Research Article Extracting Concrete Thermal Characteristics from Temperature … · 2019. 7. 31. · Research Article Extracting Concrete Thermal Characteristics from Temperature

Research ArticleExtracting Concrete Thermal Characteristics from TemperatureTime History of RC Column Exposed to Standard Fire

Jung J Kim1 Kwang-Soo Youm2 and Mahmoud M Reda Taha3

1 School of Architecture Kyungnam University Changwon-si 631-701 Republic of Korea2 GS EampC Research Institute Gyeonggi-do 449-831 Republic of Korea3 Department of Civil Engineering University of New Mexico Albuquerque NM 87131-0001 USA

Correspondence should be addressed to Kwang-Soo Youm lansing2004navercom

Received 24 February 2014 Accepted 2 July 2014 Published 11 August 2014

Academic Editor Alejandro Rodrıguez-Castellanos

Copyright copy 2014 Jung J Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A numerical method to identify thermal conductivity from time history of one-dimensional temperature variations in thermalunsteady-state is proposed The numerical method considers the change of specific heat and thermal conductivity with respectto temperature Fire test of reinforced concrete (RC) columns was conducted using a standard fire to obtain time history oftemperature variations in the column section A thermal equilibriummodel in unsteady-state conditionwas developedThe thermalconductivity of concrete was then determined by optimizing the numerical solution of the model to meet the observed time historyof temperature variations The determined thermal conductivity with respect to temperature was then verified against standardthermal conductivity measurements of concrete bricks It is concluded that the proposed method can be used to conservativelyestimate thermal conductivity of concrete for design purpose Finally the thermal radiation properties of concrete for the RCcolumn were estimated from the thermal equilibrium at the surface of the column The radiant heat transfer ratio of concreterepresenting absorptivity to emissivity ratio of concrete during fire was evaluated and is suggested as a concrete criterion that canbe used in fire safety assessment

1 Introduction

Fire safety for structural members in buildings is necessaryto be assessed as failure of structural members due to firecan cause extensive property damage or loss of life Manyresearch efforts have been conducted to assess fire safety forstructural members [1ndash3] For a reinforced concrete (RC)member exposed to fire the increase in temperature inboth steel and concrete causes a decrease in strength andstiffness of the member Therefore most design codes for RCstructures limit temperatures in RC member subject to fire[4ndash8] Alternatively effective sections can be considered forfire safety assessment of RC members such that the strengthof RCmember under fire is evaluated with a reduced strengthfor reinforcing steel based on its temperature while thestrength of concrete over 500∘C is ignored [7 8] Temperaturevariations in the RC member exposed to fire depends on thethermal properties of concrete and steel and the shape ofthe cross-sections The thermal properties of concrete and

steel are obtained from fire test or using prediction methodsWhile many methods were proposed to predict temperaturevariations in concrete members subject to fire [9ndash13] thereexists difficulty to choose the thermal characteristics espe-cially thermal conductivity of concrete This work suggests asimple method to identify thermal characteristics of concreteunder fire

The thermal conductivity of concrete is a major propertythat affects the temperature variations in concrete memberssubject to fire In normal concrete the thermal conductivityvaries from 05W(mK) to 33W(mK) depending on thetype of aggregate and the concrete mixture used [15ndash19] Asthe thermal conductivity of concrete is relatively lower thanthat of other construction materials such as 533W(mK) at20∘C for structural steel [20] RC members exhibit good per-formance in fire if extensive spalling is prevented [14 21]Thethermal conductivity of concrete has also been reported to beaffected by themeasuring techniques [19 22ndash24] Tomeasurethe thermal conductivity of concrete at a test temperature it

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 242806 10 pageshttpdxdoiorg1011552014242806

2 The Scientific World Journal

Numerical analysis

Verification

Experimental program 2

No

Yes

Thermal conductivitymeasurement ofconcrete bricks

by hot wire(ASTM C 1113)

Thermalconductivity k0RC

Thermalconductivity kfRC

kiRC

Temperatureprofiles TO(t x)

Temperature profiles TN(t x)

Thermalconductivity kCB

max[R2(TN TO)]


fire test of RC column

Figure 1 Layout of the proposed research program to extract thermal conductivity from unsteady-state test measurements and verify themeasurements using standard test methods

requires the concrete specimen to reach the status of thermalsteady-state for the test temperature [23 24] As the thermalconductivity of concrete is relatively low it takes significantlylong time for a concrete brick to reach the status of thermalsteady-state for a test temperature which is typically a hightemperature For example the soak period of concrete bricks(230mm times 114mm times 65mm) for a test temperature of 600∘Cis four days This relatively long time-period is necessaryto measure the thermal conductivity of concrete followingASTMC1113 [14] If a test time period shorter than the ASTMstandard is to be used it would be necessary to develop amethod to measure thermal conductivity from temperaturevariations in RC under unsteady-states condition

The heat flux is transferred from onemedium to the otherby conduction convection and radiation [25] In generalthe heat flux is transferred from fires to structures by bothconvection and radiation [26 27] The convective heat flux isproportional to the difference in temperatures between theconvection currents due to fire and the structural surface[26 27] The radiant heat flux of fire will be transferredto the surface of a structure by absorptivity ratio of thesurface while the structure emits the radiant heat flux inproportion to its emissivity [28] The transferred heat fluxfrom a fire to the structure is then transferred into thestructure by conduction In fire tests of RC structures thethermal steady-state condition when the temperatures of gasin fire test furnaces and the whole body of test structurebecome equal has been rarely observed [11] However if adecreasing temperature change with time is observed at thesurface of RC structure during a fire test using a standard firethe thermal radiation properties of RC structure might beestimated by considering thermal equilibrium of the surface

