v Y OAK RIDGE NATIONAL LABORATO operated by Rfc g3cc UNION CARBIDE CORPORATION NUCLEAR DiVISION LT for the U.S. ATOMIC ENERGY COMMISSION ORNL- TM~- 1647 o COPY NO. - -+ '3, DATE - October 13, 1966 GOV~ EXPERIMENTAL DYNAMIC ANALYSIS OF THE MOLTEN-SALT REACTOR EXPERIMENT* . = oUING z RELEASED FOR ANE: T. W. Kerlin and S. J. Ball ABSTRACT Dynamics tests were performed on the Molten-S5Salt Reactor Experiment (MSRE) for the full range of operating power levels to determine the power=to-reactivity frequency response. Three types of input disturbances were used: the pseudo-random binary reactivity input, the pulse reactivity input, and the step reactivity input. The frequency response of the uncontrolled reactor system displayed resonant behavior in which the frequency of oscillation and the damping increased with increasing power level. Measured periods of natural oscillation ranged from thirty minutes at 75 KW to two minutes at 7.5 MW. Thege oscillations were lightly damped at low power, but strongly damped at higher power. The measured results generally were in good agreement with predictions. The observed natural periods of oscillation and the shapes of the measured frequency response agreed very well with predictions. The absolute amplitude of the frequency response differed from predictions by a factor that was approximately constant in any test (though different tests at the same power level did not have the same bias). This bias difficulty is apparently partly due to eguipment limitations (sta.ndard MSRE control rods were used) and partly due to uncertainties in the parameters in the theoretical model. The mein conclusion is that the system has no operational stability problems and that the dynamic characteristics are essentially as predicted. ¥ Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. For presentation at the Winter Meeting of the American Nuclear Society to be held October 30-November 3%, 1966 in Pittsburgh, Pa. NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use at the Oak Ridge Nationel Laboratery. It is subject to revision or correction and therefore does not represent a final report. re LAV EE R BT R g o ceece .o Crape - [ei8 DOUUMENT B0s T o IVTRITOOEY S ey gy MO INVENTCHZ U Bl Dinoeks] 10 THE A.E.C. A;;‘;;&j{g@flaaa THEREIN, 7 ] fo — LEGAL NOTICE This report was prepared as an account of Government sponsored work, Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Mokes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this repert may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any informotion, epparatus, method, or process disclosed in this report. As used in the cbove, ‘‘parson acting on behalf of the Commission® includes any employee or contractor of the Commission, or employse of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contracter prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor. IT TIT v Vi VIiI ViIT IX Appendix A. Appendix B. iii CONTENTS IntrodUCtiOn.eeseseseessossovrsosasansnons Description of the MoRE . veeeeecrcssnves Theoretical PredictionS.esecessssencenssne A. Description of Mathematical Model... B. ResulfS.ceerecscensrsstsossscsasesnas Selection of Experimental MethodsS.seeess A, Characteristics of the MSRE RegulatingRod...l'......."‘..l.. B. Test Signals Used in the Experiments 1. Pseudo-random Binary Test..ev... 2. Pulse TestSiseeevrsarcscososncasns 3., Otep TestSeeeeeestesessssorenane Experimental ProcedureSievseeeeescessess A. TImplementation of Pseudo-random Binary TestSeecsaececosonsrecscscns B. Implementation of Pulse and StepTests-.ll...ll....l...'l...‘. Analysis ProcedUreSeceesrsecsssesssasssans A. Direct Anelysis of PRES TestS.iervaess B. Indirect Analysis of PRBS TestSeeese C. Step Response Test Analysis. ® - » L] * . » . D. DPulse Response Test AnalysiS.cacesas ResUltSieeceeessessasasercscanasssonasens A. Transient RespOnseSesssscessscsssesns B. Correlation FunctionS.eeessssesssens C. TFrequency RespONSESeecsecssccssasseass Interpretation of the Results..cevecevose ReferenCesSieieenscescsesorenessnssscoscons Potential Sources of Experimental Error Due to Equipment Limitations.. The Direct Method for Cross~Power Spectrlflfl Analysis.a-ooooc-OOOAA‘NOT‘CE LEGAL 55, OF us¢ letene rocy, COMPIETEnt pne> of any spformation, apparaty rivately owned rlg‘m.s;. ot sttes wh ' [, Assumes any liabili of mation, th respect 1© T & of any info ppat * As used in the above, ployee oF contractor of th:or i\ sm:h employee OF contrac digseminates, or Commiss mploymem with & with the O ion, OF his & qes access Oy contractoT. prov uch & O F F F P Hm 10 12 13 13 13 13 13 17 21, 21 21 22 22 32 32 45 L7 1 damages oyee of guch ©O y his emplo port, oF that the use rt may not iniringe . repott. _ n*' includes any em . to the extent that ntracior prepares, -1 I. INTRODUCTION A gseries of experiménts was performed on the Molten Salt Reactor Experiment (MSRE) to determine the frequency response of the uncontrolled reactor system. Tests were performed at eight different power levels ranging from zero to full power. Three different types of input disturbances were used to obtain the nuclear power to reacti#ity frequency response: the pseudo-random binary reactivity input, the pulse reactivity input, and the step reactivity input. Subsequent sections of this report will give a description of the system, a review of previously published theoretical predictions, a description