Question

Q1. (a) For a flow phenomenon governed mainly by viscous forces, derive the scale factors for velocity, force and power in terms of 24, , and I, where subscripts L, p and u are length, density and dynamic viscosity respectively and is the ratio of the value of "x" in the prototype to the value of "x" in the model. [4 marks) (b) Volumetric flow rate (discharge) Q through a circular pipe with diameter D can be measured by measuring the speed of rotation N of a propeller-based flowmeter fitted inside the pipe. Use dimensional analysis with P, N,D as recurring set to show that Q = kND', where the propeller coefficient k is a function of and the propeller Reynolds number D. is the D propeller diameter; pand u are fluid density and dynamic viscosity respectively. [6 marks) PND (c) A 50 mm-diameter propeller was installed in a 150 mm-diameter water pipe and the propeller speed was measured for a range of water discharge in the pipe. The water had a density and dynamic viscosity of 1000 kg/m and 0.00112 Ns/m² respectively. The measured results were as follows: Q (litres/s): 12 28 45 63 95 120 160 180 N(rps): 5 10 15 20 30 40 60 80 Plot the dependence of propeller coefficient against propeller Reynolds number. A geometrically similar propeller with diameter 100 mm was installed in a 300 mm-diameter pipe conveying oil with density 800 kg/m' and dynamic viscosity 0.007 Ns/m². Estimate the discharge in the oil pipe for measured propeller speeds of 40 rps and 90 rps. (10 marks)

Answer 1

This problem is based on dimensional analysis in process engineering problems. Proceed as follows:

I 192: Variables: velocity (9), Force (F), Power (P), viscosity (4), density (8) , & length (L). (6 variables in 3 dimensions : M, L,T). 3 dimensioners groups: Ti, Tz, &Itz. T = u Lagu T2 = FL" sou & T=PLstue ✓ a L [u] [_]* [M] 4 Lilta-3b-c=0 Lilta-b-c=0 T: -2 -C=0 > C=-3 Miltbtc=0 T: -3-C=0 >> C-3 M: 1+b+C 20 6=2] | solving for a : - [F] [L]“L£J* [u]=- [P] [_7°816) [1.7") L)(ML"]*MCT) = fever! (L.962-49*(m2* Fok424) 29 (moeghel (ME'T=1 L:2 +9-3 b-c=0 It-c=0= [C=-1 M: b+c=0 =) (b=1 solving, (a = 1 Mi= utslet solving for a : (a=0 OR r,=uLS Ta= F8/42 T3=PL821 for scaling: ł., Ma & Is should remain constant over 2 sets of experiments. i Vilisi=: Uz Lafz; Fisi Fa fa . f 'P, L. P² = Pales r. hila Am fi lag | Fa Pa MI M22 MP TH2² fa 1 L --1220 3 12 log Tung