By Richardson J.F & Harker J.H
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Extra info for Coulson and Richardson Chemical Engineering-Volume 4-Solutions to The Problems in Volume 1
5. A pitot tube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. 585 For the power-law fluid: RDk dux dy n A force balance on a fluid core of radius s in pipe of radius r gives: Rs 2 sl D or: Integrating: Rs D k dux ds n D P s2 Ps 2l dux D ds P 2kl 1/n P 2ks 1/n ux D s1/n nC1 n s n C constant nC1 SOFTbank E-Book Center Tehran, Phone: 66403879,66493070 For Educational Use. 34 Thus, the calculated and experimental values of ux /uCL agree within reasonable limits of experimental accuracy.
0 N/m2 and ( P/l) to be determined as 7860 N/m2 /m. p For case (c), the flowrate D 2Q and the pipe diameter D a 2. 52 N/m2 and P/l D 821 N/m2 /m. SOFTbank E-Book Center Tehran, Phone: 66403879,66493070 For Educational Use. 36 The rheological properties of a particular suspension can be approximated reasonably well by either a “power law” or a “Bingham plastic” model over the shear rate range of 10 to 50 s 1 . 2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham plastic model?
Solution Making a force balance over an element distance r from the axis of a pipe whose radius is a, then: P r 2 D du/dr 2 rl where u is the velocity at distance r and l is the length of the pipe. Hence: and: du D P/2 l r dr P/4 l r 2 C constant uD When r D a, u D 0, the constant D P/4 l a2 and hence u D P/4 l a2 r 2 . At a distance r from the axis, the time taken for the fluid to flow through a length l is given by 4 l2 / P a2 r 2 . The volumetric rate of flow from r D 0 to r D r is: r P/4 l a2 D r 2 2 rdr 0 D P /8 l 2a2 r 2 Volumetric flowrate over the whole pipe D and the mean velocity D r4 P a4 /8 l Pa2 /8 l.