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Nuclear magnetic resonance (NMR) two-phase mass flow measurements
Flow Meas. Instrum.. Vol. 7, No. I. pp. 25-37. 1996 Copyright (B 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 09555986I96 $15.00 + 0.00
PIT: so955-598q%)0ooo5-2
Nuclear magnetic resonance (NMR) phase mass flow measurements G. J. Kriiger, Experimental
Fluid
two-
A. Birke and R. Weiss Dynamics
European Commission,
Sector,
21020
Received 28 November
Safety Technology
lspra (VA),
1994;
Institute,
Joint Research Centre (Ispra),
Italy
in revised
75 November
form
7 995
An NMR measurement method for highly-turbulent liquid-gas two-phase flow has been developed in this laboratory [Kriiger, G.J., Haupt, j. and Weiss, R. A nuclear magnetic resonance method for the investigation of two-phase flow. In Measuring Techniques in GasLiquid Two-phase Flows (eds J.M. Delhaye and C. Cognet). Springer, Berlin, 1984, p. 4351. It allows measurement of the liquid velocity and of the fraction, both averaged over the inner volume of the NMR RF coil and over the measuring time. By signal-averaging, it is possible to extend the averaging time to mins, or even hrs or days. This time-averaging improves the signal relative to background noise as well as to fluctuations caused by the flow and hence improves the accuracy of the measurements. The influence of insufficient mixing of the spins during the polarization period is discussed. Measurements with this method have been executed in turbulent water flow and in waternitrogen two-phase flow. The calibration procedure for the velocity vNMR and liquid fraction ?? NMK is done with turbulent water all-liquid flow. The velocities ranged from 0.1 to 5 m/s and the in two-phase flow, which is liquid fraction from ca 0.5 to 1 .O. The mass flow Mf,,, proportional to the product vNMK times ?? NMK, indicates an accuracy of better than 2 5%. Copyright 0 1996 Elsevier Science Ltd.
Kevwords: nuclear magnetic resonance; two-phase prdbability distributioi; noise; fluctuat/ons ’
flow:
mass flow;
M,,
Nomenclature
iWL B,,
BP BRF 4
C SN d
EFC
f h I KC JRC k L Lfe LP
M Mi MC,
inner
cross-section
air-water
test
area of the flow
tube
MF
loop
field for NMR measurements static magnetic field for polarization magnetic RF field half the amplitude of BRF constant part of equation (25) [equation
MF, MF, MF-r
(27)l
MFMAR
static
magnetic
m
inner diameter of the flow tube efflux curve noise figure of receiver input Planck’s constant divided by 2rr nuclear spin iso-speed curve joint Research Centre Boltzmann’s constant length of NMR RF coil effective length of NMR RF field effective length of polarizing magnetic field in flow direction nuclear paramagnetic magnetization parallel to BO [equation (1)l magnetization of spins with velocity vi [equation (18)l thermal equilibrium value of M [equation
m. moo m O(”= 0) mOO,,=o) mOj m,,,,=,, N Ni N, Nl Nh.4
NMR
(1 )I 25
velocity
thermal
equilibrium
value
of Mj [equation
(1 WI sometimes used for liquid mass flow, mass flow gas mass flow liquid mass flow liquid mass flow determined by turbine flow meter liquid mass flow determined by NMR nuclear magnetization perpendicular to B,, [equation (311 maximum value of m at liquid-gas twophase flow maximum value of m at liquid flow without gas value of m, at zero velocity value of moo at zero velocity value of m. of spin ensemble with velocity v, [equation (7)l value of mOi at zero velocity number of spins in one cm3 of matter number of spins with velocity v, in one cm3 of matter number of spins in one cm3 of phase j number of spins in one cm3 of liquid number of measurements nuclear magnetic resonance
26
G. /. Krijger
number of velocities in a velocity or efflux time distribution quality factor of the RF resonance coil circuit radio frequency RF signal-to-noise ratio lequation (25)) SIN time-averaged signal-to-noise ratio lequ(SIN)‘,, ation (33)] signal-to-fluctuation ratio Iequation (34)l signal-to-fluctuation ratio time-averaged [equation (35)l absolute temperature in K spin-lattice relaxation time [equation (I )I spin-lattice relaxation time of phase j lequation (25)l spin-spin relaxation time [equation (.3)l efflux time [equations (4)-(6)l one efflux time of an efflux time distribution average efflux time of an efflux time distribution time acquisition time of signal-averaging polarization time [equation (1)I polarization time of spin ensemble with velocity \‘, Iequation (18)l inner volume of RF coil total volume of substances in RF coil (all phases together) velocity velocity of one-phase liquid flow [eyuation (16)j one velocity of a velocity distribution velocity of phase j velocity determined by NMR velocity determined by turbine flow meter average velocity of a velocity distribution average velocity of one-phase liquid flow magnetogyric ratio (frequently called gyromagnetic ratio in literature) relative error of Q (Table 2) any fraction of bulk matter liquid fraction gas (void) fraction [equation (1311 fraction of phase j liquid fraction determined by velocity v~,~,~ [equation (39)l ?? NMKE E,,,,) liquid fraction determined by magnetization m,, [equations (12), (16) and (17)l proton resonance frequency in B,, UlI proton resonance frequency in B,, vl’ Au bandwidth of the receiver with the detector static nuclear paramagnetic susceptibility x0 [equation (1 )I same of spin ensemble with velocity v, X~l, [equation (19)l N,
Q
1. Introduction Nuclear magnetic resonance (NMR) is a long-established method for the investigation of molecular structures. Besides this, it has been widely used as a powerful tool for the investigation of molecular motions, the principles of which are given by Abragam’. These motions may be all sorts of diffusive motions, but they can also be coherent motions of the
et al nuclear spins and hence of the spin bearing molecules, e.g. in all types of flow. A number of NMR flow measurement methods has been developed’. Since they work only through magnetic transitions in the energy levels of the nuclear spin system, which is very weakly coupled to all the other degrees of freedom in bulk matter, they do not disturb the flow. Recently some NMR results on two-phase flow have been reported l,J. An NMR method particularly useful in highlyturbulent, high-velocity, two-phase flow has been developed in the Magnetic Resonance Laboratory of the joint Research Centre (JRC) Ispra’. It will form the basis for our discussions of NMR flow investigations in two-phase water-gas flow. For the discussion of NMR flow measurements we consider an arrangement as shown schematically in Figure 1. The fluid passes from left to right first through a polarizing magnetic field B,, and then through a measuring magnetic field B,,. The fluid consists of only water or of a water-nitrogen mixture. The spins spend on average the polarization time t,, in the field until they reach the RF coil in the centre of the magnet. The nuclear magnetization M in direction of B,, is then given by M = M,, II pexp (pt,JT,)l = xoB,, I1 -exp
(-t,,lT,)l
where M,, is the equilibrium very long t,, and ,y,, = N-/h’/
(1) magnetization
reached at
(/+I )/(3kT)
(2)
is the static paramagnetic susceptibility with N being the number of spins I of the fluid per cm’ and y is their magnetogyric ratio, h is Planck’s constant divided by 2~r, k Boltzmann’s constant and T the absolute temperature. T, is the spin-lattice relaxation time of the nuclear spins in the fluid. In our RF coil the eight kW transmitter produces J magnetic RF field BKF with amplitude 2B, as usual (Figure 1). We use a Carr-Purcell-Gill-Meiboom pulse sequence’,“. The envelope of the spin-echo sequence created by this is, without flow, given by
m(t) = m,, exp (-tlT,) with the spin-spin
2.
Principles
relaxation
C.3) time T?.
of the measurement
method
Under flow conditions, the spins originally excited by the 90” pulse, which form the nuclear magnetization perpendicular to the B,, field, will leave the RF coil with the flow velocity. The number of transversely polarized spins in the RF coil, therefore, will diminish as time passes. Neglecting relaxation effects, the echo envelope will decay in this case in an efflux curve EFC (tiT,) due to the efflux of spins out of the RF coil’. The characteristic efflux time Tk is inversely proportional to the average velocity of the fluid and is used to determine the latter’. Now taking relaxation effects into consideration again, we write the echo envelope of the transverse magnetization m(t) = m,, EFC (tlTF) exp (-tlT,)
Nuclear magnetic resonance
27
(NMU)
I
Polarizing
Measuring
Magnet
Magnet
I ,
7 Schematic
of NMR flow
= M, [l -exp
(--&IT,)1
measurement
-
apparatus
EFC (Ur)
at a flow
(4)
exp (--t/T11
Transmitter Receiver
tube
N”
mo=Cmo;
(8)
r=l
It is easy to correct a measured curve for spin-spin this correction relaxation4. For fast flow with TE<
(5)
We have measured this curve by pulling a sealed piece of the flow tube filled with water with constant velocity through our RF coil. It must be determined in a similar way for each specific coil arrangement and then serves as a calibration curve for the measurement of unknown flows5. The iso-speed curve ISC (t/T,) allows determination of the efflux time TE5. We define an effective length Lr, of the 6, field, for which v = LfJTE
(6)
This effective field length is somewhat shorter than the physical length of the RF coil and must be determined by calibration experiments with known velocities4z5. In a real flow situation we have a distribution of velocities and a corresponding distribution of iso-speed curves. For the evaluation we use a discrete distribution of N, velocities vi with corresponding magnetizations m,, and efflux times TE,. So we rewrite the flow part of equation (4)
and the average efflux
time
N”
= x
mo,
TE;/mo
= 5
mo; ISC (t/Tr;)
(7)
(9)
r=l
The average velocity < v> = L,/<
is then
TE>
(10)
The magnetizations m,, form the TEi distribution and, according to equation (6), also the velocity probability distribution. Since, the total magnetization m, is proportional to the total number of spins [equations (1) and (211 and hence to the mass of fluid in the RF coil, we obtain for the mass flow MF=
m,,
(11)
The evaluation procedure has been outlined in our earlier papers4,5. Our method is straightforward. It results in the velocity probability distribution and its mean values. It does not need additional gradient pulses of the B,, field as have been used by other authors for twophase flow measurement?. In the case of liquid-gas two-phase flow, where the gas is normally not observable by NMR due to too small a number of spins in it, equation (1 1) is still valid for the liquid flow. We must then measure the total magnetization moo without gas in a tube full of liquid preferably at the same velocity (v) and obtain the liquid fraction El = molm,,
m(t) = m, EFC (d)
NMR Pulse Soectro‘meter
.‘.