of the testing procedures, and the experimental results. IT. DESCRIPTION OF THE MSRE The MSRE is a graphite-moderated, circulating-fuel reactor. The fuel is a mixture of the molten fluoride salts of uranium, lithium, beryllium, and zirconium.l The basic flow diagram is shown in Fig. 1. The flows and temperatures shown are nominal values which were calculated for operation at 10 MW, but heat transfer limitations at the radiator currently restrict maximum power operation to about Te5 MW, The molten fuel-bearing salt enters the core matrix at the bottom and passes up through the core in channels machined out of unclad, 2-inch graphite blocks. The heat generated in the fuel and that transferred from the graphite raise the fuel temperature about 50°F. When the system operates at reduced power, the flow rate is the same as at full power and the temperature rise through the core is smaller. The heated fuel salt travels to the primary heat exchanger, where it transfers heat to a non-fueled secondary salt before reentering the core. The heated secondary salt travels to an air-cooled radiator before returning to the primary heat exchanger. The design parameters of major importance from the standpoint of dynamics are shown in Table 1. A detailed description of the MSRE appears in Ref. 1. || PRIMARY SALT || LIF - 70% 1025° F Bef, ~ 23% . “ 2rfy — 5% I LOOF - Il They — 1% II I UF, — 1% I SECONDARY SALT | I LiF — 66% I A ou - 34% | Da ORNL-LR-DWG 56870 REACTOR CELL | 300°F AIR 167,000 cfm 100°F SPARE FILL AND FLUSH COOLANT FILL AND DRAIN TANK TANK DRAIN TANK DRAIN TANK (68 cu ft) (68 cuft) (40 cu ft) (68 cu ft) Fig. 1 MSRE Flow Diagram -3- Table 1, MSRE Design Data Nuclear Flow Heat Temperature coefficient of reactivity of the fuel, °F-1 Temperature coefficient of reactivity of the graphite, °F-1 Neutron lifetime, sec. Total delayed neutron fraction Reactivity loss due to fuel circulation, % 8K K Flow rate in the primary loop, gpm Flow rate in the secondary loop, gpnm Fuel transit time in the core, sec. Fuel transit time in external primary loop, sec. Total secondary loop transit time, sec. Transfer Fuel salt heat capacity, MW sec/°F Graphite heat capacity, MW sec/°F Heat exchanger heat capacity, MH sec/°F Bulk graphite ~ fuel salt heat transfer coefficient, MW/°F Fuel salt~heat exchanger metal heat transfer coefficient, MW/°F Heat exchanger metal, secondary salt-heat transfer coefficient, MW/°F Fraction of power generated in the fuel b7 x 1077 2,6 x lO"'5 L0002k .00666 ~0,.,212 1200 830 8.5 16.7 24 .2 L.2 3.6 0.02 0.36 Q.17 0.93 =l ITT. THEORETICAL PREDICTIONS A. Description of Mathematical Model Throughout the MSRE design effort, a wide variety of mathematical models was used to predict the dynamic behavior. We will limit our discussion here to the most up-to-date and detailed model reported, referred to in Ref. 2 as the "complete'" model. The core fluid flow and heat transfer eguations were represented by 18 fuel nodes and 9 graphite nodes. The nuclear power distribution and the nuclear importances for each node were derived from a 2-group neutron diffusion calculation. The flow rates and heat transfer coefficients for each node were determined from calculations based on full-scale hydraulic core mockup tests. The assumed flow mixing characteristics were verified by transient tests on the mockup. The neutron kinetic behavior was described by the usual space- independent equations with six delayed-neutron groups, but with modi- fications to include the dynamic effects of the circulation of precursors around the primary loop. The thermal reactivity feedback was computed by using a weighted nuclear importance for each of the 27 fuel and graphite nodes. The xenon poisoning reactivity feedback included iodine production and decay into xenon, xenon decay and burnup, and xenon absorption into the graphite. The transport of molten salt in the primary and secondary loop piping was described by a plug flow model, where heat transfer to the pipes was included. The primary heat exchanger and the salt-to-air heat exchanger were each represented by a 50-node model. B. Results of Theoretical Analysis Several different methods of solution were used on the various MSRE dynamics models, including analog and digital computer simulation (time response), frequency response analysis, and pole configuration analysis. The frequency response analyses can be directly compared to the experimental results, since the latter are readily cast in this form. Fig. 2 shows the theoretical MSRE inherent frequency response ORNL-DWG 65-9816 2 10,000 10 o0 2 5 001 2 5 o 2 5 t 2 S 10 2 5 100 FREQUENCY {rodians/sec) 90 80 70 PHASE OF E#%% 60 0 50 40 30 20 PHASE (deg) o NO FEEDBACK -70 ———m ‘ -90 00001 2 5 0001 2 5 oo 2 5 ot 2 5 1 2 5 10 FREQUENCY (radians/sec) Ffig. 2 MSRE Theoretical Frequency Response -6 characteristics for normalized neutron level response to reactivity perturbations at several power levels. It can be seen that the system becomes more oscillatory at progressively lower frequencies as the nominal power level decreases, though it is stable for all power levels of interest. An explanation of the inherent stability characteristics is given in Ref. 2. IV. SELECTION OF EXPERIMENTAL METHODS The selection of the experimental methods for the MSRE dynamics tests was based on the information required and on the capabilities of the available equipment. It