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PostPosted: May 09, 2007 10:26 am 
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Here are some documents I've found on the dynamic modeling problem for a circulating fluoride salt reactor:

ORNL-TM-3767: Hybrid Computer Simulation of the MSBR (2.5M PDF)

ORNL-TM-3102: MSBR Control Studies: Analog Simulation Program (1.2M PDF)

ORNL-TM-2927: Control Studies of a 1000-MWe MSBR (2.1M PDF)

and here are some articles about dynamic modeling of nuclear gas turbines:

Dynamic Modeling of a Cogenerating Nuclear Gas Turbine Plant—Part I: Modeling and Validation

Dynamic Modeling of a Cogenerating Nuclear Gas Turbine Plant—Part II: Dynamic Behavior and Control

Now, the nuclear gas turbine described in these documents is not EXACTLY the same as the one I want to use, and the steam power conversion coupled to the fluoride reactor in those old ORNL documents is obviously not the same as the gas turbine I want to couple to---but there's the challenge...to fit these two pieces together in the most intelligent way!


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PostPosted: May 09, 2007 10:27 am 
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In thumbing through the control system documents, I noticed something kind of interesting in one of them (ORNL-TM-3767). In the introduction of the document they state: "Due to the highly nonlinear nature of the once-through steam generator, it was deemed necessary to have a highly detailed model of this part of the system."

I thought about this for a minute and realized that modeling the fluid performance in the steam generator could be really complicated, because when the fluid (water) is liquid and subcooled (temperature below the boiling point), the addition of heat will lead to an increase in temperature and enthalpy.

Then when the fluid reaches the boiling temperature, the addition of heat energy will lead to an increase in enthalpy, BUT not an increase in temperature--boiling is isothermal. Then when the fluid has completely undergone phase change, the addition of heat will again lead to an increase of temperature and enthalpy, but with totally different fluidic properties (density, specific heat, etc.)

Now on the other hand, in the gas turbine cycle I am advocating for a modern fluoride reactor, the equivalent to the steam generator in the old MSBR is the coolant salt/helium heat exchanger. Across this heat exchanger, the helium will pick up the additional enthalpy that it will use to drive the turbines. And a basic and very important difference between the heat exchanger and the steam generator is that the helium will have very linear behavior and constant properties across the heat exchanger.

So for modeling purposes and dynamic analyses, the salt/helium heat exchangers may be much simpler and easier to model, with more accurate results, than the steam generators in the old MSBR.


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PostPosted: May 09, 2007 10:28 am 
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Here's some more good documents to add to the list:

ORNL-TM-1070: Stability Analysis of the MSRE (PDF, 4.9M)

ORNL-TM-2571: Theoretical Dynamic Analysis of the MSRE with 233U Fuel (PDF, 2.2M)

There appear to be lots of block diagrams and transient response analyses in there. I really wish I paid better attention when I was taught control systems, since I barely remember a thing!


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PostPosted: Jan 26, 2008 7:42 pm 
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I'm going to be working in a new position where I will be much more exposed to control theory than I am now, and hope to be able to do some work on the dynamic response modeling of the LFR with a gas turbine power conversion system.


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PostPosted: Feb 28, 2011 10:23 pm 
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A few days ago I spoke with Syd Ball, an ORNL retiree who wrote the dynamic model for the MSRE described in the document ORNL-TM-1070, which can be found in the document repository.

It would seem to me that a good project for some of the clever minds on this forum would be to update that dynamic model for today's modern programming languages, like MATLAB. Since MATLAB is a bit expensive for the casual user, might I suggest that we try to replicate the results of ORNL-TM-1070 in an open-source code like Scilab?

The dynamic model of the MSRE is described in detail in the appendices of TM-1070. I'm attaching a "cleaned-up" version of TM-1070 to this post.


Attachments:
ORNL-TM-1070-ocr.pdf [1.45 MiB]
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PostPosted: Feb 28, 2011 10:37 pm 
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In appendix C of TM-1070 it contains a "flow diagram" that describes the dynamic model Ball developed. Here's the diagram:
Attachment:
ORNLTM1070_fig-C1.gif
ORNLTM1070_fig-C1.gif [ 30.42 KiB | Viewed 3509 times ]

The transfer functions that describe the performance on the model are given in terms of different pieces of the overall model, also in Appendix C.

The transfer functions that describe the relationship between the inlet and outlet temperatures, and the overall neutron flux and a change in the multiplication constant are:

Attachment:
TM1070_eqsC1C2.gif
TM1070_eqsC1C2.gif [ 4.83 KiB | Viewed 4137 times ]


As far as I can tell, some of the individual transfer functions can be found defined in Appendix A.

G1 is Equation A-30, I think.
G2 is Equation A-55.
G3 is Equation A-16.
G4 is Equation A-15.
G5 is Equation A-14.
G6 is Equation A-13.
G7 and G8 are described by timing lags, and can use Equation A-47, I think?
Is G9 Equation A-42, with the core fluid being fluid 1?
Is G10 Equation A-43, with the core fluid being fluid 1 and the coolant fluid being fluid 2?
Is G11 Equation A-45, with the core fluid being fluid 1 and the coolant fluid being fluid 2?
Are G12 and G13 Equation A-43, with the coolant fluid being fluid 2?
I think G14 and G15 can also be described by timing lags.

