Journal of Chemistry
Volume 2015 (2015), Article ID 809672, 10 pages
http://dx.doi.org/10.1155/2015/809672

Research Article

Parametric Study of Time-Dependent Corrosion Product Activity due to ⁵⁶Mn, ⁵⁸Co, and ⁶⁰Co in the Primary Coolant Circuit of a Typical Pressurized Water Reactor

Muhammad Rafique,1,2 Nasir M. Mirza,³ Sikander M. Mirza,³ Kimberlee J. Kearfott,² Shahab A. Abbasi,¹ and Syed F. Naeem²

¹Department of Physics, University of Azad Jammu and Kashmir, Muzaffarabad, Azad Kashmir 13100, Pakistan
²Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109-2104, USA
³Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan

Received 4 December 2014; Revised 5 January 2015; Accepted 15 January 2015

Academic Editor: Concha Gimeno

Copyright © 2015 Muhammad Rafique et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Results of a detailed study, based on the parametric analysis of activated corrosion products, in primary coolant of a typical pressurized water reactor (PWR) are presented. The parameters influencing time dependent buildup of corrosion product activity (CPA) in primary coolant loop of PWR were identified. The computer program CPAIR was used to accommodate for time dependent corrosion rates. The behaviors of ⁵⁶Mn, ⁵⁸Co, and ⁶⁰Co were studied over the reactor operational time. During the course of normal operation of reactor, the CPA is dominated by ⁵⁶Mn, while ⁵⁸Co and ⁶⁰Co are the predominant radionuclides after reactor shutdown. Parametric study suggests that the total CPA is most sensitive to ion-exchanger removal rates. For a removal rate of 300 cm³-s⁻¹, the specific activity due to ⁵⁶Mn has the maximum value of 3.552 × 10⁴ Bq-m⁻³ after 1,000 hours of reactor operation. This value decreases drastically to 8.325 × 10³ Bq-m−3 at removal rate of 900 cm³-s⁻¹. Additionally, CPA due to ⁵⁶Mn, ⁵⁸Co, and ⁶⁰Co shows strong dependence on removal rates from the core material surfaces. Variations in the values of radionuclide removal rates from piping surface and radionuclide removal rate from deposition on pipes showed only very small effects on CPA buildup.

1. Introduction

Corrosion products activated by high neutron flux in the reactor core are the dominant contributors to the postshutdown radiation field (PSRF) for all water reactors. Pressurized water reactors (PWRs) have an order of magnitude higher PSRF values compared to gas-cooled reactors. Large PSRF values affect the plant availability factor by prolonging repair and maintenance schedules, entailing annual economic penalties of several billion US dollars [1].

In the past, several authors [2–18] carried out investigations on the formation, transport, and deposition of radioactive corrosion products in various parts of PWRs. Several models were developed to simulate corrosion product transport and the buildup of activity in the primary coolant systems of PWRs. The PACTOLE program [4] was developed for predicting the time-dependent behavior of the activation of PWR corrosion products. Based upon experimental data, the CORA code was upgraded in 1985 [19] to model the transport of corrosion products in primary coolant loop of PWRs. The MIGA code was developed [20] on a strong understanding of the chemical thermodynamics that determines the most stable oxide phase likely to exist under the chemistry and temperature conditions in each region of the primary circuit. The DISER program [21] simulates corrosion product activity (CPA), assuming that the coolant contains solute, colloids, and particulate corrosion products. Song and Lee [22] investigated the generation mechanism of crud in the long term fuel cycle, using the COTRAN code, which simulates the behavior of the crud based on double layer concept model and solubility difference. The effects of flow and power transients on CPA were investigated by Mirza et al. and Deeba et al. using the code CPAIR [23, 24]. This was further modified to account for CPA under linearly and nonlinearly rising corrosion rate models by Mirza et al. [13, 14] and Rafique et al. [15].

Malik et al. [25] studied CPA under flow rate transients, having extended burn-up cycles with time-dependent pH-level and boric acid variations. Their finding showed CPA buildup towards saturation following the mixing of boron in primary coolant and a decreasing trend following an increase in pH value. Another study by Malik et al. [26] showed that the use of enriched boric acid as chemical shim actually lowers the primary coolant activity when higher pH values are employed in coolant rather than natural boric acid.

