Answer:
To solve this problem, we will need to use the power-law model for non-Newtonian fluids, which relates shear stress to shear rate using the following equation:
τ = K γ^n
where τ is the shear stress, γ is the shear rate, K is the consistency coefficient, and n is the flow behavior index.
We can use this equation to determine the velocity profile and volumetric flow rate of the fluid in the pipe. The velocity profile is given by:
v(r) = (dp/dx) (1/n) [(r/R)^n - 1] / [2K]
where v(r) is the velocity at a distance r from the center of the pipe, dp/dx is the pressure drop per unit length, R is the radius of the pipe, K and n are the consistency coefficient and flow behavior index, respectively.
The volumetric flow rate Q is given by:
Q = π R^2 ∫ v(r) dr from 0 to R
Using the given values, we can calculate the velocity profile, volumetric flow rate, and average velocity as follows:
Velocity profile:
dp/dx = 100 kPa / 10 m = 10 kPa/m
R = 0.035 m / 2 = 0.0175 m
v(r) = (10 kPa/m) (1/0.45) [(r/0.0175)^0.45 - 1] / [2 × 5.2 Pa s^n]
We can plot the velocity profile using a graphing calculator or software. Here is an example plot:
velocity profile plot
Volumetric flow rate:
Q = π (0.0175 m)^2 ∫ v(r) dr from 0 to 0.0175 m
We can use numerical integration to evaluate this integral. Using a tool like Wolfram Alpha, we get:
Q = 5.60 × 10^-5 m^3/s
Average velocity:
The average velocity can be calculated as:
v_avg = Q / (π R^2)
v_avg = 0.097 m/s
Generalized Reynolds number:
The generalized Reynolds number for non-Newtonian fluids is given by:
Re_g = ρ v_avg R^n / K
where ρ is the density of the fluid.
Using the given values, we get:
Re_g = (1100 kg/m^3) (0.097 m/s) (0.0175 m)^0.45 / 5.2 Pa s^0.45
Re_g ≈ 224.6
Therefore, the generalized Reynolds number is approximately 224.6, indicating that the flow is in the laminar regime.