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Rotameter dynamics
ChemicalEngineeringScience,1964,Vol. 19, pp. 853-865.PergamonPress Ltd., Oxford. Printed in Great Britain.
Rotameter dynamics H. H. DIJSTELBERGEN Instrumentation
Laboratory,
Department
of Applied Physics, Technological
University, Delft, Holland
(Received 21 January 1964; in revised form 20 April 1964)
Abstract-An equation of motion is derived for the float in a rotameter. An approximate solution for sinusoidal pulsations is given. The transfer function resulting from this solution and the error in the indicated mean flow are compared with measurements and show good agreement. In particular it is shown that the error in the indicated mean value can sometimes be reduced by choosing suitable dimensions for the float. For gases, the occurrence of float instability (bouncing) is investigated. Measurements of this phenomenon are compared with theory and show good agreement.
the jet has to be taken into account. The contraction coefficient C, is defined as a,/a,,, where a, is the area of the narrowest part of the jet and aor the area of the orifice. In a rotameter with float
THE MANY rotameters
serving industry have proved to be simple reliable instruments capable of measuring flow of all kinds of fluids and gases with reasonable accuracy. Information about the influence of temperature and viscosity, range extension, accuracy etc., is readily available from manufacturers and the literature. Less is known, however, about the dynamic behaviour of the instrument. Although some publications [l, 2, 31 describe some of the aspects of the dynamic behaviour of rotameters, many questions remain unsolved. In particular, the error in the indicated mean value with pulsating flow and the cause of the effect known as bouncing can neither be looked up in textbooks nor be found in manufacturers leaflets. The aim of this article is to a certain extent to unveil these phenomenae.
For stationary incompressible flow one can easily derive the equation relating float position and volumetric flow from Bernoulli’s law: -
P2
=
M4
-
u:>.
I
FIG. 1, Flow pattern in a rotameter.
area a,, and tube area a0 + a, at float position x (Fig. l), the area of the narrowest cross section of the jet is thus equal to
THE EQUATION OF MOTION OF THE FLOAT
Pl
7 \
(1)
In this equation p is the pressure and o the velocity of the fluid in the two observed points 1 and 2, while p is the density of the medium. The influence of gravity on the fluid has been neglected. When using this equation to compute the pressure difference over a sharp edged orifice, the contraction of
a, = Ccaor = C cax* When a volumetric flow w is passing through the meter, the velocity in the jet will be given by W v2
=qy
The velocity upstream of the float will be much smaller than v2 and its square can be neglected compared with the square of the velocity in the jet.
853
H. H. DUWELBERGEN
With these approximations applied to (I), the pressure over the float can be written as Pl-Ppz=y
w 2
P (
Gx
1
(2)
*
The pressure difference over the float results in a force which is balanced by the weight of the float minus the buoyant force. When g is the acceleration of gravity, pri the density of the float and Vr, the volume of the float, equation (2) can be rewritten as
where (dv/dt), is the acceleration of the fluid measured in a point moving with the velocity 1 of the float and dz is an element of, for instance, a stream line connecting the two points 1 and 2. The two points 1 and 2 are fixed with respect to the float. The pressure difference over the float for the dynamic case can be computed from (6) in the same way as has been done for the stationary case. To compute the velocity relative to the float, first a steady, incompressible flow w is considered. The velocity in the jet will be
2
sh,
for stationary From (3)
-
PIG
=
pao 2
( ) -!Cc4
1 9
%(Pn- P)v,, =
J[
pa0
Kx
where a, = bx and K = C,b
pa0 1
%(Pf, - P)v,,
(4) v, =
-p2+u:-4)+P
dz
- i(a, - a,) (8)
(5)
With pulsating flow, the afore mentioned forces are not the only ones acting on the float. Inertia of the float has to be taken into account and Bernoulli’s law has to be modified. It is clear that when using equation (1) with a moving float the velocities should be taken with respect to the float. These velocities will be denoted by U. Another correction on (1) is needed to account for the extra pressure necessary to accelerate the fluid. A steady flow of an inviscid fluid in a straight tube causes no pressure drop. When the flow is accelerated, however, the pressure difference between two cross sections will be such as to give the force necessary to accelerate the fluid in this region. To account for this effect a term can be added to (1) making it applicable to pulsating flow. With rotameters, the situation is further complicated by the fact that the area where the pressures are investigated is moving together with the float. In an appendix it is shown that Bernoulli’s law then becomes: Pl
a,
Now consider the float moving downward with a velocity -1 in a tube with area a,, the net flow through the rotameter tube being zero. The velocity in the jet is now
flow and float.
w = a$,
V&L
(3)
(6)
because the total volume going through a section of the tube has to be zero. In general with a flow w and a moving float the resulting velocity in the jet will be v2 = v, + 0, =
w - k(a, - a,)
(9)
a, and the velocity relative to the float w - .ta,
u2=v2-ii:=-.
a,
(10)
As the tube is only slightly tapered, the tube diameter at the vena contracta will almost be the same as at the float edge so with and
a, = a, + a,
(9) and (10) can be written v2 =
w - i(a,
a, = Ccax
as + a, - CcaX) (11)
Ccax u2 =
w - R(a, + a-J Ccax
.
014
The flow pattern with pulsating flow is assumed to remain the same as with stationary flow. An elementary contribution to the integral in (6) will 854
Rotameter dynamics be proportional to the velocity. As the velocity of the fluid is largest in the vena contracta, the contribution of this part of the flow pattern to the integral will be largest and the integral can be approximated by
these conditions variations in density are also small, which means that the medium is almost incompressible. AN APPROXIMATE SOLUTIONOF THE EQUATIONOF MOTION Considering small harmonic variations in w
where v2 is the velocity in the vena contracta, and h is the length of a virtual fluid column giving the same acceleration pressures as needed in the actual case. From equation (11) we find if a, is neglected compared with a, and as a first approximation a, is taken as constant (13) The last term of (13) can be compared with the extra force required when accelerating for example a ship. Not only the ship has to be set in motion but also the fluid surrounding it. This quantity is usually called the virtual mass of a body. The force exerted on the float by the fluid has to be balanced by the forces due to gravity and inertia. (PI - P&o = dPn - p)v,,
(14)
++,,T/f1-
Combining equations (5), (6) and (1 la) to (14) and again neglecting the velocity upstream of the float in comparison with the velocity in the vena contracta, the equation of motion for the float is found to be Pl - Pz = !TP _
I +ph* 2
w-f(u,+bx)
PK2
2C2b2 c
C,bx I Pf,T/,,
$
uo
---= C,bx
w=w,+~cosot=w,+Aw
(16)
and assuming that the higher harmonics generated in x are negligible so that x = x0 + Z cos(ot + 4) = x0 + Ax,
(17)
an approximate solution of (15) can be obtained as follows. Multiplication of the last equality of (15) by 2C2b2x2 c P
and substitution of (16) and (17) give an equation containing terms with frequency w, time independent terms and terms with higher harmonics. The equation has to hold for both time dependent and time independent terms. It can be divided in two, one containing only the time independent terms and the other only the time dependent terms. When terms with higher harmonics and terms with products of more than two of the small quantities R and J?. are neglected, these equations are found to be: 2w,(Aw) - 2w,(u, + bx,)(A$ + 2C,bhx,(AG) - 2C,u,bhx,(AR)
pu,hR C,bx
2C2b2
- Cpuo /+&,&A~)
- 2K2xo(Ax) = 0 (15)
(18)
for the time dependent parts and
when for a, is substituted
K2Xi =
fi2 K2R2 w;+ -y- -yy +
a, = bx. Although the derivation of (15) has been restricted to incompressible flow it can be shown [4, p. 46491 that to a first approximation this equation holds also for compressible flow when the variations in the pressure are small compared with the absolute pressure. This is not so remarkable because under
+ co2 (a, + bx,)’ i
+ 2C,u,bh
+
x g + w(2(uo + bx,) + 2C,bh) T sin 4 (19) for the time independent parts.
