# How to Determine Losses in a Flow Field: A Paradigm Shift towards the Second Law Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Losses and the Second Law of Thermodynamics

- (1)
- The higher accuracy of the final results;
- (2)
- Detailed information about the location of losses within the flow field;
- (3)
- A direct physical interpretation of the losses in terms of exergy losses; this information is not available from drag and friction results, since it also depends on the temperature level;
- (4)
- A unique assessment quantity for losses in the flow and in the temperature field when heat transfer is also involved.

#### 2.1. The Alternative Approach in General

#### 2.2. The Alternative Approach for External Flows

#### 2.3. The Alternative Approach for Internal Flows

#### 2.4. Advantage of the Alternative Approach

conventional approach | alternative approach | |

external flow | $\text{c}}_{\text{D}}=\frac{{F}_{\text{D}}}{\frac{\varrho}{2}{u}_{\infty}^{2}A$ | $\text{c}}_{\text{D}}=\frac{T}{\frac{\varrho}{2}{u}_{\infty}^{3}A}{\dot{S}}_{\text{irr,D}$ |

internal flow | $\text{K}=\frac{\mathsf{\Delta}p}{\frac{\varrho}{2}{u}_{\text{m}}^{2}}$ | $\text{K}=\frac{T}{\frac{\varrho}{2}{u}_{\text{m}}^{3}A}{\dot{S}}_{\text{irr,D}}$ |

## 3. The Second Law Analysis (SLA) of a Flow Field

#### 3.1. Literature about Entropy and the SLA Approach

#### 3.2. Local Entropy Generation Rates and Turbulence Modeling

#### 3.3. Overall Entropy Generation Rates

**Figure 1.**Determination of the overall entropy generation rate due to a conduit component. $\mathsf{\Delta}{\dot{S}}_{\text{irr,D,u}}$ is the additional entropy generation upstream of the component, ${\dot{S}}_{\text{irr,D,c}}$ the entropy generation inside the component and $\mathsf{\Delta}{\dot{S}}_{\text{irr,D,d}}$ the additional entropy generation downstream of the component.

- for external flows:$${\dot{S}}_{\text{irr,D}}=\underset{V}{\int}\left({\dot{S}}_{\text{irr,D}}^{\u2034}-{\dot{S}}_{\text{irr,D0}}^{\u2034}\right)\text{d}V$$
- for internal flows:$$\underset{\phi \dot{m}/{T}_{\text{m}}}{\underbrace{{\dot{S}}_{\text{irr,D}}\phantom{\rule{-12.75pt}{0ex}}\phantom{{\int}_{{V}_{\text{c}}}}}}=\underset{\mathsf{\Delta}{\dot{S}}_{\text{irr,D,u}}=\mathsf{\Delta}{\phi}_{u}\dot{m}/{T}_{\text{m}}}{\underbrace{\underset{{V}_{\text{u}}}{\int}({\dot{S}}_{\text{irr,D}}^{\prime \prime \prime}-{\dot{S}}_{\text{irr,D}0}^{\prime \prime \prime})\text{d}V}}+\underset{{\dot{S}}_{\text{irr,D,c}}={\phi}_{c}\dot{m}/{T}_{\text{m}}}{\underbrace{\underset{{V}_{\text{c}}}{\int}{\dot{S}}_{\text{irr,D}}^{\prime \prime \prime}\text{d}V}}+\underset{\mathsf{\Delta}{\dot{S}}_{\text{irr,D,d}}=\mathsf{\Delta}{\phi}_{d}\dot{m}/{T}_{\text{m}}}{\underbrace{\underset{{V}_{\text{d}}}{\int}({\dot{S}}_{\text{irr,D}}^{\prime \prime \prime}-{\dot{S}}_{\text{irr,D}0}^{\prime \prime \prime})\text{d}V}}$$

#### 3.4. Entropy Generation versus Dissipation in a Flow Field

- from a thermodynamics point of view, dissipation is quantified by the entropy generation involved in this process and not vice versa;
- entropy generation is immediately linked to the loss of exergy (see Equation (4)), but dissipation is not; how much exergy is lost in a dissipative process is determined by the entropy generated;
- when further mechanisms, which are irreversible in character, are involved (like heat transfer), their irreversibility can be characterized by a further entropy generation in the flow field and added to that due to dissipation; the overall loss (of exergy) in a flow field is then uniquely accounted for by the overall entropy generation.

