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UNIVERSITY OF ILLINOIS
BULLETIN
Vol. XXXVIII June 17, 1941 No. 43
ENGINEERING EXPERIMENT STATION
BULLETIN SERIES No. 330
HEAT TRANSFER TO CLOUDS
OF FALLING PARTICLES
BY
H. FRASER JOHNSTONE
ROBERT L. PIGFORD
AND
JOHN H. CHAPIN
PRICE: SIXTY-FIVE CENTS
PUBLISHED BY THE UNIVERSITY OF ILLINOIS
URBANA
[Iasued weekly. Entered aa ecod4.cla matter December 11, 101, t the post oie at Urbana, Ilinois,
under the Actof Auguat 24, 1912. Acceptance for mailing at the specil rate of postage provided for in
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THE Engineering Experiment Station was established by act
of the Board of Trustees of the University of Illinois on De-
cember 8, 1903. It is the purpose of the Station to conduct
investigations and make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Chemical Engineering. This Staff is responsible for
the establishment of general policies governing the work of the Station,
including the approval of material for publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in cooiperation with the Research Corps,
composed of full-time research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and dis-
tributes a series of bulletins. Occasionally it publishes circulars of
timely interest, presenting information of importance, compiled from
various sources which may not readily be accessible to the clientele
of the Station, and reprints of articles appearing in the technical press
written by members of the staff and others.
The volume and number-at the top of the front cover page are
merely arbitrary numbers and refer to the general publications of the
University. Above the title on the cover is given the number of the
Engineering Experiment Station bulletin, circular, or reprint which
should be used in referring to these publications.
For copies of publications or for other information address
THE ENGINEERING EXPERIMENT STATION,
UNIVERSITY OF ILLINOIS,
UnBANA, ILLINOIS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN SERIES No. 330
HEAT TRANSFER TO CLOUDS OF
FALLING PARTICLES
BY
H. F. JOHNSTONE
PROFESSOR OF CHEMICAL ENGINEERING
ROBERT L. PIGFORD
RESEARCH ASSISTANT IN CHEMICAL ENGINEERING
JOHN H. CHAPIN
FORMERLY GRADUATE STUDENT IN CHEMICAL ENGINEERING
PUBLISHED BY THE UNIVERSITY OF ILLINOIS
PRICE: SIXTY-FIVE CENTS
3000-7-41-20999 I
CONTENTS
I. INTRODUCTION . . . .
1. Object of Work . .
2. Acknowledgments
II. THEORETICAL DISCUSSION.
3. Nomenclature
4 M~ohanmam nf HTTat Trannfer
5. Heat Transfer to Small Particles
6. Radiation to Clouds of Particles
7. Heat Transfer from Wall to Gas
III. EXPERIMENTAL WORK . . . . .
8. Introduction . . . . . .
9. Heat Transfer Measurements
10. Secondary Measurements .
IV. CORRELATION OF EXPERIMENTAL DATA.
11. Experimental Results . . . .
12. Analysis of Data . . . . .
V. CONCLUSIONS
13. Summarized Conclusions
14. Design of Large Furnace
PAGE
S . 5
S . 5
S . 6
Convection
6
6
8
9
22
28
29
29
29
32
38
S38
45
LIST OF FIGURES
NO. PAGE
1. Geometrical Representation of Problem of Heat Transfer to a Sphere . .16
2. Comparison of Equations of Heat Transfer to a Spherical Particle . . .19
3. Assumed Components of Fluid Velocity in Neighborhood of a Sphere . .21
4. Geometrical Analysis of Radiation from Walls of an Infinite Cylinder to
Cloud of Particles . . . . . . . . . . . . . . . . 24
5. Absorptivity of Clouds of Particles in Furnaces of Various Shapes . . . 27
6. Apparatus Used for Heat Transfer Measurements . . . . . . . . 30
7. Drag Coefficients for Particles Falling Through Air . . . . . . . 36
8. Calculated Effect of Feed Rate on Convection Coefficient from Wall to Gas 47
9. Graphical Representation of Heat Transfer Data for Sand Particles .. 48
10. Graphical Representation of Heat Transfer Data for Carborundum Particles 49
11. Graphical Representation of Heat Transfer Data for Aloxite Particles . . 50
12. Heat Transfer Coefficients and Calculated Heights of Rectangular Furnaces
for Calcination of Zinc Sulphite .. .... . . . . . . 54
LIST OF TABLES
NO. PAGE
1. Absorptivity of Clouds of Particles in Furnaces of Various Shapes . . . 26
2. Principal Dimensions of Furnace Used in Experimental Work . . . 29
3. Typical Temperature Distribution in Small Furnace . . . . . .. 31
4. Summary of Measurements of Diameter, Mass, and Density of Particles
Used in Heat Transfer Work . . . . . . . . . . . 33
5. Summary of Results on Drag Coefficients for Sand, Carborundum, and
Aloxite Particles . . . . . . . . . . . . . . . . 34
6. Calculated Times of Fall of Particles Through Furnaces . .. . . . 37
7. Heat Transfer to Clouds of Sand Particles ... . . . . . . . 40
8. Heat Transfer to Clouds of Carborundum Particles . . . . . . . 45
9. Heat Transfer to Clouds of Aloxite Particles . . . . . . . .. 46
10. Summary of Results on Individual Heat Transfer Coefficients . . . . 51
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
I. INTRODUCTION
1. Object of Work.-This investigation was undertaken for the
purpose of obtaining information essential to the design of equipment
for heating finely divided materials falling through a gas. The im-
mediate object of the work was the design of a flash calciner for the
decomposition of hydrated zinc sulphite, the product of one operation
in a cyclic process for the recovery of sulphur dioxide from waste
gases. A summary of the results and their application to that par-
ticular problem have been given in a previous bulletin.* Because of
the general applicability of the work, the results are presented here in
more detail along with the theoretical treatment essential to a complete
development of the problem.
There is a growing interest in space reactions of solids, or between
solids and gases, at high temperatures, because of the high rates of
heat transfer to clouds of solid particles suspended in a gas. The
method was suggested at one time for the carbonization of pulverized
coal to produce "bubble coke."t Adherence of the particles to the
retort walls, however, was not entirely prevented and the method did
not gain favor, although it was recognized that the rate of heat trans-
fer was rapid, and that conduction through the walls was actually a
limiting factor on the capacity.$ Application to the briquetting of
blast furnace dust, the smelting of copper and zinc ores and for the
production of soluble phosphates has been suggested.§ Other possible
applications include the decomposition of sodium bicarbonate and the
dehydration of clays and minerals. Several of these have been in-
vestigated in the experimental equipment to be described.
The general problem is also of interest in the design of spray
dryers. Lapple and Shepherd¶ have recently made a theoretical study
of the trajectories of the particles in such equipment. From the infor-
mation given in their paper, it should be possible to apply the basic
equations developed here for heat transfer to determine the heat re-
moved from the particles and, thus, the point at which solidification,
or crystallization is complete.
*Univ. of Ill. Eng. Exp. Sta. Bul. 324, 1940.
tA. H. White, Proc. First Internat. Conf. on Bit. Coal, Carnegie Inst., p. 419, 1926; W.
Runge, Ibid., p. 697; F. M. Gentry, Combustion, vol. 20, p. 225, 1929.
tF. B. Hobart and D. J. Demorest, Ohio State Univ. Eng. Exp. Sta. Bul. 65, 1932.
§E. C. St. Jacques, Ind. Eng. Chem. News Ed., vol. 15, p. 29, 1937.
TC. E. Lapple and C. B. Shepherd, Ind. Eng. Chem., vol. 32, p. 605, 1940.
ILLINOIS ENGINEERING EXPERIMENT STATION
2. Acknowledgments.-This work is a part of the research program
of the Engineering Experiment Station, of which DEAN M. L. ENGER
is the Director, and of the Division of Chemical Engineering, of which
PROFESSOR D. B. KEYES is the Head. The research work was used as
thesis subjects for advanced degrees in the Graduate School of the
University by the junior authors. The clerical assistance of several
students working under grants from the National Youth Administra-
tion is gratefully acknowledged.
II. THEORETICAL DISCUSSION
3. Nomenclature.-Throughout this bulletin the equations are
written in a form in which any consistent system of units may be
used. In those cases in which designation of the units has been neces-
sary, the English system has been used for quantities having immedi-
ate practical significance, and the metric system for quantities used
principally in the laboratory. This policy, rather than one of strict
uniformity, was adopted in order to secure greater clarity for the
average reader.
a = average projected area of a single particle
A = surface area; subscript o refers to the total area emitting
radiation, p to that of each particle, w to that of the in-
ternal wall of the furnace
b, B = arbitrary constants
c = specific heat of gas
C = concentration of particles at any point in the cloud, i.e.,
the number per unit volume
CD = drag coefficient, defined by Equation (60)
d = differential operator
D = internal diameter of furnace tube
D, = arithmetic average diameter of particles in the cloud
f = fraction of total area A covered by discs
F = feed rate, in general, expressed as lb./(min.) (sq. ft.)
FAE = angle-emissivity factor for net radiation transfer
g = acceleration due to gravity
h = heat transfer coefficient; subscripts have the following
meanings: m overall coefficient, r radiation coefficient,
cp convection coefficient for spherical particle surface, cw
convection coefficient at furnace wall. All coefficients ex-
cept hew are based on particle area
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
I = intensity of radiation, subscript I refers to the intensity at
point I measured from the front of the cloud, o refers to
that at which I = 0, n refers to that in the direction normal
to the surface
J1 = Bessel function of the first order and the first kind
k = thermal conductivity
K = volume-shape factor, defined by Equation (59)
I = length of a radiant beam; subscript o refers to the total
length of beam
L = length of path of particle, i.e., in general, the vertical dis-
tance along the furnace
M = a constant, defined by Equation (16)
N = number of particles present in the furnace at any instance
q = rate of heat transfer
r = radius; subscript I refers to radius of a spherical particle,
subscript o refers to that of an infinite cylinder
Re = Reynolds number, DVp,/1j
s = thickness of hypothetical tube of gas
t = temperature; subscripts have the following meanings: g the
temperature of the ambient gas, p that of the particle, s the
uniform temperature of the surface to which heat is trans-
ferred, w the temperature of the furnace wall, oo the uni-
form temperature of the gas at infinite distance from the
surface receiving heat
Aot = t, - to
T = absolute temperature, degrees Rankine; subscripts p and w
refer to that of particle surface and wall surface, respectively
u, v, w = components of fluid velocity in the x, y, z directions, re-
spectively
v, = average volume of a particle
V = velocity of particles relative to furnace wall (in a stagnant
gas); subscript t refers to terminal velocity
w, = average mass of a particle
x, y, z = coordinates of a fluid element
a = -/ thermal diffusivity, = 1/ k/cp
S= volume coefficient of thermal expansion
S= ratio of particle area to wall area
A = temperature rise
Ec = absorptivity of cloud of particles, i.e., fraction of radiation
incident on cloud that is absorbed
Ep = emissivity of particle surface
n = roots of first order Bessel function
ILLINOIS ENGINEERING EXPERIMENT STATION
X = characteristic linear dimension
p = viscosity of gas
7r = ratio of circumference to diameter
II, ' = conformal coordinates for potential flow around a sub-
merged body of revolution. Note that VII is the velocity
potential
p = density; subscript p and g refer to the particle and the gas,
respectively
0 = time; MA0 = time of fall through furnace
(p = angular coordinate
w = solid angle
4. Mechanism of Heat Transfer.-The transfer of heat from a solid
surface through a gas to falling particles takes place both by con-
vection and by radiation. The thermal resistance to convection is
composed of the resistance from the wall to the gas and that from the
gas to the particles, acting in series. Transfer of radiant energy takes
place directly from the wall to the particles. If the gas has infra-red
absorption bands, it also will receive heat directly from the wall and,
consequently, it will partially blanket the particles from radiant
energy. The net radiation transfer acts in parallel with the convec-
tion. The thermal resistance within the particle is important only if
the conductivity of the solid is low, or if the diameter is large. Con-
fining our attention to the opposite conditions for the present, the
overall coefficient of heat transfer is stated mathematically in terms of
the individual coefficients by the equation
hp
h = hr + (1)
hop
1 I --
hew
This equation applies only to point conditions. Each of the individual
coefficients and the area ratio, y, varies with the position of the particle.
In the case of particles falling through a cylindrical tube, y may be
expressed conveniently in terms of a dimensionless group which also
enters into the radiation coefficient
S 7rD'
NA ( 4-- dL) (4a)
S, - = -- -- = CDa.* (2)
A , iDdL
fThe surface area of particles having convex surfaces is four times the projected area, cf. p. 32.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
Little is known about the individual coefficients, as this problem
in heat transfer has apparently received little attention. A discussion
of the theory underlying the mechanism of heat transfer to falling
particles will prove helpful in interpreting the experimental data and
in extending the work to the practical design of large scale equipment.
5. Heat Transfer to Small Particles by Convection.-An exact
analysis of this problem in convection is exceedingly difficult, if not
impossible, because of the complicated hydrodynamics involved. Any
solution of the problem must be made on the basis of simplifying as-
sumptions which are valid at best only within certain limits. The re-
sulting equations, therefore, must be recognized as approximations,
and the nature of the assumptions should be clearly understood in
order to avoid faulty application.