Having expressions for G1-G15 allows you to construct the overall transfer functions, if I'm not mistaken.


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PostPosted: Feb 28, 2011 10:44 pm 
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I'll confess that my own understanding of control theory is somewhere between poor and non-existent, but I'm always anxious to learn about things that are important to LFTR, which is why I created this section of the forum and brought up this point of discussion.

A big part of using these dynamic models appears to be the construction of frequency-response diagrams, also called Bode plots if I'm not mistaken. I've made a few of these in MATLAB (mostly simple problems from my control theory text) but I've never done it in Scilab. Has anyone has any success with this before?


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PostPosted: Mar 01, 2011 11:03 am 
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Recently I've been able to get MATLAB to make Bode plots of nuclear reactivity equations similar to those I have seen in a reference text (Glasstone, Nuclear Reactor Engineering, 3rd ed, pg 280-281).

The transfer function for the reactivity response was:

dn(jw)/dk(jw) = n0/(jwL + jwB/(jw + lambda))

Attachment:
GNRE_eq-5-80a.gif
GNRE_eq-5-80a.gif [ 1.87 KiB | Viewed 4549 times ]


where n0 was the reference neutron flux and can be chosen arbitrarily,
jw is the Laplace transform value (equivalent to s in a typical Laplace transform)
L is the prompt neutron lifetime
B is the delayed neutron fraction (beta)
lambda is the one-group effective delayed neutron decay constant

When you put this equation in a Bode plot it comes out like this:

Attachment:
figure5-13.gif
figure5-13.gif [ 52.71 KiB | Viewed 3518 times ]


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PostPosted: Mar 01, 2011 11:14 am 
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So here's a sequence of MATLAB commands that should replicate this plot:

Code:
n0 = 1;
Beta = 0.0065;
Lambda = 0.08;
% Generate Bode Plots
figure,  L=10^(-3); bode(n0*[1 Lambda],[L (Lambda*L + Beta) 0],logspace(-2,3));
hold on; L=10^(-4); bode(n0*[1 Lambda],[L (Lambda*L + Beta) 0],logspace(-2,3));
hold on; L=10^(-5); bode(n0*[1 Lambda],[L (Lambda*L + Beta) 0],logspace(-2,3));
hold on; L=10^(-6); bode(n0*[1 Lambda],[L (Lambda*L + Beta) 0],logspace(-2,3));


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PostPosted: Mar 01, 2011 11:57 am 
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OK, I think I'm having a little more luck in Scilab. In this sequence I define four different values for the prompt neutron lifetime, just like the four curves in the figure from the text, and I make four different equations each with a different value of prompt neutron lifetime and plot them:

Code:
n0 = 1;
Beta = 0.0065;
Lambda = 0.08;
L3=0.001;
L4=0.0001;
L5=0.00001;
L6=0.000001;
s=poly(0,'s');
h3=syslin('c',n0*(s+Lambda)/(L3*s^2+(Lambda*L3 + Beta)*s));
h4=syslin('c',n0*(s+Lambda)/(L4*s^2+(Lambda*L4 + Beta)*s));
h5=syslin('c',n0*(s+Lambda)/(L5*s^2+(Lambda*L5 + Beta)*s));
h6=syslin('c',n0*(s+Lambda)/(L6*s^2+(Lambda*L6 + Beta)*s));
clf();
bode([h3;h4;h5;h6],0.001,10000,['h3';'h4';'h5';'h6'])


Here's the result:
Attachment:
scilab_bode_GNRE_fig5-13.png
scilab_bode_GNRE_fig5-13.png [ 11.54 KiB | Viewed 4503 times ]


One of the differences is that the Glasstone plot is in cycles per second, whereas the Scilab plot is in radians per second. Also the Glasstone text mentions that the value of n0 is adjusted so that the magnitude plot has a value of zero at 1 cycle per second with a prompt neutron lifetime of 10^-4 seconds, so the specific values on the magnitude plot are somewhat arbitrary. It is the shape that matters, and it appears to be the same.


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PostPosted: Jun 27, 2012 10:48 am 
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INTRA-LABORATORY CORRESPONDENCE
OAK RIDGE NATIONAL LABORATORY

February 1, 1973

MSR-73-7

A CATALOG OF DYNAMICS ANALYSES
FOR CIRCULATING-FUEL REACTORS

P.N. Haubenreich

Abstract

The effects of fuel circulation on reactor kinetics, mainly through the transport of delayed-neutron precursors, have been the subject of a dozen or so significantly different investigations over the past 25 years. Each of the methods now available for calculating the dynamics of complex molten-salt reactor power plants involves approximations that limit its applicability. Good fundamental understanding exists, however, and mathematical formulations that can be developed into more realistic, practical computations have been established. This report briefly catalogs the various efforts on the dynamics of circulating-fuel reactors and lists references that give full details.

Introduction

Analysis of the dynamics of large molten-salt reactor power plants is an area in which sufficient progress has been made and enough reassuring results are available that the MSR program, faced with the necessity of contraction because of the budget, has suspended work. A great deal remains to be done, however, before another molten-salt reactor is built and operated. Penetrating safety analyses will demand more definitive calculations of the effects of credible reactivity events. The requirements on plant equipment, instrumentation, and controls to permit stable, convenient operation must be defined by dynamics analyses yet to be done. None of the existing analysis tools are completely satisfactory for these tasks.

