Bilinear Predictive Law of the Thermal Conductivity of Tropical Woods with Respect to the Infradensity and the Moisture Content, using Schwartz Conditions

Bilinear Predictive Law of the Thermal Conductivity of Tropical Woods with Respect to the Infradensity and the Moisture Content, using Schwartz Conditions

NGOHE-EKAM Paul-Salomon 1*, TALLA André2, NJANKOUO Jacques Michel3, KANMOGNE Abraham2, NDZANA Bénoît4, TAMO TATIETSE Thomas 3, GIRARD Philippe5

1Département de Mathématiques et Sciences Physiques,Ecole Nationale Supérieure Polytechnique,Yaoundé, Cameroun 2Département des Génies Industriel et Mécanique,Ecole Nationale Supérieure Polytechnique,Yaoundé, Cameroun 3Département de Génie Civil et Urbanisme,Ecole Nationale Supérieure Polytechnique,Yaoundé, Cameroun

4Département des Génies Electrique et des télécommunications,Ecole Nationale Supérieure Polytechnique,Yaoundé, Cameroun

5Laboratoire Energie et Environnement,CIRAD РFor̻t, Montpellier, France.

"Corresponding Author: NGOHE-EKAM Paul-Salomon, Ecole Polytechnique, Yaoundé, Cameroun; Email: pasanek@yahoo.fr

Abstract An experimental study led to linear regressions of the density of tropical woods. These regressions are so forth used to compute the variations of the thermal conductivity with moisture content, by using a series-parallel model in which Wood is considered as an association of wet cells walls (cross and side ones) and air. Linear approximation is done of these variations as well as the ones of the thermal conductivity with respect to basal density (or infradensity). This double linear variation is then used to determine a bilinear law linking the moisture content and the infradensity to the thermal conductivity of tropical wood. The obtained empirical law, subjected to obey the Schwartz condition on partial derivatives, enables the prediction of the thermal conductivity of any tropical wood species, at any desired moisture content, if at least its infradensity is known.

Keywords Thermal conductivity; Tropical woods; Infradensity; Moisture content; Porosity; Schwartz conditions

1. INTRODUCTION

   Mastering thermal conductivity of materials has constantly left impression on the sensitivity of engineers and architects, all over the years, especially in this century in which man is so attentive to energy related problems, namely for thermal comfort as well as for the material conditioning. In fact, greenhouse gases (GHG) emissions, responsible for the global warming of the planet, is mainly due to the high energy consumption through the use of devices providing a

   good thermal comfort in the interior of the building, but which strongly emit carbon dioxide (CO2). This high consumption rate observed is due either to poor insulation of the building or the use of materials with high coefficient value of thermal conductivity [1]. Finally, we also know that the insulation has an important role in the thermal and acoustic applications [2]. Thus, the importance of thermal conductivity is no longer to be shown.

   Thermal conductivity can be reached by using transient methods ([3] and [4]). An experimental study, based on a permanent regime technique [5], has been used on woods, and led us to some abacus [6]. Wood, the only renewable raw material to our knowledge, has thus been the subject of several research works whose most important results have been summarized by Siau in [7]. In fact, Siau presents the variations of the thermal conductivity with porosity, but porosity which is a function of moisture content and specific weight of the species; now specific weight varies enormously for the same sample. Infradensity is a characteristic parameter of wood species used in calculations ([8] and [9]). We make an advantageous use of it in order to make thermal conductivity of tropical wood become more accessible. As Gordillo-Delgado F did in [10], the species we use are taken from the bottom, middle and top culm regions of the plants. The purpose of this study is to propose a mathematical model allowing an easier prediction of the thermal conductivity of tropical woods.