In this study we suggest an inverse methodology toidentify thermal conductivity of concrete from timehistory oftemperature variations during fire test The developed meth-odology was applied to find the thermal conductivity of RC

column exposed to a standard fire Fire test of an RC columnwas conducted following ISO-834 standard fire curve [29]The profiles of time history of temperatures at six differentlocations across the RC column section were obtained Thethermal conductivity of RC columns was then determinedby minimizing errors between the numerical solution andthe temperature observations with time The determinedthermal conductivity was compared to that observed fromexperiments followingASTMC1113 [24] for verification of theproposed method Moreover the thermal radiation proper-ties of the RC column were estimated considering the ther-mal equilibrium at the surface of the RC column in fire Theuse of the thermal radiation properties of RC columns as afire safety index is discussed

2 Experimental Methods

In this work two experimental programs were conducted asshown in Figure 1 First fire test of an RC column was pre-pared to obtain the time history of temperature variations ina section of the column The temperature variations will beused to extract the thermal conductivity of the column usinga proposed numerical method To verify the results of thethermal conductivity extracted from RC column testing thesecond experimental program to measure the thermal con-ductivity of concrete bricks was conducted following ASTMC1113 [24]

21 Fire Test of RCColumn AnRCcolumnwas tested to eval-uate the timehistory of temperature variations following ISO-834 standard fire curve [29] for 180 minutes The ISO-834standard fire curve is presented as

119879119892= 345 log (8119905 + 1) + 119879

0[∘C] (1)

The Scientific World Journal 3

C

S S

C

350

616150

50

250

250

1500

4545

350

45

16

114

114

16

45

350

20

15

175 175

1206011

12060122

114 114

16 16114 114

Section ldquoS-Srdquo

Section ldquoC-Crdquo

1206013 times 5 hole to installthermocouples in steel

Thermocouples in concrete

204 = 80 204 = 80

3003 = 900

Figure 2 Details of RC column for fire test and locations of thermocouples All dimensions are in mm

where 119879119892is the gas temperature in fire test furnace at

time 119905 in minutes 1198790is the ambient temperature The RC

column was constructed following the design specificationproposed by the Korean Concrete Institute (KCI) [30] TheRC column had the square section of 350mm by 350mmand the height of 1500mm The column was reinforcedwith 8-12060122mm steel rebars longitudinally and 12060110mm hoopbars placed at 300mm spacing along the column heightGrade 420 steel rebars which have the yield strength of420MPa were used to reinforce the column Details ofcolumn reinforcement are shown in Figure 2 A concretehaving the 28-day characteristic compressive strength of399MPa was used for the column The maximum aggregatesize of 25mm pea gravel was selected for the concrete Theslump of fresh concrete was controlled in the range from180mm to 220mmThemixture proportion of the concrete ispresented in Table 1 The temperatures in concrete followingthe standard fire curve were measured at the center sectionof the column located at 750mm from the bottom of thecolumn Across the section of the column temperature was

Table 1 Concrete mixture (kgm3)

WC Water Cement Sand Gravel055 175 320 827 939

measured at the surface 20mm 40mm 60mm 80mm and175mm from both sides respectively The temperatures inlongitudinal steel rebars were also measured for all 8 rebarsat the center section where the temperatures of concrete weremeasured The locations of the thermocouples are shown inFigure 2The column was placed in fire test furnace as shownin Figure 3 and the time histories of temperatures followingthe standard fire curve for 180 minutes were recorded

22 Thermal Conductivity Measurement of Concrete BricksTo verify the proposed inverse method to identify thermalconductivity a standard testmethod tomeasure thermal con-ductivity of concrete was used as shown in Figure 4 In thisstandardmethod the thermal conductivity of concrete bricks


Figure 3 RC columns in furnace before fire test

Figure 4 Thermal conductivity measurement of concrete bricks(230mm times 114mm times 65mm) using hot wire (platinum resistancethermometer technique) [14]

constructed with the mixture proportion presented in Table 1was measured following ASTM C1113 [24] For the main partof the harness wire platinum wire of 033mm diameter wasused The specimens consisted of three 230mm times 114mm times

65mm straight concrete bricks The thermal conductivityof concrete bricks was measured at nine test temperaturesof 50∘C 100∘C 200∘C 300∘C 400∘C 500∘C 600∘C 670∘Cand 780∘C The thermal conductivity of concrete bricks wasmeasured three times at each test temperature Heating ratefor the furnace temperature was 55∘Chour Soak periods tomake concrete bricks reach thermal steady-state with the testtemperature were different for each test temperature

3 Numerical Methods

In a thermal system without the heat generation one-dimen-sional unsteady-state heat equation considering both specificheat and thermal conductivity as functions of temperaturecan be formulated as

120588

120597 119888 (119879) 119879 (119905 119909)

120597119905

=

120597

120597119909

119896 (119906)

120597119879 (119905 119909)

120597119909

(2)

where 119879(119905 119909) is the temperature of concrete at the distance 119909from the surface at time 119905 and 120588 is the density of concrete con-sidered constant and equal to 2300 kgm3 in all calculations

119888(119879) and 119896(119879) are the specific heat and the thermal con-ductivity of concrete at temperature 119879 Equation (2) can beexpanded as

120588119888 (119879) + 119879 (119905 119909)

119889119888 (119879)

119889119879

120597119879 (119905 119909)

120597119905

= 119896 (119879)

1205972

119879 (119905 119909)

1205971199092

+

119889119896 (119879)

119889119879

(

120597119879 (119905 119909)

120597119909

)

2

(3)

For a finite difference approximation all variables in (3)except the temperature 119879 are linearly transformed to bedimensionless by putting 119909 = 119909119897

lowast = 119905119905lowast 119896(119879) = 119896(119879)119896

lowastand 119888(119879) = 119888(119879)119888

lowast Equation (3) is then rearranged as

120597119879 ( 119909)

