1.3 Numerical Methods and Measurement Error

Measurement Error and Propagation

In chemical engineering practice, physical quantities are measured with inherent uncertainties, and complex design equations often lack analytical solutions. Therefore, understanding measurement error, error propagation, and numerical methods is crucial. The FE Chemical Exam tests these concepts under the heading of numerical methods and measurement error, drawing on specific formulas from the FE Reference Handbook.

Accuracy vs. Precision

• Accuracy: How close a measured value is to the true or accepted reference value.
• Precision: How reproducible or close measurements are to each other, regardless of the true value.
• Significant Figures: Rules ensure that calculations do not imply more precision than the source data. For addition and subtraction, the result is rounded to the same number of decimal places as the least precise measurement. For multiplication and division, the result is rounded to the same number of significant figures as the term with the fewest significant figures.
• Types of Error: Systematic errors (bias) shift all measurements in one direction (e.g., poorly calibrated sensors or thermocouple zero drift). Random errors (noise) cause scatter around the mean value and are minimized by averaging multiple measurements. The standard deviation of the sample mean decreases with the square root of the number of measurements: $\sigma_{\bar{x}} = \sigma / \sqrt{N}$.

The table below summarizes the key differences between systematic and random errors:

Error Propagation

If a calculated value $f$ is a function of independent, measured variables $x, y, z, \dots$, each with an associated standard deviation (uncertainty) $\sigma_x, \sigma_y, \sigma_z, \dots$, the propagated uncertainty $\sigma_f$ is calculated using the partial derivatives of the function:

$\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 \sigma_x^2 + \left(\frac{\partial f}{\partial y}\right)^2 \sigma_y^2 + \left(\frac{\partial f}{\partial z}\right)^2 \sigma_z^2 + \dots}$

This formula represents the root-sum-square of the individual contribution of each variable to the total error. It assumes that the measurements are independent and that the uncertainties are small relative to the values of the variables.

Taylor's Series Expansion

Taylor's series represents a differentiable function as an infinite sum of terms calculated from the values of its derivatives at a single point. Expanding a function $f(x)$ about a point $x = a$ is defined as:

$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n(x)$

where $R_n(x)$ is the remainder or truncation error. When $a = 0$, the expansion is called a Maclaurin series. On the exam, Taylor's series is used to linearize non-linear algebraic equations near an operating point or to approximate transcendental functions like $e^x$, $\sin x$, or $\ln(1+x)$. Linearization is a key step in control system design, where non-linear differential equations representing process equipment are approximated by linear equations to allow the use of Laplace transfer functions.

Curve Fitting and Least-Squares Regression

Curve fitting determines the functional relationship between independent and dependent variables.

Linear Regression

For a set of $n$ data points $(x_i, y_i)$, the method of least squares finds the best-fit line $y = mx + b$ by minimizing the sum of squared residuals. The formulas for the slope $m$ and intercept $b$ are:

$m = \frac{n \sum_{i=1}^n (x_i y_i) - \sum_{i=1}^n x_i \sum_{i=1}^n y_i}{n \sum_{i=1}^n x_i^2 - \left(\sum_{i=1}^n x_i\right)^2}$ $b = \bar{y} - m\bar{x}$

where $\bar{x}$ and $\bar{y}$ are the sample means of $x$ and $y$. The correlation coefficient $r$ indicates the strength of the linear relationship, and the coefficient of determination $R^2 = r^2$ represents the proportion of variance in the dependent variable explained by the model.

In chemical engineering, non-linear relationships are frequently linearized using logarithmic transformations. For example, a power-law relationship $y = a x^b$ can be linearized as $\ln y = \ln a + b \ln x$. Linear regression can then be performed on the log-transformed variables to determine the parameters $a$ and $b$.

Newton-Raphson Method

The Newton-Raphson method is a powerful iterative algorithm used to find the roots of a non-linear equation $f(x) = 0$.

Iteration Formula

Starting with an initial guess $x_0$, the subsequent approximation $x_{n+1}$ is determined by the intersection of the tangent line at $x_n$ with the x-axis:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Convergence and Limitations

The method converges quadratically when the initial guess is close to the root. However, it can fail under specific conditions:

1. Zero Derivative: If $f'(x_n) = 0$, the denominator becomes zero, indicating a horizontal tangent line that never intersects the x-axis.
2. Oscillation/Cycling: The algorithm can oscillate between two or more values if the function has local extrema between the values.
3. Divergence: If the initial guess is far from the root, the iterations may diverge or converge to a different root.

In chemical engineering, the Newton-Raphson method is commonly used to solve the Colebrook equation for pipe friction factors or to perform bubble point and dew point calculations in vapor-liquid equilibrium (VLE) modeling.

Fourier Series

Fourier series represent periodic functions as a sum of simple sines and cosines. A periodic function $f(t)$ with period $T$ (where $\omega_0 = 2\pi / T$) can be written as:

$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t) \right]$

The Fourier coefficients are calculated using Euler's formulas:

$a_0 = \frac{2}{T} \int_0^T f(t) dt$ $a_n = \frac{2}{T} \int_0^T f(t) \cos(n \omega_0 t) dt$ $b_n = \frac{2}{T} \int_0^T f(t) \sin(n \omega_0 t) dt$

Fourier series are used to analyze heat conduction in solids subjected to periodic temperature variations (e.g., diurnal heating of soil or reactor walls) and are essential in signal processing for filtering noise from process measurements.

Worked Example: Newton-Raphson Root Finding

In fluid mechanics, the friction factor $f$ for turbulent flow in a pipe is solved using the Colebrook equation. Let us solve a simplified version $g(x) = x^3 - x - 1 = 0$ using the Newton-Raphson method with an initial guess $x_0 = 1.5$. Find the value of $x_2$ after two iterations.

Solution

First, compute the first derivative of the function:

$g'(x) = 3x^2 - 1$

First Iteration ($n=0$):

Evaluate the function and its derivative at $x_0 = 1.5$:

$g(1.5) = (1.5)^3 - 1.5 - 1 = 3.375 - 1.5 - 1 = 0.875$ $g'(1.5) = 3(1.5)^2 - 1 = 3(2.25) - 1 = 6.75 - 1 = 5.75$

Apply the Newton-Raphson formula to find $x_1$:

$x_1 = x_0 - \frac{g(x_0)}{g'(x_0)} = 1.5 - \frac{0.875}{5.75} \approx 1.5 - 0.1522 = 1.3478$

Second Iteration ($n=1$):

Evaluate the function and its derivative at $x_1 = 1.3478$:

$g(1.3478) = (1.3478)^3 - 1.3478 - 1 \approx 2.4482 - 1.3478 - 1 = 0.1004$ $g'(1.3478) = 3(1.3478)^2 - 1 \approx 3(1.8166) - 1 = 5.4498 - 1 = 4.4498$

Apply the formula to find $x_2$:

$x_2 = x_1 - \frac{g(x_1)}{g'(x_1)} = 1.3478 - \frac{0.1004}{4.4498} \approx 1.3478 - 0.0226 = 1.3252$

After two iterations, the root is approximately $1.3252$.