Sinusoidal excitation of viscous fluids in pipes

V/es. & Piping 70 (1997) 161-165 0 1997 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/97/$17.00

ht. J. Pm.

Printed PlkSO308-0161(96)00025-7

ELSEVIER

Sinusoidal excitation

of viscous fluids in pipes A. H. Elkholy

Mechanical

rind Industrial

Engineering

Department,

Kuwait

University,

P.O. Box 5969, 13060 Safat, Kuwait

(Received 21 February 1996; accepted 8 May 1996)

The concept of applying a prescribed oscillation to viscous fluids to aid or increase flow is examined. Application of this technique to fluids presents unique problems such as physical separation; control of heat and mass transfer in certain industrial applications; and improvement of some fluid process methods. The problem as stated is to obtain the velocity distribution, wall shear stress and energy expended when a pipe containing a stagnant viscous fluid is externally excited by a sinusoidal pulse, one end of the pipe being pinned. On the other hand, the effect of different parameters on the results are presented. Such parameters include fluid viscosity, frequency of oscillations and pipe geometry. It was found that the flow velocity is maximum at the pipe wall, and it decreases rapidly towards the pipe centerline. The frequency of oscillation should be above a certain value in order to obtain meaningful flow velocity. The amount of energy absorbed in the system is mainly due to pipe wall strain energy, and the fluid pressure and kinetic energies are comparatively small. 0 1997 Elsevier Science Ltd. All rights reserved.

chemical and physical processes has beneficial effects. Examples of these processes where such is the case are physical separation, liquid-liquid extraction and increase of heat and mass transfer. Many practical world-wide examples have been reported in the literature. The following are a few examples, quoted to show the wide range of application. Sonic coagulation of mists and fogs is practical in some casesId where this effect is being utilized to construct installations4 Byon et al. have investigated the breaking up of thermoplastic foams and determined the optimal frequency range for applications5 The application of controlled pulsations of several Hz has been used by Lea and Fozooni’ and Goshawk and Waters’ for the increase of efficiency in liquid-liquid extraction columns. It has been known from the work of Shuai et al.’ that natural convection could be increased by several hundred percent by application of vibrations of frequencies of the order of several tens of Hz. Cars and Grislis” found that ultrasonic vibrations improved heat transfer in viscous fluid flow by an appreciable amount. Mass transfer rates may also be improved, according to the work of Nakashima and One.” It has also been noted in the

NOMENCLATURE A, B, C, D 4, Bo

Cl, c2, Kl, K2, K3, K, ;

L, Kz, Jn i

e P(r), G(t), N(z)

Greek

Parameters defined by eqn (15) Velocity constants Constants of integration Sonic speed in pipe material Shear force Bessel functions

Square root of -1 Pipe length Functions of Y, t, z, respectively Radial, time and axial coordinates Flow velocity Flow velocity at pipe wall Parameters defined by eqn (14)

symbols

P

Ratio of frequency of oscihation to sonic speed Parameter defined in eqn (10) Dynamic viscosity Kinematic viscosity Shear stress Frequency of oscillation

processing of certain foods by a reverse osmosis that

osmosis permeation flux is improved by pulsation.“*‘2 This paper is concerned with the theoretical prediction of the velocity distribution, wall shear stress and energy dissipation for a pipeline containing a stagnant viscous fluid for which the boundary is

1 INTRODUCTION

It has been known for some time that the application of controlled oscillations or pulsations to many 161

162

A. H. Elkholy

subjected to a prescribed oscillation. The problem, as stated, is to investigate the possibility of using such a technique as a simple pump in which stagnant fluid is forced from one end of the pipe to the other end by means of oscillation. Two check valves are to be installed at each end of the pipe to prevent flow return during the second half of the oscillation cycle. This will result in fluid flow from one end of the pipe to the other end.

2 THEORETICAL

MODEL

For a horizontal pipeline containing a viscous fluid and subjected to small amplitude oscillations at one end of the pipe while holding the other end stationary, the boundary conditions at any instant are such that at z = 0,

u, = 0 and at z = f,

u, = B0 sin wt

(7)

where Y is the kinematic viscosity of the fluid and Y is measured from the centerline of the pipe. The problem resolves itself to finding a solution for eqn (7), subject to the boundary conditions in eqn (6). It will be noted that there is no initial condition to the problem since we shall only be concerned with the steady periodic part of the solution. 3 SOLUTION

(2)

where u, = velocity at (z, t) in z-direction; c = sonic speed in pipe material. The solution of eqn (2) by separation of variables is

OF MODEL

We seek a separation of variables solution of the form

(1)

The pipe wall velocity will be a solution to the wave equation: a*u 1 a*; a~---! dz2 c* at2

Navier-Stokes equation, after neglecting the convective terms, will be expressed in cylindrical coordinates as:

u(r, z, t) = AOP(r)eiwr sin pz

Which upon differentiation appropriately substitution in eqn (7) yields the following: ($+:z)A,e’w’sinpz

(8) and

+P(-P*)A,e’““sinpz -P

A,,eiwrsinPz ==0

or

u,(z, t) = H(z)G(t) where

where H(z)=K,sin[(~)z]+K2cos[(f)z]

(3)

G(r) = K, sin wt + K4 cos wt.

