**several criteria**.

**All fluid flow** is classified into one of two broad categories or regimes. These two flow regimes are:

**Single-phase Fluid Flow****Multi-phase Fluid Flow**(or**Two-phase Fluid Flow**)

This is a **basic classification**. All of the fluid flow equations (e.g. **Bernoulli’s Equation**) and relationships that were discussed in this section (Fluid Dynamics) were derived for the flow of a **single phase** of fluid whether liquid or vapor. Solution of multi-phase fluid flow is **very complex and difficult** and therefore it is usually in advanced courses of fluid dynamics.

Another usually more common classification of **flow regimes** is according to the shape and type of **streamlines**. All fluid flow is classified into one of two broad categories. The fluid flow can be either laminar or turbulent and therefore these two categories are:

**Laminar Flow****Turbulent Flow**

**Laminar flow** is characterized by **smooth** or in **regular paths** of particles of the fluid. Therefore the laminar flow is also referred to as **streamline or viscous flow**. In contrast to laminar flow, **turbulent flow** is characterized by the **irregular movement** of particles of the fluid. The turbulent fluid does not flow in parallel layers, the lateral mixing is very high, and there is a disruption between the layers. **Most industrial flows**, especially those in nuclear engineering **are turbulent**.

The flow regime can be also classified according to the **geometry of a conduit** or flow area. From this point of view, we distinguish:

**Internal Flow****External Flow**

**Internal flow** is a flow for which the fluid is confined by a surface. Detailed knowledge of behaviour of internal flow regimes is **of importance in engineering**, because circular pipes can withstand high pressures and hence are used to convey liquids. On the other hand, **external flow** is such a flow in which boundary layers develop freely, without constraints imposed by adjacent surfaces. Detailed knowledge of behaviour of **external flow** regimes is **of importance especially in aeronautics** and **aerodynamics**.

## Internal Flow

In fluid dynamics, internal flow is a flow for which the fluid is **confined by a surface**. Detailed knowledge of behaviour of internal flow regimes is of importance in engineering, because circular pipes can withstand high pressures and hence are used to convey liquids. Non-circular ducts are used to transport low-pressure gases, such as air in cooling and heating systems. The internal flow configuration is a convenient geometry for heating and cooling fluids used in energy conversion technologies such as nuclear power plants.

For internal flow regime an **entrance region** is typical. In this region a nearly inviscid upstream flow converges and enters the tube. To characterize this region the **hydrodynamic entrance length** is introduced and is approximately equal to:

The maximum hydrodynamic entrance length, at **Re _{D,crit} = 2300** (

**laminar flow**), is L

_{e}= 138d, where D is the diameter of the pipe. This is the longest development length possible. In

**turbulent flow**, the boundary layers grow faster, and L

_{e}is relatively shorter. For any given problem,

**L**has to be

_{e}/ D**checked**to see if L

_{e}is negligible when compared to the pipe length. At a finite distance from the entrance, the entrance effects may be neglected, because the boundary layers merge and the inviscid core disappears. The tube flow is then

**fully developed**.

## Hydraulic Diameter

**enlarge the range of applications**, the

**hydraulic diameter**is introduced:

**The hydraulic diameter, D _{h}**, is a commonly used term when handling flow in

**non-circular tubes and channels**. The hydraulic diameter transforms non-circular ducts into pipes of

**equivalent diameter**. Using this term, one can calculate many things in the same way as for a round tube. In this equation A is the

**cross-sectional area**, and P is the

**wetted perimeter**of the cross-section.

Most industrial flows, especially those in nuclear engineering are **turbulent**. For single straight pipe analysis, assuming unidirectional flow, geometric and kinematic pipe-design problems rely on the **Moody chart** and can be grouped as follows:

- Evaluate the necessary pump characteristics (
**Q-H characteristics**) based on the computed pressure drop Δp in order to convey a given maximum flow rate. - Calculate a specified
**pressure drop**for the pipe of diameter D, of given pipe length and flow rate. This problem requires an iterative procedure because the**Reynolds number**, and hence the friction factor f, is not known. - Calculate the
**flow rate Q**for a given pipe geometry (D, L,**ε/D**) and pressure drop, where ε/D is the relative surface roughness. This problem requires an iterative procedure because the Reynolds number, and hence the friction factor f, is not known.

See also: Power-law velocity profile

## Example: Reynolds number for a primary piping and a fuel bundle

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

**Pressurized water reactors** are cooled and moderated by high-pressure liquid water (e.g. 16MPa). At this pressure water boils at approximately 350°C (662°F). Inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m^{3}). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m^{3}) as the water flows through the core.

The primary circuit of typical PWRs is divided into **4 independent loops** (piping diameter ~ 700mm), each loop comprises a** steam generator** and one **main coolant pump**. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the **downcomer**). From the bottom of the pressure vessel, the flow is reversed up through the core, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.

Assume:

- the primary piping flow velocity is constant and equal to 17 m/s,
- the core flow velocity is constant and equal to 5 m/s,
- the
**hydraulic diameter of the fuel channel**,*D*, is equal to 2 cm_{h} - the kinematic viscosity of the water at 290°C is equal to 0.12 x 10
^{-6}m^{2}/s

See also: Example: **Flow rate through a reactor core**

Determine

- the flow regime and the Reynolds number inside the
**fuel channel** - the flow regime and the Reynolds number inside the
**primary piping**

The Reynolds number inside the primary piping is equal to:

**Re _{D}** = 17 [m/s] x 0.7 [m] / 0.12×10

^{-6}[m

^{2}/s] =

**99 000 000**

This fully satisfies the **turbulent conditions**.

The Reynolds number inside the fuel channel is equal to:

**Re _{DH}** = 5 [m/s] x 0.02 [m] / 0.12×10

^{-6}[m

^{2}/s] =

**833 000**

This also fully satisfies the **turbulent conditions.**

**Reactor Physics and Thermal Hydraulics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
- Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
- Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
- Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
- Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
- U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.
- White Frank M., Fluid Mechanics, McGraw-Hill Education, 7th edition, February, 2010, ISBN: 978-0077422417

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