Lewis et al. [27] developed a physical model describing the coolant activity behavior of ⁹⁹Tc during constant and reactor shutdown operations. Kang and coworkers [19] investigated the inventory of radioactivity in the typical 1,000 MWe Spanish PWR. Analysis was based on DORT models. Henshaw and coworkers [28] proposed a model for simulating thermal hydraulic and chemical conditions within fuel crud deposits. The unique feature of their model is that the chemistry is coupled to the thermohydraulics via the increase in the saturation temperature with the concentration of dissolved species. The model explains several crud scrape observations.

Particulate fouling due to corrosion product sedimentation plays an important role in heat exchanger material integrity and water chemistry. Zebardast et al. [29] proposed a new experimental method for the detection of magnetite particles at temperatures up to 473 K.

In-pile loops were used for the experimental study of possible improvements in the coolant chemistry for reducing the corrosion product dose rates [30, 31]. Those studies indicated that dissolution and crystallization cannot be modeled accurately enough in the absence of the required experimental coefficients for these processes. Those detailed studies revealed that the rate of corrosion in the reactor systems continues increasing as the time of reactor operation at full power increases. Access and frequent maintenance to areas near primary pumps then become very limited due to high radiation doses from primary water and pipe scale.

In this work, parametric studies of CPA under normal reactor operation were carried out. The three radionuclides ⁵⁶Mn, ⁶⁰Co, and ⁵⁸Co were considered. Sensitive parameters are identified from the models, showing their effects on CPA buildup in the primary coolant of typical PWRs.

2. Mathematical Models

The purpose of the parametric analysis of CPA in the primary coolant loop of PWRs is to study the sensitivity of the model simulations to uncertainties in the values of model input data. This work identifies sensitive parameters occurring in mathematical models used for simulating the behavior of CPA.

The governing time-dependent differential equations for modeling CPA in the primary circuits of PWRs appear in the literature [13–16, 18, 23, 24] and now are summarized. The rate of buildup of target nuclide concentration in coolant water can be written as

where

and the quantities , , , and are removal rates due to ion-exchanger, deposition on pipes, deposition on core surfaces, and removal by filters, respectively. Similarly, the leakage rate from the th leak is given by , where the term is the rate at which primary coolant loop loses water from its th leak (cm³-s⁻¹). In addition, , where is the rate at which isotopes are removed from the scale on piping (cm³-s⁻¹), and , where is the rate at which isotopes are removed from the scale on core material (cm³-s⁻¹).

In the above equations (the radionuclide removal rates due to ion-exchanger, deposition on pipes, deposition on core surfaces, and removal by filters), (leakage rate from the th leak), (radionuclide removal from piping surface), and (radionuclide removal rate from scales on core materials) are sensitive variables which change the time-dependent CPA in the primary coolant of PWR.

If is the group constant for the production of isotope from target nuclide, then , the effective group flux (neutrons-cm⁻²-s⁻¹), is defined by the equation

The values of decay constant () for ⁵⁶Mn, ⁶⁰Co, and ⁵⁸Co are provided in Table 1. The group flux is averaged over the geometry of the core and has been estimated using LEOPARD [32] and ODMUG [33] programs in the CPAIR. The times and are the core residence and loop times, respectively.

Table 1: Activated corrosion products and their reaction properties [13].

Corrosion products
Reaction and neutron energy
Activation cross section and half-life

⁵⁶Mn
⁵⁵Mn(n, ) ⁵⁶Mn (En is thermal)
13.4 b (2.58 h)
9.2 × 10¹³

⁵⁸Co
⁵⁸Ni(n, p) ⁵⁸Co (fast neutrons)
0.146 b (70.88 d)
9.02 × 10¹³

⁶⁰Co
⁵⁹Co(n, ) ⁶⁰Co
20 b (5.3 y)
9.02 × 10¹³

The core residence time is given by

where is the core height, is the flow cross-sectional area in (cm²), is the coolant density at operating temperatures, and is the time-dependent flow rate (g-s⁻¹).

The loop time is approximated as for the complete loop length . , , and stand for the target nuclide concentration in water, on piping surfaces, and on core surfaces (atoms-cm⁻³); is flow rate perturbation parameter, defined as , where is the time-dependent flow rate and is the steady state flow rate under normal operations. Because in this study transient conditions are avoided, in our subsequent study we will take value of equal to unity; that is, . is primary coolant volume.

