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Ind. Eng. Chem. Res. 2009, 48, 2236–2244
Emissivity of High-Temperature Fiber Composites Xiangning Li and William Strieder* Department of Chemical and Biomolecular Engineering, UniVersity of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, Indiana 46556
Effective emissivities of high-temperature fiber composites are calculated for several external edge surface structuressrandomly overlapping, parallel cylinders of radius a protruding out a distance δ from the composite matrix, with central axes either perpendicular to or into the direction of the composite external edge. Firstorder multiple scattering reciprocal bounds provide effective emissivity values, usually within an error of 10% or less; e.g., for carbon-carbon composites with a fiber-matrix emissivity of εs ) 0.8 and a fiber fraction of 30 vol %, the maximum error bound is 4%-5%. The fundamental behavior of the effective emissivity, as a function of protrusion depth, is examined, and significant deep-bed, blackbody radiative effects are observed, even at modest model protrusions. A maximum value for the fiber composite effective emissivity at an optimum porosity is found for any given ratio of protrusion length to fiber radius (δ/a) and fiber-matrix emissivity (εs); e.g., for a perpendicular protrusion with δ/a ) 3 and εs ) 0.5, a maximum effective emissivity enhancement of 27% is found at Φ ) 0.6, and for fibers protruding parallel to the edge with δ/a ) 3 and εs ) 0.5, a maximum enhancement of 21% is observed at Φ ) 0.6. Model equation results agree with the measured effective emissivity values of carbon-carbon brake composites. 1. Introduction The total rate of thermal radiation emitted from a unit area of the external surface of a high-temperature fiber composite can never exceed the corresponding black surface value, and the ratio, which is called the effective emissivity coefficient (εeff), is a measure of its ability to emit thermal energy by radiation. Because radiative heat transport from a fiber composite surface is significant at higher temperatures, the effective emissivity is an important thermodynamic material property. For example, carbon-fiber-carbon-matrix brakes, which are used in commercial and military jet aircraft, can exceed brake temperatures of 2000 °C during a rejected takeoff, and increasing the effective emissivity of both the fibrous friction contact surfaces and external surfaces will impact brake performance and safety.1 Optical and scanning electron microscopy (SEM) microscopic examinations of these surfaces show that they are primarily composed of distinct regions of unidirectional fibers, with all fibers in the region either parallel (denoted by the symbol “|”) or perpendicular (denoted by the symbol “⊥”) to the edge surfaces.1,2 Any further enhancement of the effective emissivity of the high-temperature reusable insulation tiles on the bottom of the National Aeronautics and Space Administration (NASA) space shuttle, (which already have a black radiative-inducing material coating over the tile fibers), will improve the radiative emission of the shearing heat (650-1260 °C) generated by reentry air friction.3 On the other hand, the fraction of the total radiation heat flux that crosses the external surface, enters the porous solid bed, and is absorbed by the local surfaces of the bed material is called the effective absorption coefficient (Reff). The potential use of carbon-fiber-carbon-matrix composites as high-temperature gas-solid heat exchangers for high-temperature furnaces,1,2 or as fusion-facing heat-transfer components for hydrogen fusion electric power plants,4 will be dependent on the composites ability to absorb very high-temperature radiant energy. This absorption will be improved by any enhancement of the effective absorptivity coefficient Reff of the outside edge of the carbon-carbon fiber composite. For the increased * To whom correspondence should be addressed. E-mail:
[email protected].