A-
Figure
Computer Controlled
(12)
and the void (gas) fraction as usual
i=l
with the total 90” pulse
magnetization
immediately
after the
Eg = 1 -El In this
case, (T,),
(13) (v) and ?? l are averaged over the
C. 1. Krtiger et al.
28
liquid volume in the RF coil. If m,,,, cannot be measured at the same velocity we use equation (l), with the polarization time (14)
t,1= L,/(v)
where L,, is the effective length of the polarizing field, which must be determined by a calibration experiment for each experimental arrangement. We therefore obtain m (ii, = 01 - mu/11 -exp
11.5)
I-L,,l((v)T,)It
and
11-exp I --L,ll((v,,)T,)Il m,,,,II -exp I-L,] CT,)ll nh
E, =
(lb)
with (v,,) being the mean velocity ot the no,,,, measurement without gas. lf this measurement is done without flow, i.e. (v,,) = 0, equation (16) reduces to El =
/JJ,,/{rJJ,,,,
11
- eX(l
(--L,J((Vi
r,,,]]
1171
The above arguments are only valid, ii the l’low is highly turbulent, to such an extent that during the polarization time t,, as well as during the efflux time r, all the spins get sufficiently mixed in $uch a way that they experience all possible velocities of the distribution in the same way. In strictly laminar flow, where each subensemble of spins maintains its velocity \‘, during the wholr> measurement time, equation (4) ot the i”’ subensemhle will read i 18)
M, = M,,, I I pexp ( t,,,j7-, j =x,,, B,, II pexp I~~L,J(\:7;)IJ where xi,) = N,, -!A’/ (/+ I ,/i 3kn
i
1‘1)
and N, is the number of spins in the sub-ensemble with velocity v, in the fluid per volume. The total magnetization is equal to that given in equation (I) (LO)
M,, = 2. M,,, = B,, Xx,), The velocity distribution function at as above is now a distribution of M, of m,,, used in equations (8) and obtain meaningful mean values we distribution by ? Y
rJJ,,,,,..,,t = MA 1 -exp
= MA1 -exp
distinct r,, values = IM,/ rather than (9). In order to must correct our
I -L,J(v,T, )I I
I~LT;,(L,,,T,)Il
(11)
Equations (8) and (1.5) become identical and change to
/=I Equation (9) remains valid if we replace m,,, by m,,,,,,,, without gas is obtained in and ml1 by ml,,,=,,,. Q,,,,=_,,, the same way as m,,,,+_,,, and with these values reduced to zero velocity we obtain: El = nJo,,,~~,,/nJ,,~,,,0,
(2.3)
as above. In two-phase flow, we normally have highly turbulent flow, and we can use our original efflux time distribution function, the mean values of which are
given by equations (8) and (9). We denote these values now by subscript (OR) (original distribution). In the (ase of low velocities, however, it can happen that the mixing of the spins during the polarization time is not sufficient, even if the flow is still turbulent. It is obvious that this problem occurs at any NMR flow measurement method whatsoever. It does not depend on the details of the method and is more pronounced if the liquid has a higher viscosity, e.g. oilJ. We denote the T,-corrected values for laminar flow as obtained by equations (9) and (22) by subscript (CR) (T,-corrected distribution). Now we introduce a percentage of mixwhich is 100% for highly turbulent flow and ing P,,,, zero for laminar flow. Using this we can get quite a good approach by setting the final value V;II f,,, = P,,, VJL,,,,
+ (1 --P,,,) Vclk( K,
(24)
where Va/ stands for m,,,, ,),, (Tt) or tl. P,,, must he determined experimentally by test and calibration experiments over a sufficiently wide range of flow situations and c-an then be taken into consideration during the computer evaluation of measured results of a specific flow situation. With our RF coil, which has a length of IO cm, we obtain efflux times T, ranging from 1 s to 20 ms at average velocities of cd 0.1 to 5 m/s. One single measurement results, therefore, in mean values of velocity and liquid fraction which are averaged over this short time r, only. If average values over a longer period of time under pseudo stationary conditions are required, the measurement must be repeated during this time of, e.g. one minute (the averaging time is, of course, not limited to one minute; NMR averaging can be done over very long periods of time of up to several days, if necessary, without problems). Then
noise and
The precision of NMR measurements is determined by the instrumental noise’. In highly turbulent flow, however, we have fluctuations due to the turbulence, and in two-phase flow there are additional fluctuations of the liquid dnd gas fractions due to the free interface between the phases. The determination of the total magnetization and the average efflux time, or else the average velocity
Nuclear
magnetic
according to equations (8)~(lo), is straightforward. It is a type of self-consistent measurement, the accuracy of which depends only on the signal to fluctuation ratio of the initial magnetization m, according to equation (8) at the end of the 90” pulse and on the fluctuations of the efflux time TE. Under the assumption of several phases j with their respective fractions in the fluid, which are defined analogously to the liquid fraction, and following the treatment of Abragam’ we obtain the signal-to-noise ratio of the NMR measurement:
(25) 1 = 7 Cs, 2
?? ,N, (1 -exp
[-Lp/(vj~~,)ll
/ where we assumed proton resonance with CSN=-
_
(27)
??? ? = noise figure of the receiver input,
N,
?? Q
T,, K ?? VT
.
??
?? Vj ?? 6, ??
vo
??
VP
?? Av
INMR)
; =f
29
c,,
L, c
(29)
F,
/ 11
i ; = f CSNL, ~
#I
(30)
T,
This approximation works sufficiently well over the whole range of velocities larger than 2 m/s and in the total temperature range of interest. If we omit the polarizing magnet (Figure l), a further simplification takes place. The polarization is in this case accomplished whilst the fluid is flowing through the first half of the measuring magnet towards the RF coil. The length of the polarizing field L, lies in the order of somewhat more than half the length of the measuring magnet in the flow direction. It depends on the geometry of this magnet and must be determined by calibration experiments at different velocities. For the polarizing field we find B, = B0 and hence v,,= vO. Thus we can write
I= 112 and
which is constant for one particular measurement. In addition to the constants and variables already discussed, we have here:
??
resonance
= number of nuclear spins per cm3 of phase j, = quality factor of the NMR resonance coil circuit, = spin-lattice relaxation time of phase j, = volume of the RF coil, = volume of the substance in the RF coil (all phases together), = average velocity of phase j, = fraction of phase j, resonance frequency of the nuclear spins in me measuring field BO, = resonance frequency of the nuclear spins in the polarizing field B,, = bandwidth of the receiver with the detector.
Under the conditions expected for two-phase watergas flow, the gas will not practically contribute to the sum due to its very low value of Nj. For only one liquid phase in two-phase flow, the sum will, therefore, practically reduce to only one term and equation (26) becomes
(28) where is now the average velocity of the liquid, T, its spin-lattice relaxation time, N, the number of spins in one cm3 of liquid and ?? l the liquid fraction. If we assume the liquid to be incompressible, S/N will not depend on the pressure. If this were not the case, the pressure would show up in N, only. This is also true for a mixture of several liquids with their respective values Nj For high velocities the exponential can be replaced by the first two terms of the power series, and equations (26) and (28) reduce to
(31) Equation (30) shows further
s
N
1
cSN_
T,
T312T,
that
(32)
if we leave all the other parameters constant. Since T, of water is well known in dependence of temperature’, we can calculate SIN in dependence of temperature and velocity for different experiments, provided it has been measured for one specific case. Moreover we find that CsN, and hence the signal-to-noise ratio, is proportional to the volume of substance V, in the coil. Thus with large flow tube diameters we get high S/N ratios even if we have high velocities and hence short polarization times. Examples will be given in the next section. The signal-to-noise ratio can be further improved by signal averaging over a longer period of time. As already mentioned in Section 2, this is done by adding together measured efflux curves point by point. All the parameters of the measurement must be kept constant during the acquisition time t,. If NM measured signals are added together, the signal-to-noise ratio of the averaged signal becomes
s
s
.“=N d(-1 N
NM
(33)
where S/N is the signal-to-noise ratio of one single measurement. In highly-turbulent flow we get additional fluctuations of the NMR signal due to velocity fluctuations. This will be even more the case for two-phase or multiphase flows, because the liquid fraction or the fractions of single components and their velocities will undergo fluctuations too. In such conditions we can distinguish two principally different cases: first, we may have pseudo-stationary conditions in one-phase liquid flow or in twophase flow in such a way that the liquid fraction remains the same throughout our whole measuring section (e.g. stratified or annular flow). In such cases one single measurement will not be affected in prin-
30
(;. 1. Kriiger
ciple. It can be evaluated in the normal way ds outlined in Section 2. There will be, however, additional fluctuations due to the flow and these can be much larger than the noise of the NMR experiment as such. A second case arises if we have a separation and hence fluctuations of the phases in the flow direction. In strongly pulsating flow this can lead to situations where our measuring section is at times completely full of liquid or at other times completely full of gas. So we will have strong fluctuations as large as our whole NMR signal. In all these cases it is difficult to calculate the signal-to-fluctuation ratio. It will strongly depend on the geometry and on the statistics of the flow. As a very crude first approach for liquid-gas two-phase flow we assume 34) where (E,) and (E,,) are the time averages of liquid fraction and gas (void) fraction, respectively. Equation (34) seems quite reasonable for E,: 5 0.5 and hence S/F 2 1. In the limiting case Ed= 0, where strong pulsation fluctuations do not exist, S/F becomes very large and the measuring accuracy is determined by the much weaker fluctuations of turbulent one-phase liquid flow and, eventually, by the signal-to-noise ratio S/N of the NMR spectrometer. Equation (34) will probably be less meaningful for E,: > 0.5. In the limiting case ?? g = 1 and cl = 0 it shows, however, that S/F= 0, and, indeed, measurement of the liquid is no longer possible. The signal-to-fluctuation ratio can be improved in the same way as the signal-to-noise ratio by time averaging of a large number of measurements N,,, where the efflux curves are added together point by point. In the case of random fluctuations we obtain
=
S-
F\NM
in analogy to equation (33). This procedure works perfectly in pseudo-stationary one-phase liquid flow and also in two-phase flow with stationary liquid fraction as defined above. In strong pulsation two-phase flow it works only if this flow on a large time scale is also pseudo-stationary and the measurements are taken in such a way that all the possible liquid fractions and velocities show up with their respective probabilities. In this case the average efflux curve gives the average flow conditions, whereas single efflux curves may not be very meaningful and can eventually only show single flow situations on a short time scale. The evaluation of time-averaged efflux curves is done exactly in the same way as that of one single EFC, only the total magnetization amplitude m,, obtained by equation (8) must afterwards be divided by NM in order to cope with the amplitude of a single measurement. 4. 4.1.
