For
pipe analysis and sizing purposes, piping is usually treated using a
1^{st} 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 = P_{2} –
P_{1} pressure drop (psi)

Q flow rate (in^{3}/sec)

l pipe length (in)

D inside pipe diameter (in)

ρ fluid density
(lb_{f}-sec^{2}/in^{4})

μ absolute viscosity
(lb_{f}-sec/in^{2})

υ kinematic viscosity
(in^{2}/sec)

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

Friction Factor Method

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

(1)

where

Re Reynolds number [dimensionless]

d internal diameter of the pipe [in]

v flow velocity [in/sec]

υ kinematic viscosity
[in^{2}/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

(2)

Where

Δp pressure drop
through pipe (p_{1} – p_{2})

*l* length

*d *internal flow
diameter

ρ fluid density

*V *fluid velocity

The volumetric flow equation is

(3)

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

(4)

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

(5)

In terms of Q

(6)

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

(7)

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

* * (8)

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

(9)

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

(10)

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 K_{t} accounts for bends, exits,
etc. Values for K_{t} 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

(11)

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,

(12)

leads to

(13)

where
ρ_{0} 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,

(14)

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

(15)

and using this value of n,

(16)

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
Q_{1} and (Δp)_{1}
condition in equation (11)

Therefore,

which is the governing flow equation for the check valve.

*Pipe Temperature
Losses*

Temperature losses through a pipe adhere to the following relationship

(17)

where

T_{1} inlet
temperature

T_{2} outlet
temperature

T_{amb} ambient
temperature outside of pipe

U_{overall} overall
heat transfer coefficient

Q volumetric flow rate

A heat transfer area

c_{p} 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. U_{overall}
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).