## 4. Examples for External Flows

#### 4.1. Drag of a Flat Plate for Laminar Boundary Layer Flow

**Figure 2.**Discretization of the near-flow field around the flat plate; the plate is located between $x/L=0$ and 1.

**Figure 3.**(

**a–c**): Distribution of ${\dot{S}}_{\text{irr,D}}^{\prime}$ along the x-axis, non-dimensionalized with the overall entropy generation rate, ${\dot{S}}_{\text{irr,D}}$, and the length of the plate, L. The location of the flat plate is between $x/L=0$ and 1. (

**d**) Drag coefficient ${\text{c}}_{\text{D}}\left(\text{Re}\right)$.

#### 4.2. Drag of a Rising Bubble

**Figure 4.**(

**a**–

**c**): Distribution of ${\dot{S}}_{\text{irr,D}}^{\prime}$ along the x-axis, non-dimensionalized with the overall entropy generation rate, ${\dot{S}}_{\text{irr,D}}$, and the diameter, D, of the bubble. (

**d**): drag coefficient.

**Figure 6.**Entropy generation ${\dot{S}}_{\text{irr,D}}^{\prime}/({\dot{S}}_{\text{irr,D}}/D)$ for $\text{Re}=128$. (

**a**) Bubble: slip at the boundary; (

**b**) sphere: non-slip at the boundary. More details can be found in [39].

## 5. Examples for Internal Flows in Conduit Components

#### 5.1. Straight Pipes and Channels

**Figure 7.**Experimental data and SLA results for three different pipes with rough walls and turbulent flow; experimental data are from [46].

#### 5.2. Ninety-Degree Bend and Bend Combinations

**Figure 9.**Geometry and K-value of double bend combinations; laminar flow; more details in [50].

**Table 2.**Comparison of K-values for three different 90-degree bend combinations with twice the K-value of a single 90-degree bend; more details in [50]; laminar flow.

Re | 2 × 90-degree bend | 0-degree double bend | 180-degree double bend | 90-degree/90-degree double bend |

4 | 44.38 | 43.76 | 43.76 | 43.51 |

8 | 22.50 | 22.20 | 22.14 | 22.05 |

16 | 11.82 | 11.67 | 11.57 | 11.59 |

32 | 6.93 | 6.71 | 6.63 | 6.76 |

64 | 4.51 | 4.35 | 4.31 | 4.60 |

128 | 4.53 | 3.15 | 3.06 | 3.53 |

256 | 4.34 | 3.06 | 2.31 | 2.90 |

512 | 4.54 | 3.18 | 2.20 | 2.68 |

#### 5.3. Branching Conduit Components

**Figure 10.**Dividing of the mean flow rate, $\dot{m}$, into ${\dot{m}}_{12}$ and ${\dot{m}}_{13}$ for the symmetric T-junction with a square cross-section $A={D}_{\text{h}}^{2}$. Corners are described by circular arcs of radius $R={D}_{\text{h}}$. ${L}_{\text{Vu}}$, ${L}_{\text{Vd12}}$, ${L}_{\text{Vd13}}$: lengths over which effects are accounted for numerically upstream and downstream.

**Figure 11.**Head change coefficient ${K}_{12}$ for the partial flow with ${\dot{m}}_{12}$; more details in [51].

**Figure 12.**Head change coefficients for engulfed and symmetric flows. Onset of engulfment at $\text{Re}>205$. Small pictures show indicator field a at the end of the combining duct.