absorption of solar radiation by the selective surfaces of solar collectors, thin carbon films are deposited on solid substrates by low-pressure direct-current (DC) magnetron sputtering.5 The carbon films generated have a characteristic microstructure that consists of cylindrical carbon columns with central axes oriented perpendicular to the substrate surface. How a thin film that is composed of columnar structures oriented perpendicularly to the solar collector surface can improve the collector’s ability to absorb incoming radiation, which is useful information for solar collector design, is treated in this paper. In each of the aforementioned cases, the effective emissivity (or effective absorptivity) results of this paper will demonstrate precisely how much structural changes of the fiber composite external edge (perpendicular fiber protrusion length, parallel fiber protrusion depth, composite edge porosity) and changes in the local fiber-matrix surface emissivity can improve the composite’s ability to emit (or absorb) radiation. The several associated fundamental phenomena that underlie the effective emissivity (or effective absorptivity) behavior will be clarified. This information can be used either directly, to improve the designed radiation emitting (or absorbing) ability of the aforementioned products, or it can be used to provide an effective thermodynamic coefficient for thermal modeling. The fiber composite edge columnar structure model is constructed from parallel, long right circular cylinders of radius a, randomly placed and allowed to freely overlap. Consistent with micrographs,1,2 the edge fiber columns are all oriented either perpendicular or parallel to the composite edge Σ0. Also, the effective emissivity edge surface model will include the effects of fiber protrusion that occurs due to matrix friction wear,1,2 or edge fiber fracture on machining, or deliberately by design,3-5 because the protrusion will increase the effective emissivity (or absorbtivity) of the fibrous surface. Both the fiber and matrix surfaces are opaque and gray bodies, each characterized by total hemispherical emissivity ε(r) values that are dependent only on the temperature and material surface at the surface point r.6,7 Diffusive emission and reflection are normally assumed. For the case of a large dispersed solid dimension ξ (where ξ is equal to π times the fiber diameter d, divided by
10.1021/ie8008583 CCC: $40.75 2009 American Chemical Society Published on Web 01/15/2009
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2237
the thermal wavelength λth), radiative heat transfer obeys geometric optics.7 For typical fiber dimensions (d ≈ 10 µm) and thermal wavelengths characteristic of very-high-temperature furnaces, fusion energy production, aircraft brake rejected takeoff temperatures, or maximum shuttle re-entry shear temperatures (λth ≈ 1 µm), πd is an order of magnitude or more greater than λth.6 The parameter ξ is well in excess of the critical value of 5 that has been suggested in the work by Siegel and Howell. (See page 568 of ref 7.) Also, any diffraction is almost entirely in the forward direction and can be neglected.6,7 Under these conditions, radiation scattering may be considered as being due to multiple optical diffuse reflections at the void/solid fiber or void/matrix interface. Kirchoff’s law is used, which equates the thermodynamic absorptivity and emissivity coefficients. Any gas within the external edge voids of the fiber composite is regarded as being transparent to the radiation. In section 2 of the paper, model geometries for regions of the fiber composite edge structure, where parallel columnar fibers are all oriented perpendicular (⊥) or parallel (|) to the fiber composite external edge are presented. The fiber and matrix surface to surface transport functions are defined, and first-order multiple scattering reciprocal bounds6 are given. Fiber-bed statistics necessary for the evaluation of these integrals, in the case of fibers perpendicular and fibers parallel to the composite external edge, are developed, and the integrals are calculated in section 3. The results of section 3 for the single fiber-matrix emissivity case are discussed in section 4. High-temperature fiber composites are almost always nonmetallic, and from maximum error bound curves, it is shown that first-order multiple scattering reciprocal bounds are sufficiently accurate in this case, for most practical situations. In section 5, a priori model predictions compare favorably with measured effective emissivities of carbon-carbon composite brake materials over a 210 °C range. A brief summary and conclusion is provided in section 6. 2. Fiber Composite Edge Surface Models and First-Order Multiple Scattering, Effective Emissivity Reciprocal Bounds The microscopic examination of the edge of a typical longfiber, high-temperature composite is usually dominated by two different types of structural regions:1,2 (i) in some regions, unidirectional fibers are mostly oriented perpendicular to the composite outer edge with the fibers sticking out somewhat from a matrix that holds the fibers together, or (ii) in other regions, the fibers are aligned parallel to the edge with the exposed edge fibers protruding out sideways from the edge. A few of the fibers are clustered, but, primarily, they are randomly arranged. Figure 1a shows a fiber-matrix composite edge model with all fiber axes oriented perpendicular to the outer edge Σ0, whereas Figure 1b shows a fiber-matrix composite edge model with all fiber axes aligned parallel to the outer edge Σ0. Fibers in a composite may typically stick together or be pressed together during manufacture, and in the model, the cylinders of radius a are parallel to each other, randomly placed, and allowed to freely overlap as shown. The fibers in either model are worn or cut at an outside cut edge Σs0 located just inside, but infinitesimally close to, the outside edge Σ0 at x ) 0. The coordinate x runs in the positive i-direction from x ) 0 to the composite matrix Σm located at x ) δ. The surface porosity Φ is defined to be the edge void fraction and sf is the area per unit volume of the nonoverlapped void/fiber interface Σf for the volume lying between the x ) 0 and x ) δ planes. The total fiber composite surface Σs is comprised of the sum of Σf, Σs0, and Σm. This paper will
Figure 1. Schematic diagrams of fiber-matrix composite models with all fiber axes (a) perpendicular and (b) parallel to the outer edge Σ0 of the composite.
