(a) Basic Considerations. The resistance thermometer is based upon the inherent characteristic of metals to change electrical resistance* when they undergo a change in temperature. The electrical resistance of verj pure metals varies with temperature from about 0.3 to 0 6% resistance — change per degree at room temperature (or about 0.17 to

‘For many metals the change is completely reversible over fairly large temperature ranges.

0.3 3% per degree Eahrenheit). Industrial resistance — thermometer bulbs are usually made of platinum, copper, or nickel.

An impurity or alloying constituent in a metal decreases the temperature dependence markedly except for a few unusual alloys. Pure platinum in a fully annealed and strain-free state has a resistance—temperature relationship that is especially stable and reproducible. For this reason, pure platinum has been chosen as the international standard of temperature measurement in the temperature range from the liquid oxygen boiling point to the antimony melting point. For the resistance element, platinum is drawn into wire with utmost care to maintain high purity, and the wire is formed into a coil that is carefully supported so that it will not be subjected to mechanical strain caused by differential thermal expansion. Rugged designs are required in military and other applications so that vibration and mechanical shocks will not give momentary or permanent detrimental strain to the platinum coil.

Pure nickel has also been widely used for industrial and many military applications where moderate temperature ranges are involved Tungsten, copper, and some other metals are also used

The fractional change m electrical resistance of a material per unit change in temperature is the temperature coefficient of resistance for the material. The coefficient is expressed as the fractional change in resistance (ohms per ohm) per degree of temperature change at a specific temperature. For most metals, the temperature coefficient is positive.

For pure metals the change in resistance with tempera­ture is practically linear, at least over a substantial range of temperature The relationship can be expressed as

Rt = R0 (1 + at) (4.1)

where Rt equals the resistance in ohms at temperature t, R0 equals the resistance in ohms at 0°C (or some other reference temperature), and the coefficient a is the tem­perature coefficient of resistance. In differential form the relationship is

1 dR

Ro dt

When the resistance does not vary linearly with the temperature, it is customary to include quadratic and cubic terms

Rt = R0 (1 + at + bt2 + ct3) (4.3)

where the coefficients a, b, and c are determined from measurements of the resistance at three or more tempera­tures uniformly spaced over the working range of tempera­ture.

The resistance—temperature relation for platinum is given by the Callendar—VanDusen equation:

Rt

Rn (4.4)

where T is the temperature in degrees Centigrade and (3 is taken as zero for T above 0°C.

(b) Comparison of Resistance Materials. In Fig. 4.10 the resistance R and dR/dT vs. temperature T for a typical platinum resistance sensing element are normalized to 1.00 ohm at 0°C.

Tables 4.6 and 4.7 give the values of resistance vs. temperature for platinum, nickel, and copper.

Platinum. As noted earlier, platinum is the standard reference material for resistance thermometers. Recently, sensors made of very thin platinum films deposited on a substrate (usually a ceramic) have come into use. This method of constructing resistance thermometers leads to small sensing elements with high impedance (resistance) values.

Copper. Copper is inexpensive and has the most nearly linear relation of known metals over a rather wide temperature range. Copper has low resistance to oxidation above moderate temperatures and has much poorer stability
and reproducibility than platinum in most applications. The low resistance of copper is a disadvantage when a high — resistance element is desired.

Nickel. Nickel has been widely used as a temperature sensing element over the range from about —100 to +300°C (—150 to 570°F), principally because of its low cost and the high value of its temperature coefficient. Above 300°C (570 F), the resistance—temperature relation for nickel changes character. Nickel is very susceptible to contamina­tion by certain materials, and the relation of resistance to temperature is not as well known nor as reproducible as that of platinum.

Tungsten. The resistance vs. temperature relation of tungsten is not as well known as that of platinum. Full annealing of tungsten is impractical, and therefore tungsten sensors have been found to be less stable than well-made platinum sensors. Tungsten has been shown to have good resistance to very high nuclear-radiation levels and com­pares with platinum in this respect. Because of its mechani­cal strength, extremely fine tungsten wires are rugged, and sensors having high resistance values can be manufactured.

Table 4.8 lists some typical characteristics of the principal resistance thermometers.

(c) Resistance-Element Structure. The elements of re­sistance thermometers can be constructed in a variety of ways, varying from a cage-like open array of resistance wires within a guard screen to a coil wound on a mandrel and encased in a rugged well. The choice of structure depends on such factors as (1) compatibility of the resis­tance material with the environment, (2) requirements for speed of response, (3) extent of immersion permitted, and (4) expected mechanical stresses to be experienced.

Figure 4.11 shows six types of resistance elements, and Fig. 4.12 shows a typical resistance element assembly in a protecting well.

(d) Resistance-Thermometer Instrumentation. The in­strument measuring the changes in resistance usually employs some form of Wheatstone bridge circuit and may be either an indicator or a recorder. The bridge may be the balanced or unbalanced type. Potentiometnc methods of measuring the resistance are used occasionally.

