Dimensional Analysis
Dimensional analysis is a mathematical technique used to check the consistency of equations and to convert units from one system to another. It involves examining the dimensions of physical quantities and using them to derive relationships between different variables. By analyzing the dimensions of various terms in an equation, engineers can ensure that their calculations are accurate and that the units of measurement are consistent.
6 Key excerpts on "Dimensional Analysis"
eBook - ePubDimensional Analysis
Practical Guides in Chemical Engineering
- Jonathan Worstell(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
...We use the concept of dimensional homogeneity as a check on our algebraic manipulations. If a physical equation involves apples and we finish with “orpels,” then we know we have made an incorrect algebraic manipulation in our study. We also use Dimensional Analysis to develop self-consistent systems of units and to establish the conversion factors between those systems of units. As mentioned several times previously, aeronautical, civil, and mechanical engineers have used Dimensional Analysis quite successfully to obtain functional solutions to problems too complex for derivation of an analytic solution. The problems these engineering disciplines have solved using Dimensional Analysis involve the smallest possible set of fundamental dimensions and a small number of variables. Some heat transfer problems have been resolved using Dimensional Analysis. Fewer such solutions exist because they involve a more complex set of fundamental dimensions and a larger number of variables. Chemical processes are so complex that few attempts have been made to use Dimensional Analysis to analyze them. This situation will change in light of free-for-use matrix calculators on the Internet. Once we have a functional solution developed from Dimensional Analysis, we can use it to gain physical insight about the problem. For example, using a functional solution, we can generate experimental data identifying the controlling regimes of the mechanism or process. Dimensional Analysis provides us with a tool for designing experimental programs. By grouping dimensional variables into dimensionless parameters, we can reduce the number of independent variables, thereby reducing the number of experiments required to define a mechanism or process. Using dimensionless parameters also allows us to succinctly present the results of a given experimental program. Lastly, Dimensional Analysis provides a firm, logical foundation for the theory of models...
eBook - ePub- Kenneth J. Valentas, J. Peter Clark, Leon Levin(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...11 Dimensional Analysis I. Introduction Dimensional Analysis is a useful and powerful tool for solving scale-up problems. Dimensional Analysis is the method of reducing the equations that describe a process into a form containing no reference to units of measurement, that is, a dimensionless form. By its very nature, Dimensional Analysis gives a relationship between the independent and dependent variables that is independent of scale. As a consequence, the analysis of a problem through Dimensional Analysis, if it is possible, will inevitably lead to a solution of the scale-up problem. At the very least, a Dimensional Analysis will lead to a clearer understanding of the problem. Unfortunately, there is a large class of problems, associated with the scale-up of food processes, to which Dimensional Analysis cannot be applied: the analysis of the effect of process variables on most product qualities. This conclusion may be simply summarized by saying there is no succinct definition of a dimensionless variable for flavor or texture. 1 The literature (8,31,34,56), although it does not discuss the type of properties with which the food technologist normally deals, states the problem as the inability of Dimensional Analysis to deal with attributes lacking the property of absolute significance of relative magnitude. The most common example of this is the use of the Celsius or Fahrenheit scales of temperature, instead of the absolute Kelvin or Rankine scales. There is no problem when dealing with temperature differences. A better example of lacking the property of absolute relative magnitudes is the hardness scales that are used to describe materials. The common Mohs scale, which assigns a value of 10 to a diamond and a value of 1 to talc, does not mean that diamonds are 10 times harder than talc particles. Similarly, a grade of 10 on an orgaoleptic scale does not imply that the product tastes 10 times better than a product that was assigned a grade of one...
eBook - ePubAdiabatic Fixed-Bed Reactors
Practical Guides in Chemical Engineering
- Jonathan Worstell(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
...We do not prove this assertion; it is proven elsewhere. 8(chps 8,10,11) A.7 Summary As with all engineering and scientific endeavors, Dimensional Analysis involves procedure. Procedures are mechanisms that help us organize our thoughts. They are outlines of what we plan to do. As such, they minimize the likelihood that we will overlook or ignore an important point of our project. In other words, procedures reduce the time we expend on a given project and increase the accuracy of our result. The procedure for using the matrix formulation of Dimensional Analysis is 1. state the problem—clearly; 2. research all available literature for published results; 3. develop the pertinent balances, i.e., momentum, heat, and mass, for the problem; 4. list the important variables of the problem; 5. develop a dimension table using the identified variables; 6. write the dimension matrix; 7. determine the rank of the dimension matrix; 8. identify the rank matrix and calculate its inverse matrix; 9. identify the bulk matrix; 10. multiply the negative of the inverse rank matrix with the bulk matrix; 11. determine whether irrelevant variables are in the dimension matrix; 12. build the total matrix; 13. read the dimensionless parameters from the total matrix; 14. rearrange the dimensionless parameters to maximize physical content interpretation. References 1. Murphy G. Similitude in engineering New York, NY: The Ronald Press Company; 1950; p. 17. 2. Bridgman P. Dimensional Analysis New Haven, CT: Yale University Press; 1922; [chapter 4]. 3. Hunsaker J, Rightmire B. Engineering applications of fluid mechanics New York, NY: McGraw-Hill Book Company, Inc.; 1947; [chapter 7]. 4. Huntley H. Dimensional Analysis New York, NY: Dover Publications, Inc; 1967; page (First published by MacDonald and Company, Ltd., 1952). 5. Barenblatt G. Scaling Cambridge, UK: Cambridge University Press; 2003; p. 17–20. 6...
