TAMU CHEN 304 - L3 (15 pages)

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Pages:
15
School:
Texas A&M University
Course:
Chen 304 - Chem Engr Fluid Ops
Chem Engr Fluid Ops Documents

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CBE341 Lecture 3 9 17 Dimensional analysis The development of fluid mechanics has depended heavily on experimental results as most flow problems of practical interest cannot be solved exactly by analytical methods As experimental work is both time consuming and expensive it is natural to examine if one can gather the desired information from as few experiments as possible while working with a suitably scaled model of the geometry of interest Dimensional analysis is an important tool to achieve this goal It also exposes dimensionless groups or parameters which can be used to correlate the experimental data which allows us to use the correlations across geometric scales Dimensional analysis also allows us to examine the exact equations of motion critically By recasting these equations in a dimensionless form one can recognize the relative importance of various terms in the model equations This may allow rational simplifications of the exact equations These simplified equations may be solved more easily and sometimes even analytically As we have not discussed the equations of motion you have to simply take my word for it at the present time The dimensional analysis can also suggest when one may be able to do away with the differential equation model for the flow and accept simple macroscopic balance solutions Nature of Dimensional Analysis Let us first consider the problem of designing suitably scaled experimental systems and correlating the data Dimensional analysis performed for these specific goals does not require as a starting point the differential equations describing the evolution of various variables Instead we simply need to remember that any equation correlation we derive must be dimensionally homogeneous We will now discuss an organized procedure for doing dimensional analysis and selecting dimensionless groups Buckingham Pi theorem which is the foundation for dimensional analysis can be stated as follows 19 Given a physical problem in which the dependent parameter q1 is a function of n 1 independent parameters q2 q3 qn we expect 8 q1 f q2 q3 qn or equivalently 9 g q1 q2 qn 0 If m is the minimum number of independent dimensions mass length time temperature electric charge etc required to specify the dimensions of all the parameters the number of independent dimensionless ratios is i i n m If r is the rank of the dimensional matrix described below then i n r That is the rank of the dimensional matrix is less than or equal to m r m Note that i is the number of independent dimensionless ratios Denoting the i dimensionless ratios as 1 2 i we write 8 9 as 10 1 G1 2 3 i or G 1 2 i 0 The Buckingham Pi theorem does not predict the functional form of G or G1 and these must be found experimentally 20 Dimensional Matrix Independent Dimensions List of Parameters Mass m Length L Time t Temperature Charge q1 q2 a1 b 1 c1 d1 e1 a2 b2 q3 qn an bn cn dn en Here ai bi ci



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