import marimo

__generated_with = "0.25.1"
app = marimo.App()


@app.cell
def _():
    import marimo as mo

    return (mo,)


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    # GUM Budgets: Orifice Flow Meter and Storage Tank
    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    ## 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.
    """)
    return
@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    # GUM Budgets: Orifice Flow Meter and Storage Tank
    """)
    return




@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    <table>
        <tr>
            <td><img src="../images/orifice_flowmeter.jpg" alt="Orifice Flowmeter" width="300"/></td>
            <td><img src="../images/orifice_scheme.png" alt="Orifice Scheme" width="300"/></td>
        </tr>
    </table>
    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    [](../images/orifice_flowmeter.jpg)
    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    ### 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 $d_1$ ($\beta$):**

    $$\beta = \frac{d_2}{d_1}$$

    **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:

    $$Q = \frac{C}{\sqrt{1 - \beta^4}} \epsilon \frac{\pi}{4} d_1^2 \sqrt{2 \rho \Delta p}$$

    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 $\text{Pa}$)
    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    ### 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).

    | Quantity ($X_i$) | Symbol | Value ($Y$) | Units | Expanded Uncertainty ($U$) | Coverage Factor ($k$) |
    | :--- | :---: | :---: | :---: | :---: | :---: |
    | Discharge coefficient | $C$ | $0.6$ | - | $0.003$ | $2.0$ |
    | Expansibility factor | $\epsilon$ | $0.997$ | - | $0.00027$ | $2.0$ |
    | Pipe diameter | $d_1$ | $0.5$ | $\text{m}$ | $0.0001$ | $1.73$ |
    | Orifice diameter | $d_2$ | $0.3$ | $\text{m}$ | $0.00001$ | $1.73$ |
    | Pressure drop | $\Delta p$ | $50.0$ | $\text{kPa}$ | $100$ | $2.0$ |
    | Density | $\rho$ | $48.7$ | $\text{kg/m}^3$ | $0.146$ | $2.0$ |

    #### Python Setup and Variable Initialization

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

    ```python
    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 $Q$.

    ```python
    # 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:

    $$u_c(Q) = \sqrt{\sum_{i=1}^{N} (c_i u_i)^2}$$

    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.

    | Quantity | Value ($Y$) | $U$ | $k$ | $u = U/k$ | $c$ | $u \cdot c$ | $(u \cdot c)^2$ |
    | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
    | $C$ | $0.6$ | $0.003$ | $2$ | $0.0015$ | $167$ | $0.25$ | $0.0628$ |
    | $\epsilon$ | $0.997$ | $0.00027$ | $2$ | $0.00013$ | $100$ | $0.0135$ | $0.0002$ |
    | $d_1$ | $0.5$ | $0.0001$ | $1.73$ | $5.78\text{E-}5$ | $-60$ | $-0.0035$ | $1.20\text{E-}5$ |
    | $d_2$ | $0.3$ | $0.00001$ | $1.73$ | $5.78\text{E-}6$ | $766$ | $0.0044$ | $1.96\text{E-}5$ |
    | $\Delta p$ | $50,000$ | $100$ | $2$ | $50$ | $0.001$ | $0.05$ | $0.0025$ |
    | $\rho$ | $48.7$ | $0.146$ | $2$ | $0.07305$ | $1.03$ | $0.075$ | $0.0057$ |
    | **$Q$** | **$100.0$** | **$0.533$** | **$2$** | **$0.267$** | **$1$** | **$0.267$** | **$0.0711$** |

    ### Calculations based on the Table

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

    ```python
    # --- 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 ($V$):**

    $$V = \frac{\pi d^2 h}{4}$$

    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 $V_{\text{meas}}$.

    $$V_{15} = \frac{V_{\text{meas}}}{(1 + \alpha(T_{\text{meas}} - 15))}$$

    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**.

    | Source | $U$ | $k$ | $u$ | $c$ | $u \cdot c$ | $(u \cdot c)^2$ |
    | :--- | :---: | :---: | :---: | :---: | :---: | :---: |
    | Calibration (Diameter) | $0.048$ | $2.00$ | $0.024$ | $39.96$ | $0.955$ | $0.920$ |
    | Determination (Diameter) | - | - | $0.021$ | $39.96$ | $0.839$ | $0.704$ |
    | Time drift (Diameter) | $0.005$ | $1.73$ | $0.003$ | $39.96$ | $0.115$ | $0.013$ |
    | Calibration (Height) | $0.100$ | $2.00$ | $0.050$ | $18.10$ | $0.905$ | $0.819$ |
    | Resolution (Height) | $0.026$ | $1.73$ | $0.015$ | $18.10$ | $0.272$ | $0.074$ |
    | Time drift (Height) | $0.010$ | $1.73$ | $0.006$ | $18.10$ | $0.104$ | $0.011$ |
    | Temperature ($T$) | $0.550$ | $2.00$ | $0.275$ | $-0.004$ | $-0.001$ | $1.21\text{E-}6$ |
    | Cubical Expansion ($\alpha$) | $2.25\text{E-}6$ | $1.73$ | $1.3\text{E-}6$ | $-1437$ | $-0.002$ | $3.49\text{E-}6$ |
    | **Volume ($V$)** | **$3.188$** | **$2.00$** | **$1.594$** | **$1$** | **$1.594$** | **$2.541$** |

    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 $2.541$).

    $$u_c(V)_{\text{uncorr}} = \sqrt{2.541} \approx 1.594$$

    The Expanded Uncertainty $U$ is then calculated with $k=2.00$:
    $$U_{\text{uncorr}} = 2.00 \times 1.594 \approx 3.188$$

    ### 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 $r_{ij}$:

    $$u_c^2(Y) = \sum_{i=1}^{N} (c_i u_i)^2 + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} c_i u_i c_j u_j r_{ij}$$

    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.

    | Source | $U$ | $k$ | $u$ | $c$ | $u \cdot c$ | $(u \cdot c)^2$ |
    | :--- | :---: | :---: | :---: | :---: | :---: | :---: |
    | Calibration (Diameter) | $0.048$ | $2.00$ | $0.024$ | $39.96$ | $\mathbf{0.955}$ | - |
    | ... | ... | ... | ... | ... | ... | ... |
    | Calibration (Height) | $0.100$ | $2.00$ | $0.050$ | $18.10$ | $\mathbf{0.905}$ | $\mathbf{3.475}$ |
    | ... | ... | ... | ... | ... | ... | ... |
    | **Volume ($V$)** | **$4.136$** | **$2.00$** | **$2.068$** | **$1$** | **$2.068$** | **$4.277$** |

    **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.
    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
 
    """)
    return


if __name__ == "__main__":
    app.run()