120597

=

119896lowast

119905lowast

120588119888lowast

(119897lowast

)2

times

[119896 (119879)(120597

2

119879 ( 119909) 1205971199092

)+(119889119896 (119879) 119889119879)(120597119879 ( 119909) 120597119909)

2

]

[119888 (119879) + 119879 ( 119909) (119889119888 (119879) 119889119879)]

(4)

Putting 119905lowast

= 120588119888lowast

(119897lowast

)2

119896lowast the constant term in the right side

of (4) becomes one A finite difference approximation of (4)will be expressed as

119879(119894+1)119895

= 119860119894119895119879119894(119895+1)

minus 2119879119894119895+ 119879119894(119895minus1)

+ 119861119894119895119879119894(119895+1)

minus 119879119894119895

2

+ 119879119894119895

(5)

where

119860119894119895=

Δ119905

Δ1199092

119896119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

119861119894119895=

Δ119905

Δ1199092

1198961015840

119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

(6)

where 119894 is the number of time steps and 119895 is the number ofdistance steps Δ119905 and Δ119909 are the step sizes for time and dis-tance respectively 119896

119894119895and

1198961015840

119894119895

are the thermal conductivityand the corresponding derivative with respect to temperatureevaluated at the temperature of 119879

119894119895while 119888

119894119895and 119888

1015840

119894119895

are thespecific heat and the corresponding derivative with respectto temperature evaluated at the temperature of 119879

119894119895 Using the

quadratic forms of the thermal conductivity and the specificheat [6] as

119896 (119879) = 120572(

119879

120

)

2

minus 20 (

119879

120

) + 200

20∘C le 119879 le 1200

∘C [W (mK)]

(7)

119888 (119879) = 120573225 + 20 (

119879

120

) minus (

119879

120

)

2

20∘C le 119879 le 1200

∘C [J(kg∘C)]

(8)


0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Surface20mm40mm

60mm80mmCenter

Figure 5 Temperature variations with respect to time in concrete

the coefficients 119860119894119895and 119861

119894119895in (5) can be calculated as

119860119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

288 times 104

minus 2400119879119894119895+ 1198792

119894119895

324 times 104

+ 4800 119879119894119895minus 31198792

119894119895

119861119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

2119879119894119895minus 2400

324 times 104

+ 4800119879119894119895minus 31198792

119894119895

(9)

The normalizing factors 119896lowast and 119888

lowast for the thermal conduc-tivity and the specific heat are selected as those evaluated at20∘C using (7) and (8) respectively While 120573 for the specificheat in (8) is fixed at 40 [6] 120572 for the thermal conductivityin (7) will be changed to find the calculated time history oftemperature variations which gives the minimum error withthe observed time history of temperature variations from theRC column exposed to the standard fire The founded 120572 willbe used to determine the thermal conductivity of concrete asa quadratic function of temperature using (7)

4 Results and Discussion

41Thermal Conductivity of Concrete After finishing the firetest of the RC column for 180 minutes it was confirmed thatthere was no spalling of the RC column The time historiesof temperatures measured in concrete at the two locationsfor the same depth from the concrete surface are averagedand presented in Figure 5 for the depths of 0mm (surface)20mm 40mm 60mm 80mm and 175mm respectivelyFor the time histories of temperatures at 40mm 60mm80mm and 175mm in Figure 5 the temperature increasesstop at the temperature of 100∘C for a while and then con-tinue This can be attributed to the effect of capillary wateron concrete thermal behavior For normal concrete it wouldhave partially saturated capillary poresThis enables concreteto conduct heat rapidly up to the temperature of 100∘CFor temperatures above 100∘C water in the capillary pores

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Corner steelMiddle steel

Figure 6 Temperature variations with respect to time in steel

evaporates and the concrete with empty pores will conductheat more slowly than that with partially saturated pores

The time histories of temperaturesmeasured in longitudi-nal steel bars located at the middle of a side and the corner ofthe square section are averaged respectivelyThe average timehistories of temperatures are presented in Figure 6 Becausethe longitudinal steel bars located at the corner of squaresection take heat flux from two sides the temperature of thecorner steel bars was always higher than the temperature ofsteel bars located at the middle of a side This observationindicates that two-dimensional heat transfer mechanism isnecessary to predict steel temperature located at the cornerof square section However the time history of temperaturein the steel located at the middle of a side is almost identicalto that in concrete at the depth of 60mm from the surface asshown in Figure 7 Therefore the temperatures in steel barslocated at the middle of a side of the square section can beestimated to be equal to concrete temperatures at the samedepth from surface Using the time histories of temperaturesat the six locations a regression surface which gives themaximum 119877

2 value of 0983 was achieved with 120572 value in (7)of 00115 as shown in Figure 8 The corresponding thermalconductivity at 20∘C 119896lowast is calculated as 226W(mK) InFigures 9(a) 9(b) and 9(c) the calculated time histories oftemperature variations using the numerical analysis with 120572 of0008 00115 and 0015 are compared with the observed timehistories of temperature variations respectivelyThe time his-tory of temperature variations in Figure 9(b) is obviously theoptimal temperature variation Amonotonic sequence can beobserved in Figure 9The increase of thermal conductivity ofconcrete enables conducting more heat to the center of thesquare section of RC column The thermal conductivity ofconcrete for RC column with respect to temperature 119896RC(119879)

is then determined using (7) as shown in Figure 10 Thethermal conductivity of concrete bricks 119896CB measured at thetest temperatures 50∘C 100∘C 200∘C 300∘C 400∘C 500∘C


0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Concrete at 40mmConcrete at 60mm