Results of NMR
flow
measurements
The NMR apparatus
Measurements have been executed using the method described so far in this paper. The NMR spectrometer
et al was a computerized BRUKER CXP Spectrometer with a solenoidal wide-bore (20 cm) measuring magnet manufactured by BRUKER ANALYTISCHE MESSTECHNIK CMBH upon our request and specifications. The arrangement at the flow tube was similar to Figure 1, but the polarizing magnet had been omitted for simplicity, which reduces the nuclear magnetic polarization and hence the sensitivity. The solenoid of the measuring magnet had a length of 40 cm in the flow direction. The homogeneity of the B,, field in the centre was ca 10 ’ over a sphere of 10 cm diameter. The RF coil had a certain similarity with a saddle coil. It consisted of six windings arranged round the flow tube in such a way as to obtain a sufficient homogeneity of the B, field over a large part of the inner crosssection of the tube”. Its length was 10 cm. The proton resonance frequency v,, was 4 MHz corresponding to a magnetic induction 6,) of ca 0.0939 T. 4.2.
All-liquid
flow measurements
The experiments were started at the air-water loop (AWL) of the JRC Ispra. This loop is a test loop for two-phase flow. The loop tube is mounted horizontally as shown in Figure 1 and has an inner diameter of 73.7 mm. The volume of substance in the RF coil is v, = 427 cm’. Our first measurements were done with water only. The velocity of the water was measured by a liquid flow turbine for comparison. We used tap water with T, = 2 s at room temperature and did not add paramagnetic species to shorten the relaxation time. Figures 2 and 3 show two examples at flow velocities of 0.966 and 2.524 m/s (determined by the turbine). The top of the figures shows the efflux curve (EFC) and the bottom the corresponding efflux time (T,,) distribution. The 90” pulse of 5 us duration is at the origin t= 0 of the time scale. In the EFC of Figure 2 each dot corresponds to the mean value of two spin echoes. The smoothed EFC, which was used for the determination of the T,, distribution, has been omitted for clarity. In the EFC of Figure 3 each dot shows the measured height of one spin echo. The solid curve is the smoothed EFC used to evaluate the rki distribution. The dots in the bottom part of the figures mark the set of efflux times TL, used for the distribution evaluation. Their respective partial magnetizations m,,, are given in the figures. It turns out that in this highlyturbulent flow most of the m,,, are zero and the efflux time distribution is rather narrow. The more efflux times we use for the calculations, the narrower the distribution gets, but it shows then small m,,, values at certain Tt, which are in fact artefacts caused by random fluctuations in the EFC. Thus, not too much attention should be paid to the details of the distribution at least in the present state of the art. The distribution should only be used to calculate the total magnetization m,,, the mean efflux time (TE) and the mean velocity (v) according to equations (8)-(10). Since the calibration procedures of Lf, and L,, are done in the same way as the measurements, all the results obtained in this way are consistent with each other. We obtained for these two case velocities of 0.968 and 2.503 m/s, which compare very well with the turbine values of 0.966 and 2.542 m/s, respectively
Nuclear
magnetic resonance (NMR)
31
mA 1200-
‘:j,
?>_, ':: -. :'<_, '%. ':._,
lOOO-
:.. 5:
'k 1:. -. x. ... I.‘.. -:‘::. ...
aooIn f3 6002 m L z 2
:., :.. ... :; :. .is. .. ... i. :; -..>, ".e, -...;. t.. '.:.... '-A... :. :.., I '-..'.Y.:;.y.,.... ~ ...%.'_...+.k%8 I
400-
200-
07 0
I 20
I 40
a0
60
100
120
lb0
120
mc
140
t ms
1200
800
0
I
0
20
-
-
7
40
-
-
I-
60
-
I
a0
T--I
*
140 -TEE
ms Figure2 NMR one-ohase time distribution ’
flow
measurment
of water at a velocity of 0.97m/s,
(this particular turbine had an accuracy of + 0.5% and was also used for the calibrations). Similar measurements have been done for velocities ranging from ca 0.1 to ca 5 m/s. The results are shown as dots in Figure 4, where the velocity obtained by NMR is plotted vs the turbine value. The agreement is excellent as can be seen by the two dashed lines which mark ? 5% of the real velocity. For low velocities of 0.1 5 v 5 0.7 m/s we can claim an accuracy of ca 2 5%. For higher velocities 0.7 5 v 5 5 m/s the accuracy seems to be much better. This is not surprising since our method has been developed on purpose for high velocities of up to 100 m/s5. The determination of the total magnetization is not quite so easy. What we obtain from the efflux time distribution is a magnetization m, at time zero of this particular measurement just after the 90” pulse according to equation (8). Now our measurement
top: efflux
curve (EFC) vs time, bottom:
efflux
software works in the following way: it adjusts the timescale of the measurement and the number of spin echoes in such a way that the efflux curve (EFC) fits into the time range registered by the spectrometer. At the same time it adjusts the attenuation at the RF receiver ahead of the phase detector of the spectrometer so as to not obtain more than 12 bit maximum initial amplitude (m,,) at the ADC output which avoids overflow of the computer memory. The criteria for these adjustments are always taken from the last measurement executed and a timescale index together with the receiver attenuation in dB are saved in the computer memory together with their respective EFC for the evaluation of the latter in real time and amplitude. It is therefore no problem to normalize the magnetization m, by taking into account the attenuation. The normalized moo obtained at liquid flow without gas could then be used for the determination
G. /. Kruger et al.
32
m
t
800
mol 800
Figure 3 Same as Figure
t
2 at 2.5 m/s
of the liquid fraction according to equation (12). This, however, is only possible if m, of two-phase flow and moo of the all-liquid flow without gas have been measured at the same temperature T, because the relaxation time T, is temperature-dependent. If the measurements are done at different temperatures, equation (15) must be used to obtain the magnetizations at zero velocity m,,,,, of two-phase flow and m,,,,,,, of all-liquid flow. In this procedure the spin-lattice relaxation time T, of the liquid must be known, of course. In our all-liquid flow measurements, the average velocities of which were plotted in Figure 4, we taking into account the actual have calculated m,,,,, attenuation and the spin-lattice relaxation time, which was measured in our loop at zero velocity. The values of moo,Go,are plotted in Figure 5 vs the turbine measured velocity. The velocity scale is the same as that of Figure 4. There is some scatter in moo,,,, at low velocities of v< 1 m/s (Figure 5). The same is true for the velocity values (Figure 4), but to a lesser extent. Look-
ing into this matter more deeply, we found that in these measurement conditions for velocities v < 1.4 m/s the mixing of the spins during polarization was not perfect and we had to use the mixing equation (24) with the mixing procedure outlined in Section 2. The velocity dependence of the mixing percentage P, has been obtained empirically to be 1 .4-v 14
P,=l-
i
.
2.5
1
(36)
A similar relation P,,,(v) must be established for each particular flow measurement situation for a specific tube and magnet geometry, and included into the computer evaluation software. This procedure works very well as can be judged by the lower velocity parts of Figures 4 and 5. At larger velocities, where the simpler equations (8)-(15) could be applied directly, the magnetizations moo,,,, agree very well over quite a large velocity range. In fact, most of them lie within the -C 5% limits marked by the dashed lines. Only at the last value of ca 5 m/s we have a deviation of
Nuclear
magnetic resonance (NMR)
33
A VNMR
m/s 5
0
I
0
1
I
I
I
3
4
5
*
VT m/s
Figure 4 Velocity
obtained
by NMR
(v +,,&
in one-phase
flow
about + 8.5%, but the mean square error bars, which are also shown in Figure 5, indicate that this is not a real deviation. We must remember at this point that the measurements were done without a polarizing magnet which leads to a rather low signal-to-noise ratio for high velocities. Calculating S/N for our two examples using equation (28) we arrive at values of 690/f and 280/f for our velocities of 0.966 and 2.524 m/s, respectively. Assuming the receiver input noise figure to be f2 2, the S/N values come to 5 390 and 5 140. The measured curves, however, show signal-to-fluctuation ratios of ca 100 and 35, respectively. This seems to indicate that fluctuations due to the flow also show up, but not to a large extent. 4.3.
Results obtained in two-phase flow
As a last example we discuss briefly the application of our method to liquid mass flow measurements of two-phase water-nitrogen flow. For these measurements we used a loop, which has been constructed on purpose. It allows measurements of mixtures of different liquids and nitrogen. The liquid mass flow into the loop is measured by a turbine flow meter as above, the gas (nitrogen) mass flow is measured by a MKS type 558A mass flow meter before the mixing of
of water vs velocity obtained
by turbine
flowmeter
(vT)
liquid and gas. Temperature and pressure are measured in the liquid and in the gas before the mixing, as well as in the liquid-gas mixture close to the measuring section. The loop can be used in two modes: either it can be a closed circulation loop, which is used for calibration measurements with liquid only, or it can be open ended for liquid-gas two-phase flow. In this case the liquid-gas mixture is lost after each measurement and the next measurement is done with a fresh liquid-gas mixture. This is necessary in order to obtain reliable measurements of the liquid and gas mass flows before the mixing. The loop tube is mounted vertically and has an inner diameter of 42 mm at the measurement section, with a volume of substance in the coil of v,140 cm3. The flow is downwards. We used the same magnet and CXP Spectrometer as above. The length of the RF coil was 10 cm too, but it had a smaller diameter in order to cope with the diameter of the loop tube and to obtain a larger filling factor for this NMR coil. Calculating the signal-to-noise ratio as above using equation (28) we get 2400/f at zero velocity. The observed value is in the order of 1000. This shows that the matching of the RF coil to the receiver input is excellent. From this latter value we calculated the signal-to-noise ratio in all-liquid flow
G. J. Krijger
et al.