855
H. H. DIJSTELBERGEN
As in this case the coefficient s is mostly small, the gain shows a maximum at the resonant frequency o. = l/Jt. Equations (21) and (5) give with (Pr* - P) = Pfl
Using complex variables Ax and Aw in (18) and supposing that to a first approximation we = Kx, the transfer function of the system can be written as
wo=
HzKAX
l+jwq - Aw = 1 + jos + (jo)2t
(20)
C bh K
a, + bxo s = s1 + s&3 = K t =
t1 +
t,x, =
Ccaobh
+
(21) C,Zb’pr,F,xo
K2
*
paoK
In a second approximation the mean value of the flow as indicated by the rotameter will differ from the true mean value. The error in the mean rotameter reading can be described by M =
K2x2 - w2 Ao2 ‘, W
dyqs
- t>2 + 2m4t2xoq2
+ W2(2t2Xo+ 42) . (1 - oZt)2 + 0J2S2 (23)
When q2 Q t,x, as is the case in most instances when x0 is not too small and q 4 s the last equation can be approximated by d(qs
M = +
- ty + 2&,X,
(1 - &)2
+ &2
*
(24)
For gases a further approximation is possible when p Q pfl. For these cases qs & t and t, 4 t,x, w4t2 + 2&
M=$
(1 - c&)2 + WV
(25)
and --= ILK&
Aw
1 1 + jws + (jc~)~t
(27)
BOUNCING
Kx, being the indicated value and w. the actual value of the mean flow. (This factor M is approximately equal to half the frequency factor b in Head’s pulsation threshold [ 11.) With (19), (20), (21) and (22) M can be written as M=$
\xol
independent of rotameter dimensions. As q is mostly small it can be seen from (20) that the float behaves to a first approximation as a second order system. As will be shown in the paragraph “measurements” the response of the float ranges from heavily damped for high density media such as fluids to underdamped for gases. For fluids the term q can obscure the second order character. The error in the rotameter reading as given by (22) and (23) shows that for high frequencies the error goes to a limit determined by the value of the coefficients q, s and t. As will be shown in the paragraph “measurements” it is possible in some cases to reduce the value of the error by choosing suitable values of q, s and t.
with q=c-
29 ii - i
v
(26)
Although for compressible flow the same equation of motion holds as for incompressible flow, a rotameter used with a compressible, low density medium may show an effect which is absent for high density incompressible fluids. This effect is known as bouncing, a spontaneous oscillating movement of the float without any forced pulsation of the flow. The occurrence of bouncing is strongly influenced by the pipes connected to the meter. Therefore, the rotameter and the elements attached to it will be described in terms of impedance, i.e. the ratio of the variations of pressure difference and flow rate. This gives all the necessary information about the stability of the system. If there is a closed loop in the system containing a source of energy and having a loop impedance with a negative real part it will be unstable. As dashpots are frequently used to prevent float bouncing, the impedance will be computed for a float having an external damping mechanism attached to it. Only the lower end of the frequency band will be considered, where wavelengths are large compared with the dimensions of the system.
856
Rotameter dynamics For such a rotameter expressed as (~1 - 0,
the force on the float can be
= pnl/f,g + P,,V,+ + 7%
Eliminating
= (.,i~)2~,v;&
+ MYAX.
(29)
(jw)2
and
;-
of Z, is drawn
K b&J
PflVfl -.
K
(33)
(30)
From (32) it can be seen that bouncing is most likely to occur for low float positions. Now consider a rotameter without external damping and with a restriction upstream. The part of the rotameter tube between the float and restriction will represent a capacitance
I
0
diagram
y >
=
CI~= ‘s.
0
+
a0
jcobl~ + (jo)2 a2 = 1jjwS + (joI)%
ImZf
can be written
The term on the right side of (32) is largest for x0 = 0 (the lowest float position) so bouncing will be completely suppressed for
s
Ap1 - APZ _ Z = a& 0 f Aw 1 + jos + (jo)2t
with tlI = +K
Pf,l/n a0
(Ax) from (26) and (29) leads to JWY I
The polar
y >
(28)
where y signifies the damping of the dashpot. For small harmonic variations Apr, Ap2 and Ax (ApI - AP&,
With (21) and (30) this condition as
CL
kp
in Fig. 2.
with V the volume, k the specific heat ratio, and p the mean pressure. When the pressure downstream of the rotameter is constant (the gas for example flowing freely into the atmosphere), the electrical analogue of the circuit can be given by Fig. 3 where R represents the restriction.
1
(1
b
=f t
R
FIG. 2. Polar diagram of the impedance Zf.
FIG. 3. Analogue of a rotameter with a restriction upstream from the float.
When the real part of Zf is negative, instability can occur under certain circumstances. An interesting question is what value of the damping coefficient has to be chosen to prevent bouncing for every possible circuit. For the real part of 2, can be written
The resistance of the restriction is considered to be very large so its influence can be neglected. Opening the circuit at points a and b and computing the impedance between these points gives
a2tw2 Re
Zf
=
(1
_
w2t)2
a2 + a1s +
&2
‘a,=
2
CT0 *
(31)
For a2 c als the real part of Zf is always positive so no bouncing is possible.
1 + jos + ( jw)2t + ( jw)3a2C (1 + jos + (jw)2t}joC
*
(35)
Considering the Hurwitz conditions for the numerator of (35) it is found that stability is assured for
st = a,C > 0 857
(36)
H. H.
With (5), (21), (30) and (34) this can be written
DIJSTELBERGEN
as (37)
when again the approximations used on the end of the preceding paragraph are made. In general the occurrence of bouncing is favoured by low pressures, small float area, large float mass, and small internal resistance of the pipes attached to the rotameter. MEASUREMENTS
Measurements have been carried out with water and with air using several tubes and floats.
-l i waste
-+-cl
mot or
puisatcr
FIG. 4. Test apparatus for measuring sinusoidal varying flow of an incompressible fluid.
