties forpredicting and optimally designing the thermal behavior of pave-ment structures and materials.

Conventionally, the laboratory testing procedure for thermalproperties of building materials is based on ASTM C-177, whichemploys a one-dimensional (1-D) steady-state method to measure

H. Li et al. / Building and Environment 59 (2013) 501e516502

the thermal conductivity. It is difficult to achieve ideal 1-D heatflow in practice; however, quasi-1-D flow is practically achievablein laboratory tests. To ensure a quasi-1-D heat flow condition, thisprocedure is limited to the flat slab specimen with thickness notexceeding one-third of the maximum linear dimension of themetered region. Besides, it also requires that the temperaturegradient within the test specimen need to be small enough toensure reasonable approximation of differential terms in theFourier equation for heat conduction. It is practically difficult tomeet these requirements of slab size and temperature gradient. Inaddition, it is difficult to reduce and consider the heat loss from theedges of the testing specimen, which will influence the accuracy.Moreover, this steady-state method needs a separate test tomeasure the heat capacity or thermal diffusivity.

Carlson et al. [32] also employed a one-dimensional (1-D)steady-state method to measure the thermal conductivity.Conventional cylinder specimens were proposed to be used toreduce the practical difficulty of obtaining a thermally thin slab ofasphalt in the laboratory. It is still difficult to reduce and considerthe heat loss from the top and bottom of the cylinder specimen ormake it thermally long enough to ensure a 1-D heat flow, and thiswill influence the testing accuracy. In addition, since it is also usinga steady-state method, a separate test is needed to measure theheat capacity or thermal diffusivity.

Somestudies [21,33,34] employed a transientmethod todeterminethe thermal conductivity and heat capacity (or thermal diffusivity) ofasphalt or concrete slab specimens from a single test. However, sincethey used 1-D heat transfer theory, 1-D heat flow condition was stillrequired in their method. This requires that the specimen be a ther-mally thin slab as other 1-D methods discussed above.

Xu and Solaimanian [35] employed a multi-dimensional tran-sient method to measure the thermal properties of asphaltconcrete. This method relaxes the requirement on the specimenshape and size. However, the authors used one-term approxima-tion of the series solutions of temperature, which increase the errorof the model. Besides, the procedure of back-calculation of thermalproperties developed by the authors caused an issue on theuniqueness of the back-calculated thermal properties. These will bedetailed in the following sections of this paper.

Nguyen et al. [36] investigated the change of temperature ofasphalt mixtures during cyclic tests on cylindrical specimens,which is created in the sample by the viscous dissipated energy thatis completely transformed into heat. Temperature is measured atthe surface and inside the specimen. From the analysis of theexperimental results using 1-D transient heat transfer method withinternal heat generation, the thermal properties were obtained.However, the thermocouple embedded into the specimen mightweaken the specimen for mechanical testing.

Table 1 summarizes some of studies on thermal properties ofasphalt and concrete materials, the thermal properties values fromthe literature and the corresponding measurement methods[21,32e40]. As noted from the table and discussion on the litera-ture, most of the existing methods of measuring thermal propertiesof asphalt or concrete materials are based on 1-D steady-state heattransfer theory. The critical challenge for these methods has beenhow to achieve a 1-D heat flow condition for the testing specimen.It is really practically difficult or even impossible for these methodsto meet thermally thin slab or thermally long cylinder criteria,epically for the common beam or cylinder specimens prepared inthe laboratory or extracted from field in-service pavements.

Therefore, an improved multi-dimensional (3-D for slab/beamspecimen and 2-D for cylinder specimen) transient method isneeded to reduce the challenge and requirement on the testingspecimen size and shape, and make it possible to accuratelymeasure the thermal properties from one single test.

1.2. Objectives of this study

The first objective of this paper is to develop a multi-dimensional transient model and a practical tool to simulate thetransient temperature at any location on a beam or cylinderspecimen of various sizes subject to the convection heat transferbetween the specimen and the surrounding airflow. This modeland the practical tool developed can be used to, if the thermalproperties are known, simulate the transient temperature andpredict the time it takes to reach a specified target temperature atany location for a specimen of various shapes and sizes, which ispreheated or precooled in the forced convection oven or temper-ature chamber for mechanical and other temperature-relatedlaboratory testing.

The second objective is to develop and validate a procedure forback-calculating the thermal properties of a specimen of variousshapes and sizes from the measured transient temperatures profileof the specimen, based on the temperature simulation modeldeveloped in the first objective.

2. Theoretical model for simulation of temperature

Themulti-dimensional (3-D for slab/beam specimen and 2-D forcylinder specimen) model to simulate the transient temperature atany location on a beam or cylinder specimen of various sizessubject to the convection heat transfer is developed based on the1-D transient heat transfer theory for infinite plate and infinite longcylinder and geometric intersection of these 1-D models. Moredetails about the fundamental theory can be found in most text-books on heat transfer such as references [41e44].

2.1. Governing equations

For the infinite plate (plane wall) with uniform and homoge-neous thermal conductivity and without internal heat generation,the transfer of heat is assumed to take place only in the longitudinaldirection (say z). This one-dimensional (1-D) heat transfer can bedescribed using the following governing equation in a Cartesiancoordinate system:

v2Tvz2

¼ rckvTvt

¼ 1a

vTvt

(1)

where k is thermal conductivity; r is density; c is specific heat; T istemperature, and t is time. a ¼ k/rc, and is thermal diffusivity.

Analogously, for the infinite long cylinder with uniform andhomogeneous thermal conductivity and without internal heatgeneration, the transfer of heat is assumed to take place only in theradial direction (say r). The governing equation can be written as1-D heat transfer in the cylindrical coordinate system:

1rv

vr

�rvTvr

�¼ v2T

vr2þ 1

rvTvr

¼ 1a

vTvt

(2)

For simplification, the dimensionless temperature q anddimensionless coordinate Z and R are defined as following,

qðZ;FoÞ ¼ Tðz;tÞ�TaT0�Ta

; qðR;FoÞ ¼ Tðr;tÞ�TaT0�Ta

; Z ¼ z=d0; R ¼ r=r0:

where d0 is the half thickness of plate; r0 is the radius of cylinder;Fo ¼ at=d20, and is Fourier number.

The 1-D heat transfer equations (1) and (2) would be changed toequations (3) and (4) for infinite plate and infinite long cylinder,respectively.

Table 1Thermal properties and test methods from literature.

