TEE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING A NUCLEAR REACTOR SIMULATOR FOR TEACHING PURPOSES L. W. Orr Wo. Kerr H. J. Gomberg February, 1956 IP-147

ACKNOWLEDGEMENT We wish to express our appreciation to the authors and the Engineering Research Institute for permission to give this prepaper limited distribution under the Industry Program of the College of Engineering.

Ab stract The nuclear power reactor furnishes an excellent illustration of many of the principles and problems involved in the field of Nuclear Engineering. The dynamics of a system which incorporates a reactor as a heat source can be treated very successfully by the use of an analog computer or simulator. A portable electronic reactor simulator suitable for teaching purposes has been constructed. Its design and operation are discussed.

A NUCLEAR REACTOR SIMULATOR FOR TEACHING PURPOSES Investigation of Reactor Power Plant Systems Perhaps the most interest provoking situation in which controlled nuclear reactions are used in an engineering application is in the nuclear power reactor. This application is a good one for teaching purposes, because it illustrates several types and complexities of nuclear measurement. The larger problem of correlating these measurements with measurements of temperature, flow, etc., and using this information to control the reactor is also worth considerable study. The nuclear power reactor is part of an integrated system in which a nuclear reactor serves as a source of power, say in a steam-electric generating station. The problem has several interesting facets, that of reactor control, that of the control of the steam-electric part of the system, and the overall problem of determining the dynamic behavior of the integrated complex. In Figure 1 is shown a block diagram of a possible simplified arrangement of such a system. The numbers indicate: (1) Reactor (2) Control Rod Drive (3) Transducer, Error Signal to Control Rod Drive (4) Neutron Measuring Device (5) Comparator (6) Heat Transfer, Reactor Core to Coolant (7) Heat Exchanger, Reactor Coolant to Steam (8) Turbine (9) Electric Generator (10) Coolant Temperature and Flow Measuring Device (11) Steam Temperature and Flow Measuring Device (12) Power Demand Input -l

Using the appropriate unclassified literature and reasonable engineering approximations, a set of equations can be written to represent the flow of energy and information in the system.lp2'3 Two items are of primary concern: nuclear kinetics in the reactor itself, and heat transfer in the reactor and the heat exchanger. In addition, since most reactors are designed in such a way that the reactivity is a function of temperature, the interrelationship of the two must be established. Investigation shows that because of this relationship between reactivity and temperature, the system has many of the characteristics of a feedback amplifier. It can thus be treated analytically by the well developed techniques that are available for the analysis of feedback amplifiers and automatic control systems. In most cases of interest the reactor kinetic equations are non-linear, hence the additional interesting problem of treating non-linear differential equations is introduced. Several useful approaches to the problem are available. One of these involves linearization of the equations, and a subsequent application of the linear theory of feedback systems, such as the Bode diagram type analysis, to investigate system stability.4 This approach is interesting not only because it leads to useful results in this situation, but because it is so widely applicable to problems in the automatic control of large systems. The effect of the non-linearity may also be treated, and appropriate unclassified publications are available.5,6 Again the method of treating the problem may be much more generally applicable than to this specific situation. Analog Computers As an alternative or as a supplement to the analytical approach, it may be desirable to use an analog computer or reactor simulator to study system performance. The use of the simulator allows the non-linearity of the system to be 2 -

treated with facility. The simulator also allows easy and rapid variation of system parameters, and exhibits the changes in system performance which result therefrom. An important advantage of an analog computer is that as one observes solutions developing on the recorder an intuitive "feel" for system performance is developed. This may be much more difficult to achieve from a purely analytical treatment. The simulator which is used to treat the reactor-power-plant study can be either very elaborate or fairly simple. Although the simpler types have less precision, they have the advantages of compactness, portability and low cost. They serve admirably as demonstration units for teaching purposes. Portable Unit A portable reactor simulator has been constructed at the University of Michigan which is of sufficient accuracy for teaching purposes. The equations simulated are, using the notation of Bell & Straus,7 dt =( )k-ff) N- (/1 + eA Z XiCi + (T-T,)K1 (1) dt kl* -"!N + S, dCi = _ iCi + (e _A/l*.) iN, (2) Q = M dT + W(T-TC) (3) dt Here in addition to the symbols defined by Bell & Straus: T = temperature of the lumped reactor To= reference temperature of reactor TC = average coolant temperature K1 = temperature coefficient of reactivity - 3 -

