## About Zirconium-Niobium Alloy

Zirconium alloys with niobium are used as claddings of fuel elements of VVER and RBMK reactors. These alloys are the basis material of assembly channel of RBMK reactor. The Zr + 1% Nb alloy of type N-1 E-110 is used for fuel element claddings, the Zr + 2.5% Nb alloy of type E-125 is applied for tubes of assembly channels.

The cladding material for the new 17×17 fuel designs is based also on the **zirconium-niobium alloys** (e.g. Optimized ZIRLO material), which has been demonstrated to have improved corrosion resistance compared with prior fuel cladding materials. The optimized tin level provides a reduced corrosion rate while maintaining the benefits of mechanical strength and resistance to accelerated corrosion from abnormal chemistry conditions.

### Summary

Name | Zirconium-Niobium Alloy |

Phase at STP | solid |

Density | 6560 kg/m3 |

Ultimate Tensile Strength | 514 MPa |

Yield Strength | 381 MPa |

Young’s Modulus of Elasticity | 99 GPa |

Brinell Hardness | 89 BHN |

Melting Point | 1850 °C |

Thermal Conductivity | 18 W/mK |

Heat Capacity | 285 J/g K |

Price | 25 $/kg |

## Density of Zirconium-Niobium Alloy

In words, the density (ρ) of a substance is the total mass (m) of that substance divided by the total volume (V) occupied by that substance. The standard SI unit is **kilograms per cubic meter** (**kg/m ^{3}**). The Standard English unit is

**pounds mass per cubic foot**(

**lbm/ft**).

^{3}Density of Zirconium-Niobium Alloy is **6560 kg/m ^{3}.**

### Example: Density

Calculate the height of a cube made of Zirconium-Niobium Alloy, which weighs one metric ton.

**Solution:**

**Density** is defined as the **mass per unit volume**. It is mathematically defined as mass divided by volume: **ρ = m/V**

As the volume of a cube is the third power of its sides (V = a^{3}), the height of this cube can be calculated:

The height of this cube is then **a = 0.534 m**.

### Density of Materials

## Mechanical Properties of Zirconium-Niobium Alloy

### Strength of Zirconium-Niobium Alloy

In mechanics of materials, the **strength of a material** is its ability to withstand an applied load without failure or plastic deformation. **Strength of materials** basically considers the relationship between the **external loads** applied to a material and the resulting **deformation** or change in material dimensions. **Strength of a material** is its ability to withstand this applied load without failure or plastic deformation.

### Ultimate Tensile Strength

Ultimate tensile strength of **Zirconium-Niobium Alloy **is about 514 MPa.

The **ultimate tensile strength** is the maximum on the engineering stress-strain curve. This corresponds to the** maximum stress** that can be sustained by a structure in tension. Ultimate tensile strength is often shortened to “tensile strength” or even to “the ultimate.” If this stress is applied and maintained, fracture will result. Often, this value is significantly more than the yield stress (as much as 50 to 60 percent more than the yield for some types of metals). When a ductile material reaches its ultimate strength, it experiences necking where the cross-sectional area reduces locally. The stress-strain curve contains no higher stress than the ultimate strength. Even though deformations can continue to increase, the stress usually decreases after the ultimate strength has been achieved. It is an intensive property; therefore its value does not depend on the size of the test specimen. However, it is dependent on other factors, such as the preparation of the specimen, the presence or otherwise of surface defects, and the **temperature** of the test environment and material. **Ultimate tensile strengths** vary from 50 MPa for an aluminum to as high as 3000 MPa for very high-strength steels.

### Yield Strength

Yield strength of **Zirconium-Niobium Alloy **is about 381 MPa.

The **yield point** is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning plastic behavior. **Yield strength** or yield stress is the material property defined as the stress at which a material begins to deform plastically whereas yield point is the point where nonlinear (elastic + plastic) deformation begins. Prior to the yield point, the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible. Some steels and other materials exhibit a behaviour termed a yield point phenomenon. Yield strengths vary from 35 MPa for a low-strength aluminum to greater than 1400 MPa for very high-strength steels.

### Young’s Modulus of Elasticity

Young’s modulus of elasticity of **Zirconium-Niobium Alloy **is about 99 GPa.

The Young’s modulus of elasticity is the elastic modulus for tensile and compressive stress in the linear elasticity regime of a uniaxial deformation and is usually assessed by tensile tests. Up to a limiting stress, a body will be able to recover its dimensions on removal of the load. The applied stresses cause the atoms in a crystal to move from their equilibrium position. All the atoms are displaced the same amount and still maintain their relative geometry. When the stresses are removed, all the atoms return to their original positions and no permanent deformation occurs. According to the** Hooke’s law, **the stress is proportional to the strain (in the elastic region), and the slope is **Young’s modulus**. Young’s modulus is equal to the longitudinal stress divided by the strain.

### Hardness of Zirconium-Niobium Alloy

Rockwell hardness of **Zirconium-Niobium Alloy **is approximately 89 HRB.

