Basic Principles of Fluid Flow

Basic Principles of Fluid Flow

Average Velocity (u)

• In fluid flow through a pipe or control volume boundaries, each fluid particle doesn't necessarily have the same velocity.
• Average velocity is the mean velocity of all fluid particles at a section or boundary, perpendicular to it.

Steady Flow

• If fluid velocity remains constant at a boundary over time, the flow is considered steady.
• Steady flow does not mean the fluid velocity is the same upstream or downstream, but that it doesn't change at a particular boundary.

Volumetric Flow Rate (V)

• Volumetric flow rate is the volume of fluid per unit time crossing a boundary.
• For steady flow: V = V/t

Relationship between Volumetric Flow Rate and Velocity

• For fluid flowing across a cross-sectional area A with constant velocity u, the distance moved by the fluid in time t is ut, and the volume moved is utA. Therefore: V = uA, and V = uA
• V = uA has units of m³/s when u is in m/s and A is in m².

Mass Flow Rate (m)

• Mass flow rate is the mass of fluid per unit time crossing a boundary, m = dm/dt
• For steady flow, mass flow rate is constant at any section or boundary being considered, m = m/t
• Density is mass per unit volume (ρ = m/V), therefore mass flow rate can also be expressed as m = ρV.
• m = pAu, where A is the cross-sectional area and u is the average velocity.

Continuity of Flow

• Mass cannot be created or destroyed, therefore under steady flow conditions, the mass flow rate into a control volume must equal the mass flow rate out (m₁ = m₂).
• For liquids with negligible density changes continuity equation becomes V₁ = V₂ or U₁A₁= U₂A₂.

Continuity Equation Applied to Branched Pipes

• Continuity equation applies to control volumes with multiple branches, as long as flow is steady.
• Sum of mass flows in must equal sum of mass flows out.
• If density changes are negligible, use volumetric flow rates: Sum of volumetric flow rates in = sum of volumetric flow rates out.

Forces Developed by Flowing Fluids

• Fluid flow generates force.
• High-velocity fluids are used for cleaning and descaling, capable of cutting through various solids.
• Free jets of fluid generate force due to changes in direction.
• Enclosed fluids generate additional forces due to pressure changes along the pipe.

The Impulse-Momentum Equation

• Whenever a free-flowing jet of fluid changes the direction or magnitude of its velocity, a force is generated.

Viscosity and Its Effect on Fluid Flow

Streamlines

• Streamlines show the direction of fluid particle motion and are tangential to the fluid's velocity.
• In steady flow, streamlines can be observed by releasing smoke or dye into the fluid stream.
• Streamlines never cross and their spacing reflects fluid velocity; closer streamlines indicate faster velocity.

Laminar and Turbulent Flow

• Laminar flow (stream flow or viscous flow) has an orderly pattern of streamlines, with layers of fluid sliding over one another.
• Turbulent flow has random fluid particle motion, with mixing and diffusion throughout the fluid stream.
• Flow can transition between laminar and turbulent, often occurring in cases where there is change in section, like at the entry to ducts and pipes.

Viscosity

• Viscosity is a fluid's resistance to flow.
• A solid can be considered a fluid with infinite viscosity
• Liquid viscosity decreases with temperature increase.

Mechanism of Viscosity

• Viscosity results from cohesiveness (primary in liquids) and momentum transfer (primary in gases).
• Cohesiveness is the tendency of molecules to adhere; heating decreases cohesiveness.
• Momentum transfer is when molecules move into regions of different velocities, setting up a resisting force
• Heating a gas increases the velocity gradient which increases momentum transfer

Viscosity Measurement

• Methods to measure viscosity include: measuring flow time through an orifice (Saybolt viscometer), using concentric cylinders with a torsion wire, or dropping a ball through the fluid.

Dynamic Viscosity (μ)

• For two parallel plates separated by a fluid film where one plate is stationary and the other moving with velocity u, the dynamic viscosity is given by μ = (F/A) / (u/x).

Kinematic Viscosity (ν)

• Kinematic viscosity is the ratio of dynamic viscosity to density (ν = μ/ρ).
• Metric units are m²/s.

Reynolds Number (Re)

• Reynolds number (Re) is ud/v = udρ/μ a dimensionless number that determines whether a flow will be laminar or turbulent.
• u is fluid velocity, d is distance or length, ρ is density, and μ is dynamic viscosity
• Turbulence is more likely with high velocity and low viscosity fluids.

Critical Reynolds Number

• The critical Reynolds number defines the transition between laminar and turbulent flow.
• Below Re = 2000, flow is laminar.
• Above Re = 4000, flow is turbulent.
• Between 2000 and 4000, flow is uncertain and can be either laminar or turbulent.
• Factors such as roughness, sharp corners, and vibration may affect the transition between the two.

Velocity Profile in a Pipe

• Depending on the Reynolds number, the flow can be laminar or turbulent
• Laminar flow has a parabolic velocity distribution, with max velocity equals twice the mean.
• Turbulent flow has a more uniform velocity profile, with a steep gradient near the wall and a max velocity is 1.2-1.4 times the mean.

Effect of Viscosity on Flow Losses

• Viscosity causes energy losses when a fluid flows
• Absence of viscosity means no loss of energy or pumping power required
• Higher viscosity results in more difficult flow and greater pumping power needed
• Regime affects loss; roughness is not a factor in laminar flow