A. Simple Concept
One of the simplest concepts of the mechanism of heat transfer
to falling particles is analogous to that recently proposed* for the
absorption of gases by spray droplets. The assumptions, while ad-
mittedly crude, lead to an expression for the heat transfer coefficient of
the same form as that given by the more rigorous attack on the
problem to be described later. Furthermore, the numerical values
calculated in this manner agree fairly well with those obtained from
the experimental data, so that, as a first approximation, the equation
has been recommended for design purposes.t The corresponding
equation for mass transfer was likewise found to agree within ten
percent with the observed values of the coefficient for the absorption
of ammonia, sulphur dioxide, hydrogen sulphide, and carbon dioxide
by falling droplets of various solutions.$
In this concept, it is assumed that each particle makes contact with
a tube of gas described by the trace of its periphery and of thickness s.
This thickness represents the distance through which all of the heat
above the temperature of the particle surface can be transmitted by
conduction while the particular layer is in contact with the particle
surface. As the velocity of the particle increases, s must decrease. In
any case, s is small compared to the diameter of the particle, so that
the volume of the tube is xD,Ls. In its simplicity, the tube of gas
represents a zone in which the temperature is reduced to that of the
surface of the particle, since complete temperature equilibrium is
*H. F. Johnstone and R. V. Kleinschmidt, Trans. Am. Inst. Chem. Engrs., vol. 34, p. 181, 1938.
tUniv. of Ill. Eng. Exp. Sta. Bul. 324, p. 79, 1940.
IH. F. Johnstone and G. C. Williams, Ind. Eng. Chem., vol. 31, p. 993, 1939.
ILLINOIS ENGINEERING EXPERIMENT STATION
established within the allotted time. The layer obviously does not cor-
respond to the so-called "laminar film," or "boundary layer" and is
purely hypothetical.
The quantity of heat transferred across the tube from the ambient
gas to the particle surface as the particle moves through a distance
of one diameter is
rD,2k Dp
(t9 - tp) --
s V
From a heat balance for this segment of the tube,
k D,
"rD2 - (t. - tp) - = rDp2spc (t, - tp) (3)
s V
whence
/ kDp
s = Vp " (4)
Vpac
The heat transfer coefficient hep is
k kVpoc
hp =- -= - (5)
s Dp
With the exception of the proportionality constant, here equal to
unity, the term on the right is indeed identical with that obtained by
Boussinesq in his classical treatment of thermal conduction to bodies
submerged in moving fluids. A brief discussion of the fundamental
equation and of the customary assumptions required for its solution
will now be given, along with a solution which avoids the most doubt-
ful approximation in the Boussinesq treatment.
b. Solution of the Fourier-Poisson Equation for a Spherical Particle
The Fourier-Poisson equation is a differential equation for thermal
conduction in moving fluids. In rectangular coordinates its form is
/ t at at at\ a / t\ a / at \ a / at\
cp -+u-+v-+- =- kI- +- k-+- k . (6)
\a0 ax ay az/ ax \ ax/ y\ ay/ az\ az
This equation itself is derived from the laws of hydrodynamics and
thermodynamics. In its simple form, as written,* it applies strictly
*The assumptions underlying the Fourier-Poisson equation and its solution for various cases
are discussed by T. B. Drew, Trans. Am. Inst. Chem. Engrs., vol. 26, p. 26, 1931.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
only to an ideal incompressible fluid in which the forces of viscosity are
negligible, and in which the kinetic energy changes are small compared
to the total heat change. These restrictions themselves do not seriously
limit the usefulness of the equation, as for many fluids, either gaseous
or liquid, moving through a heat exchange system at relatively low
velocities, the forces of viscosity are negligible and the pressure is
nearly constant.
Application of Equation (6) to convection problems is made after
determining, or assuming, the velocity distribution of the fluid, so that
u, v, and w may be evaluated at all points in the system. The simplest
case is that in which the velocity in all directions is zero. Such a con-
dition would prevail approximately in heat transfer from a quiescent
fluid to a very small particle. For the steady state in this case the
solution* of Equation (6) requires that
q = 2rkD, (t, - t,). (7)
Thus, the heat transfer coefficient is
2k
hcp =- (8)
Dp
This agrees with the recent observation made from experimental data
by Meyert that, for natural convection, the Nusselt number, h,,r/k,
approaches a constant value between 0.5 and 1.0 for small values of
the Grashof group, r3p2pgAot/p2. Meyer estimates that the heat trans-
fer coefficient for a particle 0.0016 inch in diameter surrounded by
air in a spray drier would be 240 B.t.u./(hr.) (sq. ft.) (deg. F.). With
a temperature gradient of 200 deg. F., which customarily prevails in
this apparatus, the rate of heat transfer would be 50 000 B.t.u./(hr.)
(sq. ft. of particle surface). To obtain a rate of this magnitude by
radiation alone would require a radiator temperature of about 2000
deg. F. This emphasizes the high rate of heat transfer attainable for
small particles.
a. The Boussinesq Solutions
In order to develop the Boussinesq solution of Equation (6) for
spherical particles submerged in a moving fluid, it is desirable to con-
sider first the case of the submerged flat plate. From this the
*The solution is best accomplished by writing the equation in spherical coordinates.
tP. Meyer, Trans. Inst. Chem. Engrs. London, vol. 15, p. 127, 1937.
tJ. Boussinesq, J. math. pures et appl., vol. 1, p. 285, 1905.
ILLINOIS ENGINEERING EXPERIMENT STATION
more complicated problem can be attacked by making additional
assumptions.
Let the flat plate of width x, and of infinite length be located in
the x, y plane with the leading edge at the y axis. The fluid moves in
the x direction parallel to the surface. It is supposed that v and w are
everywhere and at all times equal to zero, that u is constant and in-
dependent of z, and that k, c, and p are constants. These conditions
limit the case to streamline flow over the plate, although the results
should be valid as an approximation even when there may be turbu-
lence near the trailing edge of the plate, but when the boundary layer
as a whole is essentially laminar.
For the steady state, the foregoing conditions reduce the left side
of Equation (6) to the single term, cpuOt/ax. Boussinesq simplified
the equation further by assuming that the second partials with respect
to x and v are negligible. For a long plate it is obvious that the flow
of heat in the v direction must be negligible and, therefore, 02t/Oy2
will vanish. Neglect of the term O2t/Ox2, however, requires that the
thermal conduction in the direction of flow also be negligible. This
assumption will be shown to hold only within certain limits of the
fluid flow defined by the Reynolds number.
The simplified equation now becomes
at a2 02t
-- (9)
ax u az2
where a2 is the constant k/cp.
A solution of Equation (9) which satisfies the boundary condi-
tions, t = to for all positive values of x and for z = o, and t = ts*
for z = 0 for all values of x along the surface of the plate, is
t - too 2 f
=1 e- di (10)
ts - to 7r a
where
z u
2 a 2x
The rate at which heat is transferred from the fluid to the plate
*Although the equation may be solved for any variation of t with x, only the simple case
of constant surface temperature need be considered here.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
is found from the value of the temperature gradient at the surface.
From Equation (10)
/ at \ \~u
= -(t -- ) (11)
For the differential area dxdy located at the distance x from the
leading edge
dq = -k(- dxdy = k (t, - to) - dxdy. (12)
\z a-o \ra2x
The overall rate of heat transfer to the strip of width dy and length
X1 is
q = k (t, - ) dy 1-- dx (13)
0 &a2
= 2k (t, - t,) xidy kcpu
rXl
The average coefficient of heat transfer over the strip is
q 2 kcpu kcpu
h-- 1.13 (14)
(t, - t,)xldy N/ W Xl Xl
Before proceeding further, it is well to examine the simplifying
assumption made by Boussinesq. The assumption of negligible
conduction in the direction of flow is valid only when the ratio of
02t/ay2 to 02t/Ox2 is large.
If these derivatives are evaluated from Equation (10), the ratio
of the first to the second is found to be
/ 2x2 \2 1
\ z / 3 /2x \2 x\
-x 1 ( ) )2 (Re - Pr)-1 (1
Thus the approximation becomes more exact the larger the values of
Re - Pr and of x/z. It is not necessary to consider small values of
ILLINOIS ENGINEERING EXPERIMENT STATION
x/z, however, because most of the heat transfer takes place near the
plate. If only that region is included for consideration in which the
temperature gradient at/8z is greater than one-tenth of its value at
the surface, and if the smaller of the two derivatives may be neglected
when its absolute value is one-tenth that of the larger, the approxima-
tion is valid for values of Re - Pr greater than eight. The principal
assumption on which the Boussinesq solution is based is justified for
diatomic gases when the Reynolds group exceeds 10.8, and for water
(at ordinary temperatures) when the group exceeds 1.2.
Extension of the Boussinesq solution for Equation (6) to a cylinder
submerged in a moving fluid is readily made by transformation from
rectangular coordinates to the orthogonal curvilinear system repre-
sented by the stream function and the velocity potential of classical
hydrodynamics. The only additional assumption required is that the
motion of the fluid be irrotational, which in itself indicates that a
velocity potential exists. By a suitable choice of the origin of coordi-
nates, and after making the Boussinesq assumption of negligible con-
duction along a stream line, the form of the equation becomes identi-
cal with Equation (9). The boundary conditions are likewise the
same and the solution is, therefore, similar to that already given for
the flat plate.
For the three-dimensional case of a submerged body of revolution,
the form of Equation (6) in the new coordinate system now described
by stream lines (J = constant) and equipotential surfaces (VfI = con-
stant) is
at O2 r 2t 2 an at a2t
- = -/- + --+ y2 2. (15)
an V ant2 an 2 an 2 ax a 2 a2]
The x axis is the axis of revolution, and is parallel to the direction of
flow, while y is the radial distance from the x axis.
In order to simplify Equation (15), the first and second terms in
the bracket are assumed to vanish and the value of y is limited to its
value at the surface of the body. Neglecting a2t/laI2 represents the
usual assumption of negligible conduction along the stream lines, but
the neglect of the second term and the limitation imposed on the value
of y introduce new assumptions which must be examined. Solution of
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
the simplified equation as before gives the average heat transfer co-
efficient for the general case
jkVpc
ha, = M (16)
where M is a constant dependent on the shape of the surface and A is
a characteristic linear dimension. For the sphere, the average co-
efficient is
2 kVpc = 1.13 Vc(17)
hcp =113 (17)
\7 D, D,
We note that the coefficient is slightly higher than that calculated
from the less rigorous Equation (5). As a matter of fact, the experi-
mental data obtained in this study actually indicate that the propor-
tionality constant should be less than one. Furthermore, Leveque's*
critical examination of the simplifying assumptions mentioned in the
foregoing throws considerable doubt on the validity of the generalized
Equation (16) for the transfer coefficient. It appears that the second
term in the bracket of Equation (15) is actually negligible over most
of the region surrounding the body in which the temperature gradient
exists. On the other hand, the limitation on the value of y becomes
an allowable approximation only when
/ (II1 - n0) a2
y,, > 30J 2 (18)
Here II1 and HIo are, respectively, the values of the velocity potential
at the trailing and leading edge of the body. For potential flow
around a sphere it can be shownt that
3Dp
III - IIo = (19)
2
Also, for a sphere, ya, is approximately D,/r. Substitution of these
*A. L6vcque, Ann. mines, vol. 13, Ser. 2, p. 249, 1928.
tL. Prandtl and 0. G. Tietjens, "Fundamentals of Hydro- and Aeromechanics," p. 151,
McGraw-Hill Book Co., New York, 1934.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 1. GEOMETRICAL REPRESENTATION OF PROBLEM OF
HEAT TRANSFER TO A SPHERE
values in the inequality shows that the Boussinesq assumptions are
allowable when
DV2 -P )- > 13 400.
02 -A k
In such ranges of flow there is little doubt that the entire analysis will
break down due to the departure from ideal flow conditions.
b. Solution in Spherical Coordinates in Terms of Bessel Functions
The Boussinesq solution of the Fourier-Poisson equation for heat
transfer to a sphere assumed that a velocity potential exists for the
flow around the body. By making a slightly different assumption in
regard to the flow pattern it is possible to solve the equation directly
in terms of Bessel functions which can be evaluated.
For the steady state, Equation (6) may be written in the spherical
coordinates, r, 0, and ( (see Fig. 1) as follows:
at vo Ot v, at
v-- +-- -
Or r 90 r sin 0 9p
[ 1 9/ ot \ 1 t\ 1 02t
a2 ----r2 -+ - (sin -- ---- (21)
r2 Or r r2 sin 6 99 ( r2 sin20 ±2 (2
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
v,, ve, and v, are components of velocity in the direction of increasing
r, 0, and p, respectively.
