I have various pump performance data, and I am trying to fit curves as a combination of them.

In below image, different curves are at different pump speeds (RPM), and the Y axis is Pressure head, and X axis is flow rate.

I am trying to figure out the following functions:

pressure = func (flowrate, rpm)
flowrate = func (pressure, rpm)
rpm = func (flowrate, pressure)

Usually the pump curves follow some form of affinity laws, as

1. flow rate is proportional to rpm.
2. pressure is proportional to square of rpm.

However in reality - they do not, and there will be a difference between experimental data & theoretical data. So that is why I am looking to fit a function that looks like this: (open to other suggestions) - like 3rd degree polynomial as well.

Y = a*x1**2 + b*x1+ c*x2**2 + d*x2 + e*x1*x2 + f

However, I am getting the wrong results with scipy - curve_fit. I am not in favor of employing sklearn regression as we would not be able to figure out the coefficients.

Is there a way one can employ this in Python?

See my code below so far:

from numpy import genfromtxt
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import pandas as pd
# df = pd.read_excel('./nominal_data/pump_curve.xlsx', sheet_name =0)
df = pd.DataFrame({
    'Flow(lpm)': [128.0846942, 105.7579874, 95.11146262, 69.57902038, 53.25344504, 35.47260492, np.nan, 131.96, 110.57, 91.32, 73.02, 53.9, 20.41, np.nan, 116.06, 99.46, 83.7, 68.84, 54.47, 20.98, np.nan, 103.0, 87.6, 73.6, 57.8, 44.0, 19.49, np.nan, 86.2, 73.1, 56.0, 42.6, 28.1, 16.33, np.nan, 56.2, 47.1, 38.6, 30.9, 24.0, 12.3],
    'Speed Feedback (RPM)': [5204.0, 5208.0, 5206.0, 5206.0, 5176.0, 5175.0, np.nan, 4710.72, 4706.4, 4714.93, 4687.11, 4691.0, 4602.0, np.nan, 4103.21, 4115.26, 4147.8, 4148.14, 4141.09, 4124.72, np.nan, 3675.89, 3657.88, 3673.73, 3671.41, 3675.27, 3664.88, np.nan, 3118.66, 3186.23, 3106.92, 3107.19, 3114.69, 3090.08, np.nan, 2077.44, 2073.23, 2062.01, 2069.37, 2068.02, 2067.91],
    'dP (PSI)': [16.5, 25.34, 28.78, 35.45, 37.86, 38.87, np.nan, 8.85, 17.01, 23.42, 27.48, 30.5, 32.4, np.nan, 6.69, 11.84, 17.24, 20.16, 22.64, 25.81, np.nan, 5.2, 9.6, 13.2, 16.3, 18.1, 20.38, np.nan, 3.7, 6.5, 10.0, 12.1, 13.5, 14.54, np.nan, 1.2, 2.7, 3.7, 4.7, 5.2, 6.3]
})
flow_lpm = df['Flow(lpm)'].to_numpy()
rpm = df['Speed Feedback (RPM)'].to_numpy()
dP = df['dP (PSI)'].to_numpy()
X = (flow_lpm, rpm)
y = dP

def f(X, a, b, c, d, e, f):
    x_1, x_2 = X
    return a*x_1**2 + b*x_1 + c*x_2**2 + d*x_2 + e*x_1*x_2 + f
param, param_cov = curve_fit(f, X, y)

# Test a random data point. 
flow_test = 54
rpm_test = 5195

dp_test = param[0]*flow_test**2 + param[1]*flow_test + param[2]*rpm_test**2 + param[3]**rpm_test + param[4]*flow_test*rpm_test + param[5]
print (dp_test)

I could have multiple curves of Pressure = function (flow rate) at multiple RPMs like the figure shows. For example there is a 3rd degree curve for a 5100 RPM curve. However it only works for a given RPM. The curve is different for a 2070 RPM situation.

Seeking suggestions on proper curve fitting methodology. Open to deleting end points, to ensure the curve fit works in the middle range.

datafile_imagefile

I have various pump performance data, and I am trying to fit curves as a combination of them.

In below image, different curves are at different pump speeds (RPM), and the Y axis is Pressure head, and X axis is flow rate.