A full evaluation of the situation with regard to dynamics analyses of molten-salt reactors will require a discriminating review of what has been accomplished up to now. This report is intended to be of assistance in such a review. It is essentially an uncritical catalog of the various calculations for the MSRE and later MSR designs, preceded by an account of earlier work on fluid-fuel reactor dynamics which provided the basis from which these MSR calculations developed. Numerous references are made to sources of the fully detailed information that will be necessary for an evaluation.

When one considers the dynamics calculations for the MSRE and subsequent MSR designs, he finds that they tend to fall into two general categories. One category is comprised of what might be called control and stability analyses. The other includes the analyses of control-rod experiments and relatively large transients such as conceivably could occur under accident conditions. Although the purposes of these analyses are complementary, some of the requirements on the mathematical models and the computational tools are different. Furthermore, they were generally carried out by different people at ORNL. For these reasons the discussion of the MSR calculations proceeds along the lines of these two categories rather than chronologically.

Early Analyses of Circulating-Fuel Reactor Kinetics

Fluid fuels are nothing new---within little more than a year after the discovery of nuclear fission, Halban and Kowarski, in England, were conducting neutron-multiplying experiments in which the uranium was in the form of a U3O8-D2O slurry. In 1943, as part of the Manhattan Project, preliminary designs were developed at Chicago for several types of reactors using D2O slurry fuel, intended as alternates for the Hanford reactors. (ref 1) In 1944 chemists and physicists at the Clinton Laboratories began to study a 10-MWt aqueous homogeneous reactor that would both investigate thorium-233U breeding and provide a high neutron flux for experiments. That same year (1944) the first actual fluid-fuel reactor was built and operated at Los Alamos. This was the LOPO "water boiler", using an aqueous solution of enriched uranium. Almost from the outset it had been recognized that motion of the fuel could have a pronounced effect on reactor kinetics because of the displacement of the delayed-neutron precursor atoms in the interval between their formation and the emission of the delayed neutron. For example, it was calculated that if the fuel in a bare tank were stirred vigorously the effectiveness of the delayed neutrons would be reduced by about half, and it was recognized that irregular stirring would cause fluctuations in the power level. (ref 2) Circulation of fuel through a heat-removal system outside of the core was expected to have other important effects on reactor kinetics.

The first actual reactor in which fuel was circulated outside of the core was the 1 MWt Homogeneous Reactor Experiment, which operated at Oak Ridge in 1952 and 1953. Close behind it was a molten-salt reactor, the 2 MWt Aircraft Reactor Experiment, which operated in 1954, also at Oak Ridge. Although contemporary and in close proximity, the HRE and ARE programs had quite different goals and proceeded along separate lines. Among many important differences between the reactors were the flow patterns in the core. The HRE core was a spherical tank, with a tangential inlet and central vortex outlet. In the ARE, on the other hand, the fuel flowed in tubes through a solid moderator, making 11 passes before going to the heat exchanger. In the HRE, effects associated with the rapid expansion of the aqueous solution fuel dominated the reactor's dynamic behavior during power excursions. In the ARE, with its molten-fluoride fuel, the damping effects due to fuel circulation and the emission of delayed neutrons assumed greater significance. As a result of these differences, the kinetics calculations for the HRE and for the ARE developed along separate and somewhat different lines.

In the HRE, the expulsion from the core of the expanding fuel resulted in a very large negative temperature coefficient of reactivity (around 1 x 10-3/°C). A variety of simple calculations indicated that large reactivity insertions could be tolerated without damage and in cases of interest from the standpoint of safety, the reactor would go well beyond prompt critical. The kinetics equations used in the safety analyses therefore used simplified representations of the delayed neutrons and concentrated the limited computational capabilities of those days on the description of the inertia and friction of the fuel solution in the relief pipe between the core and the surge volume. (ref 3) HRE dynamics experiments involved reactivity ramps at rates up to 0.7% dK/K per second and produced rather large power transients. V. K. Pare and Sidney Visner, in their analysis of these experiments, (ref 4) used a combination of equations developed by J. M. Stein and by K. Fuchs to describe the early and late stages of these transients. In the equations they used an effective value of beta (the delayed neutron fraction) equal to the measured steady-state value, which included the contribution of photoneutrons and the losses due to circulation.

The analyses of the dynamics of the ARE, which were concerned more with stability than with rapid transients, took into account the expected "slug" flow around the fuel loop. W. K. Ergen observed in 1953 that in an analytical model of such a circulating-fuel reactor, power oscillations tend to be damped by the circulation of the fuel apart from any effect of delayed neutrons. (ref 5) Ergen also recognized that "in a circulating-fuel reactor some delayed neutrons are emitted outside the reacting zone and this modifies the inhour equation in a manner which depends on ... whether fuel which is present in the reactor at a given moment returns to the reactor in a "slug" or gets mixed in the outside circuit with other fuel." Ergen derived the inhour formula for a circulating-fuel reactor with slug flow and flat flux in the core, (ref 7) but unfortunately, the details of his analyses were not widely reported because of the secrecy around the aircraft reactor work. In a brief, unclassified memorandum (ref 6) issued some years after the ARE operation, he reported that "the inhour equation, both with stationary fuel and with fuel circulating, was used to calibrate the regulating rod," but gave no details.

