2.1 Units and dimensions

SI Units: Fundamental vs Derived

Fundamental SI Units

The International System of Units (SI) is the modern metric system and the standard for scientific and engineering work. It's built on seven fundamental units, each representing an independent physical quantity:

These are called "fundamental" because they can't be broken down into simpler units. Every other unit in the SI system is built from combinations of these seven.

Derived SI Units and Prefixes

Derived units are formed by combining fundamental units through multiplication or division, matching the algebraic relationships between the quantities they describe.

• Force: newton, $N = kg \cdot m/s^2$
• Pressure: pascal, $Pa = N/m^2 = kg/(m \cdot s^2)$
• Energy: joule, $J = N \cdot m = kg \cdot m^2/s^2$

Notice how each derived unit traces back to fundamental units. Being able to break a derived unit down to its fundamentals is a skill you'll use constantly in dimensional analysis.

SI prefixes let you express very large or very small values without writing out long strings of zeros:

Dimensions in Engineering Calculations

Dimensional Analysis and Homogeneity

Dimensions describe the physical nature of a quantity (length, mass, time, temperature) without reference to its numerical value or specific unit. For instance, meters, feet, and miles are all different units, but they share the same dimension: length (L).

Dimensional analysis is a method for checking whether an equation makes physical sense by verifying that the dimensions match on both sides. Here's how to apply it:

1. Write out the equation you want to check.
2. Replace each variable with its fundamental dimensions (M, L, T, Θ, etc.).
3. Simplify the dimensions on each side using algebra.
4. Confirm that both sides reduce to the same dimensional expression.

For example, take $v = d/t$. Replacing with dimensions: $[v] = L/T$, $[d] = L$, $[t] = T$. The right side gives $L/T$, which matches the left side. The equation is dimensionally consistent.

Dimensional homogeneity is the principle that 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 gives $M \cdot LT^{-2} = MLT^{-2}$. They match, so the equation is homogeneous.

If you ever get mismatched dimensions, something is wrong: a missing variable, a wrong exponent, or an incorrect formula. This makes dimensional analysis one of the quickest ways to catch errors in your calculations.

Buckingham Pi Theorem and Dimensionless Numbers

The Buckingham Pi theorem provides a systematic way to reduce the complexity of physical problems. It states that if a physically meaningful equation involves $n$ variables expressed in terms of $k$ fundamental dimensions, the equation can be rewritten using $n - k$ independent dimensionless groups (called Pi groups, $\Pi$).

For example, if a fluid flow problem involves 5 variables (velocity, density, viscosity, diameter, pressure drop) and 3 fundamental dimensions (M, L, T), the Buckingham Pi theorem tells you the problem can be described by $5 - 3 = 2$ dimensionless groups. That's a major simplification.

Dimensionless numbers are ratios where all units cancel out, leaving a pure number. Two that come up constantly in chemical engineering:

• Reynolds number: $Re = \frac{\rho v D}{\mu}$ compares inertial forces to viscous forces in a fluid. A high $Re$ (typically above ~4000 in pipe flow) indicates turbulent flow; a low $Re$ (below ~2100) indicates laminar flow.
• Prandtl number: $Pr = \frac{c_p \mu}{k}$ compares momentum diffusivity to thermal diffusivity, and it characterizes how a fluid transfers heat relative to how it transfers momentum.

The practical power of dimensionless numbers lies in similarity analysis: two systems with the same values of the relevant dimensionless numbers will behave the same way, even if they differ in size or operating conditions. This is why engineers can test a small-scale model in a lab and predict how a full-scale system will perform.

Dimensions of Physical Quantities in Chemical Engineering

Fundamental Dimensions

Every physical quantity in chemical engineering can be expressed in terms of a small set of fundamental dimensions. The four you'll encounter most often:

• Mass (M): Appears in quantities like density ($kg/m^3$, dimensions $ML^{-3}$), mass flow rate ($kg/s$, dimensions $MT^{-1}$), and mass concentration ($kg/m^3$, dimensions $ML^{-3}$).
• Length (L): Appears in distance, diameter, pipe length, and film thickness. All measured in meters (m) with dimension L.
• Time (T): Shows up in rates of all kinds. Velocity has dimensions $LT^{-1}$, acceleration has $LT^{-2}$, and reaction rate ($mol/(m^3 \cdot s)$) involves time in the denominator.
• Temperature (Θ): Absolute temperature is measured in kelvin (K). Temperature differences can be expressed in K or °C since the degree size is the same, but absolute temperature calculations (like those involving the ideal gas law) must use kelvin.

Derived Dimensions

Most quantities you'll work with in chemical engineering are derived from the fundamentals above:

• Force has dimensions $MLT^{-2}$. Related quantities include pressure ($ML^{-1}T^{-2}$), stress ($ML^{-1}T^{-2}$), and surface tension ($MT^{-2}$).
• Energy has dimensions $ML^2T^{-2}$. This covers work, heat, kinetic energy, and potential energy, all measured in joules (J).
• Power has dimensions $ML^2T^{-3}$. Heat transfer rates and shaft work rates are both expressed in watts ($W = J/s$).
• Other common derived dimensions:
  • Volume: $L^3$
  • Area: $L^2$
  • Dynamic viscosity: $ML^{-1}T^{-1}$ (units of $Pa \cdot s$)
  • Thermal conductivity: $MLT^{-3}\Theta^{-1}$ (units of $W/(m \cdot K)$)

A useful exercise: pick any equation from your textbook and verify that every term has consistent dimensions. If you can do that fluently, you've mastered the core skill of this section.