Hydraulic Systems
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   Pipe Flow - Description
  Pipe Flow - Equations
   Power Control Unit - Desc.
   Pressure Regulating Valve - Desc.
   Pressure Relief Valve - Desc.
   Priority Valve - Desc.
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Pipe Flow, Hydraulic - Equations

For pipe analysis and sizing purposes, piping is usually treated using a 1st order lumped parameter equations. Lumped parameter analysis only provides information at the inlet and outlet of a pipe. If desired, pipes can be broken down into smaller increments, but this is usually not practical or necessary.

Figure 1 shows a pipe cross section with relevant input and output parameters. Pipe parameters are inside diameter and length. Subscripts 1 and 2 refer to the pipe inlet and outlet, respectively.

Figure 1 Pipe Cross Section

In the figure,

P Pressure (psi)

Δp = P2 – P1 pressure drop (psi)

Q flow rate (in3/sec)

l pipe length (in)

D inside pipe diameter (in)

ρ fluid density (lbf-sec2/in4)

μ absolute viscosity (lbf-sec/in2)

υ kinematic viscosity (in2/sec)

For analysis and sizing, one of the following 2 methods is used:

  1. Friction Factor Method

  2. Test Data:  

The test data method for a pipe (or any component) is preferred whenever it is available.

Friction Factor Method

When using the friction factor method, the first step is to compute the Reynolds number. For pipes, the Reynolds number is computed using



Re Reynolds number [dimensionless]

d internal diameter of the pipe [in]

v flow velocity [in/sec]

υ kinematic viscosity [in2/sec]

For pipe flow, the flow regime based on Reynolds number is listed below:

Laminar:    Re < 2300

Transition:   2300 < Re < 4000/FONT>

Turbulent:   Re > 4000

The equations are different for each flow regime.

For laminar flow, the pressure drop is proportional to dynamic pressure via



Δp pressure drop through pipe (p1 – p2)

l length

d internal flow diameter

ρ fluid density

V fluid velocity

The volumetric flow equation is


where Q is the volumetric flow rate, A is the cross sectional area of the pipe and v is the velocity, and the Reynolds number equation is


where v is the flow velocity, dh is the hydraulic diameter and υ is the kinematic viscosity. Substituting (3) and (4) into (2) yields


In terms of Q


Equation (6) relates the pressure drop through a pipe to the flow rate for laminar flow. The relationship between the pressure drop, Δp, and flow rate, Q, is governed by fluid properties and flow geometry.

The equation for turbulent flow is


This equation is derived using Δp = k (1/2 ρ V2) and using a curve fit for turbulent flow from friction factor graphs. Solving (7) for Q yields


Equation (8) relates the pressure drop through a pipe to the flow rate for turbulent flow.

Test Data Method

This method can be used when test data for pipes or any other component is available. Manufacturer data is often available for valves and this method is equally applicable to them.

For incompressible flow, pressure drop can be computed using friction factor and associated K pressure drop factors. The standard equation is


where is the dynamic pressure and K is the pressure drop factor given by


In equation (10), f is the friction factor (f = 64/Re for laminar flow and f = 0.332/Re.25 for turbulent flow), L is the length of the pipe, D is the diameter and Kt accounts for bends, exits, etc. Values for Kt are in the SAE AIR1168/1 and some fluids texts. Equation (10) allows the pressure drop for numerous components to be combined. For example, pressure drop through a pipe, check valve and a pipe connected in series could be combined into a single equation using equation (10).

When test data – flow vs. pressure drop – is available for a given pipe or piping configuration, the data is usually plotted on log-log axis and a relationship of the form


can be used. In this equation

σ density ratio

K pressure drop constant

Q volumetric flow rate

n exponent (for air flow, n is usually close to 2)

Comparing this equation to the general orifice or servo flow equation,


leads to


where ρ0 is the density of the fluid when the test data was obtained and ρ is the density of the fluid for the condition of interest.

For equations of the form,


a is the y-intercept and n is the slope. Thus


and using this value of n,


An example using check valve illustrates use of the above equations. A valve manufacturer provides the following flow data for a check valve:

For the size 4 check valve, two points are read from the graph

Q1 = 1 gpm (3.85 in3/sec), (Δp)1= 5.8 psi

Q2 = 2 gpm (7.8 in3/sec), (Δp)2=20 psi

Using equation (15),

Using Q1 and (Δp)1 condition in equation (11)


which is the governing flow equation for the check valve.

Pipe Temperature Losses

Temperature losses through a pipe adhere to the following relationship



T1 inlet temperature

T2 outlet temperature

Tamb ambient temperature outside of pipe

Uoverall overall heat transfer coefficient

Q volumetric flow rate

A heat transfer area

cp specific heat coefficient at constant pressure

ρ fluid density

Equation (17) computes pipe outlet temperature based on inlet temperature, ambient temperature, fluid flow rate and pipe material. Uoverall consists of forced convection heat transfer from the fluid to the pipe wall, conduction through the pipe wall and free (still air) convection away from the pipe to the ambient. There are also conduction paths from the pipe to structure though pipe supports. The area is the heat transfer area (pipe circumference x length).