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Several fuel plants that supply nuclear research reactors need to increase their production capacity in order to meet the growing demand for this kind of nuclear fuel. After the enlargement of the production capacity of such plants, there will be the need of managing the new production level. That level is usually the industrial one, which poses challenges to the managerial staff. Such challenges come from the fact that several of those plants operate today on a laboratorial basis and do not carry inventory. The change to the industrial production pace asks for new actions regarding planning and control. The production process based on the hydrolysis of UF6 is not a frequent production route for nuclear fuel. Production planning and control of the industrial level of fuel production on that production route is a new field of studies. The approach of the paper consists in the creation of a mathematical linear model for minimization of costs. We also carried out a sensitivity analysis of the model. The results help in minimizing costs in different production schemes and show the need of inventory. The mathematical model is dynamic, so that it issues better results if performed monthly. The management team will therefore have a clearer view of the costs and of the new, necessary production and inventory levels.

The pacific use of nuclear technology is expanding worldwide [

Below is a list of the main characteristics of the production of nuclear fuels for research reactors:

・ laboratory scale,

・ exclusive for one reactor,

・ production facility located near the corresponding user, and

・ non-commercial transactions.

Nowadays several facilities that produce nuclear fuel for research reactors are having their production capacities expanded [

There are several kinds of nuclear fuels for research reactors and each fuel type has its own productive processes [

The fuel type chosen for this study is the uranium silicide dispersed in aluminum, known as dispersion type U_{3}Si_{2}-Al fuel. We adopted its maximum enrichment as 20% of the isotope ^{235}U, which is named Low Enriched Uranium (LEU) and which complies with the Reduced Enrichment for Research and Test Reactors (RERTR) Program [_{3}Si_{2}-Al fuel. We opted for LEU U_{3}Si_{2}-Al fuel because of its wide use in research reactors, its good capacity of uranium loading and its excellent performance [

The production of LEU U_{3}Si_{2}-Al fuel may be divided into two large sets of processes. The first set comprises the chemical processes and the second set, the metallurgical processes. Both processes sets may have different modes. But the chemical processes are the ones that have the bigger variability [_{6}) for the ensuing reasons:

・ that path stands out for its simplicity and relative safety;

・ it is used to produce small quantities of the intermediate products and

・ the rising demand for nuclear fuels for research reactors will probably affect facilities operating that production scheme.

The facts exposed so far underline the convenience of having a safe and reliable planning and control system for an increased production of LEU U_{3}Si_{2}-Al fuel, whose fabrication process includes UF_{6} hydrolysis. The design of such a system is an extensive task that is not within the limits of this paper. Thus, the objective of this paper is to develop the first parts of what should be the production planning and control system of a plant that produces LEU U_{3}Si_{2}-Al fuel, on the route that uses UF_{6} hydrolysis.

On the other hand, there is a plant in São Paulo, Brazil, which produces LEU U_{3}Si_{2}-Al fuel for research reactors and performs UF_{6} hydrolysis. That factory belongs to IPEN, Nuclear and Energy Research Institute, which is part of CNEN, Brazilian National Commission on Nuclear Energy. IPEN’s plant has been producing reliable nuclear fuel for the research reactor IEA-R1, Atomic Energy Institute-Reactor 1, for decades. The IEA-R1 reactor also belongs to IPEN.

Among various analytical instruments available to perform planning and control, in this study the option was made for the creation of a mathematical model for linear optimization. We used real data from IPEN’s fuel plant to run the model and to conduct its sensitivity analysis, obtaining useful results.

