244
Ind. Eng. Chem. Fundam. 1986, 25, 244-249
Radiation across and down a Cylindrical Pore Having both Specular and Diffuse Reflectance Components Dah Shyang Tsal and Wllllam Strleder’ Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
Radiation heat-transfer coefficients across and down along the axis of a long straight capillary pore are derived. The most general model is assumed: a gray-body pore wall surface which reflects radiation with both specular and diffuse components. The axial radiation coefficient is always larger than the coefficient for radiation across the axis. This difference becomes very large at lower emissivities and is further increased by specular reflections.
Introduction The transport of radiation within the void spaces of a porous solid at high temperatures will always r e q ~ some e consideration of structure. For a number of situations such as the gasification of a porous solid (Gavalas, 1980; Bhata and Perlmutter, 1981) or heat transport away from a local high temperature in a porous solid catalyst (Kummer, 1980), pore radiation heat transport may be present. In the past, when structural effects on transport or reaction were examined (Smith, 1981), a capillary has often been a useful model for a pore. Franc1 and Kingery (1954) have observed both radiation heat transport above 500 “C and pore orientation effects for cylindrical pores within refractory materials. Parallel cylindrical channels have been considered as a structure for the monolith converter (Young and Finlayson, 1976); in operation, radiation heat transfer within the channels will be significant. An analysis of radiant heat transport across and down a long straight cylindrical pore is the purpose of this paper. The most general gray-body pore wall surface will reflect a mixture of both diffuse and specular components of radiation. Corlett (1966) has performed Monte Carlo calculations of radiant heat transport down the axis of a cylinder having a specular-diffuse reflecting surface and a constant pore wall temperature. Though we include pore wall thermal gradients, this problem is similar to our axial transport case. For length-to-diameter ratios greater than unity Corlett found that Monte Carlo calculations require long computer times and give results of limited accuracy. Radiant heat transport down the axis of cylindrical geometries has also been numerically calculated by semianalytic techniques, by Perlmutter and Siegel (1963), but only for the limits of pure diffuse and pure specular surface reflection. In this paper, we will derive two simple, exact equations for the thermal radiation heat conductivity respectively for transport down and across the axis of a long cylinder with mixed specular-diffuse surface reflection. To obtain these results, in each case we sum over all possible radiation paths of successive specular wall reflections and obtain an equation applicable to both the pure diffusive and the specular surface limits, as well as intermediate combinations. At moderate to lower emissivities, typical of oxide materials, we find heat transfer down the axis is heavily favored over that across the axis. Equations of Radiant Interchange in a Cylinder for Specular-Diffuse Reflecting Surfaces The directional properties of reflectance have been investigated by Torrance and Sparrow (1965, 1966, 1967),
* To whom
correspondence should be addressed.
Birkebak (1964), and Birkebak and Eckert (1965). In general, the directional distribution of reflectance depends on the surface roughness, the wavelength, the incident angle, and the nature of the material. It is impossible and unrealistic to include all these factors in complete detail in engineering models. On actual engineering surfaces it is reasonable, as a first approximation (Sparrow and Lin, 1965; Siegel and Howell, 1981), to represent the reflectance p as being divided into diffuse 6 and mirror specular y components p = y + 6
(1)
Magnitudes of p and 6 have been reported in the literature both for metallic (Birkebak et al., 1964; Munch, 1968) and ceramic (Torrance and Sparrow, 1965) materials. In addition, we assume an opaque gray body, whose radiation is emitted diffusely according to Lambert’s cosine law (Siegel and Howell, 1981). The emitted flux depends on the absolute temperature, T, of the surface, the surface emissivity, t, and the Stefan-Boltzmann constant, u, in the combination c a p . Kirchoffs law states that the same surface element will absorb only the fraction e of the incident radiation, while reflecting the fraction p = 1- e, so that t+p+6=1