focus on the effective emissivity of high-temperature fiber composites for the case of a single fiber-matrix emissivity value εs, because this commonly occurs, and because it provides the simplest straightforward illustration of the important physical phenomena. Some instances where the local fiber and matrix surface emissivities differ are treated in the thesis cited as ref 8. The composite is isothermal at a temperature T. Because we assume that Σs is an opaque, gray, diffuse surface, the radiation is emitted and reflected diffusely, according to Lambert’s cosine law.7 The emitted radiative flux from a unit element of Σs located at the cut fiber, void-fiber, or void-matrix surface point r is dependent on the absolute temperature T of the surface, the surface emissivity ε(r) value at point r, and the Stefan-Boltzmann constant σB, in the combination ε(r)σBT4; Kirchoff’s law states that the same surface element at point r will absorb only a fraction ε(r) of the incident radiation, reflecting the portion (1 - ε(r)). The fraction K(r′, r)d2r of the total radiation from the unit surface element located at r′ on Σs, which travels a straightline free path and arrives at a second surface element d2r located at r on Σs, can be used to formulate the radiant exchange between surfaces. Note the function K(r′, r) is also defined when r′ (or r) is located on the void/solid interface but r (or r′) lies on the plane surface Σ0. Because we are assuming diffuse scattering and emission from the void/solid interface, K(r′, r) is given by the cosine law: K(r′, r) ) -
[η(r) · G][η(r′) · G] πF4
K(r′, r) ) 0
(if r′ can see r) (1a)
(otherwise)
(1b)
where η(r) and η(r′) are unit normal vectors at r and r′ on Σs pointing into the void, on Σ0 pointing in the i-direction, or on Σs0 pointing to the plane Σ0 in the -i-direction. A full derivation of the upper and lower reciprocal scattering bounds on the fiber composite effective emissivity (or effective absorptivity) for arbitrary local values of the fiber and matrix emissivities has been developed and discussed in the thesis work.8 The photon scattering paths are denumerated by a scattering index of i successive surface collisions on Σs, and the effective property is expressed as an infinite sum over this index. For manuscript brevity, only an outline of the derivation’s key steps is presented here. (1) The composite effective emissivity εeff can be written as a sum over all the fluxes from photon paths that begin with diffuse emission from a surface element on Σs, travel a zigzag
2238 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
path of i + 1 straight-line segments between emission, i successive diffusive surface scatterings on Σs, and exit across the composite edge surface Σ0. Each of the i scattering integrals contains a K term (eqs 1a and 1b) for each of the i + 1 path segments, the appropriate local emissivity factors for emission or scattering, and i + 1 surface integrals over Σs with one final integral over Σ0. The derived forms are given in the thesis work.8 Although the exact εeff requires the infinite i sum, because each i term is positive, any truncation yields a lower bound. (2) On the other hand, the composite edge absorbed radiation can be cast as the difference of the total entering radiation minus the sum over the i scattering photon path fluxes that enter across Σ0, diffusely reflect from Σs surface elements i times, and exit back across Σ0. Once again, each i term contains a K term from eqs 1a and 1b for each of the i + 1 segments, appropriate local emissivity scattering factors, and i + 2 surface integrals over Σ0, the i scattering surface elements on Σs, and again over Σ0. Again, explicit forms are available in the thesis work.8 Each of the i scattering photon path fluxes is positive, and although the exact effective absorptivity is generated by the full sum to infinity, any truncation i ) 1, 2,... gives an upper bound on the effective absorptivity. (3) Using Kirchoff’s law, the effective absorptivity (Reff) and effective emissivity (εeff) are equal, and the key steps described in steps 1 and 2 generate a reciprocal bound squeeze on the effective emissivity (or absorptivity), along with a midpoint