The first of the boundary conditions that u, = 0 at z = 0 implies that K2 = 0. The second condition u, = B. sin wt at z = e gives: K, sin

i.e.

(K3 sin wt + K4 cos wt) = B0 sin ot K4 = 0

and

B0 = KIK3 sin

(4)

Equation (9) is of a modified Bessel form: y= x2y” + xy’ - (x2 + n2)y = 0; with a solution CIZn(x) + C2Kn(r) with II = 0, 1, 2,3 thus P(r) = C,Z,(Ar) + C,K,(hr)

We note that at r = 0, K,(hr)+ 0~ and because P(r) is finite at r = 0, C2 = 0. The solution of eqn (9) becomes P(r) = C,Z,(hr)

Applying

and u, may be written as u, = A, sin(pz) . sin(&)

ay r=O

(12)

the boundary condition at r = a we obtain C, = l/Z,(ha)

(-3

where p = w/c. The boundary conditions for the fluid are thus: f3V

ZdAr) P(r) = ~ MAa)

and iwt

= 0 and

u, = A0 sin@z)sin(wt)

(11)

(6)

The flow in the pipe may be considered to be laminar due to the fact that the oscillations applied at the boundary are small. The study was done with the intention of applying this technique to fluids of general interest like water, milk, oil,. . . etc. These fluids are incompressible. Therefore, the

.

sin pz

I

(13)

Equation (13) is a unique solution as it satisfies the partial differential eqn (7) and the boundary conditions at r = a and at r = 0 where the equation will reduce to: ~(a, z, t) = A0 Im[elof sin pz] = A0 sin wt sin pz at r =.a

163

Sinusoidal excitation of viscous fluids

and Z;(k)

= hZl(Ar) = All(O) = 0 at

r=0

The behaviour of eqn (13) is qualitatively illustrated in Fig. 1. It will be noted that the diagram shows one mode for the boundary r = a, i.e. for the wave equation solution. This assumption implies that the wave propagation velocity is far greater than the excitation velocity of the boundary and that there is no interaction between reflected waves. In the range of pipe lengths and frequencies of practical interest this is a valid assumption. 4 NUMERICAL

EVALUATION

In order to evaluate eqn (13) it must be put into a more tractable form. Noting that Z,(k) =.l,(ihr) and

1 Wb) c=l+~-q~)‘+~,...,

(15c)

64 Y

D=$(;)+$(;)j32+,...,

(15d)

In order to extract the i maginary part of A + iB /C + iD we write

J,(x) = 1 - x’/4 +x4/64 - x”/2304 + , . . . ,

s]-‘+i$[l+iz]-’

Z,,(k) may be expanded

in the denominator.

in the numerator and Z,(ha) The result is as follows:

IdAr) erwrsin fiz = _____ w x w2 ____ I MAa) K

Expanding theorem

the term

n brackets

by the binomial

(14)

where

r4p4 --+,...,

64 W, = [cos wt sin Bz + i sin

I sin /?z]

Pt

+~[l-(~)2+(~)‘-(~~,...,I

(16)

For this expression to be convergent D c <1

A + iB (14) is of the form c___+ iD where

Equation

A=(coswtsinpz)

z=o ___--

____.t’.-&

II

The range of values of parameters investigated was: pipe radius, a = 0.025 m; frequency of oscillation w = 10, 20 and 30 Hz corresponding to 600, 1200 and 1800 rpm; Y = 1 X lop6 m2/s (water); pipe lengths: 10, 20 and 30m giving p = n/20, n/40, and n/60, respectively. It was found that sufficient accuracy was obtained by truncating the series for I,,( ) at the third term i.e. x4/64. A computer program was written to evaluate eqns (14) and (16). The results for different parameters are presented below.

OSCILLATION b

f

4.1 Effect of (W/Y)

Fig. I. Flow velocity parameters (r, t, z).

The results of (W/Y) = 1 X lo’, 1 X 107, 1 X lo6 and 1 x 10’ are presented for sin ot = 1; sin pz = 1; j3 = n/20; a = 0.0250 m; r = 0.0225 m in Fig. 2, for W/Y = 1.0 X 10’ to 1.0 x 108, (u/A,) is almost constant. In other words, the oscillation frequency could be changed by a factor of 100 or the kinematic viscosity changed by a factor of 100 with no

164

A. H. Elkholy

l.O-

0.6561 $ '

0.5-

cc-0.001

Fig. 2. Effect of (W/Y) on flow velocity.

