obtained under the conditions and for the first inequality and
by the conditions and for the second inequality.
The range of application of eq.1.82 is
(1.84)
obtained under the condition .
In conclusion for assigned windows diameter, dw, porosity, φ, and pore size, dp, the implicit equations 1.81 or 1.82, can be solved numerically for this spherical model which, in turn, through eqs.1.68, 1.76 and 1.77 permits to evaluate the specific surface, Sv, with the aid of eqs. 1.74 and 1.75. In this paper the numerical procedure has employed the Mathematica software [42].
However, the applicability conditions of eqs.1.83 and 1.84 and the above cited experimental observations on the geometrical profile of the strut cross-section and the conclusions at the end of the previous sections suggest that the present spherical model can likely be applicable almost in the high range of porosity (0.97 ÷ 0.99) of the eq.1.84. Unfortunately, as it will be shown in next section, the available measured values of the surface per unit volume, Sv, found in the literature, refer to porosities less than 0.9 so that the eq.1.81 and 1.75 can be used and tested only in this range: i.e. exactly that which the author estimate less coherent and adapt.
1.17 Validation of the proposed correlations to calculate specific
reported in Table 1.7 and 1.9 refer to foams whose windows diameter and not pore size is known.
The triangular correlation, eq.1.62, the triangular inner-concave eq.1.63, the spherical described in the preceding section 1.16 are first considered. The reason is simply that, as anticipated in the previous section for the spherical model, in principle and on the basis of the images of real foams (ref.[16]) so like on the basis of the conclusions of the sections 1.13-1.14 those correlations are those that should
0 1000 2000 3000 4000 5000 6000
0,0005 0,001 0,0015 0,002 0,0025
Porosity 0.70 - 0.90
Experimental -Tables 1.7, 1.8, 1.9 Sv-Spherical (this work) - Section 1.16
Sv-Inayat et al.s concave triangular - Eq.1.66 Sv-Concave Triangular (this work) - Eq.1.63 Sv-Triangular (this work) - Eq. 1.62
S
v(m
2/m
3)
d
w(m)
Fig. 1.39. Measured windows size vs. specific surface.
not fit well to the experimental data because it refers to geometrical profiles observed at more higher values of the porosity. At contrary, as confirmed by our analysis of the experimental data of the section 1.13-1.14 it is waited that, for porosity less than 0.90 how are those in the Table 1.7 and 1.9 generally the cross-section of the profile of the strut is circular, or that, as a minimum, this has not a cross-section with concave profile; which should then exclude the predictive capability of the spherical model or of the inner concave correlations of the eq.1.63 or 1.66.
0 1000 2000 3000 4000 5000 6000
400 600 800 1000 1200 1400 1600
Porosity 0.70 - 0.90
Experimental-Tables 1.7, 1.8, 1.9 Sv-Triangular (this work) - Eq. 1.62
Sv-Inayat et al.'s concave triangular - Eq.1.66 Sv-Linear regression
Sv-Spherical (this work) - Section 1.16 Sv-Concave Triangular (this work) - Eq.1.63
S
v(m
2/m
3) S
v(m
2/m
3)
1/d
w(m
-1)
Fig. 1.40. Reciprocal of the measured windows size vs. specific surface.
Experimental and predicted values of the specific surface, Sv, as a function of the window diameter, dw, for porosities less than 0.9, are reported in fig.1.39.
The specific surface area as a function of the window diameter, with experimental data taken from Table 1.7, and 1.8 and predictions of the Inayat et al.'s cylindrical model (eq.1.66), triangular model in this work (eq.1.62), concave-triangular model in this work (eq.1.63), spherical model in this work (eqs. 1.81 and 1.82)., in the 0.89÷0.94 porosity range, is reported in fig.1.39.
0 1000 2000 3000 4000 5000
400 600 800 1000 1200 1400 1600
Porosity 0.70 - 0.90
Experimental-Tables 1.7, 1.8, 1.9 Sv-Cylindrical (this work) - Eq.1.61 Sv-Inayat et al.'s cylindrical - Eq.1.65 Sv-Linear regression
Sv-Richardson et al.'s triangular - Eq.1.64 Sv-Spherical (this work) - Section 1.16
S
v(m
2/m
3)
1/d
w(m
-1)
Fig. 1.41. Reciprocal of the measured windows size vs. specific surface.
Somewhat surprisingly the spherical correlations work better than the concave correlation, which, in turn, seems to work better than the concave Inayat et al.'s ones.
However, the agreement is in all cases better than that for the triangular correlation that, it is again worthwhile to underline, is a first order approximation that does not take into account the contribution of the lumps; it can easily explain the overestimation of the specific surface area.
0 500 1000 1500 2000 2500 3000 3500 4000
400 600 800 1000 1200 1400 1600
Porosity 0.70-0.90
Experimental-Tables 1.7, 1.8, 1.9 Sv-Cylindrical (this work) - Eq.1.61 Sv-Inayat's cylindrical - Eq.1.65 Sv-Linear regression
S
v(m
2/m
3)
1/d
w(m
-1)
Fig. 1.42. Reciprocal of the measured windows size vs. specific surface.
A more detailed analysis is possible if the specific surface area is represented as a function of the reciprocal of the window diameter. Therefore, specific surface area as a function of the reciprocal of the window diameter, with experimental data taken from Table 1.7, 1.8 and predictions of the Inayat's cylindrical model (eq.1.66), triangular model in this work (eq.1.62), concave-triangular model in this work (eq.1.63), spherical model in this work (eqs. 1.81 and 1.82), the linear regression of the data, in the 0.70÷0.90 porosity range, is reported in fig.1.40.
The specific surface area as a function of the reciprocal of the window diameter, with experimental data taken from Table 1.7, 1.8 and predictions of the Inayat's cylindrical model (eq.1.65), Richardson et al.’s triangular model (eq.1.64), cylindrical model in this work (eq.1.61), spherical model in this work (eqs. 1.81 and 1.82), the linear regression of the data, in the 0.70÷0.90 porosity range, is reported in fig.1.41.
It should be pointed out that eq.1.64 makes implicit reference to triangular struts, without any limitations in the porosity range for its application. On the contrary, the other two predicting correlations for cylindrical struts apply for porosities lower than 0.9, according to Bhattacharya et al. [13] and according to the results herein obtained (see sections 1.13-1.14).
Figure 1.41 clearly shows that the triangular concave and the spherical models are the best among those chosen and work acceptably even though they were designed for larger porosities. One can also notice that all the models exhibit a nearly linear trend even though the corresponding correlations exhibit a dependence on the porosity; on the contrary, the triangular concave model shows a marked sensivity to porosity variations that alone can to explain the larger scattering of data.
Also the experimental data are largely scattered, as it was to be expected considering the difficulty in measuring the specific surface, the roughness of the surface, the irregularities of the structure and the different measuring techniques used. Again a linear regression has been proposed for these data implicitly admitting that the
dependence of the surface per unit volume Sv on the porosity is negligible according to the theoretical models considered, as fig.1.41 points out, except for the triangular model, that exhibits the worst prediction of measured data.
Figure 1.42, where the same quantities as those in fig.1.41 are reported, apart from predictions of eq.1.64 and sections 1.16, shows that the cylindrical correlation herein proposed (eq.1.61) matches very well with linear regression of the data, that is, the agreement of such correlation with experimental data is certainly the best one. Also the Inayat's cylindrical correlation works well, even though worse than the other. In any case, however, it has been already underlined that in the author opinion such a correlation has a doubtful theoretical base and gives incoherent results.