# Entropy Generation During the Interaction of Thermal Radiation with a Surface

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## Abstract

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## 1. Introduction

## 2. Radiation Entropy

#### 2.1. Equilibrium Situation

**Figure 1.**The dependency of the spectral radiation temperature ${T}_{\lambda}$ on wavelength λ for grey body radiation. The parameter is the emissivity $\epsilon =const$, the grey body temperature is $T=1000K$.

#### 2.2. Fluxes

## 3. Entropy Production upon Absorption and Emission of Radiation

**Figure 2.**(

**a**) The geometric setting for the (small) plate A used in the example calculations. Shown are the two concentric regions of the hemisphere irradiating the plate. (

**b**) Cross section of Figure 2a to clarify the situation under consideration. The plate area A is infinitesimally small as compared to the hemisphere envelope.

**Figure 3.**A graphic presentation of the relation ${L}_{\lambda}={L}_{\lambda}\left({K}_{\lambda}\right)$ using the variable $z=hc/\left(k\lambda T\right)$.

**Figure 4.**The setting for the energy and entropy balance equation. The system boundary is around the plate A, the outgoing radiation energy is ${E}_{out}={E}_{pl}+{E}_{refl,\text{\hspace{0.17em}}in}+{E}_{refl,at}$.

#### Adiabatic Plate

**Figure 5.**The spectral radiation energy flux ${E}_{\lambda}=B\left(\theta \right)\cdot {L}_{\lambda}$ as a function of wavelength λ for one specific, but arbitrary setting. The incoming radiation flux can be extrapolated from the shown reflected radiation.

**Figure 6.**The spectral radiation entropy flux ${D}_{\lambda}=B\left(\theta \right)\cdot {K}_{\lambda}$ as a function of wavelength, corresponding to Figure 5. The true outgoing entropy flux is less than the simple addition of the flux components.

**Figure 7.**The entropy production rate ${D}_{irr}$ for an adiabatic plate shown as a function of the geometry parameter ${B}_{in}$ and the emissivity of the plate ${\epsilon}_{pl}$ . The incoming radiation is black.

**Figure 8.**Cross sections of the 3-dim. plot in Figure 7. Some equilibrium temperatures of the plate are given in addition.

**Figure 9.**Extension of Figure 7 to cases were ${\epsilon}_{in}$ varies. The plate temperature at some maximum points is shown.

## 4. Entropy Production Minimization

**Figure 10.**The entropy production rate ${D}_{irr}$ for a non-adiabatic plate as a function of plate emissivity ${\epsilon}_{pl}$ and plate temperature ${T}_{pl}$.

## 5. Conclusions

## 6. Summary

## Nomenclature

A | area, m ^{2} |

B | geometry parameter related to solid angle [see Equation (17)]; sr |

B_{in} | geometry parameter of the inner radiation source; sr |

B_{at} | geometry parameter of the (atmospheric) environment |

c | speed of light in vacuum, c = 299,792,458 m/s |

D | overall hemispheric radiation entropy fluxes, W/m ^{2}/K |

D_{at} | incoming radiation entropy flux from the atmosphere, W/m ^{2}/K |

D^{b} | blackbody radiation entropy flux, W/m ^{2}/K |

D_{in} | incoming radiation entropy flux from the inner source, W/m ^{2}/K |

D_{pl,overall} | entropy flux of overall outgoing radiation, W/m ^{2}/K |

D_{irr,cond.} | entropy production rate by heat conduction |

E | overall hemispheric radiation energy flux, W/m ^{2} |

E_{at} | incoming radiation energy flux from the atmosphere, W/m ^{2} |

E^{b} | blackbody radiation energy flux, W/m ^{2} |

E_{in} | incoming radiation energy flux from the inner radiation source, W/m ^{2} |

