flowact, flow rate measurements in pipes with the pulsed-neutron ...

FLOWACT, FLOW RATE MEASUREMENTS IN PIPES WITH THE PULSED-NEUTRON ACTIVATION METHOD

THERMAL HYDRAULICS KEYWORDS: neutron activation, flow measurements, evaluation methods

PER LINDÉN,* GUDMAR GROSSHÖG, and IMRE PÁZSIT Chalmers University of Technology, Department of Reactor Physics S - 412 96 Göteborg, Sweden

Received October 6, 1997 Accepted for Publication May 1, 1998

Flow measurements were performed with pulsedneutron activation (PNA) in a specially designed test loop. A stationary neutron generator was used as a neutron source, and the detection of the induced 16 N activity in the flow was performed by two bismuth germanate detectors. Stable flow could be produced in the loop and measured with high precision (;0.5% error) by a scale and a stopwatch method, concurrent with the PNA measurement. A series of measurements have been made by varying the position of the detectors, the flow velocity, etc. The accuracy of the various time-averaging methods that are used in the evaluation of the PNA measurement could be assessed by a comparison with the flow calibration data. In particular, the dependence of the error of the different PNA evaluation methods as functions of detector spacing and flow velocity was determined. The measurements are part of a program that seeks to develop a flowmeter suitable for practical applications, which will include backing up the method with flow calculation and signal-processing methods such as neural networks for off-line calibration of the equipment.

I. INTRODUCTION Accurate flow measurements are needed in a large number of industrial applications, both for determining mass transport and for calorimetric measurements of produced heat. This is also true for power reactors, especially pressurized water reactors, where the energy generated is determined from measurements of heat balance. The lat*E-mail: [email protected] NUCLEAR TECHNOLOGY

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ter includes determination of the feedwater flow. The calorimetric measurement of the total power is also used to normalize the core neutron flux and local power calculations by in-core fuel management codes, which are the basis of calculating burnup and transuranium inventories. Thus, ultimately, flow measurements are used for determining both gross power and nuclear parameters. Flow measurements can be based on intrusive methods ~i.e., those that require permanent installations! or nonintrusive or portable ones. As a rule, the permanently installed flowmeters have a high precision, but their performance can deteriorate with time, and thus they need costly servicing and recalibration. The standard tool of feedwater flow measurement is the Venturi tube, but permanent ultrasonic flowmeters have also been developed and used. The portable flowmeters are much easier to test and maintain, but as a rule their precision is lower. A method that has the promise of being able to produce the precision of the permanent installations while itself being portable is the pulsed-neutron activation ~PNA! method. This method is based on the detection of radiation transport with the flow, where the radioactive component in the flow is used as a tracer. In water-flow measurements, the tracer consists of the 16 N nuclei that are produced by neutron activation of 16 O. Nitrogen-16 is also produced in the core; thus, the feedwater flow has an average activity. The random fluctuations in this activity can be used to determine mass flow by crosscorrelation measurements.1 However, in the PNA method, the activity is generated by pulsed irradiation; thus, deterministic methods can be used to extract the flow velocity or mass flow from the measurements. The PNA method has been tested quite extensively in the past in laboratory experiments.2–10 However, in contrast to the correlation method, which has been applied at an operating plant, it was not used in practical measurements. The reason is partly that extracting the mass 31

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flow or area-averaged velocity from the measurements is not trivial, and the questions of how to determine mass flow with the method can best be tackled and investigated under laboratory conditions. Quite promising results have been achieved, showing the potential of the method. Since the need for a highly precise flowmeter that is portable still exists, further work toward developing a flowmeter with these qualities is justified. Several of the pioneering projects in this area have already been terminated. Besides, the possibilities of unfolding unknown parameters from a distribution without having access to an explicit analytical relationship have been vastly improved recently by the development of neural network techniques. Thus, a project with the aforementioned goals was started at the Department of Reactor Physics at Chalmers University of Technology in Göteborg, Sweden, under the name FlowAct. The department has a permanent neutron generator a with a neutron yield on the order of 10 11 n0s obtained with a tritium target bombarded by deuterium ions, and with a 1-ns pulse duration. This neutron generator has been used extensively in time-of-flight measurements in the past to support development of neutron spectrometry methods for fusion plasma applications. The high yield and overall good quality of the neutron generator are good prerequisites in the FlowAct project. Test-of-principle-type measurements and model calculations for radiation transport were performed during 1994 ~Refs. 8 and 9!. A series of measurements were then made with a simple flow loop and low precision flow calibration possibilities.10 Based on these measurements, a dedicated loop was built for high-precision FlowAct measurements. The loop was built so that a very stable flow in a pipe could be generated and measured by a mechanical method with a precision better than 0.5% concurrently with the PNA measurements. The accuracy of the PNA method could thus be directly evaluated. In line with previous similar experiments, it was found that the stability or reproducibility of the FlowAct method is very good, but the main problem is the determination of the mass flow from the measured detector count rate curves. This is due to the flow properties, i.e., the existence of a flow profile, turbulent diffusion, mixing, etc., as well as to the fact that the activation is not uniform over the cross section of the pipe. To put it another way, the PNA method gives a mean transit time as a weighted average of the flow profile where the weight function is determined by the spatial distributions of the activated volume and the attenuation properties of the gamma radiation. These do not obey any angular symmetry property; thus, the mean transit time will not be equal to the area-weighted flow velocity. A further complication is the radial, axial, and azimuthal diffusion and a

The neutron generator was manufactured by SAMES, France.

32

mixing that leads to a further deviation from the optimum case. To extract the searched mean time that gives the mass flow, several methods of evaluating the PNA measurements have been suggested in the past.11–14 These evaluation methods are all based on simplified models of the flow, so they are not universally valid or best. We have evaluated them in several measurements, where various parameters of the measurements were changed. As expected, the ^10t& averaging method had the best performance, but its precision was also dependent on the actual measurement parameters. The dependence of the accuracy on the different parameters, basically flow velocity and detector positioning, is reported in this paper. The dependence of the errors on various flow parameters will be called the error functions or bias functions. These bias functions express the influence of the fluid mechanical properties on the PNA method. They will be an important tool in further development of the FlowAct method. The measurements and evaluations reported in this paper are the first step in a program whose goal is the development of a measurement method. There are several possible strategies to be followed. One is to determine the dependence of the accuracy of the method as a function of the controllable parameters, i.e., the bias functions for a very wide range of flow parameters experimentally. Then, in a given case, with a given pipe diameter and an approximate knowledge of the flow, the optimum detector placing can be selected. As will be seen in the results, the optimum detector placing theoretically results in zero error. However, the high-quality laboratory measurements that are necessary in a given case to find this value may not be possible or practical to perform. It would be a much more promising possibility to simulate FlowAct measurements with a combined application of Monte Carlo or other radiation transport calculations for the radiative part and advanced computational fluid dynamics ~CFD! methods for the transport of the activated volume. If this were possible to achieve, then, instead of selecting any of the time-averaging techniques and then using the optimum values of the adjustable parameters, modern unfolding techniques such as neural network methods could be used to obtain the mass flow rate directly and transparently from the raw data. This possibility has been explored and will be reported in the continuation of this work.15 II. GENERAL BACKGROUND The essence of the PNA technique is to irradiate the flowing fluid with a short neutron pulse whose duration is much shorter than the fluid transport time between the source and the detector. Although in certain works several detectors were used in a ring around the pipe, they were all placed at the same axial distance from the source. The pulse generation time is used as a clock start time, NUCLEAR TECHNOLOGY