The water circuit is shown schematically in Fig. 4. From a reservoir the water is conducted to cock I, which is used to adjust the mean value of the flow. A pulsator with a metal bellows generates sinusoidal variations. The pulsations are measured with a magnetic flowmeter MF which is connected to the rotameter R. The switch S connects the flow during a certain interval to a measuring vessel. This time interval is measured by means of an electronic counter, which counts a 1000 c/s voltage generated by a tuning fork oscillator. The phase of the float is measured by a light beam being intercepted by the float and thus modulating the output of a phototransistor. All tubes are manufactured by Fischer and Porter. Most of the floats were made at the workshop according to the geometrical specifications of original Fischer and Porter floats but with different materials. In Table 1 the four principal combinations are mentioned. The values of q, s and f mentioned in Table 1 are computed from the float dimensions and the measured value of the rotameter constant K. The value of h will be in first approximation equal to the distance between float head and vena contracta. Visual inspection of the flow pattern (by means of small air bubbles injected into the water) did not show a sharply defined place of the vena contracta. It appeared, however, to be in the region of the top of the float. For this reason the thickness of the float head, i.e., the distance between float edge and float top was taken for h.
Table 1. The four rotameters
tested with water
I
II
III
TLlbe Tube diameter, cm Float model Materials of float
%21-10 1.3 BSVT 44 stainless steel
Mass of float, kg Mean density of float, kg/Km3 Flowmeter constant K, mZ/s Area of float, rn2 4, set sl, set ~2, set/m fl, sec2 t2, sec2/m
14.3 x 10-3 8020 207 x 10-G 1.3 x 10-d 0.51 x IO-2 063 1.05 0.2 x 10-2 6.4 x 1O-2
B4-21-10 1.3 BSVT 44 aluminium and perspex 5.4 x 10-3 3030 111 x 10-p 1.3 x 10-4 I.0 x 10-S 1.2 2.0 0.7 x 10-2 8.9 x 10-2
B5-21-25170 2.0 BSVT 54 aluminium and perspex 7.25 x 1O-3 2090 236 x 10-e 3.25 x 10-4 1.2 x 10-2 1.4 3.0 1.2 x 10-Z 11 x 10-2
858
IV B3-27-10170 0.94 stainless steel and lead 19,3 x 10-3 10690 213 x lo-” 0.7 x 10-4 0.16 x 10-Z 0.33 0.80 004 x 10-a 5.6 x 10-a
Rotameter dynamics -----cfX 5
1
2
5
10
20
0 lIHI
8 6
I
L
.
.
g=r wO
0 .
II
XO=ZOcm d
. \h
\ !\O\
,
FIG. 5. The amplitude
characteristics and II.
of rotameter
I
-60
$)o -80
I
-100
-120
0.5
i
2
10
5
23
-f%
FIG. 6. The phase characteristics
859
of rotameters I and II.
H. H. DIISTELBERGEN
lqs
The measured and computed amplitude characteristics for I and II are shown in Fig, 5. The phase characteristics are given in Fig. 6. For high frequencies the theoretical curve returns to -90”. At least for combination II the measurements confirm this to a useful accuracy. The theoretical curve with q = 0 is shown dotted for II. Comparison of the last curve with the actual measured values and with the solid curve, shows that when the acceleration term in Bernoulli’s law (6) is taken into account the phase characteristics can be very well explained. The chosen value of h appears to be a rather good guess in this case. The error in the indicated mean value is measured as a function of frequency. As a typical example the measured and computed values of M (22), (23) are reproduced in Fig. 7. These measurements are made with tube-float arrangement I at a mean position x0 = 20 cm. From equation (24) it is found that for high frequencies
M=2
2
( t-’
and making qs/t close to 1 would result in a small error. To get an idea if improvement could be obtained, the three float-tube arrangements I, III and IV are compared in Table 2. The values of M,,, (measured) in this table are the estimated asymptotic values from the measured curves (see for example Fig. 7). Table 2. Comparison of the errors resulting from pulsating Jrow for three rotameters Tube-float arrangement x0 qs 7 calculated for -=
xomax Mmax (calculated) Mm, (measured)
relative error v
SOpercent
for z
= 1
0.25M
I
I) ’
0.20-
0.1 !j-
. -
computed
curve
0.1,O-
0.015-
OL . .
FIG. 7. The value of M as a function of frequency for rotameter I.
860
I
III
IV
0.29
0.71
0.067
025 0.22
0042 0.04
043 0.42
Rotameter
dynamics
By inspection of this table it can be seen that for these three rotameters with approximately the same capacity the maximum error ranges from 2 per cent to 20 per cent depending on float and tube dimensions. The value of the error appears to be rather closely predictable. The quantity governing the value of the error for high frequencies can be deduced from (21) and (24) qs
a&h + a,bphx,
- t = a&h + C,bp,,V&
IHI s 3s
(38)
Air flow measurements were carried out with two different circuits, one for a maximum flow rate of about 0.03 m3/s, the other for a flow rate of about 0.5 x 10e3 m3/s. The circuits will be referred to as the one for high flow rate and the one for low flow rate respectively. The circuit for high flow rate is indicated in Fig. 8. V is a blower, G a generator consisting of a disc which is periodically opening and closing the pipe end. At H a hot wire anemometer is mounted, which is used to measure the instantaneous value
FIG. 8. The measuring circuit for high flow rate.
C
FIG.9. The measuring circuit for low flow rate.
of the flow. Its output is linearized to be able to use its mean indication with pulsating flow to measure the actual mean flow. The circuit for low flow rate is shown in Fig. 9. The compressor C is connected to a volume V,. A pressure controller P keeps the pressure in the volume V, at a constant value. A very high restriction R1 maintains the flow into the rest of the circuit constant. The resistances Rz and R3 are small compared with R, but large compared with the impedance of the rotameter and following equipment. The flow is divided in two branches. One branch leads to the rotameter; the other to the sine generator S. The sine generator consists of a piece of rubber tube with some foam plastic inserted in it. The tube is periodically pinched by an eccentrically mounted wheel. The amplitude, phase and wave form of the pulsations are measured with a hot wire anemometer H connected to an oscilloscope. The pulsations obtained turned out to be reasonably sinusoidal. By adjusting R, and R, the division of the flow over the two branches and the amplitude of the flow pulsations in the rotameter can be adjusted. The mean flow is measured by a positive displacement meter of the wet gasmeter type. Between the rotameter and the gasmeter two vessels are inserted to give some damping of the pulsations.
861
H. H.
DIJSTELBERGEN
0
measurements
T
0
measurements
PI
-
computed
curveX
FIG. 10. The amplitude characteristics and VI.
of rotameters
V
0
-20 -LO rp” I
-60 measurement
-100 -80
5
-120 -140
-180 10-l
2
5
100 -
FIG.
2
5 f
11. The phase characteristic
862
10’
2
%
o rotameter
VI.