Figure 4.13 is a diagram of a typical Wheatstone bridge used for resistance-thermometer measurement a and b are ratio arms of equal resistance, and r is a variable resistance, the value of which can be adjusted to balance the bridge so that, except for lead resistance, r = x, x being the resistance of the thermometer resistor.

Copper lead wires have a temperature coefficient of the same order of magnitude as that of a thermometer resistor, and, if their resistance is appreciable in comparison with that of the thermometer resistor, the lead wires may introduce large and uncertain errors into the measurement of temperature. Since the thermometer resistor usually must be placed at a considerable distance from the bridge, the resistance of the lead wires must be compensated. Figure 4.13 illustrates one method of accomplishing this result. Three wires (A, B, and C) connect the measuring instrument and the thermometer resistor (x). Of these, A and C should be identical in size, length, and material and should be placed side by side throughout their length so as to be at the same temperature. The В wire, which is one of the battery wires, need not be similar to the others, however, it is common practice to form the three wires into a cable and make them all alike. A and C are in the thermometer resistor arm (x) and the variable resistance arm (r), respectively. Their resistance remains equal al­though their temperature conditions may change, and, hence, with a one-to-one bridge ratio, such changes have no effect on the bridge reading.

No variable contact resistances should be included in the bridge arms, because the variations in bridge balance introduced at the contacts may be sufficient to affect the reliability of the measurements. The effect of these variations, as well as those resulting from unequal lead resistances, may be reduced by using a resistor of several hundred ohms resistance in the thermometer.

WELL-TYPE ELEMENT

PLATINUM WIRES

CERAMIC COATING PLATINUM TUBE

LEAD WIRE

MINIATURE CERAMIC-COATED ELEMENT

temperatures and are relatively easy to install However, at low temperatures, higher output and higher accuracy are much in favor of resistance sensors

Some of the principal advantages of resistance sensors over thermocouples are

1 A much higher output voltage can be obtained

2 Related recording, controlling, or signal conditioning equipment can be simpler, more accurate, and much less expensive because of the higher possible bridge output signal

3 The output voltage per degree for resistance sensors can be chosen to be exactly as desired over wide limits by adjusting the excitation current and/or the bridge design

4 A reference-junction temperature or compensating device is unnecessary

5 The shape of the curve of output vs temperature can be controlled, within limits, for certain resistance sensor bridge designs

6 The output of a resistance sensor bridge can be made to vary with temperature and another variable bj causing the excitation to vary with the second variable

7 Because of the higher output voltage, more electrical noise can be tolerated with resistance sensors, therefore, longer lead wires can be used

8 Sensitivity to small temperature changes can be much greater

9 In moderate temperature ranges, absolute accuracy and calibration and stability of calibration for resistance elements can be better by a factor of 10 to 100

Thermistors. Thermistors are relatively inexpensive and are very sensitive to temperature The change in resistance per unit change in temperature is large They are available in small sizes and are available with unusually high resistance values when desired Thermistors have a particu­larly nonlinear resistance—temperature relation Because of the nonlinear relation, relatively numerous calibration points are necessary, and the expense of calibration at many points is frequently a major part of the cost of a thermistor temperature sensor

Semiconductors The resistance—temperature relation for the semiconductors consisting of alloy combinations is very complex and therefore requires many more calibra­tion points than platinum sensors At very low tempera­tures semiconductor thermometer^ consisting of doped
germanium sensors have been looked upon with much favor, at least for applied thermometry, as compared to all other methods of measuring temperature When it is necessary to make continuous measurements over the range from approximately 1 to 40°K, they can be used to good advantage

Carbon Resistors At extremely low temperatures car­bon resistors are very sensitive to temperature They have been widely used, mainly for research purposes, for temperature measurements from about 0 1 to 15 or 20°K with good results Their stability is less than might be desired

J. A. Thie

5- 1 FUNDAMENTAL CONCEPTS

5- 1.1 Time and Frequency

The dynamic behavior of systems can be considered from either of two viewpoints as a function of time (time domain) or as a function of frequency (frequency domain) This dualism is quite natural, especially to mathematicians, because it stems from the well-known Fourier theorem a function of time can be represented by the sum (or integral) of sinusoidal functions of various frequencies. In this chapter both points of view are considered, although in certain specific applications we follow historically devel­oped conventions.

Table 6.1 lists the principal functions of time and frequency used in studying dynamic behavior. The func­tions are given as equivalent pairs, i. e., if one is known, the other can be obtained by computation. Thus we can measure functions in either the time or the frequency domain, whichever is the more convenient, and sub­sequently we can compute the Fourier transform function if it is preferred for purposes of interpretation.