Study Density r

[kg m�3]Specific heat capacityc [J kg�1 �C�1]

Conductivity k[W m�1 �C�1]

Diffusivity a ¼ k/(rc)[m2 s�1] � 10�7

Material Measurement method

Carlson et al. (2008) [32] e 987 e e

e 977 e e GGACe 875 e e AR OGFCe 1016 e e PCCe 1055 e e CRPCC (80 lb

rubber per yd3)e 992 e e CRPCC (160)e 956 e e CRPCC (240)e 964 e e PF PCC (0)e 997 e e PF PCC (3% fiber

content)e 977 e e PF PCC (5)e 971 e e PF PCC (8)e e 0.896 e HMAe e 1.719 e PCC

Mrawira and Luca(2002) [21]

2440 766.6 1.75 9.36 AC 1-D transient

Luca and Mrawira(2005) [34]

2297e2450 1475e1853 1.623e2.060 4.3e5.5 AC

Mrawira and Luca(2006) [33]

2410 1630e2000 1.96e2.01 4.1e5.3 HMA w/gravel,AV 4%

2420 1480e1890 1.91e1.94 4.2e5.4 HMA w/Hornfel,AV 4%

Xu and Solaimanian(2010) [35]

2313 880 2.88 14.2 AC, AV 5.8% 2-D or 3-DTransient

Nguyen et al. (2011) [36] e 820e910 1.35 5.86e6.51 AC, AV 0.8% 1-D steady/transient withinternal heat generation

Wolfe et al. (1980) [37] e 879e963 1.003e1.747 5.16e8.26 AC 1-D steady-stateHighter and Wall

(1984) [38]e 800e1600 0.800e1.600 3.50e7.50 AC 1-D steady-state/transient

Tan et al. (1997) [39] e e 1.300e1.420 5.36e5.80 AC 1-D transientSolaimanian and Bolzan

(1993) [40]e e 0.744e2.889 e AC e

Rang (Overall) 2313e2450 767e2000 0.74e2.89 3.50e14.2 AC/PCC e

H. Li et al. / Building and Environment 59 (2013) 501e516 503

v2q2 ¼ vq

(3)

vZ vFo

1R

v

vR

�Rvq

vR

�¼ v2q

vR2þ 1Rvq

vR¼ vq

vFo(4)

2.2. Initial and boundary conditions

The specimen with a uniform initial temperature T0 is con-ducting convective heat exchange with the surrounding fluid (air)of a constant temperature of Ta. The convection heat transfercoefficient can be determined as [43e45],

h ¼ kairLNu (5-a)

where Nu¼ CRemPr1=3; Re¼ UairL=vair; Pr¼ aair=vair; aair ¼ kair=raircair; vair ¼ mair=rair.

Therefore,

h ¼ C$kairPr

1 =

3

vmair$Lm�1$Um

air: (5-b)

2.2.1. Infinite plate (plane wall)The initial condition is T(z,0) ¼ T0. The boundary conditions are

vT(0,t)/vz ¼ 0, and �kvT(d0,t)/vz ¼ h[T(d0,t) � Ta].Under the dimensionless temperature q and dimensionless

coordinate Z, the initial condition and boundary conditionswould be

qðZ;0Þ ¼ 1;vqð0; FoÞ

vZ¼ 0; �k

vqð1; FoÞvZ

¼ hqð1; FoÞ:

2.2.2. Infinite long cylinderThe initial condition is T(r,0) ¼ T0. The boundary conditions are

vT(0,t)/vr ¼ 0, and �kvT(r0,t)/vr ¼ h[T(r0,t) � Ta].Under the dimensionless temperature q and dimensionless

coordinate R, the initial condition and boundary conditions would be

qðR;0Þ ¼ 1;vqð0; FoÞ

vR¼ 0; �k

vqð1; FoÞvR

¼ hqð1; FoÞ:

3. Analytical solution for simulation of temperature distribution

The analytical solution for infinite plate and infinite longcylinder could be obtained using separation-of-variable method,which can be found in most introductory textbooks (e.g. [41e44])and some literature (e.g. [46,47]) on heat transfer and is notdetailed here.

3.1. Infinite plate (plane wall)

The analytical solution for infinite plate is

qðZ; FoÞplate ¼XNi¼1

Cicos ðxiZÞe�x2i Fo (6)

where Ci ¼ 4sin xi=2xi þ sin ð2xiÞ; xitan xi ¼ Bi; Bi ¼ hd0=k, andis Biot number.

Fig. 1. Flowchart for simulation of temperature for cylinder (a) and beam (b) specimens.

H. Li et al. / Building and Environment 59 (2013) 501e516 505

3.2. Infinite long cylinder

The analytical solution for infinite long cylinder is

qðR; FoÞcylinder ¼XNi¼1

CiJ0ðxiRÞe�x2i Fo (7)

whereCi ¼ 2J1ðxiÞ=xi½J20ðxiÞ þ J21ðxiÞ�; xiJ1ðxiÞ=J0ðxiÞ ¼ Bi; Bi ¼ hr0=k, andis Biot number.

3.3. Short cylinder

The short cylinder can be viewed as the intersection of infiniteplate and infinite long cylinder that are perpendicular. The solutionfor two-dimensional short cylinder is equal to the product of theone-dimensional solutions of infinite plate and infinite longcylinder [43,44]:

qðZ;R; FoÞshort cylinder ¼ qðZ; FoÞplate � qðR; FoÞcylinder (8-a)

The dimensionless normalized temperature q of a short cylinderat location (z, r) and at time t will be,

qðz; r; tÞshort cylinder ¼ qðz; tÞplate � qðr; tÞcylinder (8-b)

The temperature T of a short cylinder at location (z, r) at time tcan be calculated as,

Tðz; r; tÞshort cylinder ¼ qðz; r; tÞshort cylinder$ðT0 � TaÞ þ Ta (9)

3.4. Short beam

Analogously, the short cylinder can be viewed as the intersec-tion of three infinite plates that are mutually perpendicular [43,44].The solution for the three-dimensional short beam could becalculated as,

qðX; Y ; Z; FoÞshort beam ¼ qðX; FoÞplate � qðY ; FoÞplate� qðZ; FoÞplate (10-a)

The dimensionless normalized temperature q of a short beam atlocation (x, y, z) at time t will be,

qðx;y;z; tÞshort beam ¼ qðx; tÞplate� qðy; tÞplate� qðz; tÞplate (10-b)

0 1 2 3 4 5

20

25

30

35

40

45

50

55

0 1.7 3.4 5.1 6.8 8.5Fo

Time (hour)

Tem

pera

ture

(°C

)

N=1N=3N=5N=10N=30N=50N=100

0~5 hours

a

Fig. 2. Simulated temperature with different numbers of t

The temperature T of a short beam at location (x, y, z) at time tcan be calculated as,

Tðx; y; z; tÞshort beam ¼ qðx; y; z; tÞshort beam$ðT0 � TaÞ þ Ta (11)

4. Case study for simulation of temperature and sensitivityanalysis

4.1. Procedure for simulation of temperature profile

According to the theoretical model described in the previoussections, the procedure for simulation of temperature profile ofcylinder and beam specimens is shown as the flowchart of Fig. 1.The procedure was implemented by programming using the freeopen-source R language.