(2)
If a surface possesses both diffuse and specular reflection, we let H represent the radiant flux incident on a unit surface. Then for a diffusely emitting surface with a diffuse reflectance component, 6, the radiosity, B , given by B = CUP + 6H (3) represents the diffusely distributed radiant flux leaving a unit surface element. The fraction of the diffusely distributed radiation from a unit surface element, located at r’, that arrives at a second area element dA, located at r, both directly and by all possible sequences of specular interreflections, E(r,r’) dA, the exchange factor, can be employed to formulate the equations of radiant interchange between surfaces. Sparrow and Lin (1965) have stated a general reciprocity principle for the exchange factor
E(r,r’) = E(r’,r)
(4)
This relationship will be verified for the surfaces of a cylindrical pore from the analytical form of the exchange factor (13) derived later in this section. The radiant flux leaving the surface element dA’ at r’ is composed of a diffusely distributed portion B(r’) dA’ plus a specularly reflected contribution. Of the diffusely distributed radi-
0196-4313/86/1025-0244$01.50/00 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 2,
1986 245
ation leaving dA’, only the fraction B(r’) dA’E(r,r’) dA will arrive at dA either directly or by all possible sequences of specular surface collisions; then the total radiant flux H(r) incident to a unit area at r is given by the integral over all surface elements of the cylindrical pore
H(r) = JE(r,r’) B(r’) dA’
(5)
We note that when the length-to-diameter ratio of the cylinder is large, the area of the ends becomes small compared to the cylinder walls, and end contributions in (5) for all but perfectly reflecting wall surfaces can be neglected. Fourier’s law in the surrounding solid may be coupled with the void radiation equations. The thermal boundary condition (Chiew and Glandt, 1983) equating the net radiative flux from the void surface to the normal flux from the solid into the void at the surface point r is aT K- = B p H - H (6) an where aT/an is the normal temperature derivative pointing outward from the void into the solid and K is the solid thermal conductivity. That all the radiation that leaves diffusely from a unit surface at r must eventually either reflect diffusely or absorb somewhere within the internal surface of the cylinder requires a unit value for the integral
+
JE(r,r’)[l - p(r’)] dA’ = 1
(7)
Note in (7)we have used the reciprocity property (4); the exchange factor is unchanged upon interchange of r and r’. Forms of B and H derived from the simultaneous solution of (3) and (6), along with (21, can be used to express the integral equation (5) in terms of the temperature and its normal derivative at the wall surface. The resulting integrodifferential equation for the temperature is written in the more symmetric form of (8) upon multiplication of the last term on the right-hand side by (7)
(-”aTo
dA’and dA as viewed projected onto a plan perpendicular to the cylindrical axis: (-), three possible paths for two specular reflections; (- - -), one possible path for no specular reflections.
cessive surface mirror reflections, i.e., 1 + i cords running from r’ to r. Each subtended angle and cord form an isosceles triangle, and as the incidence and reflection angles are equal, so are the triangles and 1 + i cords. If i specular reflections occur between radiation leaving diffusely from dA ’and arriving at dA, more than one type of path is possible. In Figure 1for the case of two specular reflections, i.e. i = 2, the three different paths are shown. To count the total number of paths possible for the general case of i specular collisions between dA ’and dA,we define an index j that runs from 0 through positive integer values. Each value of j specifies a different path. In addition, we define the angle d i j measured counterclockwise from the radial line to dA ’in Figure 1and the radial line to the first of the i specular wall interactions of the jth path. For the special case of no specular reflections, i = 0, the only possible value of j is zero, and the angle +o,o is given in Figure 1. For the general case of i specular reflections, one of the simplest paths, designated j = 0, is made up of 1 i cords of equal length running counterclockwise from dA’ to dA. The angle between the radial lines to dA ’and to the first specular wall collision is