5
102
Rotameter dynamics Two of the float tube combinations are listed in Table 3. Amplitude and phase characteristics are Table 3. The two rotameters
tested with air V
Tube
B9-27-10 37.8 x lo+ 5.46 x 10-Z 20.7 x 10-4 3.7 x 10-Z 9.0 x 10-Z 5.1 x 10-Z
Mass of float, kg Flowmeter constant K, m2/s Area of float, m2 Sl, set ~2, xc/m t2, seS/m
Q and fo. The limit of M for high frequencies which is theoretically equal to 1)is assumed, taking into account a possible measuring error in M of 0.2-0.4 for V, and 1 for VI. The bouncing theory was checked with the apparatus shown in Fig. 13. The effective volume of the vessel can be varied by filling the vessel with water to different levels. According to the theory the stability equation (37) holds for this case. The measurement is performed in the following way: The vessel V is filled up to a certain level and the pressure p is adjusted to give maximum flow through the rotameter. Now the pressure upstream of the restriction S is diminished so that the float position becomes lower. At a certain position x0 the float will become unstable. When measuring with Fischer and Porter beaded glass tubes conditional stability was observed. When a small disturbance was introduced the float would start bouncing. Theory and measurements agreed only qualitatively for these rotameters.
VI B2-l&127/70 1.2 x 1O-3 0.19 x 1O-2 0.33 x 1O-4 1.7 x 10-Z 5.7 x 10-Z 5.1 x 10-z
given in Figs 10 and 11. The measurements agree fairly well with the curves computed from (20) up to the resonant frequency. For rotameter VI the agreement is not so good for high frequencies. No explanation could be found for this behaviour. The deviation of the mean value, M, is shown in Fig. 12. The theoretical value of M is computed from (24). The values of s and t used in this equation are derived from the measured values of 3
t
.
measurements
P
0
measurements
xr
-
computed
curve
p
---
computed
C”r”e
pI
FIG. 12. The value of M as a function of frequency for rotameters V and VI.
863
H. H. DIJSTELBERGEN
culated. This is due to the fact that at x = 0 the diameter of the tube is somewhat larger than the diameter of the float head. It turns out that the actual value of x where the diameter of the float is equal to the diameter of the tube is at x = - 1.1 cm. The theoretical curve corrected for the zero shift is shown as a dotted line. CONCLUSIQN
-------
I------II----
z----I-
_-:I
---
FIG. 13. The measuring circuit for the bouncing experiments.
Measurements with a plain glass tube manufactured by Rota (type No. 32978-54) showed better agreement. The three beads in the Fischer and Porter tubes act as guides for the float. The float in these meters is equipped with a tail piece having the same diameter as the float head. The float can thus make contact with the tube at several points at a time, whereas in the plain Rota tube the float only makes contact with the tube at one point. Probably this accounts for the observed difference in behaviour between these two meter types. The measured and calculated values of x0 (37) for the last rotameter are given in Fig. 14. The measured values are always lower than those cal-
The measurements show that the equation of motion for the float as derived in this article, describes the actual behaviour of the float to a good degree of accuracy. The response of the instrument and the value of the error in the mean float position with pulsating flow can be predicted. The latter can sometimes be reduced by choosing suitable dimensions for tube and float. The occurrence of “float bouncing” can also be explained by the differential equation. Acknowledgement-The author is deeply indebted to Prof. C. J. D. M. VERHAGENfor encouraging this work and making numerous comments and suggestions. He is also indebted to A. P. SCHNEIDERS, Prof. J. F. URY, J. H. KIUMERS and H. v. d. BIGGELAARwho left a valuable test installation and many reports to the author, and to RACHMAD MOHAMAD,J. DE GREAT and F. KOOPMANS,who all made many contributions to the measurements and theory for compressible media.
/
far
the
zero
shift
/. ‘
/ / /
2
t/
O/ /, ’
I
and calculated
l
/x
0
FIG. 14. Measured
x l
-1
2
4
6
8
-
10
12
x0 (computed)
1L
16
18
cm
values of the float position where the float becomes unstable for several float masses and capacitances.
864
Rotameter dynamics APPENDIX
NOTATION
Euler’s equation for inviscid flow can be written as
-vp av -=T$+(v*v)v.t
a0
UC
ax
(Al)
P
at
aor b c C, g h H k K M
This equation only holds for &/at measured at a fixed point. We are interested in dvldt at a point, moving with the velocity v’, the velocity of the float. can be written as V)v.
Pl. Pa API. APPZ 9 s Sl 82 t fl
(A2)
Euler’s equation in this case is + {(v - v’) *v>v. Y’ Substitution of v - v’ = u leads to
-VP -=
(A3)
t2
U
dv dt
0+
P
+
Ul, u2 V VI, v2
(II *V)u,
%I
(A4)
I)8
because v’ is independent of x, y, z. For curl free fields we can write
V’
V vn
(u *V)u = p(w).
w
(W
wo
Substituting equation (A5) in (A4) gives
B
Aw X
(A6)
x0 f
Integrating along an arbitrary line from point 1 to point 2, moving with velocity v’, we find for incompressible fluids Pl -
P2 = M4
-
4)
+ P
*dl
AX Z
AiJzl
(A7)
al aa
For the assumed one dimensional flow this leads to (6).
5
P Pfl w
t Vectors are printed in bold type,
Area of float = area of tube at x = 0 Area of the narrowest part of the jet Difference of tube. area and float area at x Area of tube = ao + az Area of orifice as/x change in tube diameter with position Capacitance Contraction coefficient = se/a,, Acceleration of gravity Virtual height of accelerated fluid column Transfer function of rotameter Specific heat ratio Rotameter constant Deviation of the mean value Pressure at point 1, 2 Small harmonic variation in p at point 1,2 Coefficient in solution of the equation of motion Coefficient in solution of the eqaution of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Velocity relative to the float (vector) Velocity relative to the float in point 1, 2 Velocity of the fluid (vector) Velocity in point 1, 2 Velocity in the jet with moving float and zero net flow Velocity in the jet with stationary float and flow w Velocity of the float (vector) Volume upstream of float Volume of float Volumetric flow Mean value of flow with pulsations Amplitude of sinusoidal flow pulsations Small harmonic variation in w Float position Mean float position with pulsating flow Amplitude of sinusoidal variation in float position Small harmonic variation in x Vertical coordinate fixed to the float Impedance of float Impedance between points a and b Coefficient in float impedance equation Coefficient in float impedance equation Damping of float Phase of float Density of medium Density of float Angular frequency
REFERENCES [l] [2] 131 [4]
HEAIJ V. P., Trans. Amer. Sot. Mech. Engrs 1956 78 1471. URY J. F., Bull. Res. Count. Istraef 1960 80 119. II ARRIWN G. S. and ARMSTRONG W. D., Chem. Engng Sci. 1960 12 253. DLJSTELBERGEN H. H., The dynamic behaviour of rotameters, doctor’s thesis Delft University 1963.