estimate (εjeff) and an associated maximum error bound (∆εeff). (4) If both upper and lower bound sums are truncated at the same i value, the bounds are physically reciprocal. They not only provide a squeeze on εeff, but the evaluated path integrals for the upper bound also can be used to calculate the lower bound. For many practical problems, the first-order i ) 1 scattering bounds have sufficient accuracy that good analytical forms of jεeff can be obtained, e.g., either for smaller ratios of protrusion length to fiber radius (δ/a) or for surface emissivities of >0.5. Generally, even for simulations, the squeeze of the multiple scattering bounds can be used to eliminate timeconsuming Monte Carlo path simulations9,10 of εeff, because only the shortest i paths must be considered and the squeeze provides a built-in measure of accuracy. (5) For most common applications, the fiber and matrix are composed of similar materials (e.g., carbon-fiber-carbon-matrix composites) and a single fiber-matrix surface emissivity εs (εs ) εf ) εm ) εs0) can be assumed. In this case, the thesis work8 has shown that the first-order reciprocal scattering bounds on εeff have the form εs + εs(1 - εs)[1 - I(0, f, 0) - I(0, m, 0) - I(0, s0, 0)] < εeff < 1 (1 - εs)[I(0, f, 0) + I(0, m, 0) + I(0, s0, 0)] (2)
with I(i, j, k) ) Σ-1 0
∫∫∫ Σi
Σj
Σk
K(r′′, r′)K(r′, r) d2r′′ d2r ′ d2r (3)
where each index i, j, or k can be designated as 0, f, m, s0, or s. 3. Evaluation of the Single Scattering Integrals for Fibers Perpendicular and Parallel to the Composite Edge To evaluate the integrals I(0, f, 0), I(0, m, 0) and I(0, s0, 0), two models are generated for the high-temperature composite edge fibers protruding perpendicular (β ) ⊥) and parallel (β ) |) to the composite edge. Both structures are the ones most commonly seen upon microscopic examinations of the edge of typical composites.1,2 In both models, a slab of finite thickness
Figure 2. (a) Schematic diagram of the ω-plane for fibers whose central axes are aligned perpendicular (β ) ⊥) to the (x ) 0) edge plane, and the exclusion area A(ζ′, ζ) for a two-step path. (b) Schematic diagram of the ω-plane for randomly overlapping fibers whose central axes are aligned parallel (β ) |) to the (x ) 0)-plane, and the exclusion area A(σ′, σ) for a two-step photon backscattering path.
δ, but infinite in the other two dimensions, is cut from an infinite bed of very long, parallel, right circular cylinders placed at random in an infinite volume and allowed for overlap freely. A point in the slab within one or more cylinders is in the solid fibers, those points that lie outside the cylinders comprise the void, and points on a cylinder surface not overlapped by any other cylinder comprise the fiber/void interface Σf. For the case of fiber axes perpendicular to the edge (β ) ⊥), the parallel cut planes x ) 0 and x ) δ are sliced perpendicular to the direction of the parallel cylinder axes, and the cut finite slab is placed on the matrix plane at x ) δ. The cut fiber surfaces at x ) 0 comprise the surface Σs0 of the composite edge, whereas those points on the plane x ) δ not covered over by a fiber comprise the matrix plane Σm (see Figure 1a). For the fibers protruding parallel to the composite edge (β ) |), the cut planes x ) 0 and x ) δ run parallel to the direction of the cylinder axes, and when the slab is laid on the x ) δ plane, the upper edge of the cut fiber at x ) 0 generates the surface Σs0, wherea those points on x ) δ not directly under a portion of cut solid fiber comprise the matrix plane Σm (see Figure 1b). The fiber bed statistics in both cases are similar. To view this, an orientation vector ω is assigned along the central axis of each cylinder, and an ω plane is defined perpendicular to the ω unit vector. The cylinders with orientation ω appear as randomly overlapping circles of radius a in the ω-plane. Figures 2a and 2b show the ω-planes for the fibers aligned both perpendicular (β ) ⊥) and parallel (β ) |) to the edge plane x ) 0. The ω-plane for β ) ⊥ (Figure 2a) is parallel to both the external plane Σ0 and x ) 0 with the unit vectors ω and i, pointing into the paper. The ω-plane for β ) | (Figure 2b) is perpendicular to the Σ0 and x ) 0 planes, with the unit vector i pointing across the ω-plane in the positive x-direction from x ) 0 to x ) δ. The probability P that no circle center lies within the area A in the ω-plane can be written in terms of the density n per unit area of circle centers in the ω-plane:11 P ) exp(-nA)