+ 5%
2x10”
- 5%
1x104
L-
-7--
1
Figure 5 Zero
velocity amplitude
m,,,,,,_(,i oi NMR
2
I
I
I
3
4
5
*
-
VT
m/s all-liquid
flow
for velocities of 142 and 332 cm/s to be 110 and 50, respectively. The measured signal-to-noise ratios at these velocities for water-only flow without gas were about 110 and 40 in these cases. This demonstrates that here with a smaller tube diameter we have less turbulence and hence little additional fluctuations for all-liquid turbulent flow. Water-nitrogen two-phase flow measurements can only be done if we have a pseudo-stationary flow situation in the two-phase flow. Only in such a pseudo-stationary state are separate measurements of liquid mass flow and gas mass flow before the mixing of liquid and gas meaningful. In such a case we can measure series of NM efflux curves under the same conditions and average them in order to reduce the fluctuations according to equation (35). This procedure works also for rather strong fluctuations, where S/F according to equation (34) is of the order of unity. The evaluations are then done on the sum EFC as explained in Section 3. In Figures 6 and 7 we show results of one example. Figure 6 is the calibration measurement in analogy to Figure 4. For the measurements we used the adding of efflux curves with NM ranging from about 30 (at low velocities) to 150 (at high velocities). Plotted is the measured velocity vNMR vs the velocity v,, determined by the turbine flow meter. This curve, measured with the closed circular loop, gives us the effective length L, of the magnetic RF-field in the flow direction. The linearity of the NMR velocity measurement is excellent. For two-phase flow measurements we used NM ranging from 70 to 100. For the determination of the liquid fraction el we have several possibilities. Firstly, we can get it by the water and gas input data to the loop. This liquid fraction
measurement
of water in dependence of turbine
velocity v,
we call eroop(we omit from now on the index I for liquid, because we are only measuring the liquid fraction which we call E, the void fraction is then 1 -E). Further we can obtain E by the measured NMR velocity in two-phase flow from the continuity equation and by the turbine. we measure the liquid mass flow MF, where The tube has the cross-section area a= nd2/4, d is the inner tube diameter at the measuring section. Thus we would have a velocity vT = MF,Ia
(37)
at the measuring section without gas. With gas the velocity will be much larger and this is the measured vNMR. The liquid mass flow is then
MF,,
= E,,av,,,.
(38)
Now we have MF, = MF, = MF since no liquid gets lost and hence we obtain the liquid fraction E, determined by the velocity E, = MFTI(avNMR) = v&+,,~.
(39)
Finally, we obtain the liquid fraction ?? NMR = E,,() in quite an independent way using equations (12), (16) and (17) of Section 2. This is really the liquid fraction determined by NMR even in cases, where no turbine mass flow measurement is executed. In reality the procedures are somewhat more complicated and have been explained elsewhere4. Figure 7 shows the liquid fraction eNMK = E,,~~ plotted vs the liquid fraction E, obtained by the velocity. There are certainly fluctuations in vNMK (where we get E, from) and m,, (which gives us eNMK). These fluctuations are not correlated and show up, therefore, in the figure. The mass flow MF,,, determined by
Nuclear
0
magnetic
1
resonance
(NMR)
35
3
2
v,
4
Ill/S
Figure 6 Velocity obtained by NMR vNMR in all-liquid flow the velocity calibration for the two-phase flow measurements
the product vNMRx ?? NMR will thus have smaller fluctuations than each of both vNMR or E,+,~. If both had the same fluctuations, the signal-to-fluctuation ratio of would be improved by fi in analogy to equaMFNMR tions (33) and (35). For E < 0.5 our loop does not work well because we get a large slip with the gas velocity much larger than the liquid velocity. This phenomenon can be seen more clearly if we compare ?? NMR and E, with elm,, obtained by the loop input data. This is shown in Table 1, which gives the input mass flows of water MF, = Mf, and gas MF,, both expressed in I/min and the latter reduced to pressure and temperature in our measuring section. Further it contains the liquid fractions eloop,E, and eNMR for the measurements shown in Figure 7. Finally, we give the measured velocity vNMR of water in the two-phase flow and the liquid mass flow obtained by equation (38) but now using the measured liquid fraction eNMRand velocity vNMR. The last row contains the ratio MF,+JMF,, which shows that our overall accuracy of the mass flow obtained by NMR is well within -+ 5% for ?? NMR > 0.6. These results have been used to calculate the gas behaviour in our two-phase flow, which is shown in Table 2. The first line gives the gas fraction eg= 1 -eNMR and the second line the relative error 8~ of cg in % calculated under the assumption that eNMR has an error of t- 5%. In the third and fourth line we show the gas velocity vg and the slip, which is the velocity ratio v~vI=vp/VNMR. The errors indicated correspond to the errors of eg, vp and vNMR. At high liquid fractions we find Ed_, = E, = cNMR and V~ = vNMR.
of water
vs velocity
obtained
by turbine
flow
meter
vP
This is
For liquid fractions of 0.6 < E,.,,,,~ < 0.8 we find E, = +MR and both are larger than E~,,,~. This is due to the slip with vp > v, at high gas fractions (Table 2). The largest slip we get at $MR = 0.49 with vJv, = 1.9 -+ 0.3. But here for ?? NMR < 0.6 we find also that eNMR > E,. This is probably due to the fact that the flow is no longer in the pseudo-stationary state, which we need for the evaluation of our results. If we increase the gas flow even more, the slip becomes larger but ENMR cannot be lowered any further in our loop.
.5. Conclusions We have a direct NMR mass flow measurement method for liquid-gas two-phase flow, which does not require magnetic field gradient pulses. It measures the mean velocity of the liquid averaged over the liquid volume in the NMR RF coil and over the measuring time. Further, it allows the determination of the liquid fraction averaged over the total measuring volume in the NMR RF coil and over the measuring time. Since the product of these two is proportional to the mass flow, the latter can also be determined. It is possible to average measurements over longer time periods of mins, or even hrs or days, in order to obtain reasonable mean values over those long time periods. Such time-averaging improves the signal-to-noise ratio as well as the signal-to-fluctuation ratio. Even with very strong fluctuations in the two-phase flow, measurements of average values thus become possible. In this paper we reported on measurements with
G. J. Kriiger et al.
36
1 +5% , ,’
E NMR
,.
,’
’
,,’ ,’
,’ ,j _’
,’ ,’
0.75
I’
’/ /’ ,,*,/’ ,/
/
,*’ -5%
/
/I’
,,’
,’ ,’
-
,,”
~ 1
/
/I
~ ’
I’’
,
0.5
,‘,/
,‘/
;/,/I.
/
’
,
.
1
_’
‘,/.
0.3
1
0.5
0.3
1
0.75 Ev
Figure 7 NMR of Figure 6
liquid
fraction
Table 1 Results of two-phase
in two-phase
flow
water-nitrogen
159
163
MF,/(l/min)
287.6
133.3
%0,, E" ENMK
v,,dm/s)
Table 2 Gas behaviour 2
in%
v&n/s) y&hhb7
in two-phase
0.510 5 6.8 i- 0.4 1.9 2 0.3
165
165
calibration
165
48.7
75.5
26.5
169 17.9
0.686 0.742 0.738
0.772 0.778 0.795
0.862 0.853 0.835
0.904 0.946 0.924
3.50
2.91
2.67
2.55
2.33
2.15
163.4 1.00
164.0
168.5
0.99
161.5
1.02
0.98
0.98
flow 10 0.324 4.9 -c 0.6 1.7 2 0.3
14 0.262 3.5 2 0.6 1.3 t- 0.3
NMR in water-nitrogen two-phase flow. Other nuclear spins can also be used. We used tap water without any addition of paramagnetic species. After careful calibration procedures we were able to reach proton
the velocity
0.550 0.675 0.676
0.89
MFw.dMF,
with
0.356 0.548 0.490
142.2
MF,,$(llmin)
?? NMK vs E, (see text) obtained
measurements
MF,/(l/min)
Liquid fractions
flow,
19 0.205 2.9 -+ 0.7 1.1 '- 0.3
25 0.165 1.9 + 0.7 0.8 + 0.4
0.076 61 2.8 -c 4.4 1.3 +- 2.2
an accuracy of the liquid (water) mass flow as determined by NMR of well within k 5% as compared to the original mass flow determined by a turbine flow meter before the mixing of water and nitrogen.
Nuclear
magnetic resonance
The gas mass flow could not be measured, since we could not use nitrogen NMR. An additional nitrogen frequency in the spectrometer could give very interesting
gas
measurements sufficiently
results, N,_, for
large.
The
provided
the
time-averaging water
vapour
number
could
due
to
too
low
a number
could of
S. Morris checked the manuscript. Messrs E. Malgarini and B. Paracchini helped with the measurements.
of
be made
not be meas-
in it. In water steam, which contains droplets of liquid water, measurements should again be possible, at least by time-averaging over longer periods of time. The same is true for foamy flows where gas bubbles are surrounded by the liquid. In one liquid-gas two-phase flow, e.g. waterour measurement method nitrogen or oil-nitrogen4, has the advantage that it requires only a low resolution magnet, which has relatively low cost and power consumption. If several liquids in a mixture need to be measured, e.g. oil-water-nitrogen, the magnet should have a higher homogeneity, which will be discussed in a forthcoming paper. The software for our measurements has been developed in the Magnetic Resonance Laboratory and the Experimental Fluid Dynamics Sector of the Safety Technology Institute of the JRC lspra for a BRUKER CXP spectrometer. ured
37
(NMR)
References
protons
of Nuclear Magnetism, 2nd Press, 1962 Caprihan, A. and Fukushima, E. Flow measurements by NMR. Physics Reports 198 (1990) 195
Abragam, A. The Principles Edn. Oxford
University
Leblond, I., Benkedda, Y., Javelot, S. and Oger, 1. Twophase flows
by pulsed
field
gradient
Thanks are due to Mrs E. Dilger who, carefully and very patiently, typed and re-typed the manuscript. Dr
NMR.
Sci. Technol
forsch. 50a (1995) 845 Kriiger, C.J., Haupt, J. and Weiss, R. A nuclear magnetic resonance method for the investigation of twophase flow. In Measuring Techniques in Gas-Liquid Symposium Nancy, France, Two-Phase Flows, IUTAM 1983, p. 435. (Eds J.M. Delhaye and C. Cognet). Springer, Berlin, Heidelberg, 1984 Meiboom, S. and Gill, D. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. Ins&. 29 (1958)
688
R. Zur Protonenrelaxation Naturforsch. 18a (1963) 1143 Kriiger, G.J. Coil for the production Hawser,
Acknowledgements
spin-echo
5 (1994) 426 Kriiger, C.J. and Birke, A. Nuclear magnetic resonance measurements in oil-nitrogen two-phase flow. Z. NaturMeas.
magnetic fields.
