DIMENSIONAL ANALYSIS IN MOMENTUM TRANSFER FT I Alda Simões
Techniques for Dimensional Analysis Fluid Dynamics: Microscopic analysis, theory Physical modelling Differential balances Limited to simple geometries Macroscopic Balances Dimensional analysis is a tool for finding correlations among physical variables basedupon its dimensions only.
Dimensions Fundamental dimensions: mass, length, time, electric current intensity, light intensity, amount of a substance. Derived dimensions: velocity, acceleration, pressure, tension
Fundamentals of Dimensional Analysis Any physical law derived analytically had to be independent of the units system used to express the physical variables. Any equation (or inequation) with physical meaning has to be dimensionally homogeneous.
Pros and cons of dimensional analysis Advantages: It allows to structure, compact and clarify physical problems. Limitations: It does not provide physical information It will correlate any parameters The starting point needs to be adequately formulated.
Advantes in the establishment of physical correlations Ex: shear stress at the wall in smooth tubes: simplification of the correlation τ w = φ v, ρ, μ, D τ w = Κ 1 v m ρ l μ p D q One constant, 4 exponents τ w vρd = K ρv2 μ n One constant, one exponents
Methods for dimensional analysis: Direct method or power product method Ex: F=f(L,v,ρ,µ) Buckingham s method
Buckingham Theorems 1st Theorem p = m-n A correlation among m variables may be expresses as correlation of m-n dimensionless groups, in which n represents the number of fundamental dimensions. D V Example. M L T Edgar Buckingham 1867-1940 Re Thus, if a Physics problem may be expressed by: f Q 1, Q 2, Q 3,, Q m = 0 then it can also be expressed as: φ Π 1, Π 2, Π 3,, Π m n = 0 In Fluid Mechanics n is normally 3 (M, L, T).
Buckingham Theorems 2nd Theorem: Each group can be obtained as a correlation of n repeating variables plus one of the remaining variables. Choice of the Repeating Variables: 1. The total number of variables should be identical to the number of fundamental dimensions (n = 3 in Fluid Dynamics). 2. Every fundamental dimension must appear at least once in the set of repeating variables. 3. It must not be possible to form a dimensionless group by combination of the repeating variables. The repeating variables should be measurable in a research work and should be highly relevant for the project.
Buckingham s Method 1. Identify the variables and their fundamental dimensions. 2. Determine the number of dimensionless groups. 3. Select the repeating variables 4. Distribute the remaining variables along the groups. 5. For each group: 6. Assign a generic exponent to each variable. 7. Write the expressions using the dimensions. 8. Analyse the exponents in order to obtain a null sum for the exponents affecting each dimension. 9. The exponents thus determined are applied to the variables. 10. Check the dimensions of the groups.
Matriz Dimensional If n>3 (e.g., in heat transport, when the temperature is a parameter, ): n: the number of independent dimensions is given by the number of lines of the largest square matrix with non-null determinant that can be extratec from the dimensional matrix. Dimensional matrix: matrix constituted by tabulation of the expnents of each dimensional, with each column corresponds to a variable and each line stands for a dimension.
Example of dimensional matrix Weight of a object (p = 3-3=0??) W m g M 1 1 0 L 1 0 1 T 2 0 2 Determinant =.zero! Let us now use a square matrix with 2*2, e.g. 1 0 1 1 = 1 0 0 p=3-2 = 1 One dimensionless group: The correlation is π = W mg W mg = C
MODELS AND SIMILITUDE
Dimensional Analysis of Differential Equations Navier-Stokes equations : This a force balance (dimensions F/M): Gravity Force Pressure force Viscous force Inertia force:
Dimensionless numbers in the Navier-Stokes equations inertia force = v2 L F gravitic F inertia = gl v 2 = 1 Fr Froude gravity force = g pressure force = p ρ = p Lρ viscous force = ν 2 v = νv L 2 F pressure F inertia = p ρv 2 = Eu F viscous F inertia = ν Lv = 1 Re Euler Reynolds
Models def.: a model is physical or intelectual construction that can be manipulated in order to predict the future consequenecs of specific actions on the real system.
What makes a model good?
The Giant (10:1) Weight Bone cros section area = V 1 ρg πr bone,1 2 Dimensionless number? height density of the body g max stress of the bones V 2 ρg 2 = 1000 V 1ρg 100 πr oss,1 πr oss,2 Solutions: Density of the body Bone resistance Planet 2
Models Mathematical / theoretical Numerical /computational Ex: F=m dv/dt Eq. Hagen-Poiseuille
Models / Similitude Complex or risky/dangerous/expensive systems Physical models
Similitude 1 Geometric similitude 2 Kinematic similitude 3 Dynamic similitude
Geometric similitude a m b m = a b a a m a m c m = a c b c b m c m Ratio of the characteristic lengths
Kinematic similitude v y,m v x,m = v y,p v x,p v p v m v z,m v x,m = v z,p v x,p Orientation of the flow vs the model and the prototype
Dynamic similitude Identical dimensionless numbers Fr = v2 gl Re = v ρ D μ Eu = p ρv 2 Eu(Fr, Re)
Example Flow of a toxic liquid. Tests in a tube identical but using a non-toxic liquid Eu, Re, r D real = 0,25 in real model D model = D real p m : given p real =? V real = 1ft/s ρ real = 1,32 slug/ft 3 μ real = 6,50 μlbf s ft 2 ρ mod = 1,94 slug/ft 3 μ mod = 2,34 10 6 lbf s ft 2