EXAMPLES OF ANALOG COMPUTER APPLICATIONS IN NUCLEAR PROCESSES by

W. D. CAMERON and G. R. TAYLOR Hanford Laboratories Hanford Atomic Products Operation General Electric Company

Richland, Washington This paper was presented at the Western and Rocky Mountain Simulation Council Joint Meeting in Seattle, Washington

on July 13, 1962.

ABSTRACT

reactor

system and inserting the simulated disturb-

ances.

The purpose of this report is to outline and explain the various methods used in simulating atomic reactors and processes, particularly the types used at the Hanford plant. This report describes the newer as well as the standard techniques used in reactor simulation. Applications are described by examples. The techniques used in a complete atomic reactor plant simulator are described. Chemical processing simulations are discussed

using specific examples. INTRODUCTION The analog computer has three main applications in the nuclear reactor field: safety, design, and control studies. Of the three, safety is the most important

application.

It is necessary to know, prior to reactor startup, the consequences of any credible accident that

would affect the reactor. Studies concerned with accident transients are generally referred to as hazards analysis or safeguards study. The results of these studies are reported to the AEC Safeguards Committee prior to startup and also from time to time after the reactor is in use. This type of study is usually made by using an analog computer to simulate the

Nuclear

being used as prime of electric generation power and as of fissionable The nuclear reproducers isotopes. actor power plant system is a complex assembly of subsystems. The design of such a plant poses many complex and interrelated problems. The response to such a system cannot be obtained using normal mathematical techniques since the system is nonlinear and highly complex. By simulating the system on an analog computer, much information can be obtained which is greatly helpful in optimizing plant design. The analog computer also gives the design engineer a realistic &dquo;feel&dquo; for the operation of the system and for effects of changes in the various movers

reactors are now

in the

parameters. Reactor operators have been responsible for the control of reactor power, temperatures, and other variables. Reactor control-system engidecide what control functions should be made automatic. The analog computer approach is valuable in doing this since it enables the engineer to synthesize proposed automatic controls and test their operation as part of a simulation of the entire system. It also enables him to tie actual controllers into the simulation and so to get a realistic &dquo;feel&dquo; for their effect on the system.

dependent neers

must

23

Single-Node

REACTOR KINETICS SIMULATION Reactor kinetics

defined

by dependent diffusion equation:

where (])

are

the time- and space-

is neutron flux

shape factor) density a By assuming point reactor, the diffusion tion can be simplified to: B is

n

buckling

(a

is neutron

where

equa-

,

k is the effective

multiplication

factor of the

reactor

8k

1)~k, the reactivity R is delayed neutron yield (when 8k = Q, is (k

1* is

There are three basic types of single-node kinetics simulators: (1) one amplifier per group of delayed neutrons, (2) single amplifier using a complex input impedance, and (3) single amplifier using a complex feedback impedance. The circuit using one amplifier per group of delayed neutrons is shown in Figure 1. Figure 1 also includes the temperature circuit. Each group of delayed neutrons is represented by an integrator. The outputs of the integrators, 2BC~ are summed with a function of the power level, n, in the following amplifiers to generate dn/dt, which is integrated to form the power level, n. The circuit using one amplifier with a complex input impedance is shown in Figure 2. The six integrators and the summer are replaced with one amplifier. The remainder of the circuits are the same as in Figure 1. Each lag network going into the summing amplifier represents one group of delayed

-

neutrons. re-

activity =11 dollar) mean

effective lifetime of prompt neutrons

Xi

is number of delayed neutron groups is decay constant of the ith group of delayed

Ci

fission prodto another state with the emission of further neutrons) is concentration of ith group of delayed neu-

m

Kinetics Simulators

This trons

amplifier is used to simulate the delayed only. The transfer function for it is:

neu-

neutron precursors (unstable ucts which will in time

decay

where

tron precursors

This now gives us an ordinary derivative to work with. This set of equations, easily solvable on an analog computer, is the set normally used in reactor simulation. Using the finite difference method, a set of kinetics equations has been derived3where finite increments have been taken in space. These equations, given below, describe a reactor in both time and one-dimensional space.

and and

be

Equation

2

Terms in

Equations

can

manipulated

5 and 6

are

to

obtain:

then

compared

to

obtain:

and, The circuit using one amplifier with a complex feedback impedance is shown in Figure 3. All the where A, D, E, F, and G

coefficients i is reference to the ith group of are

delayed j

neutron precursors is reference to the jth spatial

reactor

node

This is the set of equations that is used for multinode reactor simulators.