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Fig. 1. The principle of the FlowAct method. The passing of the induced activity is recorded at two locations downstream of the pipe. The slightly upward direction of the pipe minimizes air bubbles in the flow.

and the flow rate or flow velocity is deduced from the time distribution of the detector count rate. This principle has been used in all works so far. In the FlowAct project, another principle was also investigated, namely, the use of two gamma detectors along the flow at different distances from the source. Then

the fluid velocity could be determined from the time distribution of the two detector signals, in particular from the time difference between the two detectors. The principle of this arrangement is shown in Fig. 1. The results from a sample measurement, showing the detector counts for both detectors, are shown in Fig. 2.

Fig. 2. Decay-compensated PNA time distributions for different distances between the activation point and the detector location. NUCLEAR TECHNOLOGY

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It can be seen in Fig. 2 that both detector signals are asymmetric, as a result of the broadening of the activated volume due to the flow velocity profile and diffusion during the time when the activated volume passes the detectors. The difference in the shape of the two curves, i.e., the further broadening from detector 1 to detector 2, further indicates the effect of the flow profile and flow diffusion. In the FlowAct principle, some information is obtained on the velocity profile itself, which could be used to elaborate a method that unfolds the average velocity from these detector signal shapes. However, to make use of this information, theoretical and computational modeling is necessary to construct a suitable averaging or unfolding procedure and to verify it. This will be performed in a later stage of the project. We have empirically tested a few ways of using the two detector signals, such as peak position, mean time, etc., but have not yet interpreted them in terms of models. In this paper we confine ourselves to applying the averaging methods that were previously elaborated for the one-detector measurements. We give a summary of the principles behind those models and the suggested algorithms to extract the mean transport time. These will then be applied in Sec. IV for evaluating the measurements. A PNA measurement can be divided into three different steps that will be discussed later in some detail, namely, activation of the fluid, transport of the irradiated fluid, and detection of the induced activity at a position downstream of the activation point. In principle, all three steps concern the transport of different types of particles. Nevertheless, the timescale or velocity of these processes differs very significantly. The activation ~neutron transport! and the detection ~gamma transport! are so fast in comparison with the fluid flow that a stationary fluid can be assumed. Thus, the task is well defined and can be tackled by standard methods of radiation transport, in particular with Monte Carlo methods. The fluid dynamic

part is much more complicated, and a sufficiently accurate quantitative determination of the transport and mixing of the activated component within the flow require sophisticated methods and careful selection of calculational methods. They lie outside the scope of this paper. II.A. Activation and Detection The penetration depths of neutrons and gamma rays depend on the attenuation in the flowing fluid; hence, nonuniform and geometrically nonsymmetric activation and detection profiles are obtained. There are certain ways of affecting the shape of these profiles. The axial shape of these profiles depends on the collimation of the radiation from the source and to the detector, respectively. The radial and0or angular shape depends on the source and detector geometry and the distance between the source ~or the detector! and the pipe. All these parameters can be varied; however, the attenuation of the beams from the source into the medium and from the fluid into the detector cannot be counteracted or compensated for. A first estimate about the distribution of the activated 16 N nuclei and the origin of gamma photons that contribute to the detector counts ~the detector field of view or the gamma importance function! can be obtained by allowing for attenuation only and neglecting scattered particles.9 This is a simple but powerful method because it only needs the calculation of the uncollided flux of neutrons or gamma photons, which can be performed by analytical quadrature and evaluated numerically. This approximation yields a relatively good estimate because, first, the ~n, p! reaction has a high threshold value, 10 MeV, which is close to the source energy of the neutrons, which is 14 MeV. Second, contribution of scattered photons to the detector is low. As an illustration, Figs. 3 and 4 show, respectively, activation and detector field-of-view profiles from such attenuation calculations.

Fig. 3. The PNA profiles in a cross section of the flow with different distances between the source and the pipe: ~a! the source is positioned at the periphery of the pipe and ~b! the distance between the pipe and the source is one pipe diameter. 34

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Fig. 4. Distribution of the origin of detected gamma photons ~detector field of view! in a cross section of the flow with the detector placed at the pipe surface: ~a! no collimator and ~b! a collimator with width 0.8 * D is used.

A more accurate way to determine the activation and detection profiles is to perform Monte Carlo simulations, such as in Refs. 6, 12, 16, and 17, where the effect of scattered particles and the energy dependence in the cross sections are considered. II.B. Activity Transport The interpretation of the time distribution obtained from a PNA measurement strongly depends on the activity transport in the pipe. Because of the asymmetric activation, a detailed understanding of the transport behavior is necessary. For single-phase turbulent flow, the transport of the induced activity is affected by molecular and turbulent diffusion, as well as by a velocity profile. Turbulent flow is usually described in terms of moments of a random process in which correlations play an important role. However, making use of such correlations in the analysis of a PNA measurement would be very complicated. Clearly, a simpler but still accurate model would be rather advantageous. Such models have been used in interpreting PNA measurements. One way of dealing with the transport of a substance in a pipe is to assume a diffusive behavior. The general form of the diffusion equation for a substance with concentration C and a diffusion coefficient e is dC 5 ¹{~e{¹C! . dt

~1!

This equation describes the diffusion in a frame of reference moving with the fluid. Although Eq. ~1! has a simple structure, the difficulty lies in the fact that the diffusion coefficient e itself is a function of the flow structure. For laminar flow, e is represented by the molecular diffusion coefficient D, and for fully turbulent flow, e is the sum of D and the eddy diffusivity. Early work on the transport of soluble matter in pipes, both for laminar and turbulent flows, was done by TayNUCLEAR TECHNOLOGY

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lor.18,19 These works are widely used as the framework of the theoretical analysis of PNA measurements. The transport of a soluble substance is described by a diffusion approximation with a constant diffusion coefficient. Consider a pipe with radius R in which the velocity profile is u~r!. The equation for diffusion and convection for a substance with a concentration C 5 C~r, z, t ! in cylindrical coordinates is ] 2 C 1 ]C 1 ]C u~r! ]C { 5 { 1 { , 2 1 ]r r ]r D ]t D ]z

~2!

where u~r! is the velocity profile, and the diffusion coefficient e in Eq. ~1! is replaced with the molecular diffusion coefficient D. Here the longitudinal diffusion is assumed to be much smaller than the radial diffusion and convection, i.e., ] 2 C 1 ]C ] 2C , , 1 { . ]z 2 ]r 2 r ]r

~3!