may be seen from Fig. 2 that the most significant part of the frequency response is in the range 0.0l to 0.1 radians per second, since the amplitude peaks are in this fregquency range for the operating power levels of interest. This frequency range corresponds to long periods of natural oscillation (10 min. to 1 min.). This emphasis on low frequency results fortunately made it possible to obtain the important part of the system frequency response using the standard MSRE control rods to introduce the input reactivity perturvations. In this section, we will examine the characteristics of the MSRE regulating rod and the properties of the test signals used. A, Characteristics of the MSRE Regulating Rod The MSRE has three control rods, each with an active length of 59.4 inches. One rod is normally designated as the regulating rod and is used for fine control. The other two rods are shim rods used for coarse adjustments. The rods are actually flexible, stainless steel hoses on which are strung gadolinium oxide poison cylinders. The rods are mounted in thimbles which have two 30° offsetting bends so that the rods can be centrally located even though there was no room for the control rod drive assemblies above the central axis of the core. The maximum rod speed is ~0.5 inches/second. The three control rods are identical. Figures 3 and 4 show the control rod and :drive assembly. The position indication for each rod is obtained from two synchros geared t¢ the rod drive mechanism. Synchro number 1 is used for coarse position indication and has a ORNL-LR-DWG 67311 REVERSIBLE DRIVE MOTOR COOLANT TO DRIVE ASSEMBLY COOLING GAS INLET COOLING GAS SOLENOID ACTUATED CONTAINER RELEASE ' POSITION INDICATOR :.\ 7z SYNCHRO TRANSMITTER ‘l» FIXED DRIVE SUPPORT AND | | UPPER LIMIT ——{ | 3in. CONTAINMENT TUBE SWITCH i) —COOLANT TO orwe uwT b ;'! POISON ELEMENTS l i i % in.0.0.-3045.5.- FLEXIBLE LOWER LT ,,.! '. HOSE CABLE " I!"'," 5“"0 3 h"glth\ ORNE \ T = periodicity of the test signal Note that the input power spectrum is a real quantity since Cll(T) is always an even function of 1. The frequency response, G(jak), is given by L JNTC (7) cos @ T dt . T Jo 12 Re[G(J&k)] = T 1 = J; Cll(T) cos @ T dt l - Im[G(j&k)] = - T'J; Clg(T) Sin @ T dt 1 T T j; Cll(T) cos w T dt The magnitude ratio, MR, and the phase, 6, are given by: () = o (Releim) HF v (T, elin)])" o )T (605w ) ] e(wk) = tan fig-Tfizsa;Tj . A Fortran computer code for the IBM-T7090 or IBM-360 called CABS(Q) was prepared to carry out these computations. -21- C. Step Response Test Analysis The step response tests were analyzed using a digital computer code which implements Samulon‘s method.(5’lo) D. Pulse Response Test Analysis Although pulse response tests were attempted at power levels of ~ 0, 75 KW, 465 KW, and 1 MW, the only successful runs were the ones made at zero power. This was because the low frequency random fluctuations in heat load at low (but non-zero) powers drastically reduced the signal-to-noise ratio. At these powers, the radiator is cooled primarily by natural convection and radiation, and is consequently sensitive to atmospheric disturbances. At higher powers, where most of the cooling was due to forced convection, these fluctuations were not apparent. The zero power tests required extremely accurate core temperature control and rod positioning in order to avoid drift of the flux level. Since a zero power reactor is an integrating system, a pulse reactivity input results in a change in steady-state flux output, and Fourier transforms are valid only if the response function eventually returns to its initial value. We got around this problem by first numerically filtering the output response data through a high-pass network HP(s) with a transfer function _50s HP(s) = 55551 then performing the numerical Fourier transform, and finally compensating the resulting frequency response for the high-pass filter characteristics. The numerical Fourier transform calculations were done by a digital (12) which is mathematically equivalent computer code using a novel method to the standard technique of summing the products of f(t) ¢ cos wt and f(t) * sin wt, but which reduces the computing time by a factor of 3. VII. RESUILTS The experimental data were analyzed to give correlation functions, input power spectra and cross power spectra, and frequency responses. These results are given in this section along with the directly observable 00 transient response to system disturbances. A. Transient Responses At each power level, a transient was induced by inserting a reactivity pulse or step, or by simply allowing the system to seek equilibrium after the completion of some other test. These transient responses are informative in themselves since they demonstrate the damping and the natural frequency of oscillation of the system. Figure 11 shows the observed transient response. At 0.075 MW, it took over two hours for the flux to return to equilibrium. B. Correlation Functions The pseudo-random binary tests were analyzed by the direct method and the indirect method. Autocorrelation functions of the input and cross correlation functions of the input and output were obtained &s intermediate results by the indirect method. The correlation functions had the expected appearance in all tests until the 2-1/2 MW tests. In that test, spikes appeared in the correlation functions which have been present in