Research Article Effective Thermal Conductivity of Open Cell …downloads.hindawi.com/journals/amse/2013/125267.pdf · 2019-07-31 · Research Article Effective Thermal Conductivity

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 125267 7 pageshttpdxdoiorg1011552013125267

Research ArticleEffective Thermal Conductivity of Open CellPolyurethane Foam Based on the Fractal Theory

Kan Ankang and Han Houde

Merchant Marine College Shanghai Maritime University Shanghai 201306 China

Correspondence should be addressed to Kan Ankang ankang0537126com

Received 6 March 2013 Revised 13 September 2013 Accepted 19 November 2013

Academic Editor Martha Guerrero

Copyright copy 2013 K Ankang and H Houde This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Based on the fractal theory the geometric structure inside an open cell polyurethane foam which is widely used as adiabaticmaterial is illustrated A simplified cell fractal model is created In the model the method of calculating the equivalent thermalconductivity of the porous foam is described and the fractal dimension is calculated The mathematical formulas for the fractalequivalent thermal conductivity combined with gas and solid phase for heat radiation equivalent thermal conductivity and forthe total thermal conductivity are deduced However the total effective heat flux is the summation of the heat conduction by thesolid phase and the gas in pores the radiation and the convection between gas and solid phase Fractal mathematical equation ofeffective thermal conductivity is derived with fractal dimension and vacancy porosity in the cell body The calculated results havegood agreement with the experimental data and the difference is less than 5 The main influencing factors are summarized Theresearch work is useful for the enhancement of adiabatic performance of foam materials and development of new materials

1 Introduction

Because of distinguished adiabatic performance open cellpolyurethane foam with small density and low thermalconductivity (0018sim0032200W(msdotK)) is applied in variousfields such as building food cold storage and refrigeratingcargo transportation for heat conservation purpose Theirregular geometrical construction of open cell polyurethanefoam makes it irregular in physical properties And it makesthe theoretic research difficult especially in the accuratethermal performance Actually adiabatic materialsrsquo thermalconductivity can be measured by the plate thermal guardeddevice but it is inconvenient for science research and thepolyurethane foam development It has been a considerableresearch project for thermophysics engineering and hylologyto analyze and estimate the effective thermal conductivityof porous medium for a long time [1] While the foamporous medium material is taken as the research project tocalculate the thermal conductivity it is always supposed as theconnecting virtual medium in large-scale space that is ldquotheaverage volumerdquo in geometric distribution Whitaker [2 3]andWhitaker andChou [4] took the virtual ldquoaverage volumerdquo

method to describe the procedure of the heat and masstransfer inside the porous medium The view was taken thatthe porous medium was combined with solid phase materialliquid and gas The gas phase contains dry air and vaporSupposed that all the phases in porousmediumwere balancesof heat and the poresrsquo dimensions were according to ldquoaveragevolumerdquo a dozen of variables involved in the mathematicalformula Yu et al [5 6] also investigated experimentallytheir coupling and diffusing physical model and derived therelative mathematical formula

There are two main methods to estimate the thermalconductivity of porous medium materials nowadays Oneis that the thermal conductivity is illuminated as the com-plicated mathematical functions by the pore proportionand the microstructure parameters Lagarde [7] derived theequivalent effective thermal conductivity 120582

119890function of the

saturation porous materials The equivalent effective thermalconductivity is obtain from

120582

119890= 120593120582

119891+ (1 minus 120593) 120582

119904 (1)

where 120582119891is the fluid phase thermal conductivity (Wmminus1Kminus1)

and 120582

119904is that of the solid phase (Wmminus1Kminus1)

2 Advances in Materials Science and Engineering

Here the supposition was made that heat fluxes throughfluid in the pore and by the solid phase of the porous bodywere individual and took place simultaneously However theheat transfer was also proceeding between the fluid phase andthe solid phase simultaneously So the real model was morecomplicated than the expression in (1) SoWilliams andDawe[8] developed the function as follows

120582

119890= 120576 [120576

1120582

119891+ (1 minus 120576

1) 120582

119904] +

(1 minus 120576)

120576

2120582

119891+ (1 minus 120576

2) 120582

119904

(2)

where 120593 = 120576120576

1+ (1 minus 120576)120576

2 The factor 120576 is ratio that the

heat flux transfers along with the temperature grads to thetotal heat flux while 120576

1is the factor for inexistence of the

solid-solid connection and 120576

2is for the existence of solid-

solid connection and solid-fluid connectionActually in the microspace structure of porous medium

materials the existence of ideal even distribution of the poresin the porous body is impossible So there is a big errorbetween the ideal model mentioned above and the real bodyThe available ideal models and empirical equations for foamthermal insulating materials are only generally related topores proportion which is the approximate reflection of theapparent thermal conductivity in the macrospace But for thereal foam material whose pore distribution is irregular theavailable idealmodels and empirical equations are not relativeto the microstructure and cannot expose the actual heat andmass transfer procedure and the distribution of temperatureand humidity As a result the big error is presence in theresearch work