The corrosion source term appearing in the above equation is given by

where is the time-dependent corrosion rate (gram cm⁻²-s⁻¹), is the area of system exposed to coolant for corrosion, is Avogadro’s number (6.023 × 10²³ atoms g-mole⁻¹), and is the atomic weight of the target nuclide (g). Here, and are abundances of target nuclides and chemical elements in the system, respectively.

The rate of change of active material concentration in primary coolants for each radionuclide is given by

where , , and are the concentrations of the activated radionuclides in the primary coolant water, on the piping, and on the core surfaces, respectively, in atoms-cm⁻³.

The rate of activity buildup on core scale is given by

where and .

Here is volume of the scale on the core (cm³). And , where is removal rate from scales on core surfaces. Consider .

The rate of buildup of target nuclide concentration on the core scale, , is given by

It then follows that the rate of change of target nuclide on piping walls, , is

The rate of deposition of active materials on the pipes is expressed as

where , with equal to volume of scale on the piping (cm³). , is removal rate from piping surface (cm³).

The rate of activity buildup in ion-exchanger is expressed mathematically as

where , where is volume of ion-exchanger. is removal rate by ion-exchanger.

The rate of activity buildup in filters becomes

where , is volume of filter, and is removal rate of filters.

Using the above set of coupled differential equations ((1) through (12)), the computer program CPAIR (corrosion product activity in reactors) [15], using fourth-order Runge-Kutta method to find the activity values due to corrosion, was modified for this work.

3. Methods and Assumptions

The computer code CPAIR, utilizing specifications of a typical PWR with power of 1,000 MWe, was considered [13–15, 34]. Initial impurity concentrations were assumed to be zero. The numerical values of experimental fractional exchange rates () and resolution rates () assumed for this work are summarized in Table 2.

Table 2: Experimental values of exchange rates in a typical PWR [13–15, 34].

Rate type
Value

Deposition on core ()
80.0 cm³-s⁻¹

Deposition on piping ()
13.7 cm³-s⁻¹

Ion-exchanger removal ()
500.–781 cm³-s⁻¹

Resolution ratio for core ()
40.0 cm³-s⁻¹

Resolution ratio for piping ()
6.9 cm³-s⁻¹

Volume of primary coolant ()
1.37 × 10⁷ cm³

Volume of scale on core ()
9.08 × 10⁶ cm³

Volume of scale on piping ()
1.37 × 10⁶ cm³

Total corrosion surface ()
1.01 × 10⁸ cm²

The design data values for a typical PWR under consideration are given in Table 3 [13–15, 34]. The core averaged group fluxes have been computed using LEOPARD and ODMUG computer codes. The LEOPARD program is a zero-dimensional unit cell computer code with 54 fast and 172 thermal energy groups. Evaluated Nuclear Data (ENDF-IV) set has been used in the cross section library. In this work, equivalent cells of a typical PWR [34] have been employed to generate group constants for fuel cells and water holes. These cell-averaged group constants are then used in the one-dimensional multigroup diffusion theory based ODMUG code [33]. It calculates the group fluxes as a function of position in the reactor. These group fluxes are subsequently averaged over the core. Both LEOPARD and ODMUG are treated as subroutines of the CPAIR program.

Table 3: Typical design specifications of a PWR [13–15, 34].

Parameter
Value

Specific power (MW(th)-kg⁻¹U)
33

Power density (MW(th)-m⁻³)
102

Core height (m)
4.17

Core diameter (m)
3.37

Number of assemblies
194

Number of rods per assembly
264

Fuel type
UO₂

Clad type
Zircoloy

Lattice pitch (mm)
12.6

Fuel rod outer diameter (mm)
9.5

Average enrichment (w%)
3.0

Flow rate (Mg-s⁻¹)
18.3

Linear heat rate (kW-m⁻²)
17.5

Coolant pressure (MPa)
15.5

Inlet coolant temperature (°C)
293

Outlet coolant temperature (°C)
329

The plant surface area of ~10⁸ cm² is exposed to the primary coolant for corrosion. In the presence of corrosion inhibitors, an equilibrium corrosion rate of 2.4 × 10⁻¹³ g-cm⁻²-s⁻¹ [13–15, 33–35] exists after 1 y operation of a reactor with a primary coolant volume of 1.3 × 10⁷ cm³. The corrosion rate of 25 μg-s⁻¹ has been used as normal equilibrium rate in subsequent studies.