Taylor showed that for large distances ~L02R . 50!, the center of diffusion for a uniformly distributed substance in a pipe is moving at the mean flow velocity. This is valid for both laminar and turbulent flows. Using Eq. ~2! and performing the variable substitution z 1 5 z 2 U { t, where U is defined as the average flow velocity, the behavior showed by Taylor can be modeled. It is shown 18 that the diffusion of the mean concentration Cm 5 Cm ~z, t! in a plane moving at the velocity U can be described by the equation K{

] 2 Cm ]Cm , 2 5 ]z 1 ]t

~4!

where K is a virtual coefficient of diffusion. Taylor found for laminar flow that K 5 ~R 2{U 2 !0~48{D!, and for fully developed turbulent flow, K 5 10.1{R{v * . Here, v * is the frictional velocity given by 35

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v* 5

!

t0 , r

~5!

where t0 is the frictional stress at the pipe wall and r is the density of the fluid. As reported by Taylor, the frictional velocity for a smooth pipe can be obtained from a universal relationship for the ratio between the frictional velocity and the mean flow velocity as a function of Reynolds number Re ~see Ref. 17!: U 5 22.3602 1 1.8809{ln Re 1 0.01334{~ln Re! 2 . v* ~6! Assume a uniform concentration distribution induced in a plane at z 5 0 at time t 5 0 with A particles per unit area. The solution of Eq. ~4! should be finite at large distances, and the number of particles is assumed to be conserved ~the latter is not true for decaying activity and is discussed later!. Thus, the boundary conditions are lim Cm 5 0

~7!

Cm{dz 1 5 A .

~8!

zr`

and

E

`

0

With the preceding boundary conditions, the solution of Eq. ~4! is

S

A z 12 {exp 2 Cm ~z 1 , t ! 5 #pKt 4Kt

D

~9!

.

In the case where the induced concentration is decaying, this is accounted for by multiplying A with the factor exp~2l{t !, where l is the decay constant for the induced activity. Hence, the mean activity concentration in a plane moving at the mean flow velocity can be expressed as Cm ~z, t ! 5

S

A{exp~2l{t ! ~z 2 U{t ! 2 {exp 2 #pKt 4Kt

D

. ~10!

Equation ~10! is a first-order approximation of a PNA time distribution and requires fully developed turbulent flow as well as a constant velocity profile. This is usually obtained for large distances, but the relationship gives reasonable results even when the aforementioned requirements are not exactly fulfilled. II.C. Determining Mass Flow (Cross-Sectional Averaged Velocity) from the PNA Measurement The result of the PNA measurement is a temporal function of count rates, as seen in Fig. 2. The searched parameter of interest, on the other hand, is a single num36

ber, namely, the area-averaged velocity ~called average velocity!, which yields the mass flow rate in the pipe. Thus the task is to determine a functional, i.e., a number, of the measured count rate function that is equal to the average velocity. This process will be referred to hereafter as data reduction. All methods used so far are based on the principle of determining an average time and of using the formula v 5 L0t, or can be traced back to this. The task will then be to determine a suitable “time-averaging” or “timefitting” method that yields the correct average velocity for all combinations of the flow parameters. It is not granted that such a method exists, which is why we suggest the use of neural network techniques in the later stage of this project. The known data-reduction techniques are all inaccurate to some extent. One particular contribution of this paper is to quantify the structure of the inaccuracy of the various methods as a function of certain flow and measurement parameters. The most obvious and simple method would be to take the position of the peak of the measurement as the average time. In practice, however, this method is not effective. To quantify the peak position requires some algorithm such as curve fitting, and previous investigations show that this method is more inaccurate than the other methods described here. This method, therefore, is not investigated any further in this paper. II.C.1. Time Averaging or ^ t& Method Assuming the counts Ci in the time bins around times ti as a histogram approximating a probability density function, the average ~expected! time of the activity passing by the detector can be expressed as N

( exp~l{t !{C {t i

^t& 5

i

i51 N

( exp~l{t !{C i

i

,

~11!

i

i51

where the exponentials are used to compensate for the decay in the intensity. Then the average velocity U can be estimated as U5

L , ^t&

where L is the distance between the activation point and the detector. It can easily be confirmed that ^t&, as calculated from Eq. ~11!, is a linear function of a time shift; namely, with the substitution t ' 5 t 1 T, one obtains ^t ' & 5 ^t& 1 T .

~12!

II.C.2. 10t Averaging or the ^10t& Method In view of the formula v 5 L0t, it appears theoretically more sound if an average velocity is estimated as NUCLEAR TECHNOLOGY

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^v& 5

KL KL L t

1 t

5 L{

~13!

.

Then ^10t& is estimated as

KL 1 t

N

i

5

use either ^t& or ^10t& methods. With the ^t& method, one determines the expected values ^t1 & and ^t2 & of the two detector distributions by using Eq. ~11!. Then, the average velocity is calculated as

1

( exp~l{t !{C { t ( exp~l{t !{C i

i

.

~14!

i

i51

Equation ~14! is then used in Eq. ~13! to estimate the average velocity. It has to be stressed that the ^10t& method is more sound than the ^t& method from the point of view of estimation, i.e., statistics. However, the two methods are equivalent regarding their ability to account for the effect of fluid dynamics on the detector time distributions. In particular, Perez-Griffo et al.6,12 showed that the ^10t& method works well in the case of fully developed turbulent flow where the velocity profile is flat, and it has no effect on the transport of the activated volume. The authors also showed that in the case of laminar flow with a parabolic velocity profile, another averaging method, the so-called 10t 2 averaging technique, yields better results. This latter is defined as Ci exp~l{ti !{ 2 ( ti i51 . ^v& 5 L{ N Ci ( exp~l{ti !{ ti i51

for i 5 1,2 . . . N

where Dt is the width of the time bin. This way t1 . 0, and the singularity is automatically avoided. II.D. Other Time-Averaging Methods Another possibility is to use the time delay between the two detector signals to define an average time. This is possible in the FlowAct method because two axially separated gamma detectors are used in the measurements. Again, similar to the previous treatment, we can NUCLEAR TECHNOLOGY

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where L now is the distance between the detectors.This model has been analyzed in previous reports,10 and it was found that the accuracy in the mean flow velocity provided by this model was on the order of a few percent. With the ^10t& technique, the analogue of Eq. ~13! is used as ^v& 5

K

L t2 2 t1

L K 5 L{

1 t2 2 t1

L

~16!

