# fluids tutorial¶

## Importing¶

Fluids can be imported as a standalone library, or all of its functions and classes may be imported with star imports:

```
>>> import fluids # Good practice
>>> from fluids import * # Bad practice but convenient
```

All functions are available from either the main fluids module or the submodule; i.e. both fluids.friction_factor and fluids.friction.friction_factor are valid ways of accessing a function.

## Design philosophy¶

Like all libraries, this was developed to scratch my own itches. Since its public release it has been found useful by many others, from students across the world to practicing engineers at some of the world’s largest companies.

The bulk of this library’s API is considered stable; enhancements to functions and classes will still happen, and default methods when using a generic correlation interface may change to newer and more accurate correlations as they are published and reviewed.

To the extent possible, correlations are implemented depending on the highest level parameters. The friction_factor correlation does not accept pipe diameter, velocity, viscosity, density, and roughness - it accepts Reynolds number and relative roughness. This makes the API cleaner and encourages modular design.

All functions are designed to accept inputs in base SI units. However, any set of consistent units given to a function will return a consistent result; for instance, a function calculating volume doesn’t care if given an input in inches or meters; the output units will be the cube of those given to it. The user is directed to unit conversion libraries such as pint to perform unit conversions if they prefer not to work in SI units.

The standard math library is used in all functions except where special functions from numpy or scipy are necessary. SciPy is used for root finding, interpolation, scientific constants, ode integration, and its many special mathematical functions not present in the standard math library. No other libraries will become required dependencies; anything else is optional.

To allow use of numpy arrays with fluids, a vectorized module is implemented, which wraps all of the fluids functions with np.vectorize. Instead of importing from fluids, the user can import from fluids.vectorized:

```
>>> from fluids.vectorized import *
>>> friction_factor(Re=[100, 1000, 10000], eD=0)
array([ 0.64 , 0.064 , 0.03088295])
```

## Dimensionless numbers¶

More than 30 Dimensionless numbers are available in fluids.core:

Calculation of Reynolds and Prandtl number for water flowing in a 0.01 m diameter pipe at 1.5 m/s:

```
>>> fluids.core.Reynolds(D=0.01, rho=1000, V=1.5, mu=1E-3)
15000.0
>>> fluids.core.Prandtl(rho=1000, mu=1E-3, Cp=4200, k=0.6)
7.000000000000001
```

Where different parameters may be used with a dimensionless number, either a separate function is created for each or both sets of parameters are can be specified. For example, instead of specifying viscosity and density for the Reynolds number calculation, kinematic viscosity could have been used instead:

```
>>> Reynolds(D=0.01, V=1.5, nu=1E-6)
15000.0
```

In the case of groups like the Fourier number, used in both heat and mass transfer, two separate functions are available, Fourier_heat and Fourier_mass. The heat transfer version supports specifying either the density, heat capacity, and thermal conductivity - or just the thermal diffusivity. There is no equivalent set of three parameters for the mass transfer version; it always requires mass diffusivity.

```
>>> Fourier_heat(t=1.5, L=2, rho=1000., Cp=4000., k=0.6)
5.625e-08
>>> Fourier_heat(1.5, 2, alpha=1E-7)
3.75e-08
>>> Fourier_mass(t=1.5, L=2, D=1E-9)
3.7500000000000005e-10
```

Among the other coded dimensionless numbers are Grashof, Nusselt, Sherwood, Rayleigh, Schmidt, Weber, Mach, Knudsen, Bond, Dean, Froude, Biot, Stanton, and Euler.

## Miscellaneous utilities¶

More than just dimensionless groups are implemented in fluids.core.

Converters between loss coefficient, L/D equivalent, length of pipe, and pressure drop are available. It is recommended to convert length/diameter equivalents and lengths of pipe at specified friction factors to loss coefficients. They can all be summed easily afterwards.

```
>>> K_from_f(fd=0.018, L=100., D=.3)
6.0
>>> K_from_L_equiv(L_D=240, fd=0.02)
4.8
```

Either head loss or pressure drop can be calculated once the total loss coefficient K is known. Head loss does not require knowledge of the fluid’s density, but pressure drop does.

```
>>> dP_from_K(K=(6+4.8), rho=1000, V=3)
48600.0
```

```
>>> head_from_K(K=(6+4.8), V=3)
4.955820795072732
```

If a K value is known and desired to be converted to a L/D ratio or to an equivalent length of pipe, that calculation is available as well:

```
>>> L_equiv_from_K(3.6, fd=0.02)
180.0
>>> L_from_K(K=6, fd=0.018, D=.3)
100.0
```

Pressure and head are also convertible with the following functions:

```
>>> head_from_P(P=98066.5, rho=1000)
10.000000000000002
>>> P_from_head(head=5., rho=800.)
39226.6
```

Also implemented in fluids.core are the following:

Thermal diffisivity:

```
>>> thermal_diffusivity(k=0.02, rho=1., Cp=1000.)
2e-05
```

Speed of sound in an ideal gas (requires temperature, isentropic exponent Cp/Cv):

```
>>> c_ideal_gas(T=303, k=1.4, MW=28.96)
348.9820361755092
```

A converter between dynamic and kinematic viscosity:

```
>>> nu_mu_converter(rho=998., nu=1.0E-6)
0.000998
>>> nu_mu_converter(998., mu=0.000998)
1e-06
```

Calculation of gravity on earth as a function of height and latitude (input in degrees and height in meters):

```
>>> gravity(latitude=55, H=1E6)
6.729011976863571
```

## Friction factors¶

```
>>> epsilon = 1.5E-6 # m, clean steel
>>> fluids.friction.friction_factor(Re=15000, eD=epsilon/0.01)
0.02808790938573186
```

The transition to laminar flow is implemented abruptly at Re=2040, one of the latest experimental results which is accurate to +- 10. If the Reynolds number is in the laminar regime, the transition happens automatically and the well-known solution fd = 64/Re is given.

```
>>> fluids.friction.friction_factor(Re=150)
0.4266666666666667
```

Friction factor in curved pipes in available as friction_factor_curved. The curved friction factor is applicable for helices and coils, and to a lesser extent curved bends.

```
>>> friction_factor_curved(Re=15000, Di=.01, Dc=2.5, roughness=1.5E-6)
0.02984622907277626
```

The critical Reynolds number for curved pipes is increased compared to straight pipe flow, and is a function of the curvature of the pipe. The preferred method to calculate the transition (used by default for the automatic regime transition) is the method of Schmidt (1967).

```
>>> helical_transition_Re_Schmidt(Di=.01, Dc=2.5)
3948.7442097768603
```

Although roughness is a hard value to know without measuring it for a pipe, several hundred pipe conditions have had their roughness values measured in the literature, and they can be searched through using fuzzy matching fluids.

```
>>> nearest_material_roughness('Used water piping', clean=False)
'Seamless steel tubes, Used water piping'
>>> material_roughness('Seamless steel tubes, Used water piping')
0.0015
```

The material_roughness function can also be used directly, but in that case there is no feedback about the material which was found.

```
>>> material_roughness('glass')
1e-05
```

As fuzzy string matching is a pretty terrible solution, it is encouraged to find the desired string in the actual source code of fluids.