   GHG Greenhouse gases

   Di Infradensity or basal density (kg.m-3)

   a Side of a lumen (m2) ; ratio between the width of lumen and the total width of the c

   R Thermal resistance

   Va porosity (Va = a2)

   ' Thermal conductivity of the cells walls substance (W.m-1.K-1)

   a Thermal conductivity of the air in the lumen (W.m-1.K-1)

   Density of liquid water ( 1000 kg.m-3)

   Nomenclature

   T Transversal thermal conductivity of wood (W.m-1.K-1)

   // Parallel or Tangential or Axial thermal conduc- tivity of wood (W.m-1.K-1)

   Z Part of normal walls effective for conductive flux

   1

   W Fraction of the normal wall, adjacent to the lumen and that could be considered as effective

   for conduction

   1. specific weight

      M0 Dry mass (kg)

   2. Moisture content (%)

2. MATERIALS AND METHODS

The study is carried out on five specie (Tali, Bilinga, Sapelli, Sipo and Ayous) covering a wide range of densities of most of the woods used in Cameroon. Experimental values of thermal conductivities are obtained by means of a permanent regime method called Méthode des boîtes and whose experimental device is clearly described by NGOHE in [6], [11] and [14]; as for the working techniques, it is based on a semi-empirical model developed by Siau in [7].

1. Theoretical Equation of Thermal Conductivity of Tropical Woods.

   1. Conduction in the transversal (orthoradial) direction

      Siau [7] presents a geometric model of longitudinal

      1-a

      a

      (1) Normal walls	(2) lumen
      	(3) parallel walls

      1-a

      a

      woods cells (tracheid, fibers, vessel, parenchyma,) with the following hypotheses (see Fig. 1): the cell is everywhere unitary in dimensions viewpoint; tangential and radial (with respect to the flow) cells walls have the same

      l

      a

      a

      l

      (1-a)/2

      Fig. 2: Physical considerations of Siau (1984) for the geometric model

      with the heat flux in the transversal direction, by referring himself to Fig. 2 below.

      But this mathematical study can also be carried out using Fig. 3. The only difference in both considerations is the importance given to the conduction through parallel cells. The latter consideration is chosen because it gives more reliable results from the mathematical viewpoint, especially as the Schwartz conditions,

      l

      Fig. 1: Geometric model for a woods longitudinal cell

      1-a a

      (1) Normal walls	(2) lumen
      (3) Parallel walls

      a

      thickness , and the proportion of the cells walls thickness with respect to the diameter is the same for all the cells of a given species; the presence of transversal oriented cells, of walls on the edge of the cell, and of pit openings, are neglected, and the lumen is supposed to have a unitary length and a square section of side a.

      Siau carries out the mathematical analysis of that model,

      Fig. 3: Adopted configuration for the geometric model

      1-a

      i.e. the crossed partial derivatives of the found bilinear law, are concerned.

      This confers to the normal wall a thermal resistance given

      1 a

      The three parts (Fig. 3) have the following thermal resistances:

      by R'1 '.Z..a

      ; and the transversal thermal conductivity

      1 a 1 1

      becomes:

      R1 , R2 and R3

      Z.a(1 a)'2 [(1 a)2 Z.a] '

      a' a

      ' 1 a

      T a

      (2)

      by:

      The transversal conductivity T of the cell is then given

      a(1 a)'2 [(1 a)2 a] '

      T a (1)

      a'(1 a)a

      Where a stands for the ratio between the width of lumen

      Z.a'(1 a)a

      The works of Hart [12] and Siau [7] then appear very necessary for the evaluation of the factor Z. In fact, with a null conduction assumed in the lumen, Hart [12] proposes an empirical relationship for the calculation of the fraction W1 of the normal wall, adjacent to the lumen and that could be considered as effective for conduction as follows:

      and the total width of the cell (a2 = porosity Va according to

      (1 a)

      2.08 a

      the hypothesis of the model); ' and a are respectively the thermal conductivity of the cells walls substance and

      W1 0.48 a

      1 e

      (1a )

      (3)

      that of the air in the lumen.

      In reality, there is a concentration of thermal flux on lateral walls as shown in Fig. 4, and this makes the flux not uniform on them, as opposed to the hypothesis used to obtain equation (1).

      From the definition of W1, its easy to understand that, with a null conduction in the lumen, the widths portion of the normal wall adjacent to the lumen and considered as effective in conduction, divided by the total width (here, a) of the normal wall, is equal to W1a [7]. However, in the case of heat flow, theres a significant flux in the lumen, because of the air conductivity. Since flux concentration in

      Parallel walls

      a

      the lumen is

      '

      times that in the lateral walls, the

      Lumen

      Normal

      additional fraction of the flux moving through the width of the normal wall is totally conductive, and can be obtained by the following relationship:

      walls

      Z a.W a. a

      a (4)

      1 '

      aW1

      '

      Parallel walls

      Fig. 4: Flux flow in the cell

      Siau [7] then introduces a factor Z in order to bear in mind the fact that, because of non-uniformity, a part of normal walls is not effective for the conductive flux. With this new consideration, our physical model is modified according to Fig. 5 below.