.

The expected value ^10~t2 2 t1 !& is then estimated from the two distributions as

K

1 t2 2 t1

L

N

M

1

( ( exp~l{~t 1 t !!{C {C { t 2 t i

5

j

i51 j51 N

(

i

j

j

M

i

,

( exp~l{~ti 1 tj !!{Ci{Cj ~17!

~15!

However, since our measurements were made outside this regime, we will not consider this method any further in this paper. Note that, unlike Eq. ~12!, the expected value ^10t& is not a linear function of a time shift, and thus it is sensitive to the accurate handling of the triggering. Further, if t 5 0 is part of the distribution, theoretically, for the Eqs. ~14! and ~15! time-weighting method to be nonsingular, C~t ! must tend to zero fast enough that the integral defining ^10t& converges. In practice, when calculating the velocities by Eqs. ~14! and ~15!, one overcomes this singularity problem by taking time t as the midpoint of the time bin, i.e.,

S 21 D{Dt

L , ^t2 & 2 ^t1 &

i51 j51

N

ti 5 i 2

U5

i

i51 N


where the summation in i goes over the first detector count distribution and in j over the second one. Since only the combinations tj 2 ti enter Eq. ~17!, knowledge of the pulse time is not necessary. Similar to the pure ^10t& case, we see that the two distributions Ci and Cj must be nonoverlapping; otherwise ti 5 tj for some combinations of i and j, and the expected value diverges. Again, starting with the definition of ^10~t2 2 t1 !& as an integral over continuous probability distributions, it can be verified that the integral is convergent if it is defined as a Cauchy principal value. In the digitized form, Eq. ~17!, taking the principal value corresponds to omitting the case ti 5 tj . II.E. Taylor Fitting of Experimental Data The essence of this technique is to fit an expression of the Eq. ~10! type to the measured data and from the fit extract the average flow velocity U. The other parameter of interest, K, is also extracted from the fitting by letting these two parameters be free in the fitting procedure. Further, the number of activated particles A has been normalized to the total number of counts in the measured time distribution, and the distance between the source and the detector z is a parameter known from the measurement. Also, this method can be developed for a doubledetector case; however, in this paper, just as the ^t& and the ^10t& methods, it has only been applied to a singledetector case. 37

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III. EXPERIMENTAL SETUP III.A. Piping System Since the goal of the present measurements was to investigate the accuracy of the FlowAct method, a highquality flow and flow measurement system was needed. Thus, a system with large inherent flow stability, combined with a flow calibration possibility with an accuracy in flow velocity better than 0.5%, was designed and built. In this section, the fluid system and the inherent flow stability are discussed. Flow calibration is discussed in Sec. III.B. The stable constant flow was achieved by using gravity as the driving force and an overflow tank with a constant fluid level. In such a system, stationary flow conditions can be obtained in the container0pipe system when all the flow transients have disappeared. To achieve these conditions, a special-purpose fluid container was built. The constant fluid level container ~CFLC! is divided into two separate volumes by a wall with a height

less than the CFLC height. The constant fluid level is accomplished by letting the fluid flow over the wall into a recirculation circuit. In Fig. 5 the aforementioned stationary container0pipe system consists of the left part of the CFLC and the outlet pipe connected to the bottom of the CFLC. The pipe from the CFLC passes the PNA measuring test section and ends at the weighing tank. The pipe and weighing tank are not physically connected, to avoid weight bias from the pipings. A schematic view of the measurement layout is shown in Fig. 5. Figures 6 and 7 show the upper and the lower parts of Fig. 5 as built at the department. The pipe test section, where stationary flow conditions are assumed, passes in front of the neutron source and the gamma detectors in a slightly upward direction to minimize air bubbles in the flow ~see Fig. 1!. The distance from the nearest 90-deg bend to the activation point is ;40 pipe diameters ~D 5 0.1036 m!. Two valves were mounted on the pipe, one flow control valve and one switching valve. The flow velocity can be varied by the flow control valve. The switching valve is used to direct

Fig. 5. Schematic of the piping system. The source and the detector positions are indicated. 38

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Fig. 6. The upper part of the measurement setup. The fluid container can be seen at the left and the CFLC can be seen on the floor of the experimental hall.

Fig. 7. The lower part of the experimental setup. The measuring position, the weighing tank inlet, and the recirculation container inlet are indicated. NUCLEAR TECHNOLOGY

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the flow either into the recirculation circuit or to the weighing tank. Depending on the switching valve, the system can be operated in two different modes, a recirculation mode and a weighing mode. Before starting a PNA measurement, the system is in the recirculation mode, i.e., all fluid is directed to the recirculation circuit. This mode is maintained until a stationary flow is established in the system. Another reason to start the experiment in this mode is to get rid of entrained air bubbles in the pipes. When the transient period is over, the flow is redirected by the switching valve to the weighing tank. Then the system is in the weighing mode, and all fluid in the pipe reaches the weighing tank. However, a small amount of fluid is still recirculated due to the overflow in the CFLC. When the weighing tank is full, the flow is switched over to recirculation mode. Time recording stops, and the flow calibration experiment ends. In tests of the flow system, the upper limit of the mean velocity for water was found to be ;0.5 m 0s, which corresponds to a Reynolds number of 52 000. A number of measurements were done with a constant flow control valve setting to estimate the stability of the system. The average velocity during a single measurement was determined with the flow calibration setup described in Sec. III.B. For the measurement series, an overall average velocity could be determined with an associated variance. An average velocity of 0.12 m0s was obtained with a standard deviation of 0.143%. This stability is high enough for tests of the FlowAct method with a target precision of 0.5%. III.B. Flow Calibration The flow calibration system consists of a weighing tank, a scale, and a quartz crystal timer. The weighing is performed by three separate piezoelectric scales,b one under each leg of the weighing tank. The signals from the piezoelectric scales are connected into a circuit, summed, and followed up on-line or logged. A crucial point for the accuracy of the flow calibration equipment is that of the scale. The maximum limit for the scale is 4000 kg, and the accuracy, according to the manufacturer, is better than 0.1% of the maximum weight limit. The results from a weight calibration made by a certified manufacturer showed that the accuracy of the scale, in the working range required for these measurements, is 0.05%. Both the scale and the timer are connected to a computer ~Fig. 8!. Flow calibration measurements were performed such that the weight was recorded at equal time intervals when a constant flow was maintained. A few sample measurements are shown in Fig. 9. From the slope of this graph, a mass rate m_ is determined and recalcu-

