Mass and heat transfer to an ellipsoidal particle

COMBUSTION A N D FLAME 80:209-213 (1990)

BRIEF

209

COMMUNICATION

M a s s a n d H e a t T r a n s f e r to an E l l i p s o i d a l P a r t i c l e DANA T. GROW Department of Chemical Engineering, University Station, University of North Dakota, Grand Forks, ND 58202

qsp

NOMENCLATURE a b, c C Co C1 C2 D dp h h

semimajor axis of particle (m) semiminor axes of particle (m) concentration of oxygen (mol m -3) oxygen concentration in bulk gas stream (kg m -3) constant of integration constant of integration diffusion coefficient (m 2 s - l ) particle diameter (m) heat transfer coefficient (J m -2 s -l K - l ) average heat transfer coefficient (J m -2 s-t K - I )

hi AH

kc

-

- g')]l/2/2R

enthalpy of combustion reaction (J kg - l ) chemical rate constant (kg m -2 s -1 pa-0.5)

average diffusion mass transfer coefficient (kg m -2 s -1 Pa - l ) N mass flux at surface (kg m -2 s - l ) NU Nusselt number average mass flux (kg m -2 S - l ) Nrnax maximum mass flux at surface (kg m -2 kdif

S- l )

Nmin

minimum mass flux at surface (kg m -2

Nsp

mass flux for spherical particle at surface (kg m -2 s - l ) partial pressure of oxygen (Pa) surface pressure of oxygen averaged over the particle surface (Pa) heat flux (J m 2 s - l ) average heat flux (J m 2 s -1)

S-1 )

P02 Ps q t/

Copyright © 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010

R Rmt Rsp

heat flux to sphere (J m -2 s - l ) gas constant, (8.314 J tool - l K - l ) rate of combustion (kg carbon m -2 s - l ) rate of combustion for spherical particle (kg carbon m -2 s - l )

R~

[(~ + a2)(~ + b2)(~ + C2)] 1/2

ro

particle radius (m) gas temperature mean temperature of particle and gas (K) particle temperature (K) surface temperature averaged over the partide Surface (K) wall temperature (K) Cartesian coordinate (m) Cartesian coordinate (m) Cartesian coordinate (m)

Te Tm

rp TW X

Y Z

Greek Symbols distance far from the particle (m) emissivity X thermal conductivity of the gas (J m -I s -l K-l) 0 overall rate of combustion (kg carbon m -z S- l )

o

Stefan-Boltzmann constant (5.67 × 10 -8

J

m - 2 s-1 K -4)

ellipsoiclal coordinate (m) 7/ ellipsoidal coordinate (m) ~" ellipsoidal coordinate (m) INTRODUCTION Two- and three-wavelength optical pyrometry has become a common technique to measure the tern-

0010-2180/90/$3.50

210

D . T . GROW

perature of burning coal chars [1-3]. The overall rate of combustion, p, is computed starting from an energy balance over the particle, and previous work has assumed spherical particles. The advent of videotape recorders and high-speed video data acquisition computer programs has made possible the in situ determination of size and shape as a coal or char particle bums. In general the particles are irregular in shape and possess a distribution of sizes. Hence it is instructive to examine the heat and mass transfer situation for ellipsoidal particles as an example of a nonspherical system. ANALYSIS

The solution to the Laplace equation can be calculated by analogy to potential theory in electrostatics using ellipsoidal coordinates, ~', 7, ~, which are defined by the equations [4]

l Iz =constant

~= constanl~/.//

~ i ~

Y

~ /

constant

"-

Fig. 1. Sketch of the ellipsoidal coordinate system. Surfaces of constant ~ are ellipsoids. Surfaces of constant ~"are hyperboloids of two sheets only one of which is shown. Surfaces of constant ~/are hyperboloids of one sheet.

Integration of the equation gives

x2/(a 2 + ~) + y2/(b2 + ~) +Z2/(C 2 "4-~) = 1

C(O = C1

(~ > - c 2 ) ,

+z2/(c 2 +~/) = 1 ( - c 2 > ~ > - b 2 ) ,

C(~) --, 0

x2 /(a 2 + ~) + y2 /(b2 + ~) +z2/(c 2 +~') = 1 ( - b 2 > ~- > - a 2 ) . (1) A sketch of the coordinate system is presented in Fig. 1. The diffusion equation reduces to (2)

where C is oxygen concentration. The surface of the ellipsoidal particle is located on the surface -- 0 and is given by the equation

x2/a 2 +yE/b2 +z2/c 2 = 1,

(5)

where CI and C2 are integration constants and the upper limit ensures that the concentration is finite at large ~. The boundary conditions are

x2 /(a 2 + rl) + y2 /(b2 + rl)