4.2. Input parameters for the case study

Taken a short cylinder specimen as study case, the relatedparameters are assumed as the following,

D ¼ 100mm; H ¼ 150mm; k ¼ 2:5W=ðm �CÞ; c ¼ 920 J=ðkg �CÞ;r ¼ 2300kg=m3; T0 ¼ 25 �C; Ta ¼ 55 �C; h ¼ 15W=

�m2 �C

�:

Under the convective heat transfer from the surrounding air ata constant temperature of Ta ¼ 55 �C described in the previoussection for the short cylinder, the specimen will be heated up fromthe initial temperature of T0¼ 25 �C and reaches the equilibrium upto the air temperature of Ta ¼ 55 �C in some time t.

Using the theoretical model, we can calculate the specimentemperature T(z, r, t) for the following locations and time frames:(a) Center: z ¼ 0 mm, r ¼ 0 mm, t ¼ 0e5 h; (b) Surface: z ¼ 0 mm,r ¼ 50 mm, t ¼ 0e5 h.

4.3. Roots finding for eigenvalue function

The analytical solutions (in equations (6) and (7)) of thetemperature for given location z and r (or Z and R) and time t (or Fo)depend on the eigenvalue xi and constant Ci, which are bothdetermined by the eigenvalue functions (xitan xi ¼ Bi for infiniteplate; xiJ1ðxiÞ=J0ðxiÞ ¼ Bi for infinite long cylinder). Therefore,finding the roots of the eigenvalue functions for any given Bi toobtain the series of eigenvalue xi is one critical step.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

20

22

24

26

28

30

0 0.1 0.2 0.3 0.4 0.5Fo

Time (hour)

Tem

pera

ture

(°C

)

N=1N=3N=5N=10N=30N=50N=100

0~0.3 hours

b

erm N in the solution (center: z ¼ 0 mm, r ¼ 0 mm).

Center (z=0 mm, r=0 mm), along with comparison to solutions from 1-D infinite plate and 1-D infinite long cylinder

Surface (z=0 mm, r=50 mm), along with comparison to solutions from 1-D infinite plate and 1-D infinite long cylinder

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time (hour)

No

rm

alize

d T

em

pe

ra

tu

re

infinite plateinfinite long cylindershort cylinder

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

infinite plateinfinite long cylindershort cylinder

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time (hour)

No

rm

alize

d T

em

pe

ra

tu

re

infinite plateinfinite long cylindershort cylinder

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

infinite plateinfinite long cylindershort cylinder

a

b

Comparison of temperatures for three different locations

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time (hour)

No

rm

alize

d T

em

pe

ra

tu

re

CenterMiddleSurface

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

CenterMiddleSurface

c

Fig. 3. Simulated temperatures for short cylinder at different locations.

H. Li et al. / Building and Environment 59 (2013) 501e516506

H. Li et al. / Building and Environment 59 (2013) 501e516 507

Since it is difficult and even impossible to obtain the analyticalsolutions for the roots of the eigenvalues functions, numericalmethods were employed. Numerical root finding methods includeBisection method, Newton’s method (a.k.a. NewtoneRaphsonmethod), Secant method, etc. [48e50]. A hybrid method of

Thermal conductivity k

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

k=0.5k=1k=1.5k=2.5k=5k=10k=20k=50k=100

a b

Density [kg/m3]

Thermal diffusivity [m2/s]

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

p=300p=500p=800p=1000p=1500p=2300p=3000p=5000

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

alpha=1e-06alpha=2e-06alpha=5e-06alpha=1e-05alpha=3e-05alpha=5e-05alpha=7e-05alpha=1e-04

c d

e f

Fig. 4. Sensitivity of thermal proper

combined Bisection and Newton’s method was employed tobalance the accuracy and the speed of root finding. Bisectionmethod was used to obtain the preliminary roots with loweraccuracy in a small number of iterations (refers to heat transfertextbooks (e.g. [41e44]) and literature (e.g. [46,51]) for finding the

c [J/(kg °C)]

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)Te

mpe

ratu

re (°

C)

c=300c=500c=800c=920c=1200c=1500c=2000c=3000c=5000

Convection coefficient h [W/(m2 °C)]

h/k [1/m]

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

h=1h=5h=10h=15h=20h=30h=50h=100h=300h=500

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

h/k=1h/k=3h/k=5h/k=7h/k=10h/k=30h/k=50h/k=100

ty parameters on the solution.

Center Surface

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

Specimen Shape and Size (mm)Cylinder @ 50D x 100HCylinder @ 100D x 100HCylinder @ 100D x 50HBeam @ 380L x 63W x 50HBeam @ 50L x 63W x 50HBeam @ 100L x 100W x 50H

0 1 2 3 4 5

25

30

35

40

45

50

55

Time (hour)

Tem

pera

ture

(°C

)

Specimen Shape and Size (mm)Cylinder @ 50D x 100HCylinder @ 100D x 100HCylinder @ 100D x 50HBeam @ 380L x 63W x 50HBeam @ 50L x 63W x 50HBeam @ 100L x 100W x 50H

a b

Fig. 5. Predicted temperature profiles for different specimen shape and size.

H. Li et al. / Building and Environment 59 (2013) 501e516508

root intervals of each root); these roots was then used as initialguesses for the Newton’s method, which would give the roots withhigh accuracy.

The first 10 terms of x for about 30 different Biot numbers(0.01e50) were calculated using the hybrid method. These valueswere compared with those available from some heat transfertextbooks (e.g. [41e44]) and found correct. This verifies the hybridroot finding method and the corresponding R program developedand used this study are both valid and effective for root finding.This ensures a universal, convenient and fast method of root

Fig. 6. Flowcharts for back-calcu

finding for the eigenvalue functions for any given Bi (changingwith h/k and specimen size) to obtain the series of eigenvalue xi,which is critical for the following sections.

4.4. Influence of the number of terms N on the solution

In order to examine the influence of the number of terms N onthe solution of temperature, the temperature solution obtainedfrom different terms (N ¼ 1, 3, 5, 10, 30, 50 and 100) in the solutionwere calculated and plotted in Fig. 2 for comparison.

lation of thermal properties.

Fig. 7. Test setup for measurement of thermal properties.

Table 2Specimen parameters and testing condition.

Specimenno.

