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Numerical evaluation of entropy generation in isolated airfoils and Wells turbines

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Abstract

In recent years, a number of authors have studied entropy generation in Wells turbines. This is potentially a very interesting topic, as it can provide important insights into the irreversibilities of the system, as well as a methodology for identifying, and possibly minimizing, the main sources of loss. Unfortunately, the approach used in these studies contains some crude simplifications that lead to a severe underestimation of entropy generation and, more importantly, to misleading conclusions. This paper contains a re-examination of the mechanisms for entropy generation in fluid flow, with a particular emphasis on RANS equations. An appropriate methodology for estimating entropy generation in isolated airfoils and Wells turbines is presented. Results are verified for different flow conditions, and a comparison with theoretical values is presented.

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Change history

  • 06 May 2019

    In the original publication of the article, Eqs. (26)���(28) are incorrect due to a missing �� symbol, inadvertently omitted when reporting the equations in the article.

  • 06 May 2019

    In the original publication of the article, Eqs. (26)���(28) are incorrect due to a missing �� symbol, inadvertently omitted when reporting the equations in the article.

Abbreviations

CFD:

Computational fluid dynamics

DNS:

Direct numerical simulation

LES:

Large eddy simulation

RANS:

Reynolds Averaged Navier Stokes

\({\mathbf{a}}\) :

Acceleration (m s−2)

A :

Area (m2)

\(c_p\) :

Constant pressure specific heat (m2 s−2 K−1)

\(c_v\) :

Constant volume specific heat (m2 s−2 K−1)

dS :

Differential surface (m2)

D :

Drag force (kg m s−2)

f :

Frequency (s−1)

\({\mathbf{F}}\) :

Force (kg m s−2)

h :

Specific enthalpy (m2 s−2)

H :

Total specific enthalpy (m2 s−2)

k :

Specific turbulent kinetic energy (m2 s−2)

p :

Pressure (kg m−1 s−2)

\({\mathbf{q}}\) :

Thermal flux (kg s−3)

\(r_{tip}\) :

Turbine tip radius (m)

R :

Gas constant (m2 s−2 K−1)

s :

Specific entropy (m2 s−2 K−1)

\(S_1\) :

Near-field (surface) (m2)

\(S_2\) :

Far-field (surface) (m2)

\(\dot{S}_G\) :

Entropy generation rate (kg m2 s−3 s−1)

t :

Time (s)

T :

Temperature (K), torque (kg m2 s−2)

\(u,{\mathbf {u}}\) :

Velocity (m s−1)

\(V_a\) :

Turbine axial flow velocity (m s−1)

\(\epsilon\) :

Turbulent dissipation rate (m2 s−3)

\(\Phi\) :

Dissipation function (s−2)

\(\lambda\) :

Thermal conductivity (kg m s−3 K−1)

\(\lambda _t\) :

Turbulent conductivity (kg m s−3 K−1)

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\mu _t\) :

Turbulent dynamic viscosity (kg m−1 s−1)

\(\omega\) :

Specific turbulent dissipation rate (s−1), rotational speed (s−1)

\(\Omega\) :

Mid-field (volume) (m\(^{3}\))

\({\varvec{\varPi }}\) :

Viscous stress tensor (kg m−1 s−2)

\({\varvec{\varPi }_R}\) :

Reynolds stress tensor (kg m−1 s−2)

\(\rho\) :

Density (kg m−3)

\(\sigma _T\) :

Entropy generation rate per unit mass due to heat transfer (m2 s−3 K−1)

\(\sigma _V\) :

Entropy generation rate per unit mass due to fluid flow (m2 s−3 K−1)

\(c_d\) :

Drag coefficient

e :

Nepero constant

k :

Non-dimensional frequency

\(K_{\dot{S}_G}\) :

Non-dimensional entropy generation rate

M :

Mach number

\({\mathbf {n}}\) :

Surface normal unit vector

Re :

Reynolds number

\(\alpha\) :

Incidence angle

\(\gamma\) :

Ratio of specific heats

\(\phi\) :

Turbine flow coefficient

\(\cdot\) :

Dot product

\(\varDelta\) :

Difference

\(\frac{\partial ()}{\partial t}\) :

Partial time derivative (s−1)

\(\frac{D()}{Dt}\) :

Total time derivative (s−1)

\(\nabla\) :

Nabla operator (m−1)

\(\nabla ^2\) :

Laplacian operator (m−2)

\(\nabla ^s\) :

Sum of gradient and gradient transposed (m−1)

\(\otimes\) :

Cross product

\(\overline{()}\) :

Time average

\('\) :

Fluctuating part

eff :

Effective

\(\infty\) :

At infinity (upstream)

mf :

Mean flow

t :

Turbulent, due to turbulence

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Funding

This work has been funded by the Regione Autonoma Sardegna under Grant F72F16002880002 (L.R. 7/2007 n. 7 - year 2015).

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Authors and Affiliations

  1. Department of Mechanical, Chemical and Materials Engineering, University of Cagliari, via Marengo 2, 09123, Cagliari, Italy

    Tiziano Ghisu, Francesco Cambuli, Pierpaolo Puddu & Natalino Mandas

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

    Pranay Seshadri & Geoffrey T. Parks

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  1. Tiziano Ghisu
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Correspondence to Tiziano Ghisu.

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Ghisu, T., Cambuli, F., Puddu, P. et al. Numerical evaluation of entropy generation in isolated airfoils and Wells turbines. Meccanica 53, 3437–3456 (2018). https://doi.org/10.1007/s11012-018-0896-1

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