Fluid Mechanics Fundamentals
NPSH and Cavitation
NPSHr (Net Positive Suction Head Required): Represents the minimum suction pressure needed to prevent cavitation under specific operating conditions.
NPSHa (Net Positive Suction Head Available): It represents the actual suction pressure available at the pump inlet. It is influenced by various factors, including:
- Elevation difference between the pump and the fluid level
- Presence of fittings and valves
- Compressibility of the fluid
It can be calculated or measured using pressure gauges or manometers. We can avoid cavitation by increasing the inlet pressure and also causing the fluid to come from a higher elevation to have more energy at the entrance. With this, we are not going to reach the liquid-vapor equilibrium.
Can NPSHA and NPSHR be Negative?
NPSHA: Theoretically, the calculation for NPSHA could yield a negative number if the suction pressure is below the vapor pressure of the fluid, indicating that the fluid is vaporizing before it reaches the pump. This condition is unacceptable for pump operation and must be corrected.
NPSHR: NPSHR is a positive value determined through testing. It represents a minimum requirement. By definition, it cannot be negative because it specifies the least amount of pressure head needed to avoid cavitation.
Euler’s Approximation vs. Lagrange’s Approximation
Euler: Studies the fluid with a control volume, at the entrance and the exit.
Lagrange: Studies the trajectory of the fluid particles within the control volume.
Why are Backward Curved Blades Used Primarily in Centrifugal Pumps (β=90°)?
We use backward curved blades because they impart less energy to the fluid than forward curved blades, making them more cost-effective and reducing the risk of pump burnout. They are optimal because they provide the maximum work output, and the required power increases more slowly.
Energy Losses in Pumps
When energy is transformed, there are always energy losses. This means that when energy is converted to a different form, some of the input energy is turned into a high disordered form of energy, like heat.
- Flow Friction: As flow increases and velocity increases, head losses increase.
- Recirculation: When the flow rate (Q) is below the design point, energy is wasted (the fluid does not exit the pump).
- Incidence: Losses produced due to the difference between the blade angle and the fluid angle.
Reynolds Number and Pipe Diameter
It is true that in a constant flow pipe, if the diameter is greater, the Reynolds number will be greater?
False, it decreases.
Cavitation
What is cavitation? Why and how does it happen?
Cavitation occurs when a liquid experiences a pressure drop below its vapor pressure, leading to the formation of vapor bubbles. These bubbles travel with the liquid and collapse violently upon reaching regions of higher pressure. This collapse generates shock waves that can cause significant damage to the surrounding material, such as the walls of tubes, propellers, or pump impellers, leading to pitting, erosion, and structural weakening. Additionally, cavitation reduces the efficiency of fluid systems, increases noise and vibration, and can ultimately lead to system failure if not properly managed.
Cavitation can be avoided by increasing the inlet pressure or height to ensure there is more energy at the entrance. This prevents the pressure from dropping to the vapor pressure of the liquid. Additionally, placing pumps at a lower level than the inlet, often at floor level, helps maintain sufficient pressure and avoid cavitation. These strategies ensure that the pressure drop within the system does not reach the vapor pressure, thereby preventing the formation and collapse of vapor bubbles.
Boundary Layer in Fluid Flow
Define the concept of the boundary layer in a turbulent fluid. And in a laminar fluid.
The boundary layer is the region of fluid near a solid surface where viscous effects are significant. Within this layer, the fluid velocity transitions from zero at the solid surface, due to the no-slip condition, to the free-stream velocity at a certain distance from the surface. The thickness of the boundary layer is influenced by the Reynolds number, which is a dimensionless parameter representing the ratio of inertial forces to viscous forces.
Laminar Boundary Layer:
In a laminar boundary layer, the fluid flows in smooth, parallel layers with minimal mixing between them. The flow is orderly and stable, characterized by a velocity gradient perpendicular to the surface. Viscous forces dominate, and momentum transfer occurs mainly through molecular diffusion. Laminar boundary layers typically occur at lower Reynolds numbers.
Turbulent Boundary Layer:
In a turbulent boundary layer, the fluid exhibits chaotic and irregular motion with random changes in direction and velocity. Despite the no-slip condition causing the fluid velocity to be zero at the wall, the presence of turbulence significantly enhances the mixing of fluid particles. This mixing increases the transfer of momentum, heat, and mass across the boundary layer. Turbulent boundary layers generally occur at higher Reynolds numbers and are thicker compared to laminar boundary layers.
Laminar Flow vs. Turbulent Flow
Which are the differences between laminar flux and turbulent flux?
Laminar: The fluid particles follow a smooth path that never interferes with others. The transition to maximum velocity (Umax) is smooth and progressive.
Turbulent: Characterized by chaotic, random trajectories of particles, a more uniform velocity profile away from boundaries due to mixing, and significant effects of the no-slip condition, including the formation of a viscous sublayer near solid boundaries.
Fluid Mechanics Concepts and Equations
α (Kinetic energy coefficient): It is used to correct the Bernoulli equation to consider the velocity profiles. It is a function of the flow regime.