We supposed that the plant would produce only one product. That product is a Plate-Type Fuel Element (PTFE) containing LEU U_{3}Si_{2}-Al fuel. The raw materials and intermediate products for the production of LEU U_{3}Si_{2}-Al fuel are well known and can be found in several references [

In addition to the fuel itself, IPEN’s nuclear fuel plant also produces PTFE containing LEU U_{3}Si_{2}-Al fuel for the research reactor IEA-R1. The FE produced by IPEN’s nuclear fuel plant is exclusively used in the IEA-R1 reactor and was designed specifically for that reactor. As previously mentioned, IPEN’s nuclear fuel plant has been producing this FE for decades. Consequently, the design of that product is well established as are the production processes necessary for its fabrication. This FE is the typical PTFE used by most research reactors worldwide. It is made by the assembly of several fuel plates (FP) and other mechanical components [

The design of the PTFE for the reactor IEA-R1 is the database for the cost estimate of Section 4 of this work. That design is not published, but several references point out the main characteristics of a typical PTFE [

Many of the previously mentioned questions are studied by the area of operations management [

The area of planning and control has several ways and tools to perform its tasks. A very common way is called Operations Research (OR), which is a scientific approach for problem solving of complex systems management. Nowadays OR is used in activity fields as different as agriculture, education, industry, transportation and finance [

The methodology used in this paper is divided in two parts and each part is detailed in separated sections. Ensuing is an overview of those two parts.

Estimation of cost

The basis for the analysis conducted in this paper is the cost, due to its importance in any production process. Thus, the cost is the basis that supports this study. This is the reason for which we need to calculate the value of the production cost of a typical PTFE. However, calculating the exact cost of any product is an extensive task, which would exceed the limits of this paper. That means that we work with cost estimates. Such estimates are enough to accomplish the objectives of this study, since this study aims to be a first approach in the design of a system for production planning and control.

Data processing

After estimating the costs, we need to evaluate their behavior along the time. Thus, we created a mathematical model for the minimization of the total production cost. We used the data and assumptions from previous and posterior sections to run the model for one production scheme, as presented in Sections 5 and 6. We varied some parameters of that production scheme in order to study four other production schemes. That variation constitutes the sensitivity analysis of the model, thus providing information on the precision of the model.

In this section we present an estimation of costs for the production of one typical PTFE.

Aluminum

We start by a cost estimation of the raw materials used in one typical PTFE. The main material of that FE is aluminum and

・ The second column shows the number of pieces of each component necessary to build one PTFE [

・ We used the average density of Aluminum as 2700 kg/m^{3} [

・ The forth column is the result of multiplying columns 2 and 3;

・ The fifth column shows the applied engineering factors to account for extra material needed for the production of each component [

・ The last column is the multiplication of the fifth column by 30%, which is the average percentage of aluminum waste for all processes needed to build one PTFE [

Number | Squared components | Length, width and height (mm) | Volume (m^{3}) |
---|---|---|---|

1 | Nozzle | 265 × 70 × 70 | 0.0013 |

2 | Internal FP | 660 × 1.35 × 74 | 6.6 × 10^{−5} |

3 | External FP | 670 × 1.5 × 74 | 7.4 × 10^{−5} |

4 | Side plate | 875 × 5 × 81 | 0.00035 |

5 | FP spacer | 75 × 10 × 30 | 2.3 × 10^{−5} |

Number | Round components | Length × diameter (mm) | Volume (m^{3}) |

6 | Spacer pivot | 5 × 78 | 1.5 × 10^{−6} |

7 | Handling pin | 13 × 81 | 1.1 × 10^{−5} |

8 | Screw | 10 × 13 | 1.0 × 10^{−6} |

Component number | Number of pieces in one PTFE | Unit weight (kg) | Weight in one PTFE (kg) | Component correction factor | Final weight in one PTFE (kg) |
---|---|---|---|---|---|