There is one more way of obtaining the roughness of a clean pipe, developed by Farshad and Rieke (2006). It has been established that in commercial pipe, the larger the diameter, the larger the roughness.

```
>>> roughness_Farshad('Carbon steel, bare', D=0.05)
3.529128126365038e-05
```

Only the following types of clean, new pipe have data available:

- ‘Plastic coated’
- ‘Carbon steel, honed bare’
- ‘Cr13, electropolished bare’
- ‘Cement lining’
- ‘Carbon steel, bare’
- ‘Fiberglass lining’
- ‘Cr13, bare’

There is also a term called Transmission factor, used in many pipeline applications. It is effectively a variant on friction factor. They can be inter-converted with the transmission_factor function.

```
>>> transmission_factor(fd=0.0185) # calculate transmission factor
14.704292441876154
>>> transmission_factor(F=20) # calculate Darcy friction factor
0.01
```

## Pipe schedules¶

ASME/ANSI pipe tables from B36.10M-2004 and B36-19M-2004 are implemented in fluids.piping.

Piping can be looked up based on nominal pipe size, outer diameter, or inner diameter.

```
>>> nearest_pipe(NPS=2) # returns NPS, inside diameter, outer diameter, wall thickness
(2, 0.05248, 0.0603, 0.00391)
```

When looking up by actual diameter, the nearest pipe as large or larger then requested is returned:

```
>>> NPS, Di, Do, t = nearest_pipe(Di=0.5)
>>> Di
0.57504
>>> nearest_pipe(Do=0.5)
(20, 0.47781999999999997, 0.508, 0.01509)
```

By default, the pipe schedule used for the lookup is schedule 40. Other schedules that are available are: ‘5’, ‘10’, ‘20’, ‘30’, ‘40’, ‘60’, ‘80’, ‘100’, ‘120’, ‘140’, ‘160’, ‘STD’, ‘XS’, ‘XXS’, ‘5S’, ‘10S’, ‘40S’, ‘80S’.

```
>>> nearest_pipe(Do=0.5, schedule='40S')
(20, 0.48894, 0.508, 0.009529999999999999)
>>> nearest_pipe(Do=0.5, schedule='80')
(20, 0.45562, 0.508, 0.02619)
```

If a diameter which is larger than any pipe in the schedule is input, an exception is raised:

```
>>> nearest_pipe(Do=1)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "fluids/piping.py", line 276, in nearest_pipe
raise ValueError('Pipe input is larger than max of selected scedule')
ValueError: Pipe input is larger than max of selected scedule
```

## Wire gauges¶

The construction of mechanical systems often uses the “gauge” sytems, a variety of old imperial conversions between plate or wire thickness and a dimensionless number. Conversion from and to the gauge system is done by the gauge_from_t and t_from_gauge functions.

Looking up the gauge from a wire of known diameter approximately 1.2 mm:

```
>>> gauge_from_t(.0012)
18
```

The reverse conversion:

```
>>> t_from_gauge(18)
0.001245
```

Other schedules are also supported:

- Birmingham Wire Gauge (BWG) ranges from 0.2 (0.5 inch) to 36 (0.004 inch).
- American Wire Gauge (AWG) ranges from 0.167 (0.58 inch) to 51 (0.00099 inch). These are used for electrical wires.
- Steel Wire Gauge (SWG) ranges from 0.143 (0.49 inch) to 51 (0.0044 inch). Also called Washburn & Moen wire gauge, American Steel gauge, Wire Co. gauge, and Roebling wire gauge.
- Music Wire Gauge (MWG) ranges from 0.167 (0.004 inch) to 46 (0.18 inch). Also called Piano Wire Gauge.
- British Standard Wire Gage (BSWG) ranges from 0.143 (0.5 inch) to 51 (0.001 inch). Also called Imperial Wire Gage (IWG).
- Stub’s Steel Wire Gage (SSWG) ranges from 1 (0.227 inch) to 80 (0.013 inch)

```
>>> t_from_gauge(18, schedule='AWG')
0.00102362
```

## Tank geometry¶

Sizing of vessels and storage tanks is implemented in an object-oriented way as TANK in fluids.geometry. All results use the exact equations; all are documented in the many functions in <fluids.geometry>

```
>>> T1 = TANK(D=1.2, L=4, horizontal=False)
>>> T1.V_total, T1.A # Total volume of the tank and its surface area
4.523893421169302, 17.34159144781566
```

By default, tanks are cylinders without heads. Tank heads can be specified to be conical, ellipsoidal, torispherical, guppy, or spherical. The heads can be specified independently. The diameter and length are not required; the total volume desired can be specified along with the length to diameter ratio.

```
>>> T1 = TANK(V=10, L_over_D=0.7, sideB='conical', horizontal=False)
>>> T1.L, T1.D
(1.7731788548899077, 2.5331126498427254)
```

Conical, ellipsoidal, guppy and spherical heads are all governed only by one parameter, a, the distance the head extends out from the main tank body. Torispherical heads are governed by two parameters k and f. If these parameters are not provided, the distance the head extends out will be 25% of the size of the tank’s diameter. For torispherical heads, the distance is similar but more complicated.

```
>>> TANK(D=10., V=500, horizontal=False, sideA='ellipsoidal', sideB='ellipsoidal', sideA_a=1, sideB_a=1)
<Vertical tank, V=500.000000 m^3, D=10.000000 m, L=5.032864 m, ellipsoidal heads, a=1.000000 m.>
```

Each TANK has __repr__ implemented, to describe the tank when printed.

