Blower door building air leakage test

Building airtightness is descibed by power law relation between pressure difference and air flow rate,

\[\dot{V} = K \cdot (\Delta p)^n\]

where:

• \(\dot{V}\) is the volumetric leakage airflow rate, m³·h⁻¹;

• \(\Delta p\) - pressure difference across the building envelope, Pa;

• \(n\) - airflow exponent (0.5 ≤ \(n\) ≤ 1);

• \(K\) - air leakage coefficient, m³·h⁻¹·Pa⁻ⁿ.

The airflow exponent \(n\) and the air leakage coefficient \(K\) can be measured experimentally from data data obtained in a blower door test for air leakage.

This notebook demonstrates how to determine the coefficients \(K\) and \(n\) from the power law of flow through an orifice

\[\dot{V} = K \cdot (\Delta p)^n\]

using linear regression on log-transformed of experimental data obtained in a blower door test for air leakage. We also calculate the uncertainties in \(K\) and \(n\).

Measured data

The measured data consists of pressure differences, \(\Delta p\), and corresponding volumetric flow rates, \(\dot{V}\).

Test	\(\Delta p\) / (Pa)	\(\dot{V}\) / (m³·h⁻¹)
0	0	0
1	10	237
2	20	364
3	30	444
4	40	625
5	50	688
6	60	704

Estimation of coefficients

Step 1: Linearize the equation

The equation \(\dot{V} = K \cdot (\Delta p)^n\) is linearized by taking \(\log_{10}\) of both sides:

\[ \log_{10}(\dot{V}) = \log_{10}(K) + n \cdot \log_{10}(\Delta p) \]

This is of the form \(y = a x + b\), where:

• \(y = \log_{10}(\dot{V})\)

• \(x = \log_{10}(\Delta p)\)

• \(b = \log_{10}(K)\)

• \(a = n\)

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Step 2: Perform linear regression

We use scipy.stats.linregress to perform simple linear regression on the log-transformed data. The regression provides the slope, the intercept and the confidence intervals:

• Slope \(a\): represents \(n\).

• Intercept \(b\): represents \(\log_{10}(K)\).

• Standard error of the slope \(\sigma_a\): uncertainty in \(n\).

• Standard error of the intercept \(\sigma_b\): uncertainty in \(\log_{10}(K)\).

---

Step 3: Extract parameters

From the regression results:

• \(n = a\)

• \(K = 10^b\)

---

Step 4: Calculate uncertainties

Uncertainty in n

The uncertainty in \(n\) is the standard error of the slope:

\[ \sigma_n = \sigma_a \]

Uncertainty in K

The uncertainty in \(K\):

\[ \sigma_K = K \cdot \ln(10) \cdot \sigma_b \]

is calculated using simplified formula for propagation of uncertainty for a function of a single variable.

If \(K = f(b)\), and \(b\) has the uncertainty \(\sigma_b\), then the uncertainty in \(K\), \(\sigma_K\), is given by:

\[ \sigma_K = \left| \frac{df}{db} \right| \cdot \sigma_b \]

where:

• \(b\) is the intercept from the linear regression,

• \(\sigma_b\) is its standard error (uncertainty) of intercept \(b\).

The relationship between \(K\) and the intercept \(b\) is given by:

\[ K = 10^b \]

Step 1: Compute the derivative of \(K\) with respect to \(b\)

The function \(K = 10^b\) can be rewritten using the natural logarithm:

\[ K = e^{b \cdot \ln(10)} \]

The derivative of \(K\) with respect to \(b\) is:

\[ \frac{dK}{db} = \ln(10) \cdot e^{b \cdot \ln(10)} = \ln(10) \cdot 10^b = \ln(10) \cdot K \]

Step 2: Apply the error propagation formula

Substitute the derivative into the error propagation formula:

\[ \sigma_K = \left| \frac{dK}{db} \right| \cdot \sigma_b = \left| \ln(10) \cdot K \right| \cdot \sigma_b \]

Since \(K\) and \(\ln(10)\) are positive, the absolute value can be dropped:

\[ \sigma_K = K \cdot \ln(10) \cdot \sigma_b \]

where:

• \(K\): value calculated from \(K = 10^b\).

• \(\ln(10)\): constant factor arising from the derivative of \(10^b\).

• \(\sigma_b\): standard error of the intercept \(b\), which quantifies the uncertainty in \(b\).

