INDUSTRIAL AND ENGINEERING CHEMISTRY
698
Badger, Monrad, and Diamond (1). Here the film coefficients were determined experimentally by the use of thermocouples attached to the tube walls, and their values for the diphenyl film coefficient ranged from about 250 to 600. Their determinations showed the film coefficient on the caustic side to be much higher (500 to 1200) so that the over-all coefficient ranged from about 160 to 400. McCabe (8) also determined that the film coefficient for diphenyl vapor is from 250 to 400 and that the asphalt film coefficient varies from 35 to 50. The over-all coefficient in this report varied from 30 to 42. To summarize, we can say that for heat transfer by liquid Dowthenn the film coefficient ill be much less than for water in on the Dowtherm side w equal tube sizes and velocities. Values for clean 2-inch steel tubes by this method of computation for various velocities will be as follows: Velooity of Liquid Dowtherm at 500’ F. Ft./sec.
Film Coefficient of Heat TranRfer, hid. B . t . u . / ( h p ) ( ! q . ft.) ( O F . diff. between lzquad and lube wall) 168.5 267.5 325.0 385.0 445.0
VOL. 28, NO. 6
Heat transfer coefficients in the case of Dowtherm vapor, according to theoretical calculations, are of the same range of magnitude as those of the liquid. They are, however, much lower than with steam; the film coefficients on the Dowtherm side of the tube range from about 150 to 420 B. t. u. per square foot per O F. temperature difference between the vapor and tube wall, 300 to 400 being a reasonable value to be expected in ordinary commercial practice. Combination of the above film coefficients for the Dowtherm side of heating surfaces with proper values for the ma-, terial being heated on the other side of the dividing walls should give reliable over-all values for heat transfer.
Literature Cited (1) Badger, W. L.,Monrad, C. C., and Diamond, H . W., Trana. Am. Inst. Chem. Engrs., 24,56-78 (1930). (2) Grebe, J. J., Chem. & Met. Eng., 39, 213-16 (1932). (3) Grebe, J. J., Combustion, 3, 38-41 (1931). (4) Grebe, J. J., and Holser, E. F., Mech. Eng., 55, 369-73 (1933). (5) Heindel, R. L.,Jr., Chem. & Met. Eng., 41, 308-12 (1934). ( 0 ) Xilleffer, D.H . , IND.ENG.CHEM., 27, 10-15 (1935). (7) McAdams, W.H . , “Heat Transmission,” 1 s t ed., 1933. (8) MoCabe, W.L., Univ. Mich., Eng. Research Bull. 23 (1932).
RECEIVED March 12,
1936.
Radiation Reaction at Any Point in a Furnace Cavity
T
HE necessary equations of condition and a procedure by means of which successive approximations converge toward a solution of the equilibrium at any, and hence all, points within a furnace cavity have been developed by the author for certain classes of conditions within the cavity. This involves a special treatment of the problem which deals with the net rate of exchange by radiation between a local reference zone a t the point under observation and the enclosure, including its contents. If the temperature and concentration gradients throughout the cavity may be approximated, then a knowledge concerning the latter phase of the subject alone, when introduced in the energy equation as applied to represent the equilibrium a t the local zone, is, with other well-known relations, sufficient to furnish information concerning some important practical conditions. Thus for a local reference zone, designated henceforth as point 0 zone, or simply as point 0, the relation Qv
= QK
+ Qc + R
(1)
holds if the state is steady. Here Q v represents the thermal energy release rate in the local zone represented by point 0. QK and Qc represent the net rate of exchange between point 0 zone and immediate vicinity, rewectively, by thermal conduction and convection. Rreprisents the net rate of exchange bettween the same zone and the enclosure, including its contents, by radiation. Qu, Q K , and Qc depend only on the local conditions. Expressions representing them for many cases are well known, and their values are susceptible to determination when temperatuies and gas concentrations in the vicinitv of Doint 0 are known. For some classes of , position of p o h t 0-i. e., a t the wal
