# Pressure drop and velocity relationship with acceleration

### Flow in pipe - Bernoulli equation

ADDED NOTE In your question you ask why does pressure fall . Such acceleration must mean that a force pushes them forward, and this. Flow in pipe - diameter, velocity, Reynolds number, Bernoulli equation, friction factor. pressure drop, flow rate, pipe diameter elevation above reference level; p1,2 - absolute pressure; v1,2 - velocity; ρ1,2 - density; g - acceleration of gravity. A pressure difference is like a net force, producing acceleration of the Bernoulli's equation does, relating the pressure, velocity, and height of.

Darcy formula applies when pipe diameter and fluid density is constant and the pipe is relatively straight. Friction factor for pipe roughness and Reynolds number in laminar and turbulent flow Physical values in Darcy formula are very obvious and can be easily obtained when pipe properties are known like D - pipe internal diameter, L - pipe length and when flow rate is known, velocity can be easily calculated using continuity equation.

The only value that needs to be determined experimentally is friction factor.

**Differential Equations - Terminal Velocity Example**

In the critical zone, where is Reynolds number between andboth laminar and turbulent flow regime might occur, so friction factor is indeterminate and has lower limits for laminar flow, and upper limits based on turbulent flow conditions.

If the flow is laminar and Reynolds number is smaller thanthe friction factor may be determined from the equation: Since the internal pipe roughness is actually independent of pipe diameter, pipes with smaller pipe diameter will have higher relative roughness than pipes with bigger diameter and therefore pipes with smaller diameters will have higher friction factors than pipes with bigger diameters of the same material. Most widely accepted and used data for friction factor in Darcy formula is the Moody diagram.

On Moody diagram friction factor can be determined based on the value of Reynolds number and relative roughness. The pressure drop is the function of internal diameter with the fifth power. With time in service, the interior of the pipe becomes encrusted with dirt, scale, tubercles and it is often prudent to make allowance for expected diameter changes.

For turbulent flow, the speed and or the direction of the flow varies. In steady flow, the motion can be represented with streamlines showing the direction the water flows in different areas.

## Flow in pipe - diameter, velocity, Reynolds number, Bernoulli equation, friction factor

The density of the streamlines increases as the velocity increases. Fluids can be compressible or incompressible. This is the big difference between liquids and gases, because liquids are generally incompressible, meaning that they don't change volume much in response to a pressure change; gases are compressible, and will change volume in response to a change in pressure.

Fluid can be viscous pours slowly or non-viscous pours easily. Fluid flow can be rotational or irrotational.

- Pressure drop evaluation along pipelines
- Pressure Drop, Single-Phase

Irrotational means it travels in straight lines; rotational means it swirls. For most of the rest of the chapter, we'll focus on irrotational, incompressible, steady streamline non-viscous flow. The equation of continuity The equation of continuity states that for an incompressible fluid flowing in a tube of varying cross-section, the mass flow rate is the same everywhere in the tube. The mass flow rate is simply the rate at which mass flows past a given point, so it's the total mass flowing past divided by the time interval.

The equation of continuity can be reduced to: Generally, the density stays constant and then it's simply the flow rate Av that is constant.

Making fluids flow There are basically two ways to make fluid flow through a pipe. One way is to tilt the pipe so the flow is downhill, in which case gravitational kinetic energy is transformed to kinetic energy. The second way is to make the pressure at one end of the pipe larger than the pressure at the other end.

### Fluid dynamics and Bernoulli's equation

A pressure difference is like a net force, producing acceleration of the fluid. As long as the fluid flow is steady, and the fluid is non-viscous and incompressible, the flow can be looked at from an energy perspective. This is what Bernoulli's equation does, relating the pressure, velocity, and height of a fluid at one point to the same parameters at a second point.

The equation is very useful, and can be used to explain such things as how airplanes fly, and how baseballs curve. Bernoulli's equation The pressure, speed, and height y at two points in a steady-flowing, non-viscous, incompressible fluid are related by the equation: Some of these terms probably look familiar If the equation was multiplied through by the volume, the density could be replaced by mass, and the pressure could be replaced by force x distance, which is work.

Looked at in that way, the equation makes sense: For our first look at the equation, consider a fluid flowing through a horizontal pipe. The pipe is narrower at one spot than along the rest of the pipe. By applying the continuity equation, the velocity of the fluid is greater in the narrow section. Is the pressure higher or lower in the narrow section, where the velocity increases? Your first inclination might be to say that where the velocity is greatest, the pressure is greatest, because if you stuck your hand in the flow where it's going fastest you'd feel a big force.

The force does not come from the pressure there, however; it comes from your hand taking momentum away from the fluid. The pipe is horizontal, so both points are at the same height.