+
+i,o
I
+ (1- p)4aTo3[T(r’)- T(r)]
dA’ (8)
an In writing (€9,we have retained only terms first order in the temperature gradient. This linearization presumes the temperature variation across and down the cylinder is small compared to the average pore temperature To, is consistent with the linearization of Fourier’s law in the solid, and is a necessary step to obtain radiation thermal conductivities for the void. As the temperature variations of the various transport and surface coefficients occur only on second- and higher-order terms, linearization of the equations has the effect of neglecting their temperature dependence and evaluating them at To. The exchange factor E(r,r’) dA,the fraction of radiation originating diffusely at r’ from the area element dA’ that arrives within the area element dA located at r by all possible paths of i (= 0, 1, 2, ..., m) successive specular surface reflections, will now be derived. Radiation travels within the cylindrical pore in a helical manner, and possible paths are restricted by the cylindrical geometry. As an illustration, a view down onto the axis of the cylinder is presented in Figure 1. The cylinder walls appear as a circle of radius R , and the area elements dA‘ and dA appear as elements on the circle at r’ and r, respectively. The radiation path from r’ to r with i specular reflections projects onto the plane perpendicular to the cylindrical axis as 1 + i connected straight-line segments between suct
d A‘
Figure 1. Paths of specular reflection in cylindrical pore between
= +o,o/(l
+9
(9)
For the next path J’ = 1 the 1 + i equal cords must pass dA,circumnavigatethe circle once, and return to dA. The sum of the 1+ i angles from the sphere center subtended by each cord must equal +o,o + 2 ~ hence, ; the first angle is +i,1
= (+o,o
+ 2 ~ ) / ( 1+ i)
(10)
Then for j = 2 the cords will circumnavigate the circle twice, and for arbitrary j for the first specular collision j = 0, 1, ..., i (11) +ij = (+o,o + 27rj)/(l + i) Note that for +ii greater than P, while the specularly reflected heat transfer will travel clockwise, the +iiangles between the radial lines to dA’and to the first specular reflection and angles between the radial lines to subsequent successive specular collisions on the path are still measured counterclockwise. The j values from 0 to i in eq 11 each generate a distinct path, but j values of 1 + i or larger simply repeat paths already counted, so j stops at i in eq 11. In the direction down the cylinder we define the axial coordinate z at dA and z ’at dA ’. In traveling from dA ’ to dA in a path with i successive specular wall reflections, the radiation will advance an equal distance lz - z ’ l / ( l + i) between successive specular reflections. To arrive at an element of area dA (= R d+o,odz) after i specular reflections from the cylinder wall, the radiation in the ij path, after leaving dA‘ diffusely, must make its first wall re-
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
248
Table I. Ratios of -q/4aTo3Rcu for Various Surface Emissivities
(e)
P
and Fractions of the Total Reflection That Is Specular
(U/D
€
PIP
0.1
0.3
0.5
0.0
0.2051 2.6667 0.2053 3.1941 0.2058 4.6393 0.2064 7.0303 0.2071 11.7477 0.2082 25.4035 0.2090 50.6667
0.6486 2.6667 0.6502 3.0681 0.6536 4.0844 0.6577 5.5385 0.6627 7.7908 0.6689 11.7477 0.6728 15.1111
1.1429 2.6667 1.1462 2.9474 1.1535 3.6078 1.1618 4.4444 1.1712 5.5385 1.1822 7.0303 1.1884 8.0000
0.1 0.3 0.5
0.7 0.9 1.0
0.7 1.6970 2.6667 1.7014 2.8316 1.7106 3.1941 1.7206 3.6078 1.7314 4.0844 1.7431 4.6393 1.7493 4.9524
0.9 2.3226 2.6667 2.3253 2.7205 2.3308 2.8316 2.3365 2.9474 2.3423 3.0681 2.3483 3.1941 2.3513 3.2593
1.0 2.6667 2.6667
"The upper row, -qp/40T03Rarp = 2Xp/4uTo3R = 2e,, is given by (17) and (33), and the lower row is - q , / 4 ~ ~ ~= ~XZ/4uTo3R ~a, from (30) and (34). bTo obtain a transverse radiant heat flux across a cylindrical pore independent of the conductivity of the surrounding solid matrix, the comparison in Table I is restricted to cases where in (19) the ratio Xp/K can be neglected. The upper entry is also twice the dimensionless transverse coefficient of radiant heat conduction, while the lower entry is the dimensionless axial conductivity for radiant heat; Le., Table I compares directly transverse and axial radiant thermal conductivities.
