C4_part_3_and_C5

# C4_part_3_and_C5 - P = Cd l n1 n 2 Dimensional analysis In...

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Dimensional analysis 1 (, ,, , ) Pf d l u ρμ Δ = 2 2 , Pl f ud d u μ ρ ⎛⎞ Δ = ⎜⎟ ⎝⎠ In a dimensional form In a dimensionless form 12 ll dd = 11 1 2 2 2 du d u = 22 2 2 P P uu ρρ Δ Δ ⇒= Advantages of the dimensional analysis? 1. Reduce the number of parameters 2. Deletion of nondominant effects and simplifying mathematical models 3. Developing small-scale physical models to predict flow patterns at a larger scale n1 n2 n3 n4 n5 PC d l u Δ= Geometric similarity Dynamic similarity

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Equations of Change for Isothermal Systems Dimensional analysis of the equations of change 2 .0 Consider equations and Introducing dimensionless variables D P Dt ρ μρ ∇= = + + uu u g 00 0 0 0 0 2 0 // o / r y z / / / xx l y l z l P PP P tu t l u P P l ρμ === == = = ± ± ±± ± ± () ( ) ( ) 0 22 2 2 2 2 2 2 2 0 /// xyz l l ∇= ∇= ∂∂ + + + + δδδ ± ± ± ± Substitute the above dimensionless equations into the equations of change N N 2 0 2 0 1/ 1/Re 2 0 2 0 or Fr gl D P D tl u u g D P D u l u u g μ μμ ρρ = + + =− ∇ + + g u g cf c f ± ± dg d g ± d g d g eh e h ²³´³µ ± cfcf c f dgdg d g ± d g d g eheh e h ²³´³ µ² ³´³µ 2 1 Re Fr g 1 1 0 D P P →∞⇒ =−∇ + ⇒−∇ +∇ + = g u g u ± ± ± ± ± General solution (,,,,R e ,F r ) xyzt = ± ± ± ±
Notes : 1. All the characteristic quantities (e.g., l 0 , u 0 , P 0 ) should ideally be the maximum values of the corresponding dimensional quantities (e.g., l, u, P) within the system domain under specific conditions. A-priori knowledge of system behavior is needed to estimate these maxima. 2. The characteristic length l 0 does not necessarily, but very often, coincide with any length that characterizes the geometry of the flow domain. It should be chosen such that it characterizes spatial variations in other quantities such as velocity and pressure. For instance, falling film: l 0 =4 δ ; flow in tube: l 0 =2R; flow in annulus: l 0 =2R(1- κ ). 3. The behavior of a large-scale system could be observed and measured on a smaller-scale system, provided that the two system must be geometrically similar (i.e. having the same geometric dimensionless groups), and dynamically similar (i.e. having the same ‘dynamic’ dimensionless group such as Re and Froude numbers, respectively.) Equations of Change for Isothermal Systems Dimensional analysis of the equations of change

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Equations of Change for Isothermal Systems Illustration: Problem: prediction of the vortex depth in a large tank by means of a physical model study in a smaller geometrically similar tank, without solving equations of change Large tank Small tank r z H 1 T 1 1 D 1 r z H 2 T 2 2 Boundary conditions 11 1 0/ 2 / 2 00 1 0 on 0 ( , ) rT zz H atm pp S r z << = =< < = = = uu 1 2 / 2 2 0 ( , ) H atm S r z = < = = = In terms of dimensionless variables () ( ) 2 / 2 / 111 0 ( / , / ) on D D H D pS r D z D = < = = = ±± ± ± ( ) 22 2 / 2 / 222 0 ( / , / )
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C4_part_3_and_C5 - P = Cd l n1 n 2 Dimensional analysis In...

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