all tests since then. The spikes always appeared at points 432 sec. from each end of a period in the 511 bit tests (~ 1700 sec. period), and 34 sec. from each end in the 127 bit tests (~ 630 sec. period). Figures 12 through 19 show typical autocorrelation functions and cross-correlation functions for tests before the 2-1/2 MW test and after the 2-1/2 MW test. The reason for the appearance of the spikes is not yet known. The only significant difference noted in the input signal at 0.465 MW and at 2-1/2 MW was a change in effective pulse duration for positive and negative pulses. At 0.465 MW, the duration of a pulse above the mid- point was the same as the duration of a pulse below the midpoint. At 2-1/2 MW, the duration of pulses below the midpoint was longer than the duration of pulses above the midpoint. This was apparently due to changes in the coasting characteristics of the rod. Calculations were performed on pseudo-random binary sequences in which the pulse duration was changed for positive and negative pulses to determine whether this would cause spikes in the autocorrelation function. These calculations showed spikes for some sequence lengths but not for others. In particular, spikes were not found in these calculations for a 511 bit sequence. The conflicting indications furnished by POWER CHANGE (%) -23- ORNL-DWG 66-10284 0075 MW 0 7.5 MW 0O 2 4 6 8 10 12 14 16 {18 20 22 24 26 28 30 32 34 36 TIME (min) Fig. 11 MSRE Power Level Transients Autocorrelation Function (Arbitrary Units) ORNL DWG. 68-11078 Correlation Time (sec) Fig. 12 Input Autocorrelation Function for a 511 Bit PRBS Test at 0.465 MW Cross=Correlation Function {Arbitrary Units) ORNL DWG. 66-11079 = l,llllli?. 400 800 1200 Correlation Time (sec) Fig. 13 Cross-Correlation Function for a 511 Bit PRBS Test at 0.465 MW 1600 Autocorrelation Function (Arbitrary Units) o (‘lllll x =X =[x x x !=n x ] & F YTTrwrmwmymrTys ORNL DWG. 66-11080 —98— 160 320 480 640 Correlation Tine (sec) Fig. 14 Input Autocorrelation Function for a 127 Bit PRBS Test at 1.0 MW Cross-Correlation Function (Arbitrary Units) ORNL DWG. 4646-11081 T F I * L lklllllllflflflllflll wlx ¥ 2 X LeacertneorendT EX N x M |x x = 160 Fig. 15 Cross-Correlation Function for a 127 Bit PRBS Test at 1.0 MW 320 Correlation Time {sec) 640 -La_ Autocorrelation Function {Arbitrary Units) ORNL DWG. 66-11082 W‘—wwfiw 400 800 1200 Correlation Time (sec) Fig. 16 Input Autocorrelation Function for a 511 Bit PRES Test at 2.5 MH Crose-Carrelation Punction (Arbitrary Units) ORNL DWG. 66-11083 X XN N X ot XN X X 400 800 1200 Correlation Time (sec) Fig. 17 Cross-Correlation Function for a 511 Bit PRBS Test at 2.5 MW "63"' Autocorrelation Function (Arbitrary Units) ORNL DWG. 65-11084 0 160 320 k8o Correlation Time (sec) Fig. 18 Input Autocorrelation Function for a 127 Bit PRBS Test at 2.5 MW 640 Cross-Correlation Function (Arbitrary Unite) ORNL DWG. 66-11085 » = ::nl“’" x> ® » qufl"' : © 160 320 180 Correlation Pime (sec) Fig. 19 Cross-Correlation Function for a 127 Bit PRBS Test at 2.5 MW 640 -Tg_ ~30w these calculations have not yet been resolved. However, we should note that this unexpected feature of the correlation functions does not invalidate the final result, the freguency response. It simply means that the spectral properties of the input and output are slightly different than anticipated, but that the ratio of cross power spectrum to input power spectrum still gives a valid frequency response. C. JFrequency Responses The results of the frequency response analyses are shown in PFigures 20 through 28. The legend in each figure indicates the type of test and the analysis procedure. The theoretical curves were taken from (2) corresponded to available calculations. The curves for the other cases previously published results when the experimental power levels were obtained using the same procedures as were used to obtain the (12 the random noise in the neutron flux signal are also shown with the published results. Results obtained by Roux and Fry ) by analyzing zero-power results. These results were normalized to the theoretical results at 9 rad/sec. These points were obtained by taking the square root of the measured power spectral density (PSD) of the neutron flux signal after subtracting the background PSD. This gives a result which is proportional to the amplitude of the flux-to-reactivity frequency response, assuming that the observed noise is the result of a white noise reactivity input. The natural period of oscillation of the system was determined either by direct observation of a transient or by location of the peak of the frequency response. These results, along with theoretical predictions, are shown in Fig. 29. VIII. INTERPRETATION OF THE RESULTS The objectives of the dynamic testing program are twofold: 1) provide information on the stability and operability of the system; 2) provide information for checking procedures for making theoretical predictions so that future calculation on this and other similar systems will be improved. The results are interpreted in terms of these two objectives. ORNL-DWG 66-10286 104 ZERO POWER 5 19 BIT PRBS DIRECT ANALYSIS @ 19 BIT PRBS INDIRECT ANALYSIS & 63 BIT PRBS DIRECT ANALYSIS A 2 63 BIT PRBS INDIRECT ANALYSIS V¥ 7 sec. PULSE, TEST NO. | v 103 7 sec. PULSE , TEST NO. 2 0 NOISE ANALYSIS o 5 m M‘/ko , 102 5 2 10' 0.001 0005001 002 005 Q! 02 ©0O5 10 20 50 100 200 