The purification rate due to an ion-exchanger was considered to be large enough to regard deposition, resolution, and leakage as second-order effects. Three radionuclides, ⁵⁶Mn (soluble corrosion products, coming from impurities in the steel), ⁶⁰Co, and ⁵⁸Co (insoluble corrosion products), were considered.

4. Results and Discussion

During normal reactor operation at full power, the major contributor towards activity comes from the radionuclide ⁵⁶Mn. The ⁵⁶Mn activity saturates at about 150 h after start of the reactor. The saturation of ⁵⁶Mn activity was due to the fact that we assumed abundances of target nuclides and chemical elements in the system equal to 1 and 0.5, respectively. Its saturation value is 8.325 × 10⁻³ Bq-cm⁻³ and it makes the primary coolant as a 2.453 × 10⁷ Bq source (including coactivity) within 150–230 h of reactor operation at full power.

4.1. Effect of Ion-Exchanger Removal Rates () on CPA Buildup in Primary Coolant Loop

Specific activity due to each of the radionuclides (⁵⁶Mn, ⁵⁸Co, and ⁶⁰Co) changes significantly, in the primary coolant loop, with the assumed removal rates for the ion-exchanger.

At removal rate of 300 cm³-s⁻¹, the specific activity due to ⁵⁶Mn reaches a maximum value (saturation value) of 3.552 × 10⁴ Bq-cm⁻³ after 1,000 h of reactor operation as shown in Figure 1. Specific activities at removal rates of 400, 500, 600, 700, 800, and 900 cm³-s⁻¹ result in CPA values of (2.49, 1.868, 1.465, 1.188, 0.984, 0.832) × 10⁴ Bq-cm⁻³, respectively.

Figure 1: Variation of specific activity in primary coolant of PWR with different values of ion-exchanger removal rates.

These values indicate the fact that ion-exchanger removal rates play an important role in controlling the specific activity in PWR primary coolant. So removal rates of 800 cm³-s⁻¹ and greater are optimal values of ion-exchanger removal rate. The parameter is the most sensitive parameter for the mathematical models above. By changing values of there will be considerable change in the coolant activity.

The normalized specific activity due to ⁵⁶Mn was evaluated using a conventional removal rate of 600 cm³-s⁻¹. Specific activity due to ⁵⁶Mn at removal rates of 300, 400, 500, 700, 800, and 900 cm³-s⁻¹ was 2.42, 1.7, 1.27, 0.81, 0.67, and 0.57 times the conventional removal rate, respectively. Normalized specific activity as a function of different removal rates is shown in Figure 2.

Figure 2: Normalized specific activity due to ⁵⁶Mn as a function of different removal rates.

4.2. Effect of Radionuclide Removal Rate from Core Surface Scale () on CPA Buildup in Primary Coolant Loop

Variations in the value of change the value of CPA in primary circuit of PWR. At  cm³-s⁻¹ the specific activity due to ⁵⁶Mn in primary coolant becomes 1.21 × 10⁴ Bq-cm⁻³, which decreases at higher values of and becomes 1.16 × 10⁴ Bq-cm⁻³ at  cm³-s⁻¹. The variation of ⁵⁶Mn specific activity with is depicted in Figure 3 and saturation values are listed in Table 4.

Table 4: Specific activities of ⁵⁶Mn, ⁵⁸Co, and ⁶⁰Co in primary coolant of PWR with different values of .

(cm²/s)
 Specific activity (Bq/cm³) after 1000 hours of reactor operation

⁵⁶Mn
⁵⁸Co
⁶⁰Co

25
12118
21088
432

30
12042
17753
362

35
11965
15321
311

40a
11889
13476
273

45
11813
12026
243

50
11739
10858
219

55
11666
9899
199

Actual experimental value of resolution value for core in a typical PWR.

Figure 3: Specific activity of ⁵⁶Mn as a function of reactor operation time in primary coolant of PWR with different values of .

Normalized values of specific activities due to ⁵⁶Mn as a function of different resolutions rates from core surfaces are shown in Figure 4. A conventional value of  cm³-s⁻¹ was chosen and specific activity due to ⁵⁶Mn at resolution rates of 25, 30, 35, 45, 50, and 55 cm³-s⁻¹ was found as a factor of 1.019, 1.012, 1.006, 0.99, 0.987, and 0.981 with the conventional chosen resolution rate activity.

Figure 4: Normalized specific activity of ⁵⁶Mn as a function of .