W. J. Wohlenberg Yale University, New Haven, Conn.
Q v is zero. An exception in this respect is a wall location a t the surface of a fuel bed on a grate. Now Qv represents the rate of release of thermal energy by the combustion process a t this point. The term R, representing the net rate of exchange by radiation, depends on the conditions a t every point in the cavity which point 0 “sees.” Although the general form of the involved radiation equations is known, the particular arrangements which correctly represent the exchange for the above conditions are not well known. This involves the special treatment referred to before. Accordingly, this phase of the more general furnace equilibrium problem is dealt with in the present paper. It thus paves the way for the later presentation of the more complete procedure and, as before noted, furnishes relations by means of which some conditions of practical importance may be approximated directly without recourse to that procedure. Although the resulting relations are developed with the furnace cavity in mind, they are applicable to any cavity whatever, which falls as to class within the following specifications. Thus it is assumed that no other than thermal radiation need be considered. Then the black body is the standard of reference, and temperature is introduced on the basis of the Stefan-Boltzman law. It is also assumed that Lambert’s cosine law of radiation intensitv from surfaces approximates the actual conditions to within t h e desired- degr cy for the cases to be considered.
JUNE, 1936
INDUSTRIAL AND ENGINEERING CHEMISTRY
699
Basis of Reference for Point 0 Zones Two general types of local or point 0 reference zones are involved. The first is any local spot on the walls of the cavity, and the second is a representative sample of the mixture in the cavity. This may contain particles. Such point 0 zones as will be considered are defined in detail as follows: A . A patch of surface which is small compared to the enclosure and of which point 0 is the center. B. The surface of a particle in space so small, compared to the enclosure, that, with respect t o the radiation exchange between the particle and the enclosure plus contents, point 0 may be considered as being at the center of the particle even though radiation is from its surface. C. A sample quantity of gas which is small compared to the enclosure and which has point 0 at its center. The conditions at point 0 are considered as existing throughout the sample. D. The quantity of gas considered as associated with a particle, with point 0 at the center of the system-i. e., as at the center of the particle. d/ffwsion
zone
(A)
confamfhy 90s welyhf
fyA)
I
A solution of the dynamical processes occurring in the furnace cavity requires evaluation of the radiation reaction with respect to its net effect at each of several classes of local, or point, zones within the cavity. This involves an extension of the idea of the radiant mean value with respect to the point under observation, of conditions affecting the net radiant exchange between the point and the enclosure including its contents. The classes of required local reference zones are defined, and expressions representing the net exchange by radiation between each of these zones and its surroundings are included. General forms of the expressions apply to any cavity whatever for conditions stated. Special forms are included which apply to gas, grate, or pulverized coal-fired cavities.
t h a t the four local reference systems, A , B, C, and D, are sufficient to permit investigation of any local point 0 zone in either gas, grate, or pulverized coal-fired cavities. It is necessary now to find a way of determining the mean effective value for radiation and with respect to point 0 of any condition which affects the net radiant exchange between the point 0 zone and surroundings.