Height(mm)

Diameter(mm)

Mass(kg)

Density(kg/m3)

Air void(%)

A0 63 102 1.238 2405 3.8C0 222 145 9.095 2481 0.8Testing temperature conditionInitial specimen temperature

T0 (�C)38 Air temperature

Ta (�C)70

Temperature sensor locationSpecimen No. Sensor # #1 #2 #3 #4

A0c za (mm) 3.5 0 0 e

rb (mm) 45 37 17 e

C0 za (mm) 4 �58 82 13rb (mm) 46 35 37 20

a Axial position.b Radial distance from the specimen center.c Only three sensors are installed on the small specimen A0.

H. Li et al. / Building and Environment 59 (2013) 501e516 509

It is noted that there is no significant influence of the number ofterm on the temperature profile for the long time of 10 h (Fig. 2(a)).However, as shown in Fig. 2(b), there is significant influence of thenumber of term on the temperature at the beginning, especially foronly one term (N ¼ 1) in the solution. One-term approximation(N ¼ 1) might be used for predicting the temperature after sometime from the beginning (Fo> 0.2 as recommend bymost literaturesuch as [42,43,52]) without large errors. It indeed will cause bigerrors for the temperature at the beginning. In the study conductedby Xu and Solaimanian [35], only one-term approximation wasused to back-calculate the thermal properties. This one-termapproximation caused some error as also noted by the authors of[35] (the simulated temperatures at the beginning were far awayfrom the initial temperature). Therefore, more than one termshould be used in the solution to obtain a whole temperatureprofile that is accurate over both the beginning and the followinglong time, from which the thermal properties will be back-calculated and described in the following sections. As shown inFig. 2(b), there is no significant differencewhen the number of termin the solution is larger than 10. Therefore, 10 terms (N ¼ 10) isrecommend to be used in the solution to obtain a temperatureprofile with sufficient accuracy.

4.5. Simulation results of temperature profiles

For the short cylinder descried in the previous section, theoriginal temperature profiles at the center, middle and surface ofthe short cylinder were calculated according to the producerdescribed previously and presented in Fig. 3, along with thenormalized temperature profiles. The normalized temperature andoriginal temperature profiles of 1-D infinite plate and infinite longcylinder are also plotted in Fig. 3 for comparison. It shows that thesolutions from 1-D infinite plate and 1-D infinite long cylinder bothcause large errors compared to the multi-dimensional solution fora short cylinder. The temperature profiles at three different loca-tions (center, middle and surface) are different, especially for thesurface one. As expected, the center location need more time toreach the equilibrium temperature. This implies that the insidecenter temperature of a specimen should be considered asa thermal indictor for some laboratory testing at certain targettemperature.

4.6. Sensitivity analysis of thermal property parameters on the solution

To examine the sensitivity of thermal property parameters onthe solution, the center (z ¼ 0 mm, r ¼ 0 mm) temperature of theshort cylinder under different values of each thermal propertyparameter was calculated and shown in Fig. 4. When the value ofone thermal property parameter vary, the value of the otherthermal property parameters keep constant as the default values asgiven in previous section (i.e. k ¼ 2.5 W/(m �C), c ¼ 920 J/(kg �C),r ¼ 2300 kg/m3; h ¼ 15 W/(m2 �C)).

As mentioned previously, the short cylinder specimen will beheated up from the initial temperature of T0 ¼ 25 �C and ultimatelyreaches the equilibrium up to the air temperature of Ta ¼ 55 �C insome time te, through the convective heat transfer with thesurrounding air at a constant temperature of Ta ¼ 55 �C. Therefore,in this case the thermal property parameters will influence theshape of the temperature profile and thus the time te that is neededto reach the equilibrium, as is shown in Fig. 4.

The influences of thermal conductivity k, heat capacity c, densityr and convection coefficient h on the solution of temperature areshown in Fig. 4(a) through (d), respectively. It is clearly shown andalso easily understood that, the larger values of thermal conduc-tivity k and convection coefficient h will reduce the time te that is

needed to reach the equilibrium. In contrast, the increase in valuesof heat capacity c and density r will produce a longer time te. Theshape of the temperature profile and the time te are both sensitiveto the values of thermal conductivity k, heat capacity c, densityr and convection coefficient h.

There seems to be four fundamental thermal parameters,thermal conductivity k, heat capacity c, density r and convectioncoefficient h, which determine the solution of temperature in themodel and for the case described in this study. However, they arenot completely independent parameters; there are some relation-ships between them for determining the solution of temperature.From the model and the procedure given previously, the solution ofshort cylinder temperature T(z, r, t) for a given location (r, z) anda given time t depends on the Fourier number Fo and the eigenvaluexi and constant Ci. The Fourier number Fo is determined by thermaldiffusivity a (¼k/rc); the eigenvalue xi and constant Ci are bothobtained from the eigenvalue functions, which are determined byonly the Biot numbers Bi for both infinite plate and cylinder. TheBiot numbers Bi for both infinite plate and cylinder are commonlydetermined by the ratio of convection coefficient h to the thermalconductivity k, h/k. This implies that, if you change the values of thek, c, r and h in such a way that a and h/k keep constant, the solutionof temperature will keep unchanged. Therefore, there are only twoindependent parameters (a and h/k) for determining the solution oftemperature for both infinite plate and cylinder, thus for both shortcylinder and beam of which the temperatures are calculate fromthe solution of temperature for both infinite plate and cylinder.

Asphalt specimen A0 Concrete specimen C0

0 2 4 6 8 10 12

35

40

45

50

55

60

65

70

Time (hour)

Tem

pera

ture

(°C

)

T1T2T3Ta

0 2 4 6 8 10 12

35

40

45

50

55

60

65

70

Time (hour)

Tem

pera

ture

(°C

)

T1T2T3T4Ta

a b

Fig. 8. Measured temperature at different locations for asphalt and concrete specimens.

H. Li et al. / Building and Environment 59 (2013) 501e516510

These will give an important implication for the back-calculationmethod and uniqueness of thermal parameters, which is dis-cussed in detail in the following section.

The influences of thermal diffusivity a and the ratio of convec-tion coefficient h to the thermal conductivity k, h/k, on the solutionof temperature are shown in Fig. 4(e) and (f), respectively.

a

b

Fig. 9. Adaptive range and step length, optimized parameters an

The larger thermal diffusivity will increase the diffusion speed ofheat in the specimen, and thus reduce the equilibrium time te.Similarly, the larger ratio of convection coefficient h to the thermalconductivity k, h/k, will enhance the convection heat exchangebetween the specimen and the surrounding air, and thus reduce theequilibrium time te, too.

d RMSE for different levels of optimization (specimen A0).