The other method involves fractal theory Fractal theoryintroduced into the estimation and research work to calculatethe thermal conductivity of porous foam materials is a newway for the theory development on thermal performanceof porous medium materials Fractal theory was first putforward in 1975 byMandelbrotwhowas a professor fromHar-vard University in USA Some experts such as Pitchumani[9] Yu and Li [5] andMa et al [6] have done deep researcheson the effective thermal conductivity of granular porousmedium by fractal theory and have created correspondingmathematical equations Based on the fractal theoryThovertet al [10] Zhang et al [11] and so ondeveloped the theoreticalmodels for calculation of the effective thermal conductivityof irregular porous medium According to the concept ofSierpinski carpet model Pitchumani and Ramakrishnan [1213] created the pore distribution theoretical model but themodel and mathematical equations were very complicated infractal dimensionMa et al [6] built a mathematical model ofeffective thermal conductivity for porous medium accordingto fractal theory which manifested that the thermal conduc-tivity of porous medium was a function of the pore ratio thearea ratio the thermal conductivity ratio in components andthe thermal contact resistance all together It had nothing todowith empirical constants and less parameters and simple tocalculate in the formula However different porousmedia arenot the same with each other in internal fractal essence Andit is also difficult to estimate the thermal contact resistanceof porous medium in practice The universality of the modelstill needs to be further verifiedThovert et al [10] illuminated

the fractal porous medium by the percolation mathemat-ical model and did the solution by geometrical iterationWhereafter Adler Thovert and Thompson added empiricalconstants gotten by experiments into Adlerrsquos function Andthe function is commonly described as

120582 = 120582

0120601

120572

(3)

where 120582

0is the thermal conductivity of the liquid in

the porous material pores (Wmminus1Kminus1) And the super-script 120572 here is defined as

120572 =

119889

119904+ 119863

119891(2 minus 119889

119904)

119889

119904(3 minus 119863

119891)

(4)

where the fractal dimension factor 119863119891= 25ndash285 and spec-

tral dimension 119889

119904is used to describe the procedure of the

percolation in the poresYangsheng [14] based on the percolation theory created

the relationship between the pore diameters of variousgrain material and the thermal conductivity But the poreporosity the fractal dimension and the microstructure arenot involved in the model Pitchumani and Yao [15] cal-culated the transverse and longitudinal fractal dimensionsto illuminate the microstructure fibrous materials and thethermal conductivity was derived based on the conventionalheat transfer theory But the model only serves some certainfibrous porous materials well

So it is considerably difficult and also unpractical to builda theoretically mathematical model of effective thermal con-ductivity that is universal for porous medium Consequentlycrating amathematicalmodel of thermal conductivity for onecertain porous medium that reflects its structure characteris-tic in internality is an important developing direction for theporous medium research work

2 Microstructures of the Open CellPolyurethane Foam and Fractal Description

21 Microstructures Theopen cell polyurethane is composedof solid substrates and cells By the effect of foaming agentand cell opening agent great deals of cells are generatedand continuously distributed inside the material The cellsconnect with each other side by side and the gas in porescan flow freely through one cell to another That is reallyan advantage to expulse foaming agent and vapors thatembraced in pores Meanwhile the gas in pores can easilybe expelled as the thorough connection of cells The solidsubstrate of open cell polyurethane has the certain intensityto support thematerial and to prevent the collapse in vacuumstate So the polyurethane foam with open cell structure canbe used widely as the core material of the vacuum insulationpanel

Microstructures of the open cell polyurethane consistingof skeleton of solid substrate (the white part in the picture)and cells (the black part in the picture) is shown in Figure 1(taken by electron microscopy) Cells are generally cubedstructurally in the space and continuously distributed insection plane and the dimensions of the apertures are in

Advances in Materials Science and Engineering 3

Figure 1 Microstructure section of the open cell polyurethane foam (magnified 500 and 200 times)

range of 140ndash220120583m and the length of average skeleton is125 120583mThe cellsrsquo dimensions are various and the distributionis random and irregular

22 Fractal Description Fractal theory since it was bornhas attracted lots of scientistsrsquo interest because of its uniqueadvantages of research in irregular and complicated objectsgeometrically and success in dealing with many problems ofgeometry physics geology hylology and so on Meanwhilethe various problems in the scientific subjects also enhancedthe development of the fractal theory Fractal theory isan effective approach to describe nonlinear phenomena innature complicated structures in geometry and internalobjects and spatial distribution Fractal theory firstly com-mitted research on nonlinear complex systems and analyzedthe inner laws from the investigated subjects that were notsimplified and abstract That is essential distinction betweenfractal theory and linear way Two subjects can be treatedas the self-similarity while the fractal dimension valuesare equal according to the fractal theory Various fractalmodels have been built for porousmediamaterials by expertsand researchers and the famous models such as Sierpinskicarpet model Menger sponge model and Koch curve modelare adopted by many researchers However almost porousmedia materials in nature are not the same with the modelsmentioned above They are not strict similarity but similar inmathematical calculation

According to fractal theory it is a self-similar scalingrelationship between metric measure of objects 120575 and physi-cal quantity 119873(120575) existed in 119863

119891dimension Euclidean space

including area and volume or length of a porous fractal [16]