5- 1.2 Transfer Functions

The concept of transfer functions was introduced in reactor plant analysis because of its proven utility in electrical engineering. As defined in Table 6.1, the transfer function is the ratio of output complex amplitude to input complex amplitude, this ratio is a complex number that depends on the amplitude ratio and the phase difference between two sinusoidal signals in a system of two or more dynamically related variables, all of which are oscillating at a given frequency. We may speak of a transfer function between any two variables. However, if the input driving function is oscillatory, it is conventionally used as one of the variables, in which case the transfer function is the output amplitude per unit input amplitude of sine-wave excitation. Several zero-power transfer functions are given in Fig. 6.1

CHAPTER CONTENTS

6-1 Fundamental Concepts………………………………………………………………………………… 140

6-1.1 Time and Frequency……………………………………………………………………… 140

6-1.2 Transfer Functions……………………………………………………………………….. 140

6-1.3 Impulse Response………………………………………………………………………… 141

6-1.4 Spectral Density…………………………………………………………………………… 143

6-1.5 Cross Spectral Density……………………………………………………………….. 144

6-16 Autocorrelation……………………………………………………………………………. 145

6-1.7 Cross Correlation………………………………………………………………………… 145

6-2 Reactor Applications………………………………………………………………………………….. 145

6-2.1 Neutron Kinetics…………………………………………………………………………. 145

6-2 2 Zero-Power Measurements…………………………………………………………… 146

6-2.3 Power-Reactor Feedback……………………………………………………………… 147

6-2.4 Power-Reactor Measurements …………………………………………………….. 148

6-3 Methods of Measurement……………………………………………………………………………. 149

6-3.1 Reactor Excitation………………………………………………………………………… 149

6-3.2 Noise Methods…………………………………………………………………………….. 151

6-3 3 Comparison of Methods………………………………………………………………. 153

6-4 Reactor Excitation Equipment…………………………………………………………………… 154

6-4.1 Excitation Signal…………………………………………………………………………. 154

6-4.2 Control Device……………………………………………………………… .. 155

6-5 Transfer-Function Analyzers………………………………………………………………………. 155

6-5.1 Usage……………………………………………………………………………………………… 155

6-5 2 Null-Balance Analyzer……………………………………………………………….. 156

6-5.3 Synchronous Transfer-Function Analyzer. . . 156

6-5.4 Cross Correlators…………………………………………………………………………. 157

6-5 5 Digital Techniques……………………………………………………………………….. 157

6 Frequency Analyzers………………………………………………………………………………….. 158

6-6.1 Usage……………………………………………………………………………………………… 158

6-6.2 Spectrum Analyzers…………………………………………………………………….. 159

6-6.3 Cross-Spectrum Analyzers………………………………………………………….. 160

6-6.4 Digital Spectrum Analysis………………………………………………………….. 160

6-7 Experimental Considerations ……………………………………………………………………. 161

6-7.1 Error Sources…………………………………………………………………………………. 161

6-7.2 Frequency Limits………………………………………………………………………… 161

6-7.3 Statistical Accuracies…………………………………………………………………… 162

6-7.4 Spectral-Analysis Data Planning……………………………………………….. 162

References…………………………………………………………………………………………………………. 163

Complete specification of a transfer function involves both an amplitude value and a phase difference given as a function of frequency. A complete specification of the dynamics of a system would involve all the transfer functions between all the pairs of variables given for the

entire band of frequencies of physical interest Often, however, the two most meaningful variables are related In reactor dynamics these variables might be the reactivity and the power of a reactor.

Table 6.2 contains a simple example of two variables, x and у (one input and one output), related by a differential equation having one time constant the complex transfer function is (1 + icorc)_1 and has an amplitude (1 + co2r2)~^ and a phase arctan(—corc) or real and imaginary parts of 1/(1 + w2r2)^and —сотс/(1 + согт) respectively.

Since almost all reactor dynamics analyses involve linear systems, linear systems are assumed in this chapter. In a linear system the transfer function at a given frequency is independent of the absolute magnitude used in its measure­ment. Usually a sufficiently large amplitude will cause nonlinear behavior in any system, but these cases are not treated with the techniques discussed in this chapter.

It should be mentioned, however, that the transfer — function concept may be applied to almost-hnear systems. Smets1 has presented a “describing function” approach to nuclear-reactor dynamic measurements

Describing function = (amplitude of fundamental Fourier component of output signal)/(amplitude of sinusoidal input signal) (6 1)

where the input signal is x(t) = a sin cot and the output signal is

y(t) = A! sin(cOit + фі) + A2 sin(co2t + 02) + .. (6.2)

The describing function is thus A!/a If Ai is not linear in a, then the describing function depends on the magnitude of the input amplitude a. If the system is linear, the describing function is synonymous with the transfer function.

