Question

Glycerin (relative density 1.26, viscosity, μ, 0.9 Pa·S) is pumped through a straight 50mm diameter horizontal pipeline, 60 m long. If the pressure drop, dp, between the pipe's inlet and outlet is 30 Pa, determine the flow rate, Q. Is this flow laminar? 8, Work from first principles and derive the Hagen-Poiseuille equation for solving this problem In question 8: what would be the percentage reduction in the original flow rate if the diameter is decreased by 10% but the pressure gradient remains the same? Further, what would be the percentage increase in thepressre gradient if the original fow rate is maintained with the reduced diameter? 9.

Answer 1

(8) Solution:-

Derivation of Hazen Poisseuille equation:-

F_viscosity = - µA (∆V_x/∆y)

Faster Lamina

F_{viscosity,fast} = - 2πrµ∆x(∆V/dr)

Slower lamina

F_{viscosity,slow} = 2π(r +dr) µ∆x(dV/dr)

Puttiing it all together

0 = F_pressure + F_{viscosity,fast} + F_{viscosity,slow}

Q = ∆P πR⁴/(8µL)

Given

∆P =30 Pa

R = 25 mm

Then

Q = 30*π*25⁴/(8*60*10³ *0.9)

    Q = 85.22 mm³/s           Answer

Reynold number(Re) = ρVL/µ

                                     = 1.26*85/(π/4*50²)*60*10³/0.9

                                    = 3626

Reynold number is greater than 2000 .So flow is not laminar flow. It is transition flow.

(9) Solution:-

When diameter reduce 10 %

Then

Q = 30*π*(22.5)⁴/(8*60*10³*0.9)

      = 55.91 mm³/s

% reduce in Q = (85.22 – 55.91)/(85.22)*100

                         = 34.39 %    Answer

When diameter reduce to 10 %

Discharge remain unchanged

85.22 = ∆P*π*(22.5)⁴ /(8*60*10³ *0.9)

∆P = 45.72 Pascal

% increase in pressure(∆P) = (45.72 – 30)/30*100 = 52.4%    Answer