Fig. 8. Schematic of the flow calibration system.

lated to a corresponding mean flow velocity vS using the relationship vS 5

m_ , r{A

~18!

where A is the cross-sectional area of the pipe and r is the density of the fluid. Another way of determining the mass rate is to take the ratio between the integrated mass and the total collection time. To avoid weight biasing, no pipes are connected to the tank during a flow calibration measurement. However, a temporary connection, indicated in Fig. 5, is available for emptying the tank after a flow calibration measurement. The extremely even shape of these curves ~Fig. 9! indicates the stability of the flow. III.C. Source and Detector System The source and detector system consists of a pulsed neutron source and two separate counting lines, which are all synchronized by a trigger signal. The neutron source is a SAMES 400-kV neutron generator that has a neutron yield of ;10 11 n0s. This neutron generator has a quite advanced bunching system that is capable of producing pulses in a wide time range ~1029 to 1 s!. This wide range was, however, not necessary for use in the FlowAct measurements because a 100-ms pulse length was used during these measurements. Each counting line consists of a 2-in.-high, 2-in.diam cylindrical bismuth germanate scintillating crystal c and a multichannel scaler d ~MCS! connected via an amplifier chain ~Fig. 10!. Time distributions from single or multiple neutron pulses can be recorded with the system. This is exactly the same system that was used during the preliminary measurements reported in Ref. 10. IV. RESULTS One purpose of these measurements was to provide highly accurate flow calibration data for the PNA method. These data are needed for validation of different analysis c

b

The piezoelectric scales are manufactured by Carl Lidén.

40

The scintillating crystal is manufactured by Teledyne Brown Engineering. d The MCS used is an EG&G Ortec MCS-Plus card for personal computer for companion software ~A68-BI!. NUCLEAR TECHNOLOGY

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Fig. 9. Typical weight-time graphs from flow calibration measurements.

Fig. 10. The source and detector system.

models and for calibration of future calculations. One interest of this work is to investigate the accuracy of some data-reduction methods when varying a number of input parameters. In these measurements, mainly two parameters have been varied, namely, the reference flow velocity and the distance between the source and the detectors. The relative error of a variable can be used as an estimator of the accuracy. In our measurements, the average flow NUCLEAR TECHNOLOGY

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velocity will be obtained concurrently with the flow calibration velocity. As an accuracy estimator, the relative error R v of the average flow velocity, obtained from a FlowAct measurement, will be used. The value R v is defined as Rv 5

^v&meas 2 ^v&ref , ^v&ref

~19! 41

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where ^v&meas is the average velocity estimated from PNA data and ^v&ref is the reference velocity from the flow calibration. In this definition, the reference velocity is assumed to be exact; in practice, this definition is useful whenever R v of Eq. ~19! significantly exceeds the inaccuracy of the reference measurement. As mentioned earlier, the dependence of R v on the detector position and flow velocity, and for different ways of calculating R v ~time-averaging methods!, will be called the error or bias functions. IV.A. Raw Data The detector records the activity concentration in the pipe passing a location downstream of the activation point. Excluding background, there will be no signal until the first activity reaches the vicinity of the detector. Then the activity concentration close to the detector will increase, and after a time t, when the maximum activity concentration has passed the detector, the recorded time distribution will decrease. If the recorded time distribution is compensated for the decay of the induced activity, it can be assumed that the total amount of recorded activity, i.e., the area of the time distribution, will be constant. Due to convection, diffusion, and a velocity profile affecting the transport of the activated volume, the recorded time distribution will be broader as the detector is moved farther downstream. Thus, the maximum intensity of the decay-compensated time distribution will decrease. Typical time distributions, compensated for decay, are shown in Fig. 2, illustrating the broadening and decrease of maximum intensity. In addition to the gamma photons that are produced by the 16 N decay, gamma photons are also generated simultaneously with the neutron pulse. This yields a peak in the recorded time distribution appearing at the pulse generating time. This sharp peak can be used as a reference time

for the time distribution. In our case the reference time ~t 5 0! is defined as the leading edge of the sharp sourceinduced gamma peak. This definition introduces a time bias when using this peak in the determination of the averaged flow velocity, and it adds a constant value to the bias functions mentioned earlier; however, the shape of these functions is unaffected. The width of the “reference” peak can be used to investigate the neutron pulse length and the stability of the neutron generator. During these series of measurements, the neutron pulse length was kept constant ~100 ms!, and the width of the measured reference peak did not fluctuate significantly. Recorded time distributions are also affected by background radiation, originating from construction material close to the source and activity built up in the fluid in the surrounding tanks. It is assumed that this background is linear ~i.e., it can be represented by a straight line in the background reduction procedure!. To decrease the influence of the low-energy gammaray background and the electronic noise, an MCS lowerlevel discriminator was set corresponding to a 2-MeV gamma photon. This is a rather low value compared with the 6.1-MeV photons originating from the decay and is motivated by the fact that it is preferable to collect as much information as possible during a measurement. The signal-to-noise ratio for this discriminator setting is still acceptable, as shown in Ref. 10. Before any analysis is made, background reduction needs to be performed. The background reduction is performed by first choosing two points graphically at each side of the time distribution peak. The line intersecting these two points is determined, and the area below the line is subtracted from the recorded time distribution ~Fig. 11!. The linear assumption is in most cases a good approximation. However, for detectors close to the source, the background becomes obviously nonlinear.

Fig. 11. A PNA time distribution with indicated background reduction line. 42

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TABLE I Results from Reproducibility Tests for the 10t Weighting Method

Measurement

Estimated Average Flow Velocity ~m0s!

Standard Deviation ~m0s!

Standard Deviation ~%!

0.11956 0.11876 0.12303 0.11932

0.00053 0.00053 0.00012 0.00065

0.443 0.448 0.100 0.547

51 to 54 55 to 58 59 to 62 63 to 66

IV.B. Reproducibility

IV.C. Data-Reduction and Bias Functions

One important property for a flowmeter is reproducibility. To estimate this, four measurements were made under identical circumstances, and the velocities obtained from the “10t weighting method” were averaged, and the standard deviation based on these data was calculated. The averaged velocities and the standard deviations, both in absolute numbers and relative to the corresponding average velocity, from four independent series of measurements with different detector distance are presented in Table I. In other words, no comparison to the reference velocity determined by the flow calibration system was made. The low values of the standard deviation show that the reproducibility is good. Reproducibility is dependent on the data-reduction method, and the results from the 10t weighting method are a good representation for the level of reproducibility. Since in these series only one detector was used, located at 1 m from the activation point, only the ^t& and ^10t& averaging methods and the Taylor fitting could be used. The standard deviation of the ^10t& method was between the other methods and was therefore used as a representative value ~Table I!. It can also be seen in Table I that the standard deviation is on the order of the target precision of the FlowAct method ~0.5%!.