      Note: According to the physical model there is, in addition to W1a, no longer any remaining part of the normal wall that is adjacent to the lumen. And consequently, the factor (1-a) of Siau [7] disappears and the Z expression is the one above.

      For a and ' respectively taken, like Siau did in [7], equal to 1.0 10-4 and 10.5 10-4 cal/cm.°C.s (that is 418 and

      4389 10-4 W/m.K, knowing that 1 cal/cm.°C.s = 418

      W/m.K), equations (3) and (4) allow to get the transversal conductivity T as a function of a. For the values of a going from 0.01 to 0.99, we obtain a much stronger linear regression (R2 = 0.99) that permits us to conclude that:

      1-a a

      T 10.61 9.53a

      for 0.01 a 0.99

      (5)

      	(2) lumen
      (1) Normal walls
      (3) Parallel walls

   2. Conduction in the longitudinal direction

When the geometric model of Fig. 1 is considered in relation to the flux flow in the fibers direction, it results in

Za the following equation, giving the thermal conductivity in the longitudinal direction:

1-a

Or :

//

'' (1 a2 ) a 2

a

'' (1 V ) V

(6-a)

(6-b)

// a a a

Fig. 5: Effective physical consideration

Va still designates the porosity and a the width of the lumen;

= 21.0 10-4 cal/cm.°C.s and a = 1.0 10-4 cal/cm.°C.s respectively stand for the conductivity of the walls cells substance in the longitudinal direction and that of air in the

G M0

w .V

(8).

lumen. Now, equation (8) leads to this other one:

B Application to the Determination of the Bilinear Law for the Thermal Conductivity of Tropical Woods

B 1 Conduction in the transversal (orthoradial) direction

G .

2

H O 1

1

H 100

(9)

In order to relate T to the moisture content H and to the infradensity Di of wood, we proceed as follows:

By defining the specific weight G of a wood sample of volume V at humidity H as the ratio of its dry mass and the mass of the water it would move by immersion, it appears in

[7] that porosity Va of wood (Va = a2) can be approached

a

by the following relationship: V 1 G 0.667 H

100

NGOHE [14] gives, for a given species (so a given infradensity), the variations law of the density with respect to the moisture content. With this, it becomes possible for several values of infradensity to establish a table of values for G (using equation 9), then for a (using equation 7), and

lastly for the transversal conductivity T (equation 5), with

moisture content H and the infradensity Di. Table 1-a below presents the transversal conductivity coming from Siaus

model, while table 1-b below presents its experimental

(7)

Where, by definition:

values.

Table 1-a: Transversal conductivity of tropical wood T (10-3 W.m-1K-1) as a function of basal density and moisture content (experimental values)

Moisture content H(%)	Di (infradensity)
	0,335	0,360	0,343	0,733	0,719	0,670	0,647	0,702	0,704	0,536
0	116,7	141,2	122,9	184,0	184,0	184,6	182,9	180,2	204,8	168,6
12	140,8	157,2	139,3	226,2	230,2	233,0	223,3	237,6	256,2	200,5
15	146,9	161,2	143,5	236,8	241,8	245,1	233,5	251,9	269,0	208,5
20	156,9	167,8	150,3	254,4	261,0	265,2	250,3	275,8	290,4	221,8
40	197,1	194,4	177,7	324,8	338,0	345,8	317,7	371,4	376,0	275,0
60	237,3	221,0	205,1	395,2	415,0	426,4	385,1	467,0	461,6	328,2

