Numerical Calculation of the Irreversible Entropy Production of Additively Manufacturable Off-Set Strip Fin Heat-Transferring Structures
<p><b>Left</b>: schematic 3D model of the structures investigated with geometric parameters, and <b>right</b>: schematic 3D model of the internal fin arrangement of the investigated section.</p> "> Figure 2
<p>Three-dimensional model of the complete calculation domain with boundary conditions. Periodic conditions are also applied on the top grey fin and side wall areas. Evaluation of the in- and outlet temperatures and pressure drop is carried out at the coloured inlet and outlet locations.</p> "> Figure 3
<p>Schematic of the internal structures (hot or cold side) with locations for the wall and fluid temperatures for the determination of the heat transfer coefficient.</p> "> Figure 4
<p><b>Left</b>: three-dimensional model of the validation case with boundary conditions, and <b>right</b>: location of the fluid and wall temperatures for the determination of the heat transfer coefficient.</p> "> Figure 5
<p><b>Left</b>: infinitesimal fluid element for equation 27, <b>right</b>: simplified cross section of <a href="#entropy-25-00162-f002" class="html-fig">Figure 2</a> with the corresponding temperatures according to <a href="#entropy-25-00162-f003" class="html-fig">Figure 3</a>. For simplification, the heat flow rate over the periodic boundary condition (dotted line) is added to the overall heat flow rate <math display="inline"><semantics> <mover accent="true"> <mi>Q</mi> <mo>˙</mo> </mover> </semantics></math>.</p> "> Figure 6
<p>Colburn j-factor and Fanning f-factor for rectangular off-set strip fins (hot and cold sides). Comparison with values from correlations (Equations (16)–(21)) from Manglik and Bergles [<a href="#B36-entropy-25-00162" class="html-bibr">36</a>], Chennu [<a href="#B39-entropy-25-00162" class="html-bibr">39</a>], and Joshi and Webb [<a href="#B37-entropy-25-00162" class="html-bibr">37</a>].</p> "> Figure 7
<p>Comparison of the irreversible entropy production rate for the entire domain using the calculation method of Bejan [<a href="#B8-entropy-25-00162" class="html-bibr">8</a>], the 2nd law of thermodynamics [<a href="#B44-entropy-25-00162" class="html-bibr">44</a>], and differential equation [<a href="#B7-entropy-25-00162" class="html-bibr">7</a>,<a href="#B17-entropy-25-00162" class="html-bibr">17</a>].</p> "> Figure 8
<p>Comparison of the entropy production rate by heat conduction and shear stresses in the cold (<b>left</b>) and hot (<b>right</b>) fluids, using the method of Bejan [<a href="#B8-entropy-25-00162" class="html-bibr">8</a>] and the differential equations [<a href="#B17-entropy-25-00162" class="html-bibr">17</a>].</p> "> Figure 9
<p><b>Left:</b> Entropy production rate by heat conduction in the hot/cold fluid, the wall/fins, and overall entropy production rate by heat conduction, and <b>right</b>: relative proportions of the entropy generation by heat conduction of the hot/cold fluid and the wall. For the total entropy production rate and the entropy production rate in the wall and fins, the mean Reynolds number of the hot and cold sides is used.</p> "> Figure 10
<p>Relative proportions of the molecular and fluctuating irreversible entropy production rate by heat conduction (<b>left</b>) and shear stresses (<b>right</b>) for the hot and cold fluids.</p> "> Figure 11
<p><b>Left:</b> Entropy production rate by heat conduction and shear stresses in the fluid and the wall, and <b>right</b>: relative share of the fluid and wall entropy production rates compared to the overall entropy production rate. For the entropy production rate in the wall, the mean Reynolds number of the hot and cold sides is used.</p> "> Figure 12
<p>Bejan number for the hot and cold fluids.</p> "> Figure 13
<p>Local volumetric entropy production rate by shear stress (mean values and fluctuation) at Re = 110 (<b>left</b>) and at Re = 713 (<b>right</b>) for the cold fluid side (flow direction: bottom to top). Slicing plane for the contour plot.</p> "> Figure 14