Torispherical tanks default to the ratios specified as ASME F&D. Other standard ratios can also be used; the documentation for <TANK> lists their values. Here we implement DIN 28011’s ratios.

```
>>> TANK(D=0.01, V=0.25, horizontal=False, sideA='torispherical', sideB='torispherical')
<Vertical tank, V=0.250000 m^3, D=0.010000 m, L=3183.096137 m, torispherical heads, a=0.001693 m.>
>>> DIN = TANK(L=3, D=5, horizontal=False, sideA='torispherical', sideB='torispherical', sideA_f=1, sideA_k=0.1, sideB_f=1, sideB_k=0.1)
>>> print(DIN)
<Vertical tank, V=90.299352 m^3, D=5.000000 m, L=3.000000 m, torispherical heads, a=0.968871 m.>
```

Partial volume lookups are also useful. This is useful when the height of fluid in the tank is known, but not the volume. The reverse calculation is also implemented, and useful when doing dynamic simulation and to calculate the new height after a specified volume of liquid is removed.

```
>>> DIN.h_max
4.937742251701451
>>> DIN.h_from_V(40)
2.3760173045849315
>>> DIN.V_from_h(4.1)
73.83841540117238
```

Surface areas of the heads and the main body are available as well as the total surface area of the tank.

```
>>> DIN.A_sideA, DIN.A_sideB, DIN.A_lateral, DIN.A
(24.7496775831724, 24.7496775831724, 47.12388980384689, 96.62324497019169)
```

## Miscellaneous geometry¶

In addition to sizing all sorts of tanks, helical coils are supported and so are a number of other simple calculations.

Sphericity is implemented, requiring a calculated surface area and volume. For a cube of side length 3, the surface area is 6*a^2=54 and volume a^3=27. Its sphericity is then:

```
>>> sphericity(A=54, V=27)
0.8059959770082346
```

Aspect ratio of a rectangle 0.2 m by 2 m:

```
>>> aspect_ratio(.2, 2)
0.1
```

Circularity, a parameter used to characterize 2d images of particles, is implemented. For a rectangle, one side length = 1, second side length = 100:

```
>>> D1 = 1
>>> D2 = 100
>>> A = D1*D2
>>> P = 2*D1 + 2*D2
>>> circularity(A, P)
0.030796908671598795
```

## Atmospheric properties¶

Four main classes are available to model the atmosphere. They are the US Standard Atmosphere 1976 (ATMOSPHERE_1976), a basic but very quick model; the NRLMSISE 00 model, substantially more powerful and accurate and still the standard to this day (ATMOSPHERE_NRLMSISE00); and two models for wind speed only, Horizontal Wind Model 1993 (hwm93) and Horizontal Wind Model 2014 (hwm14). The two horizontal wind models are actually fortran codes, and are not compilled automatically on installation.

ATMOSPHERE_1976 is the simplest model, and very suitable for basic engineering purposes. It supports atmospheric temperature, density, and pressure as a function of elevation. Optionally, a local temperature difference from earth’s average can be specified to correct the model to local conditions but this is only a crude approximation.

Conditions 5 km into the air:

```
>>> atm = ATMOSPHERE_1976(Z=5000)
>>> atm.T, atm.P, atm.rho
(255.67554322180348, 54048.28614576141, 0.7364284207799743)
```

The standard also specifies simplistic formulas for calculating the thermal conductivity, viscosity, speed of sound, and gravity at a given elevation:

```
>>> atm.g, atm.mu, atm.k, atm.v_sonic
(9.791241076982665, 1.628248135362207e-05, 0.02273190295142526, 320.5455196704035)
```

Those property routines are static methods, and can be used without instantiating an atmosphere object:

```
>>> ATMOSPHERE_1976.gravity(Z=1E5)
9.505238763515356
>>> ATMOSPHERE_1976.sonic_velocity(T=300)
347.22080908230015
>>> ATMOSPHERE_1976.viscosity(T=400)
2.285266457680251e-05
>>> ATMOSPHERE_1976.thermal_conductivity(T=400)
0.033657148617592114
```

ATMOSPHERE_NRLMSISE00 is the recommended model, and calculates atmospheric density, temperature, and pressure as a function of height, latitude/longitude, day of year, and seconds since start of day. The model can also take into account solar and geomagnetic disturbances which effect the atmosphere at very high elevations if more parameters are provided. It is valid up to 1000 km. This model is somewhat slow; it is a Python port of the fortran version, created by Joshua Milas. It does not support gravity profiles or transport properties, but does calculate the composition of the atmosphere (He, O, N2, O2, Ar, H2, N2 as constituents).

1000 m elevation, 45 degrees latitude and longitude, 150th day of year, 0 seconds in:

```
>>> atm = ATMOSPHERE_NRLMSISE00(Z=1E3, latitude=45, longitude=45, day=150)
>>> atm.T, atm.P, atm.rho
(285.54408606237405, 90394.40851588511, 1.1019062026405517)
```

The composition of the atmosphere is specified in terms of individual molecules/m^3:

```
>>> atm.N2_density, atm.O2_density
(1.7909954550444606e+25, 4.8047035072477747e+24)
```

This model uses the ideal gas law to convert particle counts to mass density. Mole fractions of each species are available as well.

```
>>> atm.components
['N2', 'O2', 'Ar', 'He', 'O', 'H', 'N']
>>> atm.zs
[0.7811046347676225, 0.2095469403691101, 0.009343183088772914, 5.241774494627779e-06, 0.0, 0.0, 0.0]
```

The horizontal wind models have almost the same API, and calculate wind speed and direction as a function of elevation, latitude, longitude, day of year and time of day. hwm93 can also take as an argument local geomagnetic conditions and solar activity, but this effect was found to be so negligible it was removed from future versions of the model such as hwm14.

Calculation of wind velocity, meridional (m/sec Northward) and zonal (m/sec Eastward) for 1000 m elevation, 45 degrees latitude and longitude, 150th day of year, 0 seconds in, with both models:

```
>>> hwm93(Z=1000, latitude=45, longitude=45, day=150)
[-0.0038965975400060415, 3.8324742317199707]
>>> hwm14(Z=1000, latitude=45, longitude=45, day=150)
[-0.9920163154602051, 0.4105832874774933]
```

These wind velocities are only historical normals; conditions may vary year to year.

