Fluid flow is an important part of most industrial processes; especially those involving the transfer of heat. Frequently, when it is desired to remove heat from the point at which it is generated, some type of fluid is involved in the heat transfer process. Examples of this are the cooling water circulated through a gasoline or diesel engine, the airflow past the windings of a motor, and the flow of water through the core of a nuclear reactor. Fluid flow systems are also commonly used to provide lubrication.
Fluid flow in the nuclear field can be complex and is not always subject to rigorous mathematical analysis. Unlike solids, the particles of fluids move through piping and components at different velocities and are often subjected to different accelerations.
Even though a detailed analysis of fluid flow can be extremely difficult, the basic concepts involved in fluid flow problems are fairly straightforward. These basic concepts can be applied in solving fluid flow problems through the use of simplifying assumptions and average values, where appropriate. Even though this type of analysis would not be sufficient in the engineering design of systems, it is very useful in understanding the operation of systems and predicting the approximate response of fluid systems to changes in operating parameters.
The basic principles of fluid flow include three concepts or principles; the first two of which the student has been exposed to in previous manuals. The first is the principle of momentum (leading to equations of fluid forces) which was covered in the manual on Classical Physics. The second is the conservation of energy (leading to the First Law of Thermodynamics) which was studied in thermodynamics. The third is the conservation of mass (leading to the continuity equation) which will be explained in this module.
Properties of Fluids
A fluid is any substance that flows because its particles are not rigidly attached to one another. This includes liquids, gases, and even some materials which are normally considered solids, such as glass. Essentially, fluids are materials that have no repeating crystalline structure.
Several properties of fluids were discussed in the Thermodynamics section of this text. These included temperature, pressure, mass, specific volume, and density. The temperature was defined as the relative measure of how hot or cold a material is. It can be used to predict the direction that heat will be transferred. The pressure was defined as the force per unit area. Common units for pressure are poundsforce per square inch (psi). Mass was defined as the quantity of matter contained in a body and is to be distinguished from weight, which is measured by the pull of gravity on a body. The specific volume of a substance is the volume per unit mass of the substance. Typical units are ft3/LBM. Density, on the other hand, is the mass of a substance per unit volume. Typical units are LBM/ft3. Density and specific volume are the inverses of one another. Both density and specific volume are dependant on the temperature and somewhat on the pressure of the fluid. As the temperature of the fluid increases, the density decreases, and the specific volume increases. Since liquids are considered incompressible, an increase in pressure will result in no change in density or specific volume of the liquid. In actuality, liquids can be slightly compressed at high pressures, resulting in a slight increase in density and a slight decrease in the specific volume of the liquid.
Pascal’s Law
The pressure of the liquids in each of the previously cited cases has been due to the weight of the liquid. Liquid pressures may also result from the application of external forces on the liquid. Consider the following examples. Figure 2 represents a container completely filled with liquid. A, B, C, D, and E represent pistons of equal crosssectional areas fitted into the walls of the vessel. There will be forces acting on the pistons C, D, and E due to the pressures caused by the different depths of the liquid. Assume that the forces on the pistons due to the pressure caused by the weight of the liquid are as follows: A = 0 lbf, B = 0 lbf, C = 10 lbf, D = 30 lbf, and E = 25 lbf. Now let an external force of 50 lbf be applied to piston A. This external force will cause the pressure at all points in the container to increase by the same amount. Since the pistons all have the same crosssectional area, the increase in pressure will result in the forces on the pistons all increasing by 50 lbf. So if an external force of 50 lbf is applied to piston A, the force exerted by the fluid on the other pistons will now be as follows: B = 50 lbf, C = 60 lbf, D = 80 lbf, and E = 75 lbf. This effect of an external force on a confined fluid was first stated by Pascal in 1653. Pressure applied to a confined fluid is transmitted undiminished throughout the confining vessel of the system.
Control Volume
In thermodynamics, a control volume was defined as a fixed region in space where one studies
the masses and energies crossing the boundaries of the region. This concept of a control volume
is also very useful in analyzing fluid flow problems. The boundary of a control volume for fluid
flow is usually taken as the physical boundary of the part through which the flow is occurring.
The control volume concept is used in fluid dynamics applications, utilizing the continuity,
momentum, and energy principles mentioned at the beginning of this chapter. Once the control
volume and its boundary are established, the various forms of energy crossing the boundary with
the fluid can be dealt with in equation form to solve the fluid problem. Since fluid flow
problems usually treat a fluid crossing the boundaries of a control volume, the control volume
approach is referred to as an “open” system analysis, which is similar to the concepts studied in
thermodynamics. There are special cases in the nuclear field where fluid does not cross the
control boundary. Such cases are studied utilizing the “closed” system approach.