flection within dA,,. The angle element for the first wall reflection, d40,0/(l+ i),can be obtained from direct differentiation of @ii from (11);for fixed z'the axial dimensional element for the first reflection is &/(l+ i),and the area dAij = R dc$o,o dz/(l
+ ')i
(12)
The area formula (12) can also be justified by solid angle arguments (Tsai et al., 1985). For the ij path contribution to the exchange factor the radiation must leave dA'at r' diffusely and arrive within dAii at coordinate 4ii and ( z - z ? / ( l + i). The fraction of diffuse radiation that leaves dA'at r' and arrives at dAi, is given by the standard view factor (Strieder and Prager, 1968). Each ij path is made up of a diffusive origin from dA'at r', followed by i successive specular reflections, and arrival within dA at r. The index k is now defined to designate each successive member of the sequence of specular wall reflections for a given ij path from 1through i. The specular reflectance pi:& at the surface point of the kth specular reflection on the jth path of those paths with i specular reflections gives the fraction of incident radiation from r' that is mirror reflected at each wall interaction. As the radiation must be reflected i times in the ij path, the factor nk=lpii,k must be included. The exchange factor for diffusive origin of radiation at r' that arrives within a unit area at r directly or after a sequence of specular wall reflections includes all ij paths; hence, a sum over i and j must be performed, and
T = To + CY,*,$
+ W(R/[)~CY,*~
T
-K
-(1 E
pii,k4R2(1+ i)2 sin4 (4ij/2) [ ( z - 23' + 4R2(1 + i)' sin2 ($ij/2)l2 (13) = E(r',r)
- w)a,
+ 4aTO3R(1+ w ) a ,
=
(4i+
sin2 22
4R2(1 + i)*
(4)
with &i given by (11). Any ij path of i successive reflections between r and r' can be traveled in either direction; i.e., the reciprocal ij paths r' to r are identical. Furthermore, all cords and their subtended angles on an ij path as well as their axial displacement are equal, the first through the last; thus, the results of the reciprocal calculation (4) are given by (13). The reciprocity (4) is valid even when p varies with position over the cylinder surface. If the variation of p may be neglected as in eq 8 because it has been linearized in the temperature gradient, then
(14)
The constant w is determined by substitution of T into (8) along with the exchange factor (13) and ( 7 ) . The rigorous result of these substitutions is
'
k=l
E(r,r? = C Ci=Oj=o
Radiation Thermal Conductivities Radiation across the Cylinder. For the linearized equations we can, without loss in generality, separate the applied gradient into components perpendicular and parallel to the cylinder central axis and treat each case individually. A thermal gradient applied across the cylinder, perpendicular to the cylinder central axis, is treated fiist. From the validity of Fourier's law of heat conduction in the solid phase surrounding the cylindrical void, the temperature T within the solid can be written in terms of the uniform thermal gradient ap perpendicular to the cylinder central axis and far from the inclusion, .$,the radial vector distance drawn perpendicular from the cylinder axis, and To, the temperature at the cylinder midpoint, i.e., the average cylinder temperature
I-" I
1
I
o;m4 VlLA
where c # I ~ ~is given by (11). All surface coefficients in (15) are evaluated at the average cylinder surface temperature
TO.