This section deals with the different types of datareduction methods described earlier, and for illustration some typical time values from the different methods can be seen in Tables II and III. Mainly two types of bias functions were determined using the results from these series of measurements: the dependence of the accuracy of the data-reduction method as a function of the distance, either between the source and the detector or between the two detectors; and the dependence of the accuracy of the data-reduction method as a function of the reference velocity. One series of measurements, where a constant reference velocity of 0.12 m0s was maintained while the distance was varied, yielded the distance-dependent bias functions, and another series of measurements where the two detectors were fixed but axially separated 1.0 and 1.9 m from the source and the reference velocity was varied yielded the other type, the average flow velocity-dependent bias functions. IV.D. Bias Functions with the ^t & Method The relative error, defined in Eq. ~19!, is plotted in Fig. 12 as a function of the distance between the source and the detector for two cases, i.e., by using two different

TABLE II Typical Time Values from Some Single-Detector Time-Averaging Methods * Detector 1

*

Measurement

^t1 & ~s!

1 ^t1 & ~s21 !

37 38 39

3.27686 3.24825 2.49442

0.30517 0.30785 0.40009

Detector 2

KL ~s21 !

^t2 & ~s!

1 ^t2 & ~s21 !

0.32365 0.32539 0.43623

5.85284 5.83032 4.35143

0.17085 0.17152 0.22981

1 t1

KL 1 t2

~s21 !

0.17771 0.17798 0.23866

The axial position for detector 1 is different from that for detector 2.

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TABLE III Typical Time Values from Some Double-Detector Time-Averaging Methods Double-Detector Method

Measurement

1 ^t2 & 2 ^t1 & ~s21 !

37 38 39

0.38820 0.38729 0.53850

K

1 t2 2 t1 ~s21 !

L

0.45257 0.45721 0.59465

and more Gaussian like. Hence, for the velocity value used here, the best way to apply this method is to place the detector at a distance as far as possible from the source. For the case where the reference velocity was varied and the two detectors were fixed ~1.0 and 1.9 m from the source!, the bias functions for the two detectors will be obviously different due to the influence of convection and mixing on the activity profile in the flow. The two graphs can be seen in Fig. 13, and it is possible to distinguish two curves that show the same tendency. For this data-reduction method, the magnitude of the relative error is not .10%, and the distance-dependent and the reference velocity-dependent bias functions show a rather smooth behavior. IV.E. Bias Functions with the ^1/t & Method

detectors. It would be reasonable to assume that there is no dependence on the actual physical detector; hence, data from the different detectors would show the same trend. It can be seen in Fig. 12 that the bias function is negative in the whole measurement region, which is an effect of the asymmetry of the measured time distribution. Suppose that the recorded time distribution was symmetric, which means that the induced activity profile is uniform in the pipe, then the time average of the recorded distribution would be equal to the transit time between the source and the detector, and the average flow velocity calculated with the ^t& method would be exact. It can be seen in Fig. 2 that the measured time distribution shows a small tail on the increasing time side, yielding a toolow velocity calculated with the time-averaging method. Further, it is indicated in Fig. 12 that the effect of the asymmetry decreases as the detector is moved further from the source. This is consistent with earlier observations by Taylor.19 It can also be seen in Eq. ~10! that for large distances, the time distribution is becoming more

The distance-dependent bias functions for the ^10t& method, like the ^t& method, would not depend on the actual physical detector; hence, data from the different detectors would show the same trend. This is the case except for the shortest distance, which can be seen in Fig. 14 where the relative errors, Eq. ~19!, are plotted as a function of distance between the source and the detector for two cases, i.e., by using two different detectors. The magnitude of the error does not exceed 3%, excluding the longest distance, and some linear dependence on distance can be seen. From Fig. 14 an optimum distance between 1 and 1.5 m for these particular flow conditions can be estimated, and in a practical case it would be preferable to use this distance between the source and the detector. In Fig. 15 the graphs from the other case, the reference velocity case, can be seen. Incidentally, the two detector positions chosen were, according to the distance variation tests, the worst and the best positions for a velocity of 0.12 m0s. In the figure it is possible, for a

Fig. 12. Relative error in estimated average velocity using the ^t& method as a function of the distance between the pulsed source and the detector. 44

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Fig. 13. Relative error in estimated average velocity using the ^t& method as a function of the reference velocity.

Fig. 14. Relative error in estimated average velocity using the ^10t& method as a function of the distance between the pulsed source and the detector.

Fig. 15. Relative error in estimated average velocity using the ^10t& method as a function of the reference velocity. NUCLEAR TECHNOLOGY

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Fig. 16. Relative error in estimated average velocity using the double-detector time-averaging method as a function of the distance between the two detectors. L1 is the distance between the source and the first detector.

reference velocity .0.10 m0s, to distinguish two curves with different slopes. Extrapolating the two curves to higher velocities, it would be reasonable to assume that the two curves tend toward the same value. For the ^10t& averaging method, the magnitude of the relative error is not .5%, except for one point. IV.F. Bias Functions with the 1/~^t2 & − ^t1 &! Averaging Method So far the data-reduction techniques have been based on a single detector signal. Using the ^t& method in conjunction with two detectors at different distances, the influence of the conditions at the source can be suppressed; further, one uses the same physical quantity in

the velocity determination, namely, the time distribution of a passing activity. On the other hand, one introduces an additional parameter with which to experiment: the distance between the two detectors. In Fig. 16 the relative error as a function of the distance between the two detectors is shown for several different positions L1 of the first detector. In comparison with the singledetector ^t& method, accuracy is improved, and the tendency of the curves is similar. For distances .0.40 m between the two detectors, the magnitude of the error is comparable to the ^10t& method using only one detector. From Fig. 16 an optimum detector distance setup can be determined for these particular flow conditions. In the varying reference velocity case, Fig. 17, the shape of the curve is similar to the other reduction

Fig. 17. Relative error in estimated average velocity using the double-detector time-averaging method as a function of the reference velocity. 46

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Fig. 18. Relative error in estimated average velocity using the double-detector inverted time-averaging method as a function of the distance between the two detectors. L1 is the distance between the source and the first detector.