(4)
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The void fraction can be interpreted as the probability that a randomly chosen point in the bed falls in the void, or the probability that, in the ω-plane, no circle has its center within a distance a of the random point. Then, from eq 4, we have Φ ) exp(-nπa2)
(5)
The surface area µ, overlapped or not, of those cylinders with orientation ω within a unit total volume of cylinder bed is µ ) 2πna (6) From the product with the void fraction Φ and eq 6, we obtain the exposed surface per unit total bed volume: sf ) 2πnaΦ
(7)
The cut fiber surface Σs0 in Figures 1a and 1b are similar for both the β ) ⊥ and β ) | cases, and the integral I(0, s0, 0) is the same. All the radiation that leaves the surface element d2r′ on Σs0 must pass across to the surface Σ0 immediately adjacent to d2r′; therefore, the definition (eq 3) of I(0, s0, 0) gives I(0, s0, 0) ) 1 - Φ
(8)
where the ratio of the areas Σs0 to Σ0 is 1 - Φ, where Φ is the edge void fraction. Using the form of K from eqs 1a and 1b, we can write the other two integrals in the reciprocal bounds (eq 2), in terms of a free path function defined for either the (r′′, r′) or (r′, r) pairs as h(r ′ , r) )
{
1 (if r can see r′) 0 (otherwise)
(9)
For either geometry, the forms from eqs 1a and 1b will include the variable assignments η(r′′) ) i, η(r) ) i, G′ ) r′′ - r′ and G ) r′ - r and from the definition (eq 3), I(0, m, 0) ) Σ-1 0
h(r′′, r′)h(r′, r)δ4 2 d r′′ d2r′ d2r Σ0 [π2(FF′)4] (10)
∫∫ ∫ Σ0
Σm
rectangle in the ω-plane formed by drawing parallel lines a distance a from either side of the projected free-path vectors ′ ′ and Gωp. The net area A(Gωp , Gωp) of exclusion, shown in Gωp Figures 2a and 2b, has the same shape for β ) ⊥ and β ) |, ′ , Gωp)] is given by and the exclusion probability exp[-nA(Gωp eq 4. The first scattering integral I(0, m, 0) from eq 10 becomes I(0, m, 0) )
∫∫∫ Σ0
Σf
Σ0
h(r′′, r′)h(r′, r)(i · G′)[-η(r′) · G′] × [η(r′) · G](-i · G) 2 d r′′ d2r′ d2r (11) [π2(FF′)4]
Although the exclusion areas for the two-segment free paths A(ζ′, ζ) and A(σ′, σ) shown in Figures 2a and 2b lie in different planes, they have the same shape for both β ) ⊥ and β ) |. For the first scattering integral I(0, m, 0) defined by eq 10, the operation Σ0-1∫Σ0 d2r′′ can be interpreted as an average over the entire plane Σ0. Because the integrand of eq 10 is dependent only on G′, G, and the free- path functions h of (r′′, r′) and (r′, r), the Σ0 average counts the probability of the two-segment free path G′ and G mutliplied by the corresponding value of the integrand and the surface differentials d2G′ and d2G, summed over the (x ) δ) and (x ) 0) infinite planes, respectively. Note for the case of fibers oriented perpendicular to the composite edge that the ω-plane will be parallel to the edge plane Σ0, whereas for fibers parallel to the edge, the ω-plane will be perpendicular to Σ0 (see Figures 2a and 2b). The stipulation that the two-segment free path must be free of obstruction (i.e., h(r′′, r′) and h(r′, r), as defined by eq 9, must be unity) requires that the statistically independent cylinder centers within the ω-plane be excluded from circles of radius a about the base and tips of the vectors G′ and G, projected onto the corresponding ′ and Gωp (i.e., ζ′ and ζ for β ) ⊥ or ω-planes, respectively, Gωp σ′ and σ for β ) |). Centers must also be excluded from a
2
′
4
′ , G )] d2G′ d2G exp[-nA(Gωp ωp