Variation in the value of also changes the amount of radioactivity due to ⁵⁸Co in primary coolant. The specific activity due to ⁵⁸Co decreases with increasing the values of . For  cm³-s⁻¹ the specific activity due to ⁵⁸Co has the value 2.11 × 10⁴ Bq-cm⁻³ after 1,000 hours of reactor operation, and this value decreases to 0.989 × 10⁴ Bq-cm⁻³ at  cm³-s⁻¹ as shown in Figure 5 and Table 4. The resolution rates must also be considered a sensitive parameter.

Figure 5: Variation in specific activity of ⁵⁸Co in primary coolant of PWR with different values of .

Normalized values of specific activities due to ⁵⁸Co as a function of different resolutions rates from core surfaces are shown in Figure 6. A conventional value of  cm³-s⁻¹ was chosen and specific activity due to ⁵⁸Co at resolution rates of 25, 30, 35, 45, 50, and 55 cm³-s⁻¹ was found as a factor of 1.56, 1.31, 1.14, 0.89, 0.81, and 0.73 with the conventional chosen resolution rate activity.

Figure 6: Normalized specific activity of ⁵⁸Co as a function of .

Similarly specific activity due to ⁶⁰Co in primary coolant decreases with increasing values of . For  cm³-s⁻¹ the specific activity has the value 432 Bq-cm⁻³ after 1,000 h of reactor operation. This value decreases to 199 Bq-cm⁻³ at  cm³-s⁻¹ as shown in Figure 7 and Table 4.

Figure 7: Specific activity of ⁶⁰Co in primary coolant of PWR with different values of .

4.3. Effect of Radionuclide Removal Rate from Core Surface Scale () on CPA Buildup on Piping Surface

Activated corrosion products once produced may get deposited on piping surfaces of primary coolant loop over the period of reactor operation, thereby becoming the potential source of radiation field. On piping surfaces of primary coolant loop the value of specific activity due to ⁵⁸Co for  cm³-s⁻¹ has the value 5.81 × 10⁴ Bq-cm⁻³ after 1,000 h of reactor operation, and this value decreases radically to 2.838 × 10⁴ Bq-cm⁻³ at  cm³-s⁻¹ as shown in Figure 8.

Figure 8: Specific activity of ⁵⁸Co on primary coolant piping surfaces of PWR with different values of .

On piping surfaces of primary coolant loop the value of specific activity due to ⁶⁰Co for  cm³-s⁻¹ has the value 4847 Bq-cm⁻³ after 1,000 h of reactor operation, and this value decreases radically to 2331 Bq-cm⁻³ at  cm³-s⁻¹, as illustrated in Figure 9.

Figure 9: Specific activity of ⁶⁰Co on primary coolant piping surfaces of PWR with different values of .

4.4. Effects of Radionuclide Removal Rate from Piping Scales () on CPA Buildup in Primary Coolant Loop

While changing the value , change in CPA in primary coolant has been observed. For the CPA due to ⁵⁸Co has value 3626 Bq-cm⁻³, which decreases to 3589 Bq-cm⁻³ at as shown in Figure 10. Similarly for ⁶⁰Co specific activity has the value 270 Bq-cm⁻³ at , and this value decreases to the value 266 Bq-cm⁻³ at as shown in Figure 11 and Table 5.

Table 5: Specific activity of ⁵⁸C0 and ⁶⁰Co with various values.

(cm²/s)
 Specific activity (Bq/cm³) after 1000 hours of reactor
operation

⁵⁸Co
⁶⁰Co

6.0
3595
268

6.5
3605
269

7.0b
3612
270

7.5
3621
271

8.0
3629
271

Actual experimental value of resolution rate for piping in a typical PWR.

Figure 10: Specific activity of ⁵⁸Co in primary coolant of PWR with different values of .

Figure 11: Specific activity of ⁶⁰Co in primary coolant of PWR with different values of .

4.5. Effects of Removal Rate from Deposition on Piping Surfaces () on CPA Buildup in Primary Coolant Loop

Effect of on CPA due to ⁵⁸Co and ⁶⁰Co in primary coolant has also been studied. For ⁵⁸Co the specific activity has the value 3697 Bq-cm⁻³, and this value decreases to 3589 Bq-cm⁻³ for after 1,000 h of reactor operation, as shown in Figure 12 and Table 6. For ⁶⁰Co the specific activity has the value 277 Bq-cm⁻³, and this value decreases to 268 Bq-cm⁻³ for after 1,000 h of reactor operation, as shown in Figure 13.