119873(120575) prop 120575

119863119891 (5)

For one fractal body the fractal dimension value 119863119891is in

the range from 2 to 3 But for the microstructure open cellpolyurethane the diameters of the pores are different Thestructure is irregular and the distribution is random For theopen cell polyurethane foam the largest pore size of cells is119863max = 220120583m and the smallest one is 119863min = 140 120583m andsupposing the measure length 120575 for the pace 119863max rarr 119863minthe cell volume V(120575) can be described in the following

119881 (120575) prop 120575

119863119891 (6)

138

143

148

153

158

163

44 48 52 56 6

lnV(120575)

ln 120575

Df = 2621

Figure 2 Fractal dimension calculation for cell body of the opencell polyurethane foam

Based on fractal theory the cell distribution has self-similarity statistically for the open cell polyurethane foamEquation (6) can be replaced by the following

119881 (120575) = 119862120575

119863119891 (7)

where C is constant And taking the logarithm to (7) (8) canbe gotten as

ln119881 (120575) = ln119862 + 119863

119891ln 120575 (8)

According to Sierpinski carpetrsquos random fractal methodFigure 1 is fractal calculated and the result was shown inFigure 2 That is the open cell polyurethane foam volume inthis research has the fractal feature and the fractal dimensionvalue is 119863

119891= 2621 for the sample

However the structure in the porous medium is irregularand the distribution of the pore is also random The physicalquantity 119873(120575) the poresrsquo quantity has the relationshipwith 120575 and the pore diameter D So (5) can be rewritten as

119873(120575 gt 119863) = (

119863max119863

)

119863119891

(9)

or

119873

1015840

(120575 gt 119863min) = (

119863max119863min

)

119863119891

(10)


d

d

2

22

2

1

L

L minus 2d

Figure 3 Simplified structure model of the open cell polyurethane foam

Taking differential coefficient to (9) then

119889119873 = minus119863

119891119863

119863119891

max119863minus119863119891minus1

119889119863(11)

So combining with (10) and (12) can be gotten as

minus

119889119873

119873

1015840= 119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863 (12)

Here the pore distribution probability function 119891(119863) =

119863

119891119863

119863119891

min119863minus(119863119891+1) can be rewritten as

int

infin

minusinfin

119891 (119863) 119889119863 = int

119863max

119863min

119891 (119863) 119889119863 = 1 minus (

119863min119863max

)

119863119891

(13)

The fractal effective diameter L of the pores in the opencell polyurethane can be calculated according to the poredistribution probability function

119871 = int

119863max

119863min

119863119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863

=

119863

119891

119863

119891minus 1

119863min [1 minus (119863min119863max

)

119863119891minus1

]

(14)

Based on the inner structure of the open cell polyurethaneform we suppose that cells are cubed and well distributed asin Figure 3

3 The Equivalent Thermal Conductivity ofFractal Model

The equivalent thermal conductivity 120582119890of the porous open

cell medium materials is the function of the variablethermal conductivities of the phases the inner structureand the pores distribution [17] So the equivalent thermalconductivity 120582

119890can be illuminated in the following

120582

119890= 119891 (sum120582

119894 120593 119863

119891) (15)

R1

R2

R2

R2

R2

R3

R3

R4

Figure 4 Thermal network sketch

where 120582119894is thermal conductivity of the phase i in the porous

mediummaterials For the solid phase the conductivity is 120582119904

while 120582119892for the gas in the pores 120593 is the porosity of the

average volumeThe mathematic model for the open cell polyurethane

is developed on the basis of (15) in this paper Neglectingthe effect of heat radiation in cells and gas heat convectionwe conclude that the heat transfer in one cell of open cellpolyurethane form is only influent by adjacent cells For onecell we suppose that the structure is regular prism the fractaldiameter L the height is mentioned above in (14) while thesolid substrates height d just as in Figure 3 So the whole heattransfer procedure in the cell can be analyzed as the electricitytransfer in the electrocircuit Suppose that heat current flowfrom top to underside through the cell body then thermalresistance of the cell mainly consists of four parts

119877

1is thermal resistance of vertical pillar 1 119877

2level pillar

2 1198773gas among level pillars and 119877

4gas in cavity

The thermal resistance simplifiedmodel can be describedas in Figure 4

According to interrelated heat transfer knowledge we caneasily get that

119877

1=

119871

120582

119904119889

2

119877

2=

2119889

120582

119904119889 (119871 minus 2119889)

=

2

120582

119904(119871 minus 2119889)


119877

3=

(119871 minus 2119889)

120582

119892(119871 minus 2119889) 1198892

=

2

120582

119892119889

119877

4=

4119871

120582

119892(119871 minus 2119889)

2

119877total =4119871

120582

119890119871

2=

4

120582

119890119871

(16)

where 119877total is entire thermal resistance 120582119904is thermal con-

ductivity of foamrsquos skeleton 120582119892is thermal conductivity of gas

in cells 120582119890foam is effective thermal conductivity of the form

From the analysis above we can deduce that

119877total =119877

1119877

4(2119877

2+ 119877

3)

2119877

1119877

4+ (2119877

2+ 119877

3) (119877

1+ 119877

4)

