In previous chapters we have solved diffusion equation for various shapes of reactors. For example the solution for finite cylindrical reactor
where Bg2 is the total geometrical buckling.
It must be added there are constants A and C that cannot be obtained from the diffusion equation, because these constants give the absolute value of neutron flux or actually the reactor power.
In each nuclear reactor, there is a direct proportionality between the neutron flux and the reactor thermal power. The term thermal power is usually used, because it means the rate at which heat is produced in the reactor core as the result of fissions in the fuel. Nuclear power plants also use the total output of electrical power, but this value is due to the efficiency of conversion (usually from 30% to 40%) always smaller than the thermal power of reactor.
Back to the proportionality between the neutron flux and the reactor thermal power.
In a power reactor over a relatively short period
of time (days or weeks), the atomic number density of the fuel atoms remains relatively constant. Therefore in this short period, also the average neutron flux remains constant
, when reactor is operated at a constant power level. On the other hand, the atomic number densities
of fissile isotopes over a period of months decrease due to the fuel burnup and therefore also the macroscopic cross-sections
decrease. This results is slow increase in the neutron flux
in order to keep the desired power level.
Knowledge of the neutron flux
(the total path length
of all the neutrons in a cubic centimeter in a second) and the macroscopic cross sections
(the probability of having an interaction per centimeter path length
) allows us to compute the rate of interactions (e.g. rate of fission reactions). This reaction rate
(the number of interactions taking place in that cubic centimeter in one second) is then given by multiplying them together:
Ф – neutron flux (neutrons.cm-2.s-1)
σ – microscopic cross section (cm2)
N – atomic number density (atoms.cm-3)
The reaction rate for various types of interactions is found from the appropriate cross-section type:
- Σt . Ф – total reaction rate
- Σa . Ф – absorption reaction rate
To determine the thermal power, we have to focus on the fission reaction rate. For simplicity let assume that the fissionable material is uniformly distributed in the reactor. In this case, the macroscopic cross-sections are independent of position. Multiplying the fission reaction rate per unit volume (RR = Ф . Σ) by the total volume of the core (V) gives us the total number of reactions occurring in the reactor core per unit time. But we also know the amount of energy released per one fission reaction to be about 200 MeV/fission. Now, it is possible to determine the rate of energy release (power) due to the fission reaction. It is given by following equation:
P = RR . Er . V = Ф . Σf . Er . V = Ф . NU235 . σf235 . Er . V
P – reactor power (MeV.s-1)
Ф – neutron flux (neutrons.cm-2.s-1)
σ – microscopic cross section (cm2)
N – atomic number density (atoms.cm-3)
Er – the average recoverable energy per fission (MeV / fission)
V – total volume of the core (m3)
See also: Energy Released per Fission
The total energy released in fission can be calculated from binding energies of initial target nucleus to be fissioned and binding energies of fission products. But not all the total energy can be recovered in a reactor. For example, about 10 MeV is released in the form ofneutrinos (in fact antineutrinos). Since the neutrinos are weakly interacting (with extremely low cross-section of any interaction), they do not contribute to the energy that can be recovered in a reactor.
The total energy released in a reactor is about 210 MeV per 235U fission, distributed as shown in the table. In a reactor, the average recoverable energy per fission is about 200 MeV, being the total energy minus the energy of the energy of antineutrinos that are radiated away. This means that about 3.1⋅1010 fissions per second are required to produce a power of 1 W. Since 1 gram of any fissile material contains about 2.5 x 1021 nuclei, the fissioning of 1 gram of fissile material yields about 1 megawatt-day (MWd) of heat energy.
As can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is significant amount of energy generated outside the nuclear fuel (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated in the coolant (moderator). This phenomena needs to be included in the nuclear calculations.
For LWR, it is generally accepted that about 2.5% of total energy is recovered in the moderator. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.
A typical thermal reactor
contains about 100 tons
with an average enrichment of 2%
(do not confuse it with the enrichment of the fresh fuel
). If the reactor power is 3000MWth
, determine the reaction rate
and the average core thermal flux
The amount of fissile 235U per the volume of the reactor core.
m235 [g/core] = 100 [metric tons] x 0.02 [g of 235U / g of U] . 106 [g/metric ton]
= 2 x 106 grams of 235U per the volume of the reactor core
The atomic number density of 235U in the volume of the reactor core:
N235 . V = m235 . NA / M235
= 2 x 106 [g 235 / core] x 6.022 x 1023 [atoms/mol] / 235 [g/mol]
= 5.13 x 1027 atoms / core
The microscopic fission cross-section of 235U (for thermal neutrons):
σf235 = 585 barns
The average recoverable energy per 235U fission:
Er = 200.7 MeV/fission
Zero Power Criticality vs. Power Operation
In fact the neutron flux
can have any value and the critical reactor
can operate at any power level. It should be noted the flux shape derived from the diffusion theory is only a theoretical case in a uniform homogeneous cylindrical reactor at low power levels (at “zero power criticality
In power reactor core at power operation, the neutron flux can reach, for example, about 3.11 x 1013 neutrons.cm-2.s-1, but this values depends significantly on many parameters (type of fuel, fuel burnup, fuel enrichment, position in fuel pattern, etc.). The power level does not influence the criticality (keff) of a power reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).
At power operation (i.e. above 1% of rated power) the reactivity feedbacks causes the flattening of the flux distribution, because the feedbacks acts stronger on positions, where the flux is higher. The neutron flux distribution in commercial power reactors is dependent on many other factors as the fuel loading pattern, control rods position and it may also oscillate within short periods (e.g. as a result of spatial distribution of xenon nuclei). Simply, there is no cosine and J0 in the commercial power reactor at power operation.