500 1000 w, FREQUENCY (radians/sec) 20 ORNL—-DWG 65-8907A ZERO POWER 1O | 19 BIT PRBS—DIRECT ANALYSIS A 0 | 19 BIT PRBS—INDIRECT ANALYSIS @ 63 BIT PRBS—DIRECT ANALYSIS v 4 —10 163 BIT PRBS—INDIRECT ANALYSIS ¥ 11 -20 | 7 sec. PULSE, TEST NO. | 0 bn“ - > 30 7 sec PULSE, TEST NO. 2 A Agfi,}'f ™ Q — 2 4 A A4‘3‘F;"‘ ) w —40 s X 2 4l a® TS Y vgig < 18185 e Lil Lhvily E -50 = Y Vv—ive A/ ) _60 / =70 / ° L L4 0 —80 ’“’r ) ‘_-__-—‘l/ _90 A -100 0.001 0.01 04 1 10 w, FREQUENCY (radians/sec) 8N/N ' Q Fig. 20 PFrequency Response of at Zero Power; e Iuel Circulating . ORNL-DWG 66-10708 ZFRO POWER T=See Pulse, Test No. 1 T=Sec Pulse, Test MNo. 2 3=1/2-Sec Pulse, Test Nu. 3 3-1'/2-Bec Pulse, Test No. 4 Ncise Analysis Prequency o, red/sec ORNL -DWG 65-8909A 20 PHASE (deg) ZERO POWER 7sec PULSE , TEST NO. | e 7 sec PULSE , TEST Nu. 2 A 3.5 sec PULSE ,TESTNO. 3 & 3.5 sec PULSE ,TEST NO. 4 © 0.001 0.01 0.1 1.0 10 w, FREQUENCY (radians /sec ) BN/NO Fig. 21 Frequency Response of _B—K_TK_ at Zero Power; Fuel Static PHASE (deg) -70 0.001 ORNL-DWG 66-476T7 POWER LEVEL 0.075 MW 51 BIT PRBS DIRECT ANALYSIS O + 511 BIT PRBS INDIRECT ANALYSIS O THEORETICAL 0.001 0.002 0.005 0.0¢ 0.02 0.05 oA 0.2 0.5 1.0 FREQUENCY {radions /sec) ORNL-DWG 66~4768 POWER LEVEL 0.075 MW 511 BIT PRBS DIRECT ANALYSIS @ 511 BIT PRBS INDIRECT ANALYSIS O THEORETICAL o 0.002 0005 0.04 0.02 0.05 0. 0.2 0.5 {.0 FREQUENCY ({radians /sec) BN /N 0 Fig. 22 Frequency Response of ; Power = 0.075 MW 6K7KO -36_ ORNL-DWG 66~-4769 Ton POWER LEVEL 0.465 MW 511 BIT PRBS DIRECT ANALYSIS O 5 511 BIT PRBS INDIRECT ANALYSIS O 2 /v, THEORETICAL A 10° 5 2 102 0.00f 0.002 0.005 0.0f 0.02 0.05 0.4 0.2 0.5 {.0 FREQUENCY (radians /sec) ORNL-OWG 66— 4770 60 T T T TTTTT T T T T 50 POWER LEVEL 0.465 MW 1] 511 BIT PRBS DIRECT ANALYSIS ® 40 511 BIT PRBS INDIRECT ANALYSIS O || 30 i 20 . 10 = THEORETICAL L 0 2 ué -10 Gl 1 a ~e0 5 - fi“ Attt - T - g -V e . e e . - L R - L4711 =30 . k gi/ P; & =40 o oo - 7 o _—— o ‘&\r > ?%'.J_fl %lale -50 o7 ol 1 o o° ~-60 -70 -80 0.00f 0.002 0.005 0.04 0.02 0.05 OA 0.2 0.5 1.0 FREQUENCY (radians/sec) SN/N o Fig. 23 Frequency Response of ; Power = 0.465 MW BK;KO -37~- 4 ORNL-DWG 66-4771 POWER LEVEL 1.0 MW 127 BIT PRBS—~DIRECT ANALYSIS & 27 BIT PRBS—INDIRECT ANALYSIS O 511 BIT PRBS~DIRECT ANALYSIS © 51 BIT PRBS—INDIRECT ANALYSIS O 10 Wyl 2 SAMPLING INTERVALS: 77 INDIRECT ANALYSIS —1 sec 3 DIRECT ANALYSIS-0.25 sec 10 THEORETICAL 5 2 10° 0.00¢ 0.002 0.005 0.0¢ 0.02 0.05 o} 0.2 0.5 1.0 FREQUENCY (radions /sec) ORNL—-DWG 66-4772 POWER LEVEL 1.0 MW 5i1 BIT PRBS DIRECT ANALYSIS @ 511 BIT PRBS INDIRECT ANALYSIS A 127 BIT PRBS DIRECT ANALYSIS A PHASE (deg) THEORETICAL aAd 0 0.00¢ 0.002 0.005 0.0f 0.02 0.05 0.t 0.2 0.5 .0 FREQUENCY (radions/sec) BN/N o Fig. 24 Frequency Response of ; Power = 1.0 MW 5K;KO -38- ORNL-DWG 86-10042 POWER LEVEL - 2.5 Mw STEP TEST 127 BIT PRBS - DIRECT ANALYSIS 127 BIT PRBS - INDIRECT ANALYSIS 51t B8IT PRBS - DIRECT ANALYSIS 511 BIT PRBS - INDIRECT ANALYSIS g > O oo 0002 0005 001 002 005 Ol 02 05 10 FREQUENCY (radions/sec) ORNL-DWG €6-10038 POWER LEVEL - 25 Mw STEP TEST o 127 BIT PRBS - DIRECT ANALYSIS @ 127 BIT PRBS - INDIRECT ANALYSIS A 511 BIT PRBS ~ DIRECT ANALYSIS & 511 BIT PRBS - INDIRECT ANALYSIS v o * Y A PHASE (deg) THEORET ® A O - 48" APy g " -50 VIrayg & -70 0001 0002 0005 00! 002 005 Of 02 05 10 FREQUENCY (rodians/sec) SN/N o Fig. 25 Frequency Response of ; Power = 2.5 MW 6K7KO ORNL-DWG 66-10041 POWER LEVEL - 5.0 Mw STEP TEST o 127 BIT PRBS -DIRECT ANALYSIS @ 127 BIT PRBS - INDIRECT ANALYSIS & 51 BIT PRBS - DIRECT ANALYSIS & 511 BIT PRBS - INDIRECT ANALYSIS V MWy /i, 10 0.002 0.005 001 002 005 0! 02 03 1.0 FREQUENCY (rodions/sec) ORNL-DWG 66-10035 POWER LEVEL - 50 Mw STEP TEST o 127 BIT PRBS - DIRECT ANALYSIS e 127 BIT PRBS - INDIRECT ANALYSIS & 5i1 BIT PRBS - DIRECT ANALYSIS & 511 BIT PRBS - INDIRECT ANALYSIS ¢ A O v PHASE (deg) THEORE TICAL » 0 a 0001 0002 0005 0Ot 002 005 o1 0.2 05 10 FREQUENCY {radians/sec) &N/N Fig. 26 Frequency Response of 2 ; Power = 5 M SK;KO 40~ ORNL-OWG 66-10039 POWER LEVEL - 6.7 Mw STEP TEST NO.1 e STEP TEST NQ.2 511 BIT PRBS - INDIRECT ANALYSIS v, 5 Bk o? Q002 0005 001 002 005 o1 02 05 10 FREQUENCY (radians/sec) ORNL-DWG 66-10037 POWER LEVEL - 6.7 Mw STEP TEST NO. 1 ® STEP TEST NO.2 A 511 BIT PRBS - INDIRECT ANALYSIS v THEORETICAL PHASE (deg) 0.001 0002 0005 Q01 002 0.05 o1 02 05 10 FREQUENCY (radians/sec) SN/N o Fig. 27 Frequency Response of ; Power = 6.7 MW 5K;KO 84/ PHASE (deg) 2 10 5 102 0002 -]~ ORNL-DWG 66-10040 POWER LEVEL - 7.5 Mw STEP TEST o 127 BIT PRBS ~DIRECT ANALYSIS o 127 BIT PRBS - INDIRECT ANALYSIS A 511 BIT PRBS - DIRECT ANALYSIS & 5t BIT PRBS - INDIRECT ANALYSIS ¢ 4 e, v v v % 6’*@? i® V‘O‘ 0.005 oot 002 0.05 0l 02 05 10 FREQUENCY (rodions/sec) ORNL-DWG 66-10036 80 T -~ T '~ POWER LEVEL - 7.5 Mw 70 / o © ore o1d o % STEP TEST o[ T 60 b ol2 v A 127 BIT PRBS - DIRECT ANALYSIS el L o s . A:'b 127 BIT PRBS- INDIRECT ANALYSIS A 30— | 0{: 511 BIT PRBS - DIRECT ANALYSIS Al 40 (=2 : 511 BIT PRBS - INDIRECT ANALYSIS V|- 30 4 % 20 n v fo 10 a (—THEORET!CAL 0 AN -10 ‘&l\ / Z 20 Qfif S~ TN L -30 %99 7 —V__] o Ly ?A<7VA ¢ -40 0 M-‘“ AL - L 50 3 -60 -70 0001 0002 0005 001 002 005 04 02 05 10 FREQUENCY (radians/ec) &N/N 0 Fig. 28 Frequency Response of ; SK;KO Power = 7.5 MW PERIOD OF OSCILLATION (min) ORNL-DWG 66-10285 ° 1 ® EXPERIMENTAL & \\\ 2 \\ 10 2 THEORETICAL 5 < o \\ \. > N N \ 1 002 005 01 02 05 10 20 