Moisture content H(%)	Di (infradensity)
	0,555	0,522	0,562	0,603	0,494	0,454	0,579	0,784	0,764
0	153,3	170,4	154,6	192,7	148,8	86,8	110,2	213,3	206,6
12	183,8	203,0	189,6	216,5	163,7	118,8	146,8	263,3	253,6
15	191,4	211,2	198,4	222,4	167,4	126,9	156,0	275,9	265,4
20	204,1	224,8	213,0	232,3	173,6	140,2	171,2	296,7	285,0
40	254,9	279,2	271,4	271,9	198,4	193,6	232,2	380,1	363,4
60	305,7	333,6	329,8	311,5	223,2	247,0	293,2	463,5	441,8

Table 1-b: Transversal conductivity of tropical wood T (10-3W.m-1K-1) as a function of basal

density and moisture content (Siaus model)

Moisture content H(%)	Di (infradensity)
	0,335	0,360	0,343	0,733	0,719	0,670	0,647	0,702	0,704
0	91,6	97,9	90,6	164,4	183,3	172,6	163,7	176,4	177,7
12	102,6	109,7	101,4	187,9	211,4	198,0	188,3	203,9	205,2
15	105,4	112,7	104,2	194,0	218,7	204,6	194,6	211,1	212,4
20	110,1	117,7	108,8	204,3	231,3	215,9	205,5	223,5	224,6
40	129,6	138,5	127,9	248,9	287,6	265,0	252,9	278,5	279,4
60	150,0	160,3	147,9	301,7	362,0	325,5	310,2	349,9	350,2

Moisture content H(%)	Di (infradensity)
	0,536	0,555	0,522	0,562	0,603	0,494	0,454	0,579	0,784	0,764
0	144,0	151,2	131,5	148,1	158,6	132,0	124,3	153,0	203,0	203,2
12	162,7	170,0	150,0	167,2	178,9	148,4	139,4	171,9	234,8	233,1
15	167,4	174,8	154,8	172,1	184,1	152,6	143,2	176,7	243,2	241,0
20	175,5	182,8	162,9	180,3	192,8	159,6	149,5	184,7	257,6	254,5
40	209,0	216,4	197,2	214,9	229,6	188,5	175,9	218,3	324,9	315,7
60	246,0	253,2	235,6	253,1	270,7	219,7	203,9	255,1	433,5	404,2

It should be noted that both figures 6 and 7 represent experimental values. A linear regression of these various curves leads to laws in the following forms:

T = T(Di) = aD(H).Di + bD(H) for a fixed H (10) and

T = T(H) = aH(Di).H + bH(Di) for a fixed Di (11)

with correlations coefficients all greater than 0.8 (Fig. 6 and Fig.7).

The horizontal exploitation of table 1-b allows us to plot using points the variations of the transversal thermal conductivity of tropical woods as a function of infradensity, for the various values of moisture content (one curve by moisture content), as shown on Fig. 6.

This double linearity then suggests, for transversal thermal conductivity, a bilinear law in the following form:

On the other hand, the vertical exploitation of that table gives rise to the layout of the transversal conductivity as a function of moisture content, for several values of the infradensity (one curve by infradensity), as shown in Fig. 7.

T (H,Di) = HDi + Di + H + (12)

Now, in order to consider T as a function only of variables H and Di, one needs to first verify the following Schwartzs condition for the crossed partial derivatives of T :

2

T

HDi

2

T

DiH

(13)

Equation (13), with respect to (10) and (11), comes to simply

compare H aD

and a

H

Di

. Tables 2 and 3 give the

variations aD(H) and aH(Di) which, when plotted as in Fig. 8 and Fig.10 below, enable us to deduce the values 5.6 and 6.2 for those partial derivatives. We then conclude their near equality with a relative uncertainty of about 10%.

Table 2 : Variations of coefficients aD(H) and bD(H)

Coefficients	Moisture content H(%)
	0	12	15	20	40	60
aD(H)	202,12	280,09	299,58	332,06	462,02	591,95
bD(H)	45,79	36	33,56	29,48	13,18	-3,12

Coefficients	Di (infradensity)
	0,335	0,454	0,536	0,603	0,702	0,784
aH(Di)	2,01	2,67	1,98	2,66	4,17	4,78
bH(Di)	116,7	86,8	192,7	168,6	213,3	180,2

Table 3 : Variations of coefficients aH(Di) and bH(Di)

