2.1 Units and dimensions

Units and dimensions are the building blocks of chemical engineering calculations. They help us measure and compare physical quantities accurately. Understanding these concepts is crucial for solving problems and designing processes effectively.

SI units provide a standardized system for measurement, with seven fundamental units and many derived units. Dimensional analysis ensures equation consistency, while dimensionless numbers help simplify complex problems and identify key relationships between variables.

SI Units: Fundamental vs Derived

Fundamental SI Units

Top images from around the web for Fundamental SI Units

• 

  Table on Derived quantities and their SI units | Measurements View original

  Is this image relevant?

• 

  SI base unit - Wikipedia View original

  Is this image relevant?

• 

  Standard Units (SI Units) | Introduction to Chemistry View original

  Is this image relevant?

• 

  Table on Derived quantities and their SI units | Measurements View original

  Is this image relevant?

• 

  SI base unit - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental SI Units

• 

  Table on Derived quantities and their SI units | Measurements View original

  Is this image relevant?

• 

  SI base unit - Wikipedia View original

  Is this image relevant?

• 

  Standard Units (SI Units) | Introduction to Chemistry View original

  Is this image relevant?

• 

  Table on Derived quantities and their SI units | Measurements View original

  Is this image relevant?

• 

  SI base unit - Wikipedia View original

  Is this image relevant?

1 of 3

• The International System of Units (SI) is the modern form of the metric system and is the most widely used system of measurement
• There are seven fundamental SI units
  • meter (m) for length
  • kilogram (kg) for mass
  • second (s) for time
  • ampere (A) for electric current
  • kelvin (K) for temperature
  • mole (mol) for amount of substance
  • candela (cd) for luminous intensity

Derived SI Units and Prefixes

• Derived units are formed by combining fundamental units according to the algebraic relations linking the corresponding quantities
  • Examples include units for force (newton, N = kg⋅m/s²), pressure (pascal, Pa = N/m²), and energy (joule, J = N⋅m)
• Prefixes are used to indicate decimal multiples or fractions of SI units
  • Examples include kilo- (10³), centi- (10⁻²), and milli- (10⁻³)
  • A kilometer (km) is 1000 meters, a centimeter (cm) is 0.01 meter, and a milliliter (mL) is 0.001 liter
• Using prefixes allows for convenient expression of very large or very small quantities without using excessive zeros or decimal places

Dimensions in Engineering Calculations

Dimensional Analysis and Homogeneity

• Dimensions are the physical nature of a quantity, such as length, mass, time, or temperature, without regard to its numerical value
• Dimensional analysis is a method for checking the consistency of an equation by verifying that the dimensions on both sides of the equation are the same
  • For example, the equation $v = d/t$ is dimensionally consistent because $[v] = L/T$, $[d] = L$, and $[t] = T$, so both sides have dimensions of $L/T$
• Dimensional homogeneity is the principle that all terms in an equation must have the same dimensions for the equation to be valid
  • In the equation $F = ma$, both sides must have dimensions of force ($MLT⁻²$) for the equation to be physically meaningful

Buckingham Pi Theorem and Dimensionless Numbers

• Buckingham Pi theorem states that any physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − k dimensionless parameters, where k is the number of fundamental dimensions used
  • This theorem is useful for reducing the number of variables in a problem and identifying dimensionless groups that govern the system's behavior
• Dimensionless numbers are ratios of quantities with the same dimensions, resulting in a pure number without any units
  • Examples include Reynolds number ($Re = \frac{\rho vD}{\mu}$) and Prandtl number ($Pr = \frac{c_p\mu}{k}$)
• Dimensionless numbers are important in scaling and similarity analysis in engineering design and research
  • Two systems with the same values of relevant dimensionless numbers are said to be dynamically similar and will exhibit the same behavior

Dimensions of Physical Quantities in Chemical Engineering

Fundamental Dimensions

• Mass (M) is a fundamental dimension in SI units
  • Common quantities with dimension M include density ($kg/m³$), mass flow rate ($kg/s$), and mass concentration ($kg/m³$)
• Length (L) is a fundamental dimension
  • Common quantities with dimension L include distance ($m$), height ($m$), diameter ($m$), and thickness ($m$)
• Time (T) is a fundamental dimension
  • Common quantities with dimension T include velocity ($m/s$), acceleration ($m/s²$), frequency ($1/s$ or $Hz$), and reaction rate ($mol/(m³⋅s)$)
• Temperature (Θ) is a fundamental dimension
  • Common quantities with dimension Θ include absolute temperature ($K$) and temperature difference ($K$ or $°C$)

Derived Dimensions

• Force (F) has dimensions of $MLT⁻²$
  • Common quantities with dimension F include pressure ($Pa$ or $N/m²$), stress ($Pa$ or $N/m²$), and surface tension ($N/m$)
• Energy (E) has dimensions of $ML²T⁻²$
  • Common quantities with dimension E include work ($J$ or $N⋅m$), heat ($J$), and potential energy ($J$)
• Power (P) has dimensions of $ML²T⁻³$
  • Common quantities with dimension P include heat transfer rate ($W$ or $J/s$) and shaft work ($W$)
• Other derived dimensions include volume ($L³$), area ($L²$), viscosity ($ML⁻¹T⁻¹$), and thermal conductivity ($MLT⁻³Θ⁻¹$)