Radiant Mean Value, Z
A proper basis of reference for condition A is obviously a unit of area. For B i t may be either the surface of the particle or a unit of area of particle surface. For C it is the weight of a cubic foot of mixture entering the furnace cavity when at standard conditions of pressure and temperature. And for D it may be either the weight of gas considered as associated with a particle or the weight of gas per unit area of particle surface. If the quantity of gas, gx (Figure 1) which is contained within the diffusion zone, is considerable when compared to the total quantity, g, then it may be advantageous to replace g by (g - g?); the latter quantity is outside the diffusion zone, X, and in it the gas temperature, T,,may be assumed to be uniform. But for the usual conditions in pulverized coal firing, g h is so sinall compared to g that the net exchange b y radiation between g and surroundings may be computed as though temperature Tuwere uniform throughout both zones X and y. This is also true of the thermal conduction rates into the gases from the surface of the particle. It is found also that the diffusion equations for a syskem such as D have such a form that for the usual conditions of suspension firing of particles any variations which occur in thickness X result in practically the same combustion rates as those computed on the basis that X equals infinity. Therefore X may be assumed as having any value between, say, X = 5r, and X = 03 ,and so there is no need of dealing with a special dimension X. It may for convenience be taken either equal to (ro - rp) or equal to infinity, and the results will be practically the same. Therefore it is plain
Referring to Figure 2, let u represent the average value such as temperature over a patch of surface, AB, subtending solid angle do from a given point 0. The mean effective value of temperature TB of the boundary B with respect to point 0 is then
-
s
u coscy dw
complete enclosure
u=J
(2)
coscu dw
complete enclosure
Since that part of the radiation emanating from point 0 which spreads through the space of solid angle dw is completely defined by the conditions included within the region of this differential of solid angle, then the coordinate u in Equation 2 may represent any condition which affects the radiation. Thus it may represent conditions in the space included by the solid angle dw. It may also represent any
f /.y 2.
INDUSTRIAL AND ENGINEERING CHEMISTRY
700
combination of such quantities. The radiant mean U is then the effective mean value with respect to point 0 of this combination. The combinations here involved are obviously those in terms of which the radiant exchange between 0 and the region defined by dw may be expressed. The integration of Equation 2 is possible only in special cases. For others an approximation as close as desired may be obtained on the basis of the following weighted mean. Thus -
u
E
uiFi
+ uzFz + . . . . u,F.
Fk
=
2:". dw
(4)
dw
where F k and wb represent, respectively, the angle factor value and solid angle subtended by section IC (Figure 3) from point 0. In all cases, FI
+ .. . . . . F,
=
unity
(5)
Expressions Representing Net Radiant Energy Exchange between Any Point 0 and an Enclosure Consider that the space within the cavity (Figure 3) is filled with gases which contain particles in suspension, the transmissivity of the space for radiation being designated by 7. If the radiation is between point 0 and walls w, the transmissivit,y is tw;for radiation between 0 and section k of the walls, it is v w k . f
But on the basis of Equation 3:
(3)
The total enclosure is now divided into n sections in which the average vadues of u are (ul . . . un), respectively, for section 1 . . . n. The weights F , . . . F , are the so-called angle factor values (2, 3, 5 ) subtended respectively by sections (1 . . . n) of the cavity from point 0. They are evaluated by means of the relation, f%a
Jecf/bn W
J
1
Therefore the form of Equation 7 is correctly expressed by Rw
uo[qT4
r
~
l
(9)
-
Rw = crOo[ToP - I?,]
(10)
But Torepresents the temperature a t a point which is constant in all directions from the point and so
_ _ 7 w T o 4 = 9w5''o4
(11)
Hence for such conditions the right-hand side of Equation 11 may be substituted in Equation 9.