Meanwhile, at Brookhaven National Laboratory, studies had begun on the Liquid-Metal Fuel Reactor, in which molten U-Bi solution circulated through a graphite core and an external heat exchanger. As part of the LMFR studies, J. A. Fleck, Jr. pursued a different, basic approach to the analysis of the kinetics of circulating-fuel reactors. Fleck developed a method for computing reactor periods by finding the eigenvalues of the system of differential equations describing the reactor kinetics, first with one (ref 8), then with several groups of delayed neutrons. (ref 9) In recognition of the utility of simpler relationships, he also derived approximate inhour relations for systems with slug flow, with either sinusoidal or flat power distribution in the core section of the loop. From his studies of these relations Fleck noted the important fact that "while the flux is increasing, there are more delayed neutrons in the system than would be predicted by the fraction [at steady power]. ... As the flux rises, the circulating fuel reactor simply loses less reactivity to the external loop than when it is just critical. The system thus behaves as though it contained an additional latent reactivity whenever the power rises." He concluded that the use of an effective delayed neutron fraction with a "conventional" inhour expression is permissible only when periods are large. In analyses of delayed-neutron effects, Fleck neglected power-temperature feedback; later he did a study of the temperature-dependent kinetics of a circulating-fuel reactor with the "open-loop" or constant inlet temperature boundary condition and neglecting delayed neutron effects. (ref 10) Results of Univac computations with Fleck's equations were encouraging, but there was a lingering thought that, because of the complexity of the kinetics at high power, the response of a complex, circulating-fuel reactor would not be accurately predicted and that further studies were required to assure stability against oscillations at high power."

The line of dynamics analysis that started with the HRE was developed further by P. R. Kasten for a second aqueous homogeneous reactor, the 5 MWt Homogeneous Reactor Test*, which operated at Oak Ridge from 1957 into 1961.

*This reactor was usually referred to in ORNL publications as the HRT and in other literature as the HRE-2.

An unusual feature of this reactor was that there were no safety rods or other fast-acting reactivity controls: HRE experience and analyses of conceivable reactivity additions and peak pressures in the HRT indicated that such controls were not essential. (ref 12) The mathematical description of the HRT dynamics, developed by Kasten (ref 13) for safety studies, again stressed the hydraulics involved in the expulsion of the fuel solution from the core. Reactivity effects of temperature (fuel density) changes were approximated by simplified thermal equations. Kinetics of the chain reaction were represented by the equations for a stationary-fuel reactor, with the delayed neutron fractions reduced to account for steady-state losses due to circulation. Safety calculations on the Oracle determined reactivity steps or ramps corresponding to the maximum tolerable pressure surge, for comparison with rates of reactivity increase associated with physical events. (ref 14) Stability criteria were also established by Kasten, using different sets of linearized equations to examine high-frequency (nuclear power) and intermediate-frequency (load-demand) effects. The stability of the coupled system of core, drain tanks, and condensate storage to fuel concentration effects was analyzed by Melvin Tobias. (ref 15)

The dynamics experiments on the HRT were minimal (because there were no control rods) but considerable effort was devoted to the analysis of some dynamic phenomena observed during operation. The small (up to 1%) fluctuations in reactor power were studied extensively by Jitsuya Hirota*. Hirota showed by noise analysis techniques (implemented only by recorder charts and desk calculator) that certain characteristics of the fluctuations could be related to system conditions, particularly to the status of the fuel solution. (ref 16) Larger excursions in reactor power that occurred on an average of a few times per hour when the reactor was operated above 1 MWt were analyzed by M. W. Rosenthal, S. Jaye, and M. Tobias. They developed an IBM-704 program for computing the time variation of the reactivity associated with observed power excursions, based on conventional neutron kinetics equations and temperature feedback from a simplified thermal model. (ref 17) The indications were that the excursions were most likely caused by movement of separated uranium in the core. (A reactivity balance and physical evidence of local overheating were other indications of uranium separation by at least one of several possible processes.)

*Visiting scientist from Japan Atomic Energy Research Institute

At about the time of the HRT startup, John MacPhee of AMF Atomics published a paper (ref 18) in which he considered some simple approximations for the effects of delayed neutrons. He derived transfer functions and computed the frequency response of a model in which temperature feedback was neglected, the core was perfectly mixed, and the delayed-neutron precursors returned to the core after decaying during passage through the external loop in slug flow.

Reactivity effects produced by fluid motion in reactor cores were further elucidated by some original work done by Bertram Wolfe between 1959 and 1961. His initial interest was in "Borax"-type excursions where the fuel is stationary but the moderator water is ejected at high velocity, "dragging" neutrons with it. Using perturbation techniques, he developed expressions for the reactivity effects (which turned out to be significant). Wolfe then applied the same mathematical treatment to circulating-fuel reactors to obtain expressions for the delayed neutron lifetime and the inhour relation for systems with slug flow and sinusoidal flux distribution in the core. (ref 19) His analysis did not include temperature effects on reactivity.