When (15) is solved for the constant w , the familiar w form for a long cylinder inclusion in an infinite matrix is obtained w = ( K - X,)/(K + A,) (16) where the thermal radiation conductivity across the cyl-
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
inder pore with a specular-diffuse surface A, = 4t,aTo3R
(17)
and
_tp -E
1 - (1 - p ) F l-(l-p-t)F
This result implies that the calculation of the heat flux in the direction of the vector a, across the cylinder can be treated as a two-phase medium problem, the surrounding solid matrix with conductivity K and the cylindrical void characterized by a transverse radiant conductivity .A, Fourier's law within the solid along with (14) and (16) can be used to calculate the heat transfer from the solid surface into the cylindrical void. Summing the heat transfer from the appropriate complementary cylindrical pore wall surfaces, we obtain the flux qp per unit area per unit time across any plane in the cylindrical void perpendicular to the applied gradient a, qp
+ A,/W
= -2A,a,/(l
(19)
While the transverse thermal radiation conductivity X, will depend only on the pore wall surface properties and pore geometry, the transverse heat flux qp includes the conductivity of the solid matrix in the factor A,/K. This well-known effect arises from the local temperature in the neighborhood of the cylinder, in particular the w term in the temperature (14), and is caused by the difference in the termal conductivities of the solid K and cylindrical inclusion A., The integral F of (18) is obtained from (15) and has the form
247
interaction, at some time before and at some time after it crosses the plane. For this purpose we split the cylinder pore walls into two parts: A, for negative z and A2 for positive z. The flux across the surface can be written in terms of the sum of the paths in the positive minus the paths in the negative direction. If in addition the radiant axial flux is linearized in ar,i.e., in effect we presume the temperature difference across the capillary AT small with respect to the average pore temperature To q2 = (4aT:/.~R~)(1 - d2SS E ( r l , r 2 ) X
[ V r d - T(rJ1 dAl dAz (26) When the temperature distribution and the exchange factor (13) are substituted into (26), we obtain qz = - ( ~ U T O ~ ~ , / Tp)2 ~ )X( ~
+ [(zl - z J 2 + 4R2(1+ ')i
4R2(1 i)2 sin4 (4ii/2)(22 - 2 , ) sin2 (4ij/2)]2
dz, d81 d.22 d82 (27)
where 4ijis given by (11). In turn, (27) is integrated over 21, 22, and 82 4 2 = -
-4uT:a,(l
2r
1
- P)~RC Cp i ( l + i)ssin3 (4ij/2) d40,0 i=O j=O
0
(28) then integrated over @I~,~ and summed over j m
+
q2 = -(32/3)d'O3~,(1 - p)Rz/.~I(l i ) 2 i=O
(29)
and finally summed over i to yield 4, = -(32/3)aT,"%R(l
In the diffuse reflection limt p After integration over z
Q,
+ ~ ) / ( -1 CL)
-
0 (30) becomes
= -(32/3)d'03a,R
whereas in the specular reflection limit p (21) summation over j , followed by integration over 40,0 m
F = -Cpi/(3 i=O
+ 8i + 4i2)
(22)
and summation over i, we have
F = - [ ( p - 1)p-ll2 arctanh p1/2+ 1]/2p For the diffuse limit
p
-
0 from (171, (18), and (23)
A, = 16taTO3R/(4- E)
-
(diffuse)
whereas for a purely specular surface p A, = 2tcTo3R[2- t
(23) (24)
1- t
- t2(1- €)-,I2 arctanh (1 ~ ) l / ~ ] /-( lE) (specular) (25)
Radiation down the Cylinder. A cylindrical pore is an anisotropic structure for radiant transport and the radiant heat flux qE,per unit area of the circular cross section, along the central axis must be calculated separately. When a thermal gradient a, is applied along the axis of a long cylinder, the temperature distribution is linear. A path of the radiant heat flux across a plane at z = 0, perpendicular to the cylindrical axis, must experience a diffuse
(30)
-
q2 = -(32/3)r~T,"c~,R(2- €)/E
(31) 1- t we have
(32)
The factor (2 - ,)/e of (32) was first obtained for Knudsen gas diffusion through a long capillary with specular surfaces by Smoluchowski (1910). The derivations of expressions (191, (231, and (30) for qp and q2,both from the same three-dimensional exchange factor (13), are complementary; hence, they are presented together in this paper. It is straightforward to show, by using the three-dimensional equations (20) and (27), that in general the axial thermal gradient a, must vanish from qp and the radial component a, will drop out of qz. The result (30) for qr is a generalization of Smoluchowski's result to a surface that reflects both specularly and diffusely.