(12) where the d2G′ and d2G terms are, respectively, over the entire (x ) δ) and (x ) 0)-planes, and the pair G′ωp, Gωp are either (ζ′, ζ) for β ) ⊥ or (σ′, σ) for β ) |. For the second scattering integral I(0, f, 0), again, the operation Σ0-1∫Σ0 d2r′′ is an average over the x ) 0 plane, and because the integrand of the integral described by eq 11 is dependent only on G′, G, the surface normal η(r′) on Σf, and the free-path functions (eq 9) for the paths (r′′, r′) and (r′, r), the Σ0 average counts the probability of the events shown in Figures 2a and 2b, multiplied by the corresponding value of the integrand from eq 11 with the surface differentials. This result must then be summed over all geometrical possibilities of η(r′) and the (G′, G) paths. If d3G′ is a volume element in the protrusion region [0, δ] of x, then from the definition described by eq 6, µd3G′d2η/(2π) is the surface area, overlapped or not, within d3G′ with a surface unit normal η at r′ within d2η. The surface element on the x ) 0 plane is given by d2G. The stipulation that the two-segment free path and the surface point r′ be free of obstruction requires that the statistically independent cylinder centers lie outside of the exclusion areas A(ζ′, ζ) or A(σ′, σ) shown in Figures 2a and 2b. The exclusion ′ , Gωp)] and the second probability is given by eq 4 as exp[-nA(Gωp scattering integral becomes I(0, f, 0) )
and I(0, f, 0) ) Σ0-1
4
∫ ∫ π (Fδ F)
∫∫∫
µ(i · G′)[-η · G′][η · G](-i · G) × π2(F′F)4 ′ , G )] d3G′ d2G d2η exp[-nA(Gωp ωp (13) 2π
η·G′e0,η·Gg0
where the d3G′ integration is over the protrusion volume, infinite in two dimensions and of thickness from x ) 0 to x ) δ, the d2G integration is over the entire surface x ) 0, and the d2η integration is over all possible unit surface normal orientations subject to the conditions that -η · G′ and η · G are non-negative. These conditions ensure that the cylinder whose surface is at r′ does not block either of the two segments G′ or G of the path. For any cylinder, the orientation of the surface normal will be perpendicular to ω and lie in the planes shown in Figures 2a and 2b. The exclusion area A(G′ωp, Gωp) shown in Figures 2a and 2b can be bounded for both the β ) ⊥ and β ) | cases from above and below by ′ , G ) e A(1) A(2) e A(Gωp ωp
(14a)
with ′ +F ) A(γ) ) πa2 + γa(Fωp ωp
(for γ ) 1, 2)
(14b)
Because the integrands of both scattering integrals (eqs 12 and 13) are positive, γ ) 2 gives a lower bound and γ ) 1 gives an upper bound, when the exclusion area bounds (eqs 14a and 14b) are substituted into the scattering integrals for either β ) ⊥ or β ) |: I(0, j, 0;β;2) e I(0, j, 0;β) e I(0, j, 0;β;1) (for β ) ⊥ , | and j ) f, m) (15)
2240 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
To proceed further, to calculate the scattering integrals bounding estimates I(0, j, 0; ⊥; γ) for the εeff inequalities (eq 2) when β ) ⊥, the vectors G′ and G are respectively split into components x and -x in the ω-direction of the cylinder axes and ζ′ and ζ parallel to the ω-plane. For the first scattering integral (eq 12), the integration d2G′ ) ζ′dξ′dζ′, with limits of 0 e ξ′ e 2π and 0 e ζ′ < ∞, refers to the (x ) δ)-plane of Figure 1a, and d2G ) ζdξdζ, with similar limits of 0 e ξ e 2π and 0 e ζ < ∞, is over the (x ) 0)-plane. For β ) ⊥ with the exclusion area bounds (expressions 14a, 14b, 15, and, for Φ, eq5), the first scattering integral (eq 12) becomes 4
∫ ∫∫ ∫ ∞
∞
δ × 2 0 0 0 0 [((ζ′) + δ2)(ζ2 + δ2)]2 exp[-nγa(ζ′ + ζ)]ζ′ dξ′ dζ′ζ dξ dζ (16) Equation 16 generates a lower-bound estimate for γ ) 2 and an upper-bound estimate for γ ) 1. These integrals can be expressed directly in terms of the auxiliary functions f and g for the sine si(y) and cosine ci(y) integrals12
Φ I(0, m, 0; ⊥ ;γ) ) 2 π
2π
2π
f(y) ) ci(y) sin y - si(y) cos y
(17a)
and g(y) ) -ci(y) cos y - si(y) sin y and the analytical form
(17b)
I(0, m, 0; ⊥ ;γ) ) Φ[1 - γτf(γτ)]2 (18a) where the dimensionless protrusion depth thickness τ, in terms of the surface void fraction (eq 5), is δ ln Φ (18b) aπ In the second scattering integral (eq 13) for the case β ) ⊥, the volume integral d3G′ ) dx′ζ′dξ′dζ′ is over the fiber protrusion slab (i.e., 0 e x′ e δ, 0 e ξ′ e 2π and 0 e ζ′ < ∞) and the surface integral d2G ) ζdξdζ is over the (x ) 0)-plane, with the same limits as those for eq 16. The limits for the d2η cylinder surface normal integration are due to the conditions -η(r′) · G′ g 0 and η(r′) · G g 0 (or, in this case, - η(r′) · ζ′ g 0 and η(r′) · ζ g 0) applied to the case where η is perpendicular to the x-direction of the cylinder central axis. The coordinates are shown in Figure 2a and eq 13, with the exclusion area (eq 14b), becomes τ ) naδ ) -