Figure 2 1 shows a typical location for an out of core neutron sensor In this example, the sensor is also outside the reactor vessel The figure also shows the magnitude of the neutron flux, the gamma exposure rate, and the temperature in the out of core location typical of a boiling-water or pressurized-water reactor during operation at rated power

In today’s power reactors the neutron flux inside the core boundary is always greater than 1011 neutrons cm’2 sec”1 Consequently, it is current practice to define an out of-core sensor as one that is not exposed to a neutron flux greater than 1011 neutrons cm” sec” The tern perature and gamma exposure rate are not involved in this definition Both the temperature and the gamma exposure rate in Fig 2 1 are at least an order of magnitude less in out of core locations than they would be within the core

boundary. An out-of-core sensor, however, can be located inside the reactor pressure vessel, e. g., in the region of the thermal shield, provided the neutron flux does not exceed 10 neutrons cm-2 sec-1 .

4- 7.1 Steam-Generator Feedwater Specifications

Feedwater conditioning is required to maintain opera­tional capability of the steam generator The steam­generating surfaces remain clean and heat-transfer capa­bilities are favorable if good water quality is maintained The minimum standards given in Table 4 19 should be maintained for satisfactory feedwater quality.

4- 7.2 Usual Impurities in Water Supply

Table 4.20 lists the impurities usually found in water supplies and indicates their properties, effects, and methods for treatment and removal.

4- 7.3 Effect of Impurities in Steam-Generator Feedwater

(a) Total Solids (Dissolved and Undissolved). The total solids in the feedwater is a general indicator of how much material is collecting in the steam generator. Insoluble materials are deposited on the steam-generator surfaces. The soluble material (e. g., NaCl, NaOH, and Na2S04) is carried over in the steam with the remaining material and tends to collect on the steam-generator tubes. The types of constituents in the feedwater depend on the preboiler characteristics. The quantity of soluble material should be larger than the quantity of insoluble material. Turbidity is a measure of the undissolved constituents, and electrical conductivity is a measure of the dissolved constituents

(b) Dissolved Oxygen. Dissolved oxygen promotes cor­rosion in the steam generator and therefore should be kept as low as possible. Deaeration removes much of the dissolved oxygen. Hydrazine, a good oxygen scavenger, can eliminate the remaining oxygen. A high level of dissolved

oxygen could indicate a malfunctioning deaerator or an air leak in the area of the condenser.

(c) Total Silica. Silica should be maintained at a low level for two reasons (1) silica can concentrate in the steam generator and subsequently plate out on heat-exchange surfaces thereby reducing steam generation and (2) silica may carry over and plate out m the turbine causing turbine inefficiency. A higher than allowable silica concentration implies a demineralizer breakthrough. Switching to the spare demineralizer while the exhausted resin is regenerated or changed can probably eliminate the high silica level.

(d) Total Iron. Iron tends to build up in the steam generator and reduces its efficiency by degrading the heat-transfer characteristics. The level of iron in the feedwater affords some measure of the degree and rate of corrosion in the system.

(e) Total Copper. Copper should be avoided where possible in the feedwater system. Equipment in contact with the feedwater should be ferritic or austenitic stainless steel. Copper in the feedwater system can be carried into the steam generator in solution and plate out there. The copper plate can then make it necessary to clean the steam generator in a two-stage process one for copper and another for iron. Copper carryover in the steam can plate out on the turbine and lower its efficiency Copper alloy tubes in the condenser should be satisfactory because the temperatures and pressures are reduced and the dissolution of the copper is less likely.

(f) Total Lead. Lead in the feedwater concentrates in the steam generator This can result in problems with Inconel in oxygenated water containing lead Satisfactory instrumentation for monitoring traces of lead is not available at the present time

(g) Conductivity. Cation (positive-ion) conductivity cells can be used to monitor the feedwater Measurements

•From R. T. Kent, Mechanical Engineers’ Handbook, Power, 12th ed, p. 7 51, John Wiley & Sons, Inc, New York, 1950

should be made after removal of the ammonia that is used to regulate the pH

(h) Corrosion. The principal accelerators of corrosion are dissolved oxygen, acids, surface deposits, especially those electronegative to steel, dissimilar metals in contact, and electrolytes.

Common methods to prevent corrosion are removal of dissolved gases, especially oxygen, neutralization of acids and maintenance of desirable alkalinity and pH, periodic mechanical cleaning, and avoiding excessive salt concentra­tions.

Devices involving the principles of Figs. 6 12 and 6 13 are usually restricted to sinusoidal excitation experiments A technique that is more generally applicable in transfer — function determinations involves the use of cross correla­tion. In addition to being used with sinusoidal excitation, cross correlation is commonly used with self-induced noise or types of excitation other than sinusoidal.