1 | 1 | 3.50 | 3.50 | 1 | 4.51 |

2 | 16 | 0.18 | 2.84 | 1.2 | 4.48 |

3 | 2 | 0.20 | 0.40 | 1.2 | 0.83 |

4 | 2 | 0.96 | 1.91 | 1.2 | 3.06 |

5 | 2 | 0.06 | 0.12 | 1.2 | 0.19 |

6 | 2 | 0.01 | 0.02 | 2 | 0.06 |

7 | 1 | 0.03 | 0.03 | 2 | 0.08 |

8 | 8 | 0.01 | 0.08 | 2 | 0.19 |

Adding the final column of

Other raw materials

The values obtained from the references of

Purchased parts

Some parts of a typical PTFE are not produced in the same plant that processes the nuclear fuel and the PTFE itself. Such parts are usually made with the same aluminum of the rest of the PTFE. In order to estimate these costs, we set manufacturing factors over the aluminum content of each purchased part and expose them in

The costs presented in

Material number | Formulae | Price (US$/unit) | Reference | Quantity in one PTFE | Cost in one PTFE (US$) |
---|---|---|---|---|---|

1 | UF_{6} | 110.20/kg | [ | 2.25 kg | 247.95 |

2 | Si | 6.00/kg | [ | 3.00 kg | 18.88 |

3 | Mg | 4.00/kg | [ | 3.00 kg | 12.00 |

4 | Ni | 4.00/liter | [ | 50 liters | 200.00 |

5 | HNO_{3} | 0.80/liter | [ | 5 liters | 4.00 |

6 | SnCl | 88.00/kg | [ | 10.00 kg | 880.00 |

Component number | Component | Al weight (kg) | Manufacturing factor | Cost in one PTFE (US$) |
---|---|---|---|---|

1 | Nozzle | 4.51 | 4 | 188.34 |

7 | Handling pin | 0.19 | 8 | 15.87 |

8 | Screws | 0.08 | 12 | 10.02 |

column of

Energy

The main type of energy used in the production processes of IPEN’s fuel plant is electrical. For this reason, we will only consider that kind of energy in this paper.

The last column of

Firstly, we must address the yearly fuel demand, which we set as 100 PTFE. This value is the sum of the following three factors:

・ There is a project to build a new nuclear research reactor in Brazil in the near future. The new reactor will belong to CNEN, imposing its fuel to be supplied by IPEN’s factory according to Brazilian regulations. That new reactor will demand 60 PTFE per year;

・ IPEN’s nuclear fuel plant will continue to supply 10 PTFE per year to the IEA-R1 reactor and

・ There is the possibility of exporting approximately 30 PTFE per year.

Besides, we consider as base energy consumption the energy the factory uses independently of production, i.e., for lightning, security and office. The basic consumption exists for the whole year and is estimated in 1.2% of the energy used for production each month. Thus the base consumption in one year is 12 × 1.2 = 14.4% of the total year demand of energy for production of 100 PTFE.

Finally, we name maintenance energy consumption the energy the factory uses for the maintenance activities held during two months each year. The maintenance consumption is estimated in 3% of the energy used for production each month. Thus the maintenance energy consumption in one year is (2 × 3)/10 = 6% of the total year demand of energy for production of 100 PTFE.

The values of energy consumption for both maintenance and base refer to the production of 100 PTFE, considering production in 10 months and maintenance in two months. The addition of those two factors results in approximately 14.4% + 6% = 20% increase in the cost of energy to produce one PTFE. Thus, we added 20% to the cost of

Item | Gear | Power (kW) | Average working time (hours) | Cost for one PTFE (US$) |
---|---|---|---|---|

1 | Dryer | 5.6 | 36 | 348.77 |

2 | Crucible furnace | 32.5 | 4 | 224.90 |

3 | Induction furnace | 43.8 | 3 | 227.32 |

4 | Tempering furnace | 22.3 | 8 | 308.63 |

electrical energy for one PTFE and obtained US$ 1331.55 as the total energy cost per PTFE, named Cost 4.

Labor

The basis for our estimate of labor cost is the monthly salary of a Technician on the top of the Career of Technological Development. That salary was R$ 7902.14 [

IPEN’s nuclear fuel plant is divided in four work centers, each one having the number of employees exposed in

Adding the last column of

Since the mentioned 100 PTFE are all produced in the period of ten months within the year and since the cost of labor happens all 12 months, so there is an approximate addition of (12/10) − 1 = 20% in the cost of labor for the total yearly production. Thus, we added 20% to the labor cost for one PTFE and obtained US$ 24,385.64 as the total labor cost per PTFE, named Cost 5.