R. V. Meghreblian and D. K. Holmes, in their 1960 book on reactor analysis, devoted some 20 pages to an exposition of the kinetics of circulating-fuel reactors. (ref 20) Their presentation of the separate effects of delayed neutrons and temperature followed the general scheme of analysis used by Fleck (ref 9), but with a somewhat cruder model.

Control and Stability Analyses for Molten-Salt Reactors

MSRE

In the first investigation of MSRE dynamics, in 1960, an analog representation of the core and heat removal system set up by O.W. Burke (I & C Division) was used to determine temperatures following interruption of fuel circulation. (ref 21) Over the next year Burke analyzed various other transients to delineate the need for automatic control systems. (ref 22) Later in the design of the servo control system, S.J. Ball (I & C Division) used an analog simulation in which the core was represented by nine regions, each containing two fuel "lumps" and one graphite "lump," instead of the one-region (three-lump) representation used by Burke. Both Burke and Ball represented delayed neutrons by one group, with a total yield reduced to account for losses due to
circulation. (ref 23)

When Ball and T. W. Kerlin (Reactor Division) made their more definitive analysis of the inherent stability of the MSRE in 1965, they used the same lumped-thermal model of the core, but more realistic equations for some other parts of the reactor system, including 6 groups of delayed neutrons with approximate representation of precursor losses and around-the-loop feedback. (Their report (ref 24) includes a convenient comparison of the different models and parameter values used in the various MSRE studies.) Instead of an analog computer, Ball and Kerlin used a digital computer program, MATEXP, (ref 25) to solve the system of equations for the time response and another code to get the frequency response. The adequacy of their analysis was later verified when they measured the frequency response of the MSRE, using small input reactivity disturbances, and found that the results generally were in good agreement with their predictions. (ref 26)

Prior to operation of the MSRE with 233U, R.C. Steffy, Jr. and P.J. Wood, (Reactor Division) in 1968 predicted the frequency response of the reactor with the new fuel. Their model was the same as Kerlin's and Ball's except that a first-order time lag had been added to represent the mixing in the reactor vessel plenums that had been inferred from comparison of observed and predicted frequency response. (ref 27) Results of extensive dynamics testing with 233U fuel later confirmed their predictions. (ref 28, 29) These theoretical and experimental analyses of MSRE dynamics were eventually summarized in two papers in Nuclear Technology. (ref 30,31)

In 1965, as part of the operator training program, Ball set up two reactor kinetics simulators at the MSRE, wired directly to the reactor controls and instrumentation. (ref 32) The reactor model used in these simulators was a somewhat simplified version of that being used in the stability studies. (The core was represented by one three-lump region.)

The capability of the servo system for controlling the reactor with 233U fuel was proved in advance by Ball, Steffy and a team from the MIT practice school by an unusual experiment with the reactor during operation with 235U. (ref 33) The method consisted of modifying the flux signal to the servo system so that the response to rod motion resembled that which would result if the fuel were 233U instead of 235U. Prior to doing the test on the reactor, the method was verified by analog simulation, using a model similar to that used in the operator training simulator.

Throughout the years of MSRE operation, the neutron noise was extensively analyzed and techniques were developed by which it could be used for reactor diagnosis. D.N. Fry, R.C. Kryter, and J.C. Robinson (I & C Division) observed that changes in the amount of gas bubbles in the fuel salt could be inferred from noise measurements, (ref 34) using a mathematical model of system hydraulics and neutronics. (ref 35) (Their model, with its emphasis on hydraulics, differed somewhat from that of Ball et al.) Experiments with small induced pressure perturbations were used to test the model and to determine the best boundary conditions. (ref 36, 37) Comparison with the observed behavior indicated minor deficiencies in the hydraulic model, which Ulrich later refined. (ref 38) Noise studies were facilitated when C.D. Martin (I & C Division) programmed the on-line digital computer at the MSRE to collect the data and compute the neutron power spectral density upon demand. Following the observation that the neutron noise around 1 Hz was especially sensitive to the bubble fraction in the fuel, a continuous monitoring system was installed that displayed the magnitudes of the neutron noise and the pressure noise in the frequency range 0.5 to 2 Hz. (ref 39, 40)

MSBR

A preliminary study of the dynamics and control of a 1000-MWe, single-fluid MSBR was made by W. H. Sides, Jr. (I & C Division) in 1969, using an analog computer simulation. (ref 41) For this study he used an abbreviated, lumped-parameter model of the thermal system in which the core was represented by one region (two fuel lumps and one of graphite). Two groups of delayed neutrons were included, with approximate representation of circulation effects. Later Sides went to an expanded lumped-parameter model in which the actual 2-zone core was simulated by three core regions (each with two lumps of fuel and one of graphite) with salt flow in two parallel paths. (ref 42) Delayed neutrons again were represented by two groups, but the model allowed for the effects of variable fuel circulation rate on losses and feedback to the core inlet. Analog simulation with this model was used to study the integrated plant response to perturbations such as changes in load demand, salt flow, and reactivity. (ref 43)

After analysis of earlier results indicated the need for further refinement of the steam generator models, Burke set up a hybrid computer simulation of the reference-design MSBR. (ref 44) This latest (1972) model featured a highly-detailed representation of the steam generator on ORNL's digital-analog hybrid computer and a lumped-parameter analog model of the rest of the system like that used by Sides.