See also: Nuclear Reactor as the Antineutrino Source
In an operating power reactor
the neutron population is always large enough to generated heat. In fact, it is the main purpose of power reactors to generate large amount of heat
. This causes the temperature of the system changes and material densities change as well (due to the thermal expansion
Because macroscopic cross sections are proportional to densities and temperatures, neutron flux spectrum depends also on the density of moderator, these changes in turn will produce some changes in reactivity. These changes in reactivity are usually called the reactivity feedbacks and are characterized by reactivity coefficients. This is very important area of reactor design, because the reactivity feedbacks influence the stability of the reactor. For example, reactor design must assure that under all operating conditions the temperature feedback will be negative.
The reactivity coefficients that are important in power reactors (PWRs) are:
As can be seen, there are not only temperature coefficients that are defined in reactor dynamics. In addition to these coefficients, there are two other coefficients:
The total power coefficient is the combination of various effects and is commonly used when reactors are at power conditions. It is due to the fact, at power conditions it is difficult to separate the moderator effect from the fuel effect and the void effect as well. All these coefficients will be described in following separate sections. The reactivity coefficients are of importance in safety of each nuclear power plant which is declared in the Safety Analysis Report (SAR).
Example: Power increase – from 75% up to 100%
During any power increase the temperature, pressure, or void fraction change and the reactivity of the core changes accordingly. It is difficult to change any operating parameter and not affect every other property of the core. Since it is difficult to separate
all these effects (moderator, fuel, void etc.) the power coefficient
is defined. The power coefficient combines the Doppler, moderator temperature, and void coefficients
. It is expressed as a change in reactivity per change in percent power, Δρ/Δ% power
. The value of the power coefficient is always negative in core life but is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.
Let assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power. The reactor operator must first bring the reactor supercritical by insertion of a positive reactivity (e.g. by control rod withdrawal or borondilution). As the thermal power increases, moderator temperature and fuel temperature increase, causing a negative reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be increasing, positive reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted. The total amount of feedback reactivity that must be offset by control rod withdrawal or boron dilution during the power increase (from ~1% – 100%) is known as the power defect.
- the power coefficient: Δρ/Δ% = -20pcm/% of rated power
- differential worth of control rods: Δρ/Δstep = 10pcm/step
- worth of boric acid: -11pcm/ppm
- desired trend of power decrease: 1% per minute
75% → ↑ 20 steps or ↓ 18 ppm of boric acid within 10 minutes → 85% → next ↑ 20 steps or ↓ 18 ppm within 10 minutes → 95% → final ↑ 10 steps or ↓ 9 ppm within 5 minutes → 100%
The power distribution
Thermal Power and Power Distribution
significantly changes also with changes of thermal power of the reactor. During power changes at power operation mode
(i.e. from about 1% up to 100% of rated power) the temperature reactivity effects play very important role
. As the neutron population
increases, the fuel
and the moderator
increase its temperature
, which results in decrease in reactivity
of the reactor (almost all reactors are designed to have the temperature coefficients negative
). The negative reactivity coefficient
acts against the initial positive reactivity insertion and this positive reactivity is offset
by negative reactivity from temperature feedbacks
This effect naturally occurs on a global scale, and also on a local scale.
During thermal power increase the effectiveness of temperature feedbacks will be greatest where the power is greatest. This process causes the flattening of the flux distribution, because the feedbacks acts stronger on positions, where the flux is higher.
It must be noted, the effect of change in the thermal power have significant consequences on the axial power distribution.
In reality, when there is a change in the thermal power and the coolant flow rate remains the same, the difference between inlet and outlet temperatures must increase. It follows from basic energy equation of reactor coolant, which is below:
Power increase. Let assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power.
The inlet temperature is determined by the pressure in the steam generators, therefore the inlet temperature changes minimally during the change of thermal power. It follows the outlet temperature must change significantly as the thermal power changes. When the inlet temperature remains almost the same and the outlet changes significantly, it stands to reason, the average temperature of coolant (moderator) will change also significantly. It follows the temperature of top half of the core increases more than the temperature of bottom half of the core. Since the moderator temperature feedback must be negative, the power from top half will shift to bottom half. In short, the top half of the core is cooled (moderated) by hotter coolant and therefore it is worse moderated. Hence the axial flux difference, defined as the difference in normalized flux signals (AFD) between the top and bottom halves of a two section excore neutron detector, will decrease.
AFD is defined as:
AFD or ΔI = Itop – Ibottom
where Itop and Ibottom are expressed as a fraction of rated thermal power.
In general, we have to distinguish between three types of power outputs in power reactors.
- Nuclear Power. Since the thermal power produced by nuclear fissions is proportional to neutron flux level, the most important, from reactor safety point of view, is a measurement of the neutron flux. The neutron flux is usually measured by excore neutron detectors, which belong to so called the excore nuclear instrumentation system (NIS). The excore nuclear instrumentation system monitors the power level of the reactor by detecting neutron leakage from the reactor core. The excore nuclear instrumentation system is considered a safety system, because it provide inputs to the reactor protection system during startup and power operation. This system is of the highest importance for reactor protection system, because changes in the neutron flux can be almost promptly recognized only via this system.
- Thermal Power. Although the nuclear power provides prompt response on neutron flux changes and it is irreplaceable system, it must be calibrated. The accurate thermal power of the reactor can be measured only by methods based on energy balance of primary circuit or energy balance of secondary circuit. These methods provide accurate reactor power, but these methods are insufficient for safety systems. Signal inputs to these calculations are, for example, the hot leg temperature or the flow rate through the feedwater system and these signals change very slowly with neutron power changes.
- Electrical Power. Electric power is the rate at which electrical energy is generated by the generator. For example, for a typical nuclear reactor with a thermal power of 3000 MWth, about ~1000MWe of electrical power is generated in the generator.