2.3 Dimensional analysis

Dimensional analysis gives you a way to check whether equations make physical sense and to figure out how different quantities relate to each other. In chemical engineering, you'll use it constantly: verifying formulas, scaling up processes from lab to plant size, and building relationships between variables before you even run an experiment.

Dimensional Analysis Principles

Fundamental Concepts

Every physical quantity can be broken down into a combination of fundamental dimensions: length (L), mass (M), time (T), and temperature (Θ). Dimensional analysis works by expressing quantities in terms of these building blocks and then comparing them.

The dimensions of a quantity are written as a product of powers of the fundamental dimensions:

• Velocity has dimensions of $LT^{-1}$ (length per unit time)
• Acceleration has dimensions of $LT^{-2}$ (length per unit time squared)
• Force has dimensions of $MLT^{-2}$ (mass × acceleration)

Dimensional homogeneity is the core rule: every term in a valid equation must have the same dimensions. In $F = ma$, the left side has dimensions of force ($MLT^{-2}$), and the right side is mass ($M$) times acceleration ($LT^{-2}$), which also gives $MLT^{-2}$. If the dimensions don't match, the equation is wrong, guaranteed.

Dimensionless Quantities

A dimensionless quantity is a ratio where all the units cancel out, leaving a pure number. These show up everywhere in chemical engineering because they let you compare systems of completely different sizes.

Two important examples:

• Reynolds number (Re) characterizes fluid flow behavior. It tells you whether flow is laminar or turbulent, regardless of whether you're looking at water in a pipe or oil in a reactor.
• Mach number (Ma) is the ratio of an object's speed to the speed of sound in the surrounding medium.

Dimensionless quantities are the key to scaling. If a small lab model and a full-size plant have the same relevant dimensionless numbers, they'll behave the same way. This is called dynamic similarity, and it only holds when all relevant dimensionless quantities match between the two systems. Geometric similarity (same shape, different size) is a related but weaker condition.

Dimensional Analysis for Equation Consistency

Verifying Dimensional Consistency

To check whether an equation is dimensionally consistent, express every term in its fundamental dimensions and confirm they all match. Here's the process:

1. Write out each term in the equation separately.
2. Replace each quantity with its fundamental dimensions (L, M, T, Θ).
3. Simplify the dimensions on each side.
4. Confirm that every term reduces to the same dimensional expression.

Example: Check $v = u + at$, where $v$ and $u$ are velocities, $a$ is acceleration, and $t$ is time.

• $[v] = LT^{-1}$
• $[u] = LT^{-1}$
• $[a][t] = (LT^{-2})(T) = LT^{-1}$

All three terms have dimensions of $LT^{-1}$, so the equation is dimensionally consistent.

If the dimensions don't match, the equation is definitely wrong. You can catch a lot of algebra mistakes this way.

Limitations of Dimensional Analysis

Dimensional consistency is necessary but not sufficient. An equation can have the right dimensions and still be wrong:

• Numerical coefficients can't be determined from dimensions alone. The equation $v = 3u + at$ is dimensionally consistent but physically incorrect.
• Missing terms won't show up. If you accidentally drop a term that has the same dimensions as the others, dimensional analysis won't catch it.

Think of dimensional analysis as a filter: it catches equations that are definitely wrong, but passing the check doesn't prove an equation is correct. You still need experimental data or theoretical derivation to confirm accuracy.

Dimensional Analysis for Quantity Relationships

Deriving Relationships Using Dimensional Analysis

Beyond checking equations, dimensional analysis can help you discover the form of a relationship between quantities, even when you don't know the underlying physics in detail.

The Buckingham Pi Theorem formalizes this. It states: if you have $n$ physical quantities involving $r$ fundamental dimensions, you can reduce the problem to a relationship among $n - r$ independent dimensionless groups (called Pi groups, written as $\pi_1, \pi_2, ...$ ).

To apply the Buckingham Pi Theorem:

1. Choose $r$ repeating variables that collectively contain all the fundamental dimensions in the problem.
2. Form dimensionless Pi groups by combining each remaining variable with the repeating variables, adjusting exponents until the dimensions cancel.
3. Express the result as $\pi_1 = f(\pi_2, \pi_3, ..., \pi_{n-r})$.

The theorem doesn't tell you the exact function $f$, but it dramatically reduces the number of variables you need to investigate experimentally.

Steps for Deriving Relationships

For simpler problems, you can use dimensional analysis directly without forming Pi groups. Here's a worked example.

Problem: Find how the period $T$ of a simple pendulum depends on its length $L$ and gravitational acceleration $g$.

1. List the quantities and their dimensions:

   • $T$ (period): $T$
   • $L$ (length): $L$
   • $g$ (gravitational acceleration): $LT^{-2}$

2. Assume a power-law relationship: $T = k \cdot L^a \cdot g^b$, where $k$, $a$, and $b$ are unknowns.

3. Write dimensions for both sides and equate:

   • Left side: $T^1$
   • Right side: $L^a \cdot (LT^{-2})^b = L^{a+b} \cdot T^{-2b}$

4. Match exponents for each fundamental dimension:

   • For $L$: $0 = a + b$
   • For $T$: $1 = -2b$

5. Solve: $b = -1/2$ and $a = 1/2$

6. Write the result: $T = k\sqrt{L/g}$

Dimensional analysis gives you the form of the equation. The coefficient $k$ has to come from experiment or theory. For the pendulum, $k = 2\pi$, so the final equation is $T = 2\pi\sqrt{L/g}$.

This approach is especially powerful in chemical engineering when you're dealing with complex systems where the full governing equations are hard to solve. You can identify the key dimensionless groups, then use experiments to fill in the functional relationships.