Analytical estimation of elasticity of cyclone effectiveness factor

TECHNICAL ANALYTICAL

(First

NOTE

ESTIMATION OF ELASTICITY EFFECTIVENESS FACTOR

received 1 I June 1984 and receitied

forpublicarion

OF CYCLONE

4 October

1984)

Abstract-The elasticity of a cyclone effectiveness factor has been determined with respect to all the design and operating parameters of cyclone. Cooper’s (1983) method of numerical treatment, which essentially involves the estimation of the fractional change in the effectiveness factor per unit fractional change in the variable (keeping all other variables at the base values) has been utilized to analyse analytically. Leith-Licht and shepherd-Lapple equations are used for the purpose. The advantages of analytical treatment has been demonstrated with illustrations.

NOMENCLATURE

; B d

dP

D 4 E,

/ h

H

ki K

Kc I M n “1

n2

iv

NH

P. 4

s

T

K VH X AP P PP w fli

variable X. This definition of elasticity and the determination of elasticity of a cyclone effectiveness factor ‘q’ with respect to all the design and operational parameters has been dealt with by Cooper (1983) by numerical treatment. He has also described the significance of the effectiveness factor in the optimisation studies, the use of elasticity analysis in identifying parameters having significant influence on q,and the way the effectiveness factor changes due to changes in some of the parameters. For such analysis to be made in any design situation, analytical expressions for elasticities will be more useful. It is the aim of this paper to develop such expressions using cyclone model equations.

Ratio of inlet height to cyclone diameter Ratio of inlet width to cyclone diameter Ratio ofdust outlet diameter to cyclone diameter Ratio of cone diameter at natural length to cyclone diameter Particle size, m Cyclone diameter, m Ratio of vortex finder diameter to cyclone diameter Elasticity of q with respect to X: t$ Arbitrary function Ratio of cylinder height to cyclone diameter Ratio of overall height to cyclone diameter Empirically defined parameter Parameter in Leith-Licht equation Parameter in Leith-Licht equation Ratio of natural length to cyclone diameter Parameter in Leith-Licht equation Vortex exponent Empirically defined parameter Empirically defined parameter Parameter in Leith-Licht equation No. of velocity heads Penetration defined as (1 -Q) Effectiveness factor Ratio of vortex finder length to cyclone diameter Absolute temperature, degrees Kelvin Inlet velocity, m s- 1 Velocity heads, N me2 Variable, N m - ’ Pressure drop, N m-’ Gas density, kgm-’ Particle density, kg m - ’ Gas viscosity, kg m - ’ s- ’ frictional efficiency.

2. EXPRESSION

FOR ELASTICITY

The effectiveness factor is defined as: 4=

-ln(P,) AP

’

(2)

where Pm is the penetration defined as (1 -vi) and AP is the pressure drop. For expressing q analytically, the Leith-Licht for fractional efficiency and (1972) equation Shepherd-&apple (1940) equation for pressure drop are used in this work. The form of the two equations used here are summarized in Table 1. The expression for effectiveness factor then becomes Md$ 4=-&g-

(3)

Applying the basic definition of elasticity, the following equation is obtained after partial differentiation: , (4)

where E, is the elasticity of q with respect to any parameter x. By analytically estimating the partial derivatives and simplifying the expressions, the elasticities with respect to all variables are determined. The derived equations are summarized in Table 2.

IM-RODL’CTION

The elasticity of a function with respect to a variable is defined as

3. NUMERICAL

(1) where E, is the elasticity. It essentially provides the fractional change in the function f per unit fractional change in the 831

ILLIJSTRATIOK

The derived equations are numerically illustrated for a specific Leith-Mehta (1973) dzsign case considered earlier by Cooper (1983). The base values and the computed elasticity values are given in Table 3. The elasticity values are also

Technical Note

832

Table 1. The Sl No

(1)

formof model equations used in elasticity analysis

Model

Expressions ‘li= l-cxp(-Md”)

Leith-Licht (1972) for qi and Alexander (1949) for exponent n

I’

n = n, + n2 n, = 1-0.18385T0,3 n2 = 0.467 (1 -n,) D”.lJ

N.‘2

d=l-

(2)

g

(S+l-h)

AP = VHNH

Shepherd-&apple (1940)

N

H

-

16ab D:

Table 2. Elasticity of effectiveness factor with respect to cyclone design and operational parameters SI No

Parameters

Symbol for elasticity

(1)

D

ED

4-1.5N-0.14n2NZ

(2)

a

E,

1 -N--

(3)

b

-%

I-N--

(4)

DC

(5)

S

(6)

h

Expression for elasticity

Et&

c I

IUll 48K,

{

1 a(d’-D:)+U d2-Dt

-D:)

I

2D:(a-ZS-I)+I(d*-Di)

ED:

JL

nNa ,6K

$1

c

+d’--2D:)

(1+2d) (d-B) 3(1 -B)

(1 -d)

--

1 +d+d’ 3

Technical Note

833

Table 2. (Contd.) SI No

Parameters

Symbol for elasticity

(7)

H

EH

Expression for elasticity

EB

g

E

N/2

PP

I

Edp

N

=&2d)(H-h)

(11) (12)

Q

EQ

(N/2) - 2

P

4

(13)

P

E,

-N/t -1

Table 3. Numerical iilustration of elasticity equations Sl NO

Parameter

Units

D a

m

Ii;

b

(3) (4) (5) (6) (7) (8) (9) (10) (If) (12) (13)

D,

s,

H

d”, PP M 0

i.7

kgmm3 kgm-‘s-r kgmw3 MaS-’

obtainable by incrementing each parameter by 1 per cent and applying Equation (1). The relative simplicity of using the derived equations can be seen. 4. CONCLUSION

Analytical expressions for the elasticity ofcyclone effectiveness factor can be obtained using the Leith-Licht and Shepherd-iappie equations. The derived equations are useful for appliition in any design situation where elasticity values and performance analysis are required. Ac~~~wiedge~nt-be authors wish to acknowledge Dr H. C. Visvesvaraya, the Chairman and Director General, CRI, for his valuable guidance in the preparation of this paper.

REFERENCES

ofCyclone Design Inst. Min. Metall. (New

Alexander R. MC K. (1949) Fundamentals and Operation. Proc. Australas. Series) 152-3,203-228.

Base value 1.00

0.43 0.17 0.68 1.20 3.00 5.00 0.375 5x 10-e 1000 1.8 x 10-s I.20 1.78

Elasticity value 3.1365 0.4202 0.4528 1.0925 0.0389 0.2184 0.2307 0.0554 0.6012 0.3906 -0.3096 --LOO&3 - 1.7OtXl

Cooper D. W. (1983)Cyclone design: sensitivity,eIasticity and error analyses. Armospheric Environment 17.485-489. Leith D. and Licht W. (1972) Collection efficiency of cyclone type particle collector, a new theoretical approach. AICHE Symp. Ser. April 1971. Leith D. and Mehta D. (1973) Cyclone performance and design. Atmospheric Environment 7, 527-549. Shepherd C. B. and Lapple C. E. (1940) Flow pattern and pressure drop in cyclone dust collectors. Ind. Engng Gem. 32, 12461248.

Cement Research Institute of India M 10 South Exrensian I1 New Delhi 110049 Department Indian

ofChemicalEngineering

Institute

New Delhi

ofTechnology

R. GANAPATHY

B.

&TCHUMANl