Conditions Where Radiant Mean Value of a Product Is Equal to the Product of Radiant Means Two examples of this condition have been cited. The question arises as to whether or not a general statement may be made in this respect. Consider a number of conditions u, v, w, . . . on which the radiation exchange between point 0 and the enclosure depends. Then, for a steady state in a given furnace, the value of a condition, for radiation in a given direction, may be expressed as
The net rate of reception of radiant energy by wall section k from that radiated by a unit of surface a t point 0 is then: UoFk??wk[To4
-~ ~
Thus it is noted that net radiation exchange between walls and a unit of surface at point 0 involves in the parenthesis the difference between the radiant mean values,with respect to the point, of two products. The first of these, q,TO4,represents the net radiation reaction of the surface a t 0 when an absorbing medium is present within the enclosure, The second represents the net radiation reaction of the enclosure w with respect to a relatively small surface a t point 0 when the absorbing medium is present. If the space within the enclosure is perfectly transparent, then qW is constant at unity in all directions from point 0; therefore Equation 9 reduces to
24
Rwt =
VOL. 28, NO. 6
-
(6)
Here r,t represents the average value of Tw4 over wall section k , and T , represents the absolute wall temperature. The coefficient of radiation for the surface at point 0 is designated as co. Since this surface (unit surface) is, for all cases to be considered, small compared to the enclosure, the value uomay also be taken as that of the combined coefficient of radiation, in so far as reception of radiation a t walls w from that discharged a t point 0 is concerned. The rate of reception of radiation at the total walls of the enclosure is the summation of Equation 6 over the n sections subtended by walls w from point 0. Thiq may be stated in the form:
=
'p"(cu,
P)
(12)
where cy and j l are angles of altitude and azimuth which specify the direction of the radiation from point 0. Hence for the three conditions there are in general the three functions: (13)
If p., p?,,and pwrepresent the same function of cy and (3, or differ from each other only by a scale factor, then the radiant mean of the product is equal to the product of the radiation means. Since this relationship probably never exists, except for the case of uniform conditions, it need not be considered further a t this time. I t follows that, when, for the sake of convenience, the product of the radiant means is adopted in place of the radiant mean of the product, it may be advisable first to investigate the magnitude of the error which results from its adoption. This is the form which has been adopted in
JUNE, 1936
INDUSTRIAL AND ENGINEERING CHEMISTRY
the radiation formulas in use today; therefore, whatever error is involved on this account exists in the results of calculations based on them.
Specific Rate Referred to Unit Surface at Point 0 at Which Radiation I s Discharged from This Surf ace Consider that the absorbing medium within the space of the cavity is filled with gases containing particles, the particles being considered as small black bodies opaque to radiation. Then the net rate a t which radiation is discharged from a local region in the vicinity of point 0 is equal to the part of this direct discharge, R,, which arrives a t walls w ,plus the part RG which is absorbed by the gases, plus the part R, which is absorbed by the particles in suspension. Consider that point 0 is located a t a solid surface within the cavity-that is, either a t the surface of a particle or a t a point on the walls. Under these conditions the sum of the above three quantities per unit of surface a t point 0 assumes the form: R where
R,
=
=
Rp
+ RG f Rw
-1
U O [ ~~ ~r)prp] T O ~
- PffrG] ~ [ i w To ~17lurWI
RG
Uo bffT04
R,
(14)
The method of evaluating the indicated radiant mean quantities is as represented by Equations 8 and 3. For any section k of the cavity, the value qpk of q , represents the transmissivity of the space for radiation between point 0 and the particle in suspension. The value pGk of pa for section k represents the absorptivity of the gases in this space for radiation between point 0 and the gases cont,ained. The value rpk of r p for this space represents the average value of TPk4, where T, is the absolute temperature a t the surface of particles. ra indicates the same relations with respect to the gas in the cavity. If no particles are present, the term R, vanishes from Equation 14, and, if no radiating gaseous constituents are present, the term RG vanishes.