H. Li et al. / Building and Environment 59 (2013) 501e516 511

4.7. Influence of specimen shape and size on the solution

To examine the influence of specimen shape and size on thesolution, the predicted temperature profiles of cylinders of threesizes and beams of three sizes at the center and surface of speci-mens are shown in Fig. 5. The three sizes of cylinder are (H � D)100 � 50 mm, 100 � 100 mm and 50 � 100 mm; the three sizes ofbeams are (L � W � H) 380 � 63 � 50 mm (standard fatigue testbeam), 50 � 63 � 50 mm, and 100 � 100 � 50 mm.

The large cylinder of 100 � 100 mm takes a long time to reachthe equilibrium of 55 �C than the small cylinders of 50 � 100 mmand 100 � 50 mm. The diameter D has a large influence on thetemperature profile and the equilibrium time than the height H fora cylinder. Similarly, the small beam of 50 � 63 � 50 mm takesa short time to reach the equilibrium of 55 �C than the large beamsof 380 � 63 � 50 mm and 100 � 100 � 50 mm. The surfacetemperature of specimens of cylinders or beams reaches theequilibrium faster than the center temperature of a specimen withthe same shape and size, as expected. Therefore, the model and theprocedure of predicting temperature are also sensitive to thespecimen shape and size.

5. Procedure for back-calculation of thermal properties

From the previous section, we discussed that the model and theprocedure developed for predicting temperature of a cylinder or

0 2 4 6

40

45

50

55

60

65

70

Tim

Tem

pera

ture

(°C

)

alpha h/k

RMS N_t

0 2 4 6

40

45

50

55

60

65

70

Tim

Tem

pera

ture

(°C

)

alpha h/k

RMS N_t

0 2 4 6

40

45

50

55

60

65

70

Tim

Tem

pera

ture

(°C

)

alph h/

RMS N_t

Fig. 10. Predicted temperature with the optimized thermal properties compared with measur

beam specimen, which has convective heat exchange with thesurrounding air of a constant temperature, is sensitive to thethermal properties parameters of the specimen as well as the shapeand size of a specimen. Therefore, it is feasible to employ the modeland the procedure as a base to develop a method and procedure forback-calculating the thermal properties of a specimen from themeasured temperature profile of the specimen.

5.1. Optimization method

The concept of curve-fitting is used for back-calculating thethermal properties of a specimen, making the predicted tempera-ture profile match the measured profile as well as possible. Thethermal properties are optimized through minimizing the rootmean squared error (RMSE), to minimize the overall differencebetween predicted results and measurements of temperature asshown in the objective function in equation (12).

minfr;c;k;hg

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1�Tmi � Tpi

�2N

s(12)

where Tmi is measured temperature at ith time point; Tpi istheoretically predicated temperature at ith time point based onthe method described in the previous sections; N is the totalnumber of measurements. c, k, r and h are thermal properties tobe optimized.

8 10 12e (hour)

A0 @ Location 1

(z=3.5 mm, r=6 mm)

T_predictedT_measured

=5.7e-07=19.3

E=0.18erm=10

8 10 12e (hour)

A0 @ Location 2

(z=0 mm, r=14 mm)

T_predictedT_measured

=6.2e-07=17.3

E=0.19erm=10

8 10 12e (hour)

A0 @ Location 3

(z=0 mm, r=34 mm)

T_predictedT_measured

a=5e-06k=2.2

E=0.28erm=10

ed temperature: asphalt specimen A0 (Unit: alpha or a in m2/s; h/k in 1/m; RMSE in �C).

H. Li et al. / Building and Environment 59 (2013) 501e516512

5.2. Uniqueness of the back-calculated thermal properties

As discussed previously, theoretically predicated temperature ofa given specimen, at a given location and time under the givencondition of T0 and Ta, is ultimately determined only by the twoindependent thermal parameters: thermal diffusivity a (¼k/rc) andthe ratio of convection coefficient h to the thermal conductivity k,h/k. Therefore, if the thermal properties r, c, k and h, which are notcompletely independent for the predicted temperature, are usedfor the optimization as used by Xu and Solaimanian [35] in theirstudy, the optimized thermal properties r, c, k and h might not beunique. In other words, if the values of the r, c, k and h arecombined in such a way that a (¼k/rc) and h/k keep constant at theoptimized values, the predicted temperature profile will keepunchanged as the optimized temperature profile which is closest tothe measured one. From this point of view, we change the opti-mization variables from r, c, k and h to the independent parametersa and h/k, as shown in the new objective function in equation (13).

minfa;h=kg

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1�Tmi � Tpi

�2N

s(13)

0 2 4 6

40

4550

55

60

6570

Tim

Tem

pera

ture

(°C

)

alpha h/

RMS N_t

0 2 4 6

40

4550

55

60

6570

Tim

Tem

pera

ture

(°C

)

alpha h

RMS N_t

0 2 4 6

40

4550

55

60

6570

Tim

Tem

pera

ture

(°C

)

alpha h/

RMS N_t

0 2 4 6

3540

455055

606570

Tim

Tem

pera

ture

(°C

)

alph h/k

RMS N_t

Fig. 11. Predicted temperature with the optimized thermal properties compared with measure

where a and h/k are thermal properties to be optimized. Others aresame as the previous definition.

Once the a (¼k/rc) and h/k are determined through the opti-mization process, the two out of four parameters r, c, k and h can becalculated if only the other two are known. The density r ofa specimen can be easily obtained if the weight and the volume ofthe specimen are known. The convection coefficient h can becalculated according to the model shown in equation (5).

Since it is difficult and even impossible to analytically solve thisoptimization problem, a numerical method of trialecomparisonwas used instead to find the optimized solution, and is imple-mented through R programming. During the numerical method oftrialecomparison for optimization, all the possible combinationsof the two independent parameters a and h/k in the feasibleranges are selected with appropriate step lengths to run thetemperature simulations and calculate the corresponding RMSEsof the predicted temperatures. The combination that givesa minimal RMSE is the optimized solution of the two parameters.The whole procedure of back-calculation of thermal propertiesbased on the new objective function is listed by a flowchart inFig. 6.

8 10 12e (hour)

C0 @ Location 1

(z=4 mm, r=26.5 mm)

T_predictedT_measured

=1.4e-06k=5.7

E=0.33erm=10

8 10 12e (hour)

C0 @ Location 2

(z=-58 mm, r=37.5 mm)

T_predictedT_measured

=7.1e-06/k=1

E=0.35erm=10

8 10 12e (hour)

C0 @ Location 3

(z=82 mm, r=35.5 mm)

T_predictedT_measured

=1.1e-06k=9.9

E=0.39erm=10

8 10 12e (hour)

C0 @ Location 4

(z=13 mm, r=52.5 mm)

T_predictedT_measured

a=3e-07=93.8

E=0.34erm=10

d temperature: concrete specimen C0 (Unit: alpha or a in m2/s; h/k in 1/m; RMSE in �C).