Environmental and total costs

The CER (Critical Environmental Rate) is a coefficient for account for environmental cost for different human activities and it may vary substantially [

Mathematical models for linear optimization have wide use in agriculture, production planning and control, logistics, telecommunications, finance, transportation and many other areas [

Work Center | Employee Number | Cost (US$/hour) | Time (hour) | Cost for one PTFE (US$) |
---|---|---|---|---|

1 | 4 | 112.86 | 45.0 | 5078.58 |

2 | 3 | 84.64 | 20.8 | 1756.34 |

3 | 5 | 141.07 | 17.0 | 2398.22 |

4 | 6 | 169.29 | 65.5 | 11,088.23 |

Cost | Description | US$ |
---|---|---|

1 | Aluminum | 139.90 |

2 | Other raw materials | 1361.95 |

3 | Purchased parts | 214.23 |

4 | Electrical Energy | 1331.55 |

5 | Labor | 24,385.64 |

6 | Total | 27,433.27 |

7 | Environment | 2743.33 |

8 | Grand total | 30,716.59 |

ture [

x_{t} = quantity of PTFE to be produced in month t

That amount will never be negative, so that:

The boundary conditions of the model are presented below.

・ The plant produces only one product;

・ It is desired to plan its production for T periods of time;

・ The time period was set as one month;

・ The model was made for one year of planning, so that T = 12 months;

・ Demand is known every month;

・ The resources required for production are limited, renewable and there is enough availability of them in the beginning of each month;

・ There is the possibility of keeping inventory from one month to the other and

・ Production stops for two months per year for maintenance. This way: x_{1} = x_{7} = 0

The model’s goal is to minimize the production cost.

We supposed that the cost of holding one PTFE in inventory each month is R = 1.5% of the production cost in the same month. We adopted the number of PTFE held each month as the average of the beginning and ending inventory for the month. We also assumed that the beginning inventory in any month is equal to the ending inventory from the previous month. Thus, the inventory cost (IC) is expressed by Equation (1):

The Objective Function must reflect the goal of the model, i.e., the Objective Function must be minimizing the total annual production cost. That cost is the sum of the

Symbol | Definition | Value |
---|---|---|

d_{t} | Demand in month t | |

D | Yearly demand from Section 4 | 100 PTFE |

R | Monthly interest rate | 1.5% |

PCt | Production capacity in month t | 12 PTFE |

C | Cost to produce one PTFE | US$ 30,716.59 |

IB_{t} | Inventory in the beginning of month t | |

IE_{t} | Inventory in the end of month t | |

IB_{1} | Inventory in the beginning of the first month | 5 PTFE |

Imax_{t} | Maximum inventory in the end of month t | 6 PTFE |

Imin_{t} | Minimum inventory in the end of month t | 1 PTFE |

product of the cost to produce one PTFE to the total production plus the inventory cost. Thus the Objective Function is expressed by Equation (2):

Minimize

The Objective Function must obey following constraints, according to the previous assumptions:

Inventory constraints

IB_{t} = IE_{t−}_{1} t = 1, 2, ∙∙∙, T;

IE_{t} = IE_{t−}_{1} + x_{t} − d_{t} t = 1, 2, ∙∙∙, T and

Imin_{t} ≤ IE_{t} ≤ lmax_{t} t = 1, 2, ∙∙∙, T

Replacing the adopted values:

1 ≤ IE_{t} ≤ 6 t = 1, ..., 12;

Production capacity constraints

Production yield cannot be bigger than production capacity, i.e.:

x_{t} ≤ PC_{t} t = 1, 2, ..., T

Replacing the adopted values:

For any month x_{t} ≤ 12

For one year x_{t} ≤ 120

Demand meeting constraints

x_{t} + IE_{t−}_{1} ≥ d_{t} or x_{t} + IE_{t−}_{1} − d_{t} ≥ 0.