#### 5.4. Loss Coefficients for Compressible Flows

- (1)
- Density changes due to variations in pressure and temperature, and thus, there is no longer a fully developed flow far upstream and downstream of a conduit component.
- (2)
- The impact of a component in terms of an additional entropy generation due to the component again can be cast into the three parts: $\mathsf{\Delta}{\dot{S}}_{\text{irr,u}}$, ${\dot{S}}_{\text{irr,c}}$ and $\mathsf{\Delta}{\dot{S}}_{\text{irr,d}}$; see Figure 1. While $\mathsf{\Delta}{\dot{S}}_{\text{irr,u}}$ is a finite value, due to the gradual deviation of ${\dot{S}}_{\text{irr}}^{\u2034}$ from the undisturbed value, $\mathsf{\Delta}{\dot{S}}_{\text{irr,d}}$ is no longer finite, since ${\dot{S}}_{\text{irr}}^{\u2034}$ in the downstream part does not return to its undisturbed value. There is, however, a certain fixed shift, $\mathsf{\Delta}x$, in the streamwise coordinate by which $\mathsf{\Delta}{\dot{S}}_{\text{irr,d}}$ becomes finite again, and ${\dot{S}}_{\text{irr}}$, due to the component, is recovered.
- (3)
- For incompressible, flow K according to Equation (10) could be called the “head loss coefficient”, and it could be immediately attributed to the specific dissipation in the flow field due to the validity of Equation (9). For compressible flow, still, K according to Equation (10) can be used as a “loss coefficient”. It should be named the “exergy loss coefficient”, however, since exergy and not total head is the quantity that gets lost, and the (exergy) loss cannot be fully attributed to the specific dissipation; for details see [54].Since, however, exergy is related to the ambient temperature, a “reference exergy loss coefficient”:$${\text{K}}_{\text{RE}}=\frac{{T}_{\text{R}}}{\frac{{u}_{\text{m}}^{2}}{2}\dot{m}}{\dot{S}}_{\text{irr}}$$$${\text{K}}_{\text{E}}=\frac{{T}_{\infty}}{{T}_{\text{R}}}{\text{K}}_{\text{RE}}$$

**Figure 13.**Reference exergy loss coefficients for compressible flow; more details in [54].

#### 5.5. Loss Coefficients for Unsteady Flows

**Figure 14.**Cross-section entropy generation in a circular duct along one period.$\dot{M}$, according to Equation (37) with ${\dot{\text{M}}}_{\text{A}}=0.5$F: nondimensional frequency data taken from [55].

**Figure 15.**Unsteadiness coefficient for the mass flow rate of Figure 14.

## 6. Flow with Heat Transfer

**Figure 16.**Influence of wall roughness ${\text{k}}_{\text{S}}\le 0.5$ on the entropy generation rates; all values refer to ${\dot{S}}_{\text{irr},\circ}^{\prime}$ (at ${\text{Re}}_{\text{opt,s}}$).

## 7. Conduit Components As Part of a System

- ${\text{N}}_{i}=0$: reversible energy transfer operation;
- ${\text{N}}_{i}=1$: energy transfer operation that completely devaluates the energy (rate).

**Figure 17.**Energy devaluation by consecutive energy transfer operations illustrated by the decrease of exergy during the energy transfer operations. Progress in time for finite energies (progress in process steps for finite energy rates).

## 8. The Second Law Analysis Based on DNS Results for Turbulent Flows

**Figure 18.**The entropic potential and its use on the way the energy becomes part of the internal energy of the environment; here, the contribution of an energy transfer operation, i.

**Figure 19.**Geometrical arrangement of general regular wall roughness. H: Half channel height; h: Height of the roughness elements; b: Bottom width of the roughness elements; t: Top width of the roughness elements; s: Spacing of the roughness elements; α: Inclination angel with respect to the flow direction.

