INVESTIGATION AND CALCULATION

OF THERMOPHYSICAL

PROCESSES

PNEUM OHYDRAULIC PULSATION EXTRACTOR WITH U-SHAPED ARRANGEMENT OF THE VESSELS V. V. K a f a r o v , * V. G. Vygon, G. A. Mikh.eeva, and V. A. Rudakov

UDC 66.061.001.24

One of the main advantages of pulsation ext ract ors is the uniform supply of energy over the cr o ss section of the r e a c t or and the absence of moving parts. Nevertheless, intensive pulsation is used today only in low-capacity r e a c t o r s , where complete extraction and high-purity products are required. Large plate and packed columns operate with low pulsation intensities, the magnitude of which is far from the optimum and significantly reduces the pulsation effect. The limitations in the size of pulsation columns are due to the increase in inertia of the liquid column with increasing dimensions of the reactor, to an increased load on the column support, and to a high energy consumption to achieve pulsation. This makes it difficult to reproduce the optimum pulsation regime, established in the laboratory, on a plant scale. At p r e s e n t there is a tendency in the design of pulsation columns to use schemes with static compensation [1-3]. A design has been proposed in [4] with an U-shape d arrangement of the reactors, where the pulsation is generated by periodic p r e s s u r e changes in the air space above the liquid. The U-shaped arrangement of the r e a c t o r s (Fig. 1) decr e a ses significantly the energy consumed for pulsation (by eliminating the hydrostatic p r e s s u r e component) and the dynamic load on the basis. The energy consumption can be minimized by using oscillations at the equipment, s own frequency [5, 6]. This frequency is determined by the dimensions of the equipment and drops sharply with increasing dimensions. It is therefore necessary to r e s o r t to forced pulsations to achieve an optimum pulsation regime. *Corresponding Member, Academy of Sciences of the USSR. 9

l

/Pf

g

"TOthea t m o ~ Dpe~

I

NN,

atmosphere o

Fig. 1. Scheme of the extraction apparatus; I, II) branches of the apparatus, 1) slidevalve distribution system; 2) pulsation chamber; 3, 5) top and bottom settlingchamb e r s ; 4) working section of the column; 6) U-tubes; 1.p. is the light phase, h. p. is theheavy phase.

. < J J .4. Translated from Khimicheskoe i Neftyanoe Mashinostroenie, No. 10, pp. 15-18, October, 1974.

9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

886

ZR, m m

I50

.~o0

r

1 ,Pf = 6 kg/cm2

I

2

3

NHz

Fig. 2. P u l s a t i o n a m plitude x k as function of pulsation f r e q u e n c y at different feed p r e s s u r e s pf.

The a p p a r a t u s with a U - s h a p e d a r r a n g e m e n t o f the r e a c t o r s r e p r e s e n t s a p n e u m o h y d r a u l i c s y s t e m which has not been sufficiently investigated. The m a t h e m a t i c a l d e s c r i p t i o n of the oscillations in this pneumohydraulic s y s t e m includes an equation f o r the m o v e m e n t of the liquid in the pulsation a p p a r a t u s and an equation f o r the b e h a v i o r of a i r in the pulsation c h a m b e r s above the liquid. The d i f f e r e n t i a l equation for the oscillating m o v e m e n t of a liquid column in the a p p a r a t u s can be d e r i v e d f r o m the e n e r g y balance of the moving s y s t e m [7]. The v a r i a b l e f o r c e pvf(t) acts on the liquid s u r f a c e in the pulsation c h a m b e r ; it is r e l a t e d to the s u r f a c e a r e a of the pulsation c h a m b e r and changes following a periodic law. Under the action of the disturbing f o r c e the liquid in the pulsation c h a m b e r is shifted for a c e r t a i n distance in the t i m e t. The w o r k c a r r i e d out by the disturbing f o r c e is spent to a l t e r the kinetic and potential e n e r g i e s of the liqui d and to o v e r c o m e hydraulic r e s i s t a n c e s . C o n s i d e r i n g the whole s y s t e m , c o n s i s t i n g of n s e c t i o n s with different c r o s s s e c t i o n s , lengths, h y d r a u l i c d i a m e t e r s , r e s i s t a n c e coefficients, and d e n s i t i e s , we obtain i