Table 4Thermal properties calculated from the optimized parameters.

Asphalt specimen A0

Location # 1 2 3 4 Avga Stdeva

Thermal diffusivity a

[�10�7 m2/s]5.7 6.2 50.2 e 6.0 0.33

Ratio h/k [1/m] 19.3 17.3 2.2 e 18.3 1.4Density r [kg/m3] 2405Convection coefficient h

[W/(m2 �C)]30.02

Thermal conductivity k[W/(m �C)]

1.56 1.74 13.39 e 1.65 0.13

Heat capacity c[J/(kg �C)]

1133.6 1167.5 1108.2 e 1150.6 24.0

RMSE [�C] 0.18 0.19 0.28 e 0.19 0.01Concrete specimen C0

Location # 1 2 3 4 Avgb Stdevb

Thermal diffusivity a

[�10�7 m2/s]14.2 70.9 11.0 3.0 12.6 2.2

Ratio h/k [1/m] 5.7 1.0 9.9 93.8 7.8 3.0Density r [kg/m3] 2481Convection coefficient h

[W/(m2 �C)]28.03

Thermal conductivity k[W/(m �C)]

4.94 28.03 2.83 0.30 3.88 1.49

Heat capacity c[J/(kg �C)]

1399.9 1592.9 1033.4 405.9 1216.7 259.2

RMSE [�C] 0.33 0.35 0.39 0.34 0.36 0.04

a Average/standard deviation on locations 1 & 2.b Average/standard deviation on locations 1 & 3.

H. Li et al. / Building and Environment 59 (2013) 501e516 513

5.3. Initial range and step length of independent parameters

Now we have the objective function for this optimizationproblem shown in equation (13). However, the constrains of thevariables (independent parameters a and h/k) are still missing forthe optimization problem. Theoretically, all positive values arepossible intervals (or ranges) for the both variables. However, theseinfinite intervals are not practical, and will be impossible for thereal calculation. To reduce the calculation load (the number trialiteration), the practical interval of each independent parameter isroughly determined based on the literature (see Table 1). Forthermal diffusivity a, the initial range is set as [1 �10�7, 1 �10�5];for the ratio h/k, it is set as [1, 100]. The initial step lengths used topick up the trial values of both parameters are set as 2 � 10�7 and 5for a and h/k respectively. Using these initial intervals and steplengths, the optimized parameter values can be determined whichgive the minimal RMSE for all trial parameter values.

5.4. Adaptive range and step length (ARS) method

From the initial range and step length, a set of optimizedparameter values can be determined. However, the accuracy of theobtained optimized parameter values is highly dependent on thestep lengths used. A small step length will produce an optimizedresult of high accuracy, but it will also take a long time of compu-tation. Therefore, an adaptive range and step length (ARS) method,which is similar to the Bisection method for root finding, wasproposed to balance the accuracy and computation. This ARSmethod will have adaptive range and step length during the wholeoptimization process. After the optimized results are obtainedusing the initial range and step length, the optimization will go tonext level of optimization in which both of the new range and steplength are one half of the previous range and step length. Theiteration optimization will be continued until the either currentrange or step length goes down to a value small enough, saying1 � 10�9 and 0.1 for a and h/k, respectively.

6. Case study for back-calculation of thermal properties frommeasured temperature profiles

6.1. Laboratory test results of temperature

To demonstrate the model and the procedure for back-calculating thermal properties developed previously in this study,two cylinder specimens, one asphalt (A0) and one concrete (C0),were used to run the test. These two specimenswere cored from in-service road pavements in California. The materials are standarddense graded materials commonly used on highway, street andparking lot pavements in California and other states in US. The testsetup is as shown in Fig. 7. The specimens were first heated to

Table 3Parameters for convection coefficient h.

Parameter Unit Value

Thermal conductivity k W/(m �C) 2.63 � 10�2

Heat capacity c J/(kg �C) 1007Thermal diffusivity a m2/s 2.25 � 10�5

For asphalt specimen A0Airflow speed U m/s 6Specimen diameter D m 0.102ReD e 3.85 � 104

C e 0.0266m e 0.805Nu e 116.44h W/(m2 �C) 30.02

Note: ReD ¼ UD/v; Nu ¼ CReDmPr1/3; h ¼ kNu/D; Pr ¼ v/a.

a uniform temperature of 38 �C in a temperature chamber. Fromthis initial temperature, the specimens were heated up to 70 �Cthough forced convection heat exchange between the specimensand the surrounding air flow of a constant temperature of 70 �C.The temperature profiles of the specimens at different locationswere measured using thermocouple sensors, and the data wererecorded by a data logger (Fig. 7). The detailed specimen parame-ters and testing condition are listed in Table 2.

The measured temperature profiles (T1eT4) of the asphaltspecimen A0 and the concrete specimen C0 are shown in Fig. 8, aswell as the temperature profile of air (Ta) in the chamber.

6.2. Optimized thermal properties and predicted temperature withthe optimized parameters

The thermal properties were back-calculated according to theARS optimization method. To illustrate the ARS optimizationmethod, the adaptive range and step length, optimized parametersand RMSE for different levels of optimization of the asphalt spec-imen A0, as an example, are shown in Fig. 9. During the wholeprocess of optimization, the range [a, b] and the step length

Parameter Unit Value

Kinematic viscosity v m2/s 1.59 � 10�5

Density r kg/m3 1.1614Prandtl number Pr e 0.707For concrete specimen C0Airflow speed U m/s 6Specimen diameter D m 0.145ReD e 5.48 � 104

C e 0.0266m e 0.805Nu e 154.56h W/(m2 �C) 28.03

a b

c d

e f

Fig. 12. Influence of testing time on the optimized parameters (specimen A0).

H. Li et al. / Building and Environment 59 (2013) 501e516514

D decrease with the level of optimization; the accuracy, as shownby RMSE, of the optimized parameters (a and h/k) increase with thelevel of optimization. This verifies that the proposed ARS optimi-zation method is effective in balancing the accuracy and efficiency.

The optimized parameters (a and h/k) and the predictedtemperature under the optimized parameters are shown in Figs. 10and 11.

6.3. Thermal properties from the optimized parameters

According to the procedure developed previously, the thermalproperties k and c can be calculated from the optimized a and h/kif h and r are known. The density r is known as shown in Table 2.The convection coefficient h can be calculated from the airflowparameters (at 300 K) according to the model shown in previousequation (5). The convection coefficient h for the asphalt specimenA0 and the concrete specimen C0 are calculated and listed in

Table 3. Using the known h and r, the thermal properties k and ccan be calculated from the optimized a and h/k and listed inTable 4.

