Reynolds Number Regimes
Laminar flow. For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Red,crit = 2300.
Transitional flow. At Reynolds numbers between about 2000 and 4000, the flow is unstable due to the onset of turbulence. These flows are sometimes referred to as transitional flows.
Turbulent flow. If the Reynolds number is greater than 3500, the flow is turbulent. Most fluid systems in nuclear facilities operate with turbulent flow.
Definition of Reynolds Number
The Reynolds number is the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. It can be interpreted that when the viscous forces are dominant (slow flow, low Re), they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. Even very low Re indicates viscous creeping motion, where inertia effects are negligible. When the inertial forces dominate over the viscous forces (when the fluid flows faster and Re is larger), the flow is turbulent.
It is a dimensionless number comprised of the physical characteristics of the flow. An increasing Reynolds number indicates increasing turbulence of flow.
It is defined as:
where:
V is the flow velocity,
D is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.)
ρ fluid density (kg/m3),
μ dynamic viscosity (Pa.s),
ν kinematic viscosity (m2/s); ν = μ / ρ.
Laminar vs. Turbulent Flow
Laminar flow:
- Re < 2000
- ‘low’ velocity
- Fluid particles move in straight lines
- Layers of water flow over one another at different speeds with virtually no mixing between layers.
- The flow velocity profile for laminar flow in circular pipes is parabolic in shape, with a maximum flow at the center of the pipe and a minimum flow at the pipe walls.
- The average flow velocity is approximately one-half of the maximum velocity.
- Simple mathematical analysis is possible.
- Rare in practice in water systems.
Turbulent Flow:
- Re > 4000
- ‘high’ velocity
- The flow is characterized by the irregular movement of particles of the fluid.
- Average motion is in the direction of the flow.
- The flow velocity profile for turbulent flow is fairly flat across the center section of a pipe and drops rapidly extremely close to the walls.
- The average flow velocity is approximately equal to the velocity at the center of the pipe.
- Mathematical analysis is very difficult.
- The most common type of flow.
Reynolds Number and Pipe Flow
The configuration of the internal flow (e.g.,, flow in a pipe) is a convenient geometry for heating and cooling fluids used in energy conversion technologies such as nuclear power plants.
In general, this flow regime is important in engineering because circular pipes can withstand high pressures and hence convey liquids. Non-circular ducts are used to transport low-pressure gases, such as air in cooling and heating systems.
For the internal flow regime, an entrance region is typical. In this region, a nearly inviscid upstream flow converges and enters the tube. The hydrodynamic entrance length is introduced to characterize this region and is approximately equal to:
The maximum hydrodynamic entrance length, at ReD,crit = 2300 (laminar flow), is Le = 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 Le is relatively shorter. For any given problem, Le / D has to be checked to see if Le is negligible compared to the pipe length. The entrance effects may be neglected at a finite distance from the entrance because the boundary layers merge and the inviscid core disappears. The tube flow is then fully developed.
Power-law velocity profile – Turbulent velocity profile
The velocity profile in turbulent flow is flatter in the central part of the pipe (i.e., in the turbulent core) than in laminar flow. The flow velocity drops rapidly, extremely close to the walls. This is due to the diffusivity of the turbulent flow.
In the case of turbulent pipe flow, there are many empirical velocity profiles. The simplest and the best known is the power-law velocity profile:
where the exponent n is a constant whose value depends on the Reynolds number. This dependency is empirical, and it is shown in the picture. In short, the value n increases with increasing Reynolds number. The one-seventh power-law velocity profile approximates many industrial flows.
Hydraulic Diameter
Since the characteristic dimension of a circular pipe is an ordinary diameter D and especially reactors contain non-circular channels, the characteristic dimension must be generalized.
For these purposes, the Reynolds number is defined as:
where Dh is the hydraulic diameter:
The hydraulic diameter, Dh, 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. The wetted perimeter for a channel is the total perimeter of all channel walls that contact the flow.
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). The inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) 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). The flow is reversed up through the core from the bottom of the pressure vessel, 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, Dh, is equal to 1 cm
- the kinematic viscosity of the water at 290°C is equal to 0.12 x 10-6 m2/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:
ReD = 17 [m/s] x 0.7 [m] / 0.12×10-6 [m2/s] = 99 000 000
This fully satisfies the turbulent conditions.
The Reynolds number inside the fuel channel is equal to:
ReDH = 5 [m/s] x 0.01 [m] / 0.12×10-6 [m2/s] = 416 600
This also fully satisfies the turbulent conditions.