Discussion of Results In order to examine the results of radiant heat transport in a cylinder, values of the radiant fluxes are listed in Table I. For each row the value of p / p , the ratio of specular to total reflection, is a measure of what fraction of the reflection is specular or diffuse. The ratio p / p provides a means to compare the effects of different types of reflective surfaces. For each column the emissivity t is at the same time an indication of the surface blackness, and by Kirchoff's law 1 - c is the net fraction of incident radiation
248
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
that is reflected. In the double entries of Table I the upper entry gives the dimensionless flux -qp/4aTo3Ra, across the cylinder axis. From (19) the flux depends on the conductivity of the solid around the cylindrical pore through the term X,/K in the denominator. Except for large pores or high temperatures, A,/K is usually on the order of unity or less; for micropores the ratio is small. To obtain the transverse heat flux independent of the solid matrix of the porous medium, we err by no more than about a factor of ‘I2 if we neglect X,/K in (19) and write -qP/4aT2Ra, = 2Ap/4aT:R
(33)
No matter what the ratio X,/K, the upper entry is always twice the dimensionless transverse coefficient for radiant heat transfer across the axis, which from (17), (18), and (23) depends only on p and E. The lower entry -q,/ 4aT:Ra, gives the dimensionless axial radiant heat transport down the cylinder --qZ/4aToSRaZ = XZ/4aTo3R
(34)
The dimensionless axial radiant heat transport coefficient Xz/4aT2R from (30) depends only on p. The ratio of the upper and lower entry permits a calculation of the anisotropy of radiant heat transport across and down the cylinder for various equivalent pore wall surfaces. For any column except the black body limit E = 1.0 of no reflections, the values p / p 0 refer to the diffusive reflecting and p / p 1to the specularly reflecting surface. For a fixed emissivity both the upper and lower entries increase from a minimum to a maximum value with p / p . While it has been stressed in the literature (Siegel and Howell, 1981) that ideal surfaces do not always constitute limiting cases for radiant energy transfer, they do provide the extreme values for radiant transport across and down a cylindrical pore. Transverse radiant heat transfer across the cylinder axis, the upper elements in Table I, increases slowly with specular surface reflection for a particular emissivity. A maximum increase of 4.0% occurs at an emissivity of 0.439. This result supports the conjecture by Sarofim and Hottel(l966) and Siegel and Howell (1981) that directional effects within enclosures, i.e., across the axis, will be smaller because of the many reflections taking place between the surfaces. For radiant heat flux down the axis, the lower elements in Table I, heat transfer increases significantly with direction effects in reflection, particularly at lower emissivities. The conjecture is clearly not valid for long enclosures. That both thermal fluxes for the case p / p = 0.5 lie approximately halfway between the upper and lower extremes when emittance is high, but at lower emissivity p / p = 0.5 lies closer to the diffuse limit, has been observed in similar radiation problems (Sparrow and Lin, 1965; Tsai and Strieder, 1985). The upper and lower entries (Table I) for each e and p / p give a direct comparison of radiant heat transport across and down the axis and provide a means to examine anisotropy in radiative heat transport. For a black surface the values of the radiant heat flux per unit applied gradient across and down the axis are the same. For all values of p / p , the specular to total surface reflection ratio, the flux across the axis drops significantly as the emissivity of the pore wall surface decreases. Quite the opposite trend is observed for the flux down the axis. For purely diffuse 0 heat transport down the axis surface reflection p / p is independent of the surface emissivity. But with any amount of specular reflection, heat transport down the axis will increase with decreasing emissivity. The effect is further enhanced by increasing specular reflections to the rather large effect observed for a purely specularly re-
-
-
-
-
flecting surface, p / p 1.0. For a gray body the net anisotropy given by the ratio of the upper and lower entries is already significant for a diffusely reflecting surface and becomes largest for the specularly reflecting pore wall. In the engineering modeling of radiant heat transport in a consolidated porous medium, while the solid is isotropic with solid thermal conductivity K, the pore is anisotropic with a radial radiant thermal conductivity A, and an axial conductivity A,. Simple analytical expressions for A, and X, of a cylindrical pore with the most general case of a specularly-diffusely reflecting surface, respectively (17), (18), (23), (30), and (34), have been derived. Nomenclature A,, A2 = cylinder pore wall surface areas for z < 0 and z > 0