## Compressor sizing¶

Both isothermal and isentropic/polytropic compression models are implemented in fluids.compressible. Isothermal compression calculates the work required to compress a gas from one pressure to another at a specified temperature. This is the best possible case for compression; all actual compresssors require more work to do the compression. By making the compression take a large number of stages and cooling the gas between stages, this can be approached reasonable closely. Integrally geared compressors are often used for this purpose.

```
>>> isothermal_work_compression(P1=1E5, P2=1E6, T=300)
5743.425357533477
```

Work is calculated on a J/mol basis. If the second pressure is lower than the first, a negative work will result and you are modeling an expander instead of a compressor. Gas compressibility factor can also be specified. The lower the gas’s compressibility factor, the less power required to compress it.

```
>>> isothermal_work_compression(P1=1E6, P2=1E5, T=300)
-5743.425357533475
>>> isothermal_work_compression(P1=1E5, P2=1E6, T=300, Z=0.95)
5456.2540896568025
```

There is only one function implemented to model both isentropic and polytropic compressors, as the only difference is that a polytropic exponent n is used instead of the gas’s isentropic exponent Cp/Cv k and the type of efficiency is changed. The model requires initial temperature, inlet and outlet pressure, isentropic exponent or polytropic exponent, and optionally an efficiency.

Compressing air from 1 bar to 10 bar, with inlet temperature of 300 K and efficiency of 78%:

```
>>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=0.78) # work, J/mol
10416.873455626454
```

The model allows for the inlet or outlet pressure or efficiency to be calculated instead of the work:

```
>>> isentropic_work_compression(T1=300, P1=1E5, P2=1E6, k=1.4, W=10416) # Calculate efficiency
0.7800654085434559
>>> isentropic_work_compression(T1=300, P1=1E5, k=1.4, W=10416, eta=0.78) # Calculate P2
999858.5366533266
>>> isentropic_work_compression(T1=300, P2=1E6, k=1.4, W=10416, eta=0.78) # Calculate P1
100014.14833613831
```

The approximate temperature rise can also be calculated with the function isentropic_T_rise_compression.

```
>>> T2 = isentropic_T_rise_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=0.78)
>>> T2, T2-300 # outlet temperature and temperature rise, K
(657.960664955096, 357.96066495509604)
```

It is more accurate to use an enthalpy-based model which incorporates departure functions.

Polytropic exponents and efficiencies are convertible to isentropic exponents and efficiencies. For the above example, with k=1.4 and `eta_s`=0.78:

```
>>> eta_p = isentropic_efficiency(P1=1E5, P2=1E6, k=1.4, eta_s=0.78) # with eta_s specified, returns polytropic efficiency
>>> n = polytropic_exponent(k=1.4, eta_p=eta_p)
>>> eta_p, n
(0.8376785349411107, 1.517631868575738)
```

With those results, we can prove the calculation worked by calculating the work required using these polytropic inputs:

```
>>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=n, eta=eta_p)
10416.873455626452
```

The work is the same as calculated with the original inputs. Note that the conversion is specific to three inputs: Inlet pressure; outlet pressure; and isentropic exponent k. If any of those change, then the calculated polytropic exponent and efficiency will be different as well.

To go in the reverse direction, we take the case of isentropic exponent k =Cp/Cv=1.4, eta_p=0.83 The power is calculated to be:

We first need to calculate the polytropic exponent from the polytropic efficiency:

```
>>> n = polytropic_exponent(k=1.4, eta_p=0.83)
>>> print(n)
1.5249343832
```

```
>>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=n, eta=0.83)
10556.494602042329
```

Converting polytropic efficiency to isentropic efficiency:

```
>>> eta_s = isentropic_efficiency(P1=1E5, P2=1E6, k=1.4, eta_p=0.83)
>>> print(eta_s)
0.7588999047069671
```

Checking the calculated power is the same:

```
>>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=eta_s)
10556.494602042327
```

## Gas pipeline sizing¶

The standard isothermal compressible gas flow is fully implemented, and through a variety of numerical and analytical expressions, can solve for any of the following parameters:

- Mass flow rate
- Upstream pressure (numerical)
- Downstream pressure (analytical or numerical if an overflow occurs)
- Diameter of pipe (numerical)
- Length of pipe

Solve for the mass flow rate of gas (kg/s) flowing through a 1 km long 0.5 m inner diameter pipeline, initially at 10 bar with a density of 11.3 kg/m^3 going downstream to a pressure of 9 bar.

```
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, L=1000, D=0.5)
145.4847572636031
```

The same case, but sizing the pipe to take 100 kg/s of gas:

```
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, L=1000, m=100)
0.42971708911060613
```

The same case, but determining what the outlet pressure will be if 200 kg/s flow in the 0.5 m diameter pipe:

```
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, D=0.5, L=1000, m=200)
784701.0681827427
```

Determining pipe length from known diameter, pressure drop, and mass flow (possible but not necessarily useful):

```
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, D=0.5, m=150)
937.3258027759333
```

Not all specified mass flow rates are possible. At a certain downstream pressure, chocked flow will develop - that downstream pressure is that at which the mass flow rate reaches a maximum. An exception will be raised if such an input is specified:

```
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, L=1000, D=0.5, m=260)
Exception: The desired mass flow rate cannot be achieved with the specified upstream pressure; the maximum flowrate is 257.216733 at an downstream pressure of 389699.731765
>>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=3E5, L=1000, D=0.5)
Exception: Given outlet pressure is not physically possible due to the formation of choked flow at P2=389699.731765, specified outlet pressure was 300000.000000
```

The downstream pressure at which chocked flow occurs can be calculated directly as well:

```
>>> P_isothermal_critical_flow(P=1E6, fd=0.00185, L=1000., D=0.5)
389699.7317645518
```

A number of limitations exist with respect to the accuracy of this model:

- Density dependence is that of an ideal gas.
- If calculating the pressure drop, the average gas density cannot be known immediately; iteration must be used to correct this.
- The friction factor depends on both the gas density and velocity, so it should be solved for iteratively as well. It changes throughout the pipe as the gas expands and velocity increases.
- The model is not easily adapted to include elevation effects due to the acceleration term included in it.
- As the gas expands, it will change temperature slightly, further altering the density and friction factor.

We can explore how the gas density and friction factor effect the model using the thermo library for chemical properties.