There are some differences of the optimized parameters a andh/k and thus the k and c between locations. The results at thelocation 3 of the asphalt specimen and the locations 2 and 4 for theconcrete specimen C0 are quite different from those of otherlocations on the same specimen. The reason for that might be thatthese locations are closer to the specimen surface (with largeabsolute values of z and r, as shown in Table 2 and Fig. 10). Asnoticed during the trial testing, the measured temperature profileson the specimen surface and those from the location close to thesurface are not very smooth and might cause error when used toback-calculate the thermal parameters. Therefore, to reduce theerrors and obtain the back-calculate thermal parameters of highaccuracy, the measured temperature profiles used for back-calculation should be close to the center of a specimen as possible.

H. Li et al. / Building and Environment 59 (2013) 501e516 515

Based on this observation, the effective averages of thermalproperties of the asphalt and concrete specimens (A0 and C0) arecalculated and shown in Table 4. The thermal conductivity k andheat capacity c of the concrete specimen are larger than those of theasphalt specimen. These values are comparable to those found inliterature (Table 1). This verifies partly that the method andprocedure developed here are practical and valid for measuringthermal properties of building materials.

6.4. Influence of the length of testing time on the optimizedparameters

The optimized parameters shown in Fig. 10 through Fig. 11 andTable 4 are obtained according to the measured temperatureprofiles, which are completely developed curves for the transientheat transfer between the specimens and the surrounding airflow.The total time is over 12 h, which is a long time. Two questions iscoming up, that is, whether the length of testing time influencesthe optimized results, and what is the shortest time needed to runthe test for back-calculation of the thermal properties.

To answer these questions, taken asphalt specimen as example,the optimized parameters with the measured temperatureprofiles with different testing time lengths are back-calculatedand plotted in Fig. 12. From the plots, it is noted that, the back-calculated parameters do change with the length of testing time.When the testing time is over 4 h, the back-calculated parameterstend to be constant. Therefore, the testing time for back-calculation is recommended as at least 4 h to balance the accu-racy and cost in terms of both testing time and energyconsumption. Otherwise, some error might be caused for theback-calculated parameters.

7. Conclusions

This paper is devoted to discussion of questions related to themulti-dimensional modeling and simulation of transient temper-atures at any location on a beam or cylinder specimen of varioussizes subject to the convection heat transfer, and back-calculating ofthe thermal properties of a specimen of various shapes and sizesfrom the measured transient temperatures profile of the specimen.

The developed model and tool can accurately predict transienttemperature at any location on a beam or cylinder specimen ofvarious sizes subject to the convection heat transfer. The casestudies presented in this paper verify that it can be used to, if thethermal properties are known, simulate the transient temperatureand predict the time it takes to reach a specified target testingtemperature at any location for a specimen of various shapes andsizes, when the specimen is preheated or precooled in the forcedconvection oven or temperature chamber for mechanical or othertemperature-related testing.

Based on the developed temperature simulation model, theprocedure and tool for back-calculating thermal properties weredeveloped and validated in part by case studies on both asphalt andconcrete materials. The partly validated procedure and tool can beused to obtain the thermal properties of a specimen of buildingmaterials from the measured transient temperatures profile,regardless of the shape and size of specimen. Of course, moretesting studies on different materials are needed to further validatethe procedure and tool developed in this paper. With the fullyvalidated procedure and tool, thermal properties of novel buildingmaterials (various initiative cool materials such as porous concreteand high thermal resistance materials) can be easily measured andthen used for evaluating and modeling the thermal performance ofbuilt environment composed of these materials.

Acknowledgements

The research was supported by a grant (dissertation fellowship)from the Sustainable Transportation Center (STC) at the Universityof California Davis, which receives funding from the U.S. Depart-ment of Transportation and Caltrans, the California Department ofTransportation, through the University Transportation Centersprogram. The research activities described in this paper was alsopartly sponsored by the California Department of Transportation(Caltrans), Division of Research and Innovation. Both sponsorshipsare gratefully acknowledged. The contents of this paper reflect theviews of the authors and do not reflect the official views or policiesof the Sustainable Transportation Research Center, the U.S.Department of Transportation, the State of California or the FederalHighway Administration.

References

[1] Synnefa A, Karlessi T, Gaitani N, Santamouris M, Assimakopoulos DN,Papakatsikas C. Experimental testing of cool colored thin layer asphalt andestimation of its potential to improve the urban microclimate. Build Environ2011;46:38e44.

[2] Xu T, Sathaye J, Akbari H, Garg V, Tetali S. Quantifying the direct benefits ofcool roofs in an urban setting: reduced cooling energy use and loweredgreenhouse gas emissions. Build Environ 2012;48:1e6.

[3] Santamouris M, Gaitani N, Spanou A, Saliari M, Giannopoulou K,Vasilakopoulou K, et al. Using cool paving materials to improve microclimateof urban areas e design realization and results of the Flisvos project. BuildEnviron 2012;53:128e36.

[4] Santamouris M. Cooling the cities e a review of reflective and green roofmitigation technologies to fight heat island and improve comfort in urbanenvironments. Solar Energy in press. Available from: http://dx.doi.org/10.1016/j.solener.2012.07.003.

[5] Akbari H, Matthews HD, Donny S. The long-term effect of increasing thealbedo of urban areas. Environ Res Lett 2012;7:024004.

[6] Akbari H, Matthews HD. Global cooling updates: reflective roofs and pave-ments. Energy Buildings in press. Available from: http://dx.doi.org/10.1016/j.enbuild.2012.02.055.

[7] EPA U. Green infrastructure programs [cited 2011 Mar. 10]; Available from:http://cfpub.epa.gov/npdes/greeninfrastructure/gicasestudies.cfm; 2010.

[8] Mahmoud AHA. Analysis of the microclimatic and human comfort conditionsin an urban park in hot and arid regions. Build Environ 2011;46:2641e56.

[9] Kültür S, Türkeri N. Assessment of long term solar reflectance performance ofroof coverings measured in laboratory and in field. Build Environ 2012;48:164e72.

[10] Karlessi T, Santamouris M, Synnefa A, Assimakopoulos D, Didaskalopoulos P,Apostolakis K. Development and testing of PCM doped cool colored coatings tomitigate urban heat island and cool buildings. Build Environ 2011;46:570e6.

[11] Jo JH, Carlson JD, Golden JS, Bryan H. An integrated empirical and modelingmethodology for analyzing solar reflective roof technologies on commercialbuildings. Build Environ 2010;45:453e60.

[12] Wanphen S, Nagano K. Experimental study of the performance of porousmaterials to moderate the roof surface temperature by its evaporative coolingeffect. Build Environ 2009;44:338e51.

[13] Asaeda T, Ca VT. Characteristics of permeable pavement during hot summerweather and impact on the thermal environment. Build Environ 2000;35:363e75.