We shall assume that the velocity of the fluid around the sphere
is everywhere parallel to the surface and equal to V. As a result of
symmetry about the axis of revolution, v,, v,, and a2t/aýp2 are zero,
and ve = V. The second term in the bracket represents the ac-
cumulation of heat in an element of fluid due to heat transfer in
the direction of increasing 0. Since the streamlines have been as-
sumed to be parallel to the surface, this represents the net conduc-
tion along a streamline, and should be negligible according to the
usual Boussinesq assumption. The equation then becomes
at a2 / ( t \
- r2- . (22)
a Vr Or O r
A solution of Equation (22) may be assumed to have the form
t - t,
= F(O) G(r) (23)
to3 - t8
where F (0) is a function of 0 alone and G (r) is a function of r
alone. Differentiation of Equation (23) and substitution in Equa-
tion (22) gives
1 dF a2 2dG d'G|
SL 2- + r--]. (24)
F d VG dr 2dr2
Since each side of this equation is a function of only one independent
variable, each may be set equal to a constant, say -b2. Then the
forms of the functions F(0) and G(r) are found by solving the ordinary
differential equations:
dF
d + b2F = 0 (25)
dO
and
d2G dG b2V
r-- + 2 --+ -G = 0. (26)
dr2 dr a2
The first of these has the solution
F(O) = Be-b'o
ILLINOIS ENGINEERING EXPERIMENT STATION
and the second, 2
a 2b\ rV
G(r) = b /r J1) (28)
where J1 is the Bessel function of the first kind and of the first order.
Since the original equation is linear in G and its derivatives, the
sum of any number of particular solutions is also a solution, or
t - t o a (2bi\/ rY
= Bie-b\ J1 .( (29)
to - t, i0 b6, rTV a
The arbitrary constants Bi and bi may be determined from the
boundary conditions which require, first, that t = t, for 0 > 0 and
r = ri and, second, that t = to for 0 = 0 and r > ri. From the
first condition,
S= Ji(2b- rV ). (30)
Thus, the successive values of the constant become
112 22
bi2 = 0, , , etc.
2Re Pr' 2Re Pr
The values of 7, which are the roots of Ji(f) = 0, are given in
standard tables.*
From the second boundary condition
o a ( 2bV rV
1 =1: B ,-- Jr --. (31)
t-o bV rV \ a (31)
i=0 bi\ a
The values of Bi may now be determined by comparing the coeffi-
cients with those of similar terms of a Fourier-Bessel seriest for the
expansion of -/r :
a 2bi rV \
S__ 2bVrV ) a ). (32)
bio 2bJi, a
*E. Jahnke and F. Emde, "Funktionentafeln," Teubner, Berlin, 1909.
tF. Bowman "Introduction to Bessel Functions," p. 108-11, Longmans, Green and Co.,
London, 1938.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
3-51 -1 -
5.
FIG. 2. COMPARISON OF EQUATIONS OF HEAT TRANSFER TO A SPHERICAL PARTICLE
The final solution of Equation (22) then becomes
_2, a e-bO ( 2b/ arV \)
E- - J - -- (33)
E o 2b,/ rV \ V - . (33)
biJ2 - - a
The total rate of heat transfer across the surface of the hemi-
sphere is
q = - k r2 sin 0 dcpdO
o Jr r
1 + e--rb2
= kri (t( - t,)
i=o 1 + bi4
(34)
The average heat transfer coefficient then is
k 1 + e-b,
he, = +
D, i=o 1 + b,4
(35)
The series converges fairly rapidly. For Re Pr= 10 the sixth term
is only 0.1 per cent of the sum, while for Re-Pr= 100, seventeen
terms are required for this percentage.
ca'fi:
/0
/
t - t,
= -
too - t,8
~
~if-~Kff~--+-F
W I I Jl l
----
I
3
1
--
* CifL
]
I
t
F
C
^
^
r
*9
J>
^
*J
*]
J
fL
~qc/f/or 73
ILLINOIS ENGINEERING EXPERIMENT STATION
Equation (35) is plotted in Fig. 2 along with Equations (17) and
(36). It is interesting to note that, according to Equation (35), the
Nusselt group, hcD,/k, approaches the value of 2.0 as the Reynolds
number approaches zero, as required by Equation (8) for the heat con-
duction to very small spheres in a quiescent fluid. At high Reynolds
numbers the curve approaches the value given by
kVpc
h,, = 0.714 kVpc (36)
Dp
This is a limiting value which may be derived by integrating Equation
(13) for heat transfer to a submerged flat plate over a spherical sur-
face. Such an integration is justifiable when the boundary layer is
very thin. In such a case the surface of the sphere may be conceived
of as being made up of many narrow flat strips each of angular width
d>, and extending from the leading point to the trailing point. The
diagrammatic representation of this problem is also shown in Fig. 1.
We again assume that the velocity of the fluid elements is everywhere
constant and parallel to the surface of the sphere. From Equation (14)
the rate of heat transfer to an element of area, r 2 sin OdOdp,, located
at a distance, rO from the leading point of the sphere is
cpVk ri sin 0 dodwp
dq = (t. - t) cpk r sin (38)
V v V r1O
Integration with respect' to 0 may be performed after expansion of
sin 0 in a series. This gives
1.790 cpVk
q = (t, - t7) - 2rr2 (39)
V 2 Dp
and the average heat transfer coefficient becomes that given by
Equation (36).
Not only do the values of he, from Equation (35) agree more
nearly with the experimental results than the Boussinesq Equation
(17), but also there is reason to believe that the assumed pattern of
flow around the sphere corresponds more nearly to actual conditions
in the region near the surface in which the resistance to heat transfer
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
8=90°
vI=0 for all
via'/ues of r
Z0
. ------------------
- - -'^
0 =a
V v,- a A0=or a/!
va/ues of r
S1 1.5
------ Pofen/a/ F/ow (Assi/med hiy Bo'ss/nresq).
- - - Vi'sco us F/owv. ---- /ow Assc/med Here.
FIG. 3. ASSUMED COMPONENTS OF FLUID VELOCITY IN
NEIGHBORHOOD OF A SPHERE
exists" than that required by the theory of potential flow. The com-
ponents of velocity are compared with those calculated for potential
flow and for viscous flow around a sphere in Fig. 3. The latter are
derived from the Stokes' equationst which neglect entirely the forces
of inertia. At a point on the surface 90 degrees from the leading point,
according to the potential flow theory, the fluid has a tangential
velocity of 1.5V, while, according to the theory of viscous flow, the
*The effective thickness of a uniform fluid layer surrounding a sphere in which all of the
resistance to heat transfer may be considered to exist, expressed as a fraction of the radius of
the sphere, is approximately 2k/hcp Dp. According to Equation (35) this group is never greater
than one and at a value of Re-Pr equal to 100 the value is 0.3. Therefore the distance in
question cannot be greater than the radius of the sphere.
tLeigh Page, "Introduction to Theoretical Physics," p. 233, D. van Nostrand, Inc., New
York, 1928.
1·
0
ILLINOIS ENGINEERING EXPERIMENT STATION
velocity must be zero. Actually the velocity is zero at the surface and
increases rapidly within a narrow region to a value nearly equal to V.
The radial component is zero according to either theory. Thus, at this
location, the velocity has approximately the value assumed for the
foregoing solution. At other points on the surface the assumption of
the constant tangential velocity is not far in error.
6. Radiation to Clouds of Particles.-Heat transfer by radiation
from a solid surface to small opaque particles is similar in principle to
radiation to an absorbing gas. When a radiant beam enters a cloud of
particles, those particles near the front of the cloud have a greater
chance of intercepting the beam than have those in the interior or at the
rear of the cloud. This is equivalent to saying that the average inten-
sity of the radiation varies through the cloud. It is necessary to know
the variation of intensity with distance along the beam in order to cal-
culate the transfer of radiant energy to particle clouds of various
shapes. The problem is analogous to that arising in gas radiation. In
fact, the equations used for the calculation, with the introduction of
new constants, will be similar to those used by Nusselt* for calculating
radiation to different gas shapes.
For particles which absorb the incident radiation completely the
problem of variation of intensity with distance is that of finding at
any point the average fraction of the beam blanked out by particles
along its path. This is equivalent to finding the covering power of a
number of small discs tossed at random on a plate. The analysis has
been made by Haslam and Hottelt as follows:
When n discs, each of area a, have been tossed on a plate of area
A, the fraction covered will be f,. The probability that the (n + 1) st
disc will fall on uncovered area is 1 - f,. On the average, its effective
covering power will be a (1- f,). For n+ 1 discs the fraction
covered is
a aa\ a
fn+1 = f + (1 - f) = f, 1- - + . (39)
In a similar manner, the fraction covered by (n + 2) discs is
+ a a a 2 a a
f.+2 = f+1- 1 + A- A) - +2 -
*W. Nusselt, Forschungsarbeiten Geb. Ing., No. 264, p. 40, 1923; Zeit. Ver. Deut. Ing., vol.
70, p. 763, 1926.
tR. T. Haslam and H. C. Hottel, Trans. Am. Soc. Mech. Eng. (FSP 50-3), vol. 50, p. 9,1928.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
or
a 2a 2 a
f.+2 1=fý (l_ ) 1- -+(2a - (f- . (40)
A -AA A
By mathematical induction, for (m + n) discs
a
f+ -1 = (f -1) - - (41)
When n = 0,
a m
= 1 - - (42)
If C is the number of solid particles per unit volume in the space
above A and 1 is the thickness of the cloud above A, the number of
particles, m, projected on A is CIA. Therefore
m= 1 - 1 - - (43)
When the area of each particle is very small compared to the area
of the radiating surface,
r / a \aCIA
fm=1 Lim I 1 - -(
a/A-0 I A
= 1 - [Lim(1-" A )Ala]Cla
a/A-0 A
= 1 - ecla. (44)
From Equation (44) the relationship between the average intensity
at point I and the intensity at 1 = 0 is
II = Ioe-Cla. (45)
For particles which do not absorb completely, but which have a
high emissivity so that the second order radiation is negligible,
Equation (45) becomes
I, = Ioe-ecla.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 4. GEOMETRICAL ANALYSIS OF RADIATION FROM WALLS OF AN
INFINITE CYLINDER TO CLOUD OF PARTICLES
This equation is similar to Beer's law for the absorption of light
by liquids and gases except that here the proportionality factor may
be calculated from the size of the particle, while in Beer's law it must
be evaluated experimentally.
Equation (46) may be used for calculating the absorptivity of a
cloud of particles of a given shape and concentration. Referring to
Fig. 4, the rate at which heat radiating from a surface element dA,
strikes an element of volume dV, located in a cloud of particles at a
distance 1 from dA and subtending the solid angle do at dA is given by
dqi = In cos a e-(c'CdAdo. (47)
The average intensity of the ray at the front face of the volume ele-
ment is obtained by dividing Equation (47) by the cross sectional
area (12d,) of the ray at the point 1.
cos a e- ,Cla
I1 = I, dA. (48)
l2
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
Now consider that the element of volume, dV, is subdivided into
cylindrical elementary rays, each having a length dl and a cross sec-
tional area dA'. The quantity of radiation entering each of these ele-
mentary cylinders has the value IidA'. From Equation (46) the frac-
tion ecCadl of this amount is absorbed on passage through the cylinder.
The total amount dq2 absorbed by the volume element dV is ecCall dV,
or
ECaI, cos a e-*CladAdV
dq2 = (49)
The total radiation from a solid wall which is absorbed by a cloud
of particles is obtained by integration of Equation (49) over the area
of the radiation wall and the volume of the cloud,
f f cos a - e-'cladAdV
q = CaI 1--------. (50)
o.J area 12
The total rate at which energy is emitted in all directions from the
surface is InrA. The absorptivity of the cloud of particles is the frac-
tion of this total emission which is absorbed, that is,
q2 epCa cos a - e-',Cla
Ec = - = -- ----- dAdV. (51)
C IarA 7rA vol. area 12
The integration of Equation (51) has been performed by Nusselt*
for the case of the sphere and the case of the infinite cylinder. For the
former,
1 1 + 2eCroa
C, = 1 - + e-2cra. (52)
2 (peCroa)2 2 (ECroa)2
For the infinite cylinder
4 fi/2 f7/2 -2e,Croa cos y
4c = 1 - -e cos cos2 cos - d#dy. (53)
The integral in Equation (53) was evaluated graphically.
*W. Nusselt, Zeit. Ver. Deut. Ing. vol. 70, p. 763, 1926.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 1
ABSORPTIVITY OF CLOUDS OF PARTICLES IN FURNACES OF VARIOUS SHAPES
Absorptivity e
2 eCroa
0
0.10
0.20
0.25
0.30
0.40
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.50
3.00
Sphere
0
0.0642
0.1243
0.1520
0.1792
0.2308
0.2748
0.3835
0.4715
0.5451
0.6066
0.6596
0.7030
0.7719
0.8220
Infinite
Cylinder
0
0.0932
0.1767
0.2149
0.2512
0.3170
0.3764
0.5000
0.5957
0.6709
0.7298
0.7769
0.8142
0.8689
0.9048
Between
Infinite
Parallel
Plates*
0
0.1674
0.2961
0.3506
0.3999
0.4854
0.5568
0.6905
0.7806
0.8428
0.8866
0.9175
0.9397
0.9674
0.9822
2epCroa
Absorptivity e,
Sphere
0.8547
0.8865
0.9073
0.9232
0.9356
0.9454
0.9532
0.9595
0.9646
0.9688
0.9724
0.9753
0.9778
0.9800
1.0000
Infinite
Cylinder
0.9293
0.9460
0.9580
0.9665
0.9728
0.9776
0.9814
0.9843
0.9868
0.9885
0.9901
0.9913
0.9927
0.9940
1.0000
Between
Infinite
Parallel
Plates*
0.9901
0.9945
0.9969
0.9983
0.9989
0.9994
0.9996
0.9998
0.9999
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
*Here ro represents the half distance between the plates.