But we defined x_{t} + IE_{t−}_{1} − d_{t} = IE_{t}.

Thus to guarantee that the demand will be met, it is enough to impose:

Imin_{t} ≤ IE_{t} ≤ lmax_{t} t = 1, 2, ..., T

Therefore the complete model has the following formulation:

Minimize

Subject to:

IE_{t} = IE_{t−}_{1} + x_{t} − d_{t} t = 1, 2, ∙∙∙, 12

1 ≤ lE_{t} ≤ 6 t = 1, 2, ∙∙∙, 12;

x_{t} ≤ 12 t = 1, 2, ∙∙∙, 12;

x_{1} = x_{7} = 0

x_{t} ≥ 0 t = 1, 2, ∙∙∙, 12

The model is linear and has more equations than unknown variables, thus having many possible solutions. The most common way of finding the optimal solution is the Simplex Method, which is widely studied in the literature [

・ Column 2: Inventory at the beginning of month t, IB_{t}.

・ Column 3: Monthly production, x_{t}.

・ Column 4: Monthly demand, d_{t}.

・ Column 5: Inventory at the end of month t, IE_{t}.

Columns 2 to 5 of _{1} = x_{7} = 0 in column 3 of

Month of the year | IB_{t} | x_{t} | d_{t} | IE_{t} | Production cost (US$) | Inventory carrying cost (US$) | Accumulated cost (US$) |
---|---|---|---|---|---|---|---|

1 | 5 | 0 | 3 | 2 | 0.00 | 1612.62 | 1612.62 |

2 | 2 | 4 | 5 | 1 | 122,866.36 | 691.12 | 125,170.10 |

3 | 1 | 10 | 10 | 1 | 307,165.90 | 460.75 | 432,796.75 |

4 | 1 | 12 | 12 | 1 | 368,599.08 | 460.75 | 801,856.58 |

5 | 1 | 12 | 12 | 1 | 368,599.08 | 460.75 | 1,170,916.41 |

6 | 1 | 12 | 7 | 6 | 368,599.08 | 1612.62 | 1,541,128.11 |

7 | 6 | 0 | 5 | 1 | 0.00 | 1612.62 | 1,542,740.73 |

8 | 1 | 7 | 7 | 1 | 215,016.13 | 460.75 | 1,758,217.61 |

9 | 1 | 10 | 10 | 1 | 307,165.90 | 460.75 | 2,065,844.26 |

10 | 1 | 12 | 12 | 1 | 368,599.08 | 460.75 | 2,434,904.09 |

11 | 1 | 12 | 12 | 1 | 368,599.08 | 460.75 | 2,803,963.92 |

12 | 1 | 5 | 5 | 1 | 153,582.95 | 460.75 | 2,958,007.62 |

Total | 96 | 100 | 2,948,792.64 | 9214.98 | 2,958,007.62 |

Production Scheme | Minimum Inventory (PTFE number) | Maximum Inventory (PTFE number) | Total Yearly Cost (US$ Millions) |
---|---|---|---|

1 | 0 | 5 | 2.92 |

2 | 1 | 6 | 2.96 |

3 | 2 | 7 | 2.99 |

4 | 3 | 8 | 3.00 |

5 | 3 | 7 | 3.03 |

From

As mentioned before,

From scheme 1 to scheme 4 in

Negro, M.L.M., Durazzo, M., de Mesquita, M.A., de Car- valho, E.F.U. and de Andrade, D.A. (2016) Studies on Production Planning of Dispersion Type U_{3}Si_{2}-Al Fuel in Plate-Type Fuel Elements for Nuclear Research Reactors. World Journal of Nuclear Science and Te- chnology, 6, 217-231. http://dx.doi.org/10.4236/wjnst.2016.64023