techniques, and for velocities .0.10 m0s, the magnitude of the relative error is on the order of 3%. Also here, the relative error is comparable to the ^10t& method. The magnitude of the relative error is ,3% for this reduction method if the flow velocity is .0.10 m0s and the distance between the two detectors is .0.40 m. IV.G. Bias Functions with the ^1/~t 2 2 t1 !& Method The shape of the bias functions obtained with this reduction method is similar to the double-detector ^t& method; however, the magnitude differs radically. The range of the distance-dependent bias functions is be-

tween 270 and 30% ~Fig. 18!, which is a factor of 10 greater than all the other reduction methods. The same trend can also be seen in the reference flow velocity case, plotted in Fig. 19. Thus, this method appears to be much less practical than the other data-reduction techniques. IV.H. Taylor Fitting In Taylor theory, Eq. ~10!, there are two parameters of specific interest: the average velocity U and the virtual coefficient of diffusion K. By chi-square minimizing with two degrees of freedom, both U and K can be estimated. The minimization procedure used here is

Fig. 19. Relative error in estimated average velocity using the double-detector inverted time-averaging method as a function of the reference velocity. NUCLEAR TECHNOLOGY

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Fig. 20. Taylor theory approximation fitted to the measured PNA time distribution.

the Powell method, extensively described in Ref. 20. Results from a typical fit to measured data can be seen in Fig. 20. The average bias functions, extracted from the Taylor theory fitting, are shown in Figs. 21 and 22. They show the same trends as the results from the 10t weighting method. However, the bias functions are much smoother, whereas the magnitude range of the relative errors are larger for this case. Since the Taylor fit uses the same input data as the ^t& and ^10t& techniques, the reason for the smoother behavior of the bias functions in case of the fitting is suspected to lie in its lower sensitivity to the background reduction procedure. Taylor has introduced a virtual coefficient of diffusion K, which is a measure of the rate of diffusion for the

induced activity. This parameter provides an extended understanding of the flow behavior and can also be measured by the PNA technique. For turbulent flow, K consists of the well-known constant of 10.1 obtained by Taylor,19 hereafter referred to as the Taylor constant. For crosschecking purposes, it was interesting to estimate this constant from the Taylor fitting of the PNA measurement, although the measuring conditions were not in agreement with Taylor theory assumptions. The value of the Taylor constant, as extracted from our measurements with the fitting technique, lies between 20 and 26, as also seen in Figs. 23 and 24. This is a factor of ;2 higher than the generally accepted value of 10.1. The main reason for the fact that there is such a large deviation between the measured and true values of K, while the error in the velocity measurement is only a few percent, may be traced back to a few related reasons. First, Eq. ~10! gives the axial extension of the activated volume due to an infinitely narrow ~spatial d function! activation that is instantaneous in time. In reality the activation is not a d function either in space ~finite beam width even with collimation! or in time ~a pulse length of 0.1 s was used!. Further, the measured quantity is not this distribution either, but the decaying gamma activity, which is only equivalent to the activated volume in case of an infinitely thin collimator. For all these reasons, the detector distributions will be significantly broader than that predicted by Eq. ~10!, but their expectation value will not be altered much due to symmetry of the broadening. Now, as Eq. ~10! shows, the mean velocity is determined as the mean value of a Gaussian distribution, whereas the diffusion coefficient is given by its variance ~width!. Thus it is clear that one obtains reasonable estimates for the mean velocity but a much higher diffusion coefficient than the true value.

Fig. 21. Relative error of the extracted average velocity from a fit to the Taylor approximation for a soluble substance in a pipe, as a function of the distance between the pulsed-neutron source and the gamma detector. 48

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Fig. 22. Error of extracted average velocity from a fit to the Taylor approximation for a soluble substance in a pipe, as a function of the reference velocity.

Fig. 23. Extracted Taylor constant from a fit to the Taylor approximation for a soluble substance in a pipe, as a function of the distance between the pulsed-neutron source and the gamma detector.

Fig. 24. Extracted Taylor constant from a fit to the Taylor approximation for a soluble substance in a pipe, as a function of reference velocity. NUCLEAR TECHNOLOGY

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It was nevertheless expected that the values of the diffusion coefficient and its dependence on the distance would be identical for the two detectors. However, the K values, as seen in Fig. 23, deviate for the two detectors. One possible explanation is that the sensitive area of the detectors differs, yielding a different broadening in the transfer from the activated volume-to-detector signal distribution, as previously described. It can also be seen, however, that the experimentally determined K values decrease with increasing distance, thus indicating that a further reason for the large measured values could be the short distance between the source ~irradiation point! and the detectors. In Fig. 24 the measured Taylor constant is shown as a function of reference velocity. For small velocities, the measured diffusion parameter increases with velocity and is more or less constant above v 5 30 cm0s. In this range, a difference between the two detectors is seen again; detector 1 yields a lower value, even though it is situated a shorter distance from the source.

V. DISCUSSION AND CONCLUSION Pulsed-neutron activation measurements, with highly accurate flow calibration, have been performed and analyzed with different analysis models. For highly accurate flow calibration, a dedicated flow loop was designed, built, and tested. It was found that the flow loop was very stable and working well. The upper working limit for the flow loop, expressed in Reynolds number, is 50 000. During these measurements the Reynolds number range was 7000 to 50 000. Flow calibration could be performed with a precision better than 0.5% The precision of the different evaluation ~datareduction! methods, as functions of the detector distance and the flow velocity, was evaluated. These socalled error or bias functions constitute the main results of this paper. They show that the bias functions are not constants but are indeed functions of the flow and detector parameters; thus, none of the data-reduction techniques is universally best. Among the time-averaging techniques, the ^10t& method is the best, and among all methods investigated, this method has the smallest overall absolute error. On the other hand, the experimentally determined bias functions of the ^10t& method show a certain scattering; i.e., they are not perfectly smooth functions. As mentioned earlier, this indicates that the reproducibility of this method is not as good as that of the Taylor fitting, most likely due to greater sensitivity to background subtraction. The Taylor fitting method gave an absolute error for the average velocity that is a factor of ;2 higher than that of the ^10t& method. On the other hand, the bias functions of this method appeared to be much smoother, indicating a higher reproducibility. Since by further development, the 50

mean bias ~the expected value of the bias functions! may be reduced to zero, the ultimate precision of the method will become equal to the reproducibility. In this respect the Taylor fitting method appears to have the greatest potential. Further development is possible along two main lines. One is the development of simple analytical methods for flow description, e.g., extending Eq. ~10! to allow for finite beam size and irradiation time by convolutions. Then an improved Taylor fitting could be used to determine the average velocity. A second possible strategy is to describe ~reconstruct! the measured values by numerical simulations of neutron and gamma transport with Monte Carlo and simulation of the flow with advanced CFD codes. If the measured values can be reconstructed by numerical methods, the simulated data can be used to train a neural network to recognize ~identify! the true velocity ~flow rate! from a measurement. In this case it is insignificant that no analytical relationship exists between the average velocity and the detector curves, because neural networks can invert parametric dependence without analytical relationships. Both strategies will be explored in further work. The measurements reported here confirm the expectation that the FlowAct method can be developed into a highly accurate nonintrusive flow measurement tool. If sufficient accuracy is obtained, the nonintrusive performance of the method makes it suitable for on-site calibration of permanently installed flowmeters. Also, the applicability of the FlowAct method becomes better for higher Reynolds numbers due to a flatter velocity profile, more intensive mixing, etc. Thus, the chances of applying a refined version of the FlowAct method on flows with much higher Reynolds numbers than the present ones seem quite promising. ACKNOWLEDGMENTS We thank the technicians, R. Rydz, L. Norberg, and L. Urholm, for all their support and efforts with the experimental setup. Financial support from the Center of Nuclear Technology, Stockholm, Sweden, is also gratefully acknowledged.