With the linear forms a

(H)=

H ,

The condition of equaling crossed derivatives is what made us

bD(H)=

1H 1

D

1

, aH(Di)= 2

Di 2

1

and

finally adopt the physical model on Fig. 3 rather than that of Siau (Fig. 2) which gives a relative gap greater than 25%.

bH(Di)= 2Di 2 , equations (10) and (11) respectively give the following ones:

The procedure for getting empirical values of coefficients

(H,D ) HD

• D

  H

  (14)

  , , and of equation (12) is as follows:

  T i

  and

  1 i 1 i 1 1

  Graphics aD = aD(H), bD = bD(H), aH = aH(Di) and bH =

  (H,D ) HD

• D

  H

  (15)

  bH(Di) are represeted respectively on figures 8, 9, 10 and 11; they also show a strong linear correlation (R2>0.8) except for bH(Di) (which has R2 0,6).

  T i 2 i 2 i 2 2

  Its then needed, by identification, that the following

  factors be nearer in pairs: 1 and 2, 1 and 2, 1 and 2, and 1 and 2. We observe, from figures 8, 9, 10 and 11, the following values: 1= 5.55, 2= 6.20 , 1= 212.54, 2= 217.65, 1= -1.15, 2= -1.49, 1= 23.35 et 2= 35.87 ; These

  values lead to the following mean values for the coefficients: =(1+2)/2=5.9, =(1+2)/2 = 215.1, = (1+2)/2 = -1.29

  and =(1+2)/2 = 24.2 with respective relative gaps of 11%, 3%, 22% and 7%. We then conclude with the following expression for the bilinear model of the conductivity of tropical woods, in the transversal direction:

  T (H,Di) = 5.9 HDi + 215.1 Di 1.29 H + 24.2 (16)

  where T is in 10-3 W.m-1.K-1, H in % and Di without unit.

  Figures 12, 13, 14, 15 and 16, show the comparison of the experimental measures with both bilinear model and Siaus model above.

  These five figures show an increasing difference between the bilinear model and the experimental values when moisture content increases. We then conclude that coefficients and have to be rectified, so that the model given under the form

  1. considerably fits the experimental measures. Even the relative gaps of 11% and 22% that we got above for these coefficients could have suggested their rectification. We then require the solver of Excel 2003, in which coefficients and are variables and we choose as target, that the mean value of the sum of all relative gaps must be minimum. This gap is between the values given by the bilinear model and those got experimentally.

     The solver so generates new values of these coefficients, with the mean value of relative gap of 8%, when we also allow coefficient to vary. The rectified bilinear model is then the following:

     T (H,Di) =555.0 HDi +215.1 Di 33.7 H +42.6 (17)

     B 2 Conduction in the longitudinal direction

     The same work done for transversal conductivity is carried up for the samples cut up in the longitudinal or axial

     direction; table 4-a gives the values of

     //

     coming from

     Siaus model (equation 6-b) for several values of H and Di, and table 4-b those obtained experimentally.

     Table 4-a : Longitudinal conductivity of tropical wood // (10-3W.m-1K-1) as a function of basal density and moisture content (Siaus model)

     Moisture content H(%)	Di (infradensity)
     	0,330	0,359	0,710	0,568	0,610	0,511	0,517	0,782
     0	231,6	254,6	474,1	377,0	441,5	367,9	345,5	530,5
     12	264,3	289,7	538,7	431,8	494,0	408,2	393,6	596,5
     15	272,4	298,5	554,7	445,4	506,9	418,0	405,5	612,7
     20	286,0	313,0	581,2	468,0	528,1	434,2	425,3	639,5
     40	340,2	370,8	685,8	557,9	611,2	497,0	504,0	744,5
     60	394,2	428,3	788,9	647,1	691,9	557,6	581,9	846,8

     Tableau 4-b : Longitudinal conductivity of tropical wood // (10-3W.m-1K-1) as a function of moisture content and basal density (experimental values)

     Moisture content H(%)	Di (infradensity)
     	0,330	0,359	0,710	0,568	0,610	0,511	0,517	0,782
     0	268,8	259,5	437,7	354,9	427,6	352,2	313,7	527,7
     12	288,7	283,5	489,1	389,3	464,2	388,1	343,5	584,3
     15	293,7	289,5	501,9	398,0	473,4	397,1	350,9	598,5
     20	302,0	299,5	523,3	412,3	488,6	412,0	363,3	622,1
     40	335,2	339,5	608,9	469,7	549,6	471,8	412,9	716,5
     60	368,4	379,5	694,5	527,1	610,6	531,6	462,5	810,9