dA, dA’ = elements of surface area at points r and r’ dAl, dAz = elements of surface area within AI and A2 dAij = area element within which the first specular collision must occur for the ij path to dA B = radiosity F = defined by (20) E(r,r’) dA = exchange factor for a cylinder given by (13) H = total radiant flux incident to a unit area i = number of successive specular collisions between r’ and r j = index to specify different paths between r’ and r for a given number of specular reflections i k = index to specify each successive wall specular reflection for a given ij path K = thermal conductivity of the solid qp = radiant heat flux across the cylinder axis qz = radiant heat flux down the cylinder axis r, r’ = position vectors on the solid-void interface rl, r2 = position vectors on the cylinder pore wall surface within Al and Az R = cylinder radius T = temperature To = average pore temperature z , z‘ = coordinates along the cylinder axis at r and r’ zl, z z = coordinates along the cylinder axis at rl and r2 Greek Symbols cy,, = thermal gradient far from the cylindrical void perpendicular to its axis a, = thermal gradient along the cylindrical axis 6 = coefficient of diffuse reflectance a/an = normal derivative in the solid at the void-solid interface pointing outward t = surface emittance tp = defined by (18) 01,0 2 = angular cylindrical coordinates at rl and r2 A, = radiant thermal conductivity across the cylinder axis A, = radiant thermal conductivity down the cylinder axis = coefficient of specular reflectance F = radial vector from the cylinder axis p = hemispherical reflectance a = Stephan-Boltzmann constant & j = angle measured from a radial line to dA’and to the first wall collision of the ij path w = constant in (14) Literature Cited Bhata, S. K.; Perlmutter, D. D. AIChE J . 1981, 2 7 , 247-254. Birkebak, R. R.; Eckert, E. R. G. J. Heat Transfer 1965, 8 7 , 85-94. Birkebak, R. C.; Sparrow, E. M.: Eckert, E. R . G. J. Heat Transfer 1984, 8 8 ,
193-199. Chlew, Y. C.; Glandt, E. Ind. Eng. Chem. Fundam. 1983, 22, 276-282. Corlett, R. C. J. Heat Transfer 1986, 88, 378-382. Francl, J.; Kingety, W. D. J. Am. Ceram. SOC. 1954, 3 7 , 99-107. Gavalas, G. R. AIChE J . 1980, 26, 577-585. Kummer, J. T. Prog. Energy Combust. Sci. 1980, 6 , 177-199. Munch, B. Doctoral Dissertation, Swiss Technical College of Zurich, 1968; NASA Technical Translation F-497.1988. Perlrnutter, M.; Sigel, R. J. Heat Transfer 1963, 8 5 , 55-62. Sarofim, A. F.; Hottel, H. C. J . Heat Transfer 1966, 88, 37-44. Siegel, R.; Howell, J. R. ”Thermal Radiation Heat Transfer”; McGraw-Hill: New York, 1981;p 320.
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Ind. Eng. Chem. Fundam. 1986, 25, 249-258
Srnnh, J. M. “Chemical Engineering Kinetics”; McGraw-Hill: New York, 1981. Smoluchowski, M. Ann. Phys. (Leipig) 1910, 33, 1559-1586. Sparrow, E. M.; Lln, S . L. Int. J. Heat Mass Transfer 1985, 8 , 769-779. Strider, W.; Prager, S. J. Phys. Fluids 1988, 1 7 , 2544-2548. Torrance, K. E.: Sparrow, E. M. J. Heat Transfer 1985, 8 7 , 283-292. Torrance, K. E.; Sparrow, E. M. J. Heat Transfer IW8, 88, 223-230. Torrance, K. E.: Sparrow, E. M. J. Opt. SOC. Am. 1987, 5 7 , 1105-1114.
Tsai, D. S.;Ho, F. G.; Strieder, W. Chem. Eng. Sci. 1984, 3 9 , 775-779. Tsai, D. S.; Strider, W. Chem. Eng. S d . 1985, 4 0 , 170-173. Young, L. C.; Finlayson, B. A. AIChE J . 1978, 22, 331-353.
Received f o r reoiew July 10, 1984 Accepted May 16, 1985
Chromate Ion Exchange Mechanism for Cooling Water Arup K. Sengupia’+ and Dennis Clifford Environmental Engineering Program, University of Houston -University
Park, Houston, Texas 77004
The chromate ion exchange recovery process for a cooling tower blowdown is unique due to the early, gradual breakthrough of highly preferred Cr(V1) from fixed-bed columns for all types of anion-exchange resins. I t is shown that the early Cr(V1) breakthrough is not due to poor column kinetics but is predictable from an equilibrium model by using the appropriate exchange reaction involving both HCr0,- and Cr,O?-. At acidic pH, HCr0,- is practically the only Cr(V1) species in the aqueous phase, while in the exchanger phase both HCr0,- and Cr20,2- exist. The presence of Cr20?- in the exchanger’s solid phase causes a positively curved (concave upward) isotherm at relatively low Cr(V1) loading of the resin, and this equilibrium property is primarily responsible for the unusual, gradual breakthrough of Cr(V1). The presence of Cr,O,- in the solid phase may be viewed as the dimerization of HCr0,-, according to the Donnan equilibrium principle.