E_{pl} | radiation energy flux, emitted from the plate, W/m ^{2} |

E_{refl,at} | reflected radiation energy flux from the atmosphere, W/m ^{2} |

E_{refl,in} | reflected radiation energy flux from the inner source, W/m ^{2} |

h | Planck’s constant, h = 6.6261 × 10 ^{−34} J·s |

K^{b} | overall radiation entropy intensity of blackbody radiation, W/K/m ^{2}/sr |

K_{λ} | spectral radiation entropy intensity, W/K/m ^{2}/µm/sr |

${K}_{\lambda}^{b}$ | spectral entropy flux of blackbody radiation, W/K/m ^{2}/µm |

k | Boltzmann’s constant, k = 1.3806 × 10 ^{−23} J/K |

L^{b} | overall energy intensity of blackbody radiation, W/m ^{2}/sr |

L_{λ} | spectral radiation intensity, W/m ^{2}/µm/sr |

${L}_{\lambda}^{b}$ | spectral intensity of blackbody radiation, W/m ^{2}/µm/sr |

N_{λ} | density of number of photons, 1/m ^{3} |

${N}_{\lambda}^{eq}$ | density of number of photons (equilibrium), 1/m ^{3} |

$\overrightarrow{n}$ | normal vector of a surface |

p | pressure, N/m ^{2} |

$\dot{Q}$ | heat conduction flow to or from the plate, W/m ^{2} |

S | entropy, J/K |

S^{eq} | volume specific radiation entropy in equilibrium, J/m ^{3}/K |

${\dot{S}}_{irr}$ | entropy production rate, W/K |

s | volume specific entropy, J/m ^{3}/K |

s_{λ} | volume specific spectral radiation entropy, J/m ^{3}/K |

T | absolute temperature, K |

T_{at} | temperature of (atmospheric) environment, K |

T_{pl} | temperature of the plate, K |

T^{eq} | equilibrium temperature, K |

T_{in} | temperature of inner radiation source, K |

T_{s} | formal radiation flux temperature, K |

T_{λ} | spectral radiation temperature, K |

t | time, s |

U | internal energy, J |

${U}_{\lambda}^{eq}$ | spectral energy of cavity radiation in equilibrium, J/µm |

u | volume specific internal energy, J/m ^{3} |

u^{eq} | volume specific overall energy of cavity radiation in equilibrium, J/m ^{3} |

${u}_{\lambda}^{eq}$ | volume specific spectral energy of cavity radiation in equilibrium, J/m ^{3}/µm |

u_{λ} | volume specific spectral energy of cavity radiation, J/m ^{3}/µm |

V | Volume, m ^{3} |

x | average occupation number of the photon state in equilibrium |

X(ε) | grey body entropy function |

Ω | solid angle, sr |

ε | emissivity coefficient |

ε_{in} | emissivity coefficient of inner radiation source |

ε_{at} | emissivity coefficient of outer (atmosphere) radiation source |

ε_{pl} | emissivity coefficient of the plate |

ε_{re} | real part of complex dielectrical constant |

ε_{λ} | energy of a photon with wavelength λ |

θ | polar angle measured from normal of surface, ° |

θ_{at} | polar angle of radiation from environment, ° |

θ_{in} | upper limit of polar angle of inner radiation source, ° |

φ | azimuth angle, ° |

λ | wavelength in vacuum, µm |

σ | Stefan–Boltzmann constant, σ = 5.67 × 10 ^{−8} W/m^{2}/K^{4} |

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Kabelac, S.; Conrad, R.
Entropy Generation During the Interaction of Thermal Radiation with a Surface. *Entropy* **2012**, *14*, 717-735.
https://doi.org/10.3390/e14040717

**AMA Style**

Kabelac S, Conrad R.
Entropy Generation During the Interaction of Thermal Radiation with a Surface. *Entropy*. 2012; 14(4):717-735.
https://doi.org/10.3390/e14040717

**Chicago/Turabian Style**

Kabelac, Stephan, and Rainer Conrad.
2012. "Entropy Generation During the Interaction of Thermal Radiation with a Surface" *Entropy* 14, no. 4: 717-735.
https://doi.org/10.3390/e14040717