# What is Example of Volume – Definition

Examples of Volumes. Volume is a basic physical quantity. Here are examples with volume of atom, nucleus and reactor coolant system. Thermal Engineering

## What is Volume

Volume is a basic physical quantity. Volume is a derived quantity and it expresses the three dimensional extent of an object. Volume is often quantified numerically using the SI derived unit, the cubic metre. For example, the volume inside a sphere (that is the volume of a ball) is derived to be V = 4/3πr3, where r is the radius of the sphere. As another example, volume of a cube is equal to side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.

If a square has one side of 3 metres, the volume would be 3 metres times 3 metres times 3 metres, or 27 cubic metres.

Formulas for Calculation of Different Volumes
Volume formulas for some common three-dimensional objects:

## Example: Volume of an Atom and Nucleus

The atom consist of a small but massive nucleus surrounded by a cloud of rapidly moving electrons. The nucleus is composed of protons and neutrons. Typical nuclear radii are of the order 10−14 m. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r0 . A1/3

where r0 = 1.2 x 10-15 m = 1.2 fm

If we use this approximation, we therefore expect the volume of the nucleus to be of the order of 4/3πr3 or 7,23 ×10−45 m3 for hydrogen nuclei or 1721×10−45 m3 for 238U nuclei. These are volumes of nuclei and atomic nuclei (protons and neutrons) contains of about 99.95% of mass of atom.

## Volume of Coolant in Reactor Coolant System

In typical modern pressurized water reactors (PWRs), the Reactor Coolant System (RCS), shown in the figure, consists of:

All RCS components are located inside the containment building.

At normal operation, there is a compressed liquid water inside the reactor vessel, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C (662°F).  Inlet temperature of the water is about 290°C (554°F). The water (coolant) is heated in the reactor core to approximately 325°C (617°F) as the water flows through the core. As it can be seen, the reactor contains approximately 25°C subcooled coolant (distance from the saturation).This high pressure is maintained by the pressurizer, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water (partially with saturated steam) which is heated to the saturation temperature (boiling point) for the desired pressure by submerged electrical heaters. Temperature in the pressurizer can be maintained at 350 °C. At normal conditions, about 60% of volume of pressurizer occupies the compressed water and about 40% of volume occupies the saturated steam.

Volumes of typical PWR are in the following table.

It is an illustrative example, following data do not correspond to any reactor design.

It must be noted the volume of coolant significantly changes with the temperature of the coolant. The total mass of the coolant remains always the same, a change in water volume is not a change in water inventory. The reactor coolant volume changes with temperature because of changes in density. Most substances expand when heated and contract when cooled. However, the amount of expansion or contraction varies, depending on the material. This phenomenon is known as thermal expansion. The change in volume of a material which undergoes a temperature change is given by following relation:

where ∆T is the change in temperature, V is the original volume, ∆V is the change in volume, and αV is the coefficient of volume expansion.

The volumetric thermal expansion coefficient for water is not constant over the temperature range and increases with the temperature (especially at 300°C), therefore the change in density is not linear with temperature (as indicated in the figure).

See also: Steam Tables

At normal conditions the total volume of coolant in the reactor coolant system is almost constant. On the other hand, during transient load conditions the volume can significantly change. These changes are naturally reflected in a change in pressurizer water level. When the average temperature of reactor coolant goes gradually down, the total water volume is also decreasing, which lowers the pressurizer level. On a gradual load pick-up, the increase in reactor coolant average temperature causes the total water volume to expand, which raises the pressurizer level. These effects must be controlled by pressurizer level control system.

References:
Reactor Physics and Thermal Hydraulics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.

## See also:

Volume

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