For the case of infinite parallel plates, Jakob* obtained the equation
/2 -2e,Croa
S= 1- 2 e cos a sin a - cos a - da.
o
Here ro represents one-half of the distance between the plates. The
integral in Equation (55) may be evaluated in terms of the exponential
integral
e-x
Ei(-p) = --dx
o x
for which values are listed in standard tables.t The equation then
becomes
S= 1 - (1 - 2eCroa) e-2*,Croa+ (2epCroa)2Ei (-2e,Croa). (55)
All of these shapes are of practical importance in furnace design.
The values of Ec for the three cases are listed in Table 1 and are plotted
in Fig. 5. Inspection of the curves shows that for the same value of
*M. Jakob, "Der Chemie-Ingenieur." vol. 1, No. 1, pp. 299-303, Akad. Verlag., Leipzig, 1933.
tJahnke-Emde, "Functionentafeln," loc. cit.
---
---------
---
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
FIG. 5. ABSORPTIVITY OF CLOUDS OF PARTICLES IN FURNACES OF VARIOUS SHAPES
2eCroa, or eCDa, a cloud of particles enclosed between infinite
parallel planes has the greatest absorptivity. This may be explained
qualitatively by the fact that the average length of a radiant beam is
longest for the case of infinite parallel planes because of infinite di-
mensions in two directions. The absorptivity of the cloud enclosed
in an infinite cylinder is higher than that for a sphere because the
cylinder has infinite dimensions in one direction. The absorptivity for
other shapes of clouds encountered in practice may be estimated from
Fig. 5 by interpolating between the curves shown. For example, the
value for a cube should fall between the curve for an infinite cylinder
and that for a sphere, since the average length of a radiant beam is
less than that in the cylinder but greater than that in a sphere. The
absorptivity for a long shaft of rectangular cross section should fall
between the two upper curves.*
*In a recent paper, Hottel and Egbert show that the effect of the gas shape on radiation
of furnace gases may be approximated by using an effective average beam length. On this basis,
the absorptivity of a hemisphere of a cloud of particles receiving radiation from a spot on the
center of the base of the hemisphere of radius r is 1- e- ,cr. When the gas shape is other
than a hemisphere, the same formulation suffices provided the radius of the hemisphere is
replaced by the effective beam length, 1, which, for small values of the quantity epCra, is four
times the mean hydraulic radius, i.e., four times the gas volume divided by the area of the
bounding walls. This is exact as cc approaches zero. Comparison with the theoretical curves of
Figure 5 shows that the suggested approximation is good up to values of ec of 0.2 but above
this value considerable deviation occurs. For high values of the group analogous to cpCra in
gas radiation, Hottel and Egbert recommend using 85% of the effective average beam length
obtained as above. This appears to be satisfactory for ec greater than 0.6. While this method
of approximation has the advantage of conforming to established engineering practice in the
calculation of heat transfer from radiating gas masses, it lacks the theoretical background of
Equations (52), (53) and (54). (H. C. Hottel and R. B. Egbert, Preprint of paper presented
before Am. Soc. Mech. Eng., New York, Dec. 1940.)
Zp C-ra
ILLINOIS ENGINEERING EXPERIMENT STATION
The area-emissivity factor FAE is given by
1
FAE = (56)
1 1
-+ -1
Ew Ec
where c, is the absorptivity of the wall. Since e~ is generally small
compared to cw, FAE may be taken as equal to c,. The coefficient of
radiant heat transfer to clouds of particles expressed on the basis of
the area of the particles then is given by
0.172e, [(T,/100)4 - (T,/100)4]
hr = (57)
7 (T, - T,)
7. Heat Transfer from Wall to Gas.-When a cloud of particles
falls through a heated tube, three factors produce convection within
the tube, viz., the normal convective forces due to density differences
along the wall, the increased convection due to the introduction of cold
particles at the top of the tube, and the disturbance due to the passage
of the particles through the gas. Consequently, the rate of heat
transfer from the wall to the gas is somewhat greater than that due to
natural convection. Because of the complexity of the problem no at-
tempt has been made to reduce it to a mathematical basis.
In the summary of the work previously published,* the Rice equa-
tiont for the heat transfer coefficient for natural convection from a
long vertical pipe was recommended as a rough approximation. In
this the coefficient he, is given as a function of the Grashof group.
k D3p,2g(t - ) 113
hew = 0.115 --D---- ]. (58)
It was realized that the values of hew from Equation (58) would be
quite conservative. From the treatment of the experimental data
which follows it appears that the actual values are from two and one-
half to four times those calculated from the Rice equation. The co-
efficient is apparently independent of the wall temperature and of the
particle size but increases with the diameter of the furnace.
*Univ. of Ill. Eng. Exp. Sta. Bul. 324, p. 74, 1940.
tW. H. McAdams, "Heat Transmission," p. 254, McGraw-Hill Book Co., New York, 1933.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 2
PRINCIPAL DIMENSIONS OF FURNACES USED IN EXPERIMENTAL WORK
Small Furnace Large Furnace
Inside diameter of tube, in........................ 1.61 3.76
Outside diameter of tube, in.......................... 1.90 4.00
Inside diameter of refractory lining, in............... 2.5 5.0
Total length of tube, in............................. 54.5 120.0
Heated length of tube, in .......................... 48.0 110.0
Distance from top of calorimeter to bottom of heated
section, in................................... . 2.0 4.0
Distance from feeder to top of heated section, in...... 4.0 17.0
Location of thermocouples above bottom of tube, in.. . 8, 20, 32, and 44 27.6, 49.2, 70.8,
92.4, and 114.0
III. EXPERIMENTAL WORK
8. Introduction.-The object of the experimental work was to
provide accurate data covering a sufficient number of conditions so
that the individual heat transfer coefficients could be determined. The
agreement of these with the theoretical equations or with empirical
correlations should justify extrapolation to large scale design. To
this end, measurements were made on the rate of heat transfer to
particles of known average sizes of several materials falling through
heated tubes at known wall temperatures. The rate at which the
particles were introduced was varied over a considerable range as this
proved to be the important variable by which the data could be
analyzed to calculate the individual coefficients. The effect of changing
from air to carbon dioxide as the ambient gas was also studied.
The general method of the study consisted in adjusting the walls
of a tubular furnace to constant temperature and then introducing
the fractionally screened particles at a constant rate. The quantity of
heat transferred and the temperature of the particles were determined
by collecting the material in a calorimeter immediately below the
furnace and observing the temperature rise of the water therein. The
particles studied were sand, carborundum, and aloxite. These materials
were selected because they are inert to air at high temperatures and
do not react with water. Furthermore, their heat conductivities and
absorptivities differ considerably.
9. Heat Transfer Measurements.-Two sizes of furnaces were
used. The essential dimensions of these are given in Table 2. Details
ILLINOIS ENGINEERING EXPERIMENT STATION
Sheet /ron
7
(cr)- Sma// Furnace (io)-Top of Lar'-e Furnace
FIG. 6. APPARATUS USED FOR HEAT TRANSFER MEASUREMENTS
of the small furnace and the top of the large furnace are shown in
Fig. 6. In both cases stainless steel tubes were used and thermocouples
were peened into the walls from the outside. The smaller furnace was
heated electrically by resistance coils wound in four sections on a
corrugated alundum sleeve. Uniformity of temperature was main-
tained by adjusting the external resistance of the heating elements.
A typical temperature distribution along the inside of the tube is
shown in Table 3.
The large furnace was heated by natural gas. The stainless steel
tube was surrounded by cylindrical fire-brick tile, 1 in. thick, for pro-
tection against the direct action of the flames. The outer wall of the
furnace was constructed of 41/2-in. fire-brick with nine burner ports
arranged for tangential firing. Control of this furnace was manual.
The temperature of the tube wall indicated by the six thermocouples
was recorded by a multipoint recording pyrometer. Variation along
the tube did not exceed 25 deg. F. during a run.
The feed to the small furnace was introduced into the center
through a glass tube. The rate was controlled by an electric vibrating
feeder. The feed to the large furnace was introduced from a hopper
by means of a motor-driven screw. This discharged into a water-
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 3
TYPICAL TEMPERATURE DISTRIBUTION IN SMALL FURNACE
Distance Below Top of Temperature Distance Below Top of Temperature
Heating Zone, in. deg. F. Heating Zone, in. deg. F.
2 707 26 1047
5 992 29 1052
8 1105 32 1072
11 1130 35 1092
14 1085 38 1112
17 1075 41 1107
20 1075 44 1030
23 1067 47 812
Average temperature, omitting top and bottom points = 1075 deg. F.
Average temperature from four wall thermocouples = 1080 deg. F.
cooled section of vertical pipe fitted in the center of the stainless steel
tube through asbestos packing. This arrangement was necessary in
order to avoid heating the particles in the screw feed by conduction.
The calorimeter used was a four-liter Dewar flask fitted with a
wooden top through which a stirrer was attached. With both furnaces,
the tube extended about one inch through a hole in the top of the
calorimeter. A forty-junction copper-constantan thermocouple, with
one set of junctions in ice, was used to measure the temperature rise.
The thermocouple was carefully calibrated against a Beckmann ther-
mometer. A Leeds and Northrup students' type potentiometer with
extended lower scale was used to measure the thermocouple e.m.f.
The water equivalent of the calorimeter, thermocouple, and stirrer was
determined electrically several times during the work.
The general procedure adopted for the runs was as follows: A
known small quantity of the powder was introduced into the hopper
of the feeder. When the temperature of the furnace was uniform and
constant and the rate of temperature rise of the calorimeter was con-
stant, the feed was started. The rate was maintained constant until
the hopper had discharged to a fixed mark. In order to avoid a draft
action through the tube, the hopper was not allowed to empty com-
pletely. The elapsed time was measured with a stop watch. The net
increase in temperature of the calorimeter was noted after the rise
had again reached a constant rate. The wall temperature was taken
as the average of that observed just preceding and just after the run.
In general, these did not differ by more than 5 deg. F.
Measurements were made in this way in the small furnace at wall
temperatures from 575 to 1050 deg. F., and at feed rates from 3 to
50 lbs. per min. per sq. ft. Three particle sizes of sand and two of
carborundum and aloxite were used. Two series of runs were made
ILLINOIS ENGINEERING EXPERIMENT STATION
with sand at wall temperatures of 725 and 1030 deg. when the air
in the furnace had been replaced by carbon dioxide. In the large
furnace two series of runs were made at a wall temperature of 1000
deg. F. using two screen fractions of sand.
10. Secondary Measurements.-In order to calculate the overall
heat transfer coefficients from the measurements it was necessary to
have accurate data on the average diameter, the projected area, and
the rates of fall of the particles. This necessitated a secondary study
in which these quantities were measured accurately by improved
methods.
A. Average Particle Size
The arithmetic average "diameter" of the particles in each screen
fraction was determined by observation under a petrographic micro-
scope equipped with a movable stage. The "true average diameter"
was obtained by securing random orientation on the microscope slide
by using a method suggested by Tooley.* A thin layer of a gum arabic
solution in glycerine and water was spread uniformly over the slide.
Pin scratches in the layer were observed under the microscope until
the film reached a stiff jelly-like consistency. The material to be
measured was mixed thoroughly and a small quantity was taken from
the whole with a spatula. This was spread on a paper which was then
held a short distance above the slide and tapped lightly to transfer the
particles with a uniform distribution. The orientation of the particles
remained fixed as they fell on the gum.
Several hundred particles on each slide were traversed under the
microscope from several directions. The average diameter was found
by dividing the reading on the micrometer screw by the number of
particles measured in each traverse. The average for all the traverses
was then taken as the "true average diameter," D,.
B. Average Projected Area
Tooley* has shown that for particles having plane or convex sur-
faces the surface area available for heat transfer is four times the
average projected area obtained for all possible orientations of the
particles. The slides prepared as described were placed on a projecting
microscope and the images on a ground glass plate were traced on
paper. The projected areas of individual particles were then measured
with a planimeter. The areas of 50 to 100 particles from each sample
*F. V. Tooley, Ph.D. Thesis in Engineering, University of Illinois, 1939.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 4
SUMMARY OF MEASUREMENTS OF DIAMETER, MASS, AND
USED IN HEAT TRANSFER WORK
DENSITY OF PARTICLES
Average Projected Area a
Average Average Volume sq. in. X 10-2
Screen Diameter Particle Density Constant
Size D, Mass w, gm. per cc. K
mm. gm. X 104 Experi- Calculated
mental rD,2/4
A. Sand
30-40 0.545 1.865 2.65 0.435 2.50 2.15
40-50 0.436 0.941 .... 0.429 1.40 1.39
60-80 0.326 0.35 .... 0.431 0.904 0.904
Av. 0.432
B. Carborundum
No.40 0.502 1.214 3.21 0.290 1.92 1.98
No.60 0.344 0.386 .... 0.351 1.02 0.935
Av. 0.320
C. Aloxite
No.46 0.485 1.362 3.92 0.304 1.88 1.85
No. 60 0.344 0.581 .... 0.344 0.949 0.935
Av. 0.324
were averaged. This method of determining the surface area is
obviously not applicable to particles having concave surfaces. With
none of the materials studied was this condition serious.
