Non-Newtonian Fluid Model Determination
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Non-Newtonian fluid model determination using experimental flow rate values measured under different applied pressure gradients in a horizontal pipe of radius a=0.01 m.

Reference: Problem 15 Chapter 11 Page 553 of Fluid Mechanics for Chemical Engineers, J. O. Wilkes, Prentice Hall, 1999.

Determination of Bingham Fluid Parameters:(Subscript[τ, 0])=tau0 and μ=visc using experimetal data.

Expression of the Volumetric Flow Rate for Bingham Fluids

QBingham[PressureGradient_] := 
      Pi a^4/(2 visc) (PressureGradient) (1/
              4.  - β[PressureGradient]/3 + β[PressureGradient]^4/
              12)

β[PressureGradient_] := 
      2 tau0/(a PressureGradient)

Experimental data

mydata = {{10^4, 5.37 10^-5}, {2 10^4, 
            2.64 10^-4}, {3 10^4, 6.89 10^-4}, {4 10^4, 
            1.29 10^-3}, {5 10^4, 2.35 10^-3}, {6 10^4, 
            3.36 10^-3}, {7 10^4, 4.87 10^-3}, {8 10^4, 
            7.13 10^-3}, {9 10^4, 9.12 10^-3}, {10^5, 
            1.11 10^-2}};

Calculation of the Sum of Squares

sumOfSquares[data_] := 
      Apply[Plus, 
        Apply[Plus, 
          Map[{(QBingham[#[[1]]]  -  #[[2]])^2} &,  
            data]]]

myEvaluater[tau0_, visc_,  data_] := sumOfSquares[data]

Pipe Radius

a = 0.01;

We find (Subscript[τ, 0]) = 77.55 and μ = 0.0326

paramBingham = 
      FindMinimum[
        myEvaluater[tau0, visc, mydata], {tau0, 0.1, 90}, {visc,  .01, 1000}, 
        MaxIterations -> 300]

Determination of Power Law Fluid Parameters: n and k using experimetal data.

a =. ;

Expression of the Volumetric Flow Rate for Power Law Fluids

QPowerLaw[
        PressureGradient_] := (PressureGradient/(2 k))^(1/
              n) Pi n a^(3 + 1/n)/(1 + 3 n)

Calculation of the Sum of Squares

sumOfSquares[data_] := 
      Apply[Plus, 
        Apply[Plus, 
          Map[{(QPowerLaw[#[[1]]]  -  #[[2]])^2} &,  
            data]]]

myEvaluater[n_, k_,  data_] := sumOfSquares[data]

Pipe Radius

a = 0.01;

We find n = 0.437  and k = 6.708

param1 = 
      FindMinimum[myEvaluater[n, k, mydata], {n, 0.42, 0.5}, {k, 6, 10}, 
        MaxIterations -> 100000]

Conclusion Power Law model fits better experimental data