Compute the downstream pressure of 50 kg/s of natural gas flowing in a 0.5 m diameter pipeline for 1 km, roughness = 5E-5 m:

```
>>> from thermo import *
>>> from fluids import *
>>> D = 0.5
>>> L = 1000
>>> epsilon = 5E-5
>>> S1 = Stream('natural gas', P=1E6, m=50)
>>> V = S1.Q/(pi/4*D**2)
>>> Re = S1.Reynolds(D=D, V=V)
>>> fd = friction_factor(Re=Re, eD=epsilon/D)
>>> P2 = isothermal_gas(rho=S1.rho, fd=fd, P1=S1.P, D=D, L=L, m=S1.m)
>>> 877852.8365849017
```

In the above example, the friction factor was calculated using the density and velocity of the gas when it enters the stream. However, the average values, at the middle pressure, and more representative. We can iterate to observe the effect of using the average values:

```
>>> for i in range(10):
>>> S2 = Stream('natural gas', P=0.5*(P2+S1.P), m=50)
>>> V = S2.Q/(pi/4*D**2)
>>> Re = S2.Reynolds(D=D, V=V)
>>> fd = friction_factor(Re=Re, eD=epsilon/D)
>>> P2 = isothermal_gas(rho=S2.rho, fd=fd, P1=S1.P, D=D, L=L, m=S1.m)
>>> print(P2)
868992.832357
868300.621412
868246.236225
868241.961444
868241.625427
868241.599014
868241.596938
868241.596775
868241.596762
868241.596761
```

As can be seen, the system converges very quickly. The difference in calculated pressure drop is approximately 1%.

## Gas pipeline sizing: Empirical equations¶

In addition to the actual model, many common simplifications used in industry are implemented as well. These are equally capable of solving for any of the following inputs:

- Mass flow rate
- Upstream pressure
- Downstream pressure
- Diameter of pipe
- Length of pipe

None of these models include an acceleration term. In addition to reducing their accuracy, it allows all solutions for the above variables to be analytical. These models cannot predict the occurrence of chocked flow, and model only turbulent, not laminar, flow. Most of these models do not depend on the gas’s viscosity.

Rather than using mass flow rate, they use specific gravity and volumetric flow rate. The volumetric flow rate is specified with respect to a reference temperature and pressure. The defaults are 288.7 K and 101325 Pa, dating to the old imperial standard of 60° F. The specific gravity is with respect to air at the reference conditions. As the ideal gas law is used in each of these models, in addition to pressure and specific gravity the average temperature in the pipeline is required. Average compressibility factor is an accepted input to all models and corrects the ideal gas law’s ideality.

The full list of approximate models is as follows:

- Panhandle_A
- Panhandle_B
- Weymouth
- Oliphant
- Fritzsche
- Muller
- IGT
- Spitzglass_high
- Spitzglass_low

As an example, calculating flow for a pipe with diameter 0.34 m, upstream pressure 90 bar and downstream pressure 20 bar, 160 km long, 0.693 specific gravity and with an average temperature in the pipeline of 277.15 K:

```
>>> Panhandle_A(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
42.56082051195928
```

Each model also includes a pipeline efficiency term, ranging from 0 to 1. These are just empirical correction factors, Some of the models were developed with theory and a correction factor applied always; others are more empirical, and have a default correction factor. 0.92 is the default for the Panhandle A/B, Weymouth, and Oliphant models; the rest default to a correction of 1 i.e. no correction at all.

The Muller and IGT models are the most accurate and recent approximations. They both depend on viscosity.

```
>>> Muller(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5, Tavg=277.15)
60.45796698148659
>>> IGT(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5, Tavg=277.15)
48.92351786788815
```

These empirical models are included because they are mandated in many industrial applications regardless of their accuracy, and correction factors have already been determined.

A great deal of effort was spent converting these models to base SI units and checking the coefficients used in each model with multiple sources. In many cases multiple sets of coefficients are available for a model; the most authoritative or common ones were used in those cases.

## Drag and terminal velocity¶

A number of spherical particle drag correlations are implemented.

In the simplest case, consider a spherical particle of diameter D=1 mm, density=3400 kg/m^3, traveling at 30 m/s in air with viscosity mu=1E-5 Pa*s and density 1.2 kg/m^3.

We calculate the particle Reynolds number:

```
>>> Re = Reynolds(V=30, rho=1.2, mu=1E-5, D=1E-3)
>>> Re
3599.9999999999995
```

The drag coefficient Cd can be calculated with no other parameters:

```
>>> drag_sphere(Re)
0.3914804681941151
```

The terminal velocity of the particle is easily calculated with the v_terminal function.

```
>>> v_terminal(D=1E-3, rhop=3400, rho=1.2, mu=1E-5)
8.971223953182939
```

Very often, we are not interested in just what the velocity of the particle will be at terminal conditions, but on the distance it will travel and the particle will never have time to reach terminal conditions. An integrating function is available to do that. Consider that same particle being shot directly down from a helicopter 100 m high.

The integrating function, integrate_drag_sphere, performs the integral with respect to time. At one second, we can see the (velocity, distance travelled):

```
>>> integrate_drag_sphere(D=1E-3, rhop=3400., rho=1.2, mu=1E-5, t=1, V=30, distance=True)
(10.561878111154627, 15.60790417764922)
```

After integrating to 10 seconds, we can see the particle has travelled 97 meters and is almost on the ground.

```
>>> integrate_drag_sphere(D=1E-3, rhop=3400., rho=1.2, mu=1E-5, t=10, V=30, distance=True)
(8.971223987066322, 97.13276290361276)
```

For this example simply using the terminal velocity would have given an accurate estimation of distance travelled:

```
>>> 8.971223953182939*10
89.7122395318294
```

Many engineering applications such as direct contact condensers do operate far from terminal velocity however, and this function is useful there.

## Pressure drop through packed beds¶

Twelve different packed bed pressure drop correlations are available. A meta function which allows any of them to be selected and automatically selects the most accurate correlation for the given parameters.

Pressure drop through a packed bed depends on the density, viscosity and velocity of the fluid, as well as the diameter of the particles, the amount of free space in the bed (voidage), and to a lesser amount the ratio of particle to tube diameter and the shape of the particles.

