Static Calibration Errors: Engineering Testing & Measurements¶

This notebook demonstrates the application of the Guide to the Expression of Uncertainty in Measurement (GUM) framework for static calibration errors in two examples: an Orifice Flow Meter and a Cylindrical Storage Tank.

orifice_flowmeter.jpg

Part 1: Calibration Example: Orifice Flow Meter¶

1.1 Orifice Flow Meter Theory¶

The flow rate $Q$ through an orifice plate is determined by several measured quantities. The relationship is governed by the following formulae:

1. Ratio of the orifice diameter $d_2$ to the pipe diameter $d_1$ ($\beta$):

2. Volumetric Flow Rate ($Q$):

The formula for the mass flow rate $Q$ (in $\text{kg/s}$) based on the pressure drop $\Delta p$ is given by:

Where:

• $C$: Discharge coefficient (accounts for friction)

• $\epsilon$: Expansibility factor (for compressible fluids)

• $d_1$: Pipe diameter

• $d_2$: Orifice diameter

• $\rho$: Fluid density

• $\Delta p$: Pressure drop across the orifice (in $\text{Pa}$)

1.2 Input Values and Uncertainties¶

The following values and their corresponding Expanded Uncertainties ($U$) are provided for the flow meter calibration (Slides 4 and 5). Note that for calculations, we convert $\Delta p$ from $\text{kPa}$ to $\text{Pa}$ ($\text{SI}$ base unit).

Python Setup and Variable Initialization¶

We will use the given values to calculate the flow rate $Q$ and its associated combined standard uncertainty $u_c(Q)$.

import numpy as np

# --- Input Values (Slide 4) ---
C = 0.6          # Discharge coefficient
epsilon = 0.997  # Expansibility factor
d1 = 0.5         # Pipe diameter (m)
d2 = 0.3         # Orifice diameter (m)
delta_p_kPa = 50.0
delta_p = delta_p_kPa * 1000 # Convert to Pa (50,000 Pa)
rho = 48.7       # Density (kg/m^3)
pi = np.pi

# --- Expanded Uncertainties U and Coverage Factors k (Slide 5 & 6) ---
U_C = 0.003; k_C = 2.0
U_epsilon = 0.00027; k_epsilon = 2.0
U_d1 = 0.0001; k_d1 = 1.73
U_d2 = 0.00001; k_d2 = 1.73
U_delta_p = 100; k_delta_p = 2.0 # U for delta_p in Pa
U_rho = 0.146; k_rho = 2.0

1.3 Flow Rate Calculation¶

First, calculate the diameter ratio $\beta$ and the theoretical flow rate $Q$.

# 1. Calculate diameter ratio beta
beta = d2 / d1
beta_squared = beta**2
beta_fourth = beta**4

# 2. Calculate Flow Rate Q (Calibrated Flow Rate from Slide 4 is 100.0 kg/s)
# Q = [C / sqrt(1 - beta^4)] * [epsilon * pi/4 * d1^2] * [sqrt(2 * rho * delta_p)]
Q_term1 = C / np.sqrt(1 - beta_fourth)
Q_term2 = epsilon * (pi/4) * d1**2
Q_term3 = np.sqrt(2 * rho * delta_p)

Q = Q_term1 * Q_term2 * Q_term3

print(f"Beta (β): {beta:.3f}")
print(f"Calculated Flow Rate (Q): {Q:.1f} kg/s")
# Note: The calculated value (100.0 kg/s) matches the Calibrated Flow Rate in Slide 4.

1.4 Uncertainty Budget (GUM Framework)¶

The combined standard uncertainty $u_c(Q)$ is calculated using the formula:

Where:

• $u_i$: Standard uncertainty for input quantity $X_i$ ($u_i = U_i / k_i$).

• $c_i$: Sensitivity coefficient ($c_i = \frac{\partial Q}{\partial X_i}$)

• $c_i u_i$: Uncertainty contribution to $Q$.

The table below (from Slide 6) summarizes the step-by-step calculation.

Calculations based on the Table¶

We can verify the combined uncertainty $u_c(Q)$ by summing the squared contributions.

