Glossary

2.3 Dimensional analysis

4 min read•august 14, 2024

Dimensional analysis is a powerful tool in chemical engineering, helping us understand relationships between physical quantities. It's all about breaking down complex problems into their fundamental dimensions like , , and .

By analyzing dimensions, we can check if equations make sense and even derive new relationships between quantities. This skill is crucial for solving real-world engineering problems and scaling up processes effectively.

Dimensional Analysis Principles

Fundamental Concepts

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  • Dimensional analysis is a method for analyzing the relationships between physical quantities using their dimensions or units
  • Dimensions are the fundamental physical properties of a quantity, such as length (L), mass (M), time (T), and temperature (Θ)
  • The dimensions of a physical quantity are expressed 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)
  • states that all terms in an equation must have the same dimensions for the equation to be valid
    • In the equation F=maF = ma, both sides must have dimensions of force (MLT⁻²)

Dimensionless Quantities

  • Dimensionless quantities are ratios of quantities with the same dimensions, resulting in a pure number without any units
    • (Re) is a dimensionless quantity used in fluid mechanics to characterize flow behavior
    • Mach number (M) is a dimensionless quantity representing the ratio of the speed of an object to the speed of sound in the surrounding medium
  • Dimensionless quantities are useful for scaling and comparing physical systems with different sizes or operating conditions
    • Geometric ensures that dimensionless quantities are equal for two systems (model and prototype)
    • Dynamic similarity is achieved when all relevant dimensionless quantities are equal between two systems

Dimensional Analysis for Equation Consistency

Verifying Dimensional Consistency

  • To verify the of an equation, express each term in the equation in terms of its fundamental dimensions (L, M, T, Θ)
  • For an equation to be dimensionally consistent, all terms must have the same dimensions after cancellation of common dimensions
    • In the equation v=u+atv = u + at, both sides must have dimensions of velocity (LT⁻¹)
    • [v]=[u]+[a][t]=LT−1+(LT−2)(T)=LT−1[v] = [u] + [a][t] = LT⁻¹ + (LT⁻²)(T) = LT⁻¹
  • If the dimensions of the terms in an equation do not match, the equation is inconsistent and cannot be correct

Limitations of Dimensional Analysis

  • Dimensional analysis can be used to check the plausibility of an equation, but it does not guarantee its accuracy or validity
    • An equation may be dimensionally consistent but still incorrect due to incorrect coefficients or missing terms
    • Dimensional analysis cannot determine the numerical values of coefficients or constants in an equation
  • Dimensional analysis is a necessary but not sufficient condition for the validity of an equation
    • Further experimental or theoretical analysis is required to confirm the accuracy of an equation

Dimensional Analysis for Quantity Relationships

Deriving Relationships Using Dimensional Analysis

  • Dimensional analysis can be used to determine the form of an unknown relationship between physical quantities based on their dimensions
  • The Pi Theorem states that a dimensionally homogeneous equation involving n physical quantities can be reduced to a relationship between n - r independent dimensionless quantities, where r is the number of fundamental dimensions
  • The provides a method for determining the dimensionless quantities (Pi groups) in a problem
    • Choose r repeating variables that include all fundamental dimensions
    • Form dimensionless Pi groups by combining the remaining variables with the repeating variables
    • Express the relationship between the Pi groups as a function π1=f(π2,π3,...,πn−r)π_1 = f(π_2, π_3, ..., π_{n-r})

Steps for Deriving Relationships

To derive a relationship using dimensional analysis:

  1. List all relevant physical quantities and their dimensions
  2. Express the desired quantity as a product of powers of the other quantities
  3. Equate the dimensions of both sides of the equation and solve for the unknown exponents
    • For the period of a simple pendulum, T=kLagbT = k L^a g^b, where TT is the period, LL is the length, gg is the acceleration due to gravity, and kk, aa, and bb are unknown constants
    • [T]=[L]a[g]b=La(LT−2)b=La+bT−2b[T] = [L]^a [g]^b = L^a (LT⁻²)^b = L^{a+b} T^{-2b}
    • Equating the dimensions: T=La+bT−2bT = L^{a+b} T^{-2b}, which gives a=1/2a = 1/2 and b=−1/2b = -1/2
    • The resulting equation is T=kL1/2g−1/2T = k L^{1/2} g^{-1/2} or T=kL/gT = k \sqrt{L/g}
  4. The resulting equation gives the form of the relationship between the physical quantities, with the coefficients determined experimentally or through further analysis
    • For the simple pendulum, the coefficient kk is found to be 2π2π, giving the final equation T=2πL/gT = 2π \sqrt{L/g}

Key Terms to Review (19)