Figure 7 Comparison between temperature evolution in concreteand steel

600∘C 670∘C and 780∘C following ASTM C1113 [24] arealso presented in Figure 10 At each test temperature thethermal conductivity of concrete bricks was measured threetimes and the results were averagedThe thermal conductivitycomputation with respect to temperature of normal concreteproposed by the Euro code [6] (119896code) presented in (7) with120572 of 0008 is plotted in Figure 10 As shown in Figure 10 thestandard thermal conductivity 119896CB is less than 119896RC until thetemperature reaches 600∘C It can also be observed that thedifference between the two measurements gets reduced asthe temperature increases from 200∘C to 600∘C This mightbe attributed to the fact that 119896CB is the thermal conductivityfor concrete with empty pores because water in the capillarypores evaporates during the soak period to enforce thethermal steady-state at test temperature However 119896RC is thethermal conductivity for concrete with partially saturatedcapillary pores which might be more preferable for practicaluse The observations of 119896CB over the temperature of 600∘Cagree well with 119896RC Moreover the thermal conductivity 119896codeproposed by the design code is less than 119896RC Therefore itmight be necessary to consider the effect of partially saturatedcapillary pores on the thermal conductivity of concrete for aconservative estimate of RC membersrsquo thermal behavior infire

42 Radiant Heat Transfer Ratio of Concrete Examining thetemperature difference between the surface of RC columnand ISO-834 standard fire curve shown in Figure 11 it canbe observed that the RC column is still in thermal unsteady-state At the surface of RC column the heat flux is transferredfromfire to the columnby both convection and radiationThethermal equilibrium at the surface is formulated as

119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue

Thermal conductivity at 20∘C klowast (WmmiddotK)

The maximum R2 = 0983 klowast = 226W(mK) (120572 = 00115)

Figure 8 Regression analysis to identify 120572 giving the maximum 1198772

value (ie minimum error)

where 119902119888and 119902119903are the heat flux components due to convec-

tion and radiation respectively 119881 and 119878 are the volume andthe exposure area of the RC column respectively Accord-ing to the amount of transferred heat flux and the surface tem-perature the time rate of increase of temperature 119889119879(119905 0)119889119905

will be determined The amount of heat flux by convectivetransfer is given by

119902119888= ℎ119888(119879119892minus 119879119888) (11)

where ℎ119888is the coefficient of convective heat transfer 119879

119892and

119879119888are the gas temperature in fire test furnace and the surface

temperature of the RC column in Kelvin By assuming thatthe surrounding gas in fire test furnace is a black body andthe RC column is a grey body the amounts of heat flux byradiant transfer are formulated with the thermal propertiesof concrete as

119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)

where 120590 is Stefan Boltzmannrsquos constant of 567 times 10minus8W(m2K4) 120572

119888is the thermal absorptivity of concrete for the

gas in fire test furnace and 120576119888is the thermal emissivity of

concrete toward the gas As indicated in (12) the radiant heattransfer keeps until 1198794

119888

is equal to (120572119888120576119888) 1198794

119892

Therefore when(119879119888119879119892)4 is equal to 120572

119888120576119888 the radiant heat transfer stops We

name the 120572119888120576119888ratio here as the radiant heat transfer ratio

Considering this aspect (119879119888119879119892)4 with time is formulated

in Figure 12 As shown in Figure 12 the change of (119879119888119879119892)4

with time decreases as it tends to converge An exponentialequation can be used to approximate the convergence valueof (119879119888119879119892)4 such as

(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)

Using regression analysis of (13) with the formulation of(119879119888119879119892)4 with time the radiant heat transfer ratio 120572

119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200

Depth from surface (mm)

Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)

Figure 9 Temperature surfaces according to (a) 120572 = 0008 (b) 120572 = 00115 the optimal surface with the thermal conductivity 119896RC at 20∘C of226W(mK) which give the maximum 119877

2 value of 0983 and (c) 120572 = 00150

0587 and 120591 of 592 minutes is determined as shown inFigure 12 Considering that the radiant heat transfer ratio120572119888120576119888represents the ratio of radiant heat flux inflow to outflow

of the RC column the radiant heat transfer ratio can be usedas a reference value to assess fire resistance of RC elementsConcrete with a low the radiant heat transfer ratio 120572

119888120576119888

will be capable of preventing thermal radiant heat transferto the entire structure when exposed to fire It is interestingto note that although the radiant heat transfer ratio cannotbe directly compared with that for solar radiation becauseof the difference in wave lengths of thermal energy in firetest furnace from that of solar energy the ratio determinedin this study (0587) is close to the value of 0568 reportedfor concrete and based on solar absorptivity of concrete05 over the emissivity of concrete as 088 [31] There have

been extensive studies in the literature about the ratio of thesolar absorptivity over emissivity for selecting appropriatecoating materials for spacecraft [28] The regression analysisto find the radiant heat transfer ratio 120572

119888120576119888will be effective

when the time rate of temperature increase 119889119879(119905 0)119889119905 in(10) significantly decreases such as in Figure 13 It is notablethat the radiant heat transfer ratio 120572

119888120576119888is developed due to

the black body assumption of the gas surrounding the RCcolumn in fire test furnace Moreover the thermal radiationof compartment walls making up the fire test furnace isalso neglected [32] Therefore if the information about thecombined emissivity 120601 of gas and compartment walls in testfire furnace is available [32] the radiant heat transfer ratio120572119888120576119888can be divided by 120601 and can be used as a criterion for

concrete for fire safety assessment


08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB

Figure 10 Comparison of thermal conductivity 119896RC (proposedmethod) 119896CB (concrete brick testing) and 119896CODE (Korean code ofpractice) respectively

0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180

Applied temperature (ISO-834 standard fire curve)Concrete surface

Tem

pera

ture

T(∘

C)

Time t (min)

Figure 11 The difference in ISO-834 fire curve and the observedtemperature on the surface of RC column confirming the unsteady-state conditions of the test measurements

