Question

If a solution with 50% one fluid 50% another flowing through a pipe, with a given temperature, and internal diameter of pipe and pressure gradient per metre.

How would you confirm the flow is laminar ?

Calculate the volumetric flow rate ?

And calculate the local velocity of liquid at a given perpendicular distance from the inner wall of the pipe ?

Answer 1

(a) To determine laminar flow:

Let average density of the mixture = ? and length of the pipe = L

The average density of mixture can be found by the equation, 1/? = (0.5/?₁ + 0.5/?₂) = 0.5(1/?₁ + 1/?₂); where ?₁ and ?₂ are densities of 1^st and 2^nd fluids respectively.

Internal diameter = d, Pressure gradient = dp/dl

For laminar flow, average velocity of fluid inside pipe, u_avg = -(1/8?)*(dp/dL)*(d/2)² … (i)

Where, ? = viscosity of fluid.

From equation (i) the value of u_avg can be found out.

Reynold’s number, R_e = (?*u_avg*d)/? …(ii)

From equation (ii) the value of R_e can be found out. If R_e < 2000, then the flow is laminar.

(b) Calculating the volumetric flow rate:

Area of pipe cross section, A = (?/4)*d²

Hence, volumetric flow rate, q = u_avg*A = u_avg*(?/4)*d²

(d) Calculating local velocity at any distance from inner wall of pipe:

For laminar flow, velocity of fluid at a perpendicular distance ‘r’ from the inner wall of the pipe,

u_r = -(1/4?)*(dp/dL)*[(d/2)² – r²]…(iii)

The equation (iii) will provide the local velocity of liquid at a given perpendicular distance from the inner wall of the pipe.