C. Average Mass and Volume of Particles
The average mass of a particle was found by counting and weigh-
ing from one thousand to three thousand particles out of each fraction.
The average volume of the particles was calculated from the
average mass and the density. The latter was found by displacement
of water in a volumetric flask.
From the average volume and diameter of the particle the "volume-
shape factor" K was calculated from
K
Dp3
Values of the factor were approximately constant for different sizes
of the three materials studied but were much larger for the sand than
TI
ILLINOIS ENGINEERING EXPERIMENT STATION
for the carborundum or aloxite particles. In all cases the coefficient
was lower than 7r/6 which would be the value for spheres.
A summary of the data obtained in these measurements is given in
Table 4.
TABLE 5
SUMMARY OF RESULTS ON DRAG COEFFICIENTS FOR SAND,
CARBORUNDUM, AND ALOXITE PARTICLES
Average Diameter Observed Terminal Drag Reynolds
Screen Size of Particles Dp Velocity Vt Coefficient Number*
mm. ft. per sec. CD DpVIp,/p
A. Sand
30-40 0.534 12.1
0.534 12.4 0.933 128
0.568 12.7 0.941 139
0.583 13.5 0.900 149
0.594 14.6 0.875 162
0.584 15.9 0.671 173
40-60 0.401 8.85
0.427 9.35 1.36 75.8
0.435 9.88 1.25 81.5
0.465 10.2 1.24 90.8
0.490 10.8 1.18 100
0.518 12.1 0.966 120
60-80 0.313 5.50
0.308 6.05 2.54 34.9
0.319 6.57 2.12 39.3
0.335 7.08 1.92 44.5
0.387 7.57 1.81 52.4
B. Carborundum
No. 40 0.464 8.48
0.490 9.00 1.52 79.8
0.490 9.70 1.33 85.5
0.483 10.1 1.17 89.1
0.499 10.9 1.07 97.8
0.523 11.5 0.988 109
0.535. 12.3 0.893 119
No. 60 0.272 5.65
0.302 6.12 2.08 33.1
0.326 7.18 1.75 40.5
0.365 8.06 1.49 51.9
0.381 8.79 1.27 59.9
C. Aloxite
No. 46 0.480 10.5...
0.494 11.5 1.18 101
0.507 12.5 1.02 113
0.524 13.7 0.880 128
0.591 15.7 0.791 162
No. 60 0.331 7.56
0.348 8.50 1.56 51.6
0.371 9.58 1.31 62.5
0.410 10.7 1.16 77.2
0.430 12.3 0.941 92.2
*Based on physical properties of air at 85 deg. F.
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
D. Terminal Velocity and Drag Coefficient of Particles
The terminal velocity of the particles was measured by air elutria-
tion tests using a method similar to that of Martin.* The pro-
cedure was as follows: About 25 grams of the material from each
screen fraction was placed in a brass cone at the base of the elutriation
tube. This was a glass tube 1 in. in diameter and 51 in. high. Air was
blown into the cone through an inverted nozzle. The velocity was
measured by means of a calibrated Venturi meter. Beginning at a low
velocity the air velocity was gradually increased until a few particles
were thrown a short distance above the top of the elutriator tube. At
this velocity, the flow of air was maintained constant until only a very
few particles were being ejected, and these were rising only a short
distance above the tube. The air was then shut off and the material
collected outside of the tube was weighed. An effort was made to
adjust the air velocity so that each fraction had approximately the
same weight. Several fractions were collected from each sample placed
in the elutriator, and the average diameter of the particles in each
fraction was determined as described in the foregoing. In calculating
the drag coefficients, the arithmetic average of the air velocities of
two successive sub-fractions was taken as the terminal velocity of the
particle having the average diameter of the second sub-fraction. The
coefficients were calculated from the equation
8K pp - p Dg (60)
CD = (60)
7r pg Vt2
It may be noted that in this equation the projected area of the particle
on which the drag resistance is based is taken as 7D,2 /4. There may
be some question as to whether this is correct, since it has already
been shown that the volume-shape factor was less than 7r/6 for all of
the materials studied. For convex surfaces, however, the error in-
volved in computing the projected area as that of a sphere having the
diameter D, is less than that in computing the volume on this basis.
Furthermore, the projected area calculated in this way agrees closely
with that measured directly, as is shown in Table 4.
Values of the drag coefficient calculated for the various materials
are shown in Table 5. The table also shows the variation in the size
of the particles in each screen fraction used in the heat transfer
*Geoffrey Martin, Trans. Inst. Chem. Engrs., London, vol. 4, pp. 164-78, 1926.
ILLINOIS ENGINEERING EXPERIMENT STATION
JoCIOKes L -al',_
4C 4
___ ^- /?e
4
3
2
1~
Aliens >
Eq'ua'f/on, >
1,
t
,,_,
o -Sand
o- Carborund'um
a- A/oxife
* -.C7 d ,f-//t/
t
L
t
L
'/O 20 30 40 60 80 /00 200 300 400 600
RetWno/ds A/umber, Re
FIG. 7. DRAG COEFFICIENTS FOR PARTICLES FALLING THROUGH AIR
measurements. The drag coefficients are also plotted in Fig. 7. The
data agree excellently with Martin's results on carefully sized fractions
of sand particles. When the volume-shape factors were not included,
the points representing the data on the three materials were scattered
considerably. Since inclusion of the factors brings the data together,
the surface roughness evidently is not important in determining the
frictional coefficient. The carborundum particles were considerably
rougher than the sand, as shown by the smaller shape factor. Under
the microscope also these particles appeared to have much more ir-
regular shapes than the sand and aloxite particles. Martin's volume-
shape factor for sand was 0.284, this indicating very rough grains.
It is probable, however, that he did not take precautions to prevent
preferred orientation of the particles on the microscope slide and that
the lower value may be explained on this basis.
From the drag coefficients, the time of fall was calculated from the
familiar equation of accelerated motion,*
dV 7rCD p, V2
- = g --(61)
dO 8K pp Dp
*A slight error is recognized in this calculation since the effective mass of a body is greater
for accelerated motion than for uniform motion due to the inertia of the ambient fluid. The
mass is increased by approximately one-half of that of the displaced fluid.
- 4IRe
0.8
0.6
0.4
C
r= -7/
/VeWTrQ/S Law, LP V.('
-N
*~1
0
$
---
'--'' ~ "
(
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 6
CALCULATED TIMES OF FALL OF PARTICLES THROUGH FURNACES
Time in Heated Section of Tube, sec.
Screen Size Ambient Gas
Small Furnace Large Furnace
A. Sand
30-40 Air 0.474 0.823
40-60 Air 0.518 0.976
40-60 CO2 0.605 ..
60-80 Air 0.613 1.258
B. Carborundum
No. 40 Air 0.511 .....
No. 60 Air 0.619 .....
C. Aloxite
No. 46 Air 0.485 .....
No. 60 Air 0.552
Equation (61) was integrated graphically, or by a series approxima-
tion in the range between Reynold's number of 10 and 200 where
CD = 12.8Re-°03. (62)
The distance of fall as a function of velocity was likewise determined
by a second series integration. The time required for a particle to pass
through the heated zone was then found from graphs of the velocity-
time and distance-velocity functions.
It may be observed that the product CDg is nearly independent
of temperature and, therefore, for any one gas, the time of fall calcu-
lated for one temperature may be safely used at other temperatures
not too far removed. The viscosity of air varies approximately as the
0.7 power of the absolute temperature. Since the density varies as the
-1.0 power, the Reynolds number varies as the -1.7, and the drag co-
efficient as the 0.9 power. Thus CDP, must depend only on the -0.1
power of the absolute temperature. In the experimental work, the
maximum variation of the "film" temperature of the particles was
from approximately 300 deg. F. to 800 deg. F. This would cause a
variation in CDP, of only 5.5 per cent.
Table 6 shows the calculated average times of fall for the various
materials studied in the two furnaces.
ILLINOIS ENGINEERING EXPERIMENT STATION
IV. CORRELATION OF EXPERIMENTAL DATA
11. Experimental Results.-The complete experimental data in
the heat transfer measurements are given in Tables 7, 8, and 9. The
runs are classified according to the furnace wall temperature, particle
size, quantity of material fed, size of furnace, and nature of the
ambient gas. It will be observed that the basic quantities measured
by the calorimetric observations give the final particle temperature as
well as the total heat transferred. The logarithmic mean temperature
difference between the wall and the particle was calculated and, from
this, the rate of heat transfer per degree difference was computed.
On the basis of the feed rate, surface area of each particle, and the
time of exposure, the overall heat transfer coefficient, designated as hm,
was then calculated. The radiation coefficient, hr, was estimated on
the basis of Equation (57). The overall convection coefficient, based
on the particle area, is equal to (hm - hr).
A. Sample Calculation
A sample calculation of the data, based on Run 95, is shown below.
Weight of feed ....................................................100 gm.
D uration of feed ............................. .................. 80.3 sec.
161 0.0141 sq. ft.
Cross-sectional area of furnace. (- ( = 0.0141 sq. ft.
100 X 60 X 1
F, feed rate .................... = 11.7 (lbs.)/(min.)(sq. ft.)
453 X 80.3 X 0.0141
A, increase in particle temperature................... ...............361 deg. F.
Mean specific heat of quartz ............................................0.205
100
Heat transferred during run ....................- X 0.205 X 3.5 = 16.6 B.t.u.
453
16.6 X 3600
Rate of heat transfer ............................ = 745 B.t.u./hr.
80.3
t., average wall temperature ................... ................... 858 deg. F.
(to-t,l), temperature difference at top of furnace........... 858 - 75 = 783 deg. F.
(t.-tz), temperature difference at bottom of furnace. .858 - (75+361) = 422 deg. F.
783 - 422
At,,, logarithmic mean temperature difference ........... = 585 deg. F.
783
In--
422
OFDaA 0.475 X 11.7 X 1.61 X 2.65 X 10-6
y, area ratio.... CDa = = 0.0198
60w1L 60 X 4.12 X 10-7 X 48
Aw, wall area ............................................. vDL = 1.686 sq. ft.
NA,, total particle area in tube ............ yA, = 1.686 X 0.0198 = 0.0334 sq. ft.
16.6 X 3600
h,, overall heat transfer coefficient .........----. -
80.3 X 585 X 0.0334
= 38.2 B.t.u./(hr.)(sq. ft.)(deg. F.)
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
e,, absorptivity of quartz*.................. 0.5; (eCDa) = 0.5 X 0.0198 = 0.0099
e~, absorptivity of cloud (Table 1) ..................................... 0.0099
h,, radiation coefficient, based on arithmetic average particle temperature.....
[ 858 + 460 4 255 + 460
0.172 X 0.0099 100 ) 100 3.6
= 3.6
0.0198 585
(h, - hr), net transfer coefficient due to conduction-convection exchange alone..
38.2 - 3.6 = 34.6 B.t.u./(hr.)(sq. ft.)(deg. F.)
B. Validity of Assumptions Made in Calculation of Data
In the calculation of the heat transfer coefficient as described in the
foregoing the following basic assumptions were made: (1) that a
steady state was rapidly established between the wall and the gas,
(2) that the average temperature of the particle as determined from
the calorimeter measurements was equal to the surface temperature
and (3) that the overall rate of heat transfer is proportional to the
logarithmic mean temperature diffeence between the wall and the
particle. These assumptions will now be investigated.
The first assumption appears t6 be logical on account of the small
heat capacity of the gas in the furiatie and the high heat conductivity
of the walls. As proof that a steady state between the wall and the
gas was reached, several series of tfun under various conditions were
made at approximately constant irajt of feed for varying lengths of
time. These are shown in Table tf io 71, inclusive. When the ob-
served heat transfer coefficients are corrected for slight variations in
the feed rate, the results indicate that the steady state was reached in
less than ten seconds, as no signi~bhnt variation in the overall co-
efficient was found over periods from ten seconds to four minutes even
at high rates of feed.
The second assumption requirbe hiat the unsteady radial flow of
heat within the particle be rapid o..copared with the rate of heat
transmission up to the particle surfite The ratio of the thermal re-
sistance outside the particle to that tisile is equal to 2k/hcpD. Mc-
Adamst shows that when this ratio is jrmter than about 6, the temper-
ature gradient within the particle may b6 Yieglected. It will be shown
later that the experimental values of the SUrface coefficient vary from
about 50 B.t.u./(hr.) (sq. ft.) (deg. F.) for the largest particles to about
30 for the smallest. Using the value 4.2 it.t./(hr.) (sq. ft.) (deg. F./ft.)
for the thermal conductivity of quartz, which has the lowest value of
any of the three materials studied, the vAlties of the thermal resistance
*K. Hild, Mitt. Kaiser Wilhelm Inst. Eisenforscht, (bflnseldorf), vol. 14, Part 200, pp.