t Fpc Pvf

0

d x p c dt : dt

2 I=

t

•] F j L j d t

fz ~-

dt

t

-

F1 h

0

dt~ d t

0

j=o

pgj - ~ - dr,

(1)

o

where Fpc and Fj is the c r o s s sectional a r e a of the pulsation c h a m b e r and the j - t h s e c t i o n of the column in m 2, r e s p e c t i v e l y , Xpc and xj is the liquid shift in the pulsation c h a m b e r and the j - t h section of the column in m, r e s p e c t i v e l y , t is the t i m e in sec, g is the a c c e l e r a t i o n of g r a v i t y in m / s e e 2, 7j and 7pc is the liquid density at the j - t h s e c t i o n of the column and in the pulsation c h a m b e r in k g / c m 2, r e s p e c t i v e l y , Lj is the length of the j - t h s e c t i o n of the column in m, and Apgj is the p r e s s u r e drop at the j - t h section of the column in k g / m 2. It is a s s u m e d that the a r i t h m e t i c m e a n values of the coefficients of local r e s i s t a n c e ~j for d i r e c t and r e v e r s e m o v e m e n t a r e substituted in Eq. (1). Equation (1) can be m a d e l i n e a r due to the sloping c h a r a c t e r of the c u r v e r e p r e s e n t i n g the hydraulic r e s i s t a n c e of plate and packed columns as function of the t h e o r e t i cal flow r a t e [7]

a p ~ = Z~-g w ~j+xj G /

(2)

,

where vj is the liquid velocity at the j-th column section in m/see, vj is the mean velocity at the j-th section of the column in m / s e e ; }j is the coefficient of local resistance at the j-th column section, M is the friction coefficient at the j-th column section, and Dj is the hydraulic diameter of the j-th columnJ section in m. The mean liquid velocity can be represented as half of the maximum velocity, determined from the expression dxj ~ -

= tllax

to v T a x " "

)

assuming a sinusoidal character of the m o v e m e n t , where co is the angular frequency in r a d / s e c and x ? ax is the pulsation amplitude at the j-th column section in m. Equation (2) then takes the form A pgj =

~;v] 4g

.max ~1+ ~J

~o x]

9

(3)

The equation for the continuity of flow m a k e s it p o s s i b l e to e x p r e s s the shifts in all sections of the a p p a r a t u s by the shifts x k in the working section. Substitution of/Xpgj f r o m Eq. (3) in Eq. (1) and d i f f e r e n tiation of the equation obtained gives

887

j=l

(4)

Fi

where F K is the free c r o s s - s e c t i o n of the column in m 2. The p r e s s u r e on the liquid in both a r m s of the apparatus (see Fig. 1) is equal to the p r e s s u r e diff e r e n c e in the pulsation c h a m b e r s :

II [ Pvf = Ppc --Ppc"

(5)

Let us e x p r e s s in Eqs. (4) and (5) the v a r i a b l e s by i n c r e m e n t s and linearize the functions entering this equation with r e g a r d to the m o m e n t to, which c o r r e s p o n d s to the m o m e n t when the liquid level in both a r m s of the apparatus is the s a m e . C o m p a r i s o n o f the equations obtained gives the equation for the m o v e meat of the liquid in the a p p a r a t u s : I,

" ,

h P p c , h Ppc --

2F~ ~ g

1=1

1 L.d~AxK F:'~ ' ~ ~i 1 dr2 + - -2g

~1 Q L_~j) dAx~ FK - -2 ~1+ X1 ~ - c ~pc &rK. + 2 -FP J~i F]

Let us now consider the behavior of a i r in the pulsation c h a m b e r . above the liquid in the c h a m b e r is given by the e x p r e s s i o n d_q_Q= oi dt

(6)

The change in the amount of air

od,

(7)

where Q is the weight of air in the pulsation c h a m b e r in kg, and Gi and Gd is the air flow rate in the intake and d i s c h a r g e lines in k g / s e e , r e s p e c t i v e l y . Using t h e relation between the a i r flow rate through the slide valve, the p r e s s u r e in the pulsation chamber, and the slide valve resistance, it follows that