X NEGATIVE

Cr 0.2

0.4

VELOCITIES

I 0.8

0.6

1.0

Fig. 4. Flow velocity variation along pipe length.

corresponding velocity change. For O/Y 5 1 X lo’, the flow velocity is negligibly small. This suggests that, in order to obtain a reasonable flow velocity, the frequency of oscillation must be above a certain value or the fluid viscosity be small. 4.2 Effect of (r/a)

A typical set of data for sin @z = 1, /3 = ~120 i.e. a 10 m long pipe of diameter = 0.05 m is plotted in Fig. 3. It is clear from the figure that the velocity is zero at the centreline of the pipe, and it gradually builds up radially. Typical values of u/AO, at r/a equals 0.2, 0.4, 0.6, 0.8, 1.0, are O-002, 0.026, 0.130, 0.410, 1.0; respectively. This shows that flow movement occurs mainly at the layers adjacent to the wall due to viscous effect. It is also clear that the flow will be moving forward and backward in one complete cycle according to pipe oscillation. If the flow movement is required to be in only one direction, then proper check valves should be installed at both ends of the pipe. Obviously, these valves should be chosen to be able to respond to the frequency of oscillation under consideration. 4.3 Effect of pipe length

The effect of pipe length is contained within the factor pz. The distribution of velocity along the pipe is

wt:270.

shown in Fig. 4 for r/a = 0.9. We note that the values are cyclic over the range (wt)O-360 repeating in magnitude between 180 and 360 but having negative values. At z = 0 (the pinned end of the pipe) the velocity is everywhere z&o. 4.4 Evaluation

of shear stress

Shear stress and total force exerted on the wall of the fluid may be determined by first determining the velocity gradient at each point along the boundary and summing the product of r=a

i.e.

F=kp$dA, 1

at

r=a

This

necessitates an accurate evaluation of 1 ure 5 shows velocity gradients close to F’g the wall at wt = 90”; pz = 90”, a = 0.025, W/Y = 1 X 107. In order to determine F accurately, a linear interpolation was used between the values of (u/A(,) corresponding to (r/a) = 1.0 and 0.99. This procedure was carried out at (r/a) = 0.99 along a pipe 10 m long for wt = 90” and wt = 0”. Each incremental area is dA = 0.157 m2. The results are plotted graphically in (duldr)l,,,.

,ot=O.l80,wt=90

T a

-1.0

vlAo c---t

+I.0

Fig. 3. Flow velocity variation across pipe radius.

(17)

Fig. 5. Flow velocity gradient close to wall.

16.5

Sinusoidal excitation of viscous fluids REFERENCES "E i

loo-

r' m B E In

/'

1. Davidson, G. A. and Jagar W., Turbulence and aerosol coagulation in high intensity sound fields. Journal of Sound and Vibration, 1980, 72(l), 123-126. 2. Schetter, A., Bernhard, B., Funcke, C. and Joachim, E., Aggolomeration of the disperse phase of aerosols by strong sound fields. Forschung im Zngenieurwesen, 1990,

,

lo-

3 5

r 0.2

0.4

0.6

0.8

1.0

z/l

Fig. 6. Shear stress variation along pipe length.

Fig. 6. The amount of energy may be calculated by integrating the value of Jpv*/2 dv. This gives the kinetic energy.

5 CONCLUSIONS

The system absorbs energy in two ways: (a) Strain energy in the walls of the pipe. This is the energy dissipated during the elastic deformation of the pipe during oscillation. The energy is readily caIcuIated from the resilience of the pipe material.

(b) Pressure and kinetic energy in the fluid. For example, in a pipe 10m long which is extended 1 cm (corresponding to a (v)~=~ maximum = 20.34 m/s which for steel is still within the elastic range (strain = 0.1%) the strain energy is calculated to be 716.9 N m. There is thus no question that the overwhelming amount of energy is contained in the pipe wall as strain energy. The superposition of such a boundary movement however has a marked effect on the velocity distribution in the fluid.

56(5)148-149. 3. Hnatkow, R., Increase of efficiency of cyclone by means of a powerful acoustic field. Acustica, 1987, 62(3), 229-23.5. 4. Statnikov, Y. G. and Shirokova, N. L., Calculation of the efficiency of acoustic aerosol coagulation. Soviet Physics, Acoustics, 1974, 20(l), 68-69.

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convective heat transfer by pulsed flow in a viscous fluid. Canadian Journal of Chemical Engineering, 1994 72(3), 468-475. 9. Cars, A. and Grislis, V., Heat transfer during surface vibration in viscous fluids. Latuijas PSR Zinatnu Akademijas Vestis, Fizikas un Tehnisko Zinatnu Ser., 1970, 1, 91-94.

10. Nakashima, M. and Ono K., A finite difference scheme for solving the incompressible viscous fluid flow around a vibrating body. Transactions of the Japan Society of Mechanical 766-772.

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