59-70, 1932.
tW. H. McAdams, "Heat Transmission," p. 37, McGrkw-Hill Book Co., New York, 1933.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 7
HEAT TRANSFER TO CLOUDS OF SAND PARTICLES
h, hr h, - hr
F
Run 0 lb./(min.) t" A algF NAr
Run s. ^A(sq .) deg.F. deg.F. de.glF. sq. ft.
sec. (sq. ft.) de. deF. d . s B.t.u./(hr.) (sq. ft. particle
surface) (deg. F.)
A. Small Furnace; Nominal Wall Temperature 1050 deg. F.
a. Particle size-30-40 mesh; 100 gm. fed
136 39.6 23.7 1074 414 775 0.0694 33.0 5.7 27.3
137 37.0 25.4 1072 410 775 0.0745 32.6 5.7 26.9
138 35.3 26.6 1073 413 774 0.0780 33.0 5.7 27.3
139 38.0 24.7 1073 405 780 0.0724 32.0 5.7 26.3
140 323 2.92 1066 530 693 0.0856 48.5 6.2 42.3
141 76.7 12.4 1058 444 739 0.0364 37.1 5.7 31.4
142 86.7 10.8 1042 428 733 0.0316 36.3 5.5 30.8
143 210.8 4.45 1064 524 697 0.0130 48.0 6.1 41.9
144 254.8 3.68 1057 522 690 0.0108 48.0 6.1 41.9
145 125.8 7.46 1054 468 722 0.0218 40.7 5.8 34.9
b. Particle size-40-60 mesh; 100 gm. fed
126 25.4 37.9 1092 436 780 0.1330 28.2 5.4 22.8
127 40.4 23.2 1075 454 750 0.0835 30.8 6.0 24.8
128 45.2 20.7 1069 463 738 0.0745 32.1 5.9 26.2
129 127.8 7.35 1072 570 672 0.0264 44.6 6.5 38.1
130 197.0 4.76 1067 600 638 0.0171 49.8 6.7 43.1
131 108.6 8.65 1062 560 669 0.0312 43.8 6.3 37.5
132 53.2 17.6 1054 475 715 0.0634 34.1 5.7 28.4
133 147.8 6.35 1045 580 636 0.0228 48.0 6.3 41.7
134 73.8 12.7 1044 509 684 0.0457 38.6 5.9 32.7
c. Particle size-60-80 mesh; 100 gm. fed
146 184.0 5.10 1051 686 583 0.038 29.8 6.8 23.0
147 76.4 12.3 1045 593 618 0.092 24.8 6.3 18.5
148 250.2 3.75 1039 678 558 0.028 31.8 6.9 24.9
149 54.6 17.2 1027 558 633 0.128 22.3 5.9 16.4
150 25.0 37.5 1067 461 738 0.280 15.4 5.7 9.7
151 51.6 18.2 1053 547 667 0.136 20.7 6.0 14.7
152 50.4 18.6 1043 578 636 0.139 23.2 6.1 17.1
153 221.4 4.24 1033 639 580 0.032 28.4 6.6 21.8
154 122.5 7.66 1027 616 591 0.057 26.4 6.3 20.1
B. Small Furnace; Nominal Wall Temperature 850 deg. F.
a. Particle size-30-40 mesh; 100 gm. fed
94 34.4 27.2 865 315 620 0.0796 30.2 3.9 26.3
95 80.3 11.7 858 361 585 0.0343 38.2 3.6 34.6
96 148.0 6.34 852 385 563 0.0186 41.7 4.0 37.7
97 128.5 7.30 849 371 569 0.0214 40.8 4.0 36.8
98 226.9 4.13 846 420 533 0.0121 48.8 4.2 44.6
99 140.0 6.70 833 376 549 0.0196 41.8 3.9 37.9
100 86.0 10.9 827 371 545 0.0320 41.3 4.1 37.2
101 100.4 9.34 823 351 555 0.0274 38.0 3.8 34.2
102 82.6 11.3 816 342 551 0.0332 37.4 3.7 33.7
103 104.6 8.97 858 340 596 0.0263 34.0 4.0 30.0
105 30.4 30.8 846 302 609 0.0903 29.2 3.7 25.5
106 40.0 23.4 839 302 602 0.0685 29.6 3.7 25.9
107 28.8 32.6 835 300 598 0.0955 29.5 3.7 25.8
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 7 (CONTINUED)
HEAT TRANSFER TO CLOUDS OF SAND PARTICLES
Shm h h, h, - h,
F
Run lb. /(mn.) Ah. NA
Run sec. b./(min.) d F. deg.F. sq. ft
(sq. ft.) F. de F. s. B.t.u./(hr.)(sq. ft. particle
surface) (deg. F.)
b. Particle size-40-60 mesh; 100 gm. fed
108 32.6 28.8 839 339 581 0.1037 28.4 3.8 24.6
108b 36.4 25.8 832 340 572 0.0929 28.8 3.7 25.1
109 46.3 20.2 827 360 554 0.0727 32.0 3.8 28.2
110 121.4 7.72 824 414 514 0.0278 40.6 3.7 36.9
111 101.9 9.21 868 239 545 0.0332 40.4 4.4 36.0
112 191.0 4.91 860 468 517 0.0177 46.1 4.5 41.6
114 102.4 9.15 848 417 540 0.0330 38.8 4.2 34.6
115 85.5 10.9 843 422 548 0.0394 38.6 4.1 34.5
116 53.4 17.6 859 378 572 0.0634 32.8 4.1 28.7
c. Particle size-60-80 mesh; 100 gm. fed
117 72.3 12.9 853 451 521 0.0961 21.2 4.2 17.0
118 76.7 12.2 842 452 518 0.0924 21.1 4.2 16.9
118b 173.6 5.40 841 505 486 0.0402 26.8 4.5 22.3
119 200.4 4.67 841 532 447 0.0348 29.9 4.7 25.2
120 150.5 6.23 853 540 455 0.0464 29.9 4.8 25.1
121 116.4 8.05 848 501 481 0.0600 26.0 4.5 21.5
122 43.8 21.4 842 428 526 0.1596 19.8 4.1 15.7
123 114.0 8.23 833 499 466 0.0614 26.6 4.5 22.1
124 61.6 15.2 824 439 498 0.1133 21.4 4.0 17.4
125 27.9 33.6 826 351 558 0.250 14.9 3.7 11.2
C. Small Furnace; Nominal Wall Temperature 700 deg. F.
a. Particle size-30-40 mesh; 100 gm. fed
180 359.2 2.61 723 346 455 0.0076 45.7 3.2 42.5
181 142.0 6.60 718 318 467 0.0193 40.4 3.1 37.4
182 250.0 3.75 716 332 456 0.0110 43.7 3.1 40.6
183 99.6 9.42 714 300 475 0.0276 37.3 3.1 34.2
184 62.2 15.1 715 286 485 0.0442 34.4 3.0 31.4
185 32.0 29.3 714 249 506 0.0858 28.2 2.9 25.3
186 28.6 32.8 714 246 507 0.0960 27.9 2.8 25.1
195 51.0 18.4 698 257 484 0.0539 30.6 2.8 27.8
b. Particle size-40-60 mesh; 100 gm. fed
170 58.7 16.0 698 318 447 0.0575 35.0 3.0 32.0
171 89.8 10.5 695 340 426 0.0378 38.6 3.1 35.5
172 59.7 15.7 693 310 446 0.0565 33.4 2.9 30.5
173 66.8 14.0 693 320 438 0.0504 35.6 3.0 32.6
174 150.0 6.25 695 384 412 0.0225 43.5 3.3 40.2
175 312.7 3.0 693 382 396 0.0108 48.0 3.2 44.8
176 216.7 4.35 691 375 399 0.0157 46.5 3.2 43.3
177 112.8 8.32 697 342 429 0.0299 39.1 3.0 36.0
178 30.5 30.7 697 260 482 0.1105 25.5 2.8 22.7
179 33.8 27.8 698 280 270 0.1000 28.3 2.9 25.4
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 7 (CONTINUED)
HEAT TRANSFER TO CLOUDS OF SAND PARTICLES
F
Run s lb./(min.) t', te F'
see. (sq. ft.) deg. F. deg. F. deg. F.
h. hr h. - hr
.t.u./(hr.) (sq. ft. particle
surface) (deg. F.)
c. Particle size-60-80 mesh; 100 gm. fed
187 224.0 4.21 743 460 395 0.0314 28.6 3.8 24.8
188 89.6 10.5 737 418 420 0.0783 23.9 3.4 20.4
189 222.0 4.22 732 451 389 0.0314 28.5 3.8 24.7
190 39.7 23.6 728 347 457 0.176 17.9 3.1 14.8
191 36.4 25.8 724 320 472 0.192 15.8 3.1 12.7
192 26.8 35.0 709 324 454 0.261 16.6 2.9 13.7
193 36.0 26.0 706 336 442 0.194 18.0 3.0 15.0
194 70.0 13.4 702 368 417 0.0998 21.0 3.2 17.8
D. Small Tube Furnace; Nominal Wall Temperature 575 deg. F.
Particle size-40-60 mesh; 100 gm. fed
155 28.0 33.5 586 218 392 0.1207 25.7 2.2 23.5
156 92.4 10.1 584 298 340 0.0366 42.0 2.3 39.7
157 50.4 18.6 586 260 366 0.0670 33.6 2.2 31.4
159 202.8 4.62 578 308 326 0.0166 45.5 2.3 43.2
160 77.2 12.1 578 278 346 0.0436 38.2 2.2 36.0
161 38.9 24.1 576 239 368 0.0868 30.3 2.2 28.1
162 72.3 13.0 579 274 350 0.0468 37.0 2.2 34.8
163 31.5 29.8 582 215 390 0.1073 25.2 2.1 23.1
164 70.9 13.2 576 270 350 0.0475 36.6 2.2 34.4
165 36.1 26.0 570 220 376 0.0936 27.2 2.1 25.1
166 55.8 16.8 566 251 352 0.0605 33.4 2.1 31.3
167 128.8 7.29 563 292 320 0.0262 43.6 2.2 41.4
168 89.8 10.4 563 272 334 0.0374 38.8 2.2 36.6
169 176.2 5.32 561 294 317 0.0192 44.3 2.2 42.1
ratio, 2k/heDp, vary from 94 to 246. Therefore, the assumption of
negligible temperature gradient through the particle is satisfactory for
the particle sizes studied in this investigation.
The validity of the" assumption that the overall rate of heat transfer
is proportional to the logarithmic mean temperature difference between
the wall and the particle for a particle falling with accelerated motion
through a tube at constant wall temperature may be proven as follows:
The heat balance on a single particle requires
wcdt = hA, (t, - 1,) dO.
(63)
If tw is constant, this equation may be integrated between limits to
give
tw - t"p hmAp
In = --- A,. (64)
tw - tp2 WpC
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 7 (CONTINUED)
HEAT TRANSFER TO CLOUDS OF SAND PARTICLES
F h, h, h.- h,
Weight 0 lb./ . A A N, NA,
Run Feed . I
gm. see. (min.) deg.F. degF. deg.F. sq. ft.
gm. (sq. ft.) B.t.u./(hr.)(sq. ft. particle
surface)(deg. F.)