Gi

P f --.P.pc (t) -

R~ (t)

] ;

(8)

ppc (t) Od -- Rsd (t)

i where pf is the feed p r e s s u r e at the inlet to the slide valve distribution m e c h a n i s m in k g / c m x, and R s and R ds is the slide valve r e s i s t a n c e at the intake and d i s c h a r g e lines in s e e / m 2, r e s p e c t i v e l y . Let us e x p r e s s in E q s . (8) the v a r i a b l e s by i n c r e m e n t s , a s s u m i n g that in the moment t o the r e s i s tances in the intake and d i s c h a r g e lines are the same, and the r e s i s t a n c e i n c r e m e n t s in these lines have the s a m e absolute values but opposite signs. Equation (7) then takes the f o r m d(aQ) 2hPpc+ P f ~ A d 2aPpc Pf AR~ ,"

o

- -

-

,

(9)

where R~s is the m e a n r e s i s t a n c e of the slide valve in s e e / e r a 2. Let us integrate Eq. (9) and write it for each a r m of the apparatus, assuming that r e s i s t a n c e changes in both pulsation c h a m b e r s have the same absolute value but are shifted in phase by 180 deg: ,~QI =

__

-- Pf

- A R s dt;

o

t [ __

AQ[I=

II

2APpc o - + Pf

AR ] dr,

]

(lo)

o

where AQI and AQ II is the change in the amount of a i r in the pulsation c h a m b e r s of the a r m s I and II of the apparatus in kg, r e s p e c t i v e l y , and AR s is the change in r e s i s t a n c e of the slide valve d i s t r i b u t o r in see / e r a 2, F r o m the above a s s u m p t i o n s it follows that the m e a n p r e s s u r e s in the pulsation c h a m b e r s of both a r m s a r e equal to each other and equal to half of the feed p r e s s u r e

OI .Oil __ 1 P p c = V p c - - 2 el 9 Knowing the air p r e s s u r e and its volume in the pulsation c h a m b e r we can d e t e r m i n e the air m a s s f r o m the C l a p e y r o n - M e n d e l e e v equation

888

1

mm/min , 7000

9 Pf = 11 kg/cm2

,~ mm/min

l

8000

. .~RP/6 (g--~ ~,7

5~

0008

5000

400#

3000

,700~

0

\\

4000

3000

,,~ ~

,k ss6l

zooo .Z /

100o

7000

'

5008 ~ . . ~ RsO=278sec/cE

k ~ .18

zooo

2000

max, k,~. -/cm~ Pf ,;Ppc

; /,mm/min

!

Lmz

'

f=O,7SHz

= 1.25 k~/cm - ~ l 2 3 4 5f, Hz a)

0

1

Z

b)

0

3 f , Hz

I

Z

J

';' f, Hz

c)

1#

!

6

o~

?

I.x, M

Fig. 4

Fig. 3

Fig. 3. P u l s a t i o n intensity I as function of pulsation frequency f: a) at different feed p r e s s u r e s pf; b) at different packing r e s i s t a n c e s Rp in the working section of the column; c) at different a v e r a g e r e s i s t a n c e s R ~ of the slide valve d i s t r i b u t o r . max Fig. 4. Feed p r e s s u r e pf and m a x i m u m p r e s s u r e in the pulsation c h a m b e r Ppe as function of height L K of the working section of the column at a pulsation intensity of 1500 r a m / r a i n and d i f f e r e n t f r e q u e n c i e s f: ) pf = f(LK), - - - - ) p ~ a x = f(LK). QOl_ M -- R T Q0[I =

_ol o[ FPc VPc;

M

(11)

_0II V011

R'--'~/Jpc "pc '

w h e r e M is the a v e r a g e m o l e c u l a r weight of a i r in k g / k ~ - m o l e , R is the gas constant in kg . m / k g - m o l e dchgamTeirSs Y t h ~ r ::mmPse;::dei~nidegmK'