Test Case | $h/H$ | $h/s$ | $h/b$ | $t/b$ | α | Ref. | |
---|---|---|---|---|---|---|---|

SW | - | - | - | - | - | smooth wall | [66] |

R | $0.034$ | $0.14$ | 1 | 1 | 90 degrees | ||

obstacles | |||||||

B1 | $0.01$ | $0.2$ | 10 | 1 | 0 degrees | [67] | |

B2 | 0.02 | $0.4$ | 20 | 1 | 0 degrees | ||

B3 | 0.03 | $0.6$ | 30 | 1 | 0 degrees | ||

B4 | 0.04 | $0.8$ | 40 | 1 | 0 degrees | ||

B5 | 0.05 | $0.4$ | 20 | 1 | 0 degrees | ||

blades | |||||||

W1 | $0.02$ | $0.4$ | $0.866$ | 0 | 0 degrees | ||

W2 | 0.02 | $0.4$ | $1.866$ | 0 | 0 degrees | ||

wedges |

**Figure 20.**Conventionally rough wall: local change of entropy generation rate minus this quantity for a smooth wall (Re = 5,600); further details in [66].

**Figure 21.**Shark skin-textured walls: Local change of entropy generation rates minus this quantity for a smooth wall (Re = 5,600); further details in [67].