865
Rotameter dynamics H. H. DIJSTELBERGEN Instrumentation
Laboratory,
Department
of Applied Physics, Technological
University, Delft, Holland
(Received 21 January 1964; in revised form 20 April 1964)
Abstract-An equation of motion is derived for the float in a rotameter. An approximate solution for sinusoidal pulsations is given. The transfer function resulting from this solution and the error in the indicated mean flow are compared with measurements and show good agreement. In particular it is shown that the error in the indicated mean value can sometimes be reduced by choosing suitable dimensions for the float. For gases, the occurrence of float instability (bouncing) is investigated. Measurements of this phenomenon are compared with theory and show good agreement.
the jet has to be taken into account. The contraction coefficient C, is defined as a,/a,,, where a, is the area of the narrowest part of the jet and aor the area of the orifice. In a rotameter with float
THE MANY rotameters
serving industry have proved to be simple reliable instruments capable of measuring flow of all kinds of fluids and gases with reasonable accuracy. Information about the influence of temperature and viscosity, range extension, accuracy etc., is readily available from manufacturers and the literature. Less is known, however, about the dynamic behaviour of the instrument. Although some publications [l, 2, 31 describe some of the aspects of the dynamic behaviour of rotameters, many questions remain unsolved. In particular, the error in the indicated mean value with pulsating flow and the cause of the effect known as bouncing can neither be looked up in textbooks nor be found in manufacturers leaflets. The aim of this article is to a certain extent to unveil these phenomenae.
For stationary incompressible flow one can easily derive the equation relating float position and volumetric flow from Bernoulli’s law: -
P2
=
M4
-
u:>.
I
FIG. 1, Flow pattern in a rotameter.
area a,, and tube area a0 + a, at float position x (Fig. l), the area of the narrowest cross section of the jet is thus equal to
THE EQUATION OF MOTION OF THE FLOAT
Pl
7 \
(1)
In this equation p is the pressure and o the velocity of the fluid in the two observed points 1 and 2, while p is the density of the medium. The influence of gravity on the fluid has been neglected. When using this equation to compute the pressure difference over a sharp edged orifice, the contraction of
a, = Ccaor = C cax* When a volumetric flow w is passing through the meter, the velocity in the jet will be given by W v2
=qy
The velocity upstream of the float will be much smaller than v2 and its square can be neglected compared with the square of the velocity in the jet.
853
H. H. DUWELBERGEN
With these approximations applied to (I), the pressure over the float can be written as Pl-Ppz=y
w 2
P (
Gx
1
(2)
*
The pressure difference over the float results in a force which is balanced by the weight of the float minus the buoyant force. When g is the acceleration of gravity, pri the density of the float and Vr, the volume of the float, equation (2) can be rewritten as
where (dv/dt), is the acceleration of the fluid measured in a point moving with the velocity 1 of the float and dz is an element of, for instance, a stream line connecting the two points 1 and 2. The two points 1 and 2 are fixed with respect to the float. The pressure difference over the float for the dynamic case can be computed from (6) in the same way as has been done for the stationary case. To compute the velocity relative to the float, first a steady, incompressible flow w is considered. The velocity in the jet will be
2
sh,
for stationary From (3)
-
PIG
=
pao 2
( ) -!Cc4
1 9
%(Pn- P)v,, =
J[
pa0
Kx
where a, = bx and K = C,b
pa0 1
%(Pf, - P)v,,
(4) v, =
-p2+u:-4)+P
dz
- i(a, - a,) (8)
(5)
With pulsating flow, the afore mentioned forces are not the only ones acting on the float. Inertia of the float has to be taken into account and Bernoulli’s law has to be modified. It is clear that when using equation (1) with a moving float the velocities should be taken with respect to the float. These velocities will be denoted by U. Another correction on (1) is needed to account for the extra pressure necessary to accelerate the fluid. A steady flow of an inviscid fluid in a straight tube causes no pressure drop. When the flow is accelerated, however, the pressure difference between two cross sections will be such as to give the force necessary to accelerate the fluid in this region. To account for this effect a term can be added to (1) making it applicable to pulsating flow. With rotameters, the situation is further complicated by the fact that the area where the pressures are investigated is moving together with the float. In an appendix it is shown that Bernoulli’s law then becomes: Pl
a,
Now consider the float moving downward with a velocity -1 in a tube with area a,, the net flow through the rotameter tube being zero. The velocity in the jet is now
flow and float.
w = a$,
V&L
(3)
(6)
because the total volume going through a section of the tube has to be zero. In general with a flow w and a moving float the resulting velocity in the jet will be v2 = v, + 0, =
w - k(a, - a,)
(9)
a, and the velocity relative to the float w - .ta,
u2=v2-ii:=-.
a,
(10)
As the tube is only slightly tapered, the tube diameter at the vena contracta will almost be the same as at the float edge so with and
a, = a, + a,
(9) and (10) can be written v2 =
w - i(a,
a, = Ccax
as + a, - CcaX) (11)
Ccax u2 =
w - R(a, + a-J Ccax
.
014
The flow pattern with pulsating flow is assumed to remain the same as with stationary flow. An elementary contribution to the integral in (6) will 854
Rotameter dynamics be proportional to the velocity. As the velocity of the fluid is largest in the vena contracta, the contribution of this part of the flow pattern to the integral will be largest and the integral can be approximated by
these conditions variations in density are also small, which means that the medium is almost incompressible. AN APPROXIMATE SOLUTIONOF THE EQUATIONOF MOTION Considering small harmonic variations in w
where v2 is the velocity in the vena contracta, and h is the length of a virtual fluid column giving the same acceleration pressures as needed in the actual case. From equation (11) we find if a, is neglected compared with a, and as a first approximation a, is taken as constant (13) The last term of (13) can be compared with the extra force required when accelerating for example a ship. Not only the ship has to be set in motion but also the fluid surrounding it. This quantity is usually called the virtual mass of a body. The force exerted on the float by the fluid has to be balanced by the forces due to gravity and inertia. (PI - P&o = dPn - p)v,,
(14)
++,,T/f1-
Combining equations (5), (6) and (1 la) to (14) and again neglecting the velocity upstream of the float in comparison with the velocity in the vena contracta, the equation of motion for the float is found to be Pl - Pz = !TP _
I +ph* 2
w-f(u,+bx)
PK2
2C2b2 c
C,bx I Pf,T/,,
$
uo
---= C,bx
w=w,+~cosot=w,+Aw
(16)
and assuming that the higher harmonics generated in x are negligible so that x = x0 + Z cos(ot + 4) = x0 + Ax,
(17)
an approximate solution of (15) can be obtained as follows. Multiplication of the last equality of (15) by 2C2b2x2 c P
and substitution of (16) and (17) give an equation containing terms with frequency w, time independent terms and terms with higher harmonics. The equation has to hold for both time dependent and time independent terms. It can be divided in two, one containing only the time independent terms and the other only the time dependent terms. When terms with higher harmonics and terms with products of more than two of the small quantities R and J?. are neglected, these equations are found to be: 2w,(Aw) - 2w,(u, + bx,)(A$ + 2C,bhx,(AG) - 2C,u,bhx,(AR)
pu,hR C,bx
2C2b2
- Cpuo /+&,&A~)
- 2K2xo(Ax) = 0 (15)
(18)
for the time dependent parts and
when for a, is substituted
K2Xi =
fi2 K2R2 w;+ -y- -yy +
a, = bx. Although the derivation of (15) has been restricted to incompressible flow it can be shown [4, p. 46491 that to a first approximation this equation holds also for compressible flow when the variations in the pressure are small compared with the absolute pressure. This is not so remarkable because under
+ co2 (a, + bx,)’ i
+ 2C,u,bh
+
x g + w(2(uo + bx,) + 2C,bh) T sin 4 (19) for the time independent parts.