The method consists in determining the cross­correlation function, Сху(т), over a range of the time-lag r Either the digital definition, Eq 6.9, or the continuous integral (Table 6.1) definition may be used, depending on the experimental approach It is convenient to handle only the fluctuating parts of variables, i. e., x(t) — x and y(t) — y, such as in Eq. 6.10 After Сху(т) is known, either from an on-line or off-line device, it must be Fourier analyzed to obtain the cross spectrum, Pxy(f) (see Table 6 1), and the transfer function,

PXy(f)

G(f) = T^0 (6 35)

where Px(f) is the spectrum of the input variable, x(t)

The three operations of time delay, multiplication, and integration required to obtain Cxy(r) are indicated in Fig 6 14 Here x(t) may be either a fluctuating signal in the system or an excitation signal The operations shown have been done on-line4 1 for pseudorandom excitation by using “0” and “1” signals (read at the appropriate values of r from a tape containing the input sequence) in a simple switching multiplier

Frequently, however, the two signals x(t) and y(t) are recorded on a frequency-modulation (f-m) magnetic-tape system.40 95 1 00 Off-line playback is carried out using tape heads that are displaced to give the y(t) and x(t — r) input to the multiplier—integrator combination of F’lg 6 14 Analog-computer components are typically used to perform the operations required to give Cxy(r) In the special case where x(t) = y(t), the autocorrelation function may be obtained m this manner.

Valid statistical data on the performance of out-of core neutron and gamma sensors are scarce However it may be inferred from examination of sensor designs and from general knowledge of the field that reliability has been good The most frequently reported difficulties have been from spurious signals attributable to microphomc and electrical effects (These are discussed in Chap 10 ) Since these effects can usually be observed in the debugging period, they can be rectified by modifying the system design

Ionization detectors generally have long lives The life of a neutron detector is, however, limited by consumption of sensitive materials Since the consumption is directly related to the neutron fluence, the loss may be calculated and compensated for by recalibration In early designs, flaking of sensitive material with subsequent deposition in insensitive parts of the chamber volume was occasionally experienced Generally, flaking is not a problem at present As might be expected, gamma detectors have an in­definitely long life since they contain no sensitive materials

In a number of ionization chamber detectors using special gas mixtures, gradual degradation results from radiologically induced changes in gas composition Since this type detector is normally used only for very special purposes, it is expected that the user would be alert to any possible difficulties Certain proportional counters fall in this category and are normally used only at moderate or low radiation levels

Scintillation and solid-state detectors are useful at low radiation levels only and can withstand only a limited total radiation exposure

Proper tables must be used for interconnecting nuclear instrumentation to reduce noise and to transmit the best possible signal to the readout equipment. See Chap. 10 for
additional information on grounding, shielding, and selec­tion of cables

Noise-free cables must be used in the start-up instru­mentation Noise pulses will be amplified and counted as neutron pulses. Besides the inaccuracies in counting, noise bursts can cause period and level scrams at low reactor power levels

When a vacuum-tube preamplifier is used, the sensor and preamplifier are usually connected together as a unit or with a few feet of coaxial cable The cable used must have an impedance that matches the preamplifier input. The signal cable used between the vacuum-tube amplifier and the LCRM should be coaxial and match the impedances of the two units If the cable is routed through high-noise areas, a tnaxial cable should be used with the outer braid connected to ground.

Power-supply cables between the power supply and vacuum-tube preamplifier should be shielded conductors to minimize the noise level The power-supply ripple voltage should not exceed 10 mV for satisfactory operation of the sensor and attached preamplifier.

The greater the distance between the detector and the preamplifier, the more important is the cable quality. For

LOCAL

METER

OUTPUT

RECORDER

OUTPUT

COMM.

REMOTE

METER

-15V +15V

ТС TRIM

HIGH TRIP

Fig. 5.12—Solid-state alarm (trip) circuit.

solid-state preamplifiers the importance of the cable cannot be overemphasized. There is one positive procedure for noise reduction complete the installation, and, using one or more of the methods outlined in Chap. 10, experiment until the noise has been reduced to a minimum. Neutron sensors for use at high temperatures are under development Cables and connectors for use with these sensors must operate at high temperature without producing noise pulses Radiation-resistant cables should be used to increase the time between replacement, to reduce maintenance costs, and to improve availability. Cables and connectors are available for high-temperature use, but extreme care must be exercised in their use Avoid use of high-tempera­ture components if at all possible, they are subject to changes in resistance which adversely affect both signals and high-voltage cables

A triaxial cable should be used to minimize the noise introduced between the sensor and solid-state preamplifier The outer braid of the triaxial cable should be tied at both ends to the inner braid. The impedance of the cable must match the input impedance of the preamplifier to reduce pulse reflections on the cable.