     The bilinear model is established, this time, with the coefficients written with a superscript << ' >> ; that is for example

     //

     //

     (D ) a' D (H)D

• b' D (H) for a fixed H. Tables 5 and 6 give the variations of these new coefficients respectively

  i

  i

  Coefficients	Moisture content H (%)
  	0	12	15	20	40	60
  a'D(H)	564,9	642,7	662,19	694,62	824,34	954,06
  b'D(H)	57,99	51,38	49,73	46,97	35,96	24,95

  as a function of the moisture content and the infradensity. Table 5 : Variations of coefficients aD(H) and bD(H)

  Coefficients	Di (infradensity)
  	0,33	0,359	0,511	0,517	0,568	0,61	0,71	0,782
  a'H(Di)	1,66	2	2,99	2,48	2,87	3,05	4,28	4,72
  b'H(Di)	268,8	259,5	352,2	313,7	354,9	427,6	437,7	527,7

  Table 6 : Variations of coefficients aH(Di) and bH(Di)

  The cross derivatives of the longitudinal conductivity are equal almost with a relative gap of 3%, and we obtain their mean values as follows: = 6.5, = 564.9, = -0.55 and = 58, with relative gaps of less than 5%.

  Then the variation law of the thermal conductivity in the longitudinal direction is given by:

  // (H,Di) = 6.5 HDi +564,9 Di -0.55 H +58 (18)

  with // in 10-3 W.m-1.K-1, H in % and Di without unit.

  A rectification of this model with the Excel 2003s solver

  seems not necessary, since the mean relative gap obtained from the values given by the model of equation(18) with respect to the experimental ones is lower than 9%. Figures 21 to 25 below show its various comparisons with the experimental values and those coming from the Siaus model.

  1. RESULTS AND DISCUSSION

     The significant set of results to which we end consists of two mathematical empirical models presented on equations

     1. and (18). Their validation is made by comparison with values of tables 1 to 4. The graphics coming from this are presented in figures 12 to 16 for the transversal conductivity and figures 21 to 25 for the longitudinal one.

     From these graphics the following comments can be done:

     • The model of equation (18) represents the longitudinal thermal conductivity of tropical woods in a very good way, up to 60 % moisture content and with basal density covering almost all the tropical woods: we observe, in fact, a relative deviation always less than 9%.

     • The model of equation (17) is enough representative of the transversal thermal conductivity of tropical woods, for moisture content going up to 60% and basal density covering almost all tropical woods: the relative deviation is less than 16%; this high deviation compared to the one obtained for the longitudinal direction, is probably coming from the fact that the radial and orthoradial directions have been combined to constitute the transversal one.

     • The graphical representations of the proposed models show once more, a higher conduction (2 to 2.5 times) more in the longitudinal direction than in the transversal one, a result which fits with the observations of Mac Lean in [13] and Ngohe-Ekam in [14] and [15].

On the other hand, with comparison to the model of Siau

[7] given by equation (5) and which requires, at any moisture content, to first compute the specific weight of the sample before getting its porosity and after that its conductivity, the bilinear models presented on equations (17) and (18) have the advantage that, as soon as the basal density of the sample is known, one can directly predict its conductivity at any moisture content.

1. CONCLUSION

The work presented below aimed to propose a mathematical model allowing an easier prediction of the thermal conductivity of tropical woods both in the axial and the transversal directions (i.e. parallel and perpendicular directions to the grain). It has led to a bilinear law whose comparison to the calculations stem from experimental values has given very low deviations, allowing us then to confirm the validity of the presented model, for tropical woods, in the ranges of all basal density and moisture contents up to 60%.

ACKNOWLEDGMENT

We thank our colleague Julius NASHIPU, a native English speaker, who carefully read and corrected this paper.

ETHICS

This article is original and contains unpublished materials. The corresponding author confirms that all of the other authors have read and approved the manuscript and there are no ethical issues involved.

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