REFERENCES 1. S. HORÁNYI, D. PALLAGI, T. HARGITAI, and S. TÖZSÉR, “Experience with the Operation of an ON-LINE Primary Coolant Flowmeter System Based on N-16 Noise Analysis at PAKS PWR,” Prog. Nucl. Energy, 15, 709 ~1985!. 2. C. R. BOSWELL and T. B. PIERCE, “Flow Rate Determination by Neutron Activation,” Modern Development in Flow Measurements, Conf. Publ., Vol. 10, p. 264, Peter Peregrinus Ltd. ~1972!. 3. H. A. LARSON, C. C. PRICE, R. N. CURRAN, and J. I. SACKETT, “Flow Measurement in Sodium and Water Using NUCLEAR TECHNOLOGY

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Pulsed-Neutron Activation: Part 1, Theory,” Nucl. Technol., 57, 264 ~1982!.


12. M. L. PEREZ-GRIFFO, R. C. BLOCK, and R. T. LAHEY, Jr., “Nonuniform Tagging and Flow Structure Effects in PNA Measurements,” Trans. Am. Nucl. Soc., 35, 637 ~1980!.

4. C. C. PRICE, H. A. LARSON, R. N. CURRAN, and J. I. SACKETT, “Flow Measurement in Sodium and Water Using Pulsed-Neutron Activation: Part 2, Experiment,” Nucl. Technol., 57, 272 ~1982!.

13. P. R. BARRETT, “An Examination of the Pulsed-Neutron Activation Techniques for Fluid Flow Measurements,” Nucl. Eng. Des., 74, 2, 183 ~Feb. 1983!.

5. K. G. A. PORGES, S. A. COX, C. HERZENBERG, and C. KAMPSCHOER, “Flow Speed Measurement and Rheometry by Pulsed Neutron Activation,” Trans. ASME—J. Fluid Eng., III, 3, 337 ~Sep. 1989!.

14. J. L. ACHARD and J. M. DELHAYE, “Modeling Aspects of the PNA Technique for Flow Rate Measurements,” Proc. 2nd Int. Topl. Mtg. Thermal-Hydraulics of Nuclear Reactors, Santa Barbara, California, January 11–14, 1983, Vol. 2, p. 1456, American Nuclear Society ~1982!.

6. M. L. PEREZ-GRIFFO, R. C. BLOCK, and R. T. LAHEY, Jr., “Measurement of Flow in Large Pipes by the Pulsed Neutron Activation Method,” Nucl. Sci. Eng., 82, 19 ~1982!. 7. P. KEHLER, “Pulsed Neutron Measurement of Single and Two-Phase Liquid Flow,” IEEE Trans. Nucl. Sci., 26, 1, Pt. 2, 1627 ~Feb. 1979!. 8. K. DROZDOWICZ, “Experiments for Water-Flow Measurements by Pulsed Neutron Activation,” CTH-RF-106, Chalmers University of Technology, Gothenburg, Sweden ~Aug. 1994!. 9. G. GROSSHÖG and I. PÁZSIT, “FlowAct, a Neutron Activation Method for the Measurement of Feed-Water Flow,” Proc. SMORN VII, Symp. Nuclear Reactor Surveillance and Diagnostics, June 19–23, 1995, Avignon, France.

15. P. LINDÉN and I. PÁZSIT, “Study of the Possibility of Determining Mass Flow of Water from Neutron Activation Measurements with Flow Simulations and Neural Networks,” to appear in Kerntechnik, 1998. 16. R. P. GARDNER, W. HAQ, and K. VERGHESE, “A Monte Carlo Model for the Pulsed Neutron Activation Method of Flow Measurement,” Trans. Am. Nucl. Soc., 55, 711 ~1987!. 17. R. P. GARDNER, C. L. BARRETT, W. HAQ, and D. E. PEPLOW, “Efficient Monte Carlo Simulation of 16 O Neutron Activation and 16 N Decay Gamma-Ray Detection in a Flowing Fluid for On-Line Oxygen Analysis or Flow Rate Measurement,” Nucl. Sci. Eng., 122, 326 ~1996!. 18. G. I. TAYLOR, “Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube,” Proc. R. Soc. A, 219, 186 ~1953!.

10. P. LINDÉN, “Results from Preliminary FlowAct Measurements During June 1995,” CTH-RF-126, Chalmers University of Technology, Gothenburg, Sweden ~Feb. 1997!.

19. G. I. TAYLOR, “Dispersion of Soluble Matter in Turbulent Flow Through a Pipe,” Proc. R. Soc. A, 223, 446 ~1954!.

11. R. C. BLOCK, M. PEREZ-GRIFFO, U. N. SINGH, and R. T. LAHEY, Jr., “ 16 N Tagging of Water for Transient Flow Measurements,” Trans. Am. Nucl. Soc., 27, 682 ~1977!.

20. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, and W. T. VETTERLING, Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge ~1986!.

Per Lindén ~MSc, engineering physics, Chalmers University of Technology, Göteborg, Sweden, 1992! is a PhD student in the Department of Reactor Physics, Chalmers University of Technology. His research interests are radiation methods for flow measurements and neutron measurements techniques. Gudmar Grosshög ~Dr, reactor physics, Chalmers University of Technology, Göteborg, Sweden, 1970! is now retired from the Department of Reactor Physics, Chalmers University of Technology. His research interest is neutron measurements, especially for nuclear and fusion reactors. Imre Pázsit ~PhD, physics, ELTE University of Sciences, Budapest, Hungary, 1975; DSc, Academy of Sciences, Budapest, 1985!, is a professor and head of the Department of Reactor Physics, Chalmers University of Technology, Göteborg, Sweden. His research interests include transport theory; zero and power reactor noise and noise diagnostics; atomic collisions, sputtering, and radiation and correlation methods in flow measurements and fusion plasma physics.

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