Nondimensionalization of Nondimensionalization of the Wall Shear Formulathe Wall Shear Formula

John GradyJohn Grady

BIEN 301BIEN 301

2/15/072/15/07

C5.1 Problem StatementC5.1 Problem Statement

• For long circular rough pipes in turbulent flow, wall shear stress can be written as a function of ρ, μ, ε, d, and V

RequiredRequired

• Rewrite the function for wall shear in dimensionless form

• Use the formula found to plot the given data of volume flow and shear stress and find a curve fit for the data

Given InformationGiven Information

Q (gal/min)

1.5 3.0 6.0 9.0 12.0 14.0

τw (Pa) 0.05 0.18 0.37 0.64 0.86 1.25

d = 5 cm

ε = 0.25 mm

AssumptionsAssumptions

• Incompressible

• Turbulent

• Long circular rough pipe

• Viscid liquid

• Constant temperature

Pi EquationsPi Equations

• The function for wall shear contained 6 variables

• The primary dimensions for these variables were found to include M,L,T

• Therefore, we should use 3 scaling parameters

• Scaling parameters were ρ, V, and d

Pi EquationsPi Equations

• Now use these parameters plus one other variable to find pi group by comparing exponents.

• Π1 = ρa Vb dc μ-1 = (ML-3)a (LT-1)b (L)c (ML-1T-1) -1 = M0L0T0

• This leads to a = 1, b = 1, and c = 1

Pi GroupsPi Groups

• So, the first pi group is:→ Π1 = ρ1 V1 d1 μ-1 = (ρVd) / μ = Re

• The other pi groups were found to be:→ Π2 = ε / d

→ Π3 = τ / (ρV2) = Cτ

Dimensionless FunctionDimensionless Function

• The three pi groups were used to find the dimensionless function for wall shear

→ Π3 = fcn (Π1,Π2)

→ Cτ = fcn (Re,ε/d)

Data ConversionData Conversion

• The volume flow rates given were converted to SI units

• The values for ρ and μ for water @ 20°C were looked up

• The value for the average velocity, V, was found by using the following formula→ V = Q / A

Data ManipulationData Manipulation

• Next, the values for V, d, ρ, and μ were used to calculate the Reynolds number for each Q given

• Then, Cτ was calculated for each Reynolds number (ε/d was not used since it was constant for all values of Q)

• Finally, Cτ vs. Re was plotted using Excel.

Graph of Graph of CCττ vs. Re vs. Re

Shear Stress vs. Reynolds Number log-log y = 3.6213x-0.6417

R2 = 0.9532

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

Reynolds Number

Sh

ea

r S

tre

ss

(P

a)

Analysis of GraphAnalysis of Graph

• A power curve fit was used to find a formula for shear stress as a function of the Reynolds number

→ τ = 3.6213(Re)-0.6417

→ R2 = 0.9532

ConclusionConclusion

• It was determined that the formula for wall shear can be reduced to an equation using a single variable, Re

• This can save a lot of time and money testing different flows

Biomedical ApplicationBiomedical Application

• An application of this problem could involve the flow of various fluids into a subject using an IV or catheter

• Wall shear would need to be factored in to determine the correct flow rate of the fluid

• Questions??