855
H. H. DIJSTELBERGEN
As in this case the coefficient s is mostly small, the gain shows a maximum at the resonant frequency o. = l/Jt. Equations (21) and (5) give with (Pr* - P) = Pfl
Using complex variables Ax and Aw in (18) and supposing that to a first approximation we = Kx, the transfer function of the system can be written as
wo=
HzKAX
l+jwq - Aw = 1 + jos + (jo)2t
(20)
C bh K
a, + bxo s = s1 + s&3 = K t =
t1 +
t,x, =
Ccaobh
+
(21) C,Zb’pr,F,xo
K2
*
paoK
In a second approximation the mean value of the flow as indicated by the rotameter will differ from the true mean value. The error in the mean rotameter reading can be described by M =
K2x2 - w2 Ao2 ‘, W
dyqs
- t>2 + 2m4t2xoq2
+ W2(2t2Xo+ 42) . (1 - oZt)2 + 0J2S2 (23)
When q2 Q t,x, as is the case in most instances when x0 is not too small and q 4 s the last equation can be approximated by d(qs
M = +
- ty + 2&,X,
(1 - &)2
+ &2
*
(24)
For gases a further approximation is possible when p Q pfl. For these cases qs & t and t, 4 t,x, w4t2 + 2&
M=$
(1 - c&)2 + WV
(25)
and --= ILK&
Aw
1 1 + jws + (jc~)~t
(27)
BOUNCING
Kx, being the indicated value and w. the actual value of the mean flow. (This factor M is approximately equal to half the frequency factor b in Head’s pulsation threshold [ 11.) With (19), (20), (21) and (22) M can be written as M=$
\xol
independent of rotameter dimensions. As q is mostly small it can be seen from (20) that the float behaves to a first approximation as a second order system. As will be shown in the paragraph “measurements” the response of the float ranges from heavily damped for high density media such as fluids to underdamped for gases. For fluids the term q can obscure the second order character. The error in the rotameter reading as given by (22) and (23) shows that for high frequencies the error goes to a limit determined by the value of the coefficients q, s and t. As will be shown in the paragraph “measurements” it is possible in some cases to reduce the value of the error by choosing suitable values of q, s and t.
with q=c-
29 ii - i
v
(26)
Although for compressible flow the same equation of motion holds as for incompressible flow, a rotameter used with a compressible, low density medium may show an effect which is absent for high density incompressible fluids. This effect is known as bouncing, a spontaneous oscillating movement of the float without any forced pulsation of the flow. The occurrence of bouncing is strongly influenced by the pipes connected to the meter. Therefore, the rotameter and the elements attached to it will be described in terms of impedance, i.e. the ratio of the variations of pressure difference and flow rate. This gives all the necessary information about the stability of the system. If there is a closed loop in the system containing a source of energy and having a loop impedance with a negative real part it will be unstable. As dashpots are frequently used to prevent float bouncing, the impedance will be computed for a float having an external damping mechanism attached to it. Only the lower end of the frequency band will be considered, where wavelengths are large compared with the dimensions of the system.
856
Rotameter dynamics For such a rotameter expressed as (~1 - 0,
the force on the float can be
= pnl/f,g + P,,V,+ + 7%
Eliminating
= (.,i~)2~,v;&
+ MYAX.
(29)
(jw)2
and
;-
of Z, is drawn
K b&J
PflVfl -.
K
(33)
(30)
From (32) it can be seen that bouncing is most likely to occur for low float positions. Now consider a rotameter without external damping and with a restriction upstream. The part of the rotameter tube between the float and restriction will represent a capacitance
I
0
diagram
y >
=
CI~= ‘s.
0
+
a0
jcobl~ + (jo)2 a2 = 1jjwS + (joI)%
ImZf
can be written
The term on the right side of (32) is largest for x0 = 0 (the lowest float position) so bouncing will be completely suppressed for
s
Ap1 - APZ _ Z = a& 0 f Aw 1 + jos + (jo)2t
with tlI = +K
Pf,l/n a0
(Ax) from (26) and (29) leads to JWY I
The polar
y >
(28)
where y signifies the damping of the dashpot. For small harmonic variations Apr, Ap2 and Ax (ApI - AP&,
With (21) and (30) this condition as
CL
kp
in Fig. 2.
with V the volume, k the specific heat ratio, and p the mean pressure. When the pressure downstream of the rotameter is constant (the gas for example flowing freely into the atmosphere), the electrical analogue of the circuit can be given by Fig. 3 where R represents the restriction.
1
(1
b
=f t
R
FIG. 2. Polar diagram of the impedance Zf.
FIG. 3. Analogue of a rotameter with a restriction upstream from the float.
When the real part of Zf is negative, instability can occur under certain circumstances. An interesting question is what value of the damping coefficient has to be chosen to prevent bouncing for every possible circuit. For the real part of 2, can be written
The resistance of the restriction is considered to be very large so its influence can be neglected. Opening the circuit at points a and b and computing the impedance between these points gives
a2tw2 Re
Zf
=
(1
_
w2t)2
a2 + a1s +
&2
‘a,=
2
CT0 *
(31)
For a2 c als the real part of Zf is always positive so no bouncing is possible.
1 + jos + ( jw)2t + ( jw)3a2C (1 + jos + (jw)2t}joC
*
(35)
Considering the Hurwitz conditions for the numerator of (35) it is found that stability is assured for
st = a,C > 0 857
(36)
H. H.
With (5), (21), (30) and (34) this can be written
DIJSTELBERGEN
as (37)
when again the approximations used on the end of the preceding paragraph are made. In general the occurrence of bouncing is favoured by low pressures, small float area, large float mass, and small internal resistance of the pipes attached to the rotameter. MEASUREMENTS
Measurements have been carried out with water and with air using several tubes and floats.
-l i waste
-+-cl
mot or
puisatcr
FIG. 4. Test apparatus for measuring sinusoidal varying flow of an incompressible fluid.