~72C = O,

d~/R~ --F-C2,

C(O ~ C0

as ~ ~ 0,

(6)

as ~1/2 __> ~,

(7)

where 6 is far from the particle. The first boundary condition states that the surface concentration is zero, which is strictly true only for an infinitely fast reaction. The second states that the oxygen concentration is constant far from the particle. The boundary conditions permit evaluation of the integration constants. Application of the first boundary condition gives

(3)

where the semiaxes have lengths a, b, and c. Concentric ellipsoids are given by the formula = constant. Equation 2 is written in ellipsoidal coordinates and it is assumed that the concentration is a function of the coordinate ~ alone. Thus Laplace's equation becomes

0 = Ci

(O/O0(R~ cqC/cq~) = 0, where R~ = ((~ + a2)(~ + b2)(~ + c2)) 1/2.

A look at the situation for large ~ shows that R~ approaches ~3/2 and the concentration far away

(4)

d~/R~ + C2.

(8)

Therefore, C(~) becomes

C(O -- CI

d~/R~

-

C~

d~/R~.

(9)

BRIEF COMMUNICATION

211

from the particle becomes

m-3). The assumption of burning to CO at the surface yields

C(~)=Cl [2/~l/2- fo~d~/R~].

(10)

Furthermore, at very large distances, ~1/2~ r ~ / 5 , and C --+ Co and the bulk concentration is given by the expression

Co--C~

( :5 2/~-

)

d~/R~ ,

(ll)

which allows computation of C1. Finally C(O = Co

(

(2/8) -

/0

d~/R~

):

d~/R~. (12)

The mass flux at the surface, N, is the mass of oxygen that diffuses to the surface per unit area per unit time and is calculated from the normal derivative,

N = -D(Tm)OC/On = (-D(Tm)/h 1)(tgC/0~ )/~---0,

(13)

where hi = ((~ - ~)(~ - ~))1/2/2R~, and D(Tm) is the diffusion coefficient evaluated at the mean temperature of the particle and the gas, Tin. Thus the flux becomes

N = -2D(Tm)Co x

(

(2/~) -

:o

d~/R~

)1

(~.)U2.

(14)

Finally, using the definitions of ~/and ~" the mass flux at the surface is given by the expression

N =-2D(Tm)Co ((2/tS)- fo~d~/R~) -1 × (abc)-l(x2/a 4 + y2/b4 + Z2/C4)-1/2

Rmt = (O.048D(Tm)/RTg)(Pch) × ( ( 2 / b ) - f o ~ d ~ / R ~ ) - ' ( a b c ) -1 × [X2/a4 +y2/b4 "FZ2/C4]-1/2

(16)

where Pth is the partial pressure of oxygen in the hulk (Pa), Tg is the gas temperature (K), R is the gas constant, 8.314 J mo1-1 K -1 . Particle temperatures measured by optical pyrometry may be used to compute the overall combustion rate, p, from an energy balance over the particle

p A H = h(Tp - Tg) + ea(Tp 4 - Tw4).

(17)

The balance accounts for the heating of the particle by reaction and transfer of the heat from the particle by conduction and radiation [1, 5]. Here A H is the enthalpy of the combustion reaction (J k g - l ) and is computed from heat capacity and heat of formation data. The heat transfer coefficient is h (J m -2 s -1 K -1) and for spherical particles is given by h = Nu(~/dp), where Nu is the Nusselt number, k is the thermal conductivity (J m - l s -1 K - l ) , and dp is the particle diameter. The emissivity, assumed to be unity, is denoted by and o is the Stefan-Boltzmann constant. The wall temperature is given by Tw (K). The heat transfer coefficient is the ratio of the heat flux at the surface to the temperature difference between the surface and the ambient gas. By similar analysis the heat flux, q, is found by the solution of the equation V2T = 0 to determine the temperature profile, followed by evaluation of the flux at the surface

q = ( -)~(Tm)/h 1)(OT/OO~_-o.