24

amplifiers

in

Figure

1

are

replaced

fier. The temperature feedback and are the same as in Figure 1. The transfer function for the

with

one

ampli-

reactivity

circuits

amplifier is:

Equations 1 function, given

and 2 here:

are

solved for their transfer

where T; =11/Xi. Equations 7 and 8 are then pared to solve for component values:

com-

The three circuits described here each have their place in nuclear reactor studies. The circuit in Figure 1 is generally used when sufficient equipment is available or when it is desired to put initial conditions in the kinetics simulation. The circuit in Figure 3 is used when it is necessary to save amplifiers and only a single node reactor is being studied. This is because dn /dt is not explicitly available. The circuit in Figure 2 is generally used when it is necessary to save amplifiers and dn dt must be available, such as in a multinode reactor simulation. LARGE RANGE REACTOR SIMULATORS

The power level of a nuclear reactor can increase by as much as a factor of 101? from source level to peak flux levels during startup. It is necessary to simulate startup for certain studies, e.g., nuclear hazards and safety analysis. Since it is impractical to cover more than two decades in a single run on an analog computer, several alternate methods may be considered.

Logarithmic

1

Plutonium Recycle Test Reactor Neutron Kinetics (above and right)

Figure 2 Reactor Kinetics Simulator

FOLD OUT

Complex Input lmpedance (Foldout, lower left)

―――――――~*

Simulation

substitution of variables, the kinetics equabe solved for the logarithm of a power level rather than the power level itself. This method will permit the power level to vary over a range of 100 decades and remain within the computer lim- ,

By

tions

Figure

a

can

3 Reactor Kinetics Simulator

Figure

Complex Feedback Impedance (Foldout, lower right)

&dquo; .

FOLD OUT -

itations.

This method of simulation has been discussed by disadvantages in this type of simulation are the large number of multipliers that must be used (one multiplier for each group of delayed neutrons) and the high accuracy needed in the multipliers to obtain satisfactory results. Franz and Simcic.4 The

Exponential Transformation To solve the kinetics equations for large excursions, the following transformation can be made.55

is chosen large enough, the solution for a step of input k1(’,y = 8 or 10 dollars can be easily obtained. a may be either positive or negative, so that the transformation is equally useful for scaling up the solution for large positive or negative values of k,,,, (reactivity). Since eat is a continually increasing or decreasing scale factor, the transformation is most usefulI in scaling solutions of the reactor kinetics equations which continually increase or decrease; a bounded solution will eventually exceed the range of the

If

a

computer.

25

Linear Simulation An accurate method of simulating reactors to many decades is to manually rescale the problem. The simulator is started from source level with the variables equal to calculated initial conditions. Each succeeding range of two decades is treated as a separate problem with its initial conditions decover

Figure 4 Reactivity Simulator

termined by the computed values of the variables at the end of the previous two decades of operation. If many decades of power are to be covered, it is evident that this can be a time consuming process. In addition, the many manual operations involved greatly increase the probability of error by the

Circuit for Six-Node

Reactor Simulator

operator.

.

(Foldout, left) Point

Figure 5 Delayed-Group

Circuit for jth Node of Six-Node Reactor Simulator

The point storage

Figure 6 Temperature Circuit for jth Node of Six-Node Reactor Simulator

(Above)

Figure 7 (Below)