Table 6: Specific activity of ⁵⁸Co and ⁶⁰Co in primary coolant of PWR with different values of .

(cm²/s)
Specific activity (Bq/cm³) after 1000 hours of reactor operation

⁵⁸Co
⁶⁰Co

10.7
3589
268

11.7
3606
269

12.7
3624
271

13.7c
3642
273

14.7
3660
274

15.7
3678
276

16.7
3697
277

Actual experimental value of resolution rate from deposition on piping.

Figure 12: Variation in specific activity of ⁵⁸Co in primary coolant of PWR with different values of .

Figure 13: Specific activity of ⁶⁰Co in primary coolant of PWR with different values of .

Some of the results of CPAIR have been compared with the code CRUDSIM/MIT [31]. The same data were used to simulate the CPA using CPAIR program, resulting in 2.02 × 10⁴ Bq-cm⁻³ estimated for the stainless steel letdown filters, whose function is to retain corrosion products ⁵⁸Co, ⁶⁰Co, and ⁵⁶Mn. Because solubility is a function of temperature and the temperature of the filters is less than that of the bulk coolant, the filter activity concentrations are not necessary representatives of actual coolant activity concentrations. The filter results, however, are very similar to the value of 1.85 × 10⁴ Bq-cm⁻³ reported elsewhere [31].

5. Conclusions

Corrosion products are strong contributors to radiation levels produced in personnel working environment of PWR, and thus the estimation of corrosion product activity (CPA) is important. Higher burn-up, longer-lived reactor cores make the problem more critical. In this work, a parametric study of CPA in primary coolant of a typical PWR was carried out using the CPAIR computer code. Conclusions based upon those numerical simulations include the following.

(i)During normal operation ⁵⁶Mn dominates the CPA, while ⁵⁸Co and ⁶⁰Co are the main contributors after shutdown.(ii)The CPA shows a strong sensitive dependence on .(iii)The variation in other parameters, including , , and , only weakly affects the CPA.(iv)Excellent agreement was found between predictions of CPA using a modified version of CPAIR and CRUDSIM/MIT findings.

Nomenclature

: | Target nuclide concentration in primary coolant water, atoms-cm⁻³ ((1), (8), and (9))
: | Target nuclide concentration on inner walls of the piping, atoms-cm⁻³ ((1) and (9))
: | Target nuclide concentration on core surfaces, atoms-cm⁻³ ((1) and (8))
: | Radionuclide concentration primary water, atoms-cm⁻³ ((6), (7), (10), (11), and (12))
: | Radionuclide concentration on the piping, atoms-cm⁻³ ((6) and (10))
: | Radionuclide concentration on the core surface, atoms-cm⁻³ ((6) and (7))
: | Removal rate due to ion-exchanger (1)
: | Removal rate from deposition on pipes (1)
: | Removal rate from deposition on core surfaces (1)
: | Removal rate by filters (1)
: | Primary coolant loop water loss rate from the th leak, cm³-s⁻¹ (1)
: | Radionuclide removal rate from pipe scale, cm³-s⁻¹ (1)
: | Radionuclide removal rate from core surface scale, cm³-s⁻¹ (1)
: | Corrosion source term (5)
: | Time-dependent corrosion rate, g-cm⁻²-s⁻¹ (5)
: | Area of system exposed to coolant for corrosion, 10⁸ cm² (5)
: | Avogadro’s number, 6.023 × 10²³ atoms-g⁻¹-mole⁻¹ (5)
: | Target nuclide atomic weight (5)
: | Target nuclide abundance of target nuclide in system (5)
: | Chemical element abundance of chemical element in system (5)
: | Volume of the scale on the core, cm³
: | Volume of scale in the piping, cm³
: | Coolant water volume, cm³ ((1) and (5))
: | Thermal neutron flux averaged over the core geometry, neutrons-cm⁻²-s⁻¹ (3)
: | The group constant for radionuclide production from target nuclides
: | The effective group flux, neutrons-cm⁻²-s⁻¹
: | Steady state flow rate under normal operations (4)
: | Time-dependent flow rate (4).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Muhammad Rafique is grateful to the Higher Education Commission of Pakistan for providing postdoctoral fellowship in the form of Grant 2-6(22)/PDFP/HEC/2013/14.

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