5 Conclusions

An inverse methodology to find the thermal conductivityfrom time history of temperature variations in the thermalunsteady-state is proposed A square RC column with crosssection of 350mm by 350mm and the height of 1500mmwastested in order to evaluate the time history of temperaturevariations following ISO-834 standard fire curve Using anumerical solution for thermal equilibrium model in theunsteady-state condition the thermal conductivity of con-crete for the RC column was determined in quadratic formas 119896(119879) = 00115(119879120)

2

minus 20(119879120) + 200 in W(mK)for the temperature range from 20∘C to 1200∘C Thermalconductivity of concrete was conservatively estimated using

0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)

Figure 12 Regression analysis of (119879119888

119879119892

)4 formulation versus time

0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)

Figure 13 Time rate of temperature increase 119889119879(119905 0)119889119905 versustime

the proposed method when compared with the thermal con-ductivity of concrete bricksmeasured usingASTM standardsMoreover another aspect of fire safety assessment was evalu-ated from the difference of the applied temperature (ISO-834standard fire curve) and the surface temperature in fire testfurnaceThe radiant heat transfer ratio representing concreteabsorptivity over emissivity 120572

119888120576119888ratio was determined from

regression analysis of (119879119888119879119892)4 formulation with time It can

be observed that concrete with relatively low radiant heattransfer ratio can protect structures during fire by preventingthermal radiant heat transfer to the structures from fire It issuggested that the radiant heat transfer ratio shall be used asa concrete criterion for fire safety assessment

Nomenclatures

119888(119879) Specific heat at the temperature of 119879ℎ119888 The coefficient of convective heat transfer

119894 The number of time steps


119895 The number of distance steps119896(119879) Thermal conductivity at the

temperature of 119879119902119888 Heat flux components due to

convection119902119903 Heat flux components due to radiation

119878 The exposure area of RC column119888(119879) Specific heat at the temperature of 119879119896(119879) Thermal conductivity at the

temperature of 1198791198790 Ambient temperature

119879119888 The surface temperature of RC column

119879119892 Gas temperature in fire test furnace

119879(119905 119909) Temperature of concrete at the distance119909 from the surface at time 119905

119881 The volume of RC column120572 The variable for the thermal

conductivity in(7)120572119888 Thermal absorptivity of concrete for the

gas in fire test furnace120573 The coefficient for the specific heat in

(8) which is fixed at 40120576119888 Thermal emissivity of concrete toward

the gasΔ119905 The step sizes for timeΔ119909 The step sizes for distance120588 Density of concrete120590 Stefan Boltzmann constant which is

567 times 10minus8W(m2K4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thefirst author would like to acknowledgeNational ResearchFoundation (NRF) Grant funded by the Korean government(MOE) (no 2013R1A1A2062784) The corresponding authorwould like to acknowledge a Grant (code no 101584009 RampD A01)from Cutting-Edge Urban Development Program fundedby the Ministry of land Transport and Maritime Affairs ofKorean Government

References

[1] N Pinoteau P Pimienta T Guillet P Rivillon and S RemondldquoEffect of heating rate on bond failure of rebars into concreteusing polymer adhesives to simulate exposure to firerdquo Interna-tional Journal of Adhesion and Adhesives vol 31 no 8 pp 851ndash861 2011

[2] H Sen Z Liaojun Z Hanyun and L Longzhong ldquoAnalysis ofthermal stress for bent column of concrete filled steel tube inexposure to firerdquo Water Resources and Power vol 30 no 1 pp177ndash179 2012

[3] S J Choi S Kim S Lee R Won and J Won ldquoMix proportionof eco-friendly fireproof high-strength concreterdquo Constructionand Building Materials vol 38 pp 181ndash187 2013

[4] ACI Committee 216 Standard Method for Determining FireResistance of Concrete and Masonry Construction AssembliesAmerican Concrete Institute Detroit Mich USA 1997

[5] Canadian Standards Association Code for the Design of Con-crete Structures for Buildings CAN3-A233-M02 CanadianStandards Association Toronto Canada 2002

[6] CEN DAFT ENV Euro code 2 Design of Concrete Structures1993

[7] Ministry of Land Infrastructures and Transport The legal pro-posal of fire safety of beams and columns made of highstrength concrete Notice of Ministry of Land Infrastructuresand Transport 2008-334 Sejong Republic of Korea 2008

[8] AS3600 Concrete Structures Standards Australia 2009[9] T T Lie and R J Irwin ldquoMethod to calculate the fire resistance

of reinforced concrete columns with rectangular cross sectionrdquoACI Structural Journal vol 90 no 1 pp 52ndash60 1993

[10] Z Huang A Platten and J Roberts ldquoNon-linear finite elementmodel to predict temperature histories within reinforced con-crete in firesrdquo Building and Environment vol 31 no 2 pp 109ndash118 1996

[11] J-T Yu Z-D Lu and Q Xie ldquoNonlinear analysis of SRC col-umns subjected to firerdquo Fire Safety Journal vol 42 no 1 pp1ndash10 2007

[12] Y Xu andBWu ldquoFire resistance of reinforced concrete columnswith L- T- and +-shaped cross-sectionsrdquo Fire Safety Journalvol 44 no 6 pp 869ndash880 2009

[13] V K R Kodur B Yu and M M S Dwaikat ldquoA simplifiedapproach for predicting temperature in reinforced concretemembers exposed to standard firerdquo Fire Safety Journal vol 56pp 39ndash51 2013

[14] M B Dwaikat and V K R Kodur ldquoHydrothermal model forpredicting fire-induced spalling in concrete structural systemsrdquoFire Safety Journal vol 44 no 3 pp 425ndash434 2009

[15] K Kim S Jeon J Kim and S Yang ldquoAn experimental studyon thermal conductivity of concreterdquo Cement and ConcreteResearch vol 33 no 3 pp 363ndash371 2003

[16] S B Tatro ldquoSignificance of tests and properties of concrete andconcrete-makingmaterialsrdquo Tech Rep STP 169D ASTM Inter-national 2006