## 9. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Munson, B.; Young, D.; Okiishi, T. Fundamentals of Fluid Mechanics, 5th ed.; John Wiley & Sons, Inc.: New York, NY, USA, 2005. [Google Scholar]
- Moran, M.; Shapiro, H. Fundamentals of Engineering Thermodynamics, 3rd ed.; John Wiley & Sons, Inc.: New York, NY, USA, 1996. [Google Scholar]
- Baehr, H. Thermodynamik, 14th ed.; Springer-Verlag: Berlin, Germany, 2009. [Google Scholar]
- Herwig, H.; Kautz, C. Technische Thermodynamik; Pearson Studium: München, Germany, 2007. [Google Scholar]
- Schmandt, B.; Herwig, H. Internal Flow Losses: A Fresh Look at Old Concepts. J. Fluids Eng.
**2011**, 133, 051201. [Google Scholar] [CrossRef] - Schmandt, B.; Herwig, H. Diffuser and Nozzle Design Optimization by Entropy Generation Minimization. Entropy
**2011**, 13, 1380–1402. [Google Scholar] [CrossRef] - ASME. Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer; The American Society of Mechanical Engineers: New York, NY, USA, 2009. [Google Scholar]
- Atkins, P. The Second Law; Scientific American Books - W.H. Freeman and Company: New York, NY, USA, 1984. [Google Scholar]
- Goldstein, M.; Goldstein, I. The Refrigerator and the Universe; Harvard University Press: Cambridge, MA, USA, 1993. [Google Scholar]
- Dugdale, J. Entropy and its Physical Meaning; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Falk, G.; Ruppel, W. Energie und Entropie; Springer-Verlag: Berlin, Heidelberg; New York, NY, USA, 1976. [Google Scholar]
- Herwig, H.; Wenterodt, T. Entropie für Ingenieure: Erfolgreich das Entropie-Konzept bei energietechnischen Fragestellungen anwenden; Springer: Berlin, Germany, 2011. [Google Scholar]
- Lieb, E.; Yngvason, J. A fresh look at entropy and the second law of thermodynamics. Phys. Today
**2000**, 11, 106. [Google Scholar] - Beretta, G.; Ghoniem, A.; Hatsopoulos, G. Meeting the entropy challenge. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2008. [Google Scholar]
- Bejan, A. The concept of irreversibility in heat exchanger design: counter-flow heat exchangers for gas-to-gas applications. J. Heat Transf.
**1977**, 99, 274–380. [Google Scholar] [CrossRef] - Sekulic, D. Entropy generation in a heat exchanger. Heat Transf. Eng.
**1986**, 7, 83–88. [Google Scholar] [CrossRef] - Gaggioli, R. Second law analysis for process and energy engineering. In Efficiency and Costing, Second Laws Analysis of Processes; American Chemical Society: Washington, DC, USA, 1983. [Google Scholar]
- Bejan, A. A study of entropy generation in fundamental convective heat transfer. J. Heat Transf.
**1979**, 101, 718–725. [Google Scholar] [CrossRef] - Bejan, A. Entropy Generation through Heat and Fluid Flow; John Wiley & Sons: New York, NY, USA, 1982. [Google Scholar]
- Bejan, A. Entropy Generation Minimization; CRC Press: Boca Raton, New York, NY, USA, 1996. [Google Scholar]
- Hesselgreaves, J. Rationalisation of second law analysis of heat exchangers. J. Heat Mass Transf.
**2000**, 43, 4189–4204. [Google Scholar] [CrossRef] - Herwig, H.; Kock, F. Direct and indirect methods of calculating entropy generation rates in turbulent convective heat transfer problems. Heat Mass Transf.
**2007**, 43, 207–215. [Google Scholar] [CrossRef] - Anand, D. Second law analysis of solar powered absorption cooling cycles and systems. J. Sol. Energy Eng.
**1984**, 106, 291–298. [Google Scholar] [CrossRef] - Nuwayhid, R.; Moukalled, F.; Noueihed, N. On entropy generation in thermoelectric devices. Energy Conv. Management
**2000**, 41, 891–914. [Google Scholar] [CrossRef] - Assad, E. Thermodynamic analysis of an irreversible MHD power plant. Int. J. Energy Res.
**2000**, 24, 865–875. [Google Scholar] [CrossRef] - Shiba, T.; Bejan, A. Thermodynamic optimization of geometric structure in the counterflow heat exchanger for an environmental control system. Energy
**2001**, 26, 493–511. [Google Scholar] [CrossRef] - Saidi, M.; Yazdi, M. Exergy model of a vortex tube system with experimental results. Energy
**1999**, 24, 625–632. [Google Scholar] [CrossRef] - Adeyinka, O.B.; Naterer, G.F. Entropy-based metric for component-level energy management: application to diffuser performance. Int. J. Energy Res.