E. Large Furnace; Nominal Wall Temperature 1000 deg. F.
a. Particle size-30-40 mesh; varying amounts fed as indicated
13 448 30.0 25.8 992 492 614 0.712 29.4 5.5 23.9
14 378 45.0 14.5 992 559 567 0.400 36.8 5.8 31.0
15 434 90.0 8.32 981 600 524 0.230 43.2 6.1 37.1
16 334 120.0 4.80 973 632 492 0.1327 49.0 6.1 42.9
17 443 25.0 30.5 984 467 622 0.842 27.4 5.4 22.0
18 342 180.0 3.28 997 672 486 0.445 37.1 5.9 31.2
b. Particle size--40-60 mesh; varying amounts fed as indicated
20 452 25.0 31.2 985 508 596 0.1155 23.4 5.5 17.9
21 406 45.0 15.6 995 623 522 0.577 33.8 6.3 27.5
22 328 180.0 3.14 998 745 420 0.1161 52.3 7.7 44.6
23 356 25.0 24.6 983 547 566 0.910 26.8 5.7 21.1
24 401 50.0 13.8 967 625 490 0.511 36.2 6.1 30.1
25 278 60.0 8.00 977 700 437 0.296 46.4 6.9 39.5
26 233 90.0 4.46 984 746 402 0.1652 54.2 7.7 46.5
F. Small Furnace; Nominal Wall Temperature 1050 deg. F.
a. Particle size-30-40 mesh; varying amounts fed as indicated
196 18.8 30.0 5.88 1050 484 706 0.0168 44.4 5.7 38.8
197 28.5 60.0 4.45 1045 533 670 0.0127 52.5 5.8 46.7
198 75.5 120.0 5.90 1044 508 685 0.0169 48.1 5.7 42.4
199 102.8 180.0 5.35 1041 513 680 0.0153 49.2 5.7 43.5
200 180.7 240.0 7.05 1039 489 692 0.0202 45.7 5.6 40.1
201 41.2 30.0 12.9 1045 452 722 0.0369 40.0 5.6 34.4
202 81.6 60.0 12.8 1043 456 720 0.0366 40.4 5.6 34.9
203 116.0 90.0 12.1 1042 460 714 0.0346 41.3 5.5 35.8
204 165.9 120.0 13.0 1037 448 715 0.0372 39.8 5.4 34.4
205 242.7 180.0 12.7 1031 448 719 0.0363 39.7 5.5 34.2
b. Particle size-40-60 mesh; varying amounts fed as indicated
207 15.1 15 9.44 1035 553 644 0.0330 47.6 5.8 41.8
208 31.2 30 9.76 1035 550 647 0.0341 46.0 5.8 40.2
209 43.9 45 9.15 1034 562 638 0.0320 47.8 5.8 42.0
210 68.5 75 8.56 1031 548 644 0.0300 45.7 5.7 40.0
G. Small Furnace; Nominal Wall Temperature 725 deg. F.
Particle size--40-60 mesh; varying amounts fed as indicated
228 255 60.0 39.9 718 268 502 0.140 26.1 2.8 23.3
229 152 40.0 35.6 717 279 490 0.125 27.8 2.8 25.0
230 84.5 20.2 39.2 719 270 500 0.137 26.6 2.9 23.7
231 42.7 10.0 40.1 719 274 495 0.140 27.0 2.9 24.1
232 48.9 120.0 3.82 721 395 418 0.0134 48.1 3.2 44.9
233 90.5 240.0 3.54 721 408 409 0.0124 51.3 3.2 48.1
234 8.9 30.0 2.78 722 411 407 0.0097 52.1 3.2 48.9
235 37.4 100.0 3.51 721 409 408 0.0123 51.6 3.2 48.4
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 7 (CONCLUDED)
HEAT TRANSFER TO CLOUDS OF SAND PARTICLES
Weight
Run Feed
gm.
Shm h, h,.m - hr
lb./ t' A Ati, NA,
(min.) deg. F. deg. F. deg. F. sq. ft. -,
(smi) deg. F. deg F. deg. F. s. ft B.t.u./(hr.)(sq. ft. particle
surface) (deg. F.)
H. Small Furnace; Nominal Wall Temperature 1050 deg. F.
Particle size-40-60 mesh; varying amount fed as indicated; atmosphere of CO0
211 S9.9 75 11.3 1035 576 629 0.0427 45.9 6.2 39.7
212 35.9 45 7.48 1038 592 621 0.0282 48.2 6.5 41.7
213 9.5 15 5.94 1041 649 582 0.0224 57.5 6.7 50.8
214 21.3 30 6.65 1042 637 593 0.0251 54.9 6.7 48.2
215 94.4 120 7.38 1034 617 599 0.0279 52.3 6.5 45.8
I. Small Furnace; Nominal Wall Temperature 725 deg. F.
Particle size--40-60 mesh; varying amounts fed as indicated; atmosphere of CO,
224 39.6 10 37.8 732 292 497 0.1407 26.8 3.1 23.7
225 76.1 20 36.3 730 302 490 0.1350 26.8 2.9 23.9
226 164.4 40 39.2 727 288 495 0.1460 26.4 3.0 23.4
227 240.3 60 38.2 723 283 495 0.1421 26.0 3.0 23.0
Here AOf is the time of fall. The true average temperature difference
is seen to be the logarithmic mean if the expression for the total heat
transferred in time is written
WpC (tp2 - t1l) = hmA, (t, - t,)a,.AOf.
By comparison with Equation (64) it is seen that
tp2 - tpl
(t - tp)av. =
tw - tp
tw - tp2
= (tw - tp)lm.
The derivation of Equation (64) is based on the assumption of a
constant value of h,. Numerical integration of Equation (63) using
for h,, theoretical values obtained from individual coefficients calcu-
lated by means of Equations (36), (57), and (58) showed that under
typical conditions hm might increase from 30 to 36 B.t.u./(hr.) (sq. ft.)
(deg. F.), and remain constant at the latter value over the lower
half of the furnace. This change was too small to affect the validity of
the logarithmic mean assumption.
(66)
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
TABLE 8
HEAT TRANSFER TO CLOUDS OF CARBORUNDUM PARTICLES
hA hr h. - h,
Run lb./(min.) d" a ag NA,
(sq. ft.) de F. de F. de.F. sq. f. B.t.u./(hr.)(sq. ft. particle
surface) (deg. F.)
A. Small Furnace; Nominal Wall Temperature 1050 deg. F.
a. Particle size-No. 40; 100 gm. fed
C19 496.0 1.89 1052 627 611 0.0069 55.1 6.8 48.3
C20 193.0 4.86 1046 577 641 0.0179 47.4 6.4 41.0
C21 116.2 8.09 1039 539 658 0.0296 42.9 6.2 36.7
C22 72.2 13.0 1033 506 674 0.0475 39.0 5.9 33.1
C23 58.2 16.1 1032 498 676 0.0593 38.0 5.8 32.2
C24 38.0 24.7 1020 452 692 0.0905 33.2 5.5 27.7
C25 49.2 19.0 1017 473 675 0.0700 35.8 5.6 30.2
C26 29.0 32.4 1020 429 706 0.1185 30.6 5.4 25.2
b. Particle size-No. 60; 100 gm. fed
C27 286.0 3.28 1047 697 553 0.0238 33.4 7.2 26.2
C28 146.4 6.40 1035 662 565 0.0465 32.0 6.9 25.1
C29 118.0 7.95 1028 641 578 0.0576 30.2 6.6 23.6
C30 78.0 12.0 1022 613 590 0.0870 28.3 6.3 22.0
C31 59.2 15.8 1021 583 586 0.1146 25.5 6.1 19.4
C32 42.0 22.3 1022 548 635 0.1618 22.7 6.0 16.7
C33 34.9 28.9 1026 543 643 0.1950 22.4 5.9 16.5
C34 27.9 33.6 1028 512 666 0.244 20.2 5.7 14.5
B. Small Tube Furnace; Nominal Wall Temperature 725 deg. F.
a. Particle size-No. 40; 100 gi. fed
C1 45.2 20.8 720 290 475 0.0765 26.8 3.0 23.8
C2 21.8 43.0 714 283 495 0.1583 23.5 2.8 20.7
C3 27.3 24.4 707 289 486 0.1267 24.8 2.9 21.9
C4 39.7 23.6 705 282 475 0.0869 26.6 2.9 23.7
C5 273.0 3.44 709 375 420 0.0127 44.0 3.3 36.7
C6 91.0 15.4 705 317 454 0.0566 31.9 3.0 28.9
C7 93.7 10.0 745 364 464 0.0368 35.0 3.3 31.7
C8 180.6 5.19 743 393 443 0.0191 41.9 3.5 38.4
C9 155.0 6.05 738 385 442 0.0222 41.5 3.5 38.0
b. Particle size-No. 60; 100 gm. fed
C10 207.8 4.52 732 463 380 0.0328 29.2 3.7 25.5
C11 375.7 2.50 728 471 368 0.0181 30.4 3.7 26.7
C12 124.8 7.52 725 434 394 0.0545 26.4 3.5 22.9
C13 97.3 9.65 721 415 403 0.0700 24.5 3.4 21.1
C14 70.2 13.4 712 385 414 0.0970 22.1 3.2 18.9
C15 46.2 20.3 712 358 434 0.1472 18.5 3.1 15.4
C16 26.4 33.0 710 320 455 0.239 16.4 2.9 13.5
C17 20.8 45.1 707 315 488 0.327 15.1 2.8 12.3
C18 39.6 24.3 707 343 440 0.1762 18.4 3.0 15.4
12. Analysis of Data.-When a cloud of particles falls through a
heated gas space under the influence of gravity, the time of contact of
a single particle with the hot gas is essentially independent of the rate
of feed. On the other hand, the ratio of the area of the particles to
the area of the wall, y, should be directly proportional to the feed rate.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 9
HEAT TRANSFER TO CLOUDS OF ALOXITE PARTICLES
hm h, hm - h,
F
Run 0 lb./(min.) t Atl NA,
se. (sq. ft.) de F. deg F. de. F. . . B.t.u./(hr.) (sq. ft. particle
surface) (deg. F.)
A. Small Furnace; Nominal Wall Temperature 1050 deg. F.
a. Particle size-No. 46; 100 gm. fed
A16 395.0 2.38 1032 563 634 0.0076 52.5 6.1 46.4
A17 187.2 5.62 1031 525 660 0.0179 47.5 5.9 41.6
A18 92.6 10.1 1027 454 702 0.0323 37.8 5.6 31.2
A19 214.8 4.36 1073 550 692 0.0140 50.5 6.5 44.0
A20 151.0 6.21 1072 537 693 0.0199 48.3 6.4 41.9
A21 103.8 9.05 1070 508 713 0.0290 44.1 6.2 37.9
A22 68.8 13.6 1070 467 740 0.0435 39.3 6.0 33.3
A23 46.3 20.2 1072 425 769 0.0646 34.0 5.8 28.2
A24 36.2 25.9 1070 400 780 0.0829 31.1 5.7 25.4
A25 30.2 31.0 1062 380 773 0.0992 29.4 5.5 23.9
A26 24.2 38.8 1082 356 800 0.1240 26.4 5.4 21.0
b. Particle size-No. 60; 100 gm. fed
A27 197.2 4.75 1043 613 613 0.0220 44.0 6.6 37.4
A28 252.0 3.72 1038 631 593 0.0172 48.4 6.9 41.5
A29 90.6 10.3 1069 555 681 0.0524 32.6 6.4 26.2
A30 54.2 17.3 1062 508 704 0.0799 31.2 6.1 25.1
A31 64.0 14.6 1055 527 685 0.0679 33.2 6.1 27.1
A32 35.6 26.3 1051 453 726 0.1214 31.8 6.1 25.7
A33 31.6 29.7 1043 425 737 0.1371 29.3 5.8 23.5
B. Small Furnace; Nominal Wall Temperature 700 deg. F.
a. Particle size-No. 46; 100 gm. fed
A8 289.2 3.24 716 340 450 0.0104 43.5 3.2 40.3
A9 111.6 8.41 714 306 470 0.0269 37.2 3.1 34.1
A10 184.0 5.10 713 320 461 0.0163 40.6 3.2 37.4
All 67.4 13.9 712 280 483 0.0445 32.8 3.0 29.8
A12 32.8 28.6 711 230 513 0.0915 27.6 2.8 24.8
A13 43.6 21.4 711 247 502 0.0688 27.8 2.9 24.9
A14 21.5 43.6 712 205 527 0.1396 21.4 2.7 18.7
A15 17.6 53.3 715 196 530 0.1704 20.5 2.7 17.8
b. Particle size-No. 60; 100 gm. fed
Al 178.0 5.26 711 382 418 0.0244 36.6 3.3 33.3
A2 225.6 4.15 704 393 403 0.0192 39.4 3.3 36.1
A3 74.0 12.7 698 334 435 0.0582 30.4 3.0 27.4
A4 103.2 9.08 692 354 417 0.0420 33.4 3.1 30.3
A5 29.2 31.1 687 270 465 0.1483 22.8 2.8 20.0
A6 36.8 25.5 722 288 490 0.1179 23.1 3.0 20.1
A7 51.0 18.8 718 313 470 0.0869 25.2 3.0 22.2
Now if we may assume as an approximation that the individual coef-
ficients, he, and hew, are essentially independent of the feed rate, it
follows that the ratio of the convective resistance at the wall to that
at the particle, NA,hcp/Awhc, must likewise be proportional to the
feed rate. As shown in Tables 7, 8, and 9 for each series of runs the
feed rate was varied over a range of nearly fifteen-fold while other
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
1.5 -------
1.6 -- ---- -- - ---- -------
69if -- - -- -- -- - - -- - - - - -
0.7 . = ._
0.6
0=015
' 2 3 4 5 6 7 8 9/0 20 30 ,40
Feed Ra7e, F
FIG. 8. CALCULATED EFFECT OF FEED RATE ON CONVECTION
COEFFICIENT FROM WALL TO GAS
conditions were maintained constant. The corresponding variation in
the resistance ratio thus permits estimation of the individual convec-
tion coefficients. For this purpose either of the following equations
might be used:
1 1 7
= + -- (67)
h, - hr cp ,p hw
1 1 1
= - + - (68)
7 (hm - hr) hew YhcP
For the first equation a plot of 1/(h,, - h,) against F is required,
from which the reciprocals of h,, and hw, respectively, may be deter-
mined from the slope and the intercept at F = 0. For the second equa-
tion, 1/y(hm - hr) is plotted against 1/F. The slope of the line now
gives 1/hcp, and extrapolation to the zero abscissa, representing infinite
feed rate, gives 1/hw,. It should be noted that in this equation
y(h,, - h,) represents the overall convection coefficient based on the
wall area. Consequently, any error in computing the time of fall and
area of the particles will not affect the value of he, obtained by the
extrapolation. Possibly for this reason, and also because the assump-
tion of the independence of the coefficients of feed rate apparently holds
better at high feed rates, the second method of plotting the data was
more satisfactory than the first. Before examining the data it is well to
consider the validity of the foregoing assumption as far as possible
from the theoretical standpoint.