(17)

From (3) and (17) (18) can be easily gotten

120582

119890foam =

4120582

119904119889

2

119871

2+

120582

119892(119871 minus 2119889)

2

119871

2+

4120582

119892120582

119904(119871 minus 2119889) 119889

119871 (2120582

119892119889 + 119871120582

119904minus 2120582

119904119889)

(18)

where 120582

119890foam in (18) is the effective thermal conductivitywhen there is static gas in pores of the open cell polyurethane

The conception of porosity 120593 for porous polyurethanewould be introduced here Generally it is the ratio of thesummation of the vacancy volume 119881

119894to the whole material

block volume 119881 With the calculating methods by the fractaltheory the porosity 120593 can be easily illuminated as [18]

120593 =

sum

119899

119894=1119881

119894

119881

= (1 minus

2119889

119871

)

119863119891

(19)

Combining (18) with (19) the effective thermal conductivitywill be gotten

120582

119890foam = 120582

119904(1 minus 120593

1119863119891)

2

+ 120582

119892120593

2119863119891

+ 2

120582

119892120582

119904(1 minus 120593

1119863119891) 120593

1119863119891

120582

119892(1 minus 120593

1119863119891) + 120582

119904120593

1119863119891

(20)

From (20) we can conclude that the effective thermalconductivity of the open cell polyurethane form has relation-shipwith the phases of the cell body and the fractal dimensionand the cell structure that is the porosity

The thermal conductivity would decrease with fractaldimensionrsquos increase of cells volume and increase of poreporosity and that is in accordance with heat conductingperformance The bigger the fractal dimension and porosityare the less the solid substrates are and the worse the heatconducting property is

4 The Effective Thermal Conductivity ofThermal Radiation

Heat radiation is an important factor for the open cellpolyurethane foam It can be treated as a gray-body medium

to estimate the radiation heat flow in cells [10] So the rate ofradiation heat flow for a cell is

119902

119903= minus

4120590 (119879

4

1minus 119879

4

2)

3120573119871

(21)

where 120590 is Stefn-Boltzmann constant 120590 = 56697 times

10

minus8W(K4sdotm2) 120573 is radiation extinction coefficient forporous medium and 119879

1and 119879

2are separately heat flowrsquos

temperature of entrance and exitSo we can get the equivalent radiation thermal

conductivity 120582119890119903

for a porous medium

120582

119890119903=

4120590 (119879

2

1+ 119879

2

2) (119879

1+ 119879

2)

3120573

(22)

5 The Comparison between Results ofTheoretical Calculation and Experiment

The entire equivalent thermal conductivity 120582119890can be

obtained in (23) on the condition of integrating the heatconducting and radiation-conduction heat transfer together

120582

119890= 120582

119890foam + 120582

119890119903 (23)

The certain open cell polyurethane foam above is selectedas a sample to test in experiments and its thermal conductiv-ity of solid substrates is 120582

119904= 05832W(msdotK) the thermal

conductivity of gas in pore is 120582119892= 00229W(msdotK) and the

decay coefficient tested is 120573 = 445mminus1 The measurementway to test thermal conductivity of the sample is heat guardedplate method And the standard of test refers to GBT3399-2009 The results are collected in Table 1

6 Conclusion

It can be found from Table 1 that there is little differencebetween the results calculated by the theoretical modelpresent above and the experimental ones Conclusions fromthe research work are as follows

There is a good consistency between experimental andtheoretical calculations presented in this paper Error is lessthan 5 Especially when taking the open cell polyurethanefoam as the core of vacuum insulation panels the gasthermal conductivity in (18) can be ignored and calculationssimplified and more accurate results can be obtained

The effective thermal conductivity of the open cellpolyurethane foam has a relationship with the material prop-erties inner microstructure and the service environmentaltemperature Thermal conductivity during heat conductionin entire effective thermal conductivity is predominant innormal temperature while the effective thermal conductivityduring radiation is a little undulating but the value is notprimary So increasing the porosity of the body can enhanceits entire heat insulating property on conditioning that itsstructural strength is enough for the open cell polyurethanefoam

The research work has manifestly established a connec-tion between a thermophysical property and the internal


Table 1 The comparative table between results of calculation and experiment

Sample Densitykgm3

Porosity

Fractaldimension

Averagetemperature K

120582W(msdotK) Difference

120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32

microstructure of porous media by fractal theory The the-oretical work would be an important reference in enhancingheat insulating of porousmedia and useful in developing newmaterial for environmental protection and energy conserva-tion

Nomenclature

C Constant value119863min The smallest bore size in dimension119863max The biggest bore size in dimension119863

119891 Fractal dimension factor

119889

119904 Spectral dimension

d Width of the model pillarL Length of the model pillar119873(120575) Physical quantityR Thermal resistance (m2sdotKW)T Temperature (K )V Volume (m3)

Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)

120590 Stefn-Boltzmann constant120590 = 56697 times 10minus8W(K4sdotm2)

120573 Radiation extinction coefficient120582 Thermal conductivity (W(msdotK))120575 Variable measure length(m)120593 Pore porosity in the average volume