US
Patent 4231008,
des Wassers.
Z.
of homogeneous 1980
PIT: so955-598q%)0ooo5-2
Nuclear magnetic resonance (NMR) phase mass flow measurements G. J. Kriiger, Experimental
Fluid
two-
A. Birke and R. Weiss Dynamics
European Commission,
Sector,
21020
Received 28 November
Safety Technology
lspra (VA),
1994;
Institute,
Joint Research Centre (Ispra),
Italy
in revised
75 November
form
7 995
An NMR measurement method for highly-turbulent liquid-gas two-phase flow has been developed in this laboratory [Kriiger, G.J., Haupt, j. and Weiss, R. A nuclear magnetic resonance method for the investigation of two-phase flow. In Measuring Techniques in GasLiquid Two-phase Flows (eds J.M. Delhaye and C. Cognet). Springer, Berlin, 1984, p. 4351. It allows measurement of the liquid velocity and of the fraction, both averaged over the inner volume of the NMR RF coil and over the measuring time. By signal-averaging, it is possible to extend the averaging time to mins, or even hrs or days. This time-averaging improves the signal relative to background noise as well as to fluctuations caused by the flow and hence improves the accuracy of the measurements. The influence of insufficient mixing of the spins during the polarization period is discussed. Measurements with this method have been executed in turbulent water flow and in waternitrogen two-phase flow. The calibration procedure for the velocity vNMR and liquid fraction ?? NMK is done with turbulent water all-liquid flow. The velocities ranged from 0.1 to 5 m/s and the in two-phase flow, which is liquid fraction from ca 0.5 to 1 .O. The mass flow Mf,,, proportional to the product vNMK times ?? NMK, indicates an accuracy of better than 2 5%. Copyright 0 1996 Elsevier Science Ltd.
Kevwords: nuclear magnetic resonance; two-phase prdbability distributioi; noise; fluctuat/ons ’
flow:
mass flow;
M,,
Nomenclature
iWL B,,
BP BRF 4
C SN d
EFC
f h I KC JRC k L Lfe LP
M Mi MC,
inner
cross-section
air-water
test
area of the flow
tube
MF
loop
field for NMR measurements static magnetic field for polarization magnetic RF field half the amplitude of BRF constant part of equation (25) [equation
MF, MF, MF-r
(27)l
MFMAR
static
magnetic
m
inner diameter of the flow tube efflux curve noise figure of receiver input Planck’s constant divided by 2rr nuclear spin iso-speed curve joint Research Centre Boltzmann’s constant length of NMR RF coil effective length of NMR RF field effective length of polarizing magnetic field in flow direction nuclear paramagnetic magnetization parallel to BO [equation (1)l magnetization of spins with velocity vi [equation (18)l thermal equilibrium value of M [equation
m. moo m O(”= 0) mOO,,=o) mOj m,,,,=,, N Ni N, Nl Nh.4
NMR
(1 )I 25
velocity
thermal
equilibrium
value
of Mj [equation
(1 WI sometimes used for liquid mass flow, mass flow gas mass flow liquid mass flow liquid mass flow determined by turbine flow meter liquid mass flow determined by NMR nuclear magnetization perpendicular to B,, [equation (311 maximum value of m at liquid-gas twophase flow maximum value of m at liquid flow without gas value of m, at zero velocity value of moo at zero velocity value of m. of spin ensemble with velocity v, [equation (7)l value of mOi at zero velocity number of spins in one cm3 of matter number of spins with velocity v, in one cm3 of matter number of spins in one cm3 of phase j number of spins in one cm3 of liquid number of measurements nuclear magnetic resonance
26
G. /. Krijger
number of velocities in a velocity or efflux time distribution quality factor of the RF resonance coil circuit radio frequency RF signal-to-noise ratio lequation (25)) SIN time-averaged signal-to-noise ratio lequ(SIN)‘,, ation (33)] signal-to-fluctuation ratio Iequation (34)l signal-to-fluctuation ratio time-averaged [equation (35)l absolute temperature in K spin-lattice relaxation time [equation (I )I spin-lattice relaxation time of phase j lequation (25)l spin-spin relaxation time [equation (.3)l efflux time [equations (4)-(6)l one efflux time of an efflux time distribution average efflux time of an efflux time distribution time acquisition time of signal-averaging polarization time [equation (1)I polarization time of spin ensemble with velocity \‘, Iequation (18)l inner volume of RF coil total volume of substances in RF coil (all phases together) velocity velocity of one-phase liquid flow [eyuation (16)j one velocity of a velocity distribution velocity of phase j velocity determined by NMR velocity determined by turbine flow meter average velocity of a velocity distribution average velocity of one-phase liquid flow magnetogyric ratio (frequently called gyromagnetic ratio in literature) relative error of Q (Table 2) any fraction of bulk matter liquid fraction gas (void) fraction [equation (1311 fraction of phase j liquid fraction determined by velocity v~,~,~ [equation (39)l ?? NMKE E,,,,) liquid fraction determined by magnetization m,, [equations (12), (16) and (17)l proton resonance frequency in B,, UlI proton resonance frequency in B,, vl’ Au bandwidth of the receiver with the detector static nuclear paramagnetic susceptibility x0 [equation (1 )I same of spin ensemble with velocity v, X~l, [equation (19)l N,
Q
1. Introduction Nuclear magnetic resonance (NMR) is a long-established method for the investigation of molecular structures. Besides this, it has been widely used as a powerful tool for the investigation of molecular motions, the principles of which are given by Abragam’. These motions may be all sorts of diffusive motions, but they can also be coherent motions of the
et al nuclear spins and hence of the spin bearing molecules, e.g. in all types of flow. A number of NMR flow measurement methods has been developed’. Since they work only through magnetic transitions in the energy levels of the nuclear spin system, which is very weakly coupled to all the other degrees of freedom in bulk matter, they do not disturb the flow. Recently some NMR results on two-phase flow have been reported l,J. An NMR method particularly useful in highlyturbulent, high-velocity, two-phase flow has been developed in the Magnetic Resonance Laboratory of the joint Research Centre (JRC) Ispra’. It will form the basis for our discussions of NMR flow investigations in two-phase water-gas flow. For the discussion of NMR flow measurements we consider an arrangement as shown schematically in Figure 1. The fluid passes from left to right first through a polarizing magnetic field B,, and then through a measuring magnetic field B,,. The fluid consists of only water or of a water-nitrogen mixture. The spins spend on average the polarization time t,, in the field until they reach the RF coil in the centre of the magnet. The nuclear magnetization M in direction of B,, is then given by M = M,, II pexp (pt,JT,)l = xoB,, I1 -exp
(-t,,lT,)l
where M,, is the equilibrium very long t,, and ,y,, = N-/h’/
(1) magnetization
reached at
(/+I )/(3kT)
(2)
is the static paramagnetic susceptibility with N being the number of spins I of the fluid per cm’ and y is their magnetogyric ratio, h is Planck’s constant divided by 2~r, k Boltzmann’s constant and T the absolute temperature. T, is the spin-lattice relaxation time of the nuclear spins in the fluid. In our RF coil the eight kW transmitter produces J magnetic RF field BKF with amplitude 2B, as usual (Figure 1). We use a Carr-Purcell-Gill-Meiboom pulse sequence’,“. The envelope of the spin-echo sequence created by this is, without flow, given by
m(t) = m,, exp (-tlT,) with the spin-spin
2.
Principles
relaxation
C.3) time T?.
of the measurement
method
Under flow conditions, the spins originally excited by the 90” pulse, which form the nuclear magnetization perpendicular to the B,, field, will leave the RF coil with the flow velocity. The number of transversely polarized spins in the RF coil, therefore, will diminish as time passes. Neglecting relaxation effects, the echo envelope will decay in this case in an efflux curve EFC (tiT,) due to the efflux of spins out of the RF coil’. The characteristic efflux time Tk is inversely proportional to the average velocity of the fluid and is used to determine the latter’. Now taking relaxation effects into consideration again, we write the echo envelope of the transverse magnetization m(t) = m,, EFC (tlTF) exp (-tlT,)
Nuclear magnetic resonance
27
(NMU)
I
Polarizing
Measuring
Magnet
Magnet
I ,
7 Schematic
of NMR flow
= M, [l -exp
(--&IT,)1
measurement
-
apparatus
EFC (Ur)
at a flow
(4)
exp (--t/T11
Transmitter Receiver
tube
N”
mo=Cmo;
(8)
r=l
It is easy to correct a measured curve for spin-spin this correction relaxation4. For fast flow with TE<
(5)
We have measured this curve by pulling a sealed piece of the flow tube filled with water with constant velocity through our RF coil. It must be determined in a similar way for each specific coil arrangement and then serves as a calibration curve for the measurement of unknown flows5. The iso-speed curve ISC (t/T,) allows determination of the efflux time TE5. We define an effective length Lr, of the 6, field, for which v = LfJTE
(6)
This effective field length is somewhat shorter than the physical length of the RF coil and must be determined by calibration experiments with known velocities4z5. In a real flow situation we have a distribution of velocities and a corresponding distribution of iso-speed curves. For the evaluation we use a discrete distribution of N, velocities vi with corresponding magnetizations m,, and efflux times TE,. So we rewrite the flow part of equation (4)
and the average efflux
time
N”
= x
mo,
TE;/mo
= 5
mo; ISC (t/Tr;)
(7)
(9)
r=l
The average velocity < v> = L,/<
is then
TE>
(10)
The magnetizations m,, form the TEi distribution and, according to equation (6), also the velocity probability distribution. Since, the total magnetization m, is proportional to the total number of spins [equations (1) and (211 and hence to the mass of fluid in the RF coil, we obtain for the mass flow MF=
m,,
(11)
The evaluation procedure has been outlined in our earlier papers4,5. Our method is straightforward. It results in the velocity probability distribution and its mean values. It does not need additional gradient pulses of the B,, field as have been used by other authors for twophase flow measurement?. In the case of liquid-gas two-phase flow, where the gas is normally not observable by NMR due to too small a number of spins in it, equation (1 1) is still valid for the liquid flow. We must then measure the total magnetization moo without gas in a tube full of liquid preferably at the same velocity (v) and obtain the liquid fraction El = molm,,
m(t) = m, EFC (d
NMR Pulse Soectro‘meter
.‘.
