Research Papers: Fundamental Issues and Canonical Flows

Loss Coefficients for Periodically Unsteady Flows in Conduit Components: Illustrated for Laminar Flow in a Circular Duct and a 90 Degree Bend

[+] Author and Article Information
Bastian Schmandt

e-mail: Bastian.Schmandt@tuhh.de

Heinz Herwig

e-mail: h.herwig@tuhh.de
Institute of Thermo-Fluid Dynamics,
Hamburg University of Technology,
21073 Hamburg, Germany

Manuscript received June 15, 2012; final manuscript received November 16, 2012; published online February 22, 2013. Assoc. Editor: Chunill Hah.

J. Fluids Eng 135(3), 031204 (Feb 22, 2013) (9 pages) Paper No: FE-12-1295; doi: 10.1115/1.4023196 History: Received June 15, 2012; Revised November 16, 2012

Losses in a flow through conduit components of a pipe system can be accounted for by head loss coefficients K. They can either be determined experimentally or from numerical solutions of the flow field. The physical interpretation is straight forward when these losses are related to the entropy generation in the flow field. This can be done based on the numerical solutions by the second law analysis (SLA) successfully applied for steady flows in the past. This analysis here is extended to unsteady laminar flow, exemplified by a periodic pulsating mass flow rate with the pulsation amplitude and the frequency as crucial parameters. First the numerical model is validated by comparing it to results for unsteady laminar pipe flow with analytical solutions for this case. Then K-values are determined for the benchmark case of a 90 deg bend with a square cross section which is well-documented for the steady case already. It turns out that time averaged values of K may significantly deviate from the corresponding steady values. The K-values determined for steady flow are a good approximation for the time-averaged values in the unsteady case only for small frequencies and small amplitudes.

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Fig. 1

90 deg bend and entropy generation rates due to it (left) geometrical details; (right) entropy generation rate S· in the volume V˜ due to the 90 deg bend; V˜ = Vu + Vc + Vd

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Fig. 2

Axial velocity during one period at 10 different radii for F = 100 (f = 400) and M·A = 0.5

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Fig. 3

Comparison of numerical and analytical solutions for the velocity profile in a circular duct; velocity amplitude ratio (left) and phase lag (right) at discrete radii. Symbols: numerical solutions for different values of r; lines: analytical solutions for the corresponding values of r.

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Fig. 4

Nondimensional instantaneous entropy generation rate for a pulsating pipe flow. Mass flow rate according to Eq. (13) with M·A= 0.5. Symbols: numerical solutions for different values of r; lines: analytical solutions for the corresponding values of r.

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Fig. 5

Unsteadiness coefficient Cun according to Eq. (28) for a pulsating pipe flow with the mass flow rate (Eq. (13)) and M·A = 0.5

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Fig. 6

Head loss coefficient Kϕ,st for steady flow through a 90 deg bend (benchmark case); validation of the numerical model, taken from Ref. [6]. Line: numerical result; symbols: experiments.

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Fig. 7

Nondimensional instantaneous entropy generation rate for a pulsating flow through a 90 deg bend, M·A= 0.1 ; -: steady case

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Fig. 8

Nondimensional instantaneous entropy generation rate for a pulsating flow through a 90 deg bend, M·A = 0.5; -: steady case

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Fig. 9

Unsteadiness coefficient (Eq. (30)), note the different Cun scales

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Fig. 10

Simulated instantaneous entropy generation per length along the centerline of a circular duct at F = 1000. Developed flow after an adjustment length shorter than 1Dh.

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Fig. 11

Grid in the upper halve of a symmetric square cross section 90 deg bend



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