[17] TD Brown andMY Javaid ldquoThe thermal conductivity of freshconcreterdquoMateriaux et Constructions vol 3 no 6 pp 411ndash4161970

[18] K-Y Shin S-B Kim J-H Kim M Chung and P-S JungldquoThermo-physical properties and transient heat transfer ofconcrete at elevated temperaturesrdquo Nuclear Engineering andDesign vol 212 no 1ndash3 pp 233ndash241 2002

[19] T S Yun Y J Jeong T-S Han and K-S Youm ldquoEvaluation ofthermal conductivity for thermally insulated concretesrdquo Energyand Buildings vol 61 pp 125ndash132 2013

[20] CEN DAFT ENV Euro code 3 Design of Steel Structures 1993[21] V K R Kodur and L Phan ldquoCritical factors governing the

fire performance of high strength concrete systemsrdquo Fire SafetyJournal vol 42 no 6-7 pp 482ndash488 2007

[22] H K Kim J H Jeon and H K Lee ldquoWorkability andmechan-ical acoustic and thermal properties of lightweight aggregateconcrete with a high volume of entrained airrdquo Construction andBuilding Materials vol 29 pp 193ndash200 2012

[23] ASTM C177 Standard Test Method for Steady-State Heat FluxMeasurements andThermal Transmission Properties byMeans ofthe Guarded-Hot-Plate Apparatus American Society for Testingand Materials 2010


[24] ldquoStandard test method for thermal conductivity of refractoriesby hot wire (platinum resistance thermometer technique)rdquoASTMC1113 American Society for Testing andMaterials 2009

[25] T L Bergman F P Incropera A S Lavine and D P DeWittFundamentals of Heat and Mass Transfer John Wiley amp SonsNew York NY USA 7th edition 2011

[26] H E Anderson ldquoHeat transfer and fire spreadrdquo USDA ForestService Research Paper INT-69 International Forest andRangeExperiment Station Forest Service US Department of Agri-culture Ogden Utah USA 1969

[27] J I Ghojel ldquoA new approach to modeling heat transfer incompartment firesrdquo Fire Safety Journal vol 31 no 3 pp 227ndash237 1998

[28] M BWong and J I Ghojel ldquoSensitivity analysis of heat transferformulations for insulated structural steel componentsrdquo FireSafety Journal vol 38 no 2 pp 187ndash201 2003

[29] ldquoISO-834 Fire resistance test-elements of building construc-tionrdquo International Standard ISO384 Geneva Switzerland1975

[30] Korean Concrete Institute Standard Specification of ConcreteMinistry of Land Infrastructures and Transport 2009

[31] M M Elbadry and A Ghali ldquoTemperature variations in con-crete bridgesrdquo Journal of Structural Engineering vol 109 no 10pp 2355ndash2374 1983

[32] B Boslashhm and S Hadvig ldquoCalculation of heat transfer in fire testfurnaceswith specific interest in the exposure of steel structuresA technical noterdquo Fire Safety Journal vol 5 no 3-4 pp 281ndash2861983

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Numerical analysis

Verification


No

Yes

Thermal conductivitymeasurement ofconcrete bricks

by hot wire(ASTM C 1113)

Thermalconductivity k0RC

Thermalconductivity kfRC

kiRC

Temperatureprofiles TO(t x)

Temperature profiles TN(t x)

Thermalconductivity kCB

max[R2(TN TO)]


fire test of RC column

Figure 1 Layout of the proposed research program to extract thermal conductivity from unsteady-state test measurements and verify themeasurements using standard test methods

requires the concrete specimen to reach the status of thermalsteady-state for the test temperature [23 24] As the thermalconductivity of concrete is relatively low it takes significantlylong time for a concrete brick to reach the status of thermalsteady-state for a test temperature which is typically a hightemperature For example the soak period of concrete bricks(230mm times 114mm times 65mm) for a test temperature of 600∘Cis four days This relatively long time-period is necessaryto measure the thermal conductivity of concrete followingASTMC1113 [14] If a test time period shorter than the ASTMstandard is to be used it would be necessary to develop amethod to measure thermal conductivity from temperaturevariations in RC under unsteady-states condition

The heat flux is transferred from onemedium to the otherby conduction convection and radiation [25] In generalthe heat flux is transferred from fires to structures by bothconvection and radiation [26 27] The convective heat flux isproportional to the difference in temperatures between theconvection currents due to fire and the structural surface[26 27] The radiant heat flux of fire will be transferredto the surface of a structure by absorptivity ratio of thesurface while the structure emits the radiant heat flux inproportion to its emissivity [28] The transferred heat fluxfrom a fire to the structure is then transferred into thestructure by conduction In fire tests of RC structures thethermal steady-state condition when the temperatures of gasin fire test furnaces and the whole body of test structurebecome equal has been rarely observed [11] However if adecreasing temperature change with time is observed at thesurface of RC structure during a fire test using a standard firethe thermal radiation properties of RC structure might beestimated by considering thermal equilibrium of the surface

In this study we suggest an inverse methodology toidentify thermal conductivity of concrete from timehistory oftemperature variations during fire test The developed meth-odology was applied to find the thermal conductivity of RC

column exposed to a standard fire Fire test of an RC columnwas conducted following ISO-834 standard fire curve [29]The profiles of time history of temperatures at six differentlocations across the RC column section were obtained Thethermal conductivity of RC columns was then determinedby minimizing errors between the numerical solution andthe temperature observations with time The determinedthermal conductivity was compared to that observed fromexperiments followingASTMC1113 [24] for verification of theproposed method Moreover the thermal radiation proper-ties of the RC column were estimated considering the ther-mal equilibrium at the surface of the RC column in fire Theuse of the thermal radiation properties of RC columns as afire safety index is discussed