The water circuit is shown schematically in Fig. 4. From a reservoir the water is conducted to cock I, which is used to adjust the mean value of the flow. A pulsator with a metal bellows generates sinusoidal variations. The pulsations are measured with a magnetic flowmeter MF which is connected to the rotameter R. The switch S connects the flow during a certain interval to a measuring vessel. This time interval is measured by means of an electronic counter, which counts a 1000 c/s voltage generated by a tuning fork oscillator. The phase of the float is measured by a light beam being intercepted by the float and thus modulating the output of a phototransistor. All tubes are manufactured by Fischer and Porter. Most of the floats were made at the workshop according to the geometrical specifications of original Fischer and Porter floats but with different materials. In Table 1 the four principal combinations are mentioned. The values of q, s and f mentioned in Table 1 are computed from the float dimensions and the measured value of the rotameter constant K. The value of h will be in first approximation equal to the distance between float head and vena contracta. Visual inspection of the flow pattern (by means of small air bubbles injected into the water) did not show a sharply defined place of the vena contracta. It appeared, however, to be in the region of the top of the float. For this reason the thickness of the float head, i.e., the distance between float edge and float top was taken for h.
Table 1. The four rotameters
tested with water
I
II
III
TLlbe Tube diameter, cm Float model Materials of float
%21-10 1.3 BSVT 44 stainless steel
Mass of float, kg Mean density of float, kg/Km3 Flowmeter constant K, mZ/s Area of float, rn2 4, set sl, set ~2, set/m fl, sec2 t2, sec2/m
14.3 x 10-3 8020 207 x 10-G 1.3 x 10-d 0.51 x IO-2 063 1.05 0.2 x 10-2 6.4 x 1O-2
B4-21-10 1.3 BSVT 44 aluminium and perspex 5.4 x 10-3 3030 111 x 10-p 1.3 x 10-4 I.0 x 10-S 1.2 2.0 0.7 x 10-2 8.9 x 10-2
B5-21-25170 2.0 BSVT 54 aluminium and perspex 7.25 x 1O-3 2090 236 x 10-e 3.25 x 10-4 1.2 x 10-2 1.4 3.0 1.2 x 10-Z 11 x 10-2
858
IV B3-27-10170 0.94 stainless steel and lead 19,3 x 10-3 10690 213 x lo-” 0.7 x 10-4 0.16 x 10-Z 0.33 0.80 004 x 10-a 5.6 x 10-a
Rotameter dynamics -----cfX 5
1
2
5
10
20
0 lIHI
8 6
I
L
.
.
g=r wO
0 .
II
XO=ZOcm d
. \h
\ !\O\
,
FIG. 5. The amplitude
characteristics and II.
of rotameter
I
-60
$)o -80
I
-100
-120
0.5
i
2
10
5
23
-f%
FIG. 6. The phase characteristics
859
of rotameters I and II.
H. H. DIISTELBERGEN
lqs
The measured and computed amplitude characteristics for I and II are shown in Fig, 5. The phase characteristics are given in Fig. 6. For high frequencies the theoretical curve returns to -90”. At least for combination II the measurements confirm this to a useful accuracy. The theoretical curve with q = 0 is shown dotted for II. Comparison of the last curve with the actual measured values and with the solid curve, shows that when the acceleration term in Bernoulli’s law (6) is taken into account the phase characteristics can be very well explained. The chosen value of h appears to be a rather good guess in this case. The error in the indicated mean value is measured as a function of frequency. As a typical example the measured and computed values of M (22), (23) are reproduced in Fig. 7. These measurements are made with tube-float arrangement I at a mean position x0 = 20 cm. From equation (24) it is found that for high frequencies
M=2
2
( t-’
and making qs/t close to 1 would result in a small error. To get an idea if improvement could be obtained, the three float-tube arrangements I, III and IV are compared in Table 2. The values of M,,, (measured) in this table are the estimated asymptotic values from the measured curves (see for example Fig. 7). Table 2. Comparison of the errors resulting from pulsating Jrow for three rotameters Tube-float arrangement x0 qs 7 calculated for -=
xomax Mmax (calculated) Mm, (measured)
relative error v
SOpercent
for z
= 1
0.25M
I
I) ’
0.20-
0.1 !j-
. -
computed
curve
0.1,O-
0.015-
OL . .
FIG. 7. The value of M as a function of frequency for rotameter I.
860
I
III
IV
0.29
0.71
0.067
025 0.22
0042 0.04
043 0.42
Rotameter
dynamics
By inspection of this table it can be seen that for these three rotameters with approximately the same capacity the maximum error ranges from 2 per cent to 20 per cent depending on float and tube dimensions. The value of the error appears to be rather closely predictable. The quantity governing the value of the error for high frequencies can be deduced from (21) and (24) qs
a&h + a,bphx,
- t = a&h + C,bp,,V&
IHI s 3s
(38)
Air flow measurements were carried out with two different circuits, one for a maximum flow rate of about 0.03 m3/s, the other for a flow rate of about 0.5 x 10e3 m3/s. The circuits will be referred to as the one for high flow rate and the one for low flow rate respectively. The circuit for high flow rate is indicated in Fig. 8. V is a blower, G a generator consisting of a disc which is periodically opening and closing the pipe end. At H a hot wire anemometer is mounted, which is used to measure the instantaneous value
FIG. 8. The measuring circuit for high flow rate.
C
FIG.9. The measuring circuit for low flow rate.
of the flow. Its output is linearized to be able to use its mean indication with pulsating flow to measure the actual mean flow. The circuit for low flow rate is shown in Fig. 9. The compressor C is connected to a volume V,. A pressure controller P keeps the pressure in the volume V, at a constant value. A very high restriction R1 maintains the flow into the rest of the circuit constant. The resistances Rz and R3 are small compared with R, but large compared with the impedance of the rotameter and following equipment. The flow is divided in two branches. One branch leads to the rotameter; the other to the sine generator S. The sine generator consists of a piece of rubber tube with some foam plastic inserted in it. The tube is periodically pinched by an eccentrically mounted wheel. The amplitude, phase and wave form of the pulsations are measured with a hot wire anemometer H connected to an oscilloscope. The pulsations obtained turned out to be reasonably sinusoidal. By adjusting R, and R, the division of the flow over the two branches and the amplitude of the flow pulsations in the rotameter can be adjusted. The mean flow is measured by a positive displacement meter of the wet gasmeter type. Between the rotameter and the gasmeter two vessels are inserted to give some damping of the pulsations.
861
H. H.
DIJSTELBERGEN
0
measurements
T
0
measurements
PI
-
computed
curveX
FIG. 10. The amplitude characteristics and VI.
of rotameters
V
0
-20 -LO rp” I
-60 measurement
-100 -80
5
-120 -140
-180 10-l
2
5
100 -
FIG.
2
5 f
11. The phase characteristic
862
10’
2
%
o rotameter
VI.