**2005**, 29, 1007–1024. [Google Scholar] [CrossRef] - San, J.; Jan, C. Second law analysis of a wet crossflow heat exchanger. Energy
**2000**, 25, 939–955. [Google Scholar] [CrossRef] - Ko, T.; Ting, K. Entropy generation and optimal analysis for laminar forced convection in curved rectangular ducts: a numerical study. Int. J. Therm. Sci.
**2006**, 45, 138–150. [Google Scholar] [CrossRef] - Lakshminarayana, B. Fluid Dynamics and Heat Transfer of Turbomachinery; Wiley-Interscience Publication: New York, NY, USA, 1996. [Google Scholar]
- Kis, P.; Herwig, H. A critical analysis of turbulent natural and forced convection in a plane channel based on direct numerical simulation. Int. J. Comp. Fluid Dyn.
**2011**, 25, 387–399. [Google Scholar] [CrossRef] - Kock, F.; Herwig, H. Entropy production calculation for turbulent shear flows and their implementation in cfd codes. Int. J. Heat Fluid Flow
**2005**, 26, 672–680, CHT04. [Google Scholar] [CrossRef] - Menter, F. Improved Two-Equation k-omega Turbulence Models for Aerodynamic Flows. NASA Tech. Memo.
**1992**, 103975. [Google Scholar] - Kock, F.; Herwig, H. Local entropy production in turbulent shear flows: a high-Reynolds number model with wall functions. Int. J. Heat Mass Transf.
**2004**, 47, 2205–2215. [Google Scholar] [CrossRef] - Miller, D.S. Internal Flow Systems, 2nd ed.; BHRA, 1978; reprinted 1990. [Google Scholar]
- Van Dyke, M. Perturbation Methods in Fluid Mechanics; Parabolic Press: Stanford, CA, USA, 1975. [Google Scholar]
- Messiter, A. Boundary Layer Interaction Theory. J. Appl. Mech
**1983**, 50, 1104–1113. [Google Scholar] [CrossRef] - Herwig, H.; Schmandt, B. Drag with external and pressure drop with internal flows: a new and unifying look at losses in the flow field based on the second law of thermodynamics. Fluid Dyn. Res.
**2013**, 45, 055507. [Google Scholar] [CrossRef] - Peters, F.; Gaertner, B. Scaling parameters of bubbles and drops; interpretation and case study with air in silicon oil. Acta Mech.
**2011**, 219, 189–202. [Google Scholar] [CrossRef] - Clift, R.; Grace, J.; Weber, M. Bubbles, Drops and Particles; Academic Press: New York, San Francisco, London, 1978. [Google Scholar]
- Levich, V. Physiochemical Hydrodynamics; Prentice Hall: Englewoog Cliffs, NJ, USA, 1962. [Google Scholar]
- Moody, L. Friction factors for pipe flow. Trans. ASME
**1944**, 66, 671. [Google Scholar] - Nikuradse, J. Strömungsgesetze in rauhen Rohren. Forschung auf dem Gebiet des Ingenieurwesens
**1933**, 361, 1–22. [Google Scholar] - Herwig, H.; Gloss, D.; Wenterodt, T. A new approach to understand and model the influence of wall roughness on friction factors for pipe and channel flows. J. Fluid Mech.
**2008**, 613, 35–53. [Google Scholar] [CrossRef] - Schiller, L. Über den Strömungswiderstand von Rohren verschiedenen Querschnitts- und Rauhigkeitsgrades. Z. Angew. Math. Mech.
**1923**, 3, 2–13. [Google Scholar] [CrossRef] - Herwig, H.; Gloss, D.; T., W. Flow in Channels with Rough Walls - Old and New Concepts. In Proceedings of the Sixth International Conference on Nanochannels, Microchannels and Minichannels, Darmstadt, Germany, 23-25 June 2008. No. ICNMM2008-26064.
- Gloss, D.; Herwig, H. Micro channel roughness effects: a close-up view. Heat Transf. Eng.
**2009**, 32, 62–69. [Google Scholar] [CrossRef] - Gloss, D.; Herwig, H. Wall roughness Effects in Laminar Flows: An often ignored though significant Issue. Exp. Fluids
**2010**, 49, 658–665. [Google Scholar] [CrossRef] - Herwig, H.; Schmandt, B.; Uth, M.F. Loss Coefficients in Laminar Flows: Indispensable for the Design of Micro Flow Systems. In 8th International Conference on Nanochannels, Microchannels, and Minichannels: Parts A and B, Montreal, Quebec, Canada, August 15, 2010; pp. 1517–1528. [CrossRef]
- Schmandt, B.; Herwig, H. Performance Evaluation of the Flow in Micro Junctions: Head Change Versus Head Loss Coefficients. In 11th International Conference on Nanochannels, Microchannels, and Minichannels, Sapporo, Japan, June 1619, 2013. [CrossRef]