<p>Local volumetric entropy production rate by heat conduction (mean values and fluctuating values) at Re = 110 (<b>left</b>) and at Re = 713 (<b>right</b>) for the cold fluid side.</p> "> Figure 15
<p>Cross-sectional view of the local volumetric entropy production rate by shear stress at Re = 110 (<b>left</b>) and at Re = 713 (<b>right</b>) for the cold fluid side. Slicing plane for the contour plot.</p> "> Figure 16
<p>Cross-sectional view of the local volumetric entropy production rate by heat conduction at Re = 110 (<b>left</b>) and at Re = 713 (<b>right</b>) for the cold fluid side.</p> "> Figure 17
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side, (<b>b</b>) entropy production number due to shear stresses for the hot side for different fin heights, (<b>c</b>) entropy production number due to heat conduction in the hot/cold fluid and the walls/fins, (<b>d</b>) overall entropy production number for different fin heights, and (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nusselt number of the hot/cold side for different fin heights.</p> "> Figure 18
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different fin spacings<b>.</b> (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different fin spacings. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nusselt number of the hot/cold side for different fin spacings.</p> "> Figure 18 Cont.
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different fin spacings<b>.</b> (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different fin spacings. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nusselt number of the hot/cold side for different fin spacings.</p> "> Figure 19
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different fin lengths. (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different fin lengths. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nu number of the hot/cold side for different fin lengths.</p> "> Figure 19 Cont.
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different fin lengths. (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different fin lengths. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nu number of the hot/cold side for different fin lengths.</p> "> Figure 20
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different longitudinal fin displacement. (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different longitudinal fin displacement. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nusselt number of the hot/cold side for different longitudinal fin displacement.</p> "> Figure 20 Cont.
<p>(<b>a</b>) Entropy production number due to shear stresses for the cold side. (<b>b</b>) Entropy production number due to shear stresses for the hot side for different longitudinal fin displacement. (<b>c</b>) Entropy production number due to heat conduction in the hot/cold fluid and the walls/fins. (<b>d</b>) Overall entropy production number for different longitudinal fin displacement. (<b>e</b>,<b>f</b>) j-factor, f-factor, and Nusselt number of the hot/cold side for different longitudinal fin displacement.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Investigated Structures and Simulation Parameters
2.2. Basic Partial Differential Equations for Mass, Momentum, Energy and Turbulence Modelling
2.3. Calculation of Heat Transfer and Pressure Drop
2.4. Validation
2.5. Entropic Evaluation of the Structures
2.5.1. Partial Differential Equations for the Irreversible Entropy Production Rate
2.5.2. Method of Bejan [8] for Calculating the Irreversible Entropy Production Number
2.5.3. Second Law Evaluation for Comparison of Both Methods
3. Results and Discussion
3.1. Validation
3.2. Results of the Inclined Off-Set Structures
3.2.1. Comparison of the Calculation Methods of Bejan [8] and Kock [17]