Analyses of Large Transients in Molten-Salt Reactors

MSRE

The analog system that was set up at the very outset of the MSRE design effort to examine system dynamics was incapable of representing extremely large power excursions, such as might result from some postulated accidents. Therefore C.W. Nestor, Jr. (Reactor Division) was assigned to develop a digital computer program for MSRE kinetics. The first result was a program, called MURGATROYD, (ref 45) which was a revised version of a program used previously for aqueous homogeneous reactor analysis, extended to include in a very approximate way the effects of separate fuel and moderator in the molten-salt reactor core. Using MURGATROYD, P.N. Haubenreich and J.R. Engel (Reactor Division) analyzed several conceivable reactivity accidents, including cold slugs, filling accidents and uncontrolled rod withdrawal. (ref 46) These results, plus an analysis of fuel pump stoppage effects that was done on Burke's MSRE analog, provided the basis for the discussion of nuclear safety in the preliminary hazards report (second addendum).

MURGATROYD gave unrealistic results in many cases because the reactivity effects of fuel temperature rise were simply taken to be proportional to a weighted mean of the core inlet and outlet temperatures. To eliminate this approximation, Nestor proceeded to develop another computer program, ZORCH, (ref 47) which included computation of time-dependent axial distributions of fuel and graphite temperatures. The spatial distributions of temperatures were multiplied by a nuclear importance function and integrated over the core volume to obtain "nuclear average" temperatures that produced reactivity feedback. Neither in MURGATROYD nor in ZORCH was explicit account taken of the system outside of the core except for its effect on core pressure. (There was no provision for around-the-loop feedback to the core inlet.)

Six groups of delayed neutrons were included in both MURGATROYD and ZORCH, with equations of the same form as in fixed-fuel reactors but with fission yields reduced to allow for emission outside of the core. Both codes included a simple calculation of steady-state delayed-neutron losses from input data on loop transit times, using an equation appropriate for a flat flux in the core. In preparation for accident analyses with ZORCH, Haubenreich computed "effective" values for yields of 6 groups of delayed neutron during steady-power operation, taking into account the effect on leakage probabilities of the initial energies and the spatial distribution of precursors in the MSRE core. (ref 48) The input data for ZORCH were then adjusted so that the kinetics calculations were done with these effective delayed-neutron fractions. The analysis of various accidents with ZORCH was discussed in some detail in the MSRE nuclear analysis report (ref 23) and results were considered in the safety analysis report. (ref 49)

When the MSRE was started up in June 1965, preliminary analysis of the zero-power physics experiments revealed needs for improvement in the calculation of delayed neutrons---the decrease in reactivity due to loss of delayed neutrons was significantly less than had been calculated and the rod-bump period experiments with fuel circulating were not adequately described by the conventional inhour equation with reduced delayed-neutron fractions. To overcome these deficiencies B. E. Prince (Reactor Div.) developed an analytical model which described the effects of fuel circulation on delayed neutron source distribution while the power is changing on a stable asymptotic period and accounted for the delayed neutrons emitted in the fuel plenums above and below the core. (ref 50) When the inhour-type equation that Prince derived was used to calculate rod worths from the periods measured with the fuel circulating, values were obtained that agreed satisfactorily with those calculated with the fuel stationary. (ref 51) The reactivity loss due to circulation predicted by Prince's model was also in good agreement with the observations, reflecting the significant effect of the neutrons emitted in the upper plenum, which had been neglected in the earlier calculation.

In his development of the expressions needed for analyzing the MSRE experiments, Prince took a basic approach, starting with the general time-dependent reactor equations written to include the transport of delayed-neutron precursors. (ref 50) He carried through to a practical computation (by an IBM-7090 program) only for the cases of a just-critical reactor or one on a stable asymptotic period. In the report describing this work, however, he pointed out the desirability of analyzing the transient delayed-neutron modes and mentioned that he had begun some preliminary work on this subject. These extended studies, unreported until recently, (ref 52) show that, with inclusion of the above-mentioned fluid transport terms in the general equations for circulating-fuel reactors, the application of the usual adjoint-weighting and integration techniques used to derive the point-kinetics equations from these general equations do not result in the usual set of time-dependent ordinary differential equations associated with "point-kinetics" of stationary fuel reactors. However, a time-dependent integrodifferential equation describing the kinetics of the neutron population can still be obtained. In reference 52, Prince describes how the kernels of this equation which are associated with weighted precursor and temperature distributions may be calculated, and discusses the potential advantages of using this type of formulation in future kinetics analyses of circulating-fuel reactors.

In 1967, when operation of the MSRE with 233U fuel was being considered, the relation developed to analyze the 235U experiments was used to calculate the reactivity increments necessary to produce a given stable period with the new fuel. It was found, as expected, that because of the smaller delayed neutron yields from 233U fission, the reactivity-period relationship was more sensitive to circulation. (ref 53) An approximation that used effective yields of delayed neutrons in steady-power operation in the inhour equations for stationary fuel, although differing significantly from the exact relationship, suggested that ZORCH calculations with the same effective yields should be adequate for the necessary safety analyses. Therefore ZORCH was used to calculate the response of the 233U-fueled MSRE to two credible reactivity accidents---uncontrolled rod withdrawal and re-suspension of separated uranium. (ref 54) Results of these calculations, which included effects of rod scram on level but not on period, were considered in the review of safety with 233U fuel. (ref 55)

Analyses of MSRE transients with 233U fuel were subsequently carried out independently by Burke and F.H.S. Clarke (I & C Division), using an analog simulation of the MSRE that had been set up primarily for use in determining the changes that might be necessary or desirable in the MSRE servo controller characteristics or the control rod trip points. (ref 56) Their delayed-neutron representation included 6 groups, with approximate effects of fuel circulation on removing precursors from the core and feeding them back after decay in the external loop. In the precursor equations the core was represented as one well-mixed lump, but for calculation of thermal effects nine core regions (each with two lumps of fuel and one of graphite) were used. The servo controller and the rod trip circuits were realistically represented in this model.