26

jth Node of Six-Node

the autocomputed variables at the end of a range, and use of these stored values to provide the necessary rescaling. For each variable in the simulation which requires rescaling in order for the reactor power level to progress through consecutive decades, a separate &dquo;point storage&dquo; circuit is used. The circuit consists of an integrator having operational modes the inverse of those of a normal computing integrator. For example, when the computing integrators are in the &dquo;Initial Condition&dquo; (IC) mode, the point storage integrators are in &dquo;Compute&dquo; mode and vice versa. The inverse operation can be achieved by the individual control of the IC relay for each point storage integrator. Normally, the IC relays for all integrators are operated from a common IC control bus. The IC relay coils of the integrators selected for point storage use are removed from the IC control bus and connected to spare points on the patch bay. The operational modes of these integrators can then be inverted through the use of a simple external relay circuit which derives its actuating signal from the IC control bus. An integrator operating as a point storage element receives the variable being stored in the IC input, the normal inputs not being used. As a result, the &dquo;Hold-Operate&dquo; relay condition is of no consequence and it may be allowed to follow its normal sequence of operations. It is obvious from the foregoing discussion that the point storage technique can be used to provide automatically the proper initial conditions for the delayed neutron concentrations at the beginning of each succeeding two decades of operation of the simulation of the reactor kinetics. The power level variable, n, can be made to cover the two-decade range of one volt to 100 volts by

technique

matic storage of values of the

(Foldout, lower left)

Kinetics Circuit for

Storage Technique’

Reactor

is based

on

using an initial condition of one volt and terminating each two-decade segment at 100 volts. Since the range is the same for all segments of the run, point storage is not required. However, the range covered by the delayed-neutron concentrations, C~, is not the same for each two-decade segment of the power level curve. Therefore, it is necessary to provide each a delayed-neutron concentration its output an initial condition of 1/100th of the final value attained by that concentration at the termination of the previous two-decade segment. The point storage technique is used to produce this

integrator having

as

effect. Each point storage integrator has

one of the six concentrations connected to delayed-neutron group its initial condition input. When the computing integrators are in the compute mode, the point storage integrators are in the initial condition mode.

Thus, the outputs of the integrators will &dquo;track&dquo; or follow the outputs of their respective computing integrators. At the end of

a run segment of two both the decades, computing and the storage intehave attained the same values. Due to the grators time constant of the initial condition network of the integrators, the storage integrator reaches its final value about two seconds later than the computing integrator. Consequently, the computer is placed in &dquo;Hold&dquo; mode of operation for two or three seconds before returning to IC mode. The outputs of the storage integrators are connected through coefficient potentiometers (set for an attenuation factor of 1/100) to the initial condition inputs of their associated computing integrators. At the moment the computer is placed in IC mode, the storage integrators switch to &dquo;Compute&dquo; mode. Since the normal inputs to the storage integrators are not used, these integrators maintain an output equal to the final value attained during the &dquo;Compute&dquo; cycle. These outputs, attenuated by a factor of 1/100, serve as the new initial condition of the computing integrators for the next run segment of two decades. It is necessary to place the computer in &dquo;Hold&dquo; mode for two or three seconds before starting the next higher two-decade run segment. This allows sufficient time for storage integrators to adjust their outputs to the new output values of the computing integrators. They are then in the proper condition to track their respective computing integrators through another run segment.

satisfactory for a heterogeneous Assuming a single-node reactor permits using a simple kinetics model, leaving most of the analog equipment to simulate temperature and other effects. The multinode reactor model produces a good simulation of a heterogeneous reactor, such as a Hanford type production reactor. This type of kinetics model uses proportionally larger amounts of equipment, leaving less to simulate the temperacircumstances is reactor.

and other effects. There are three basic one-dimensional ways in which the reactor can be divided for model simulations : top-to-bottom, side-to-side, and front-to-rear. The basic simulation for each of these models is identical. The only variations are in the coefficients and rod functions. If sufficient equipment is available, a two-, or even three-, dimensional simulator could be designed using the same techniques as for a one-dimensional model. Assuming the reactor flux distribution to be symmetrical about the center requires that only half the reactor be simulated. The other half would then be a mirror image of the simulated half. A typical six-node reactor simulation is shown in Figures 4, 5, 6, and 7. Figure 4 shows the circuitry for generating the total reactivity, 8k,. The inputs to this circuit are the flux, N,, and the metal temperature, Ti, The rod functions are generated in Figure 4. The differential relays (K,-K,) determine when the rods enter each node. The circuit in Figure 5 generates the delayed neutron contributions. The inputs to the circuits are the respective flux levels, N,, in the six nodes. The outputs are fed back into the kinetics to contribute to the flux generation. This circuit is for only three groups of delayed neutrons. Three rather than six or more groups were used because of the limited amount of equipment available. The metal temperature simulation is shown in Figure 6. The inputs are the respective flux levels and integrated power after time of the incident considered in this study. The temperatures, T,, are fed back to the reactivity circuit in Figure 4 to generate the reactivity contributions from the metal temperature

tures.