I(0, f, 0; ⊥ ;γ) )
( )∫ ∫ ∫ µ 2π3
δ
0
∞
0
2π
0
∫ {∫ ∫ ∞
0
π
0
π⁄2
ξ′-(π⁄2)
+
}
(x′)2(ζ′)2ζ2 cos η cos(ξ - η) × π 3π⁄2 [(x′)2 + (ζ′)2]2[(x′)2 + ζ2]2 exp[-n{πa2 + γa(ζ′ + ζ)}] dx′ dζ′ dξ′ dζ dξ dη (19) All the integrations in eq 19 can be performed analytically, except for the finite one-dimensional integration over x′; in that case, eq 19 becomes, in terms of the dimensionless protrusion depth thickness τ (eq 18b),
∫ ∫ 2π
ξ′+(π⁄2)
( )∫
2Φ γτ I(0, f, 0; ⊥ ;γ) ) [f(x) - xg(x)]2 dx (20) πγ 0 where eqs 5 and 6 have been used, respectively, for Φ and µ, and eqs 17a and 17b have been used to define f and g, respectively. This second scattering integral (eq 20) can be readily evaluated numerically and used with eqs 2, 8, 15, and 18a to obtain values of εeff for the fiber composite edge model with fibers oriented perpendicular to the edge (β ) ⊥). In circumstances where the ratio of fiber protrusion to fiber cylinder radius (δ/a) is not large, and the integral I(0, f, 0) values
are limited, simpler analytical bounds can be used in place of eq 20 for the effective emissivity calculations. From the Schwartz inequality, expressions 5, 7, and 18b, and the definitions given in eqs 1a and 1b, general fully analytical bound forms on the scattering integral (eq 3) for I(0, f, 0) can be developed: 2πΦτI2(0, f) e I(0, f, 0) e I(0, f)
(21)
where I(0, f) ) Σ-1 0
∫ ∫ K(r′, r) d r′d r 2
Σ0
Σf
2
(22)
This scattering integral (eq 22) for fibers oriented perpendicular to the composite edge (β ) ⊥) is evaluated in a manner similar to I(0, f, 0; ⊥; γ), as I(0, f; ⊥ ) ) 2Φτf(2τ) (23) When this analytical result (eqs 21 and 23) is substituted in place of the numerical calculation of integral (eq 20) in the reciprocal bounds (eq 2), along with expressions 8, 15, and 16, a fully analytical set of reciprocal bounds is found for the effective emissivity. For carbon-carbon composite surfaces with a local surface emissivity of εs ) 0.8, a fiber volume fraction of 30% (i.e., 70% surface porosity) and a ratio of δ/a ) 2, the analytical forms (eqs 21 and 23) only increase the maximum error bound of a midpoint estimate of εeff from the single scattering inequalities (eq 2) by ∼2%. For smaller protrusions, the error would be even less. To evaluate the scattering integrals (eq 15) when the aligned overlapping fibers are parallel to the composite edge (β ) |) (see Figure 2b), the vectors G′ and G are respectively split into components z′ and z in the ω-direction of the cylinder axes and σ′ and σ parallel to the ω-plane. For cylinders parallel to the outside edge plane Σ0, the bed is invariant in the z′- and z-directions with infinite limits -∞ e z′ e ∞ and -∞ e z e ∞ and the x′-direction points away from the Σ0 plane into the composite. For the scattering integral (eq 12), the vectors σ′ and σ can be further split into a scalar component δ across to the (x ) δ)-plane and vertical components y′ and y with limits -∞ e y′ e ∞ and -∞ e y e ∞. With this coordinate system, using the exclusion area bounds (eqs 14a, 14b, and 15, and, for Φ, eq 5), we can write the scattering integral (eq 12) for β ) | as
()∫
∞ ∞ ∞ ∞ δ4 Φ [(z′)2 + δ2 + 2 -∞ -∞ -∞ -∞ π (y)2]-2[z2 + δ2 + y2]-2 × exp{-nγa[(δ2 + y2)1⁄2 + (δ2 + (y)2)1⁄2]} dz′ dz dy′ dy (24)
I(0, m, 0; | ;γ) )
∫ ∫ ∫
This integral (eq 24) can, in turn, readily be shown to be the repeated integral of the modified Bessel function of the second type, K012 I(0, m, 0; | ;γ) ) Φ[Ki2(γτ)]2
(25)
where τ is given by eq 18b. The second scattering integral (eq 13) for β ) | is most conveniently written in terms of the coordinates shown in Figure 2b. The differential elements of the integral described by eq 13 can be written as d3G′ ) dz′σ′dσ′dθ′ and d2G ) dzσ′ cos θ′ (cos θ)-2 dθ, where the angles θ of -σ and θ′ of σ′ are measured from the i-vector to the x-direction of Figure 2b, counterclockwise positive. The angle η between -σ′ and the cylinder surface normal η(r′) at r′ lies in the ω-plane and is also positive counterclockwise. In terms of these coordinates, the exclusion
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2241
area bounds (expressions 14a, 14b, and 15, and, for Φ, eq 5) the scattering integral (eq 13) for β ) | becomes I(0, f, 0; | ;γ) )