According to Equation (36), which is valid for the range of
Reynolds numbers covered in the experimental work, he, is affected by
the feed rate indirectly through the temperature of the gas, since for
4ý
ILLINOIS ENGINEERING EXPERIMENT STATION
3
- I
I
/
0
FIG. 9. GRAPHICAL REPRESENTATION OF HEAT TRANSFER
DATA FOR SAND PARTICLES
any wall temperature the gas temperature is decreased as the feed rate
is increased. The effect, however, is negligible since, for air, the group
V/ kVpc/D varies only about one per cent for each one hundred
degrees Fahrenheit.
In the case of hew the effect of the feed rate is not so obvious. If
only natural convection from the wall is considered, for a constant wall
temperature, the Grashof group is expected to increase as the feed rate
is increased. The approximate extent of this effect is shown in Fig. 8
in which the calculated values of the coefficient are plotted against the
feed rate. For the conditions chosen the relationship is seen to be
nearly exponential. The slope of the logarithmic line, however, is only
0.15, so that, from this standpoint, the approximation is nearly true.
The graphical representation of the complete data according to
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
/
F-
FIG. 10. GRAPHICAL REPRESENTATION OF HEAT TRANSFER
DATA FOR CARBORUNDUM PARTICLES
Equation (68) is shown in Figs. 9, 10, and 11. It will be observed
that, for a given furnace, the points fall very near a single straight
line regardless of the wall temperature. The change in slope from one
furnace to another is due principally to the change in the ratio between
y and F. Using the method of Least Squares for the linear equation
the coefficients, hew and he,, were calculated for each of the series of
runs. The results are shown in Table 10. It is apparent that neither of
the coefficients varies with the wall temperature beyond the experi-
mental error. Consequently, average values are justified for each
particle size.
In Table 10 the values of hew and he, calculated from Equations
(58) and (35), respectively, are also given. The observed values of h,,
range from 2.1 to 3.9 B.t.u./(hr.) (sq. ft) (deg. F.) for the small
furnace and from 3.8 to 4.1 for the large furnace. The average coeffi-
cient of natural convection calculated from the Rice equation is
about 1.0, and is independent of the diameter of the furnace. Undoubt-
edly this discrepancy is due to the increased convection caused by the
particles themselves, and by the cooling of the gas at the top of the
furnace. In a closed furnace there must be a circulation of the gas up the
walls and down the center of the furnace. The convection at the walls
apparently increases with the size of the furnace. Unfortunately, two
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 11. GRAPHICAL REPRESENTATION OF HEAT TRANSFER
DATA FOR ALOXITE PARTICLES
furnace sizes are not sufficient to establish even an empirical relation-
ship for this effect. For values of the Grashof group between 10* and
10s, a rough approximation might be made by multiplying the coeffi-
cient for natural convection by a factor equal to the diameter of the
furnace in inches. For values of the group greater than 10s it is
probably better to omit the correction factor entirely, and the overall
coefficient would then be conservative. The latter range includes most
cases of practical importance for which the furnace is large and heat
transfer by radiation predominates, so that even large errors in the
calculated value of he, can be neglected. Furthermore, in many
furnaces heat is not transmitted through the walls but is introduced
directly in hot gases so that he, does not enter into the calculations.
The observed values of he, agree within 20 per cent with those
calculated from Equation (35) for all cases except for the smallest
particles where they are about 50 per cent low. Since the observed
values depend directly on the calculated time of fall, a possible expla-
nation of the discrepancy lies in the downward motion of the gases at
the center of the tube which would result in a shorter time of contact
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES 51
TABLE 10
SUMMARY OF RESULTS ON INDIVIDUAL HEAT TRANSFER COEFFICIENTS
A. Small Furnace
Sand
Sand
Sand
Carborundum
1050
30-40 850
700
Average
1050
40-60 850
700
575
Average
1050
60-80 850
700
Average
1050
No. 40 700
Average
1050
Carborundum No. 60 700
Average
1050
Aloxite No. 46 700
Average
1050
Aloxite No. 60 700
Average
2.4 0.64 to 0.98
2.3 0.70 to 1.02
2.3 0.63 to 1.01
2.3
2.5 0.75 to 1.04
2.8 0.74 to 1.02
2.8 0.69 to 1.05
2.8
2.9
2.6
3.0
2.8
3.5
3.0
3.2
3.9
3.0
3.4
2.1
2.5
2.3
2.4
3.2
2.8
u.5 to 1.O .
0.72 to 1.10
0.76 to 1.05
0.73 to 1.05
0.61 to 1.02
0.72 to 1.08
0.69 to 1.02
0.65 to 1.04
0.63 to 1.05
0.68 to 1.10
0.72 to 1.03
0.72 to 1.06
B. Large Furnace
Sand 30-40 1000 4.1 0.73 to 1.03 49 54
Sand 40-60 1000 3.8 0.75 to 1.04 52 54
than that calculated. The error would be greater for the smaller
particles than for the larger ones. It is unlikely that an error in the
measurement of the particle areas would be positive, thus resulting
in small coefficients, since concave surfaces are known to exist in all
of the particles to some extent.
The values of h,, for carbon dioxide at a wall temperature of
1050 deg. F. are definitely greater than those obtained for air, as shown
in Fig. 9, but the values at 700 deg. F. are essentially the same for
both gases. If it is assumed that the radiation interchange between the
48
48
47
48
52
49
55
52
52
30
34
32
32
44
46
45
30
27
28
53
45
49
47
51
49
54
53
53
59
59
58
58
67
66
65
55
54
65
63
57
56
64
62
I
ILLINOIS ENGINEERING EXPERIMENT STATION
gas and the suspended particles is negligible, and therefore that hep is
the same for the two gases, the coefficient of heat transfer for radia-
tion from the furnace wall to the carbon dioxide can be estimated by
means of the usual equations for gas radiation. The values obtained
in this way for the two temperatures are, respectively, 1.0 and 0.25
B.t.u./(hr.) (sq. ft. wall area) (deg. F.). The higher value would
represent an increase of about 40 per cent in the value of hew obtained
without superimposed radiation. While the data are insufficient to
determine the exact values of the individual coefficients by plotting,
it appears that the larger values of h, obtained when carbon dioxide
is the ambient gas may be explained on the basis of the radiant inter-
change between the wall and the gas.
V. CONCLUSIONS
13. Summarized Conclusions.-The foregoing discussion may be
summarized as follows:
(1) The rate of heat transmission from a hot furnace wall to
clouds of falling particles is rapid. When the diameter of the furnace
is small, the transmission takes place principally by convection from
the wall to the gas and from the gas to the particles. The fraction
transmitted by radiation increases rapidly with the size of the furnace
and the temperature of the wall (q. infra). The factor producing the
most important effect on the overall transmission is the rate at which
the particles are fed to the furnace. When radiation is small, the main
effect of varying the feed rate is to change the ratio of particle area
to wall area, thus changing the relative magnitudes of the thermal
resistances at the wall and at the particle without altering appreciably
the respective convection coefficients themselves.
(2) The coefficient of heat transfer by convection from the gas
to the particles is a function of the particle size and the rate of fall,
and is nearly independent of the temperature of the ambient gas. This
coefficient may be calculated from fundamental considerations of heat
transfer to a submerged body in a moving fluid. The general equation is
k m 1 + e-b,~
hep = - (35)
D i 1 + b(4
where the constants bi are defined by the relationship: 2bi2Re Pr=
the squares of the roots of the first order Bessel function. For particles
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
having a specific gravity of about 3, and larger than 400 microns, fall-
ing through air, the foregoing equation may be simplified to
kVpc
h,, = 0.714 kp . (36)
D,
(3) The coefficient of heat transfer by convection from the furnace
wall to the gas is several times larger than the values predicted for
natural convection alone, and is apparently a function of the size of
the furnace. For furnaces less than eight inches in diameter and at
wall temperatures less than 1200 deg. F., it is recommended that the
natural convection coefficient be multiplied by a constant equal to the
diameter of the furnace in inches. For larger furnaces, or for higher
wall temperatures, the heat transmission takes place primarily by
radiation, and the correction may be neglected.
(4) The radiation coefficient is a function of the concentration of
particles in the cloud and of the shape of the furnace. The coefficient
may be calculated from the equation
0.172e6 [(T/100)4 - (T,/100)4]
hr = ----------(57)
7h (7T - T,)
Values of the absorptivity of the cloud, c., are given in Table 1 for
spherical, cylindrical, and rectangular furnaces.
14. Design of Large Furnace.-In order to illustrate the applica-
tion of the foregoing equations an example will be taken from the
previous publication.* The problem is the design of a flash calciner
for decomposing 83.2 tons of zinc sulphite hydrate per day into zinc
oxide, water vapor, and sulphur dioxide. The conditions are as follows:
Feed, dry ZnSOs212HO enters at 70 deg. F.
Particle size, 200 microns, found by experiment to be the arith-
metic mean of particles just passing 100 mesh standard screen.
Dehydration temperature, 200 deg. F.
Decomposition temperature, 500 deg. F.
Specific heat of feed, 0.24 B.t.u./(lb.) (deg. F.).
Heat of dehydration, 40 800 B.t.u./lb. mole.
Heat of decomposition, 58 100 B.t.u./lb. mole.
Emissivity of particles, 0.5.
*Univ. of Ill. Eng. Exp. Sta. Bul. 324.
ILLINOIS ENGINEERING EXPERIMENT STATION
I I i I I I
__Y 30 Inches
_ -40 - -- -- .O---
__ __ __ . .-- " Y 20
30 ___ i=30
7/^, 4- 1
20
/0
/0I _ _- ---__ - -
/0
f ,,Overa// ----- W// Tmperlh/Tre, /200°F
SHeA Transfer , ----. W/ Temperaf/re, i/OOO°
" \ Coefficienrs
6 \ < Y=/o ncheh
SY=30--
Heat Transfer -----__ Y=30
^ z -- - - -I-- - - - - - - -
c o o_ _ _ _ _ __ _ _ _ _ _ _ _
4 8 /2 /6 Ze q4
X, Disfa'nce Between Wa/Is, In /nches
FIG. 12. HE4T TRANSFER COEFFICIENTS AND CALCULATED HEIGHTS OF
RECTANGULAR FURNACES FOR CALCINATION OF ZINC SULPHITE
A feed rate of 11.7 lbs. per min. per sq. ft. is assumed so that the
total cross-sectional area of the furnace will be 10 sq. ft. The furnace
is to be constructed in the form of rectangular retorts with walls of
high heat conductivity. Silicon carbide tile are available, 11/2 inch
thick, tongue and groove, with a heat conductivity of 105.5 B.t.u./(hr.)
(sq. ft.) (deg. F./in.) at 1110 deg. F. and 112.5 B.t.u./(hr.) (sq. ft.)
(deg. F./in.) at 2900 deg. F. The retorts may be placed in a row with
0 " " ^ "
u
G
il
|'s
rtc
HEAT TRANSFER TO CLOUDS OF FALLING PARTICLES
the common walls formed of slotted tile. The walls will be sealed by
compression with suitably spaced springs. The furnace will be heated
by horizontal gas muffles along each side, the flames traveling in several
passes from bottom to top. The outside walls will be constructed of
fire-brick and will be insulated.
Calculations were made to determine the effect of the shape and
size of the retorts on the height required for a heat input sufficient to
completely dehydrate and decompose the particles. These calculations
were made by applying Equations (5), (57), and (58) to successive
short sections, taking into consideration the change in the velocity,
mass, and temperature of the particles and of the gas. The tempera-
ture of the furnace wall was considered to be constant. The evolved
gases were assumed to leave at the bottom of the furnace so that there
was an increase in the mass velocity of the gases in the two reaction
zones. The reactions were considered to take place only when the
particles reached the temperatures at which the respective decompo-
sition pressures are one atmosphere, and at these temperatures the
rate of reaction depended only on the rate of heat input.
The results of these calculations are plotted in Fig. 12. The fol-
lowing conclusions may be drawn from this study.
(1) With a uniform wall temperature of 1000 deg. F. the height of
the calciner of rectangular cross section 30 inches long and 12 to 20
inches wide approaches 40 feet. Nothing is to be gained by using
smaller cross sections, because the length does not decrease as rapidly
as the area of the cross section decreases and, therefore, the increased
number of units required more than offsets the decrease in height.
(2) An increase in wall temperature to 1200 deg. F. decreases the
height by nearly 50 per cent.
(3) In these large rectangular furnaces the heat transfer is ninety
per cent by radiation. If the source of heat is located at the bottom,
a greater heat flux can be obtained than is shown by the curves,
because of the great effect of the temperature level on radiant heat
transfer. Furthermore, since the greatest part of the heat is required
at the bottom in the decomposition zone, upward flow of the combus-
tion gases is highly desirable.
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ILLINOIS ENGINEERING EXPERIMENT STATION
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UNIVERSITY OF ILLINOIS
Colleges and Schools at Urbana
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and mining engineering.
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