50 100 POWER LEVEL (MW) Fig. 29 MSRE Natural Periods of Oscillation 43~ The linear stability of the system is certainly adequate. The frequency response shows a resonance which shifts to higher frequencies and lower amplitudes as power increases. This means that the transient response to a disturbance at low power will display a lightly damped, low frequency return to equilibrium (period greater than ten minutes for powers less than 500 KW). At higher power the system response is much more strongly damped and much faster. For instance, a disturbance at 7.5 MW causes a transient which is essentially completed in 1-1/2 minutes. These observations are in good agreement with prior predictions. A detailed guantitative check of the theoretical predictions by experimental tests is much more difficult than a comparison of more general dynamic features such as stability, location of resonance peaks and the changes expected in these performance measures with power level. Early attempts to fit parameters in the theoretical model to give agreement in the absolute amplitude of the frequency response were abandoned bacause of uncertainties in the measured amplitudes caused by equipment limitations. While all of the tests at a given power level give results with the same shape, there is a difference in the absolute magnitude ratios. ..Figure 27 clearly shows this bias effect. PFurthermore, the portion of the frequency response above 0.3 rad/sec should be the same for all power levels since feedback effects are small in this frequency range and the zero power frequency response should dominate. The experimental results for various power levels show the same shape in this frequency region, but different absolute ampiitudes. This further indicates a bias problem. This bias problem is not surprising in view of the equipment characteristics discussed in Section IV. In spite of the blas difficulties, one feature of the theoretical model i1s shown to be incorrect by experimental results. At high power (greater than 5 MW) the theoretical magnitude ratio curve has a dip at 0.2 rad/sec. This is due to the reappearance of a fuel salt temperature slug in the core after traveling around the primary loop. ©Since this dip was not observed experimentally, there must be =Ll more mixing and heat transfer in the primary locop than was included in the theoretical model. Because the predicted frequency response has the correct shape and location on the frequency axis, we feel that the model used for the MSRE dynamic analysis is a good representation of the system. The only discrepancy observed in predicted and observed shapes is the predicted dip at 0.2 rad/sec, just discussed. The apparent bias in the measured amplitude unfortunately prohibits detailed parameter fitting. 1. lo. 11. 12, 13, REFERENCES R. C. Robertson, MSRE Design and Operations Report, Part 1: Description of Reactor Design, USAEC Report ORNL-TM-728, Oak Ridge National Laboratory, January 1965, S. J. Ball and T. W. Kerlin, Stability Analysis of the Molten-Salt Reactor Experiment, USAEC Report ORNL-TM-1070, Oak Ridge National Laboratory, December 1965. T. W. Kerlin, The Pseudo-Random Signal for Frequency Response Testing, USAEC Report ORNL-TM-1662, Oask Ridge National Laboratory, September 1966. J. 0. Hougen and P. A. Walsh, Pulse Testing Method, Chem. Eng. Prog., 57(3): 69-79 (March 1961). H. A. Samulon, Spectrum Analysis of Transient Respounse Curves, Proc. IRE, 39, 175-186 (1951). . 8. J. Ball, Instrumentation and Controls Div. Annual Progr. Rept. Sept. 1, 1965, USAEC Report ORNL-3875, pp. 126-7, Oak Ridge National Laboratory. 5.5.L. Chang, Synthesis of Optimum Control Systems, Chap. 3, McGraw-Hill, New York, 1961. B. D. Van Deusen, Analysis of Vehicle Vibration, ISA Trans. 3, T. W. Kerlin and J. L. Lucius, CABS — A Fortran Computer Program for Calculating Correlation Functions, Power Spectra, and the Frequency Response from Experimental Data, USAEC Report ORNL-TM- 166%, Oak Ridge National Laboratory, September 1966, E. M. Grabbe, S. Ramo, and D, E. Wooldridge (Editors), Handbook of Automation, Computation, and Control, Vol. 1, Chap. 22, J. Wiley and Sons, New York, 1958. S. J. Ball, A Digital Filtering Technique for Efficient Fourier Transform Calculations, USAEC ORNL-TM report (in preparation). D. P. Roux, and D. N. Fry, ORNL Instrumentation and Controls Division, personal communication. J. E. Gibson, Nonlinear Automatic Control, Chap. 1, McGraw-Hill, New York, 1963. 1h. 15. 16, l?' . 18. 19. =46 - G. C. Newton, L. A. Gould, and J. F. Kaiser, Analytical Design of Linear Feedback Controls, pp. %6—381, J. Wiley and Sons, New York, 1957. C. T. Morrow, Averaging Time and Data-Reducing Time for Random Vibration Spectra, J. Acoust. Soc. Am., 30, 456 (1958). T. J. Karras, Equivalent Noise Bandwidth Analysis from Transfer Functions, NASA-TN-D2842, November 1965. L. D. Enochson, Frequency Response Functions and Coherence Functions for Multiple Input Linear Systems, NASA-CR-32, April 196k, H. M. Paynter and J. Suez, Automatic Digital Setup and Scaling of Analog Computers, Trans. ISA, 3, 55-64, January 196k. S. J. Ball and R. K. Adams, MATEXP, A General Purpose Digital Computer Program for Solving Nonlinear Ordinary Differential Equations by the Matrix Exponential Method, USAEC ORNL report (in preparation), Osk Ridge National Laboratory. -L7- APPENDIX A Potential Sources of Experimental Error Due to Equipment Limitations As discussed in Section IV.A, there were a number of factors that could have had adverse effects on the experimental results, and they are listed below: 1) Friction in the bends in the thimbles. Rollers are mounted in the bends of the thimble, but there is still considerable friction. (Tests show that the rods fall with an acceleration of only about 0.4 g.) This suggests that part of the motion at the rod drive might go into boewing of the flexible hose rather than into motion of the bottom of the poison section. 