Q = reactor heat output, directly proportional to N M = thermal capacity of reactor fuel elements W = heat transfer coefficient from reactor to coolant Basic Circuit The basic circuit used to simulate the reactor proper is essentially that of Bell & Straus. This circuit is an analog of the one group, thermal, point reactor equations. Five neutron delay groups are included.8 In addition, the effect of temperature rise in the reactor core caused by increased power level is also simulated. The position of the control rods, or 5keff, the coolant flow rate, the "source" magnitude and the temperature coefficient of reactivity, K1, are under the control of the operator from the front panel. A block diagram of the circuit is shown in Figure 2. A push-pull high gain dc amplifier, A, is connected as an integrator with feedback capacitor C1. The output voltages +N and -N are proportional to the time integral of the input current a, while the input' voltage remains substantially constant. The panel voltmeter Vn indicates the neutron flux N at any instant. A current source io simulates the production of prompt neutrons, the rate being determined by the setting of the 5keff control and the level of N. Current sources i1...... i5 represent the delayed neutrons, and the time constants of the delay networks, A1... A5, are set equal to the mean lives of their precursors. The time constants are fixed at the values: 0.62, 2.20, 6.52, 31.6 and 80.2 seconds. The delay fraction of each source is adjustable individually. The total delay fraction or per cent of delayed neutrons represents about 0.75 per cent of the total neutron production under steady-state conditions, There is a primary fixed source is which is independent of level N, but which is manually adjustable. A current K1NT is also added to the integrator input representing the effect of temperature on reactivity.

The loss due to neutron absorption and diffusion out of the reactor core region is given by id. This current is proportional to the level N. The sum Li of all currents including the negative current id denotes the excess neutron production and determines the rate ~of change of reactivity. When Zi- 0 for example, the reactor neutron population is constant. A current proportional to the +N voltage is fed to a capacitor C. This represents the heat input to the reactor, and C represents its thermal capacity. A control, simulating the coolant flow rate, determines the current flow rate out of capacitor C, and represents the rate of removal of heat from the reactor by the coolant. It is assumed that heat loss by radiation and conduction is small compared to that removed by the coolant. The voltage Vt on this capacitor represents the mean reactor temperature and is indicated by the panel voltmeter Vt. As mentioned above, the effect of temperature variation on the reactor is obtained by feeding an additional source of neutrons K1NT to the integrator input. K1 is the temperature coefficient of reactivity, and is negative in practical controlled reactors. Thus the current contribution is actually negative. Since this contribution is proportional to the product of flux level N and temperature T, an electronic multiplier M is used to obtain the required product. The output of the current generator K1 in the diagram is the required current, and is added to the integrator input. The magnitude of temperature coefficient of reactivity K1 is under the control of the operator. Circuit Details The circuits used to accomplish the desired result are grouped into two sections. The first section contains the prompt and delayed neutron sources, the source balance control, the 'keff control, the electronic integrator and the N meter. The circuitry associated with this section is quite similar to the original Bell & Straus7 circuit, except that the 6keff control is equipped with a vernier adjustment - 5 -

The second section contains the temperature circuit, coolant flow rate control, T meter, the N voltage converter, electronic multiplier and the K1 control. Temperature Circuit and Coolant Flow Rate Control Figure 3 is a simplified version of the temperature circuit showing the coolant flow rate control and T meter. The current il represents the heat generated in the reactor, and this flows into capacitor C100 representing the thermal capacity of the reactor. This current is proportional to the voltage difference between the +N voltage, and the capacitor voltage which is always small compared to +N. The current i2 flowing out of C100 through R102 and R104 represents the heat removed by the coolant, and this is proportional to the capacitor voltage. The capacitor voltage represents the mean reactor temperature and is indicated by the meter T. Spatial variation of temperature throughout the volume of the reactor is not represented in this simulator, since a lumped reactor is simulated. When the net capacitor current i! - i2 is zero, the mean reactor temperature remains constant. Changing the setting of the coolant flow rate control, R104 varies the current i2, representing a change in the coolant flow rate. The temperature changes, changing reactivity until a new stable operating power level is established. To give a suitable output to the electronic multiplier, an adjusted T voltage is required. This is furnished by the tap on R106. For the temperature corresponding to the stable shutdown value, this control is adjusted to give an adjusted T voltage of -8.0 volts. N Voltage Converter The N voltage converter is a 330 kc Hartley oscillator. This is shown in Figure 4 at the left side. The N voltage is decoupled, and used as the B+ supply voltage to the oscillator. The circuit constants and bias values are so arranged