The conventional control rod for water cooled reactors is made of stainless steel with neutron absorbing material either supported and clad by the rod structure or alloyed with the rod material The neutron absorbing material absorbs thermal neutrons, which reduces the effective multiplication constant, k, below unity The most com monly used neutron absorbing or poison materials in control rods are cadmium, boron, and hafnium Other less commonly used materials are silver, europium, and indium Factors determining the usefulness of a control rod material include not only thermal-neutron absorption properties but also availability, cost, and structural and machinabihty properties The material must be fabricated into various shapes and must not be affected appreciably by the temperature or pressure of the particular environment in which it is to be used Its susceptibility to corrosion and nuclear-radiation damage must be low In some instances

Table 7 2—Control Worth for Various Materials*

Relative effectiveness

Material in a water-cooled reactor

alloys of the poison materials with other materials improve their suitability for control-rod application

Table 7 2 indicates the relative worth of commonly used control-rod absorber materials

6- 3.4 Rod Shape

The shape, dimensions, and number of reactor control rods are dependent on the core mechanical design and the amount of negative reactivity needed for shutdown To function efficiently as a neutron poison, a control-rod material must have sufficient thickness to absorb most of the flux at the rod surface In light-water reactors the slowing-down length, і e, the distance required for a fission neutron to be reduced to thermal energy, is short Therefore, to be effective in absorbing thermal neutrons, the poison material must be physically close to the fuel surface and have a high surface-to-volume ratio A cruci form shape with thin wide blades of poison material fulfills these requirements Figure 7 2 shows typical control-rod configurations for power reactors Since a light-water reactor has a neutron spectrum with an appreciable fraction of epithermal neutrons, materials with large absorption cross sections for these energies, such as hafnium and indium, are used in addition to the usual thermal neutron absorbers, such as cadmium or boron

The design of instrumentation systems for a nuclear power plant must take into account the specific properties of the reactor for that plant. Of particular importance is the kinetic behavior of the reactor Many textbooks and monographs have been written on nuclear reactor kinetics, the reader is referred, for example, to Refs. 2 through 6 for details The following paragraphs summarize basic material particularly relevant to instrumentation systems in nuclear power reactors.

1-3.1 Point Kinetics Without Delayed Neutrons

The symbol n (neutrons/cm3) is used to designate the neutron density at a given position in a nuclear fission chain reactor. If the reactor is just critical, the effective multipli­cation factor, k, is exactly 1 and the neutron density, n, is constant. If the effective multiplication factor is increased by 6k = к — 1 (with 5k > 0), then n increases with time.

The rate of increase of n, dn/dt, is the number of extra neutrons in the next generation, n 5k, divided by the time between generations /

dn = n 5k = n(k — 1)

dt l l ( ’

Integrated, this is

п = п0е<«к//>‘ (12)

where n0 is the neutron density at t = 0 The reciprocal of the first factor, 5 k//, in the exponential has the dimensions of time and is known as the reactor period

To introduce the effect of delayed neutrons on the nuclear chain reaction, we consider the effective multiplica­tion factor to be the sum of two terms

к = (multiplication factor for prompt neutrons) + (multiplication factor for delayed neutrons)

(1 3)

T = //5k

These equations have been developed on the assumption that there is a single characteristic time between generations in a nuclear fission chain reaction This is the same as assuming that only prompt neutrons participate m the chain reaction

= k(l — j3) + k/3 (14)

where (3 is the delayed neutron fraction, or the number of delayed neutrons per fission divided by the total prompt and delayed neutrons per fission The delayed-neutron

Table 1.1—Delayed-Neutron Half-Lives and Yields in Thermal-Neutron Fission’

Thermowells are protective devices for the sensors of temperature indicating, recording, and controlling instru­ments As used in out-of core locations in a nuclear power plant, temperature sensors may be exposed to a wide range of pressures and temperatures and to a variety of poten­tially corrosive materials

This section includes a description of the basic types of thermowells and their materials of construction, a summary of methods for ensuring that the thermowell design will survive the mechanical stresses met in service, and a guide to the selection of thermowell materials

(a) Connection to Process Vessel. A thermowell is usually secured to a process vessel by threads, flanges, or welding (Fig 4 14)

III

The threaded connection, normally using standard-taper pipe threads, is most popular owing in large measure to its simplicity and low cost Standard threaded well connec­tions range in size from */2 in to 1 ln NPT, with specials % in to 2 in NPT meeting most requirements

Flanged assemblies of any size and/or pressure rating are available Normal means of well mounting are provided by ASME-approved welding techniques, with follow-up

machining to provide any standard sealing-face configura­tion. Flanges are commonly used to seal long thermowells or those wells which are inserted into large vessels. An alternate flange type well is the nonwelded Van Stone well with integral flange, using a lap-joint flange to hold it in place. Also available is the ground-joint type with a machined ball that mounts in a socket between a pair of mating flanges. These latter two designs have an advantage in that as thermowell replacement becomes necessary, flanges may be reused with the new assembly.