A-
Figure
Computer Controlled
(12)
and the void (gas) fraction as usual
i=l
with the total 90” pulse
magnetization
immediately
after the
Eg = 1 -El In this
case, (T,),
(13) (v) and ?? l are averaged over the
C. 1. Krtiger et al.
28
liquid volume in the RF coil. If m,,,, cannot be measured at the same velocity we use equation (l), with the polarization time (14)
t,1= L,/(v)
where L,, is the effective length of the polarizing field, which must be determined by a calibration experiment for each experimental arrangement. We therefore obtain m (ii, = 01 - mu/11 -exp
11.5)
I-L,,l((v)T,)It
and
11-exp I --L,ll((v,,)T,)Il m,,,,II -exp I-L,] C
E, =
(lb)
with (v,,) being the mean velocity ot the no,,,, measurement without gas. lf this measurement is done without flow, i.e. (v,,) = 0, equation (16) reduces to El =
/JJ,,/{rJJ,,,,
11
- eX(l
(--L,J((Vi
r,,,]]
1171
The above arguments are only valid, ii the l’low is highly turbulent, to such an extent that during the polarization time t,, as well as during the efflux time r, all the spins get sufficiently mixed in $uch a way that they experience all possible velocities of the distribution in the same way. In strictly laminar flow, where each subensemble of spins maintains its velocity \‘, during the wholr> measurement time, equation (4) ot the i”’ subensemhle will read i 18)
M, = M,,, I I pexp ( t,,,j7-, j =x,,, B,, II pexp I~~L,J(\:7;)IJ where xi,) = N,, -!A’/ (/+ I ,/i 3kn
i
1‘1)
and N, is the number of spins in the sub-ensemble with velocity v, in the fluid per volume. The total magnetization is equal to that given in equation (I) (LO)
M,, = 2. M,,, = B,, Xx,), The velocity distribution function at as above is now a distribution of M, of m,,, used in equations (8) and obtain meaningful mean values we distribution by ? Y
rJJ,,,,,..,,t = MA 1 -exp
= MA1 -exp
distinct r,, values = IM,/ rather than (9). In order to must correct our
I -L,J(v,T, )I I
I~LT;,(L,,,T,)Il
(11)
Equations (8) and (1.5) become identical and change to
/=I Equation (9) remains valid if we replace m,,, by m,,,,,,,, without gas is obtained in and ml1 by ml,,,=,,,. Q,,,,=_,,, the same way as m,,,,+_,,, and with these values reduced to zero velocity we obtain: El = nJo,,,~~,,/nJ,,~,,,0,
(2.3)
as above. In two-phase flow, we normally have highly turbulent flow, and we can use our original efflux time distribution function, the mean values of which are
given by equations (8) and (9). We denote these values now by subscript (OR) (original distribution). In the (ase of low velocities, however, it can happen that the mixing of the spins during the polarization time is not sufficient, even if the flow is still turbulent. It is obvious that this problem occurs at any NMR flow measurement method whatsoever. It does not depend on the details of the method and is more pronounced if the liquid has a higher viscosity, e.g. oilJ. We denote the T,-corrected values for laminar flow as obtained by equations (9) and (22) by subscript (CR) (T,-corrected distribution). Now we introduce a percentage of mixwhich is 100% for highly turbulent flow and ing P,,,, zero for laminar flow. Using this we can get quite a good approach by setting the final value V;II f,,, = P,,, VJL,,,,
+ (1 --P,,,) Vclk( K,
(24)
where Va/ stands for m,,,, ,),, (Tt) or tl. P,,, must he determined experimentally by test and calibration experiments over a sufficiently wide range of flow situations and c-an then be taken into consideration during the computer evaluation of measured results of a specific flow situation. With our RF coil, which has a length of IO cm, we obtain efflux times T, ranging from 1 s to 20 ms at average velocities of cd 0.1 to 5 m/s. One single measurement results, therefore, in mean values of velocity and liquid fraction which are averaged over this short time r, only. If average values over a longer period of time under pseudo stationary conditions are required, the measurement must be repeated during this time of, e.g. one minute (the averaging time is, of course, not limited to one minute; NMR averaging can be done over very long periods of time of up to several days, if necessary, without problems). Then
noise and
The precision of NMR measurements is determined by the instrumental noise’. In highly turbulent flow, however, we have fluctuations due to the turbulence, and in two-phase flow there are additional fluctuations of the liquid dnd gas fractions due to the free interface between the phases. The determination of the total magnetization and the average efflux time, or else the average velocity
Nuclear
magnetic
according to equations (8)~(lo), is straightforward. It is a type of self-consistent measurement, the accuracy of which depends only on the signal to fluctuation ratio of the initial magnetization m, according to equation (8) at the end of the 90” pulse and on the fluctuations of the efflux time TE. Under the assumption of several phases j with their respective fractions in the fluid, which are defined analogously to the liquid fraction, and following the treatment of Abragam’ we obtain the signal-to-noise ratio of the NMR measurement:
(25) 1 = 7 Cs, 2
?? ,N, (1 -exp
[-Lp/(vj~~,)ll
/ where we assumed proton resonance with CSN=-
_
(27)
??? ? = noise figure of the receiver input,
N,
?? Q
T,, K ?? VT
.
??
?? Vj ?? 6, ??
vo
??
VP
?? Av
INMR)
; =f
29
c,,
L, c
(29)
F,
/ 11
i ; = f CSNL, ~
#I
(30)
T,
This approximation works sufficiently well over the whole range of velocities larger than 2 m/s and in the total temperature range of interest. If we omit the polarizing magnet (Figure l), a further simplification takes place. The polarization is in this case accomplished whilst the fluid is flowing through the first half of the measuring magnet towards the RF coil. The length of the polarizing field L, lies in the order of somewhat more than half the length of the measuring magnet in the flow direction. It depends on the geometry of this magnet and must be determined by calibration experiments at different velocities. For the polarizing field we find B, = B0 and hence v,,= vO. Thus we can write
I= 112 and
which is constant for one particular measurement. In addition to the constants and variables already discussed, we have here:
??
resonance
= number of nuclear spins per cm3 of phase j, = quality factor of the NMR resonance coil circuit, = spin-lattice relaxation time of phase j, = volume of the RF coil, = volume of the substance in the RF coil (all phases together), = average velocity of phase j, = fraction of phase j, resonance frequency of the nuclear spins in me measuring field BO, = resonance frequency of the nuclear spins in the polarizing field B,, = bandwidth of the receiver with the detector.
Under the conditions expected for two-phase watergas flow, the gas will not practically contribute to the sum due to its very low value of Nj. For only one liquid phase in two-phase flow, the sum will, therefore, practically reduce to only one term and equation (26) becomes
(28) where
(31) Equation (30) shows further
s
N
1
cSN_
T312T,
that
(32)
if we leave all the other parameters constant. Since T, of water is well known in dependence of temperature’, we can calculate SIN in dependence of temperature and velocity for different experiments, provided it has been measured for one specific case. Moreover we find that CsN, and hence the signal-to-noise ratio, is proportional to the volume of substance V, in the coil. Thus with large flow tube diameters we get high S/N ratios even if we have high velocities and hence short polarization times. Examples will be given in the next section. The signal-to-noise ratio can be further improved by signal averaging over a longer period of time. As already mentioned in Section 2, this is done by adding together measured efflux curves point by point. All the parameters of the measurement must be kept constant during the acquisition time t,. If NM measured signals are added together, the signal-to-noise ratio of the averaged signal becomes
s
s
.“=N d(-1 N
NM
(33)
where S/N is the signal-to-noise ratio of one single measurement. In highly-turbulent flow we get additional fluctuations of the NMR signal due to velocity fluctuations. This will be even more the case for two-phase or multiphase flows, because the liquid fraction or the fractions of single components and their velocities will undergo fluctuations too. In such conditions we can distinguish two principally different cases: first, we may have pseudo-stationary conditions in one-phase liquid flow or in twophase flow in such a way that the liquid fraction remains the same throughout our whole measuring section (e.g. stratified or annular flow). In such cases one single measurement will not be affected in prin-
30
(;. 1. Kriiger
ciple. It can be evaluated in the normal way ds outlined in Section 2. There will be, however, additional fluctuations due to the flow and these can be much larger than the noise of the NMR experiment as such. A second case arises if we have a separation and hence fluctuations of the phases in the flow direction. In strongly pulsating flow this can lead to situations where our measuring section is at times completely full of liquid or at other times completely full of gas. So we will have strong fluctuations as large as our whole NMR signal. In all these cases it is difficult to calculate the signal-to-fluctuation ratio. It will strongly depend on the geometry and on the statistics of the flow. As a very crude first approach for liquid-gas two-phase flow we assume 34) where (E,) and (E,,) are the time averages of liquid fraction and gas (void) fraction, respectively. Equation (34) seems quite reasonable for E,: 5 0.5 and hence S/F 2 1. In the limiting case Ed= 0, where strong pulsation fluctuations do not exist, S/F becomes very large and the measuring accuracy is determined by the much weaker fluctuations of turbulent one-phase liquid flow and, eventually, by the signal-to-noise ratio S/N of the NMR spectrometer. Equation (34) will probably be less meaningful for E,: > 0.5. In the limiting case ?? g = 1 and cl = 0 it shows, however, that S/F= 0, and, indeed, measurement of the liquid is no longer possible. The signal-to-fluctuation ratio can be improved in the same way as the signal-to-noise ratio by time averaging of a large number of measurements N,,, where the efflux curves are added together point by point. In the case of random fluctuations we obtain
=
S-
F\NM
in analogy to equation (33). This procedure works perfectly in pseudo-stationary one-phase liquid flow and also in two-phase flow with stationary liquid fraction as defined above. In strong pulsation two-phase flow it works only if this flow on a large time scale is also pseudo-stationary and the measurements are taken in such a way that all the possible liquid fractions and velocities show up with their respective probabilities. In this case the average efflux curve gives the average flow conditions, whereas single efflux curves may not be very meaningful and can eventually only show single flow situations on a short time scale. The evaluation of time-averaged efflux curves is done exactly in the same way as that of one single EFC, only the total magnetization amplitude m,, obtained by equation (8) must afterwards be divided by NM in order to cope with the amplitude of a single measurement. 4. 4.1.