Introduction Related Studies and Gradual Breakthrough. In spite of very high selectivity of Cr(V1) anions by commercial organic anion-exchange resins, recovery of chromate from cooling tower blowdown is not yet commercially popular. Possible oxidation of the resins by Cr(V1) and consequent decrease in the resin-exchange capacity have been traditionally regarded as the prime obstacles to more frequent application of the process. Nevertheless, this recovery process is potentially important in light of the fact that other inorganic and/or organic corrosion inhibitors for cooling water systems are not as efficient as synergistic corrosion control by zinc and hexavalent chromate. In recent years there have been significant improvements in manufacturing processes for anion-exchange resins, especially in their ability to withstand physical attrition and chemical oxidation. Yamamoto et al. (1975) observed only a 5% decrease in the exchange capacity of anion resins after 12 months of operation for chromate recovery from cooling tower water. In open, recirculating cooling water systems, sulfuric acid is normally added to the make-up water to avoid concentration of scale-forming bicarbonate and carbonate ions. Sulfate is, therefore, without exception, the most predominant anion in cooling water (500-4000 mg/L) followed by chloride. Chromate, on the other hand, is present only to the extent of 5-20 mg/L Cr and as such may be regarded as a trace species in the cooling tower blowdown. Despite the severe competition from sulfate and chloride, which are present in concentrations several orders of magnitude higher than Cr(VI), several authors (Kunin, 1976; Yamamoto et al., 1975; Newman and Reed, 1980; Richardson et al., 1968; Miller, 1978) have confirmed the viability of the chromate-exchange process at acidic pH, primarily due to the
‘Present address: Environmental Engineering, Fritz Engineering Laboratory, Lehigh University, Bethlehem, PA 18015.
trememdously high affinity of chromate for anion-exchange resins. The regeneration process has also been found to be highly efficient (Kunin, 1976; Newman and Reed, 1980) for both weakly basic and strongly basic anion resins. Hexavalent chromium, Cr(VI), may exist in several different anionic forms as will be shown later. For convenience in discussion, we will represent the total of all the chromate species in water as Cr(V1) or “chromate” while each individual species will be represented by its true chemical formula. In all the above-mentioned studies, the phenomenon of gradual &(VI) breakthrough at acidic pH during fiied-bed column experiments has not been addressed. Figure 1 shows Cr(V1) breakthrough in a typical column run at pH 3.9. Note that Cr(V1) breakthrough is very gradual, i.e., non-self-sharpening. This same gradual breakthrough has been observed by all the above-mentioned researchers in fixed-bed column tests. In fact, in order to overcome this problem, a “merry-go-round”system for treating cooling tower blowdown has been proposed by Kunin (1976). During column runs, a preferred species, in general, shows sharp breakthrough characteristics as demonstrated by Clifford (1982). In a binary system, the preferred species is the one for which the ratio of the equivalent fraction distribution between the exchanger phase and the aqueous phase ( y / x ) is higher than unity. Various experiments conducted in our laboratories for the past 30 months provide sufficient evidence that such early Cr(V1) breakthrough is not due to poor column kinetics or co-ion invasion or channeling in the column. From the application viewpoint, the phenomenon of gradual breakthrough bears significant importance because a column run is always terminated at some chosen value of Cr(V1) exit concentration (normally less than 0.5 mg/L Cr). Therefore, the total available chromate removal capacity of the resin cannot be fully utilized in single fixed-bed runs. Another questionable aspect of the previous studies, including the study of Arden and Giddings (1961), is the
0196-4313/86/1025-0249$01.50/00 1986 American Chemical Society