2) Bends in the hose. It was observed that an old MSRE control rod used for out-of-pile testing did not hang straight when suspended from the top. It had gradual bends that could be worked out by hand, but which were not pulled out by the weight of the rod (6 .to 8 1b). If such bends exist in the MSRE rod used for the tests, then the motion of the bottom of the rod will not be the same as the motion at the top of the rod if a bend in the hose is passing over a roller in the bend of the thimble. | 3) Restricted twisting. The test rod showed a tendency to turn when inserted into a mockup of the MSRE thimble. This twisting is pre- vented in the reactor since the top of the rod is rigidly connected to the chain drive (see. Fig. 3). If a tendency to twist is prevented, the hose will bow and cause a difference in axial motion between upper and lower sections. 4) Sprocket chain meshing. The action of the drive motor is trans- mitted to the drive chain by a sprocket. This sprocket has a diameter of 1.282 in. and the length of the links in the chain is 1/4 in. The fact that the flat links cannot exactly follow the circular contour of the sprocket means that some of the sprocket motion is taken up by a lateral motion of the chain as well as the desired vertical motion. 5) Sticking of poison beads. The control rod thimble contains 1,8 vanes for centering the control rod. The vanes are held in place by circular spacers located 4 in. apart. If the vanes became warped, they could touch links in the poison chain in certain sections without touching links in nearby sections. ©Since there is slack in the threading of the poison cylinders on the central hose, this friction could hold up the movement of certain poison elements. 6) Indicated poisition errors. It was necessary to use the coarse synchro signal (5 deg of turn per inch of rod motion) for logging on the MSRE computer and for subsequent data analysis. Since the l/2-in. rod motion used in the tests corresponds to only 2.5 deg rotation of the synchro, sizeable percentage error could be caused by only a few tenths of a degree of deadband in the gears leading to the synchro. Periodic calibrations of the logged rod position against the fine synchro, how- ever, indicated that the error is less than +5% for a 1/2-in. rod travel. The maximum deadband in the logger signal corresponded to about +0.008 in. of rod motion. -49_ APPENDIX B The Direct Method for Cross-Power Spectrum Analysis Each of the terms used in the cross-power spectrum analysis is computed in the following manner: i(t) a(t) e Hy (J0) Input { v , e 1 - £(d) : a(t) b(t 10 msipty 0B L (a1 | - o(t) ] () | el () o Output W where H, and H, are either H(jw) or 3% H(jw)} and f(¢IO) (depending on which combination of H; and Hs are used) is related either to the real (COPOWER) or the imaginary (QUAD POWER) part of the cross-power spectral density (CPSD) ¢IO' Four combinations of the basic computation shown above are used in the CPSD analysis as shown in Fig. 9. To convert filtered time domain functions to frequency domain functions, we make use of Parseval's theorem,'® which is j'a;(t) b(t) dt = %E j'aé(jw) A(=jw) dw , (1) =00 = where B(jw) = Fourier transform of b(t), and A(—jw) = complex conjugate of the Fourier transform of a(t). _50_ Considering that only a finite integrating time T is available to us: T 1 o J ale) () at ® == [ B(jw) Al—jw) du . (2) 0 ~® Noting that A(Jw) = H3(Ju) I(Jw) , (3) B(Jjw) = Ha(jw) 0(jw) , (L) where I(jw) = Fourier transform of input, i(t), 0(jw) = Fourier transform of output, o(t). From the definition of the complex conjugate, it can be shown that A(=jw) = Hy(=jw) I(=jw) , (5) H B(=jw) = Ho(=jw) 0(=jw) . (6) Hence (2) can be rewritten in terms of the Fourier transforms of the ~ input and output signals T o fo a(t) b(t) at % &= [ Ha(dw) Hy(=ju) I(=jw) 0(jw) dw . (7) =00 Since the cross-power spectral density is defined as” oy _ o iim 1., . . 9o (dw) = L, FlI(=jw) 0(iw)] (8) it behooves us to operate on (7) in order to be able to incorporate ¢Io(jw). Taking the limit of both sides of (7) as T-» and dividing by T, we get lim1l T 1ore . oy fiim1 o, . ron T J S08) 208) 86 = 5 | W) aloge) 320 5 1) (g} 0w 9 = ) Substituting (8) into (9): _5]__ . T o0 1 [ a0 500 0t = b [ el () S @ Qo o0 Case 1 For the case where Hy (Jw) = Ho(Jw) = H(jw) and defining T TORIO E,%if % [ a(t) n(t) at , ' 0 Eq. (10) becomes a(t) b(t) = %E j'“fi(jw) H(-jw) 9;,(Jw) dw = %E m[H(jw)le ¢, (Jw) dv . - (11) = Since the input and output signals are filtered identically, it should be evident that this operation will yield information only about the in-phase relationship, or the real part of ¢Io(jw). We can show this by noting that since a(t) b(t) = b(t) alt) and b(t) a(t) = %; m[H(jw)la ¢ p(iw) dw , (12) the two integrals in Eqs. {11) and (12) must be equal; thus L J TG0 2 (3or(a0) = brolau) Y aw = 