The kinetics simulation is shown in Figure 7. Flux levels, N,, (i = 0, 1, 2, 3, 4, 5) are generated by the inputs $k~, ~ ~~~C2~, f (Ni-1), and f(Ni+l). The circuits shown in the lower part of Figure 7 are controllers

that Multinode Reactor Simulation

(or have been) normally studied assuming point reactor. This assumption is good when the reactor is homogeneous, and under some Reactors

a

are

are

used

only

in

initializing

the

problem. The

initial flux level in each of the six nodes is set in each of the controllers. The reactivity is then adjusted until zero output is obtained from each of the controllers. The reactor system is now ready to insert whatever disturbance is to be analyzed.

27

REACTOR CONTROL

The control theory that is applied to other systems can also ~be applied to nuclear reactors. The problem is in being able to adequately describe the

system. There are two basic approaches to the problem. One is to consider the reactor as a single-variable control element, which is valid for physically small reactors. The other approach is to consider the reactor as a multivariable control element, which is valid for any reactor.

Single-Node

Reactor Control

The methods of studying control of single-node are well covered in presently available texts.7 The discussion here is centered on control simulations and can serve as a basis for comparison in the section on multinode reactor control. The reactor kinetics equations, Equations 1 and 2, have been presented previously. The simplified reactor transfer function can be derived by proper manipulation of these equations and is given in Equation 8, repeated here for easy reference. reactors

This is the transfer function used in most of the literature on reactor control. With this transfer function, only the total reactor power can be controlled. System analyses can be conducted using this transfer function and simulating the reactor on an analog computer. The simulation is shown in Figure 1 and has been discussed previously. An example of a single-node reactor control problem is a study that was made on the Plutonium Recycle Test Reactor. During the design stage of the reactor, a complete analysis of the system was made current

using

an

analog computer.&dquo;

When the actual controller was received in plant, it was wired into an analog computer. The computer was programmed to have the same characteristics as the actual reactor, and the controller then controlled the computer. Only those connections on the control panel were made that were necessary for the evaluation. Relays and recorders were replaced by switches and electrical loads to make the system operate properly. The controller components used in the study were: the operator control panel, BTU compensator, error former, low-level period control amplifier, two-period-control amplifiers, automatic valve controller, and all associated power supplies and slide wire assemblies.9

28

A block diagram of the system is shown in Figure 8. The controller is shown as a single block with power and period as the input signals, and control valve position as the output. The transducers are indicated by blocks marked &dquo;C.L.&dquo;. The remainder of the diagram is self-explanatory. All blocks shown in the diagram, with the exception of the controller block, were simulated on the computer. The analog circuits for the transducers are shown in

Figure

9.

Multinode Reactor Control Hanford reactor, the problem consists of than controlling power; the neutron flux distribution must also be controlled. A large graphite moderated reactor is susceptible to spatial neutron flux oscillations or &dquo;cycles&dquo;. Although the total power of the reactor may be as desired, the power can be peaked at some locations within the core. This phenomenon is explained by the reactor diffusion equation, a partial differential equation involving time and space: In

a

more

To manually derive a reactor transfer function from the reactor diffusion equation would be an exceedingly difficult task. However, to adequately describe a large reactor for control purposes, knowledge of this spatial transfer function is needed. This problem lends itself to the analog computer method of solution. If a reactor is considered a black box, the inputs are reactivity and the outputs are flux. There would be many inputs (e.g., control rods) and an infinite number of outputs (corresponding to each flux point in the reactor). With little error, the number of outputs could be represented by a finite number, and for convenience the number of outputs could equal the number of inputs. A convenient and realistic representation of a reactor is a finite number of nodes, each with an input and output, and each interrelated according to the diffusion equation. The computer circuit for such a representation is shown in Figures 4, 5, 6, and 7 which have been discussed previously. This is a six-node model and is considered to be adequate for many types of studies. Representing the reactor by a nodal model does not solve the problem of automatically controlling the multivariable system. This model does, however, give us something to work with in solving the problem of automatic control.