eBook - ePubBasic Math Concepts
For Water and Wastewater Plant Operators
- Joanne K. Price(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...This choice depends on which other units are part of the calculation and how these units might divide out. WHEN TO USE Dimensional Analysis Dimensional Analysis is a very valuable tool used to check the accuracy of the mathematical set-up for a particular problem. It is recommended that Dimensional Analysis be used to verify a mathematical set-up rather than to construct one. All examples and practice problems presented in this chapter have been adapted from calculations found throughout the applied math texts. DIVIDE OUT SIMILAR UNITS In Dimensional Analysis units in the numerator are divided out with similar units in the denominator, such as: 4 ft × 12 in. ft These terms may be divided out just as factors are divided out when making calculations. In order to complete the division of units, it is important that all units be written in the same format: horizontal or vertical gal/cu ft gal cu/ft The vertical format is used almost exclusively in Dimensional Analysis. In addition, any abbreviations should be written in the vertical format, such as: gpm → gal min or psi → lbs sq in. Example 1: (Dimensional Analysis—Basics) □ Check the mathematical set-up given below, using Dimensional Analysis. Do the desired terms of the answer match with the math set-up as shown? (0.785) (40 ft) (40 ft) = ft 2 Analyze only the units of the problem: Example 2: (Dimensional Analysis—Basics) □ The mathematical set-up for a problem is given below. Do the units of the problem result in the desired units of the answer? (1.2 cfs) (60 sec/min) (7.48 gal/cu ft) = gpm First express all units in a vertical format, then evaluate units: Example 3: (Dimensional Analysis—Basics) □ Verify the following problem set-up using Dimensional Analysis. Is the set-up correct according to the desired units of the answer? (127, 170 ft 2) (20 ft) 0.785 = ft 2 Review the units of the problem: After reconsidering the math set-up, the error was detected...
eBook - ePub- Keith J. Moss(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...That is to say, the units of density are kg/m 3 where mass is quantified in kg and volume is quantified in m 3. Thus the units of a term define, in addition, the system of measurement being used. A term’s dimensions, on the other hand, are not confined to any system of measurement and therefore Dimensional Analysis is universal and common to all systems of units. There are three systems of units in use in the Western world namely the System International (SI), FPS or foot pound second system and MKS or metre kilogram second system. The United Kingdom has for many years adopted the SI. The United States of America is slowly changing from the FPS system to SI and likewise with Germany from the MKS system of measurement. Dimensional Analysis is adopted to undertake three discrete tasks which are • to check that an equation has been correctly formed • to establish the form of an equation relating to a number of variables • to assist in the analysis of empirical formulae in experimental work. This chapter will focus on checking some equations used in this book by Dimensional Analysis to show that they are correctly formed. AI.2 Dimensions in use It is necessary to identify the dimensions which will be used in validating a formula. There are five dimensions which are used in heat and mass transfer, and they are those for mass M, length L, time T, energy H and thermodynamic temperature θ. The dimensions for force, for example, can now be established and since force = mass × acceleration, the units of the terms are: force = kg × m/s 2. Therefore, the dimensions of the terms are: force = M × L × T −2 =MLT −2. The unit of force in the SI is the Newton (N), thus the dimensions of the Newton are MLT −2. The unit of the dimension for heat energy H is the Joule. This is the same as the units for mechanical energy which are the (Nm). So the dimensions for energy H are (MLT −2) × (L)...
- Terry Lake, Nicola Green(Authors)
- 2016(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 5 Dimensional Analysis Learning Objectives • What is Dimensional Analysis? • How to identify conversion factors • How to set up an equation using Dimensional Analysis What is Dimensional Analysis? Dimensional Analysis sounds very technical but it is actually a very simple and logical method of converting units of measurement from one form to another. In chemistry courses, you may have used this method under the name factor label method or unit factor method. In the medical sciences, we often meet problems where we are dealing with different types of measurement, and this method allows us to change the measurement type to fit the situation. Consider the problem of giving a dog an antibiotic tablet. Somehow we have to convert the size of the dog to an equivalent size of tablet (Fig. 5.1) based on a dosage. The first piece of information we need is the weight of the dog – we call this the starting factor. It is the unit of measurement that has to be converted to another unit. Secondly, we need to identify the units of measurement we wish to end up using – we call this the answer unit. In this case it is the number of tablets to give to the dog. Fig 5.1 Tablet strength and patient size are interrelated How do we arrive at the answer unit from the starting factor? We use one or a series of conversion factors. Conversion factors are ratios of units of measurement that have a true relationship and are expressed as fractions with a numerator and denominator. An example of a conversion factor is 3 feet = 1 yard. As a ratio we would show this as 3 feet 1 yard Another example is: ‘There are 1000 mg in 1 g’: 1000 mg 1 g Let’s go back to our example of a dog requiring an antibiotic tablet. If our dog weighs 10 kg, we use this as our starting factor. The answer unit will be the number of tablets: To get from our starting factor to our answer unit, we need to multiply the starting factor by one or more conversion factors that serve as bridges...