0 . (1) =00 If we assume that ¢IO(jw) and ¢OI(jw) do not change much over the effective bandwidth of the filter, then ¢Io(jw) must equal ¢OI(jw). But since 0o (Jw) = @7 (=dw) , (14) _52- this means that the imeaginary part of ¢Io(jw) mist be zero, or at least no information about Im[¢IO(jw)] is present in the output. For case 1, then L %2 : a(t) b(t) = T [H(Jw) | Re[d)IO(Jw)] dw . (15) -0 If we assume that Re[¢IO(jw)] does not change much over the effective bandwidth of H(jw), o1 il e\ (2 . a(t) b(t) & [fi-f [E(Jw) | dwjl l:Re[¢IO(Jw) ]] . (16) - ; For the present study, we used a filter with the following transfer function: Jw H(jw) = . (17) wi + jw 2§wo - wF The filter "area" term can be evaluated using a table of integrals®® %; !; IH(jm)IE dw = h%wo (rad/sec) . (18) Thus Re[d>IO(jw)] N b a(t) b(t) . (19) Cage 2 For this case Yo Hy (Jw) = o H(jw) (20) and Ho(jw) = H(jw) . Since oy E=) Hl( jw) = Ziw ’ (21) =535~ Eq. (10) becomes wy o H(Jw) H(=jw) 955(jw) a(t) o(t) = 5= | : dw (22) Since the input signal's filter has 90 deg more phase lag than the output signal's filter, we should expect that this operation will yield information only about the quadrature relationship, or the imaginary part of ¢Io(jm). We can show this as follows: In this case, revising the order of integration of the inputs makes a difference, i.e., from (2) T o [ v(t) a(t) at = %; [ A(30) B(=jw) dw (23) 0 we can use (%), (6), (19), and (23} to get 0 0 . s (t) a(t) —‘5% EIJ&%Z?I—QEQ-I(jw) O(=jw) dw . (24) -0 Again, since a{t) b(t) = b(t) a(t), from (22) and (24) we can conclude i that ¢1o(dw) _ 9or(dw) (25) —Jjw Jw if we use the same argument as we did for case 1. Thus 9 ro(Jw) = — $5r(Jw) = = ¢p4(=dw) (26) which is true only if the real part of ¢IO = 0, or at least if no information about Re[¢IO(jw)] is present in the output. Hence, we can substitute jIm[¢Io(jw)] in for ¢Io(jw) in Eq. (22). For case 2, then W e |H(jw)]2 AT 00 = - 2 [ =00 = In[9;,(jw)] dw . (27) If we assume that Im[¢10(jw)] does not change much over the W effective bandwidth of either H(jw) or 33 H(jw), then -54_ alt) blt) & [— ;% m'lfifgfllf-dw} [Im[¢IO(jw)]} . (28) - For the particular filter used (Eq. 17): W o0 IH(jw)IE = do = grg (29) Thus Im[qSIO(,jw)]z - bw_ a(t) o(t) . (30) Case 5 The case where Yo Hy(jw) = Ho(Jw) = 3o H( jw) (31) can be developed similarly to case 1 as far as Eq. (16), since we could redefine H(jw) as being equal to the expression in (31). The integral to be evaluated is 1 4 E(Jw) H(=jw) WS o [H(Jw)|® S J Hlw) mu-go) au - 2 [ SR Bl e - 0 ) o au () -0 - o0 eomo0 7 W For the filter of Egq. (17), this integral is again equal %o héw s SO o Re[ézo(jw)] n Mgwo a(t) p(t) (%3) assuming in this case, however, that Re[¢IO(jw)] does not change much W over the effective bandwidth of 3% H(jw). Case L The case where iy (3) = H(3w) (54) Ha(J0) = 2 B(30) (55) can be developed similarly to case 2. Equation (10) becomes ...55.. W 0 . s a(t) b(t) = 5% Elgeljgi_lfil ¢ (Jw) dw . (36) -0 The expression on the right hand side is the hegative of that for Eq. (22), case 2; hence for case 4, we get the expression corresponding to Eq. (3) when the filter of Eq. (17) is used: | Im[¢IO(jw)]z btw_ a(t) b(t) (37) assuming again that Im[¢IO(jw)£ does not change much in the effective bandwidths of either H(jw) or 3% H(jw). The power spectral density (PSD) of the input function is required for calculating the system transfer function G(jw). This is obtained by squaring the outputs of both the in-phase and quadrature filters and integrating the sum of the squares. In both cases, for the filter of Eq. (17): 9. (Jw) & hlw_ a(t)? (38) assuming that ¢II(jw) does not change much in the effective pass bands of H(jw) and W, H(jw)/ jw. The system transfer function G(jw) is then computed from Re[¢..(Jw)] + § Im[d_ (jw)] G(jw) = 10 ¢II(Jw) 10 ’ (39) where each of the three terms on the right hand side of (39) are computed using the sum of two estimates. (Note that all terms have the same gain factor, hgwo.) The reason for the better accuracy of this method as compared to using a single estimate of each term lies in the fact that since the effective pass-bands of the in~phase and quadrature filters are different, there is a bias in each of the quadrature, or imaginary term, estimates. ©Since the two imaginary term estimates are of different sign, this bias tends to be cancelled out. Calculations of the percent standard deviations of both input and output (PSD) estimates are made using (L0):15 -56 - T o 4 SD = 100 ¢ _ 100 , (40) *xZ /BT where 0 = standard deviation of mean square value, x2 = mean square value, B = equivalent noise bandwidth, rad/sec, T = integration time, sec. The equivalent noise bandwidthl® for the H(jw) filter used in this study is B = nlw_ » rad/sec. (41) The coherence function Y& is also computed: 19 2 _ 1°10] = W . (42) 72 The coherence function is useful for estimating expected errors in transfer function calculations when the input and output signals are random.r’ For periodic signals, however, such as the PRBS, the ex- pressions for error estimates in the literature have been found to be wildly pessimistic. The calculation of the response of the digital filters is based on Pzynterts matrix exponential method,18,1% and gives virtually exact time-response solutions very efficiently. -~ 9 ' e H O OCO—T O\ £\ o 22, R NO o A AN 6. 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