2 Experimental Methods

In this work two experimental programs were conducted asshown in Figure 1 First fire test of an RC column was pre-pared to obtain the time history of temperature variations ina section of the column The temperature variations will beused to extract the thermal conductivity of the column usinga proposed numerical method To verify the results of thethermal conductivity extracted from RC column testing thesecond experimental program to measure the thermal con-ductivity of concrete bricks was conducted following ASTMC1113 [24]

21 Fire Test of RCColumn AnRCcolumnwas tested to eval-uate the timehistory of temperature variations following ISO-834 standard fire curve [29] for 180 minutes The ISO-834standard fire curve is presented as

119879119892= 345 log (8119905 + 1) + 119879

0[∘C] (1)


C

S S

C

350

616150

50

250

250

1500

4545

350

45

16

114

114

16

45

350

20

15

175 175

1206011

12060122

114 114

16 16114 114





204 = 80 204 = 80

3003 = 900














3 Numerical Methods


120588

120597 119888 (119879) 119879 (119905 119909)

120597119905

=

120597

120597119909

119896 (119906)

120597119879 (119905 119909)

120597119909

(2)



120588119888 (119879) + 119879 (119905 119909)

119889119888 (119879)

119889119879

120597119879 (119905 119909)

120597119905

= 119896 (119879)

1205972

119879 (119905 119909)

1205971199092

+

119889119896 (119879)

119889119879

(

120597119879 (119905 119909)

120597119909

)

2

(3)


lowast = 119905119905lowast 119896(119879) = 119896(119879)119896

lowastand 119888(119879) = 119888(119879)119888


120597119879 ( 119909)

120597

=

119896lowast

119905lowast

120588119888lowast

(119897lowast

)2

times

[119896 (119879)(120597

2

119879 ( 119909) 1205971199092

)+(119889119896 (119879) 119889119879)(120597119879 ( 119909) 120597119909)

2

]

[119888 (119879) + 119879 ( 119909) (119889119888 (119879) 119889119879)]

(4)


= 120588119888lowast

(119897lowast

)2



119879(119894+1)119895

= 119860119894119895119879119894(119895+1)

minus 2119879119894119895+ 119879119894(119895minus1)

+ 119861119894119895119879119894(119895+1)

minus 119879119894119895

2

+ 119879119894119895

(5)

where

119860119894119895=

Δ119905

Δ1199092

119896119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

119861119894119895=

Δ119905

Δ1199092

1198961015840

119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

(6)


119894119895and

1198961015840

119894119895


119894119895while 119888

119894119895and 119888

1015840

119894119895


119894119895 Using the


119896 (119879) = 120572(

119879

120

)

2

minus 20 (

119879

120

) + 200

20∘C le 119879 le 1200

∘C [W (mK)]

(7)

119888 (119879) = 120573225 + 20 (

119879

120

) minus (

119879

120

)

2

20∘C le 119879 le 1200

∘C [J(kg∘C)]

(8)


0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Surface20mm40mm

60mm80mmCenter




119860119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

288 times 104

minus 2400119879119894119895+ 1198792

119894119895

324 times 104

+ 4800 119879119894119895minus 31198792

119894119895

119861119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

2119879119894119895minus 2400

324 times 104

+ 4800119879119894119895minus 31198792

119894119895

(9)





0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)








0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)






119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue








119902119888= ℎ119888(119879119892minus 119879119888) (11)


119892and



119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)





119888

is equal to (120572119888120576119888) 1198794

119892







(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)


119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C

S S

C

350

616150

50

250

250

1500

4545

350

45

16

114

114

16

45

350

20

15

175 175

1206011

12060122

114 114

16 16114 114





204 = 80 204 = 80

3003 = 900














3 Numerical Methods


120588

120597 119888 (119879) 119879 (119905 119909)

120597119905

=

120597

120597119909

119896 (119906)

120597119879 (119905 119909)

120597119909

(2)



120588119888 (119879) + 119879 (119905 119909)

119889119888 (119879)

119889119879

120597119879 (119905 119909)

120597119905

= 119896 (119879)

1205972

119879 (119905 119909)

1205971199092

+

119889119896 (119879)

119889119879

(

120597119879 (119905 119909)

120597119909

)

2

(3)


lowast = 119905119905lowast 119896(119879) = 119896(119879)119896

lowastand 119888(119879) = 119888(119879)119888


120597119879 ( 119909)

120597

=

119896lowast

119905lowast

120588119888lowast

(119897lowast

)2

times

[119896 (119879)(120597

2

119879 ( 119909) 1205971199092

)+(119889119896 (119879) 119889119879)(120597119879 ( 119909) 120597119909)

2

]

[119888 (119879) + 119879 ( 119909) (119889119888 (119879) 119889119879)]

(4)


= 120588119888lowast

(119897lowast

)2



119879(119894+1)119895

= 119860119894119895119879119894(119895+1)

minus 2119879119894119895+ 119879119894(119895minus1)

+ 119861119894119895119879119894(119895+1)

minus 119879119894119895

2

+ 119879119894119895

(5)

where

119860119894119895=

Δ119905

Δ1199092

119896119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

119861119894119895=

Δ119905

Δ1199092

1198961015840

119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

(6)


119894119895and

1198961015840

119894119895


119894119895while 119888

119894119895and 119888

1015840

119894119895


119894119895 Using the


119896 (119879) = 120572(

119879

120

)

2

minus 20 (

119879

120

) + 200

20∘C le 119879 le 1200

∘C [W (mK)]

(7)

119888 (119879) = 120573225 + 20 (

119879

120

) minus (

119879

120

)

2

20∘C le 119879 le 1200

∘C [J(kg∘C)]

(8)