MSBR

After the adoption of the single-fluid MSBR as the focus for the MSR program in 1968, O.L. Smith (Reactor Division) began to develop a mathematical model especially designed for analysis of transients such as would be of concern in safety analysis. Particular emphasis was placed on realistic treatment of the distribution of delayed neutron precursors along a fuel channel. He prepared a program for solution of his dynamics equations on a digital computer, and in 1969 published some preliminary results of studies of the sensitivity of MSBR dynamics to fuel and graphite temperature coefficients of reactivity. (ref 57) A report describing Smiths model was drafted and reviewed in 1970 but suggested revisions have not been made and the report has not been issued. The computer program is operable, however: in June 1972, Smith made the minor changes necessary to conform to the current ORNL computer systems and ran a trial case of a reactivity step in the reference MSBR.

References

1. J. A. Lane, "Homogeneous Reactors and Their Development," Chap 1 in Fluid Fuel Reactors, J. A. Lane, H. G. MacPherson and F. Maslan, Eds., Addison-Wesley, Reading, Mass. (1958).

2. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, p. 598, Univ. of Chicago Press (1958).

3. W. C. Sangren, Kinetic Calculations for Homogeneous Reactors, ORNL-1205 (April 1952).

4. V. K. Pare and S. Visner, Experiments on the Kinetics of the HRE, ORNL-2329 (1957).

5. W. K. Ergen, "Kinetics of the Circulating-Fuel Nuclear Reactor," J. Appl. Phys. 25(6), 702-711 (June 1954).

6. W. K. Ergen, The Physics of the Fused-Salt Reactor Experiment, ORNL internal memorandum CF-57-2-130 (Feb. 1957).

7. W. K. Ergen, The Inhour Formula for a Circulating-Fuel Nuclear Reactor
with Slug Flow, ORNL internal memorandum CF-53-12-108 (Dec. 1953).

8. J. A. Fleck, Jr., "Kinetics of Circulating Reactors at Low Power," Nucleonics, 12(10), 52-55 (Oct. 1954).

9. J. A. Fleck, Jr., Theory of Low-Power Kinetics of Circulating-Fuel Reactors with Several Groups of Delayed Neutrons, BNL-334 (April 1955).

10. J. A. Fleck, Jr., The Temperature-Dependent Kinetics of Circulating Fuel Reactors, BNL-357 (July 1955).

11. J. Chernick, "Reactor Physics for Liquid Metal Reactor Design," Chap. 19 in Fluid Fuel Reactors, J. A. Lane, H. G. MacPherson, and F. Maslan, Eds., Addison-Wesley, Reading, Mass. (1958).

12. S. E. Beall and S. Visner, Homogeneous Reactor Test Summary Report for the Advisory Committee on Reactor Safeguards, pp. 100-116, ORNL-1834 (Jan. 1955).

13. P. R. Kasten, Dynamics of the Homogeneous Reactor Test, ORNL-2072 (June 1956).

14. P. R. Kasten, Operational Safety of the Homogeneous Reactor Test, ORNL-2088 (July 1956).

15. M. Tobias, HRP Quarterly Progr. Rep. Jan. 31, 1955, ORNL-1853, pp. 23-29.

16. J. Hirota, Statistical Analysis of Small Power Oscillations in the HRT, ORNL internal memorandum CF-60-1-107 (Jan. 1960).

17. M. W. Rosenthal, S. Jaye, and M. Tobias, Power Excursions in the HRT, ORNL-2798 (Feb. 1960).

18. J. MacPhee, "The Kinetics of Circulating-Fuel Reactors," Nucl. Sci. Engrg 4, 588-597 (1958).

19. B. Wolfe, "Reactivity Effects Produced by Fluid Motion in a Reactor Core," Nucl. Sci. Engrg 13, 80-90 (1962).

20. R. V. Meghreblian and D. K. Holmes, Reactor Analysis, pp. 590-610, McGraw-Hill (1960).

21. MSR Program Quarterly Report, July 31, 1960, ORNL-3014, pp. 46-53.

22. MSR Program Semiannual Progress Report, Feb. 28, 1962, ORNL-3282, pp. 10-14.

23. P.N. Haubenreich, J. R. Engel, B. E. Prince, and H. C. Claiborne, MSRE Design and Operations Report, Part III -- Nuclear Analysis, ORNL-TM-730 (Feb. 1964).

24. S. J. Ball and T. W. Kerlin, Stability Analysis of the MSRE, ORNL-TM-1070 (Dec. 1965).

25. S. J. Ball and R. K. Adams, MATEXP - A General Purpose Digital Computer Program for Solving Ordinary Differential Equations by the Matrix Exponential Method, ORNL-TM-1933 (Aug. 1967).

