Viscosity of dusty gases

Chemical Engineering Science, 197 1, Vol. 26, pp. 1019-1024.

Pergamon Press.

Printed in Great Britain.

Viscosity of dusty gases PREM CHAND Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India (Received

25 September

1970)

Abstract- An attempt has been made to give an explanation of the apparent reduction in the viscosity of dusty gases when compared with uncontaminated clean gases. It is shown that the velocity gradient rather than viscosity of dusty gases changes when dusty gases are allowed to flow through two concentric rotating cylinders. The effects of various parameters influencing the apparent viscosity of dusty gases have been discussed. INTRODUCTION

was probably one of the first to report that a decrease in pressure energy is required to maintain a constant flow when dust is added in clean air flowing through a pipe and this reduction of pressure is quite appreciable. Similar observations also were made by the author when he was working in a steel factory where pulvarised coal was mixed with air and fed to a kiln. It was seen that the pressure drop along the pipe length decreased when the pulvarised coal was introduced in the clean air flowing through the pipe. The explanation put forward by Sproull is that the addition of dust decreases the viscosity of gases. This contradicts the well established Einstein’s equation of viscosity which states that the addition of foreign particles should increase the viscosity of gases. Sproull conducted experiments in a rotating cylinder viscometer and found that the torque necessary for holding the inner cylinder in position is more for uncontaminated gases than that for dusty gases. The solid-gas mixture enters the viscometer from the bottom and moves upward. This was done to avoid settling of the particles. After the mixture was blown for some time, the outer cylinder was rotated at a certain rate and the torque necessary to hold the inner cylinder SPROULL[~]

Gas phase

was measured. The details of the set-up are given in the paper published by Sproull[ I]. The result of Sproull’s experiment drew the attention of Saffman[21 who offered another explanation. He concluded that as particles move outward, they carry momentum with them. This changes the distribution of tangential velocity and alters the torque on the cylinder. This certainly seems to be a better explanation than the first. The purpose of the present study is to investigate the effect of different parameters on the tangential velocity distribution of the gas phase. A knowledge of such flows is very important from a design point of view because of applications in a number of industrial units such as a pulvarised coal feeder, pneumatic conveying of materials like cement, powder etc. The basis of the investigation is the two phase Navier--Stokes equation which has been used to find out the velocity gradient at the inner cylinder. Reduction of this velocity gradient is responsible for the reduction in torque necessary to keep the inner cylinder in position. THEORETICAL

ANALYSIS

The equations of motion in the r, 8, z directions for the two phases (gas and solid) are given by

P.CHAND

+

-a2ve a22 (2)

)

Particle phase

av,,

av,, ar

av,, vh ae

!!.!L$) = _pp(v+J ( ( vrp!Y$E+!+?!c$+!!r++ vzp%)= -pp(ve~v”) pp~+v~p!?++~~+y av,,=( PP

-+

at

v,,--+

vrp--7-

(4)

vz

aVep pp __+

ZP

az

)

(5) (6)

PP(vz>~vz)e

In the above equations, the term p,(V,-V/ represents the drag force on the particles where T, is the momentum equilibrium time defined by Singleton[3] and Michael[4 and 51 T,)

T = 2 p,02 m 9/L.’ ( 1 The following assumptions are made for the formulation of the problem. (a) The flow takes place between two concentric rotating cylinders with the inner cylinder stationary and the outer rotating at a fixed rate. The axial velocity of the solid-gas mixture is assumed negligible in comparison to the tangential velocity of the cylinder and therefore of the fluid. (b) All particles are of the same size and they are homogeneous. (c) The difference between the solid velocity and the fluid velocity is not much so that the Stokes equation could be used for the drag force. (d) No heat transfer from or to the wall is considered. (e) The physical properties of the fluid are assumed to be constant during the flow. (f) The flow does not vary in the longitudinal direction of the cylinder.

The three dimensional problem is two-dimension by assuming that the the axial direction (2 direction) is Besides this the velocity of gas in direction is zero. Therefore, the Eqs. simplified to

reduced to velocity in negligible. the radial (l)-(2) are

( 12!P+pp (!+ >

p -y

m

(7)

and

pp(ve;--,“)+p(~+~$+~) =0.

(8)

The continuity equation for the gas phase is automatically satisfied and so also are Eqs. (3) and (6). For the particle phase Eqs. (4) and (5) reduce to v

av,, !%=_&! --

TPar

r

T,

(9)

and

1020

VTP% + vepp ar

= -(ve~;ve).

(10)

Viscosity of dusty gases

Since we have assumed earlier that V,, is very small in comparison to V,,, Eq. (9) takes the form !!L!LP. r

T,’

(11)

From the equation of continuity for the particle phase we get

After rearranging Eq. (16) we obtain rz$J$+r$+J(l+Bf)=--AP.

