pybemt¶

pybemt.solver module¶

Module for the solver class.

class pybemt.solver.Solver(config_path)¶

Bases: object

The Solver object loads the config file and contains functions for running a single simulation, parameter sweeps and optimization.

Parameters: config_path (string) – Path to config file

brute_solve(sec, v, omega, n=3600)¶

Solve by a simple brute force procedure, iterating through all possible angles and selecting the one with lowest residual.

Parameters

sec (Section) – Section to solve for
v (float) – Axial inflow velocity
omega (float) – Tangential rotational velocity
n (int) – Number of angles to test for, optional

Returns

Inflow angle with lowest residual

Return type

float

optimize_pitch()¶

Optimize rotor pitch for either maximum thrust (propeller) or maximum power (turbine) using a genetic evolution algorithm.

This is intended as an example of how optimization can be done using the scipy.optimize package. The overall procedure can be readily modified to optimize for other parameters, e.g. a parametrized function for the pitch, or for a parameter sweep instead of a single parameter set.

return: Array of optimized pitches

rotor_coeffs(T, Q, P)¶

Dimensionless coefficients for a rotor.

\[\begin{split}\text{J} = \frac{V_\infty}{nD} \\ C_T = \frac{T}{\rho n^2 D^4} \\ C_Q = \frac{Q}{\rho n^2 D^5} \\ C_P = 2\pi C_Q \\ \eta = \frac{C_T}{C_P}J \\\end{split}\]

Parameters

T (float) – Thrust
Q (float) – Torque
P (float) – Power

Returns

Advance ratio, thrust coefficient, torque coefficient, power coefficient and efficiency

Return type

tuple

run()¶

Runs the solver, i.e. finds the forces for each rotor.

Returns: Calculated thrust, torque, power and DataFrame with properties for all sections.
Return type: tuple

run_sweep(parameter, n, low, high)¶

Utility function to run a sweep of a single parameter.

Parameters

parameter (string) – Parameter to sweep, must be a member of the Solver class.
n (int) – Number of runs
low (float) – Minimum parameter value
high (float) – Maximum parameter value

Returns

DataFrame of results and list of sections for each run

Return type

tuple

slipstream()¶

For coaxial calculations. Calculates slipstream radius and velocity for the upper rotor according to momentum theory. Currently only the static case is included.

\[\begin{split}r_s = \frac{R}{\sqrt{2}} \\ v_s = C_s\sqrt{\frac{2 T}{\rho A}} \\\end{split}\]

Returns: Radius and velocity of the slipstream
Return type: tuple

solve(rotor, twist, rpm, v_inflow, r_inflow)¶

Find inflow angle and calculate forces for a single rotor given rotational speed, inflow velocity and radius.

Parameters

rotor (Rotor) – Rotor to solve for
twist (float) – Angle to adjust rotor pitch
rpm (float) – Rotations per minute
v_inflow (float) – Inflow velocity
r_inflow (float) – Inflow radius (equal to blade radius for single rotors)

Returns

Calculated thrust, torque and power for the rotor

Return type

tuple

turbine_coeffs(T, Q, P)¶

Dimensionless coefficients for a turbine.

\[\begin{split}\text{TSR} = \frac{\Omega R}{V_\infty} \\ C_T = \frac{2T}{\rho A V_\infty^2} \\ C_P = \frac{2P}{\rho A V_\infty^3} \\\end{split}\]

Parameters

T (float) – Thrust
Q (float) – Torque
P (float) – Power

Returns

Tip-speed ratio, power coefficient and thrust coefficient

Return type

tuple

pybemt.airfoil module¶

Module for holding airfoil data, and providing drag and lift coefficients to the solver.

Airfoil data is stored in the folder pybemt/airfoils. Currently, the Aerodyn format is supported, with only a single airfoil table. Data must be available from -180 to 180 degrees, and a quadratic function is built to interpolate between data in the airfoil table.

This module can also be executed from the command-line to plot drag and lift coefficients for a single airfoil, e.g.

python airfoil.py NACA_4412

class pybemt.airfoil.Airfoil¶

Bases: object

Class for storing airfoil drag and lift coefficients. Should be initialized using the load_airfoil() function.

Cd(alpha)¶

Provide drag coefficent for a given angle of attack.

Parameters: alpha (float) – Angle in radians
Returns: Drag coefficient
Return type: float

Cl(alpha)¶

Provide drag coefficent for a given angle of attack.

Parameters: alpha (float) – Angle in radians
Returns: Drag coefficient
Return type: float

plot(color='k')¶

Plot lift and drag coefficients.

Parameters: color (string) – Matplotlib color key. Default value is k, i.e. black.

pybemt.airfoil.load_airfoil(name)¶

Load airfoil data from data file into an Airfoil object.