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Given data
delta_p = np.array([0, 10, 20, 30, 40, 50, 60])        # Pa
V_dot = np.array([0, 237, 364, 444, 625, 688, 704])    # m³/h

# Linearize the equation using log10 (exclude the (0, 0) point)
log_delta_p = np.log10(delta_p[1:])
log_V_dot = np.log10(V_dot[1:])

# Perform linear regression
result = stats.linregress(log_delta_p, log_V_dot)

a = result.slope
b = result.intercept
sigma_a = result.stderr
sigma_b = result.intercept_stderr

r_square = result.rvalue**2

# Extract parameters
n = a
K = 10**b
sigma_n = sigma_a
sigma_K = K * np.log(10) * sigma_b

# Output results
print(f"n = {n:.4f} ± {sigma_n:.4f}")
print(f"K = ({K:.4f} ± {sigma_K:.4f}) m³·h⁻¹·Pa⁻ⁿ")
print(f"R² = {r_square:.4f}")

n = 0.6434 ± 0.0442
K = (53.3956 ± 8.1475) m³·h⁻¹·Pa⁻ⁿ
R² = 0.9815

# Plot
# =====
# Create a range of delta_p values for plotting the fitted curve
# Range of delta_p values, including 0
delta_p_range = np.linspace(0, 60, 100)
# Corresponding log10 values, excluding 0
log_delta_p_range = np.log10(delta_p_range[1:])

# Calculate the fitted curve and regression line
# Fitted curve: V_dot = K * delta_p^n
V_dot_fitted = K * delta_p_range**n

# Regression line: log10(V_dot) = log10(K) + n * log10(delta_p)
log_V_dot_fitted = b + a * log_delta_p_range

# Uncertainty bounds in the fitted curve
V_dot_upper = (K + sigma_K) * delta_p_range**(n + sigma_n)
V_dot_lower = (K - sigma_K) * delta_p_range**(n - sigma_n)

# Uncertainty bounds n the regression line
log_V_dot_upper = (b + sigma_b) + (a + sigma_a) * log_delta_p_range
log_V_dot_lower = (b - sigma_b) + (a - sigma_a) * log_delta_p_range

# Create the plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# First pane: Original data and fitted curve with uncertainty
ax1.scatter(delta_p, V_dot,
            color='blue',
            label='Measured data')
ax1.plot(delta_p_range, V_dot_fitted,
         color='red',
         label=f'Fitted curve: $\dot{{V}} = K \cdot \Delta p^n$\n'
         f'$K = {K:.2f} \pm {sigma_K:.2f}$, $n = {n:.2f} \pm {sigma_n:.2f}$')
ax1.fill_between(delta_p_range, V_dot_lower, V_dot_upper,
                 color='red', alpha=0.2,
                 label='Uncertainty in fitted curve')
ax1.set_xlabel('$\Delta p$ / (Pa)')
ax1.set_ylabel('$\dot{V}$ / (m³·h⁻¹)')
ax1.set_title('Original data and fitted curve')
ax1.legend()
ax1.grid(True)

# Second pane: Linearized data and regression line with uncertainty
ax2.scatter(log_delta_p, log_V_dot,
            color='green',
            label='Linearized data')
ax2.plot(log_delta_p_range, log_V_dot_fitted, 
         color='orange',
         label=f'Regression line: '
         f'$lg(\dot{{V}}) = lg(K) + n \cdot lg(\Delta p)$\n'
         f'$lg(K) = {b:.2f} \pm {sigma_b:.2f}$, $n = {a:.2f} \pm {sigma_a:.2f}$'
         f'\n$R^2 = {r_square:.4f}$\n')
ax2.fill_between(log_delta_p_range, log_V_dot_lower, log_V_dot_upper, 
                 color='orange', alpha=0.2,
                 label='Uncertainty in regression line')
ax2.set_xlabel('$\log_{10}(\Delta p$ / (Pa))')
ax2.set_ylabel('$\log_{10}(\dot{V}$ / (m³·h⁻¹))')
ax2.set_title('Linearized data and regression line')
ax2.legend()
ax2.grid(True)

# Show the plot
plt.tight_layout()
plt.show()

References

Joint Committee for Guides in Metrology. (2008). Evaluation of measurement data—Guide to the expression of uncertainty in measurement. JCGM, 100(2008), 1-116.