Subscripts and Superscripts

119890 Effective119903 Radiationg Residual gaseous phase in the poref The fluid phaseS The solid phasetotal Total valuetest The value gotten from experiments

Acknowledgment

Thisworkwas financially supported by ScienceampTechnologyProgram of Shanghai Maritime University no 20120091 Weare grateful to Professor Wenzhe Sun and Professor DanCao for their advices and suggestions for this project Theauthors also acknowledge Dr Wenzhong Gao with valuable

discussion and contributions in mounting the experimentaland installing the data acquisition devices

References

[1] Y Chen and M Shi ldquoDetermination of effective thermalconductivity for porousmedia using fractal techniquesrdquo Journalof Engineering Thermophysics vol 20 no 5 pp 608ndash615 1999

[2] SWhitaker ldquoDiffusion anddispersion in porousmediardquoAIChEJournal vol 13 no 6 pp 1066ndash1085 1967

[3] S Whitaker ldquoAdvances in the theory of fluid motion in porousmediardquo Chemical Engineering vol 61 pp 14ndash28 1969

[4] S Whitaker and W T-H Chou ldquoDrying granular porousmedia-theory and experimentrdquo Drying Technology vol 1 no 1pp 3ndash33 1983

[5] B Yu and J Li ldquoSome fractal characters of porous mediardquoFractals vol 9 no 3 pp 365ndash372 2001

[6] Y Ma B Yu D Zhang and M Zou ldquoA self-similarity modelfor effective thermal conductivity of porous mediardquo Journal ofPhysics D vol 36 no 17 pp 2157ndash2164 2003

[7] A Lagarde ldquoConsideration sur le transfert de chaleur en milieuporeuxrdquo Institut Francais Du Petrole vol 2 pp 383ndash446 1965

[8] J K Williams and R A Dawe ldquoFractalsmdashan overview ofpotential applications to transport in porous mediardquo Transportin Porous Media vol 1 no 2 pp 201ndash209 1986

[9] R Pitchumani ldquoEvaluation of thermal conductivities of disor-dered composite media using a fractal modelrdquo Journal of HeatTransfer vol 121 no 1 pp 163ndash166 1999

[10] J FThovert F Wary and P M Adler ldquoThermal conductivity ofrandom media and regular fractalsrdquo Journal of Applied Physicsvol 68 no 8 pp 3872ndash3883 1990

[11] D Zhang H Yang and M Shi ldquoImportant problems of fractalmodel in porous mediardquo Journal of Southeast University vol 32no 5 pp 692ndash697 2002

[12] R Pitchumani andB Ramakrishnan ldquoA fractal geometrymodelfor evaluating permeabilities of porous preforms used in liquidcomposite moldingrdquo International Journal of Heat and MassTransfer vol 42 no 12 pp 2219ndash2232 1999

[13] B Ramakrishnan and R Pitchumani ldquoFractal permeationcharacteristics of preforms used in liquid composite moldingrdquoPolymer Composites vol 21 no 2 pp 281ndash296 2000

[14] Z Yangsheng Coupling Effect and Engineering Reflection ofPorous Media Materials Science Press Beijing China 2010

[15] R Pitchumani and S C Yao ldquoCorrelation of thermal conduc-tivities of unidirectional fibrous composites using local fractaltechniquesrdquo Journal of Heat Transfer vol 121 no 1 pp 788ndash7961991

[16] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983


[17] A K Kan T T Zhang and H J Lou ldquoFractal study of effectivethermal conductivity of fiber glass materialsrdquoChinese Journal ofVacuum Science and Technology vol 33 pp 654ndash660 2013

[18] Z Donghui J Feng and S Mingheng ldquoHeat conduction infractal porous mediardquo Journal of Applied Sciences vol 21 no3 pp 253ndash2257 2003

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Here the supposition was made that heat fluxes throughfluid in the pore and by the solid phase of the porous bodywere individual and took place simultaneously However theheat transfer was also proceeding between the fluid phase andthe solid phase simultaneously So the real model was morecomplicated than the expression in (1) SoWilliams andDawe[8] developed the function as follows

120582

119890= 120576 [120576

1120582

119891+ (1 minus 120576

1) 120582

119904] +

(1 minus 120576)

120576

2120582

119891+ (1 minus 120576

2) 120582

119904

(2)

where 120593 = 120576120576

1+ (1 minus 120576)120576

2 The factor 120576 is ratio that the

heat flux transfers along with the temperature grads to thetotal heat flux while 120576

1is the factor for inexistence of the

solid-solid connection and 120576

2is for the existence of solid-

solid connection and solid-fluid connectionActually in the microspace structure of porous medium

materials the existence of ideal even distribution of the poresin the porous body is impossible So there is a big errorbetween the ideal model mentioned above and the real bodyThe available ideal models and empirical equations for foamthermal insulating materials are only generally related topores proportion which is the approximate reflection of theapparent thermal conductivity in the macrospace But for thereal foam material whose pore distribution is irregular theavailable idealmodels and empirical equations are not relativeto the microstructure and cannot expose the actual heat andmass transfer procedure and the distribution of temperatureand humidity As a result the big error is presence in theresearch work