26. T. W. Kerlin and S. J. Ball, Experimental Dynamic Analysis of the MSRE, ORNL-TM-1647 (Oct. 1966).

27. R. C. Steffy, Jr. and P. J. Wood, Theoretical Dynamic Analysis of MSRE with 233U Fuel, ORNL-TM-2571 (July 1969).

28. R. C. Steffy, Jr., Frequency-Response Testing of the MSRE, ORNL-TM-2823 (March, 1970).

29. R. C. Steffy, Jr., Experimental Dynamic Analysis of the MSRE with 233U Fuel, ORNL-TM-2997 (April, 1970).

30. T. W. Kerlin, S. J. Ball, and R. C. Steffy, "Theoretical Dynamics Analysis of the MSRE," Nuclear Technology 10, 118 (Feb. 1971).

31. T. W. Kerlin, S. J. Ball, R. C. Steffy, and M. R. Buckner, "Experiences with Dynamic Testing Methods at the MSRE," Nuclear Technology 10, 103 (Feb. 1971).

32. S. J. Ball, Simulators for Training MSRE Operators, ORNL-TM-1445 (April 1966).

33. MSR Program Semiannual Progress Report, Feb. 29, 1968, ORNL-4254, p. 32.

34. D. N. Fry, R. C. Kryter, and J. C. Robinson, Measurement of Helium Void Fraction in the MSRE Fuel Salt Using Neutron-Noise Analysis, ORNL-TM-2315 (Aug. 1968).

35. J. C. Robinson, "Analytical Determination of the Neutron Flux-to-Pressure Frequency Response: Application to the MSRE," Nucl. Sci. and Eng. , 42 (3) , 382 (Dec. 1970) .

36. J. C. Robinson and D. N. Fry, Determination of the Void Fraction in the MSRE Using Small Induced Pressure Perturbations, ORNL-TM-2318 (Feb. 1969).

37. J. C. Robinson and D. N. Fry, "Experimental Neutron Flux-to-Pressure Frequency Response for the MSRE: Determination of Void Fraction in Fuel Salt," Nucl. Sci. and Eng., 42(3), 397 (Dec. 1970).

38. W. C. Ulrich, An Extended Hydraulic Model of the MSRE Circulating Fuel-System, ORNL-TM-3007 (June 1970).

39. MSR Program Semiannual Progress Report, Aug. 31, 1969, ORNL-4449, p. 36.

40. MSR Program Semiannual Progress Report, Feb. 28, 1970, ORNL-4548, p. 7.

41. W.H. Sides, Jr., MSBR Control Studies, ORNL-TM-2489, (June 1969).

42. W.H. Sides, Jr., MSBR Control Studies: Analog Simulation Program, ORNL-TM-3102, (May 1971).

43. W.H. Sides, Jr., Control Studies of a 1000-MW(e) MSBR, ORNL-TM-2927, (May 1970).

44. O.W. Burke, Hybrid Computer Simulation of the MSBR, ORNL-TM-3767,. (May 1972).

45. C. W. Nestor, Jr., MURGATROYD - An IBM 7090 Program for the Analysis of the Kinetics of the MSRE, ORNL-TM-0203 (April, 1962).

46. P. N. Haubenreich and J. R. Engel, Safety Calculations for MSRE, ORNL-TM-0251, (May 1962).

47. C. W. Nestor, Jr., ZORCH - An IBM 7090 Program for the Analysis of Simulated MSRE Power Transients with a Simplified Space-Dependent Kinetics Model, ORNL-TM-0345 (Sept. 1962).

48. P. N. Haubenreich, Prediction of Effective Yields of Delayed Neutrons in MSRE, ORNL-TM-0380, (Oct. 1962).

49. S. E. Beall, P. N. Haubenreich, R. B. Lindauer, and J. R. Tallackson, MSRE Design and Operations Report, Part V, Reactor Safety Analysis Report, ORNL-TM-0732 (Aug. 1964).

50. B.E. Prince, Period Measurements on the Molten Salt Reactor Experiment During Fuel Circulation: Theory and Experiment, ORNL-TM-1626 (Oct. 1966).

51. B.E. Prince, et,.al., Zero-Power Physics Experiments on the Molten-Salt Reactor Experiment, ORNL-4233, (Feb. 1968).

52. B.E. Prince, Improved Representation of Some Aspects of Circulating-Fuel Reactor Kinetics, ORNL-TM- in publication (Jan. 1973).

53. MSR Program Semiannual Progress Report, Period ending Aug. 31, 1967, ORNL-4191, p. 58.

54. MSR Program Semiannual Progress Report, Period ending Feb. 29, 1968, ORNL-4254, p. 37.

55. P.N. Haubenreich et al., MSRE Design and Operations Report, Part V-A, Safety Analysis of Operation with 233U, ORNL-TM-2111, (Feb. 1968).

56. O.W. Burke and F. H. S. Clark, Analyses of Transients in the MSRE System with 233U Fuel, ORNL-4397, (June 1969).

57. MSR Program Semiannual Progress Report, Period ending Aug. 31, 1969, ORNL-4449, p. 66.


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