:e:dp:c~t~v:ld. V ~ ' c ~ m : : : 2 d ? : : : : i g : i ~

tch:a:bleT:i:~ e

the s a m e , it follows f r o m these a s s u m p t i o n s that Q0I = QOII. E x p r e s s i n g in E q s . (11) the v a r i a b l e s by inc r e m e n t s and using the equation of flow continuity, we can r e l a t e the change in a i r volume in the pulsation c h a m b e r to the shift of the liquid column in the working section of the column: AQ I - MFKx~aXks A Z M p f FKkS._AXe; RT Ppc -2RT

AQII -

.max

-

MFKx~ ks

ST

(12)

A II Mpf FKk s Poc + 2RT A xx,

w h e r e k s is the safety f a c t o r for the pulsation c h a m b e r v o l u m e . The t e r m FkXm k a Xks d e t e r m i n e s the volume of the pulsation c h a m b e r : Vpc = 2Fpex~caXks = 2FkX~ ax k s. T h e safety f a c t o r k s is selected as that liquid cannot be e j e c t e d f r o m the pulsation c h a m b e r into the a i r line, and a i r cannot e n t e r the u p p e r settling c h a m b e r . C o m p a r i n g E q s . (10) and (12), and using L a place t r a n s f o r m a t i o n , the equations obtained a r e solved relative t o ApI c and ApIIc: Mt'f FKks i x K - 2RT MFKxmax ks RT

....

I1

APpc =

Pf

2 +-SROs

M p f FKks AXK 2RT

Pf

(13) -AR s

S (R V

MF~x~ ax ks RT

=~Xi"s

2

'

+ SROs

w h e r e S is the L a p l a c e o p e r a t o r . The parameters__ Ax k and App c a r e the d e t e r m i n i n g c h a r a c t e r i s t i c s of the s y s t e m , The pulsation amplitude Ax k of the liquid column in the column allows to judge whether a given pulsation r e g i m e can be r e a l i z e d in an a p p a r a t u s of given d i m e n s i o n s for a given l i q u i d - l i q u i d s y s t e m . The p r e s s u r e in the p u l s a tion c h a m b e r d e t e r m i n e s the s t r e n g t h c h a r a c t e r i s t i c s of the a p p a r a t u s .

889

Laplace t r a n s f o r m a t i o n of Eq. (6) and substitution of AppI c and AplpIc f r o m E q. (13) leads to the expression

-

-

- -

(14)

j

--z-

\,%J where Wx is the t r a n s f e r function of the s y s t e m via slide valve r e s i s t a n c e - liquid shift in the c o l u m n , pf Fpc; j = ~ ~= ~'= 2-~pcVpc Vpcks; Vpc_FFpc ,~.'l]

2RT

;

~j+Xj L]

The amplitude-frequency c h a r a c t e r i s t i c of the s y s t e m via slide valve r e s i s t a n c e - liquid shift in the column can be written as follows: %

Ax(~) --

.

Using this e x p r e s s i o n we can a s s e s s at a given pulsation frequency the effect of various p a r a m e t e r s of the s y s t e m on pulsation amplitude and intensity, and s e l e c t the feed p r e s s u r e r e q u i r e d to establish a given pulsation r e g i m e in the column. The r e s p o n s e of an U-shaped a r r a n g e m e n t of the r e a c t o r s to v a r i o u s p a r a m e t e r s was calculated by using a Minsk-22 c o m p u t e r . Figure 2 shows the amplitude frequency c h a r a c t e r i s t i c s of a s y s t e m with the following basic p a r a m e t e r s (see Fig. 1): Height of c o l u m n sections in m: . . . . . . . . . . . . . . . . . . . working section, L k . . . . . . . . . . . . . . . . . . . . . . . . . U-tube, Rst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . top settler, Lts . . . . . . . . . . . . . . . . . . . . . . . . . . . . bottom settler, Lbs . . . . . . . . . . . . . . . . . . . . . . . . . D i a m e t e r of column sections in In: working section, Dk U-tube, Dst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . top settler, Dts . . . . . ....................... bottom s e t t l e r , Dbs . . . . . . . . . . . . . . . . . . . . . . . . . pulsation chamber, Dpc . . . . . . . . . . . . . . . . . . . . . . . Density in k g / m 3 : light phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . heavy phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liquid in the U-tube . . . . . . . . . . . . . . . . . . . . . . . . . . F r e e c r o s s section of plate in % . . . . . . . . . . . . . . . . . . . Phase flow rate ratio . . . . . . . . . . . . . . . . . . . . . . . . . . T e m p e r a t u r e in deg C Molecular weight of the gas (air) in k g / k g - m o l e . . . . . . . . Feed p r e s s u r e by c o m p r e s s e d gas in k g / c m 2 . . . . . . . . . . R e s i s t a n c e coefficient of the packing (for 1 m length) . . . . . Average r e s i s t a n c e of the slide valve d i s t r i b u t o r in s e c / m ~.