that the amplitude of the oscillator output is approximately proportional to the N voltage over a fairly wide range of values, as indicated by the curve in Figure 5 This converted N voltage is used to feed the input of the electronic multiplier. Electronic Multiplier The product NT is required for simulating the effect of temperature upon reactivity. The circuit performs this by generating a source of "negative" neutrons proportional to K1NT as described above. Since K1 is generally negative in actual reactors, the current K1NT is bled off from the integrator input, as indicated in the block diagram, Figure 2. The circuit performing the multiplication is accurate to several per cent over the range of useful operation, which although perhaps not adequate for accurate computation is satisfactory for demonstration purposeso It is based on the principle that certain variable gain tubes have a transconductance vs grid voltage curve closely approximating a straight line. The 6SK7, operated at a fixed screen voltage of 100 volts gives an almost straight line relationship between transconductance and control grid voltage in the region of control grid voltages from -8 to -2 volts, The circuit in Figure 4 shows the method of obtaining zero gain at ec - 8v. by adding the neutralizing resistor R117. The converted N voltage is fed to the grid of the 6sK7 through 0109g while the adjusted T voltage, which is the grid bias, is fed in through R113o The plate load is a resonant tank tuned to o-scillator frequency, and the output is taken from the 6SK7 plate. This output is zero for e,. = 8v., and for other values it is proportional to the product NT. The multiplier output is rectified, and fed to a dc amplifier and inverter stage having as its output a current proportional to K1NTo The multiplier -7

response is indicated in Figure 6 for three values of N voltage. The output current is negative for a negative setting of K1 which is the normal mode of operation, and this current is drawn from the integrator input as described above. Acknowledgement is made of the interest and financial assistance of the Dow Chemical Company in the construction of this simulator. - 8 -

REFERENCES 1. Hurwitz, H.; Nucleonics, 5, No. 1, 61, (July 1949) 2,0 Schultz, M. A.; AECD-3163, November 6, 1950. 3. Stone, J. J. and Mann, E. R.; ORNL-1632, April 20, 1954. 4. Siegel, R. and Hurwitz, H.; ORNL-1632, April 20, 1954. 5. Chernick, J.; BNL-173, December 20, 1951. 6. Ergen, W. K. and Weinberg, A. M.; Physica, 20, 413, (1954). 7. Bell, P. R. and Straus, H. Ao; RSI, 21, 760 (August, 1950)> 8. Glasstone, S. and Edlund, Mo; The Elements of Nuclear Reactor Theory, D. Van Nostrand, New York, 1952.

3.IBLC 1 R PW P )2 6~~~~~ aIG. hi: r-~CK DIAGRAM, jEACTOR POWER PNa FIG. I BLOCK DIAGRAM, REACTOR POWER PLANT

COOLANT FLOW RATE T ~ ~ ~ ~~~0 dT~V MT HEAT kl PER I — ld- N INE~~~~~WIEUT i ~~~~KI KI NT +N-N CIK A A2 3 4 A5 FIG. 2 BLOCK DIAGRAM OF REACTOR SIMULATOR

+ 300 V C 100 1.0 RHO 1.2 M 12 AX7 r ' - R ~RIII 4-N RI _ lOOK OM R 102 C 220K + R 109 R 04 R 105 -300 V 08 FIG. 3 FLOW RATE CONTROL FOW RATE0 K FLOW RATE CONTROL

+ N ADJUSTED T VOLTAGE VOLTAGE + 300 V REG 4.7 K 4.7 K.005 k ~.005 NEUTRALIZING 470 K.01' RESISTOR R113 1L2.001 MULTIPLIER R117 H-O OUTPUT TO 220 K.001.001 RECTIFIER. (12 AU7) -H.001 R 115 22K C109 + 100 V.0008 REG. L50 100 K00 R116 IL I 220 _ K _ CONVERTED N VOLTAGE FIG. 4 N VOLTAGE CONVERTER AND ELECTRONIC MULTIPLIER

30 25 20 RMS OUTPUT VOLTS 15 10 100 200 DC N VOLTAGE - INPUT FIG. 5 N VOLTAGE CONVERTER RESPONSE

ADJUSTED T VOLTAGE OR 6 SK7 GRID BIAS -8 -6 -4 -2 RECTIFIED 5 MULTIPLIER OUTPUT CURRENT \ AMPS -10 - -20 FIG. 6 MULTIPLIER RESPONSE