The other method involves fractal theory Fractal theoryintroduced into the estimation and research work to calculatethe thermal conductivity of porous foam materials is a newway for the theory development on thermal performanceof porous medium materials Fractal theory was first putforward in 1975 byMandelbrotwhowas a professor fromHar-vard University in USA Some experts such as Pitchumani[9] Yu and Li [5] andMa et al [6] have done deep researcheson the effective thermal conductivity of granular porousmedium by fractal theory and have created correspondingmathematical equations Based on the fractal theoryThovertet al [10] Zhang et al [11] and so ondeveloped the theoreticalmodels for calculation of the effective thermal conductivityof irregular porous medium According to the concept ofSierpinski carpet model Pitchumani and Ramakrishnan [1213] created the pore distribution theoretical model but themodel and mathematical equations were very complicated infractal dimensionMa et al [6] built a mathematical model ofeffective thermal conductivity for porous medium accordingto fractal theory which manifested that the thermal conduc-tivity of porous medium was a function of the pore ratio thearea ratio the thermal conductivity ratio in components andthe thermal contact resistance all together It had nothing todowith empirical constants and less parameters and simple tocalculate in the formula However different porousmedia arenot the same with each other in internal fractal essence Andit is also difficult to estimate the thermal contact resistanceof porous medium in practice The universality of the modelstill needs to be further verifiedThovert et al [10] illuminated

the fractal porous medium by the percolation mathemat-ical model and did the solution by geometrical iterationWhereafter Adler Thovert and Thompson added empiricalconstants gotten by experiments into Adlerrsquos function Andthe function is commonly described as

120582 = 120582

0120601

120572

(3)

where 120582

0is the thermal conductivity of the liquid in

the porous material pores (Wmminus1Kminus1) And the super-script 120572 here is defined as

120572 =

119889

119904+ 119863

119891(2 minus 119889

119904)

119889

119904(3 minus 119863

119891)

(4)

where the fractal dimension factor 119863119891= 25ndash285 and spec-

tral dimension 119889

119904is used to describe the procedure of the

percolation in the poresYangsheng [14] based on the percolation theory created

the relationship between the pore diameters of variousgrain material and the thermal conductivity But the poreporosity the fractal dimension and the microstructure arenot involved in the model Pitchumani and Yao [15] cal-culated the transverse and longitudinal fractal dimensionsto illuminate the microstructure fibrous materials and thethermal conductivity was derived based on the conventionalheat transfer theory But the model only serves some certainfibrous porous materials well

So it is considerably difficult and also unpractical to builda theoretically mathematical model of effective thermal con-ductivity that is universal for porous medium Consequentlycrating amathematicalmodel of thermal conductivity for onecertain porous medium that reflects its structure characteris-tic in internality is an important developing direction for theporous medium research work

2 Microstructures of the Open CellPolyurethane Foam and Fractal Description

21 Microstructures Theopen cell polyurethane is composedof solid substrates and cells By the effect of foaming agentand cell opening agent great deals of cells are generatedand continuously distributed inside the material The cellsconnect with each other side by side and the gas in porescan flow freely through one cell to another That is reallyan advantage to expulse foaming agent and vapors thatembraced in pores Meanwhile the gas in pores can easilybe expelled as the thorough connection of cells The solidsubstrate of open cell polyurethane has the certain intensityto support thematerial and to prevent the collapse in vacuumstate So the polyurethane foam with open cell structure canbe used widely as the core material of the vacuum insulationpanel

Microstructures of the open cell polyurethane consistingof skeleton of solid substrate (the white part in the picture)and cells (the black part in the picture) is shown in Figure 1(taken by electron microscopy) Cells are generally cubedstructurally in the space and continuously distributed insection plane and the dimensions of the apertures are in








119873(120575) prop 120575

119863119891 (5)



119881 (120575) prop 120575

119863119891 (6)

138

143

148

153

158

163

44 48 52 56 6

lnV(120575)

ln 120575

Df = 2621



119881 (120575) = 119862120575

119863119891 (7)


ln119881 (120575) = ln119862 + 119863

119891ln 120575 (8)




119873(120575 gt 119863) = (

119863max119863

)

119863119891

(9)

or

119873

1015840

(120575 gt 119863min) = (

119863max119863min

)

119863119891

(10)


d

d

2

22

2

1

L

L minus 2d



119889119873 = minus119863

119891119863

119863119891


119889119863(11)


minus

119889119873

119873

1015840= 119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863 (12)


119863

119891119863

119863119891


int

infin

minusinfin

119891 (119863) 119889119863 = int

119863max

119863min

119891 (119863) 119889119863 = 1 minus (

119863min119863max

)

119863119891

(13)


119871 = int

119863max

119863min

119863119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863

=

119863

119891

119863

119891minus 1

119863min [1 minus (119863min119863max

)

119863119891minus1

]

(14)






120582

119890= 119891 (sum120582

119894 120593 119863

119891) (15)

R1

R2

R2

R2

R2

R3

R3

R4







119877


2level pillar


4gas in cavity



119877

1=

119871

120582

119904119889

2

119877

2=

2119889

120582

119904119889 (119871 minus 2119889)