(18)

(15)

Therefore the heat flux becomes The flux varies over the surface and is greatest at the point (a, 0, 0) if the lengths of the semiaxes are in the order a > b > c. The rate of combustion in mass carbon per unit area per unit time is found by using the ideal gas law for the bulk oxygen concentration, Co (kg

q = (2),)(Tp - Tg)

(

(2/6) -

/0

d~/R~

)1

× (abc)-l(x2/a 2 + y2/b4 + z2/c4) -1/2, (19)

212

D. T. GROW TABLE 1 Relative Mass Transfer Flux for Ellipsoidal and Spherical Particles for Diffusion Limited Burning a (#m)

b (/~m)

Nm.~/N~p

Nmi./N~p

lq/N~p

200 200 200 200 200

190 175 150 100 50

1.03 1.09 1.21 1.57 2.39

0.98 0.96 0.91 0.78 0.60

1.01 1.01 1.01 1.03 1.11

100 100 100 100 100

90 80 60 50 40

1.07 1.16 1.40 1.57 1.80

0.97 0.93 0.84 0.78 0.72

1.00 1.00 1.02 1.03 1.05

50 50 50 50

40 35 25 10

1.16 1.27 1.57 2.71

0.93 0.89 0.78 0.54

1.00 1.01 1.03 1.16

where X is the thermal conductivity evaluated at the mean temperature and Tp and Tg are the temperatures of the particle surface and the gas, respectively. FOr spherical particles a = b = c -- r0 and Eqs. 16 and 19 for infinitely large 6 reduce to [1]

Rsp = [O.024D(Tm)/(roRT)](Po2)

(20)

and qsp = (X/ro)(Tp - Tg).

(21)

For an ellipsoid where semiminor axes are equal (b = c), the mass flux averaged over the surface is found by integration to be

) -1

---- - 4 D ( T m ) C o

2/5) -

kc = p / ( P g - p/kdif) 1/2,

d~/R~

× [b 2 + (ab/e)arcsin(e)] -1

aspect ratio (a/b) increases the ratio Nmax/Nsp increases from 1 to 2.4 for an aspect ratio of 4 and the ratio Nmin/Nsp decreases to 0.6. However, the ratio of the average flux to that of a sphere varies little from unity as the aspect ratio changes. In addition, particle shape may be important in determining the ignition behavior because burning could begin at points where the mass flux is greatest. The goal of the optical pyrometry experiments is to determine Arrhenius parameters for the chemical rate constant, kc, and it is of interest to examine the effect of shape on kc, which is computed from the formula [5]

(22)

where e is the eccentricity, (a 2 - b 2 ) l / 2 / a . RESULTS The ratios of the maximum, minimum, and average mass fluxes to the flux for a sphere of the same external surface area are given in Table 1 for a number of examples of diffusion-limited burning. The integral f ~ d~/R~ is computed numerically and the average flux is found from Eq. 22. As the

(23)

where p is the overall rate and kdif is the mass transfer coefficient. For a sphere the rate is found from the energy balance, Eq. 17, using a heat transfer coefficient, h, of 2h/dp. The mass transfer coefficient is 0.048D/dpRTra. For an ellipsoid the rate is found from the energy balance, Eq. 17, using an average heat transfer coefficient

h = 4X

(/0

d~/R~

× [b 2 + (ab/e)arcsin(e)] -t.

(24)

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213

Here the surface temperature is an average value, Ts, and the average heat flux ~/is h ( T g - T s ) . The mass transfer coefficient is /~dif = ( O . 0 4 8 / R T m ) D ( T m )

(/5 -

X [b 2 q-(ab/e)arcsin(e)] - l .

d~/R~

)' (25)

Here the pressure at the surface is an average value, /~s, and the average mass flux, /V, is kaif(P(h - P s ) . For a gas temperature of 1507 K, a particle temperature of 1650 K, and a bulk oxygen concentration of 6%, typical values measured by Mitchell [5], the chemical rate constants differ by 1.3% for an ellipsoid with a -- 50 /zm and b = 40 gm and for a sphere of the same surface area, r0 = 45 gm. For an ellipsoid with a = 50 gm and b = 25/~m the rate constants calculated using the two assumptions differ by 1.7%. CONCLUSIONS The rate of diffusion limited combustion for ellipsoidal particles has been calculated by solution of

the Laplace equation. The average combustion rate is only slightly higher than that for a sphere having the same surface area, and chemical rate constants calculated using the ellipsoidal expressions are only slightly changed from those computed assuming spherical symmetry. The results indicate that overall combustion rates for ellipsoidal particles may be computed from temperature measurements using procedures similar to those for spherical particles. REFERENCES 1. Mitchell,R. E., and Mclean, W. J., Proc. Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, pp. 1113-1122. 2. Young,B. C., and Niksa, S., Fue167:155 (1988). 3. Young,B. C., McCollor, D., Weber, B. J., and Jones, M. L., Fuel 67:40 (1988). 4. Stratton,J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941. 5. Mitchell,R. E., Combust. Sci. Technol. 53:165 (1987). Received 24 February 1988; revised 5 June 1989