Buckingham Pi Theorem: The Buckingham Pi Theorem is a key principle in dimensional analysis that provides a systematic method for deriving dimensionless parameters from physical equations. It states that if a physical problem involves a certain number of variables, the number of independent dimensionless groups can be determined by the difference between the total number of variables and the number of fundamental dimensions involved. This theorem helps simplify complex equations and facilitates the comparison of different physical systems.
Conversion Factors: Conversion factors are numerical ratios used to express one set of units in terms of another, facilitating the transformation of measurements from one unit system to another. They are essential tools in dimensional analysis, allowing for the accurate conversion and comparison of quantities with different units. By applying conversion factors, calculations can be performed seamlessly across diverse measurement systems, ensuring precision in scientific and engineering contexts.
Dimensional consistency: Dimensional consistency refers to the requirement that equations and expressions must have the same dimensions on both sides to be physically meaningful. This principle is essential in ensuring that calculations and relationships in engineering maintain coherence, as it allows for the verification of formulas and the identification of potential errors in unit conversions or algebraic manipulations.
Dimensional Formula: A dimensional formula expresses a physical quantity in terms of the fundamental dimensions such as mass, length, time, and temperature. It serves as a way to represent how a particular quantity relates to these base dimensions, making it essential for understanding dimensional analysis and ensuring consistency in equations.
Dimensional Homogeneity: Dimensional homogeneity refers to the principle that equations must be dimensionally consistent, meaning that all terms in an equation must have the same dimensions. This concept ensures that physical laws and relationships are correctly expressed in terms of measurable quantities, allowing for meaningful interpretations of equations. It connects closely with the analysis of units and dimensions, providing a framework for validating the correctness of mathematical formulations in scientific contexts.
J. M. Keenan: J. M. Keenan is a prominent figure in chemical engineering known for his contributions to dimensional analysis and thermodynamics. His work laid the foundation for understanding how physical quantities can be expressed in terms of fundamental dimensions, which is crucial for solving complex engineering problems and ensuring consistency in equations and calculations.
Length: Length is a measure of distance, typically used to quantify how long an object is or the distance between two points. In various scientific fields, including engineering and physics, length is fundamental for understanding the physical dimensions of objects, space, and relationships between different quantities. It's crucial for calculations involving volume, area, and other derived measurements, where precise dimensional analysis ensures accuracy in the application of formulas and equations.
Lord Rayleigh: Lord Rayleigh, also known as John William Strutt, was a prominent British scientist who made significant contributions to various fields, including acoustics, optics, and fluid dynamics. His work laid the foundation for important concepts in dimensional analysis, emphasizing the role of dimensionless parameters in understanding physical phenomena.
Mass: Mass is a fundamental property of matter that quantifies the amount of substance in an object, typically measured in kilograms or grams. It is an essential concept in physics and engineering, influencing how objects interact through forces and their behavior in various processes. Understanding mass is crucial for analyzing systems in terms of dimensions and units, which are foundational to many scientific calculations.
Model Testing: Model testing refers to the process of validating and verifying the accuracy and reliability of mathematical or physical models used to represent real-world systems. It ensures that these models can predict system behavior under various conditions, which is crucial for accurate simulations and analyses in engineering disciplines.
Modeling fluid flow: Modeling fluid flow refers to the mathematical and computational techniques used to simulate and analyze how fluids move through various environments and systems. This process helps engineers predict behavior under different conditions, optimize designs, and understand complex phenomena such as turbulence and pressure changes. It is deeply connected to dimensional analysis, which provides the framework for understanding the relationships between different physical quantities in fluid dynamics.
Non-Dimensional Numbers: Non-dimensional numbers are ratios that express a physical quantity without any units, enabling comparisons and simplifications in engineering and physics. They play a critical role in dimensional analysis by allowing the reduction of complex relationships into simpler forms, which helps in scaling models and predicting behaviors across different systems.
Nusselt Number: The Nusselt number is a dimensionless quantity used in heat transfer calculations, representing the ratio of convective to conductive heat transfer across a boundary. It helps quantify how effectively heat is being transferred by convection compared to conduction, and is essential for analyzing heat transfer in fluids and designing heat exchangers.
Rayleigh's Method: Rayleigh's Method is a technique used in dimensional analysis to derive relationships between physical quantities by considering their dimensions and finding a functional form that is dimensionally consistent. This method helps simplify complex physical problems by identifying key variables and reducing the number of dimensions involved, which can lead to easier calculations and predictions in engineering and physics applications.
Reynolds Number: The Reynolds number is a dimensionless quantity used to predict flow patterns in fluid dynamics. It helps determine whether a flow is laminar or turbulent, which is crucial in analyzing the behavior of fluids in various engineering applications, from pipe flow to mass transfer processes. This number arises from the balance between inertial forces and viscous forces in the fluid, making it a key parameter in understanding fluid behavior under different conditions.
Scaling Laws: Scaling laws describe how certain physical quantities change with size, shape, or other parameters when a system is scaled up or down. They provide crucial insights into the behavior of systems across different scales and help in predicting outcomes based on dimensional analysis, allowing engineers to simplify complex problems by focusing on the most relevant variables.
SI units: SI units, or the International System of Units, is a standardized system used globally for measuring physical quantities. It provides a consistent framework for measurement, which is essential for scientific communication and data comparison. SI units are fundamental in various fields, especially in science and engineering, and they ensure that calculations and conversions are precise and universally understood.
Similarity: Similarity in dimensional analysis refers to the concept that different physical systems can be compared through dimensionless parameters, which allow for the scaling of results from one system to another. This principle enables engineers to predict the behavior of new systems based on existing data and relationships, promoting efficiency in experimentation and design. It is a foundational concept that ties together various physical phenomena by highlighting their proportional relationships.
Time: Time is a fundamental dimension that measures the duration of events and the intervals between them. In the context of dimensional analysis, time is crucial for understanding how different physical quantities relate to one another, allowing engineers to evaluate processes and systems effectively. By incorporating time into equations and models, one can analyze dynamic behaviors and predict how changes occur over specific durations.
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Glossary
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2.4 Significant figures and rounding