Modification of Equation 14 When Point 0 I s at the Center of a Small Volume of Gas For a small gaseous body the net radiation discharge in any direction, when considered as emanating from the center of volume (point 0),depends only on the number of radiating gaseous molecules of a given kind. Hence it is independent of the shape chosen to represent such a volume. For the spherical volume the normal radiation intensity through the surface from its interior is a constant a t all points, provided the distribution in the interior may be assumed as symmetrical with respect to the center of volume, and provided the enclosure is a t a uniform temperature. Uniformity is the special case of the latter and applies here. When the normal intensity is constant over the surface of a sphere of given radius, r,, the relative emissivity, b, (relative blackness), is also a constant a t all points. Let py represent the black surface equal in radiating power to the total superficial surface, 4rrU2,of the reference gas sphere, g. Then if u represents the coefficient of radiation for a black body, u.pP(Tp4- T B ~ = ) ~b,(4rr,~)(Tp~ - TB‘)
or
p,
=
(4~r,~).b,
(15)
By application of Charles’s law for perfect gases, Equation 15 may be stated in the form: pp
=
0.0294b,(B,.g.T,)
’’’
(16)
70 1
Here Bo and g represent, respectively, the value of the gas constant and weight of gases contained, both for conditions existing within the reference sphere. Thus it becomes apparent that in this case the radiation discharge rate a t point 0 should be referred to some gas weight, g. A logical and convenient basis is to take g as the weight of a unit of volume of mixture entering the cavity, when a t standard conditions of pressure and temperature. Then g is a constant. In most cases the gas constant, B,, may also be taken as constant a t its mean value. The value of bo is, among other things, dependent on the mean effective thickness, , of the reference gas sphere. For radiation through the surface of the gas sphere from its interior, this is equal to two-thirds its diameter. By application of the law for perfect gases this may be stated in the form:
The concentration, Y, of radiating gaseous constituents a t point 0, temperature Tua t point 0, and temperature T Bare the other quantities involved in evaluating b,. When all of these conditions are known, the value of bo may be approximated by referring to tche total radiation charts for gases (4, 6,6). Since the value of b,, and hence that of p,, depends on the boundary temperature TB, then for a given location of point 0, p q is, in general, a function of the direction of the radiation discharge. Hence it must be included as an additional factor in the radiant means of products shown in Equation 14. Since the term R, does not appear in this case, Equation 14 as applied now takes the form: R = Ra f Rw
(18A)
I n these equations the symbol To4has been replaced by the temperature head Tu4existing a t point 0. The temperatures involved for the determination of a value of p , for any section, k , of the cavity are those represented in the term in which it appears. Thus for R G they are Tuk and r G k , and for R, they are Tu, and Twk, each pair applying to section k of the cavity as indicated by the second subscript.
Modification of Equation 14 When the Reference Gas Sphere Contains a Particle at I t s Center In this case (Figure 1) the gas weight, g, may be taken as that considered as associated with the particle on the basis of a uniform initial distribution of particles within the entering mixture. Then
where N1 = number of particles per pound of fuel G, = pounds air supplied per pound fuel rpl,r p = initial and later radii of particle 61, 6 = initial and later densities of particle substance In Equation 19 the product &rP3generally changes in value with time, if combustion is occurring a t the surface of the particle. Hence, unless the combustion rate is zero, the reference gas weight is, in this case, a variable. For the radiation exchange between gas body g and all particles, the particles must be considered as in two classes of positions in this case-that is, the single particle of radius r,, inside the gas sphere of radius r, and all other particles,
INDUSTRlAL AND ENGINEERING CHEMISTRY
702
the latter group being cutside of this sphere. The total exchange between gas quantity g and all particles is equal to that between it and the central particle plus that between it and the group outside of it. Let b,, represent the relative emissivity of the gas sphere of radius r, for radiation through the surface adjacent the central particle. The net exchange by radiation between this gas sphere and the central particle is then: where T,,
=
b,C(47i.TP2) (To4 - TPJ abs. temp. a t surface of central particle
(20)
VOL. 28, NO. 6
bodies. In each case the symbol letter such as P in P, indicates the ultimate region or boundary a t which the involved radiation absorption occurs. The subscript, indicates the boundary to be employed in computing the mean effective thickness of the gas column. Thus for P, the ultimate absorption boundary and that used for computing the effective thickness of the gas column are the same. It is obvious that this is the case also for coefficient W,. wa//
(w)
Neglecting the influence of the -size of the central particle, the mean effectiveness thickness, l,,, of the gas column is in hhis care r,. Application of the gas law in this case leads to
i,,
= 0.041 (B,.g.TJ/*
(21)
On this ba4s the term R, which appears on the right of the equation, R
=
R, iRG i- R,
(22)
assumes the form:
The terms RG and R , which appear in Equation 22 differ from the corresponding expressions (18B and 18C) only in the value to be used for gas weight, g, on which the value of pg is based. In this case the value of pQ is based on a value of g as determined by Equation 19 for each of the terms R,, R,, and R,. The resulting expressions then represent the radiation discharge rate from gas sphere g referred to the gas associated with a particle or also referred to the superficial surface, 4arP2,of the particle. In some cases it may prove more convenient for the sake of consistency with other terms in the energy equation to have a unit of particle surface as the basis of reference. This transformation is accomplished very simply by dividing the right-hand side of each ol the expressions representing R,, R,, and R, by the superficial surface, 4arP2,of the particle. On this new basis of reference the gas weight, g, is that per unit of superficial particle surface.