=

2

120582

119904(119871 minus 2119889)


119877

3=

(119871 minus 2119889)

120582

119892(119871 minus 2119889) 1198892

=

2

120582

119892119889

119877

4=

4119871

120582

119892(119871 minus 2119889)

2

119877total =4119871

120582

119890119871

2=

4

120582

119890119871

(16)





119877total =119877

1119877

4(2119877

2+ 119877

3)

2119877

1119877

4+ (2119877

2+ 119877

3) (119877

1+ 119877

4)

(17)


120582

119890foam =

4120582

119904119889

2

119871

2+

120582

119892(119871 minus 2119889)

2

119871

2+

4120582

119892120582

119904(119871 minus 2119889) 119889

119871 (2120582

119892119889 + 119871120582

119904minus 2120582

119904119889)

(18)

where 120582





120593 =

sum

119899

119894=1119881

119894

119881

= (1 minus

2119889

119871

)

119863119891

(19)


120582

119890foam = 120582

119904(1 minus 120593

1119863119891)

2

+ 120582

119892120593

2119863119891

+ 2

120582

119892120582

119904(1 minus 120593

1119863119891) 120593

1119863119891

120582

119892(1 minus 120593

1119863119891) + 120582

119904120593

1119863119891

(20)






119902

119903= minus

4120590 (119879

4

1minus 119879

4

2)

3120573119871

(21)


10


1and 119879



conductivity 120582119890119903

for a porous medium

120582

119890119903=

4120590 (119879

2

1+ 119879

2

2) (119879

1+ 119879

2)

3120573

(22)




120582

119890= 120582

119890foam + 120582

119890119903 (23)





6 Conclusion







Sample Densitykgm3

Porosity

Fractaldimension



120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32


Nomenclature



119889



Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)





Acknowledgment



References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










119873(120575) prop 120575

119863119891 (5)



119881 (120575) prop 120575

119863119891 (6)

138

143

148

153

158

163

44 48 52 56 6

lnV(120575)

ln 120575

Df = 2621



119881 (120575) = 119862120575

119863119891 (7)


ln119881 (120575) = ln119862 + 119863

119891ln 120575 (8)




119873(120575 gt 119863) = (

119863max119863

)

119863119891

(9)

or

119873

1015840

(120575 gt 119863min) = (

119863max119863min

)

119863119891

(10)


d

d

2

22

2

1

L

L minus 2d



119889119873 = minus119863

119891119863

119863119891


119889119863(11)


minus

119889119873

119873

1015840= 119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863 (12)


119863

119891119863

119863119891


int

infin

minusinfin

119891 (119863) 119889119863 = int

119863max

119863min

119891 (119863) 119889119863 = 1 minus (

119863min119863max

)

119863119891

(13)


119871 = int

119863max

119863min

119863119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863

=

119863

119891

119863

119891minus 1

119863min [1 minus (119863min119863max

)

119863119891minus1

]

(14)






120582

119890= 119891 (sum120582

119894 120593 119863

119891) (15)

R1

R2

R2

R2

R2

R3

R3

R4







119877


2level pillar


4gas in cavity



119877

1=

119871

120582

119904119889

2

119877

2=

2119889

120582

119904119889 (119871 minus 2119889)

=

2

120582

119904(119871 minus 2119889)


119877

3=

(119871 minus 2119889)

120582

119892(119871 minus 2119889) 1198892

=

2

120582

119892119889

119877

4=

4119871

120582

119892(119871 minus 2119889)

2

119877total =4119871

120582

119890119871

2=

4

120582

119890119871

(16)





119877total =119877

1119877

4(2119877

2+ 119877

3)

2119877

1119877

4+ (2119877

2+ 119877

3) (119877

1+ 119877

4)

(17)


120582

119890foam =

4120582

119904119889

2

119871

2+

120582

119892(119871 minus 2119889)

2

119871

2+

4120582

119892120582

119904(119871 minus 2119889) 119889

119871 (2120582

119892119889 + 119871120582

119904minus 2120582

119904119889)

(18)

where 120582





120593 =

sum

119899

119894=1119881

119894

119881

= (1 minus

2119889

119871

)

119863119891

(19)


120582

119890foam = 120582

119904(1 minus 120593

1119863119891)

2

+ 120582

119892120593

2119863119891

+ 2

120582

119892120582

119904(1 minus 120593

1119863119891) 120593

1119863119891

120582

119892(1 minus 120593

1119863119891) + 120582

119904120593

1119863119891

(20)






119902

119903= minus

4120590 (119879

4

1minus 119879

4

2)

3120573119871

(21)


10


1and 119879



conductivity 120582119890119903

for a porous medium

120582

119890119903=

4120590 (119879

2

1+ 119879

2

2) (119879

1+ 119879

2)

3120573

(22)




120582

119890= 120582

119890foam + 120582

119890119903 (23)





6 Conclusion







Sample Densitykgm3

Porosity

Fractaldimension



120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32


Nomenclature



119889



Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)





