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Research Papers: Fundamental Issues and Canonical Flows

# Internal Flow Losses: A Fresh Look at Old Concepts

[+] Author and Article Information
Bastian Schmandt

Institute of Thermo-Fluid Dynamics, Hamburg University of Technology, 21073 Hamburg, GermanyBastian.Schmandt@tuhh.de

Heinz Herwig

Institute of Thermo-Fluid Dynamics, Hamburg University of Technology, 21073 Hamburg, Germanyh.herwig@tuhh.de

Exergy: maximum theoretical work obtainable from the energy interacting with the environment to equilibrium.

Anergy: energy minus exergy.

The influence of the so-called second or bulk viscosity is neglected here, since a completely incompressible (model) fluid is assumed.

Taking the mean temperature in Eqs. (6) and (7) neglects the small effect of the temperature distribution that may exist in the cross section.

To account for losses in the curved parts of the bend it is necessary to perform an integration of $S·′′′$ over the volume first, using the actual discrete volumes of the grid cells localized upstream of a clipping plane at a certain position $sc$ leading to $S·(sc)$. A subsequent derivation $S·′(sc)=dS·(sc)/dsc$ then gives the cross section averaged value $S·′$. Thus an integration $∫sc,asc,bS·′(sc)dsc$, i.e., the “area below the $S·′/S·0′$ -line” in Fig. 6, between $sc,a$ and $sc,b$ always represents the amount of losses between the two cross sections located at $sc,a$ and $sc,b$.

Meshes in OpenFOAM are stored as unstructured meshes anyway, regardless of their topology.

J. Fluids Eng 133(5), 051201 (May 31, 2011) (10 pages) doi:10.1115/1.4003857 History: Received October 25, 2010; Revised March 23, 2011; Published May 31, 2011; Online May 31, 2011

## Abstract

Losses in a flow field due to single conduit components often are characterized by experimentally determined head loss coefficients K. These coefficients are defined and determined with the pressure as the critical quantity. A thermodynamic definition, given here as an alternative, is closer to the physics of flow losses, however. This definition is based upon the dissipation of mechanical energy as main quantity. With the second law of thermodynamics this dissipation can be linked to the local entropy generation in the flow field. For various conduit components K values are determined and physically interpreted by determining the entropy generation in the component as well as upstream and downstream of it. It turns out that most of the losses occur downstream of the components what carefully has to be taken into account when several components are combined in a flow network.

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## Figures

Figure 1

Determination of the overall entropy generation rate due to a conduit component; Δϕu: additional specific dissipation upstream of the component; ϕc: specific dissipation in the component; Δϕd: additional specific dissipation downstream of the component

Figure 2

Laminar radial flow with one rough wall. Figures and data from Ref. [15]

Figure 3

Moody chart (full lines for k/Dh=const,Dh=4A/P) and special results for a Loewenherz thread roughness; broken lines: second law analysis, symbols: experiment. Figure and data from Ref. [12].

Figure 4

K values of a 90 deg bend with circular cross section and r/Dh=1; +: Ito, see Ref. [16]; ○: Hofmann (losses of a straight tube with length of the bends centerline added, values for f taken from the same source), see Ref. [17]; - Idelchik (piecewise correlation for Re<1×104,1×104<Re<2×105,andRe>2×105), see Ref. [2]; --: Miller, see Ref. [18]

Figure 5

Cross sectional distribution of local entropy generation rates S·′′′=(S·′′′−)+(S·′′′)′ for the highest Reynolds number Re=150000, normalized with maximum values, logarithmic scale; dark: low values, light: high values

Figure 6

Cross section averaged entropy generation S·′ along the 90deg -bend’s centerline; dark: losses within the bend, light: additional losses in the upstream and downstream tangents, up to Lu and Ld

Figure 7

Numerically determined K-values of a 90deg bend with circular cross section and r/Dh=1 compared to those in Figure 4; ⊙ numerical values from (8); +: Ito, see Ref. [16]; ○: Hofmann, see [17]; -: Idelchik (correl.), see Ref. [2]; --: Miller, see Ref. [18]

Figure 8

Cross sectional averaged entropy generation S·′ along the 90deg -bend centerline for a square cross section; dark: losses inside the bend, light: additional losses in the upstream and downstream tangents up to Lu and Ld

Figure 9

Cross sectional averaged entropy generation S·′ along the U-shaped bend combination centerline; dark: losses inside the bends, light: additional losses in the upstream and downstream tangents up to Lu and Ld, hatched: downstream 90 deg bend

Figure 10

Cross sectional averaged entropy generation S·′ along the S-shaped bend combination centerline; dark: losses inside the bends, light: additional losses in the upstream and downstream tangents up to Lu and Ld, hatched: downstream 90deg bend

Figure 11

Streamlines of the time averaged flow for the square cross sectioned bend combinations for Re=100000 in the symmetry plain, background shaded according the entropy generation rate S·′′′ (logarithmic scale), light: maximum values

Figure 12

Typical cross section of the numerical grids

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