Transfer Coefficients 7 and
p
In view of the definition of ,uo and f,, coefficient G, represents the fractional part of the exchange between unit black jurface a t To and an enclosure a t T , which is absorbed by gas columns extending without interruption by particles from point 0 through to walls w. These columns as shown in Figure 4 have a thickness I,. Coefficient G, is the same thing but is for the part of the gas through which radiation passes from point 0 to the particles in suspension-that is, for the sum total of gas cones with tips a t point 0 and bases on particles in suspension. These have altitude or length equal Thus, G, arid G, are absorptibn coefficients but, with to I,. this in mind, the same conventions as above are followed with respect to the meaning of the symbol letter G and subscripts w and p .
Basis of Reference for Evaluating the Coefficients Consider the radiation spectrum in Figure 5 . Consider also that a unit black surface a t temperature TO radiates into an enclosure filled with gas of given chemical composition and a t temperature TG. Then the width b‘ and position of a band in which the gas absorbs radiation is single-valued with respect to the temperature TGof the gas.
Relations representing these coefficients may be set up in the form:
- P,) - jp(Gw- G,) (1 - fJ(1 - W d
?I,
=f P U
PG
=
7, =
G,
t
(24)
Here f, represents the fraction of rays emanating from point 0 which are intercepted by particles in suspension. This coefficient,is thus based on pure geometrical properties. For radiation from point 0 into section k of the cavity (Figure 3), the value f p k of f, is as follows ( 1 ): fpk - 1
-
e-vplc.ak.Zwh
(25)
where v p b and uk represent mean values, respectively, of concentrations (number per unit volume) and cross-sectional areas of particles, both for the space defined by solid angle wk; lW1,is the mean effective thickness of the space for radiation between point 0 and wall w for section k. Since rp is by definition a transmissivity, then P, represents the fractional part by which the net radiant exchange between a unit of black surface a t point 0 and particles in suspension is reduced by the presence of the gas. Coefficient W , is the same thing with respect to the net radiant exchange between a unit of black surface a t point 0 and walls w. Hence, these are really emissivity coefficients of the involved gaseous
Wove Len@
+
The area E‘ = A’ B’ represents the rate of radiant exchange for the above conditions when the gas is perfectly transparent to radiation. The portion of E‘ which is available for absorption by the gas when it is a t temperature TGis A’. Of this, the part A’.r’ is absorbed, where c’ is the average probability that a quantum of radiation discharged in wave band b’ from a source a t T owill be absorbed by a gas molecule. This probability is obviously a function of the concentration Y of radiating gaseous constituents, the radiant mean beam
INDUSTRIAL AND ENGINEERING CHEMISTRY
JUNE, 1936
length i between point 0 and boundary B, and of an absorption coefficient IC’ which includes the absorbing properties of the involved gas molecules. Hence:
r’
= (D(k’, Y ,
i)
(26)
This function is known as the thickness concentration function and would in form be similar to Equation 25 if k’ were independent of wave length and temperature of the radiation. This is not the case, and so the true form of Equation 26 is quite complex but needs no detailed specification for our purposes. Since A‘.[‘ represents the absorbed energy for band b’, then for several bands, b’, b”. . ., .... A’r’ A’y’
+
+
(27)
~ ( T o 4- Ta4)
where u is the coefficient of black radiation, represents the fraction of specific rate of black radiation between To and TG which is absorbed. Thus it represents coefficients G, and G, in general form. The net reduction in specific rate of black radiation between To and TB because of the presence of the gases is equal to their specific rate of emission as measured above the boundary temperature, TB. For bands b‘, b” . . . the latter quantity is plainly (B’l’’ B”{” . . .). Hence,
+
+
represents the fractional part by which black emission between TOand TB is reduced by the presence of the gas. Thus i t represents coefficients P, and Mr, in general form. Let qopl,.rl qpp,pop, rlaW,,,I( qWwrepresent the specific rate of total radiation as taken from the radiation charts for gases (4, 5 , 6). Here the first and second subscripts denote, respectively, the involved temperature and radiant mean beam length. An approximate form for the numerator of Equation 27 is then: (qop - pop) for coefficient G,. . . (qow pa), for coefficient G,, . .