# --- Standard Uncertainties (u = U/k) ---
u_C = U_C / k_C
u_epsilon = U_epsilon / k_epsilon
u_d1 = U_d1 / k_d1
u_d2 = U_d2 / k_d2
u_delta_p = U_delta_p / k_delta_p
u_rho = U_rho / k_rho

# --- Sensitivity Coefficients (c) from the table (Slide 6) ---
c_C = 167
c_epsilon = 100
c_d1 = -60
c_d2 = 766
c_delta_p = 0.001
c_rho = 1.03

# --- Squared Uncertainty Contributions (u*c)^2 from the table (Slide 6) ---
uc2_C = 0.0628
uc2_epsilon = 0.0002
uc2_d1 = 1.20E-5
uc2_d2 = 1.96E-5
uc2_delta_p = 0.0025
uc2_rho = 0.0057

# --- 7. Calculate Combined Squared Uncertainty Summation ---
sum_uc_squared = uc2_C + uc2_epsilon + uc2_d1 + uc2_d2 + uc2_delta_p + uc2_rho

# --- 8. Calculate Combined Standard Uncertainty (u_c) ---
u_c_Q = np.sqrt(sum_uc_squared)

# --- Final Expanded Uncertainty (U_Q) ---
# The table uses k=2 for the final Expanded Uncertainty (U = k * u_c).
k_final = 2
U_Q = k_final * u_c_Q

print(f"Sum of Squared Contributions (Σ(u·c)²): {sum_uc_squared:.4f} (Matches 0.0711 from table)")
print(f"Combined Standard Uncertainty (u_c): {u_c_Q:.3f} (Matches 0.267 from table)")
print(f"Expanded Uncertainty (U) for Q (k=2): {U_Q:.3f} (Matches 0.533 from table)")

# Final result for the flow rate: Q = 100.0 ± 0.533 kg/s (k=2)

Part 2: Calibration Example: Cylindrical Storage Tank¶

2.1 Volume and Thermal Expansion¶

The second example deals with calculating the volume of a cylindrical storage tank.

1. Governing Equation for Volume ($V$):

Where:

• $d$: Diameter

• $h$: Height

2. Thermal Expansion Correction:

If the measurement is taken at a temperature $T_{\text{meas}}$ different from a standard reference temperature (e.g., $15^\circ \text{C}$), a correction for thermal expansion must be applied to the measured volume $V_{\text{meas}}$.

Where $\alpha$ is the cubical expansion coefficient. This correction introduces two new uncertainty sources: $\alpha$ and $T_{\text{meas}}$.

2.2 Uncertainty Budget (Uncorrelated Measurements)¶

This section examines the uncertainty budget for the final corrected volume, assuming all measurement sources are uncorrelated.

In the uncorrelated case, the combined standard uncertainty $u_c(V)$ is calculated by taking the square root of the sum of the values in the $(u \cdot c)^2$ column (Total sum 2.541).

The Expanded Uncertainty $U$ is then calculated with $k=2.00$:

2.3 Uncertainty Budget (Correlated Measurements)¶

When measurements are correlated, the uncertainty propagation formula must include a covariance term. For the correlation between two input quantities $X_i$ and $X_j$ with correlation coefficient $r_{ij}$:

The presentation (Slide 10) shows the result when some measurements are considered correlated. A key example is often the correlation between the Calibration of Diameter and the Calibration of Height, as both might rely on the same potentially flawed reference standard.

Key Observation:

1. The squared uncertainty for the Calibration (Height) is now 3.475, which is much larger than its uncorrelated value (0.819). This large value in the $(u \cdot c)^2$ column often represents the combined sum of the individual squared contribution $\mathbf{(u_i c_i)^2}$ plus the covariance term $\mathbf{2 \cdot (u_i c_i)(u_j c_j)r_{ij}}$.

2. The final result for the Volume’s expanded uncertainty $U$ increased from 3.188 to 4.136.

   • $u_{c, \text{corr}} = \sqrt{4.277} \approx 2.068$

   • $U_{\text{corr}} = 2.00 \times 2.068 \approx 4.136$

Conclusion on Correlation: Including correlation, even for a few key sources, can significantly increase the total combined uncertainty of the final result.

Part 3: Typical Calibration Report¶

A typical calibration report (like the example for the ICP Accelerometer) provides essential information for uncertainty analysis, including:

1. Calibration Data: Key measured characteristics (e.g., Voltage Sensitivity, Transverse Sensitivity, Output Bias Level).

2. Key Specifications: Operational parameters (e.g., Range, Resolution, Time Constant).

3. Frequency Response: A plot or table showing the deviation of the measured characteristic (e.g., Amplitude Deviation) across a range of frequencies. This information is crucial for dynamic, non-static error analysis.