(17)

Equation (17) is the modified Bessel’s equation. Two cases will be taken up now. CaseI:(A=O,B=O)

(, 2)

This is a case of flow of uncontaminated gases (one phase flow). The Eq. (17) for such cases reduces to

or V 7v.T = C(0,Z).

But as V,., and r are functions of r only V Tp.T = C’ = constant.

(18)

(13)

(14)

which under the boundary conditions (at r = rl, V, = 0 and at r = r2, V, = V,,) yields the well known result

From Eq. (10) we get v@?@=J$.

V,=

(15)

The terms in Eq. ( 10) are of the order of 6. Introducing the expression for V, from Eq. (15) into Eq. (8) and understanding that V, is function of r alone we get

C,r+$

or

(19) or

Vo’A& rz

rl

(r_2. 1 r

cm

Case 11: (A s 0, B > 0)

or

(16)

The complete given by Vo = C~Z,(k%)

where

solution[7-81

+ C$,(Xbr)

of Eq. (17) is

-$TLl(lh-). (21)

(1W

and (I6b)

where the first two terms represent the complementary function and the last term is the particular integral. In the Eq. (21) I, and K1 are the modified Bessel’s function of first and second kind respectively, and L, is the modified Struve function. 1021

P. CHAND

The two constants of integration Ci and CL are evaluated using the same boundary conditions as mentioned for Case I.

can say that the reduction in torque necessary to hold the inner cylinder as observed by Sproull was due to reduction in velocity gradient at the

The velocity gradient dV,Jdr at the inner cylinder for the two cases may be obtained by differenting V, with respect to r. For Case A

inner cylinder rather than the reduction in viscosity. Now, the velocity gradients at the inner cylinder will be computed using the Eq. (25) for different conditions and that will be compared with the velocity gradient for one phase flow using Eq. (24).

_c,

dVe_dr

’

(24)

fl

DISCUSSION

In Table 1, the computed values of (dVJdr),.=,,

and for Case B Z,(d&)+L

TORQUE

z 04%)

K,(tir)

om

NECESSARY TO HOLD CYLINDER IN POSITION

The torque, necessary to hold cylinder in position is given by

INNER

the

T = 2rrr,lr,( T,&=,,

inner

(26)

where I is the length of cylinder and (T,&_.,

= shear stress at inner cylinder

+T$f)} --$${L,(\/Br) r

dVe T = 2mr12Cr.x T=T,. ( >

(27)

It could be shown using Einstein’s equation for viscosity that the change in viscosity of air, p for finer particles is not much. Therefore we

r

4

for different values of p,,, ps, u, p, C’, r, and Ve, are given. Also in the last column of Table I, the reduction in the values of velocity gradient, (dV,/dr),.=,, are given. This reduction is with respect to fresh air without any solid particles entrained in it. Now we can study the effect of different parameters on the velocity gradients and we will compare the computed values of reduction in velocity gradient with the reduction in the apparant viscosity of dusty gases observed by Sproull. Effect of peripheral

When the inner cylinder is stationary, the torque necessary to hold the inner cylinder fixed is given by

--L, g}]

1

velocity of outer cylinder

From Table 1 (S.N. 8 and 9) it can be seen that due to the increase in peripheral velocity of the outer cylinder, the gradient (dl/,/dr),=,, changes, but the reduction in (dl/,/dr),=,,, when compared to one phase flow for both the cases remains the same, i.e. 14 per cent though the peripheral velocity of the outer cylinder has been increased from 34.5 cm/set to 69*0cm/sec. Sproull also has observed that the reduction in viscosity of dusty gas does not depend upon the rotation of the outer cylinder. The test Nos. 6 and 7 of his

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Viscosity of dusty gases Table I. Computed values of reduction in velocity gradient for different conditions Reduction in S. N. p,,x 1O-4

1 2 3 4 5 6 7 8 9

pa

fJ

CL

(p/cm?

(dcm3)

(cm)

(PI

(cm%ec)

(cm)

(cm/se@

0.5 5.0 0.5 0.5 0.5 0.5 0 5.0 5.0

5.0 5.0 10.0 5.0 5.0 5.0 0 0 0

0.01 0.01 0.01 0.005 0.01 0.01 0 0.01 0.01

OG)O181 0~000181 0~000181 0~000181 0.00181 OWO181 OXI 0~000181 0~000181

0.5

0.5 0.5 0.5 0.5 5.0 0 0 0

5.75 5.75 5.75 5.75 5.75 5.75 5.75 5.5 5.5

34.5 34.5 34.5 34.5 34.5 34.5 34.5 34.5 69.00

C’