I am trying to figure out the following functions:

pressure = func (flowrate, rpm)
flowrate = func (pressure, rpm)
rpm = func (flowrate, pressure)

Usually the pump curves follow some form of affinity laws, as

1. flow rate is proportional to rpm.
2. pressure is proportional to square of rpm.

However in reality - they do not, and there will be a difference between experimental data & theoretical data. So that is why I am looking to fit a function that looks like this: (open to other suggestions) - like 3rd degree polynomial as well.

Y = a*x1**2 + b*x1+ c*x2**2 + d*x2 + e*x1*x2 + f

However, I am getting the wrong results with scipy - curve_fit. I am not in favor of employing sklearn regression as we would not be able to figure out the coefficients.

Is there a way one can employ this in Python?

See my code below so far:

from numpy import genfromtxt
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import pandas as pd
# df = pd.read_excel('./nominal_data/pump_curve.xlsx', sheet_name =0)
df = pd.DataFrame({
    'Flow(lpm)': [128.0846942, 105.7579874, 95.11146262, 69.57902038, 53.25344504, 35.47260492, np.nan, 131.96, 110.57, 91.32, 73.02, 53.9, 20.41, np.nan, 116.06, 99.46, 83.7, 68.84, 54.47, 20.98, np.nan, 103.0, 87.6, 73.6, 57.8, 44.0, 19.49, np.nan, 86.2, 73.1, 56.0, 42.6, 28.1, 16.33, np.nan, 56.2, 47.1, 38.6, 30.9, 24.0, 12.3],
    'Speed Feedback (RPM)': [5204.0, 5208.0, 5206.0, 5206.0, 5176.0, 5175.0, np.nan, 4710.72, 4706.4, 4714.93, 4687.11, 4691.0, 4602.0, np.nan, 4103.21, 4115.26, 4147.8, 4148.14, 4141.09, 4124.72, np.nan, 3675.89, 3657.88, 3673.73, 3671.41, 3675.27, 3664.88, np.nan, 3118.66, 3186.23, 3106.92, 3107.19, 3114.69, 3090.08, np.nan, 2077.44, 2073.23, 2062.01, 2069.37, 2068.02, 2067.91],
    'dP (PSI)': [16.5, 25.34, 28.78, 35.45, 37.86, 38.87, np.nan, 8.85, 17.01, 23.42, 27.48, 30.5, 32.4, np.nan, 6.69, 11.84, 17.24, 20.16, 22.64, 25.81, np.nan, 5.2, 9.6, 13.2, 16.3, 18.1, 20.38, np.nan, 3.7, 6.5, 10.0, 12.1, 13.5, 14.54, np.nan, 1.2, 2.7, 3.7, 4.7, 5.2, 6.3]
})
flow_lpm = df['Flow(lpm)'].to_numpy()
rpm = df['Speed Feedback (RPM)'].to_numpy()
dP = df['dP (PSI)'].to_numpy()
X = (flow_lpm, rpm)
y = dP

def f(X, a, b, c, d, e, f):
    x_1, x_2 = X
    return a*x_1**2 + b*x_1 + c*x_2**2 + d*x_2 + e*x_1*x_2 + f
param, param_cov = curve_fit(f, X, y)

# Test a random data point. 
flow_test = 54
rpm_test = 5195

dp_test = param[0]*flow_test**2 + param[1]*flow_test + param[2]*rpm_test**2 + param[3]**rpm_test + param[4]*flow_test*rpm_test + param[5]
print (dp_test)

I could have multiple curves of Pressure = function (flow rate) at multiple RPMs like the figure shows. For example there is a 3rd degree curve for a 5100 RPM curve. However it only works for a given RPM. The curve is different for a 2070 RPM situation.

Seeking suggestions on proper curve fitting methodology. Open to deleting end points, to ensure the curve fit works in the middle range.

datafile_imagefile

• python
• scipy
• regression
• curve-fitting

• Why don't you fit a quadratic like P = aQ^2 + bQ +c (where P is pressure or head and Q is flow rate) for each rotation rate N. Then you can separately fit coefficients a, b and c as functions of N. BTW, your understanding of the affinity laws for speed-scaling is not entirely accurate. – lastchance Commented Mar 6 at 22:52
• I added the data from the link into the post, using df.to_dict(orient='list'). This is makes the post more useful to future users in case the link breaks at some point. – Nick ODell Commented Mar 7 at 17:09
• "I am not in favor of employing sklearn regression as we would not be able to figure out the coefficients." You can get those coefficients using LinearRegression.coef_: scikit-learn./stable/modules/generated/… – Nick ODell Commented Mar 7 at 17:22