0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Surface20mm40mm

60mm80mmCenter




119860119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

288 times 104

minus 2400119879119894119895+ 1198792

119894119895

324 times 104

+ 4800 119879119894119895minus 31198792

119894119895

119861119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

2119879119894119895minus 2400

324 times 104

+ 4800119879119894119895minus 31198792

119894119895

(9)





0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)








0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)






119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue








119902119888= ℎ119888(119879119892minus 119879119888) (11)


119892and



119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)





119888

is equal to (120572119888120576119888) 1198794

119892







(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)


119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














3 Numerical Methods


120588

120597 119888 (119879) 119879 (119905 119909)

120597119905

=

120597

120597119909

119896 (119906)

120597119879 (119905 119909)

120597119909

(2)



120588119888 (119879) + 119879 (119905 119909)

119889119888 (119879)

119889119879

120597119879 (119905 119909)

120597119905

= 119896 (119879)

1205972

119879 (119905 119909)

1205971199092

+

119889119896 (119879)

119889119879

(

120597119879 (119905 119909)

120597119909

)

2

(3)


lowast = 119905119905lowast 119896(119879) = 119896(119879)119896

lowastand 119888(119879) = 119888(119879)119888


120597119879 ( 119909)

120597

=

119896lowast

119905lowast

120588119888lowast

(119897lowast

)2

times

[119896 (119879)(120597

2

119879 ( 119909) 1205971199092

)+(119889119896 (119879) 119889119879)(120597119879 ( 119909) 120597119909)

2

]

[119888 (119879) + 119879 ( 119909) (119889119888 (119879) 119889119879)]

(4)


= 120588119888lowast

(119897lowast

)2



119879(119894+1)119895

= 119860119894119895119879119894(119895+1)

minus 2119879119894119895+ 119879119894(119895minus1)

+ 119861119894119895119879119894(119895+1)

minus 119879119894119895

2

+ 119879119894119895

(5)

where

119860119894119895=

Δ119905

Δ1199092

119896119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

119861119894119895=

Δ119905

Δ1199092

1198961015840

119894119895

119888119894119895+ 1198881015840

119894119895

119906119894119895

(6)


119894119895and

1198961015840

119894119895


119894119895while 119888

119894119895and 119888

1015840

119894119895


119894119895 Using the


119896 (119879) = 120572(

119879

120

)

2

minus 20 (

119879

120

) + 200

20∘C le 119879 le 1200

∘C [W (mK)]

(7)

119888 (119879) = 120573225 + 20 (

119879

120

) minus (

119879

120

)

2

20∘C le 119879 le 1200

∘C [J(kg∘C)]

(8)


0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Surface20mm40mm

60mm80mmCenter




119860119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

288 times 104

minus 2400119879119894119895+ 1198792

119894119895

324 times 104

+ 4800 119879119894119895minus 31198792

119894119895

119861119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

2119879119894119895minus 2400

324 times 104

+ 4800119879119894119895minus 31198792

119894119895

(9)





0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)








0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)






119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue








119902119888= ℎ119888(119879119892minus 119879119888) (11)


119892and



119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)





119888

is equal to (120572119888120576119888) 1198794

119892







(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)


119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)

Surface20mm40mm

60mm80mmCenter




119860119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

288 times 104

minus 2400119879119894119895+ 1198792

119894119895

324 times 104

+ 4800 119879119894119895minus 31198792

119894119895

119861119894119895=

120572

120573

Δ119905

Δ1199092

119888lowast

119896lowast

2119879119894119895minus 2400

324 times 104

+ 4800119879119894119895minus 31198792

119894119895

(9)





0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)








0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)






119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue








119902119888= ℎ119888(119879119892minus 119879119888) (11)


119892and



119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)





119888

is equal to (120572119888120576119888) 1198794

119892







(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)


119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160 180Time (min)

Tem

pera

ture

(∘C)






119902119888+ 119902119903= 120588119888 (119879) + 119879 (119905 0)

119889119888 (119879)

119889119879

119889119879 (119905 0)

119889119905

(

119881

119878

) (10)

094

095

096

097

098

099

1

15 17 19 21 23 25 27 29

R2-v

alue








119902119888= ℎ119888(119879119892minus 119879119888) (11)


119892and



119902119903= 120590 (120572

1198881198794

119892

minus 1205761198881198794

119888

) (12)





119888

is equal to (120572119888120576119888) 1198794

119892







(

119879119888

119879119892

)

4

=

120572119888

120576119888

(1 minus 119890119905120591

) (13)


119888120576119888of


0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(a) 120572 = 0008 1198772 = 0925 119896lowast = 1574W(mK)

Time (min) 0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Tem

pera

ture

(∘C)

(b) 120572 = 00115 1198772 = 0984 119896lowast = 226W(mK)

0 25 50 75 100 125 150 175060

120180

0

200

400

600

800

1000

1200


Time (min)

Tem

pera

ture

(∘C)

(c) 120572 = 00150 1198772 = 0946 119896lowast = 295W(mK)


2 value of 0983 and (c) 120572 = 00150



119888120576119888









08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










08

1

12

14

16

18

2

22

24

0 200 400 600 800 1000 1200Temperature u (∘C)

Ther

mal

cond

uctiv

ityK

(Wm

middotK)

kRC (T) = 00115(T120)2 minus 023(T120) + 23

kcode = 0008(T120)2 minus 016(T120) + 16

kCB


0100200300400500600700800900

10001100

0 20 40 60 80 100 120 140 160 180


Tem

pera

ture

T(∘

C)

Time t (min)


5 Conclusions


2


0

01

02

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180Time t (min)

(T cTg)4

120572c120576c

= 0587

(TcTg

)4

= 0587(1 minus et592)


119879119892


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180Time t (min)

dTcdt(∘

Cmin

)






Nomenclatures




















Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Extracting Concrete Thermal Characteristics from Temperature … · 2019. 7. 31. · Research Article Extracting Concrete Thermal Characteristics from Temperature