Welded connections are normally used where process pressures are too great for flanged or threaded assemblies or where long-term inexpensive connections are desirable The welded-in type is commonly used in conjunction with high-pressure, high-velocity steam lines. This type well is frequently furnished with close tolerance limits on outside diameters in the area to be welded. These are tapered-stem wells with greater wall thickness in the weld area but with relatively low mass at the end to improve response with tip-sensitive temperature-measuring devices.

(b) Length, Bore, and Wall Thickness. Overall well length is determined not only by desired — insertion length but also by external extension of the connection end. Most threaded connection wells require an additional 2 in. of nonimmersed length to provide threads and wrenching surface. Welded or flanged wells normally require at least 1.25 in. of extra length for instrument-connection thread­ing and welding surface. Where there are layers of thermal insulation, a lagging extension should be added between the process connection and the instrument connection.

Bore size (both length and diameter) depends on the thermal sensing element to be used. The fit between the sensor and the inner wall of the thermowell must be good if accuracy and rapid response are to be achieved [Sec. 4-2.1 (i)]. Care should be taken to prevent heat loss to surroundings and to avoid variations caused by stratifica­tion of process fluids. Where clearances between measuring element and bore are minimal and welding must be performed in the field, a counter bore of 10 to 20 mils greater diameter than the bore should be made. This counter bore should be carried sufficiently far past the welded area to avoid distortion in the bore due to heat of welding.

To withstand mechanical stresses, the thermowell wall should be thick. However, to provide rapid response to process-temperature changes, the wall should be thin (and the immersed well mass should be minimum). These conflicting requirements have been met by using tapered thermowells, in which the tip has a thin wall for optimum heat transfer and a thick mounting for improved strength. The design of these wells is discussed in the next section.

(c) Design of Power Test Code Thermowells. The American Society of Mechanical Engineers recommends a standardized Power Test Code thermometer well, as shown m Fig. 4.15. Wells of this design, with 6 in. minimum wall thickness, are expected to satisfy 95% of the present needs.

Fig. 4 15 —Power Test Code thermometer well (From Sci­entific Apparatus Makers Association Standard RC 21-4-1966.)

The following design procedure enables a user to determine if a well selected for thermometry considerations is strong enough to withstand specific application conditions of temperature, pressure, velocity, and vibration. This design procedure does not allow for effects due to corrosion or erosion. If corrosion or erosion is anticipated, additional wall thickness must be allowed in all exposed sections to prevent premature well failure.

The nominal size of the sensing element is considered here to vary between % in. (6.35 mm) and ?8in. (22.225 mm). For this range the dimensions of the thermo­well are assumed to be those given in Table 4.9.

A thermometer well must be able to withstand (at the operating temperature) the static stress associated with the maximum operating pressure of the process vessel. The maximum allowable pressure is computed from the formula

P = KjS

where P = maximum allowable static gage pressure (psig)

Kj = a stress constant depending on thermowell geom­etry

S = allowable stress for thermowell material at the operating temperature as given in the ASME Boiler and Pressure Vessel or Piping Codes (psi)

For wells constructed as shown in Fig. 4.15 with dimen­sions as given in Table 4.9, the stress constant has the values listed in Table 4.10. For wells of other dimensions, the stress constant is given by (4.6) where (see Fig. 4.15) В is the minimum outer diameter (inches) at the well tip and Fg is a factor varying between 2 0 and 1.0 as shown in Table 4.11.

Thermometer wells rarely fail in service from the effects of temperature and pressure. Since a thermowell is essen­tially a cantilevered beam, vibrational effects are of critical importance If the well is subjected to periodic stresses that have frequency components matching the natural fre­quency of the well, then the well can be vibrated to destruction. In nuclear power plants the temperature of high-velocity fluid streams (steam, water, etc.) must be measured Thermowells immersed in these streams (thermo­well axis transverse to flow direction) are subject to periodic stresses attributable to the cyclic production of
vortices in the wake of the flowing fluid, the “von Karman vortex.” The frequency of these stresses, fw, is

fw = 2.64^ (in Hz) (4.7)

D

where V = fluid velocity (ft/sec)and В = well diameter at tip (in ), see Fig. 4.15. The natural frequency of the thermo­well (cantilever structure) is

fn = Kf (0 (in Hz) (4 8)

where E = elastic modulus of well material at the operating temperature (psi)

7 = specific weight of well material (lb/in.3)

L = length of well (in.) (see Fig 4.15)

Kf = a factor depending on well dimensions (Table 4.12)

The wake frequency fw should not go above 80% of the natural well frequency, fn,

^<08

In

If the ratio r is over 0.8, the well will tend to vibrate to failure.

In any practical situation, the fluid velocity, V, is fixed, and the parameters under the instrumentation engineer’s control are the well dimensions. Once the size of the sensing element is decided on (e. g., on the basis of speed of response, ruggedness, etc.), the thermometer-well outer diameter В is fixed (Table 4.9), and the wake frequency (Eq 4.7) is determined. The only well parameter remaining (except materials of construction, see next section) is the well length, L. Since fn decreases with increasing length (Eq. 4.8), the requirement for fw/fn to be less than 0 8 imposes a limitation on the length, L.