[14] Scholz M, Grabowiecki P. Review of permeable pavement systems. BuildEnviron 2007;42:3830e6.

[15] Tan Siew A, Fwa Tien F. Pavement evaluation for thermal/glare comfort duringfootdrills. Build Environ 1997;32:257e69.

[16] Tan S-A, Fwa T-F. Influence of pavement materials on the thermal environ-ment of outdoor spaces. Build Environ 1992;27:289e95.

[17] Starke P, Wallmeyer C, Rölver S, Göbel P, Coldewey WG. Development ofa new laboratory evaporation measurement device as decision support forevaporation-optimized building. Build Environ 2011;46:2552e61.

[18] Wong NH, Kwang Tan AY, Chen Y, Sekar K, Tan PY, Chan D, et al. Thermalevaluation of vertical greenery systems for building walls. Build Environ 2010;45:663e72.

[19] Wang DC, Wang LC, Cheng KY, Lin J. Benefit analysis of permeable pavementon sidewalks. Int J Pavement Res Technol 2010;3:207e15.

[20] Yamagata H, Nasu M, Yoshizawa M, Miyamoto A, Minamiyama M. Heat islandmitigation using water retentive pavement sprinkled with reclaimed waste-water. Water Sci Technol 2008;57:763e71.

[21] Mrawira DM, Luca J. Thermal properties and transient temperature responseof full-depth asphalt pavements. Des Rehabil Pavements 2002;2002:160e71.

[22] Schindler AK, Ruiz JM, Rasmussen RO, Chang GK, Wathne LG. Concretepavement temperature prediction FHWA HIPERPAV and case studies with themodels. Cement Concr Compos 2004;26:463e71.

H. Li et al. / Building and Environment 59 (2013) 501e516516

[23] Hermansson A. Mathematical model for calculation of pavement tempera-tures e comparison of calculated and measured temperatures. Assess EvaluatPavements 2001:180e8.

[24] Li H, Harvey J. Numerical simulation and sensitivity analysis of asphaltpavement temperature and near-surface air temperature using integratedlocal modeling (11-3125). In: Transportation Research Board 90th annualmeeting. Washington, D.C., 2011.

[25] Hermansson A. Simulation model for calculating pavement temperaturesincluding maximum temperature. Pavement Manag Monit 2000:134e41.

[26] Gui J, Phelan PE, Kaloush KE, Golden JS. Impact of pavement thermophysicalproperties on surface temperatures. J Mater Civil Eng 2007;19:683e90.

[27] Tapkın S. The effect of polypropylene fibers on asphalt performance. BuildEnviron 2008;43:1065e71.

[28] Ibrahim H, Wahhab A-A, Hasnain J. Laboratory study of asphalt concretedurability in Jeddah. Build Environ 1998;33:219e30.

[29] Abo-Qudais S, Al-Shweily H. Effect of antistripping additives on environ-mental damage of bituminous mixtures. Build Environ 2007;42:2929e38.

[30] Roy SK, Poh KB, Northwood Do. Durability of concrete d acceleratedcarbonation and weathering studies. Build Environ 1999;34:597e606.

[31] Ramadhan RH, Al-Abdul Wahhab HI. Temperature variation of flexible andrigid pavements in eastern Saudi Arabia. Build Environ 1997;32:367e73.

[32] Carlson JD, Bhardwaj R, Phelan PE, Kaloush KE, Golden JS. Determiningthermal conductivity of paving materials using cylindrical sample geometry.J Mater Civil Eng 2010;22:186e95.

[33] Mrawira DM, Luca J. Effect of aggregate type, gradation, and compaction levelon thermal properties of hot-mix asphalts. Can J Civil Eng 2006;33:1410e7.

[34] Luca J, Mrawira D. New measurement of thermal properties of superpaveasphalt concrete. J Mater Civil Eng 2005;17:72e9.

[35] Xu QW, Solaimanian M. Modeling temperature distribution and thermalproperty of asphalt concrete for laboratory testing applications. ConstructBuilding Mater 2010;24:487e97.

[36] Nguyen QT, Di Benedetto H, Sauzéat C. Determination of thermal properties ofasphalt mixtures as another output from cyclic tensionecompression test.Road Mater Pavement Des 2012;13:85e103.

[37] Wolfe RK, Heath GL, Colony DC. University of Toledo time temperature modellaboratory and field validation. Rep. no. FHWA/OH-80/006. Toledo, Ohio:Dept. of Industrial Engineering, Univ. of Toledo; 1980.

[38] Highter WH, Wall DJ. Thermal properties of some asphaltic concrete mixes.Transport Res Board 1984:38e45.

[39] Tan SA, Fwa TF, Chuai CT, Low BH. Determination of thermal properties ofpavement materials and unbound aggregates by transient heat conduction.J Test Eval 1997;25:15e22.

[40] Solaimanian M, Bolzan P. Analysis of the integrated model of climatic effectson pavements. Strategic Highway research program. Washington, D.C.: U.S.Federal Highway Administration, Research Rep.; 1993.

[41] Nellis G, Klein SA, Ebrary Inc.. Heat transfer. Cambridge, New York: CambridgeUniversity Press; 2009. p. xxxvii, 1107 pp.

[42] Annaratone D. Engineering heat transfer. Heidelberg: Springer; 2010.[43] Arpaci VS, Kao S-H, Selamet A. Introduction to heat transfer. Upper Saddle

River, NJ: Prentice Hall; 1999.[44] Rathore MM, Kapuno RR. Engineering heat transfer. Sudbury, Mass.: Jones &

Bartlett Learning; 2011.[45] Bejan A. Convection heat transfer. Hoboken, NJ: Wiley; 2004.[46] Wang ZH, Tan KH. Temperature prediction for multi-dimensional domains in

standard fire. Commun Numer Meth Eng 2007;23:1035e55.[47] Wang ZH, Tan KH. Temperature prediction of concrete-filled rectangular

hollow sections in fire using Green’s function method. J Eng Mech 2007;133:688e700.

[48] Epperson JF. An introduction to numerical methods and analysis. Hoboken,NJ: Wiley-Interscience; 2007.

[49] Sauer T. Numerical analysis. Boston: Pearson; 2012.[50] Kyurkchiev NV. Initial approximations and root finding methods. Berlin, New

York: Wiley-VCH; 1998.[51] Wang ZH, Tan KH. Temperature prediction for contour-insulated concrete-

filled CHS subjected to fire using large time Green’s function solutions.J Construct Steel Res 2007;63:997e1007.

[52] Wang ZH, Tan KH. Green’s function solution for transient heat conduction inconcrete-filled CHS subjected to fire. Eng Struct 2006;28:1574e85.