Results of NMR
flow
measurements
The NMR apparatus
Measurements have been executed using the method described so far in this paper. The NMR spectrometer
et al was a computerized BRUKER CXP Spectrometer with a solenoidal wide-bore (20 cm) measuring magnet manufactured by BRUKER ANALYTISCHE MESSTECHNIK CMBH upon our request and specifications. The arrangement at the flow tube was similar to Figure 1, but the polarizing magnet had been omitted for simplicity, which reduces the nuclear magnetic polarization and hence the sensitivity. The solenoid of the measuring magnet had a length of 40 cm in the flow direction. The homogeneity of the B,, field in the centre was ca 10 ’ over a sphere of 10 cm diameter. The RF coil had a certain similarity with a saddle coil. It consisted of six windings arranged round the flow tube in such a way as to obtain a sufficient homogeneity of the B, field over a large part of the inner crosssection of the tube”. Its length was 10 cm. The proton resonance frequency v,, was 4 MHz corresponding to a magnetic induction 6,) of ca 0.0939 T. 4.2.
All-liquid
flow measurements
The experiments were started at the air-water loop (AWL) of the JRC Ispra. This loop is a test loop for two-phase flow. The loop tube is mounted horizontally as shown in Figure 1 and has an inner diameter of 73.7 mm. The volume of substance in the RF coil is v, = 427 cm’. Our first measurements were done with water only. The velocity of the water was measured by a liquid flow turbine for comparison. We used tap water with T, = 2 s at room temperature and did not add paramagnetic species to shorten the relaxation time. Figures 2 and 3 show two examples at flow velocities of 0.966 and 2.524 m/s (determined by the turbine). The top of the figures shows the efflux curve (EFC) and the bottom the corresponding efflux time (T,,) distribution. The 90” pulse of 5 us duration is at the origin t= 0 of the time scale. In the EFC of Figure 2 each dot corresponds to the mean value of two spin echoes. The smoothed EFC, which was used for the determination of the T,, distribution, has been omitted for clarity. In the EFC of Figure 3 each dot shows the measured height of one spin echo. The solid curve is the smoothed EFC used to evaluate the rki distribution. The dots in the bottom part of the figures mark the set of efflux times TL, used for the distribution evaluation. Their respective partial magnetizations m,,, are given in the figures. It turns out that in this highlyturbulent flow most of the m,,, are zero and the efflux time distribution is rather narrow. The more efflux times we use for the calculations, the narrower the distribution gets, but it shows then small m,,, values at certain Tt, which are in fact artefacts caused by random fluctuations in the EFC. Thus, not too much attention should be paid to the details of the distribution at least in the present state of the art. The distribution should only be used to calculate the total magnetization m,,, the mean efflux time (TE) and the mean velocity (v) according to equations (8)-(10). Since the calibration procedures of Lf, and L,, are done in the same way as the measurements, all the results obtained in this way are consistent with each other. We obtained for these two case velocities of 0.968 and 2.503 m/s, which compare very well with the turbine values of 0.966 and 2.542 m/s, respectively
Nuclear
magnetic resonance (NMR)
31
mA 1200-
‘:j,
?>_, ':: -. :'<_, '%. ':._,
lOOO-
:.. 5:
'k 1:. -. x. ... I.‘.. -:‘::. ...
aooIn f3 6002 m L z 2
:., :.. ... :; :. .is. .. ... i. :; -..>, ".e, -...;. t.. '.:.... '-A... :. :.., I '-..'.Y.:;.y.,.... ~ ...%.'_...+.k%8 I
400-
200-
07 0
I 20
I 40
a0
60
100
120
lb0
120
mc
140
t ms
1200
800
0
I
0
20
-
-
7
40
-
-
I-
60
-
I
a0
T--I
*
140 -TEE
ms Figure2 NMR one-ohase time distribution ’
flow
measurment
of water at a velocity of 0.97m/s,
(this particular turbine had an accuracy of + 0.5% and was also used for the calibrations). Similar measurements have been done for velocities ranging from ca 0.1 to ca 5 m/s. The results are shown as dots in Figure 4, where the velocity obtained by NMR is plotted vs the turbine value. The agreement is excellent as can be seen by the two dashed lines which mark ? 5% of the real velocity. For low velocities of 0.1 5 v 5 0.7 m/s we can claim an accuracy of ca 2 5%. For higher velocities 0.7 5 v 5 5 m/s the accuracy seems to be much better. This is not surprising since our method has been developed on purpose for high velocities of up to 100 m/s5. The determination of the total magnetization is not quite so easy. What we obtain from the efflux time distribution is a magnetization m, at time zero of this particular measurement just after the 90” pulse according to equation (8). Now our measurement
top: efflux
curve (EFC) vs time, bottom:
efflux
software works in the following way: it adjusts the timescale of the measurement and the number of spin echoes in such a way that the efflux curve (EFC) fits into the time range registered by the spectrometer. At the same time it adjusts the attenuation at the RF receiver ahead of the phase detector of the spectrometer so as to not obtain more than 12 bit maximum initial amplitude (m,,) at the ADC output which avoids overflow of the computer memory. The criteria for these adjustments are always taken from the last measurement executed and a timescale index together with the receiver attenuation in dB are saved in the computer memory together with their respective EFC for the evaluation of the latter in real time and amplitude. It is therefore no problem to normalize the magnetization m, by taking into account the attenuation. The normalized moo obtained at liquid flow without gas could then be used for the determination
G. /. Kruger et al.
32
m
t
800
mol 800
Figure 3 Same as Figure
t
2 at 2.5 m/s
of the liquid fraction according to equation (12). This, however, is only possible if m, of two-phase flow and moo of the all-liquid flow without gas have been measured at the same temperature T, because the relaxation time T, is temperature-dependent. If the measurements are done at different temperatures, equation (15) must be used to obtain the magnetizations at zero velocity m,,,,, of two-phase flow and m,,,,,,, of all-liquid flow. In this procedure the spin-lattice relaxation time T, of the liquid must be known, of course. In our all-liquid flow measurements, the average velocities of which were plotted in Figure 4, we taking into account the actual have calculated m,,,,, attenuation and the spin-lattice relaxation time, which was measured in our loop at zero velocity. The values of moo,Go,are plotted in Figure 5 vs the turbine measured velocity. The velocity scale is the same as that of Figure 4. There is some scatter in moo,,,, at low velocities of v< 1 m/s (Figure 5). The same is true for the velocity values (Figure 4), but to a lesser extent. Look-
ing into this matter more deeply, we found that in these measurement conditions for velocities v < 1.4 m/s the mixing of the spins during polarization was not perfect and we had to use the mixing equation (24) with the mixing procedure outlined in Section 2. The velocity dependence of the mixing percentage P, has been obtained empirically to be 1 .4-v 14
P,=l-
i
.
2.5
1
(36)
A similar relation P,,,(v) must be established for each particular flow measurement situation for a specific tube and magnet geometry, and included into the computer evaluation software. This procedure works very well as can be judged by the lower velocity parts of Figures 4 and 5. At larger velocities, where the simpler equations (8)-(15) could be applied directly, the magnetizations moo,,,, agree very well over quite a large velocity range. In fact, most of them lie within the -C 5% limits marked by the dashed lines. Only at the last value of ca 5 m/s we have a deviation of
Nuclear
magnetic resonance (NMR)
33
A VNMR
m/s 5
0
I
0
1
I
I
I
3
4
5
*
VT m/s
Figure 4 Velocity
obtained
by NMR
(v +,,&
in one-phase
flow
about + 8.5%, but the mean square error bars, which are also shown in Figure 5, indicate that this is not a real deviation. We must remember at this point that the measurements were done without a polarizing magnet which leads to a rather low signal-to-noise ratio for high velocities. Calculating S/N for our two examples using equation (28) we arrive at values of 690/f and 280/f for our velocities of 0.966 and 2.524 m/s, respectively. Assuming the receiver input noise figure to be f2 2, the S/N values come to 5 390 and 5 140. The measured curves, however, show signal-to-fluctuation ratios of ca 100 and 35, respectively. This seems to indicate that fluctuations due to the flow also show up, but not to a large extent. 4.3.
Results obtained in two-phase flow
As a last example we discuss briefly the application of our method to liquid mass flow measurements of two-phase water-nitrogen flow. For these measurements we used a loop, which has been constructed on purpose. It allows measurements of mixtures of different liquids and nitrogen. The liquid mass flow into the loop is measured by a turbine flow meter as above, the gas (nitrogen) mass flow is measured by a MKS type 558A mass flow meter before the mixing of
of water vs velocity obtained
by turbine
flowmeter
(vT)
liquid and gas. Temperature and pressure are measured in the liquid and in the gas before the mixing, as well as in the liquid-gas mixture close to the measuring section. The loop can be used in two modes: either it can be a closed circulation loop, which is used for calibration measurements with liquid only, or it can be open ended for liquid-gas two-phase flow. In this case the liquid-gas mixture is lost after each measurement and the next measurement is done with a fresh liquid-gas mixture. This is necessary in order to obtain reliable measurements of the liquid and gas mass flows before the mixing. The loop tube is mounted vertically and has an inner diameter of 42 mm at the measurement section, with a volume of substance in the coil of v,140 cm3. The flow is downwards. We used the same magnet and CXP Spectrometer as above. The length of the RF coil was 10 cm too, but it had a smaller diameter in order to cope with the diameter of the loop tube and to obtain a larger filling factor for this NMR coil. Calculating the signal-to-noise ratio as above using equation (28) we get 2400/f at zero velocity. The observed value is in the order of 1000. This shows that the matching of the RF coil to the receiver input is excellent. From this latter value we calculated the signal-to-noise ratio in all-liquid flow
G. J. Krijger
et al.