The maximum length of a thermometer well for a given service depends not only on the vibratory stresses imposed by the flowing limit but also on the steady-state stresses

(drag) of the flowing fluid. These stresses limit the well length according to the following formula
(c) Frequency ratio (Eq. 4.9) r = 986/1305 = 0.755 <08 (satisfactory) Step 4 Maximum length calculation (a) Magnification factor (Eq. 4.11) 0.755 Fm'(l — 0.7552) J 5
v(S-K3P0) 1 + Fm
к? V
(4.10)
(in in.)
where V = fluid velocity (ft/sec) v = specific volume of fluid (ft3/lb) S = allowable stress for well material at operating temperature per codes (psi) Po = static operating pressure (psig) K2, K3 = stress constants (Table 4 10) The factor Fm is a “magnification factor” dependent on the ratio r of wake frequency to the natural frequency of the well
(b) Maximum length (Eq. 4.10)
0.4134(9725 — 0.389 X 2000)
46.8 350
(1 + 1.325)
5 3 3 in. > 4У2 in. (satisfactory)
Conclusion 1 he well selected is satisfactory for the application. An example of the installation of a typical thermo­couple or lesistance thermometer is shown in Fig 4 16 The thermowell shown is a heavy-duty weld-in well.
w n
r
(d) Materials Used in Thermowells. Because of their ability to withstand chemical attack from process fluids, stainless steels are most frequently used in thermowells Customarily, stainless steels are put in three groups martensitic, ferritic, and austenitic. The martensitic steels contain 11 5 to 18% Cr, <2.57% N1, and 0.06 to 1.20% C They have a ferritic structure when annealed but take on the properties of a martensite when cooled. They can be heat-treated, hardened, and tempered to provide a wide range of mechanical properties for use m abrasive environments or where particularly high strength is required. Martensitic steels are in the 400-senes stainless steels, excepting the ferritic grades, and are strongly magnetic. Examples are the AISI types 403, 410, 414, 416, 420, 431, and the 440 letter series. The ferritic steels contain 11.5 to 28% Cr, no N1, and 0 06 to 0.35% C. They are always magnetic and do not respond to heat treatment. They are strong and ductile when properly annealed and are generally more corrosion — resistant than the martensitics. Examples are the AISI types 405, 430, 430F, 442, and 446. The austenitic grades are chrome—nickel alloy steels with a maximum carbon content of 0.25% and with 7 to 30% Cr and 6 to 36% N1. They are nonmagnetic in a fully annealed condition but become slightly magnetic with cold working They are generally tougher and more ductile than martensitic and ferritic steels and have a much higher corrosion resistance The austenitic steels belong to the 300-series. Selection of a stainless steel requires consideration of which properties are most desirable for the application corrosion resistance, strength at operating temperature, oxidation resistance, particularly at elevated temperatures, availability in a form suitable for fabrication, or ease of fabrication (machinability, weldability)
Example To clarify the use of the above formulas, consider the following example It has been determined that a 4V2-m. well is required to accommodate a % 6-in. sensing element that will measure the temperature of superheated steam at 2000 psia, 1050°F, and flowing at a velocity of 350 ft/sec. If the thermometer well is to be made of type 316 stainless steel dimensioned according to Table 4.9, will the well be safe5 Step 1 Obtain the necessary data as follows
v Specific volume of steam L Modulus of elasticity at 1050° F 7 Specific weight of metal at 70° F S Allowable stress at 1050° F	0 4134 ASME Steam ft3/lb Tables, 1967 22 35 B31 1 0 1967, x 106 psi App C 0 290 Ib/in 3 9725 psi ASME Code, Sec VIII
Step 2 Maximum static pressure (Eq 4.5) P = 0.223 X 9725 = 2170 psig > 2000 psia (satisfactory)
Step 3 Frequency calculations (a) Natural frequency (Eq 4.8)
(Use of dimensions and specific weight values for 70°F instead of for 1050°F is partially compensatory and causes no significant error.) (b) Wake frequency (Eq. 4.7)

RTD CONNECTIONS Зо 4n 5 о Co 5o Зо ЗО 5o I T red’ r ; X X X X CD RTD 5 $ 5 RTD 5 3 TERMINAL 4 TERMINAL

T/C CONNECTIONS + — + — + I-C T/C C-A T/C C-C T/C

Fig. 4.16—Typical thermocouple and resistance element installed in heavy-duty thermowell. (Courtesy Bailey Meter Co.)

The principal properties of commonly used grades of stainless steels are summarized in Table 4.13.

Other materials may be used m thermowells. Tables 4.14 and 4.15 give the recommended allowable stress values and maximum operating temperatures for a number of thermowell materials.