5 0.9 0.5 0.5 0.5 0.2 0.1 0.6 0.6 0.2 861 1050 1000 20 0.1 20 0.029 3 12.66 556

The calculations c o n f i r m the s h a r p drop in pulsation amplitude, observed in practice, with i n c r e a s ing frequency and at constant feed p r e s s u r e applied by c o m p r e s s e d gas. At a given pulsation frequency the amplitude f i r s t r i s e s sharply with i n c r e a s i n g feed p r e s s u r e , p a s s e s through a m a x i m u m , and then slowly d e c r e a s e s . Such a change in amplitude is explained by the fact that the ratio APpc / (ppf/2), which indicates the efficiency of e n e r g y utilization, d e c r e a s e s with i n c r e a s i n g feed p r e s s u r e a n d h e n c e with i n c r e a s i n g a v e r a g e p r e s s u r e in the pulsation chamber 9 Since the pulsation intensity is the product of pulsation frequency and amplitude, the relationship between pulsation intensity and frequency has an e x t r e m e c h a r a c t e r (Fig. 3 a ) , whereby the e x t r e m e shifts towards higher frequencies with i n c r e a s i n g feed p r e s s u r e . The p r e s e n c e of a m a x i m u m indicates that, at

890

a given feed p r e s s u r e , the s a m e pulsation intensity can be obtained with two different combinations of a m plitude and frequency. However, f r o m the standpoint of b e t t e r e n e r g y utilization f o r the e s t a b l i s h m e n t of pulsation, the pulsation r e g i m e with the higher frequency and l o w e r amplitude is to be p r e f e r r e d . An e x t r e m e is a l s o p r e s e n t in the curve, r e p r e s e n t i n g the r e l a t i o n s h i p between pulsation intensity and f r e q u e n c y in the c a s e when hydraulic r e s i s t a n c e is s e l e c t e d as the v a r i a b l e (Fig. 3b). At low f r e quencies when the m o v e m e n t of liquid is slow an i n c r e a s e in the e n e r g y component spent to o v e r c o m e h y d r a u l i c r e s i s t a n c e , c a u s e s only a s m a l l d e c r e a s e in pulsation intensity. At h i g h e r f r e q u e n c i e s the i n e r t i a component is so strong that the r e s i s t a n c e p r a c t i c a l l y c e a s e s to have an effect on the pulsation intensity. F i g u r e 3c shows the pulsation intensity as function of frequency at different v a l u e s of the a v e r a g e r e s i s t a n c e in the supply line for c o m p r e s s e d air, which includes the r e s i s t a n c e of the slide valve d i s t r i b u t o r and of the connecting p i p e s . The c h a r a c t e r of the r e l a t i o n s h i p indicates that a d e c r e a s e in the r e s i s tance leads to an i n c r e a s e in pulsation intensity at all f r e q u e n c i e s , with the total e n e r g y c o n s u m p t i o n r e maining the s a m e . An i t e r a t i v e a l g o r i t h m was developed for the s e l e c t i o n of the feed p r e s s u r e r e q u i r e d to obtain an optim u m pulsation intensity at a given frequency, if the d i m e n s i o n s of the a p p a r a t u s and the c h a r a c t e r i s t i c s of the s y s t e m w e r e known. With this calculation it is p o s s i b l e to p r e d i c t how much the feed p r e s s u r e with c o m p r e s s e d a i r m u s t be i n c r e a s e d to m a i n t a i n an o p t i m u m pulsation r e g i m e in a l a r g e r column. Figure 4 shows the feed p r e s s u r e and m a x i m u m p r e s s u r e in the pulsation c h a m b e r as function of the height of the working section of the column at a pulsation intensity of 1500 m m / m i n and different f r e q u e n c i e s . The c u r v e s show that at all f r e q u e n c i e s i n v e s t i g a t e d the p r e s s u r e m u s t be i n c r e a s e d by a f a c t o r of ~ 1.5 when the height of the working section of the column is i n c r e a s e d . max It h a s b e e n shown that the volume of the pulsation c h a m b e r is given by the condition Vpc = 2FkXk k s. T h e a v e r a g e a i r volume of the pulsation c h a m b e r is equal to h a l f the c h a m b e r volume and at a given intens i t y and f r e q u e n c y it depends on the safety f a c t o r . Calculations have c o n f i r m e d that, when i n c r e a s i n g the a i r v o l u m e in the pulsation c h a m b e r , the e n e r g y consumption m u s t also be significantly i n c r e a s e d in o r d e r to m a i n t a i n a given pulsation r e g i m e , since the a i r h a s a damping effect on the oscillations of the liquid. T h e m a t h e m a t i c a l t r e a t m e n t p r e s e n t e d allows to obtain the a m p l i t u d e - f r e q u e n c y c h a r a c t e r i s t i c of the s y s t e m via slide valve r e s i s t a n c e - p r e s s u r e in the pulsation c h a m b e r . T h i s c h a r a c t e r i s t i c m a k e s it p o s sible to a s s e s s the pulsation amplitude of the p r e s s u r e in the pulsation c h a m b e r , which is r e q u i r e d for s t r e n g t h c a l c u l a t i o n s when designing the equipment, taking into account the v a r i a b l e load on the column walls. T h e second eouation of the s y s t e m (13) is r e a r r a n g e d as follows, with the denotations as used above:

a--P"

'V~s "~ VpcS Ax--,, xmax

R~

B a s e d on e x p r e s s i o n (14) the t r a n s f e r function of the s y s t e m f o r the combination of the v a r i a b l e s c o n s i d e r e d c a n be w r i t t e n as follows:

[-K-~s ~ ---s

l + i'~ Vpc

Rs ] w h e r e Wp is the t r a n s f e r function of the s y s t e m via slide v a l v e r e s i s t a n c e - pulsation c h a m b e r p r e s s u r e . Expanding Wx(iW) and Wp(iW) into an i m a g i n a r y and a r e a l t e r m , we obtain:

Wp(i o) =

Rep(o~)+ l Imp(o,)

1 -- ~ Vpc o IO Vpc (0 Rex(o) -- [ms (<~)].__ i ~ Vpc r [I + Rex(co)+ Imx (o)l

(15)

By m e a n s of Eq. (15) it is p o s s i b l e to calculate the a m p l i t u d e - f r e q u e n c y c h a r a c t e r i s t i c of the s y s t e m and to d e t e r m i n e the m a x i m u m and m i n i m u m p r e s s u r e s in the pulsation c h a m b e r . LITERATURE 1~ 2.

CITED

US Patent No. 3174831 (1965). A. F. G a l e e v and A. I. Gur, yanov, A u t h o r ' s C e r t i f i c a t e No. 162503, Byulleten' I z o b r e t e n i i i T o v a r nykh Znakov, No. 10 (1964). 891

3.

4. 5. 6, 7.

892

G. M. Veksler and Yu. I. Kipriyanov, Author, s Certificate No. 165674, Byulleten, Izobretenii i Tovarnykh Znakov, No. 20 (1964). M. H. I. Baird, wWater blow pulsation,, Brit. Chem. Eng., 12, No. 12, 1877-1886 (1967), M. H. L Baird, A. R. Gloyne, and M.A.N. Meghani, ,Solven't-extraction in an air-pulsed packed columr.,, Can. J. Chem. Eng., 46, No. 8, 242-252 (1968). M I4:. I. Baird, ,A self-triggere'd-reSonant pulse colunm,, A. I. C h . E . -I. Chem. Eng. Symp, No. 6, 53-59 (1965). S, M. Karpacheva, E. I. Zakharov, S. M. Raginskii, and V. M. Muratov, Pulsating Extractors [in Russian1, Atomizdat, Moscow (1964), p. 298.