Acknowledgment



References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




d

d

2

22

2

1

L

L minus 2d



119889119873 = minus119863

119891119863

119863119891


119889119863(11)


minus

119889119873

119873

1015840= 119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863 (12)


119863

119891119863

119863119891


int

infin

minusinfin

119891 (119863) 119889119863 = int

119863max

119863min

119891 (119863) 119889119863 = 1 minus (

119863min119863max

)

119863119891

(13)


119871 = int

119863max

119863min

119863119863

119891119863

119863119891

min119863minus(119863119891+1)

119889119863

=

119863

119891

119863

119891minus 1

119863min [1 minus (119863min119863max

)

119863119891minus1

]

(14)






120582

119890= 119891 (sum120582

119894 120593 119863

119891) (15)

R1

R2

R2

R2

R2

R3

R3

R4







119877


2level pillar


4gas in cavity



119877

1=

119871

120582

119904119889

2

119877

2=

2119889

120582

119904119889 (119871 minus 2119889)

=

2

120582

119904(119871 minus 2119889)


119877

3=

(119871 minus 2119889)

120582

119892(119871 minus 2119889) 1198892

=

2

120582

119892119889

119877

4=

4119871

120582

119892(119871 minus 2119889)

2

119877total =4119871

120582

119890119871

2=

4

120582

119890119871

(16)





119877total =119877

1119877

4(2119877

2+ 119877

3)

2119877

1119877

4+ (2119877

2+ 119877

3) (119877

1+ 119877

4)

(17)


120582

119890foam =

4120582

119904119889

2

119871

2+

120582

119892(119871 minus 2119889)

2

119871

2+

4120582

119892120582

119904(119871 minus 2119889) 119889

119871 (2120582

119892119889 + 119871120582

119904minus 2120582

119904119889)

(18)

where 120582





120593 =

sum

119899

119894=1119881

119894

119881

= (1 minus

2119889

119871

)

119863119891

(19)


120582

119890foam = 120582

119904(1 minus 120593

1119863119891)

2

+ 120582

119892120593

2119863119891

+ 2

120582

119892120582

119904(1 minus 120593

1119863119891) 120593

1119863119891

120582

119892(1 minus 120593

1119863119891) + 120582

119904120593

1119863119891

(20)






119902

119903= minus

4120590 (119879

4

1minus 119879

4

2)

3120573119871

(21)


10


1and 119879



conductivity 120582119890119903

for a porous medium

120582

119890119903=

4120590 (119879

2

1+ 119879

2

2) (119879

1+ 119879

2)

3120573

(22)




120582

119890= 120582

119890foam + 120582

119890119903 (23)





6 Conclusion







Sample Densitykgm3

Porosity

Fractaldimension



120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32


Nomenclature



119889



Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)





Acknowledgment



References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




119877

3=

(119871 minus 2119889)

120582

119892(119871 minus 2119889) 1198892

=

2

120582

119892119889

119877

4=

4119871

120582

119892(119871 minus 2119889)

2

119877total =4119871

120582

119890119871

2=

4

120582

119890119871

(16)





119877total =119877

1119877

4(2119877

2+ 119877

3)

2119877

1119877

4+ (2119877

2+ 119877

3) (119877

1+ 119877

4)

(17)


120582

119890foam =

4120582

119904119889

2

119871

2+

120582

119892(119871 minus 2119889)

2

119871

2+

4120582

119892120582

119904(119871 minus 2119889) 119889

119871 (2120582

119892119889 + 119871120582

119904minus 2120582

119904119889)

(18)

where 120582





120593 =

sum

119899

119894=1119881

119894

119881

= (1 minus

2119889

119871

)

119863119891

(19)


120582

119890foam = 120582

119904(1 minus 120593

1119863119891)

2

+ 120582

119892120593

2119863119891

+ 2

120582

119892120582

119904(1 minus 120593

1119863119891) 120593

1119863119891

120582

119892(1 minus 120593

1119863119891) + 120582

119904120593

1119863119891

(20)






119902

119903= minus

4120590 (119879

4

1minus 119879

4

2)

3120573119871

(21)


10


1and 119879



conductivity 120582119890119903

for a porous medium

120582

119890119903=

4120590 (119879

2

1+ 119879

2

2) (119879

1+ 119879

2)

3120573

(22)




120582

119890= 120582

119890foam + 120582

119890119903 (23)





6 Conclusion







Sample Densitykgm3

Porosity

Fractaldimension



120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32


Nomenclature



119889



Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)





Acknowledgment



References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Sample Densitykgm3

Porosity

Fractaldimension



120582

119890 foam 120582

119890119903120582

119890120582test

1 45 81 263 300 02804 00022 02826 0280minus093

355 02804 00028 02832 0287 13

2 60 72 253 300 03186 00022 03208 0330 28355 03186 00028 03214 0332 32


Nomenclature



119889



Greek Symbols

120572 120572 = (119889

119904+ 119863

119891(2 minus 119889

119904))119889

119904(3 minus 119863

119891)





Acknowledgment



References



























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials













CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Effective Thermal Conductivity of Open Cell …downloads.hindawi.com/journals/amse/2013/125267.pdf · 2019-07-31 · Research Article Effective Thermal Conductivity