experimental data[l] show that an increase in rev/min of the outer cylinder from 25 to 50 does not affect the reduction in viscosity of the dusty gases. Effect of annulus width The percentage reduction in velocity gradient for 7.5 mm annulus and 5 mm annulus are 32 and 14 per cent respectively as can be seen in table 1 (S.N. 2 and 8) Sproull also found a greater reduction in viscosity for wider annulus. But his explanation is different. He states that the coarse lime stone could not be forced through the appal;ttus fast enough to prevent excessive settling. Unfortunately his experimental data does not support this argument since both the lime stone and talc were of the same size, i.e. 10/L. Effect of size and concentration of the particles Sproull on the basis of his experiments states that the concentration of the particle is a very important factor and that the particle size and bulk density of the dust are relatively unimportant. In the light of the present analysis it can be seen that when the concentration of the particle is increased from 0.00005 glee to 0@005 g/cc the reduction in (dV,Jdr),=,, increases from 4.05 to 32-2 per cent (See S.N. 1 and 2 of Table 1). Theoretically it is seen that a reduction in particle size, with all other parameters unchanged, increases the reduction in (dV,Jdr),_.,.

r2 (set-‘) 47.19 33.29 48.187 41.79 47.19 47.19 49.209 60.326 120.653

4.05 32.2 2,07 15.1 4.05 4.05 0.0 14.0 14.0

The viscosity of the fluid does not affect the velocity gradient (dl/,/dr),,,, as can be seen in the above mentioned table (S.N. 1 and 5).

CONCLUSIONS

Sproull has observed the reduction in torque necessary to keep the inner cylinder in position after the dust particles were introduced in clean gas. From his experiments, he concludes that the viscosity of dusty gases is less than that of clean gas. The present analysis shows that the velocity gradient rather than the viscosity of air is responsible for the reduction of torque. Because the torque necessary to keep the inner cylinder in position is a function of viscosity, CL,and the velocity gradient, (dl/,ldr),=,, both explanations lead to same conclusion, but the method given here seems more logical and is based on fundamental equation of fluid flow. Besides that it does not contradict the Einstein’s equation of viscosity which states that the viscosity of dusty gas should increase (though not much for this case) rather than decrease as concluded by Sproull. The quantitative decrease in the velocity gradients could be compared fairly well with the decrease in so called viscosity of dusty gases.

Acknowledgment-The

author gratefully acknowledges the help rendered by Prof. B. M. Belgaumkar, Head, Mech. Engng. Dept. through discussions and suggestions.

1023

P. CHAND NOTATlON

C’ CT

r

P PS PP VT VI9 V.Z VW

azumuthal velocity of particle phase, cm set-’ axial velocity of particle phase, cm set-’ VZP peripheral velocity of inner cylinder, cm V0, set-’ V0* peripheral velocity of outer cylinder, cm set-’ t time, set P viscosity of clean air w angular velocity of the outer cylinder V@P

a constant defined by Eq. ( 14) size of the dust particles, cm variable radius, cm density of air, g cmm3 density of particles, g cmV3 concentration of particles, g cme3 radial velocity of gas phase, cm set-’ azumuthal velocity of gas phase, cm see-’ axial velocity of gas phase, cm set-’ radial velocity of particle phase, cm set-’

REFERENCES [I] SPROULL W. T., Nature, Lond. 1961 190 976. [2] SAFFMAN P. G., Nature, Lond. 1962 193 463. 131 SINGLETON E., Z. a&w. Math. Phys. 195 1 18 42 1. [4] MITCHAEL D. H. and-MILLER D. A., Mathematics 1966 13 97. 151 MICHAELD.H..J.FluidMech. 196831(l) 175. i6] SCHLICTING H., Boundary Layer Theo&: McGraw-Hill 1960. [7] LUKE Y., Integrals of Bassel Functions. McGraw-Hill 1962. [8] RELTON F. E., Applied Bessel Functions. Blackie &Son 1946. [9] MITRA C., B. Tech. Thesis, Dept. of Me& Engng. May 1960. R&me-Une tentative a Bte faite en vue d’expliquer la reduction apparente de la viscosite de gaz pousiereux par rapport a des gaz propres non contamin&. On montre que le gradient de velocid, plutot que la viscosite des gaz poussiereux, change quand on laisse les gaz poussiereux s’ecouler dans deux cylindres concentriques rotatifs. On discute de I’effet des divers parametres influencant la viscosite apparente des gaz poussiereux. Zusammenfaasung- Es wurde versucht eine Erklartmg zu geben fur die scheinbare Verminderung in der Zahigkeit verstaubter Gase im Vergleich mit nicht verunreinigten, sauberen Gasen. Es wird gezeigt, dass es weniger die Ziihigkeit der verstaubten Gase ist, die sich andert, sondem vielmehr der Geschwindigkeitsgradient, wenn man verstaubte Gase durch zwei konzentrische, rotierende Zylinder stromen 1Psst. Die Wirkung verschiedener, die scheinbare Zahigkeit verstaubter Gase beeinflussender Parameter wird erortert.

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