Consider 0.8 mm pebbles with 40% empty space with water flowing through a 2 m column creeping flow at a superficial velocity of 1 mm/s. We can calculate the pressure drop as follows:

```
>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, L=2)
2876.565391768883 # Pa
```

The method can be specified manually as well, for example the commonly used Ergun equation:

```
>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, L=2, Method='Ergun')
2677.734374999999
```

Incorporation of the tube diameter will add wall effects to the model.

```
>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, L=2, Dt=0.01)
2510.3251325096853
```

Models can be used directly as well. The length of the column is an optional input; if not provided, the result will be in terms of Pa/m.

```
>>> KTA(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) # A correlation standardized for use in pebble reactors
1440.409277034248
```

If the column diameter was 0.5 m, the flow rate would be:

```
>>> .001*(pi/4*0.5**2) # superficial_velocity*A_column
0.00019634954084936208 # m^3/s
```

The holdup (total volume of the column holding fluid not particles) would be:

```
>>> (pi/4*0.5**2)*(2)*0.4 # A_column*H_column*voidage
0.15707963267948966 # m^3
```

Not all particles are spherical. There have been correlations published for specific shapes, but what is often performed is simply an adjustment of particle diameter by its sphericity in the correlation, with the effective dp used as the product of the actual dp and the sphericity of the particle. The less spherical the particles, the higher the pressure drop. This is supported in all of the correlations.

```
>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, L=2, Dt=0.01, sphericity=0.9)
3050.419598116882
```

While it is easy to measure the volume of particles added to a given column and determine the voidage experimentally, this does not help in the design process. Several authors have methodically filled columns with particles of different sizes and created correlations in terms of sphericity and particle to tube diameter ratios. Three such correlations are implemented in fluids, one generally using sphericity, one for spheres, and one for cylinders.

1 mm spheres in a 5 cm diameter tube:

```
>>> voidage_Benyahia_Oneil_spherical(Dp=.001, Dt=.05)
0.3906653157443224
```

1 mm diameter cylinder 5 mm long in a 5 cm diameter tube:

```
>>> V_cyl = V_cylinder(D=0.001, L=0.005)
>>> D_sphere_eq = (6*V_cyl/pi)**(1/3.)
>>> A_cyl = A_cylinder(D=0.001, L=0.005)
>>> sph = sphericity(A=A_cyl, V=V_cyl)
>>> voidage_Benyahia_Oneil_cylindrical(Dpe=D_sphere_eq, Dt=0.05, sphericity=sph)
0.3754895273247688
```

Same calculation, but using the general correlation for all shapes:

```
>>> voidage_Benyahia_Oneil(Dpe=D_sphere_eq, Dt=0.05, sphericity=sph)
0.4425769555048246
```

## Pressure drop through piping¶

It is straightforward to calculate the pressure drop of fluid flowing in a pipeline with any number of fittings using the fluids library.

15 m of piping, with a sharp entrance and sharp exit, two 30 degree miter bends, one rounded bend 45 degrees, 1 sharp contraction to half the pipe diameter and 1 sharp expansion back to the normal pipe diameter (water, V=3 m/s, Di=0.05, roughness 0.01 mm):

```
>>> Re = Reynolds(V=3, D=0.05, rho=1000, mu=1E-3)
>>> fd = friction_factor(Re, eD=1E-5/0.05)
>>> K = K_from_f(fd=fd, L=15, D=0.05)
>>> K += entrance_sharp()
>>> K += exit_normal()
>>> K += 2*bend_miter(angle=30)
>>> K += bend_rounded(Di=0.05, angle=45, fd=fd)
>>> K += contraction_sharp(Di1=0.05, Di2=0.025)
>>> K += diffuser_sharp(Di1=0.025, Di2=0.05)
>>> dP_from_K(K, rho=1000, V=3)
37920.51140146369
```

If the diameter of the piping varies, not all of the loss coefficients will be with respect to the same diameter. Each loss coefficient must be converted to one standard diameter before the total pressure drop can be calculated. The following example is solved with the optional pint unit compatibility module.

40 m piping, beveled entrance (10 mm length, 30 degrees, into 5 cm ID pipe) , then a 30 degree miter bend, a sharp contraction to half the pipe diameter (5 m long), a 30 degree miter bend, a rounded 45 degree bend, a sharp expansion to 4 cm ID pipe (15 more meters), and a sharp exit:

```
>>> from fluids.units import *
>>> from math import *
>>> material = nearest_material_roughness('steel', clean=True)
>>> epsilon = material_roughness(material)
>>> Q = .01*u.m**3/u.s
>>> rho = 1000*u.kg/u.m**3
>>> mu = 1E-4*u.Pa*u.s
>>> D1 = 5*u.cm
>>> D2 = 2.5*u.cm
>>> D3 = 4*u.cm
>>> L1 = 20*u.m
>>> L2 = 5*u.m
>>> L3 = 15*u.m
>>> V1 = Q/(pi/4*D1**2)
```

```
>>> Re = Reynolds(V=V1, D=D1, rho=rho, mu=mu)
>>> fd = friction_factor(Re, eD=epsilon/D1)
>>> K = K_from_f(fd=fd, L=L1, D=D1)
>>> K += entrance_beveled(Di=D1, l=10*u.mm, angle=30*u.degrees)
>>> K += bend_miter(angle=30*u.degrees)
>>> K += contraction_sharp(Di1=D1, Di2=D2)
```

```
>>> V2 = Q/(pi/4*D2**2)
>>> Re2 = Reynolds(V=V2, D=D2, rho=rho, mu=mu)
>>> fd2 = friction_factor(Re2, eD=epsilon/D2)
```

```
>>> K += change_K_basis(K_from_f(fd=fd2, L=L2, D=D2), D1=D2, D2=D1)
>>> K += change_K_basis(K1=bend_miter(angle=30*u.degrees), D1=D2, D2=D1)
>>> K += change_K_basis(K1=bend_rounded(Di=D2, angle=45*u.degrees, fd=fd2), D1=D2, D2=D1)
```

```
>>> V3 = Q/(pi/4*D3**2)
>>> Re3 = Reynolds(V=V3, D=D3, rho=rho, mu=mu)
>>> fd3 = friction_factor(Re3, eD=epsilon/D3)
```

```
>>> K += change_K_basis(K_from_f(fd=fd3, L=L3, D=D3), D1=D3, D2=D1)
>>> K += diffuser_sharp(Di1=D2, Di2=D3)
```

```
>>> dP_from_K(K, rho=rho, V=V1)
<Quantity(608471.881547, 'pascal')>
```

## Control valve sizing: Introduction¶

The now internationally-standardized methods (IEC 60534) for sizing liquid and gas valves have been implemented. Conversion factors among the different types of valve coefficients are implemented as well.