Parameters: name (string) – name of the airfoil to load, e.g. ‘NACA_4412’.
Returns: Airfoil object
Return type: pybemt.Airfoil

pybemt.rotor module¶

Module for storing rotor properties and calculation of induction factors and forces for the airfoil sections.

class pybemt.rotor.Rotor(cfg, name, mode)¶

Bases: object

Holds rotor properties and a list of all airfoil sections.

Parameters

cfg (configparser.SafeConfigParser) – Configuration object
name (string) – Name of rotor
mode (string) – Solver mode

precalc(twist)¶

Calculation of properties before each solver run, to ensure all parameters are correct for parameter sweeps.

Returns: None

sections_dataframe()¶

Creates a pandas DataFrame with all calculated section properties.

Returns: DataFrame with section properties
Return type: pd.DataFrame

class pybemt.rotor.Section(airfoil, radius, width, pitch, chord, rotor, mode)¶

Bases: object

Class for calculating induction factors and forces according to the BEM theory for a single airfoil section.

Parameters

airfoil (Airfoil) – Airfoil of the section
radius (float) – Distance from center to mid of section
width (float) – Width of section
pitch (float) – Pitch angle in radians
chord (float) – Chord length of section
rotor (Rotor) – Rotor that section belongs to
mode (string) – Solver mode

airfoil_forces(phi)¶

Force coefficients on an airfoil, decomposed in axial and tangential directions:

\[\begin{split}C_T = C_l\cos{\phi} - CC_d\sin{\phi} \\ C_Q = C_l\sin{\phi} + CC_d\cos{\phi} \\\end{split}\]

where drag and lift coefficients come from airfoil tables.

Parameters: phi (float) – Inflow angle
Returns: Axial and tangential force coefficients
Return type: tuple

forces(phi, v_inf, omega, fluid)¶

Calculation of axial and tangential forces (thrust and torque) on airfoil section.

The definition of blade element theory is used,

\[\begin{split}\Delta T = \sigma\pi\rho U^2C_T r\Delta r \\ \Delta Q = \sigma\pi\rho U^2C_Q r^2\Delta r \\ U = \sqrt{v^2+v'^2} \\ v = (1 + Ca)V_\infty \\ v' = (1 - Ca')\Omega R \\\end{split}\]

Note that this is equivalent to the momentum theory definition,

\[\begin{split}\Delta T = 4\pi\rho r V_\infty^2(1 + Ca)aF\Delta r \\ \Delta Q = 4\pi\rho r^3 V_\infty\Omega(1 + Ca)a'F\Delta r \\\end{split}\]

Parameters

phi (float) – Inflow angle
v_inf (float) – Axial inflow velocity
omega (float) – Tangential rotational velocity
fluid (Fluid) – Fluid

Returns

Axial and tangential forces

Return type

tuple

func(phi, v_inf, omega)¶

Residual function used in root-finding functions to find the inflow angle for the current section.

\[\begin{split}\frac{\sin\phi}{1+Ca} - \frac{V_\infty\cos\phi}{\Omega R (1 - Ca')} = 0\\\end{split}\]

Parameters

phi (float) – Estimated inflow angle
v_inf (float) – Axial inflow velocity
omega (float) – Tangential rotational velocity

Returns

Residual

Return type

float

induction_factors(phi)¶

Calculation of axial and tangential induction factors,

\[\begin{split}a = \frac{1}{\kappa - C} \\ a' = \frac{1}{\kappa' + C} \\ \kappa = \frac{4F\sin^2{\phi}}{\sigma C_T} \\ \kappa' = \frac{4F\sin{\phi}\cos{\phi}}{\sigma C_Q} \\\end{split}\]

Parameters: phi (float) – Inflow angle
Returns: Axial and tangential induction factors
Return type: tuple

precalc()¶

Calculation of properties before each solver run, to ensure all parameters are correct for parameter sweeps.

Returns: None

stall_delay_model(phi, alpha, Cl, Cd)¶

The 3D correction model based on Chaviaropoulos and Hansen ref:

\[Cl_3D = Cl_2D + a (c / r)^h \cos^n{twist} (Cl_inv - Cl_2d) \ Cl_inv = \sqrt{Cl_2d^2 + Cd^2}\]

where: a = 2.2, h = 1.3 and n = 4

Parameters: phi (float) – Inflow angle
Returns: Lift coefficient with 3D correction
Return type: float

tip_loss(phi)¶

Prandtl tip loss factor, defined as

\[\begin{split}F = \frac{2}{\pi}\cos^{-1}e^{-f} \\ f = \frac{B}{2}\frac{R-r}{r\sin\phi}\end{split}\]

A hub loss is also caluclated in the same manner.

Parameters: phi (float) – Inflow angle
Returns: Combined tip and hub loss factor
Return type: float

pybemt.fluid module¶

Module for holding and calculating fluid properties. Currently, only viscosity and density are included.

class pybemt.fluid.Fluid(cfg)¶

Bases: object

Class for loading fluid properties from configuration file and providing them to the solver.

Parameters: cfg (configparser.SafeConfigParser) – Configuration object