( Φµ 8π )∫ ∫ {∫ δ
0
∫ ∫ π⁄2
θ′
π⁄2
θ-θ′-(π⁄2)
π⁄2
θ′
-π⁄2
-π⁄2
∫
(π⁄2)-θ′+θ
-π⁄2
+
} cos η cos(θ - θ′ - η)
×
( cosx′ θ′ + cosx′ θ )] dx′ dθ′ dθ dη (26)
[
exp -γna
The η and x′ integrations in eq 26 can be performed analytically; however, to obtain the tightest bounds, the angular integrations θ′ and θ are done numerically for the γτ values of interest: I(0, f, 0; | ;γ) )
( 4γΦ )∫
π⁄2
-π⁄2
∫
π⁄2
θ′
[(π - θ + θ′) cos(θ - θ′) +
cos θ′cos θ × cos θ′ + cos θ 1 1 + 1 - exp -γτ dθ′ dθ (27) cos θ cos θ′ where τ is defined by eq 18b. As with the fibers perpendicular to the composite edge, when the fiber protrusion δ/a is not large, the analytical bounds (eqs 21 and 22) on the scattering integral I(0, f, 0) can be used to obtain simpler fully analytical reciprocal bounds on εeff from the reciprocal inequalities (see eq 2) for the case of aligned fibers oriented parallel to the fiber composite edge (β ) |). The scattering integral I(0, f) from eq 22 for β ) | can be evaluated using procedures that are similar to those used for the integrals described in expressions 24, 25, 26, and 27 to obtain sin(θ - θ′)]
{
[ (
I(0, f; | ) ) Φ[1 - Ki2(2τ)]
)]}
(28)
where Ki2 is the second repeated integral of the modified Bessel function K0 of the second type12 and τ is given by eq 18b. For carbon-carbon composite surfaces with a local surface emissivity εs ) 0.8, a fiber volume fraction 30% (i.e., 70% surface porosity), and a protrusion ratio of δ/a ) 2, using the analytical forms described by eqs 21 and 28 for I(0, f, 0) in the inequality described by eq 2 increases the maximum error bound from the midpoint estimate by 0.40, for fixed small δ/a, will decrease as the porosity increases, opposite behavior to the increase in effective emissivity with porosity at large δ/a values, and these constant porosity curves must then cross over each other. As a result, we observe, in each of Figures 3a-d, at various intermediate δ/a values, the maximum enhancement will occur at an intermediate porosity. For example from Figure 3a at δ/a ) 3, εs ) 0.5 a maximum enhancement of 27% occurs at Φ ) 0.6 for perpendicular fibers, and, similarly, for δ/a ) 3, εs ) 0.8, a maximum enhancement of 8% occurs at Φ ) 0.6 for fibers parallel to the composite edge. Fiber composite effective emissivities are important, primarily for high-temperature fiber composite applications of the types discussed in the Introduction. Generally, at these high temperatures, metal fibers and metal matrices do not hold up well in service, and high-temperature composites are usually composed of nonmetals. The local surface emissivities of nonmetallic materials14,15 are >0.6, and, therefore, the application εs values for the model are selected to be >0.5. The reciprocal bounds (eq 2) with expressions 8, 15, 18a, 18b, and 20 for β ) ⊥, and expressions 8, 15, 18b, 25, and 27 for β ) | give two results: the midpoint values jεeff in Figures 3a-d and, for each jεeff, a maximum error bound ∆εeff. The effective emissivity reciprocal bounds are always valid; however, for many important cases, the jεeff values can be very accurate. In Figures 4a and 4b, the maximum percent error bound (100∆εeff/εjeff) curves for the midpoint jεeff are shown for various surface void fractions (Φ), fiber-matrix surface emissivities (εs),
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2243
protrusion depth ratios (δ/a), and, respectively, fibers oriented perpendicular and parallel to the composite edge. Each δ/a curve is the 10% error locus for that protrusion depth. As noted in the discussion of Figures 3a-3d, for any finite δ/a, the reciprocal bound midpoint gives the correct value (i.e., εs) for either limit Φ f 0 and Φ f 1, so that the 10% error curves decrease to zero at either side of Figures 4a and 4b. However, in the limit δ/a f ∞, fiber wall scattering will remain important until Φ is exactly unity, and no fibers are present. As a result. the 10% error curves for δ/a f ∞ seems to be discontinuous at Φ ) 1. Note that the larger δ/a curve (δ/a ) 15) has already begun to show impending signs of this behavior. Above the δ/a curve, the error is 0.6, and the midpoint estimate maximum errors above the εs ) 0.6 line in Figures 4a and 4b are usually