3.2.2. Detailed Consideration of the Irreversible Entropy Production Rate on the Basis of the Inclined Reference Structure
3.2.3. Irreversible Entropy Production Number and Heat-Transferring Parameters for Different Geometric Parameters
3.2.4. Interim Conclusion of the Structural Evaluation
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
heat transfer area, m2 | |
thermal diffusivity, m2/s | |
flow cross section, m2 | |
specific isobaric heat capacity, J/kg K | |
hydraulic diameter, m | |
specific internal energy, J/kg | |
F1 | blending function in k-w SST model |
Fanning friction factor, | |
gravitational acceleration, m/s2 | |
constant in k-w SST model | |
fin height, m | |
Colburn j-factor | |
turbulent kinetic energy, m2/s3 | |
length (overall or fin length), m | |
mass flow rate, kg/s | |
entropy production number | |
Nusselt number | |
pressure, Pa | |
Prandtl number | |
heat flow per unit length, W/m | |
heat flow, W | |
individual gas constant, J/kg K | |
Reynolds number | |
(fin) spacing, m | |
entropy flow, W/K | |
constant in k-w SST model | |
fin thickness or time, m, s | |
Temperature, K | |
velocity in x-, y-, and z-directions, m/s | |
volume of domain (fluid, wall), m3 | |
coordinate, m | |
constant in k-w SST model | |
Greek Letters | |
heat transfer coefficient, W/m2K | |
dimensionless fin parameter | |
pressure drop, Pa | |
isotropic dissipation rate, m2/s3 | |
thermal conductivity, W/mK | |
dyn. viscosity, Pa s | |
turbulent dyn. viscosity, Pa s | |
kinematic viscosity, m2/s | |
turbulent kinematic viscosity, m2/s | |
density, kg/m3 | |
shear stress tensor, kg/m s2 | |
constant in k-w SST model | |
inclination angle, rad | |
specific dissipation rate, 1/s | |
Subscripts and Superscripts | |
heat transfer | |
cold | |
C | conduction |
dissipation | |
effective | |
fin, fluid | |
hot | |
heat conduction | |
index (hot:h, cold:c, and wall: w) | |
inlet | |
irr | irreversible |
laminar | |
logarithmic | |
(longitudinal) fin displacement | |
mean | |
outlet | |
plate | |
partial differential equations | |
reference structure | |
s | solid |
SS | Shear Stress |
t | turbulent |
validation | |
wall | |
dimensionless fin parameter | |
volumetric | |
RANS mean values | |
RANS fluctuating values | |
Joshi and Webb, Manglik and Bergles, and Chennu | |
2nd law analysis |
Abbreviations
RANS | Reynolds-averaged Navier–Stokes equations |
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[mm] | [mm] | [mm] | [mm] | [mm] | [mm] | |||
---|---|---|---|---|---|---|---|---|
8 | 0.5 | 0.2 | 1.2 | 4 | 1.994 | 0.15 | 0.1667 | 0.05 |
Number of Elements (Mio. Elements) | [Pa] | [K] | [Pa] | [K] |
---|---|---|---|---|
2.2 | 84.9 | 454.86 | 54.56 | 395.11 |
3.0 | 101.9 | 454.89 | 64.45 | 395.06 |
3.9 | 102.5 | 454.9 | 64.52 | 395.05 |
Mesh Number | [Pa] | [K] | [W/K] | [W/K] |
---|---|---|---|---|
1 | 89.15 | 452.84 | 1.132 | 1.101 |
2 | 89.25 | 452.85 | 1.131 | 1.110 |
3 | 89.55 | 452.83 | 1.108 | 1.117 |
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Fuchs, M.; Lubos, N.; Kabelac, S. Numerical Calculation of the Irreversible Entropy Production of Additively Manufacturable Off-Set Strip Fin Heat-Transferring Structures. Entropy 2023, 25, 162. https://doi.org/10.3390/e25010162
Fuchs M, Lubos N, Kabelac S. Numerical Calculation of the Irreversible Entropy Production of Additively Manufacturable Off-Set Strip Fin Heat-Transferring Structures. Entropy. 2023; 25(1):162. https://doi.org/10.3390/e25010162
Chicago/Turabian StyleFuchs, Marco, Nico Lubos, and Stephan Kabelac. 2023. "Numerical Calculation of the Irreversible Entropy Production of Additively Manufacturable Off-Set Strip Fin Heat-Transferring Structures" Entropy 25, no. 1: 162. https://doi.org/10.3390/e25010162
APA StyleFuchs, M., Lubos, N., & Kabelac, S. (2023). Numerical Calculation of the Irreversible Entropy Production of Additively Manufacturable Off-Set Strip Fin Heat-Transferring Structures. Entropy, 25(1), 162. https://doi.org/10.3390/e25010162