- Hoffmann, M.; Schlüter, M.; Räbiger, N. Experimental Investigation of Liquid-Liquid Mixing in T-Shaped Micro-Mixers Using μ-LIF and μ-PIV. Chem. Eng. Sci.
**2006**, 61, 2968–2976. [Google Scholar] [CrossRef] - Schmandt, B.; Iyer, V.; Herwig, H. Determination of head change coefficients for dividing and combining junctions: A method based on the second law of thermodynamics. Chem. Eng. Scie.
**2014**, 111, 191–202. [Google Scholar] [CrossRef] - Schmandt, B.; Herwig, H. Loss Coefficients for Compressible Flows in Conduit Components Under Different Thermal Boundary Conditions. In Submitted to The 15th International Heat Transfer Conference (IHTC-15), 2014.
- Schmandt, B.; Herwig, H. Loss Coefficients for Periodically Unsteady Flows in Conduit Components: Illustrated for Laminar Flow in a Circular Duct and a 90 Degree Bend. J. Fluids Eng.
**2013**, 135, 031204. [Google Scholar] [CrossRef] - Herwig, H.; Wenterodt, T. The Role of Entropy Produvtion in Momentum and Heat Transfer. In 7th. International Symposium on Heat Transfer, Beijing, China, October 26-29, 2008.
- Herwig, H.; Wenterodt, T. Evaluation of Heat Transfer Enhancement Devices in Compact Heat Exchangers by a Second Law Analysis. In Workshop on Compact Heat Exchangers for Aerospace Applications, Bangalore, India, 8-9 January, 2010.
- Herwig, H.; Wenterodt, T. Second law analysis of momentum and heat transfer in unit operations. Int. J. Heat Mass Transf.
**2011**, 54, 1323–1330. [Google Scholar] [CrossRef] - Herwig, H.; Wenterodt, T. Heat Transfer and its Assessment. In Heat Transfer- Theoretical Analysis, Experimental Investigations and Industrial Systems; InTech: Winchester, UK, 2011. [Google Scholar]
- Herwig, H. The Role of Entropy Generation in Momentum and Heat Transfer. J. Heat Transf.
**2011**, 134, 031003–1–11. [Google Scholar] [CrossRef] - Redecker, C.; Herwig, H. Assessing heat transfer processes: a critical view at criteria based on the second law of thermodynamics. Forsch. Ing.
**2012**, 76, 77–85. [Google Scholar] [CrossRef] - Bünger, F.; Herwig, H. An extended similarity theory applied to heated flows in complex geometries. Z. Angew. Math. Phys.
**2009**, 60, 1095–1111. [Google Scholar] [CrossRef] - Jin, Y.; Herwig, H. Application of the similarity theory including variable property effects to a complex benchmark problem. Z. Angew. Math. Phys.
**2010**, 61, 509–528. [Google Scholar] [CrossRef] - Jin, Y.; Herwig, H. Efficient methods to account for variable property effects in numerical momentum and heat transfer solutions. Int. J. Heat Mass Transf.
**2011**, 54, 2180–2187. [Google Scholar] [CrossRef] - Herwig, H.; Wenterodt, T. Second Law Analysis for Sustainable Heat and Energy Transfer: The Entropic Potential Concept. In Procedings of the International Conference on Applied Energy (ICAE2013), Pretori, South Africa, July 1–4, 2013.
- Jin, Y.; Herwig, H. From single obstacles to wall roughness: some fundamental investigations based on DNS results for turbulent channel flow. Z. Angew. Math. Phys.
**2013**, 64, 1337–1351. [Google Scholar] [CrossRef] - Jin, Y.; Herwig, H. Turbulent flow and heat transfer in channels with shark skin surfaces: Entropy generation and its physical significance. Int. J. Heat Mass Transf.
**2014**, 70, 10–22. [Google Scholar] [CrossRef]

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Herwig, H.; Schmandt, B. How to Determine Losses in a Flow Field: A Paradigm Shift towards the Second Law Analysis. *Entropy* **2014**, *16*, 2959-2989.
https://doi.org/10.3390/e16062959

**AMA Style**

Herwig H, Schmandt B. How to Determine Losses in a Flow Field: A Paradigm Shift towards the Second Law Analysis. *Entropy*. 2014; 16(6):2959-2989.
https://doi.org/10.3390/e16062959

**Chicago/Turabian Style**

Herwig, Heinz, and Bastian Schmandt. 2014. "How to Determine Losses in a Flow Field: A Paradigm Shift towards the Second Law Analysis" *Entropy* 16, no. 6: 2959-2989.
https://doi.org/10.3390/e16062959