+ 5%
2x10”
- 5%
1x104
L-
-7--
1
Figure 5 Zero
velocity amplitude
m,,,,,,_(,i oi NMR
2
I
I
I
3
4
5
*
-
VT
m/s all-liquid
flow
for velocities of 142 and 332 cm/s to be 110 and 50, respectively. The measured signal-to-noise ratios at these velocities for water-only flow without gas were about 110 and 40 in these cases. This demonstrates that here with a smaller tube diameter we have less turbulence and hence little additional fluctuations for all-liquid turbulent flow. Water-nitrogen two-phase flow measurements can only be done if we have a pseudo-stationary flow situation in the two-phase flow. Only in such a pseudo-stationary state are separate measurements of liquid mass flow and gas mass flow before the mixing of liquid and gas meaningful. In such a case we can measure series of NM efflux curves under the same conditions and average them in order to reduce the fluctuations according to equation (35). This procedure works also for rather strong fluctuations, where S/F according to equation (34) is of the order of unity. The evaluations are then done on the sum EFC as explained in Section 3. In Figures 6 and 7 we show results of one example. Figure 6 is the calibration measurement in analogy to Figure 4. For the measurements we used the adding of efflux curves with NM ranging from about 30 (at low velocities) to 150 (at high velocities). Plotted is the measured velocity vNMR vs the velocity v,, determined by the turbine flow meter. This curve, measured with the closed circular loop, gives us the effective length L, of the magnetic RF-field in the flow direction. The linearity of the NMR velocity measurement is excellent. For two-phase flow measurements we used NM ranging from 70 to 100. For the determination of the liquid fraction el we have several possibilities. Firstly, we can get it by the water and gas input data to the loop. This liquid fraction
measurement
of water in dependence of turbine
velocity v,
we call eroop(we omit from now on the index I for liquid, because we are only measuring the liquid fraction which we call E, the void fraction is then 1 -E). Further we can obtain E by the measured NMR velocity in two-phase flow from the continuity equation and by the turbine. we measure the liquid mass flow MF, where The tube has the cross-section area a= nd2/4, d is the inner tube diameter at the measuring section. Thus we would have a velocity vT = MF,Ia
(37)
at the measuring section without gas. With gas the velocity will be much larger and this is the measured vNMR. The liquid mass flow is then
MF,,
= E,,av,,,.
(38)
Now we have MF, = MF, = MF since no liquid gets lost and hence we obtain the liquid fraction E, determined by the velocity E, = MFTI(avNMR) = v&+,,~.
(39)
Finally, we obtain the liquid fraction ?? NMR = E,,() in quite an independent way using equations (12), (16) and (17) of Section 2. This is really the liquid fraction determined by NMR even in cases, where no turbine mass flow measurement is executed. In reality the procedures are somewhat more complicated and have been explained elsewhere4. Figure 7 shows the liquid fraction eNMK = E,,~~ plotted vs the liquid fraction E, obtained by the velocity. There are certainly fluctuations in vNMK (where we get E, from) and m,, (which gives us eNMK). These fluctuations are not correlated and show up, therefore, in the figure. The mass flow MF,,, determined by
Nuclear
0
magnetic
1
resonance
(NMR)
35
3
2
v,
4
Ill/S
Figure 6 Velocity obtained by NMR vNMR in all-liquid flow the velocity calibration for the two-phase flow measurements
the product vNMRx ?? NMR will thus have smaller fluctuations than each of both vNMR or E,+,~. If both had the same fluctuations, the signal-to-fluctuation ratio of would be improved by fi in analogy to equaMFNMR tions (33) and (35). For E < 0.5 our loop does not work well because we get a large slip with the gas velocity much larger than the liquid velocity. This phenomenon can be seen more clearly if we compare ?? NMR and E, with elm,, obtained by the loop input data. This is shown in Table 1, which gives the input mass flows of water MF, = Mf, and gas MF,, both expressed in I/min and the latter reduced to pressure and temperature in our measuring section. Further it contains the liquid fractions eloop,E, and eNMR for the measurements shown in Figure 7. Finally, we give the measured velocity vNMR of water in the two-phase flow and the liquid mass flow obtained by equation (38) but now using the measured liquid fraction eNMRand velocity vNMR. The last row contains the ratio MF,+JMF,, which shows that our overall accuracy of the mass flow obtained by NMR is well within -+ 5% for ?? NMR > 0.6. These results have been used to calculate the gas behaviour in our two-phase flow, which is shown in Table 2. The first line gives the gas fraction eg= 1 -eNMR and the second line the relative error 8~ of cg in % calculated under the assumption that eNMR has an error of t- 5%. In the third and fourth line we show the gas velocity vg and the slip, which is the velocity ratio v~vI=vp/VNMR. The errors indicated correspond to the errors of eg, vp and vNMR. At high liquid fractions we find Ed_, = E, = cNMR and V~ = vNMR.
of water
vs velocity
obtained
by turbine
flow
meter
vP
This is
For liquid fractions of 0.6 < E,.,,,,~ < 0.8 we find E, = +MR and both are larger than E~,,,~. This is due to the slip with vp > v, at high gas fractions (Table 2). The largest slip we get at $MR = 0.49 with vJv, = 1.9 -+ 0.3. But here for ?? NMR < 0.6 we find also that eNMR > E,. This is probably due to the fact that the flow is no longer in the pseudo-stationary state, which we need for the evaluation of our results. If we increase the gas flow even more, the slip becomes larger but ENMR cannot be lowered any further in our loop.
.5. Conclusions We have a direct NMR mass flow measurement method for liquid-gas two-phase flow, which does not require magnetic field gradient pulses. It measures the mean velocity of the liquid averaged over the liquid volume in the NMR RF coil and over the measuring time. Further, it allows the determination of the liquid fraction averaged over the total measuring volume in the NMR RF coil and over the measuring time. Since the product of these two is proportional to the mass flow, the latter can also be determined. It is possible to average measurements over longer time periods of mins, or even hrs or days, in order to obtain reasonable mean values over those long time periods. Such time-averaging improves the signal-to-noise ratio as well as the signal-to-fluctuation ratio. Even with very strong fluctuations in the two-phase flow, measurements of average values thus become possible. In this paper we reported on measurements with
G. J. Kriiger et al.
36
1 +5% , ,’
E NMR
,.
,’
’
,,’ ,’
,’ ,j _’
,’ ,’
0.75
I’
’/ /’ ,,*,/’ ,/
/
,*’ -5%
/
/I’
,,’
,’ ,’
-
,,”
~ 1
/
/I
~ ’
I’’
,
0.5
,‘,/
,‘/
;/,/I.
/
’
,
.
1
_’
‘,/.
0.3
1
0.5
0.3
1
0.75 Ev
Figure 7 NMR of Figure 6
liquid
fraction
Table 1 Results of two-phase
in two-phase
flow
water-nitrogen
159
163
MF,/(l/min)
287.6
133.3
%0,, E" ENMK
v,,dm/s)
Table 2 Gas behaviour 2
in%
v&n/s) y&hhb7
in two-phase
0.510 5 6.8 i- 0.4 1.9 2 0.3
165
165
calibration
165
48.7
75.5
26.5
169 17.9
0.686 0.742 0.738
0.772 0.778 0.795
0.862 0.853 0.835
0.904 0.946 0.924
3.50
2.91
2.67
2.55
2.33
2.15
163.4 1.00
164.0
168.5
0.99
161.5
1.02
0.98
0.98
flow 10 0.324 4.9 -c 0.6 1.7 2 0.3
14 0.262 3.5 2 0.6 1.3 t- 0.3
NMR in water-nitrogen two-phase flow. Other nuclear spins can also be used. We used tap water without any addition of paramagnetic species. After careful calibration procedures we were able to reach proton
the velocity
0.550 0.675 0.676
0.89
MFw.dMF,
with
0.356 0.548 0.490
142.2
MF,,$(llmin)
?? NMK vs E, (see text) obtained
measurements
MF,/(l/min)
Liquid fractions
flow,
19 0.205 2.9 -+ 0.7 1.1 '- 0.3
25 0.165 1.9 + 0.7 0.8 + 0.4
0.076 61 2.8 -c 4.4 1.3 +- 2.2
an accuracy of the liquid (water) mass flow as determined by NMR of well within k 5% as compared to the original mass flow determined by a turbine flow meter before the mixing of water and nitrogen.
Nuclear
magnetic resonance
The gas mass flow could not be measured, since we could not use nitrogen NMR. An additional nitrogen frequency in the spectrometer could give very interesting
gas
measurements sufficiently
results, N,_, for
large.
The
provided
the
time-averaging water
vapour
number
could
due
to
too
low
a number
could of
S. Morris checked the manuscript. Messrs E. Malgarini and B. Paracchini helped with the measurements.
of
be made
not be meas-
in it. In water steam, which contains droplets of liquid water, measurements should again be possible, at least by time-averaging over longer periods of time. The same is true for foamy flows where gas bubbles are surrounded by the liquid. In one liquid-gas two-phase flow, e.g. waterour measurement method nitrogen or oil-nitrogen4, has the advantage that it requires only a low resolution magnet, which has relatively low cost and power consumption. If several liquids in a mixture need to be measured, e.g. oil-water-nitrogen, the magnet should have a higher homogeneity, which will be discussed in a forthcoming paper. The software for our measurements has been developed in the Magnetic Resonance Laboratory and the Experimental Fluid Dynamics Sector of the Safety Technology Institute of the JRC lspra for a BRUKER CXP spectrometer. ured
37
(NMR)
References
protons
of Nuclear Magnetism, 2nd Press, 1962 Caprihan, A. and Fukushima, E. Flow measurements by NMR. Physics Reports 198 (1990) 195
Abragam, A. The Principles Edn. Oxford
University
Leblond, I., Benkedda, Y., Javelot, S. and Oger, 1. Twophase flows
by pulsed
field
gradient
Thanks are due to Mrs E. Dilger who, carefully and very patiently, typed and re-typed the manuscript. Dr
NMR.
Sci. Technol
forsch. 50a (1995) 845 Kriiger, C.J., Haupt, J. and Weiss, R. A nuclear magnetic resonance method for the investigation of twophase flow. In Measuring Techniques in Gas-Liquid Symposium Nancy, France, Two-Phase Flows, IUTAM 1983, p. 435. (Eds J.M. Delhaye and C. Cognet). Springer, Berlin, Heidelberg, 1984 Meiboom, S. and Gill, D. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. Ins&. 29 (1958)
688
R. Zur Protonenrelaxation Naturforsch. 18a (1963) 1143 Kriiger, G.J. Coil for the production Hawser,
Acknowledgements
spin-echo
5 (1994) 426 Kriiger, C.J. and Birke, A. Nuclear magnetic resonance measurements in oil-nitrogen two-phase flow. Z. NaturMeas.
magnetic fields.
US
Patent 4231008,
des Wassers.
Z.
of homogeneous 1980