**Rockwell hardness test** is one of the most common indentation hardness tests, that has been developed for hardness testing. In contrast to Brinell test, the Rockwell tester measures the depth of penetration of an indenter under a large load (major load) compared to the penetration made by a preload (minor load). The minor load establishes the zero position. The major load is applied, then removed while still maintaining the minor load. The difference between depth of penetration before and after application of the major load is used to calculate the **Rockwell hardness number**. That is, the penetration depth and hardness are inversely proportional. The chief advantage of Rockwell hardness is its ability to **display hardness values directly**. The result is a dimensionless number noted as **HRA, HRB, HRC**, etc., where the last letter is the respective Rockwell scale.

The Rockwell C test is performed with a Brale penetrator (**120°diamond cone**) and a major load of 150kg.

### Example: Strength

Assume a plastic rod, which is made of Zirconium-Niobium Alloy. This plastic rod has a cross-sectional area of 1 cm^{2}. Calculate the tensile force needed to achieve the ultimate tensile strength for this material, which is: UTS = 514 MPa.

Solution:

**Stress (σ)** can be equated to the load per unit area or the force (F) applied per cross-sectional area (A) perpendicular to the force as:

therefore, the tensile force needed to achieve the ultimate tensile strength is:

**F** = UTS x A = 514 x 10^{6} x 0.0001 = **51 400 N**

## Thermal Properties of Zirconium-Niobium Alloy

**Thermal properties** of materials refer to the response of materials to changes in their temperature and to the application of heat. As a solid absorbs energy in the form of heat, its temperature rises and its dimensions increase. But **different materials react** to the application of heat **differently**.

Heat capacity, thermal expansion, and thermal conductivity are properties that are often critical in the practical use of solids.

### Melting Point of Zirconium-Niobium Alloy

Melting point of **Zirconium-Niobium Alloy **is around 1850°C.

In general, **melting** is a **phase change** of a substance from the solid to the liquid phase. The **melting point** of a substance is the temperature at which this phase change occurs. The **melting point **also defines a condition in which the solid and liquid can exist in equilibrium.

### Thermal Conductivity of Zirconium-Niobium Alloy

Zirconium alloys have lower thermal conductivity (about 18 W/m.K) than pure zirconium metal (about 22 W/m.K).

The heat transfer characteristics of a solid material are measured by a property called the **thermal conductivity**, k (or λ), measured in **W/m.K**. It is a measure of a substance’s ability to transfer heat through a material by conduction. Note that **Fourier’s law** applies for all matter, regardless of its state (solid, liquid, or gas), therefore, it is also defined for liquids and gases.

The **thermal conductivity** of most liquids and solids varies with temperature. For vapors, it also depends upon pressure. In general:

Most materials are very nearly homogeneous, therefore we can usually write **k = k (T)**. Similar definitions are associated with thermal conductivities in the y- and z-directions (ky, kz), but for an isotropic material the thermal conductivity is independent of the direction of transfer, kx = ky = kz = k.

### Example: Heat transfer calculation

Thermal conductivity is defined as the amount of heat (in watts) transferred through a square area of material of given thickness (in metres) due to a difference in temperature. The lower the thermal conductivity of the material the greater the material’s ability to resist heat transfer.

Calculate the rate of __heat flux__ through a wall 3 m x 10 m in area (A = 30 m^{2}). The wall is 15 cm thick (L_{1}) and it is made of Zirconium-Niobium Alloy with the __thermal conductivity__ of k_{1} = 18 W/m.K (poor thermal insulator). Assume that, the indoor and the outdoor __temperatures__ are 22°C and -8°C, and the __convection heat transfer coefficients__ on the inner and the outer sides are h_{1} = 10 W/m^{2}K and h_{2} = 30 W/m^{2}K, respectively. Note that, these convection coefficients strongly depend especially on ambient and interior conditions (wind, humidity, etc.).

Calculate the heat flux (**heat loss**) through this wall.

**Solution:**

As was written, many of the heat transfer processes involve composite systems and even involve a combination of both __conduction__ and __convection__. With these composite systems, it is often convenient to work with an__ overall heat transfer coefficient__**, **known as a **U-factor**. The U-factor is defined by an expression analogous to **Newton’s law of cooling**:

The **overall heat transfer coefficient** is related to the total thermal resistance and depends on the geometry of the problem.

Assuming one-dimensional heat transfer through the plane wall and disregarding radiation, the **overall heat transfer coefficient** can be calculated as:

The **overall heat transfer coefficient **is then: U = 1 / (1/10 + 0.15/18 + 1/30) = 7.06 W/m^{2}K

The heat flux can be then calculated simply as: q = 7.06 [W/m^{2}K] x 30 [K] = 211.77 W/m^{2}

The total heat loss through this wall will be: **q _{loss} **= q . A = 211.77 [W/m

^{2}] x 30 [m

^{2}] =

**6352.94 W**