Area-Velocity Relationship in Compressible Fluid Flow: Derivation and Explanation

Posted on May 23, 2024 in Physics

Area-Velocity Relationship for Compressible Fluid Flow

To derive the area-velocity relationship for a compressible fluid flow, we use the principles of conservation of mass (continuity equation) and the concept of isentropic flow (assuming the flow is adiabatic and reversible). This relationship helps us understand how changes in the cross-sectional area of a flow affect the velocity in a compressible fluid flow.

1. Continuity Equation

For a compressible fluid, the continuity equation (conservation of mass) is given by:

where

• ρ is the fluid density
• v is the fluid velocity
• A is the cross-sectional area

In steady flow, this simplifies to:

ρAv = constant

Taking the differential form of this equation:

d(ρAv) = 0 expanding it we get

Avdρ + ρvdA + ρAdv = 0

2. Isentropic Flow Relationship

For isentropic flow of a compressible fluid, the relationship between density and velocity can be derived from the conservation of energy and thermodynamic principles. For an isentropic process, we have the relation:

Energy Losses in Pipe Flow

Losses of energy in the flow through a pipe can be attributed to two main factors:

1. Major Losses: These are caused by frictional resistance as the fluid flows through the pipe. Factors contributing to major losses include the roughness of the pipe walls, the velocity of the fluid, and the length of the pipe.
2. Minor Losses: These losses occur due to changes in the flow direction, sudden contractions or expansions in the pipe diameter, and the presence of fittings like valves or elbows. Minor losses lead to energy dissipation in the form of turbulence and eddies within the flow.

Therefore, losses of energy in the flow through a pipe are primarily due to major losses caused by frictional resistance and minor losses resulting from changes in flow direction and pipe geometry.

Bernoulli’s Theorem

Bernoulli’s theorem states that in a steady flow of an incompressible and inviscid fluid, the total energy per unit mass remains constant along a streamline.

Assumptions of Bernoulli’s Equation

1. The flow is steady.
2. The fluid is incompressible.
3. The fluid is inviscid.
4. The flow is along a streamline.
5. There are no external forces acting on the fluid.

Limitations of Bernoulli’s Equation

1. It assumes no energy losses due to friction, heat transfer, or other dissipative processes.
2. It is not applicable to compressible fluids.
3. It does not account for changes in potential energy due to elevation changes.
4. It does not consider the effects of viscosity on the fluid flow.

Effect of Temperature on Viscosity

Water (a Liquid)

For liquids, such as water, viscosity decreases with an increase in temperature. This is because as temperature rises, the kinetic energy of the water molecules increases, causing them to move more vigorously and overcome intermolecular forces more easily. As a result, the resistance to flow (viscosity) decreases.

• At low temperatures: Water molecules have less kinetic energy, leading to stronger intermolecular attractions and higher viscosity.
• At high temperatures: Water molecules have more kinetic energy, reducing intermolecular attractions and decreasing viscosity.

For example, the viscosity of water at 0°C is higher than its viscosity at 25°C.

Air (a Gas)

For gases, such as air, viscosity increases with an increase in temperature. This is because, in gases, viscosity is primarily determined by the momentum exchange between molecules. As temperature increases, the molecules move faster and collide more frequently, which enhances the momentum transfer and results in higher viscosity.

• At low temperatures: Air molecules move slower, resulting in less frequent collisions and lower viscosity.
• At high temperatures: Air molecules move faster, leading to more frequent collisions and higher viscosity.

For example, the viscosity of air at 0°C is lower than its viscosity at 100°C

Derivation

Consider a small fluid element moving along a streamline in a steady flow. Let ds be the length element along the streamline, v be the fluid velocity, ρ be the fluid density, and P be the pressure.

Newton’s Second Law

According to Newton’s second law, the net force acting on the fluid element is equal to the mass of the element times its acceleration:

ΣF = m * a

Forces Acting on the Fluid Element

1. Pressure Forces: The difference in pressure across the fluid element provides a force. If P is the pressure at the upstream side and P + dP is the pressure at the downstream side.

Euler’s Equations of Motion

F = m.ax
In the fluid flow, the following forces are present:
(i) Fg, gravity force.
(ii) Fp, the pressure force.
(iii) Fv force due to viscosity.
(iv) Ft, force due to turbulence.
(v) Fc force due to compressibility

.

If the flow is assumed to be ideal, viscous force (Fv) is zero and the equations of motion are known as Euler’s equations of motion.