There are two forms of loss coefficient used for vales, an imperial and a metric variable called “valve flow coefficient”. Both can be converted to the standard dimensionless loss coefficient.

If one knows the actual loss coefficient of a valve, the valve flow coefficient can be calculated in either metric or imperial forms as follows. The flow coefficients are specific to the diameter of the valve.

```
>>> K_to_Kv(K=16, D=0.016)
2.56
>>> K_to_Cv(K=16, D=0.016)
2.9596140245853606
```

If Kv or Cv are known, they can be converted to each other with the proportionality constant 1.156, which is derived from a unit conversion only. This conversion does not require valve diameter.

```
>>> Cv_to_Kv(12)
10.379731865307619
>>> Kv_to_Cv(10.37)
11.988748998027418
```

If a Cv or Kv is obtained from a valve datasheet, it can be converted into a standard loss coefficient as follows.

```
>>> Kv_to_K(Kv=2.56, D=0.016)
16.000000000000004
>>> Cv_to_K(Cv=3, D=0.016)
15.57211586581753
```

For a valve with a specified Kv and pressure drop, the flow rate can be calculated easily for the case of non-choked non-compressible flow (neglecting other friction losses), as illustrated in the example below for a 5 cm valve with a pressure drop 370 kPa and density of 870 kg/m^3:

```
>>> Kv = 72.5
>>> D = 0.05
>>> dP = 370E3
>>> K = Kv_to_K(D=D, Kv=Kv)
>>> rho = 870
>>> V = (dP/(.5*rho*K))**0.5 # dP = K*0.5*rho*V^2
>>> A = pi/4*D**2
>>> Q = V*A
>>> Q
0.04151682468778643
```

Alternatively, the required Kv can be calculated from an assumed diameter and allowable pressure drop:

```
>>> Q = .05
>>> D = 0.05
>>> dP = 370E3
>>> rho = 870
>>> A = pi/4*D**2
>>> V = Q/A
>>> K = dP/(.5*rho*V**2)
>>> K_to_Kv(D=D, K=K)
87.31399925838778
```

The approach documented above is not an adequate procedure for sizing valves however because chocked flow, compressible flow, the effect of inlet and outlet reducers, the effect of viscosity and the effect of laminar/turbulent flow all have large influences on the performance of control valves.

Historically, valve manufacturers had their own standards for sizing valves, but these have been standardized today into the IEC 60534 methods.

## Control valve sizing: Liquid flow¶

To rigorously size a control valve for liquid flow, the inlet pressure, allowable pressure drop, and desired flow rate must first be known. These need to be determined taking into account the entire pipe network and the various operating conditions it needs to support; sizing the valves can be performed afterward and only if no valve with the desired performance is available does the network need to be redesigned.

To illustrate sizing a valve, we borrow an example from Emerson’s Control Valve Handbook, 4th edition (2005). It involves a flow of 800 gpm of liquid propane. The inlet and outlet pipe size is 8 inches, but the size of the valve itself is unknown. The desired pressure drop is 25 psi.

Converting this problem to SI units and using the thermo library to calculate the necessary properties of the fluid, we calculate the necessary Kv of the valve based on an assumed valve size of 3 inches:

```
>>> from scipy.constants import *
>>> from fluids.control_valve import size_control_valve_l
>>> from thermo.chemical import Chemical
>>> P1 = 300*psi + psi # to Pa
>>> P2 = 275*psi + psi # to Pa
>>> T = 273.15 + 21 # to K
>>> propane = Chemical('propane', P=(P1+P2)/2, T=T)
>>> rho = propane.rho
>>> Psat = propane.Psat
>>> Pc = propane.Pc
>>> mu = propane.mu
>>> Q = 800*gallon/minute # to m^3/s
>>> D1 = D2 = 8*inch # to m
>>> d = 3*inch # to m
```

The standard specifies two more parameters specific to a valve:

- FL, Liquid pressure recovery factor of a control valve without attached fittings
- Fd, Valve style modifier

Both of these are factors between 0 and 1. In the Emerson handbook, they are not considered in the sizing procedure and set to 1. These factors are also a function of the diameter of the valve and are normally tabulated next to the values of Cv or Kv for a valve.

```
>>> Kv = size_control_valve_l(rho, Psat, Pc, mu, P1, P2, Q, D1, D2, d, FL=1, Fd=1)
109.39701927957765
```

The handbook states the Cv of the valve is 121; we convert Kv to Cv:

```
>>> Kv_to_Cv(Kv=Kv)
126.47380957330982
```

The example in the book calculated Cv = 125.7, but doesn’t actually use the full calculation method. Either way, the valve will not carry the desired flow rate; we need to try a larger valve size. The 4 inch size is tried next in the example, which has a known Cv of 203.

```
>>> d = 4*inch # to m
>>> Kv = size_control_valve_l(rho, Psat, Pc, mu, P1, P2, Q, D1, D2, d, FL=1, Fd=1)
>>> Kv_to_Cv(Kv=Kv)
116.17550388277834
```

The calculated Cv is well under the valve’s maximum Cv; we can select it.

This model requires a vapor pressure and a critical pressure of the fluid as inputs. There is no clarification in the standard about how to handle mixtures, which do not have these values. It is reasonable to calculate vapor pressure as the bubble pressure, and the mixture’s critical pressure through a mole-weighted average.

For actual values of Cv, Fl, Fd, and available diameters, an excellent resource is the Fisher Catalog 12.

## Control valve sizing: Gas flow¶

To rigorously size a control valve for gas flow, the inlet pressure, allowable pressure drop, and desired flow rate must first be known. These need to be determined taking into account the entire pipe network and the various operating conditions it needs to support; sizing the valves can be performed afterward and only if no valve with the desired performance is available does the network need to be redesigned.

To illustrate sizing a valve, we borrow an example from Emerson’s Control Valve Handbook, 4th edition (2005). It involves a flow of 6 million ft^3/hour of natural gas. The inlet and outlet pipe size is 8 inches, but the size of the valve itself is unknown. The desired pressure drop is 150 psi.