5
102
Rotameter dynamics Two of the float tube combinations are listed in Table 3. Amplitude and phase characteristics are Table 3. The two rotameters
tested with air V
Tube
B9-27-10 37.8 x lo+ 5.46 x 10-Z 20.7 x 10-4 3.7 x 10-Z 9.0 x 10-Z 5.1 x 10-Z
Mass of float, kg Flowmeter constant K, m2/s Area of float, m2 Sl, set ~2, xc/m t2, seS/m
Q and fo. The limit of M for high frequencies which is theoretically equal to 1)is assumed, taking into account a possible measuring error in M of 0.2-0.4 for V, and 1 for VI. The bouncing theory was checked with the apparatus shown in Fig. 13. The effective volume of the vessel can be varied by filling the vessel with water to different levels. According to the theory the stability equation (37) holds for this case. The measurement is performed in the following way: The vessel V is filled up to a certain level and the pressure p is adjusted to give maximum flow through the rotameter. Now the pressure upstream of the restriction S is diminished so that the float position becomes lower. At a certain position x0 the float will become unstable. When measuring with Fischer and Porter beaded glass tubes conditional stability was observed. When a small disturbance was introduced the float would start bouncing. Theory and measurements agreed only qualitatively for these rotameters.
VI B2-l&127/70 1.2 x 1O-3 0.19 x 1O-2 0.33 x 1O-4 1.7 x 10-Z 5.7 x 10-Z 5.1 x 10-z
given in Figs 10 and 11. The measurements agree fairly well with the curves computed from (20) up to the resonant frequency. For rotameter VI the agreement is not so good for high frequencies. No explanation could be found for this behaviour. The deviation of the mean value, M, is shown in Fig. 12. The theoretical value of M is computed from (24). The values of s and t used in this equation are derived from the measured values of 3
t
.
measurements
P
0
measurements
xr
-
computed
curve
p
---
computed
C”r”e
pI
FIG. 12. The value of M as a function of frequency for rotameters V and VI.
863
H. H. DIJSTELBERGEN
culated. This is due to the fact that at x = 0 the diameter of the tube is somewhat larger than the diameter of the float head. It turns out that the actual value of x where the diameter of the float is equal to the diameter of the tube is at x = - 1.1 cm. The theoretical curve corrected for the zero shift is shown as a dotted line. CONCLUSIQN
-------
I------II----
z----I-
_-:I
---
FIG. 13. The measuring circuit for the bouncing experiments.
Measurements with a plain glass tube manufactured by Rota (type No. 32978-54) showed better agreement. The three beads in the Fischer and Porter tubes act as guides for the float. The float in these meters is equipped with a tail piece having the same diameter as the float head. The float can thus make contact with the tube at several points at a time, whereas in the plain Rota tube the float only makes contact with the tube at one point. Probably this accounts for the observed difference in behaviour between these two meter types. The measured and calculated values of x0 (37) for the last rotameter are given in Fig. 14. The measured values are always lower than those cal-
The measurements show that the equation of motion for the float as derived in this article, describes the actual behaviour of the float to a good degree of accuracy. The response of the instrument and the value of the error in the mean float position with pulsating flow can be predicted. The latter can sometimes be reduced by choosing suitable dimensions for tube and float. The occurrence of “float bouncing” can also be explained by the differential equation. Acknowledgement-The author is deeply indebted to Prof. C. J. D. M. VERHAGENfor encouraging this work and making numerous comments and suggestions. He is also indebted to A. P. SCHNEIDERS, Prof. J. F. URY, J. H. KIUMERS and H. v. d. BIGGELAARwho left a valuable test installation and many reports to the author, and to RACHMAD MOHAMAD,J. DE GREAT and F. KOOPMANS,who all made many contributions to the measurements and theory for compressible media.
/
far
the
zero
shift
/. ‘
/ / /
2
t/
O/ /, ’
I
and calculated
l
/x
0
FIG. 14. Measured
x l
-1
2
4
6
8
-
10
12
x0 (computed)
1L
16
18
cm
values of the float position where the float becomes unstable for several float masses and capacitances.
864
Rotameter dynamics APPENDIX
NOTATION
Euler’s equation for inviscid flow can be written as
-vp av -=T$+(v*v)v.t
a0
UC
ax
(Al)
P
at
aor b c C, g h H k K M
This equation only holds for &/at measured at a fixed point. We are interested in dvldt at a point, moving with the velocity v’, the velocity of the float. can be written as V)v.
Pl. Pa API. APPZ 9 s Sl 82 t fl
(A2)
Euler’s equation in this case is + {(v - v’) *v>v. Y’ Substitution of v - v’ = u leads to
-VP -=
(A3)
t2
U
dv dt
0+
P
+
Ul, u2 V VI, v2
(II *V)u,
%I
(A4)
I)8
because v’ is independent of x, y, z. For curl free fields we can write
V’
V vn
(u *V)u = p(w).
w
(W
wo
Substituting equation (A5) in (A4) gives
B
Aw X
(A6)
x0 f
Integrating along an arbitrary line from point 1 to point 2, moving with velocity v’, we find for incompressible fluids Pl -
P2 = M4
-
4)
+ P
*dl
AX Z
AiJzl
(A7)
al aa
For the assumed one dimensional flow this leads to (6).
5
P Pfl w
t Vectors are printed in bold type,
Area of float = area of tube at x = 0 Area of the narrowest part of the jet Difference of tube. area and float area at x Area of tube = ao + az Area of orifice as/x change in tube diameter with position Capacitance Contraction coefficient = se/a,, Acceleration of gravity Virtual height of accelerated fluid column Transfer function of rotameter Specific heat ratio Rotameter constant Deviation of the mean value Pressure at point 1, 2 Small harmonic variation in p at point 1,2 Coefficient in solution of the equation of motion Coefficient in solution of the eqaution of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Coefficient in solution of the equation of motion Velocity relative to the float (vector) Velocity relative to the float in point 1, 2 Velocity of the fluid (vector) Velocity in point 1, 2 Velocity in the jet with moving float and zero net flow Velocity in the jet with stationary float and flow w Velocity of the float (vector) Volume upstream of float Volume of float Volumetric flow Mean value of flow with pulsations Amplitude of sinusoidal flow pulsations Small harmonic variation in w Float position Mean float position with pulsating flow Amplitude of sinusoidal variation in float position Small harmonic variation in x Vertical coordinate fixed to the float Impedance of float Impedance between points a and b Coefficient in float impedance equation Coefficient in float impedance equation Damping of float Phase of float Density of medium Density of float Angular frequency
REFERENCES [l] [2] 131 [4]
HEAIJ V. P., Trans. Amer. Sot. Mech. Engrs 1956 78 1471. URY J. F., Bull. Res. Count. Istraef 1960 80 119. II ARRIWN G. S. and ARMSTRONG W. D., Chem. Engng Sci. 1960 12 253. DLJSTELBERGEN H. H., The dynamic behaviour of rotameters, doctor’s thesis Delft University 1963.
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