2 Answers 2

With the following dataset you provided:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import optimize

data = pd.DataFrame({
    'Flow(lpm)': [128.0846942, 105.7579874, 95.11146262, 69.57902038, 53.25344504, 35.47260492, np.nan, 131.96, 110.57, 91.32, 73.02, 53.9, 20.41, np.nan, 116.06, 99.46, 83.7, 68.84, 54.47, 20.98, np.nan, 103.0, 87.6, 73.6, 57.8, 44.0, 19.49, np.nan, 86.2, 73.1, 56.0, 42.6, 28.1, 16.33, np.nan, 56.2, 47.1, 38.6, 30.9, 24.0, 12.3],
    'Speed Feedback (RPM)': [5204.0, 5208.0, 5206.0, 5206.0, 5176.0, 5175.0, np.nan, 4710.72, 4706.4, 4714.93, 4687.11, 4691.0, 4602.0, np.nan, 4103.21, 4115.26, 4147.8, 4148.14, 4141.09, 4124.72, np.nan, 3675.89, 3657.88, 3673.73, 3671.41, 3675.27, 3664.88, np.nan, 3118.66, 3186.23, 3106.92, 3107.19, 3114.69, 3090.08, np.nan, 2077.44, 2073.23, 2062.01, 2069.37, 2068.02, 2067.91],
     'dP (PSI)': [16.5, 25.34, 28.78, 35.45, 37.86, 38.87, np.nan, 8.85, 17.01, 23.42, 27.48, 30.5, 32.4, np.nan, 6.69, 11.84, 17.24, 20.16, 22.64, 25.81, np.nan, 5.2, 9.6, 13.2, 16.3, 18.1, 20.38, np.nan, 3.7, 6.5, 10.0, 12.1, 13.5, 14.54, np.nan, 1.2, 2.7, 3.7, 4.7, 5.2, 6.3]
})

Remove nans:

data = data.dropna()

Define your model as follow:

def model(x, a, b, c, d, e, f):
    return a * x[:,0] ** 2 + b * x[:,0] + c * x[:,1] ** 2 + d * x[:,1] + e * x[:,0] * x[:,1] + f

Then optimize:

popt, pcov = optimize.curve_fit(model, data.iloc[:,:2].values, data.iloc[:,2].values)

Now we can regress the model on a finer grid:

x0lin = np.linspace(data.iloc[:,0].min(), data.iloc[:,0].max(), 200)
x1lin = np.linspace(data.iloc[:,1].min(), data.iloc[:,1].max(), 200)
X0, X1 = np.meshgrid(x0lin, x1lin)

Z = model(np.array([X0.ravel(), X1.ravel()]).T, *popt).reshape(X0.shape)

It renders as follow:

fig, axe = plt.subplots(subplot_kw={"projection": "3d"})
axe.scatter(*data.values.T)
axe.plot_surface(X0, X1, Z, cmap="viridis", alpha=0.35)

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

df = pd.read_excel('pump_curve.xlsx', sheet_name =0)
columns = ['Flow(lpm)', 'Speed Feedback (RPM)', 'dP (PSI)']
df = df[columns].dropna()
columns2 = ['lpm', 'rpm', 'psi']
df = df.rename(columns=dict(zip(columns, columns2)))

def f(X, a, b, c, d, e, f):
    x_1, x_2 = X
    return a*x_1**2 + b*x_1 + c*x_2**2 + d*x_2 + e*x_1*x_2 + f

fig, ax = plt.subplots(1, 3, figsize=(15, 5))
for idx, name in enumerate(columns2):
    y = df[name].to_numpy()
    name1, name2 = columns2[:idx]+columns2[idx+1:]
    X = (df[name1].to_numpy(), df[name2].to_numpy())
    param, param_cov = curve_fit(f, X, y)
    ax[idx].plot(f(X, *param), label='prediction');
    ax[idx].plot(y, label='data');
    ax[idx].legend();
    ax[idx].set_title(f'{name} = f({name1}, {name2})');
plt.show();