Figure

8- Plutonium

Recycle Test

Reactor Controller Evaluation Block

Diagram

Figure 9-Controller Transducer Circuits

29

PLANT SIMULATION

simulate the entire

The aircraft industry has used plant simulators for many years in the design, testing, and operation of aircraft and missiles. The concept of building a plant simulator has carried over into the nuclear field.&dquo; The simulator acts as a pilot plant, a desirable item that it is otherwise not practical to construct when working with reactors. A simulator for the Enrico Fermi Atomic Power Plant was used in the design and development stages and is now being used for operator training.ll At this facility they have gone to the extent of simulating the reactor control panel to make it possible to teach the operator to feel at home controlling the reactor, even before he sees the actual control panel. The facility, designed to train operators for normal and emergency plant operation, consists of the simulator, an instruction console, and two control consoles. The purposes of a typical reactor simulator are: modification and variation of design, control-systems development, investigation of the effects of changing reactor constants, evaluation and debugging of components, observation of effects of control interrelations and of effects of disturbances, routine training of operators, training of operators to handle emergencies, simulation of various troubles to show their effect, predicting the performance of man-machine systems, making highspeed predictions, and analyzing hazards. The major application of a reactor plant simulation facility is that of operator training. Because of the complex nature of the control and operation of a nuclear plant, it is expected that retraining of operators might be beneficial even for normal operating conditions. One important feature is that it could train teams of operators to perform simultaneous tasks, and thus familiarize them with the team work required. Included in operator training is plant operation during abnormal and dangerous conditions, which can be simulated safely, but which could not be demonstrated in an actual operating plant. Another major application of a reactor-plant simulator is that of supervisory control. Alternate operating procedures could be observed on the simulator to determine which is the most desirable. The most desirable one may be the one which involves the least cost, or the highest safety, or the highest production or any combination of the above. Also, an important aspect of supervisory control concerns prediction of optimum times for action. One example is the case where xenon poisoning affects the startup. Because of the long-term effects of xenon, it is desirable to change the time scale and

30

xenon

transient in

a

short time;

predict the xenon effects long before the effects actually appear. A third major application is that of engineering studies. Detailed studies are required to analyze hazards and to determine operating characteristics of various components under specified conditions. thus,

we can

In addition to the hazard studies, it is desirable during maintenance to be able to observe the correct (simulated) action of various controllers and components in the system. A reactor simulator should include three components in addition to the building and associated services: a simulation of the plant, a simulation of

the control room, and a problem generator. The simulation of the plant would be composed mostly of general-purpose analog computing components; the control room simulation would have mockups of control consoles, panels, and flow diagrams. It should also contain many actual controllers, along with indicating and recording instruments. The reactor plant can be divided into three separate sections: (1) the reactor kinetics, (2) the primary loop, and (3) the secondary loop. The kinetics include the generation of power (or flux) in each of the four regions (or nodes), six groups of delayed neutrons per node, control rods, source, heat transfer to water and metal, xenon generation, and all reactivities. The primary loop includes the heat exchangers (primary to secondary), coolant transports from reactor to heat exchangers to reactor, coolant pumps, and primary loop pressurizer. The secondary loop includes the heat exchangers (primary to secondary), turbine generators and other loading effects, dump condensers (transfer heat from secondary loop to river), surge tanks, and condensate pumps. A block diagram of the kinetics model is shown in Figure 10. This diagram shows the interrelations of the various parameters in a four-node kinetics

model.

Figure 11 is a block diagram of the primary loop. This shows the flow of coolant from the reactor to the steam generator and back to the reactor. The secondary loop is illustrated in Figure 12. In the model chosen, it was decided to lump parameters so that three steam generators, three dump condensers, and two surge tanks would be simulated. This lumping of parameters is considered adequate to study interactions between system components.

Figure

10-Kinetics

Figure 11 - Primary-Loop Block Test Reactor

(Four-Node

Reactor) Block Diagram

Figure

12-Reactor Simulator

Diagram

for Plutonium

Recycle

Secondary Loop

31

Xenon

Poisoning The buildup and decay of xenon and the poisoning attributable to the xenon can be easily studied using an analog computer.&dquo; The equations describing xenon are:

Since

is less than 1015 and

a,