- α = 1 → Ideal regime (initial viscosity = 0)
- α = 1.08 → Turbulent regime
- α = 2 → Laminar regime
Wa = Work available, due to the turbomachines. Theoretical energy that a pump can deliver. It is a function of the design.
Wp: Work provided, due to the pumps.
Important Fluid Properties
- Density (ρ): Mass of fluid divided by the fluid volume. It is a function of pressure and temperature (kg/m³).
- Viscosity (μ): The resistance exerted by a fluid to being deformed. It strongly depends on temperature (higher temperature, lower viscosity) (kg/m·s).
- Buoyancy: The ability of something to float in water or other fluid. It follows Archimedes’ principle that states that a body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces. F = ρL(h·A)·g = ρL·VL·g
- Shear (τ): The force per unit area exerted by the fluid against a surface, parallel to that surface. τ = μ·du/dy [Pa]
- Surface tension (γ): Molecular force between two immiscible fluids, with different densities, that appears when they are in contact (it is a repulsive force). Fluids with high surface tension adopt spherical shapes to minimize surface area [N/m].
- No-slip condition: The no-slip condition states that at a solid boundary (such as a wall), the fluid velocity is zero relative to the surface of the boundary. This means that the fluid particles in direct contact with the solid surface have zero velocity with respect to that surface. As you move away from the boundary into the bulk fluid, the velocity of the fluid particles increases gradually until it reaches the free-stream velocity.
- Pressure: The force divided by an area.
Newtonian and Non-Newtonian Fluids
- Newtonian: Viscosity is independent of the shear rate and the time for which it is applied.
- Non-Newtonian: Viscosity is dependent on the shear rate and the time for which it is applied.
- Dilatant: Viscosity increases with shear rate (e.g., cornstarch slurry).
- Pseudoplastic: Viscosity decreases with shear rate (e.g., ketchup, paint).
- Rheopectic: Viscosity increases over time (e.g., gypsum paste).
- Thixotropic: Viscosity decreases over time (e.g., yogurt).
True or False
- Cinematic viscosity of a gas is independent of temperature. (FALSE)
- Shear stress is the multiplication of a constant and the velocity gradient. (TRUE)
- The friction factor of a pipe is independent of fluid velocity. (FALSE)
- In a pseudoplastic fluid, the viscosity increases as we increase velocity. (FALSE)
- The pressure applied to an infinitesimal volume of a static incompressible fluid is equal in all directions. (TRUE)
- The first law of thermodynamics says that the variation of work + heat is 0. (TRUE)
- The amount of fluid entering a control volume is always equal to the amount leaving the control volume. (TRUE)
- In inviscid fluids, normal tensions at the walls of an infinitesimal control volume are equal to the pressure with a negative sign. (TRUE)
- In a pipe, with a constant flow rate, if we increase the diameter, the Reynolds number increases. (FALSE)
- The pressure applied to a body is independent of the direction. (TRUE)
- The kinematic viscosity is equivalent to the density divided by the viscosity. (FALSE) It is viscosity divided by density.
Navier-Stokes Equation
In the Navier-Stokes equation for an infinitesimal volume of fluid, identify the different terms.
In the conservation of momentum of a control volume equation, identify the different terms.
Local Acceleration vs. Advective Acceleration
What is the local acceleration and the advective acceleration?
Local acceleration: Local acceleration refers to the rate of change of velocity of a fluid particle with respect to time as it moves through space.
Advective acceleration: It represents the change in velocity of the fluid particle as it moves from one location to another within the flow field.
Simplifying the Navier-Stokes Equations
Which are the approximations to simplify Navier-Stokes equations?
- Incompressible Flow
- Steady State
- Laminar flow
- Low Reynolds number
CC: Contraction coefficient: It is the relation between the area of the opening and the transversal section of the fluid jet that leaves the opening. Its values are around 0.6 to 0.7.
Cv: Velocity coefficient: Relationship between the real discharge velocity and the theoretical velocity, based on Bernoulli’s equation and neglecting friction. Its values are around 1.
Common Industrial Equipment
Define and discuss the common industrial equipment mentioned in class:
- Pipes: Transport fluids throughout the plant.
- Characteristics:
- Diameter (critical)
- Materials (critical for roughness)
- Insulation
- Accessories
- Characteristics:
- Valves:
- Control valves: Regulate flow and prevent water hammer.
- Stop valves: They allow flow in one direction but prevent fluid from coming from the opposite direction.
- Safety valves: Used to release pressure if there’s an excess. When there is an excess, it means that there’s a problem, and the liquid is burned in a torch.
- Turbomachinery:
- Rotodynamic pumps: The fluid enters parallel to the floor and then goes up thanks to the impellers, to change from mechanical force to pressure force.
- Fans: The direction of the flow is 0°. The design can be radial or axial.
- Turbines: They extract energy from the fluid to convert it to electrical energy. It enters at 90° and leaves at 0°.
- Positive Displacement Pumps:
- Alternative pumps (Piston):
- Hydraulic activators: Extract energy to measure things.