-
i
(29)
The approximate form for the numerator of Equation 28 is: (QG,
(qow
- qwp) for coefficient P,. . . - pww)for coefficient W,. . .
5
(30)
In applying Equation 28, the boundary temperature T B which appears in the denominator is replaced by T , when the coefficient P, is evaluated and by T, when coefficient W , is evaluated. Although as shown in Figure 5, TO> To> T,, the above procedure applies whatever the relation between the magnitude of temperatures a t these stations. It applies also for radiation from point 0 throughout the space of any section k (Figure 3) of the cavity if the (I values are based on the average conditions, such as gas concentrations, radiant mean beam lengths, and temperatures which exist in that section. In form the above procedure is correct for gas absorption spectra in which the band areas, A‘ . . ., B‘ . . ., their posi-
703
tions, and absorption coefficients k‘ . . . do not vary for the temperature ranges involved in the determination of the differences in q values in Equations 29 and 30. Real gas absorption spectra are not absolutely independent of temperature in this way and so an error is involved. The correct procedure is indicated by the forms of Equations 27 and 28, but unfortunately there are a t present insufficient experimental data for evaluation of coefficients on this basis. Hence for the present it is necessary to resort to the approximate method. It is simple and the involved error is probably well within that of other necessary assumptions for most conditions.
Involved Radiant Mean Beam Lengths The radiant beam lengths lpk and lwk for a given section k of the cavity are arrived a t by application of Equation 2 or its approximate equivalent Equation 3. I n such cases the variable, u, to a patch of boundry is replaced by the length b, between point 0 and this patch of boundary. The integration now covers only the space of solid angle wk subtended by the part of the boundary included in section IC. It is to be noted that when the particles form this boundary, then it is composed of a series of spots scattered throughout the space of section k , each having an area, a, equal to the cross section of a particle when in the position of the spot. The solid angle Aw subtended by such a spot from point 0 is a/P, and the frequency of the spots with a uniform initial distribution of particles relative to the total weight of incoming mixture varies inversely with the absolute temperature of the gas. Hence:
f Vk
and so l p k =
O J
1 cos
(Y
LVk
-2
TG.L* a
~-
COS CY
a
__ dV
Twl
1V
In this expression V represents volume, and, in the integration? point 0 is taken as the origin of the system. Approximations to 1,k, when integration is not possible, may be arranged with the above relation in mind. Methods of evaluating lwk are well known and need no further comment.
Literature Cited (1) Haslam, R. T., and Hottel, H. C., Trans. Am. SOC. Mech. Engrs.,
Fuels Steam Power, 50, No. 3 (1928). (2) Hottel, H. C., Mech. Eng., 52, 699 (1930). (3) Hottel, H. C., Trans. Am. SOC.Mech. Engrs., Fuels Steam Power, 53, 19b (1931). (4) Hottel, H. C., and Mangelsdorf, H. G., Trans. Am. Inst. Chem. Engrs., 31, No. 3, 517 (1935). (5) McAdams, “Heat Transmission,” pp. 61-73, New York, McGraw-Hill Book Co., 1933. (6) Schack, Goldschmidt, and Partridge, “Industrial Heat Transfer,’’ pp. 182-207, New York, John Wiley & Sons, 1933. RECEIVED March 2 0 , 1936.