Regardless of the nature of the flow, all flow situations are found to be subject to the established
basic laws of nature that engineers have expressed in equation form. Conservation of mass and
conservation of energy are always satisfied in fluid problems, along with Newton’s laws of
motion. In addition, each problem will have physical constraints, referred to mathematically as
boundary conditions, that must be satisfied before a solution to the problem will be consistent
with the physical results.
Flow Regimes
All fluid flow is classified into one of two broad categories or regimes. These two flow regimes
are laminar flow and turbulent flow. The flow regime, whether laminar or turbulent, is important
in the design and operation of any fluid system. The amount of fluid friction, which determines
the amount of energy required to maintain the desired flow, depends upon the mode of flow.
This is also an important consideration in certain applications that involve heat transfer to the
fluid.
Laminar Flow
Laminar flow is also referred to as streamlined or viscous flow. These terms are descriptive of
the flow because, in laminar flow, (1) layers of water flow over one another at different
speeds with virtually no mixing between layers, (2) fluid particles move indefinitely and
observable paths or streamlines, and (3) the flow is characteristic of viscous (thick) fluid or is
one in which the viscosity of the fluid plays a significant part.
Turbulent Flow
Turbulent flow is characterized by the irregular movement of particles of the fluid. There is no
definite frequency as there is in a wave motion. The particles travel in irregular paths with no
observable pattern and no definite layers.
Viscosity
Viscosity is a fluid property that measures the resistance of the fluid to deforming due to a shear
force. Viscosity is the internal friction of a fluid that makes it resist flowing past a solid
surface or other layers of the fluid. Viscosity can also be considered to be a measure of the
resistance of a fluid to flowing. Thick oil has a high viscosity; water has a low viscosity. The
unit of measurement for absolute viscosity is:
_μ = absolute viscosity of fluid (lbfsec/ft2).
_
The viscosity of a fluid is usually significantly dependent on the temperature of the fluid and
relatively independent of the pressure. For most fluids, as the temperature of the fluid increases,
the viscosity of the fluid decreases. An example of this can be seen in the lubricating oil of
engines. When the engine and its lubricating oil are cold, the oil is very viscous, or thick. After
the engine is started and the lubricating oil increases in temperature, the viscosity of the oil
decreases significantly and the oil seems much thinner.
Ideal Fluid
An ideal fluid is one that is incompressible and has no viscosity. Ideal fluids do not actually
exist, but sometimes it is useful to consider what would happen to an ideal fluid in a particular
fluid flow problem in order to simplify the problem.
Reynolds Number
The flow regime (either laminar or turbulent) is determined by evaluating the Reynolds number
of the flow (refer to figure 5). The Reynolds number, based on studies of Osborn Reynolds, is
a dimensionless number comprised of the physical characteristics of the flow. Equation 37 is
used to calculate the Reynolds number (NR) for fluid flow.
_NR = ρ vD/μgc
_
where:
NR = Reynolds number (unitless)
v = average velocity (ft/sec)
D = diameter of pipe (ft)
μ = absolute viscosity of fluid (lbfsec/ft2)
ρ = fluid mass density (lbm/ft3)
gc = gravitational constant (32.2 ftlbm/lbfsec2)
For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. If it is
greater than 3500, the flow is turbulent. Flows with Reynolds numbers between 2000 and 3500
are sometimes referred to as transitional flows. Most fluid systems in nuclear facilities operate
with turbulent flow. Reynolds numbers can be conveniently determined using a Moody Chart;
an example of which is shown in Appendix B. Additional detail on the use of the Moody Chart
is provided in subsequent text.
Laminar and Turbulent Flow Summary

**Laminar Flow
**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 in the center of the pipe and a minimum flow at the pipe
walls.
The average flow velocity is approximately onehalf of the maximum velocity. 
**Turbulent Flow
**The flow is characterized by the irregular movement of particles of the fluid.
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. 
Viscosity is the fluid property that measures the resistance of the fluid to deformation
due to a shear force. For most fluids, temperature and viscosity are inversely
proportional. 
An ideal fluid is one that is incompressible and has no viscosity.

An increasing Reynolds number indicates an increasing turbulence of flow.\