Converting this problem to SI units and using the thermo library to calculate the necessary properties of the fluid, we calculate the necessary Kv of the valve based on an assumed valve size of 8 inches.

```
>>> from scipy.constants import *
>>> from fluids.control_valve import size_control_valve_g
>>> from thermo.chemical import Chemical
>>> P1 = 214.7*psi
>>> P2 = 64.7*psi
>>> T = 16 + 273.15
>>> natural_gas = Mixture('natural gas', T=T, P=(P1+P2)/2)
>>> Z = natural_gas.Z
>>> MW = natural_gas.MW
>>> mu = natural_gas.mu
>>> gamma = natural_gas.isentropic_exponent
>>> Q = 6E6*foot**3/hour
>>> D1 = D2 = d = 8*inch # 8-inch Fisher Design V250
```

The standard specifies three more parameters specific to a valve:

- FL, Liquid pressure recovery factor of a control valve without attached fittings
- Fd, Valve style modifier
- xT, Pressure difference ratio factor of a valve without fittings at choked flow

All three of these are factors between 0 and 1. In the Emerson handbook, FL and Fd are not considered in the sizing procedure and set to 1. xT is specified as 0.137 at full opening. These factors are also a function of the diameter of the valve and are normally tabulated next to the values of Cv or Kv for a valve. Performing the calculation:

```
>>> Kv = size_control_valve_g(T, MW, mu, gamma, Z, P1, P2, Q, D1, D2, d, FL=1, Fd=1, xT=.137)
>>> Kv_to_Cv(Kv)
1563.4479874210986
```

The 8-inch valve is rated with Cv = 2190. The valve is adequate to provide the desired flow because the rated Cv is higher. The calculated value in their example is 1515, differing slightly due to the properties used.

The example next goes on to determine the actual opening position the valve should be set at to provide the required flow. Their conclusion is approximately 75% open; we can do better using a numerical solver. The values of opening at different positions are obtained in this example from the valve’s datasheet.

Loading the data and creating interpolation functions so FL, Fd, and xT are all smooth functions:

```
>>> from scipy.interpolate import interp1d
>>> from scipy.optimize import newton
>>> openings = [.2, .3, .4, .5, .6, .7, .8, .9]
>>> Fds = [0.59, 0.75, 0.85, 0.92, 0.96, 0.98, 0.99, 0.99]
>>> Fls = [0.9, 0.9, 0.9, 0.85, 0.78, 0.68, 0.57, 0.45]
>>> xTs = [0.92, 0.81, 0.85, 0.63, 0.58, 0.48, 0.29, 0.14]
>>> Kvs = [24.1, 79.4, 153, 266, 413, 623, 1060, 1890]
>>> Fd_interp = interp1d(openings, Fds, kind='cubic')
>>> Fl_interp = interp1d(openings, Fls, kind='cubic')
>>> xT_interp = interp1d(openings, xTs, kind='cubic')
>>> Kv_interp = interp1d(openings, Kvs, kind='cubic')
```

Creating and solving the objective function:

```
>>> def to_solve(opening):
>>> Fd = float(Fd_interp(opening))
>>> Fl = float(Fl_interp(opening))
>>> xT = float(xT_interp(opening))
>>> Kv_lookup = float(Kv_interp(opening))
>>> Kv_calc = size_control_valve_g(T, MW, mu, gamma, Z, P1, P2, Q, D1, D2, d, FL=Fl, Fd=Fd, xT=xT)
>>> return Kv_calc - Kv_lookup
>>>
>>> newton(to_solve, .8) # initial guess of 80%
0.7495109870213784
```

We see the valve should indeed be set to almost exactly 75% open to provide the desired flow.

## Electric motor sizing¶

Motors are available in standard sizes, mostly as designated by the National Electrical Manufacturers Association (NEMA). To easily determine what the power of a motor will actually be once purchased, motor_round_size implements rounding up of a motor power to the nearest size. NEMA standard motors are specified in terms of horsepower.

```
>>> motor_round_size(1E5) # 100 kW motor
111854.98073734052 # 11.8% larger than desired
>>> from scipy.constants import hp
>>> motor_round_size(1E5)/hp # convert to hp
150.0
```

Motors are designed to generate a certain amount of power, but they themselves are not 100% efficient at doing this and require more power due to efficiency losses. Many minimum values for motor efficiency are standardized. The Canadian standard for this is implemented in fluids as CSA_motor_efficiency.

```
>>> CSA_motor_efficiency(P=5*hp)
0.855
```

Most motors are not enclosed (the default assumption), but those that are closed are more efficient.

```
>>> CSA_motor_efficiency(P=5*hp, closed=True)
0.875
```

The number of poles in a motor also affects its efficiency:

```
>>> CSA_motor_efficiency(P=5*hp, poles=6)
0.875
```

There is also a schedule of higher efficiency values standardized as well, normally available at somewhat higher cost:

```
>>> CSA_motor_efficiency(P=5*hp, closed=True, poles=6, high_efficiency=True)
0.895
```

A motor will spin at more or less its design frequency, depending on its type. However, if it does not meet sufficient resistance, it will not be using its design power. This is good and bad - less power is used, but as a motor drops under 50% of its design power, its efficiency becomes terrible. A function has been written based on generic performance curves to estimate the underloaded efficiency of a motor. Just how bad efficiency drops off depends on the design power of a motor - higher power motors do better operating at low loads than small motors.

```
>>> motor_efficiency_underloaded(P=1E3, load=.9)
1
>>> motor_efficiency_underloaded(P=1E3, load=.2)
0.6639347559654663
```

This needs to be applied on top of the normal motor efficiency; for example, that 1 kW motor at 20% load would have a net efficiency of:

```
>>> motor_efficiency_underloaded(P=1E3, load=.2)*CSA_motor_efficiency(P=1E3)
0.5329404286134798
```

Many motors have Variable Frequency Drives (VFDs) which allow them to vary the speed of their rotation. The VFD is another source of inefficiency, but by allowing the pump or other piece of equipment to vary its speed, a system may be designed to be less energy intensive. For example, rather than running a pump at a certain high frequency and controlling the flow with a large control valve, the flow rate can be controlled with the VFD directly.

The efficiency of a VFD depends on the maximum power it needs to be able to generate, and the power it is actually generating at an instant (load). A table of typical modern VFD efficiencies is implemented in fluids as VFD_efficiency.

```
>>> VFD_efficiency(1E5) # 100 kW
0.97
>>> VFD_efficiency(5E3, load=.2) # 5 kW, 20% load
0.8562
```