Entropy Generation and Integral Inequalities

Xiaowei Tian^1,2 and Liqiu Wang^1,2,*

¹ Department of Mechanical Engineering, the University of Hong Kong, Hong Kong

² HKU-Zhejiang Institute of Research and Innovation (HKU-ZIRI), Hangzhou, China

*Correspondence: lqwang@hku.hk

 

Abstract: As a typical irreversible process, the heat conduction in rectangles results in entropy generation. As time tends to infinity,this entropy generation evolves into a finite value when the heat conduction comes from the initial temperature distribution, but into infinity whenever a positively-averaged heat source is involved. An application of the second law of thermodynamics to this process leads to nineintegral inequalities which are important for studying heat-conduction equations and for uncovering some basic features of the total multiplicity and the Boltzmann entropy. The work correlates the second law of thermodynamics in thermodynamics and integral inequalities in mathematics, and inspires the future work in offering some fundamental insights into our future.

Table of Content

The work correlates the second law of thermodynamics in thermodynamics and integral inequalities in mathematics, and inspires future work with fundamental insights into our future.

Keywords: Entropy generation; Integral inequalities; Second law of thermodynamics; Heat conduction.

---

1. Introduction
Energy is defined as the ability or the potential of system to cause changes.^{1, 2} It can exist in numerous forms such as thermal, mechanical, kinetic, potential, electric, magnetic, chemical, and nuclear; their sum constitutes the total energy of a system.^{1, 2} During every practical process, energy is conserved following the first law of thermodynamics, but degraded according to the second law of thermodynamics.^{1, 2} The conservation in energy quantity is represented by the heat-conduction equation for heat-conduction processes in which the classical Fourier’s law of heat conduction is normally used as the constitutive relation of heat flux density.^3-5  The degradation comes from the entropy generation occurring in any practical process like heat conduction because of irreversibilities^{.1, 2, 6} 
     The solution of the heat-conduction equation with its initial and boundary conditions yields the temperature field, the temperature as the function of location and time in heat conduction.^{3, 4} The Fourier’s law of heat conduction is then applied to find heat-transfer rate and the way to enhance/reduce it.^{3, 4} With the known temperature field, the entropy generation in heat conduction becomes available as well by using the thermodynamic relations from the first and second laws of thermodynamics.^{1, 2, 6} With the available entropy generation, we can then develop integral inequalities and examine features of heat-conduction equations by applying the second law of thermodynamics in the form that the entropy generation Sgen of a system during heat conduction always increases, or, in the limiting case of a reversible process, remains constant, i.e., dSgen/dt ≥ 0 with t being the time and  Sgen(t2) ≥Sgen(t1) for all t2 ≥ t1.^{1, 2, 6} As the entropy represents the part of system energy that cannot be used to do useful work,⁷ any entropy generation will thus downgrade the energy quality. It is therefore important to study the entropy generation in general, and its boundedness in particular. 
We examine dSgen/dt and limt→∞ Sgen and derive mathematical inequalities with heat conduction in rectangles. This kind of analysis is recently initiated with heat conduction in cylinders^,8 is limited in the literature⁸ and differs from conventional second-law analysis that has been made mainly for performance evaluation, weak-component identification or performance optimization^.9-30 
 
2. Temperature field, entropy generation and integral inequalities

 

        Fig. 1 Heat conduction in adiabatic rectangles and Cartesian coordinate system.

    Consider heat conduction in a rectangle D of width a and height b subject to specified temperature gradient atthe rectangle boundary. Material properties are assumed to be constant. Note that the effect of nonhomogeneous boundary condition is representable by source and initial terms.^{3, 6} We thus limit our attention to the following initial-boundary value problem under homogeneous boundary conditions in Cartesian coordinates, shown in Fig. 1, without loss of the generality:^{3, 4, 31}

                               

 D:0 < x < a, 0< y< b

where t and T are time and temperature, respectively. a_0^2 is the thermal diffusivity. φ ( x, y ) is the initial temperature distribution over the rectangle. f ( x, y, t ) is the rate of volumetric heat generation inside the rectangle per unit specific capacity of the material. The heat generation may be due to chemical, electrical,gammy-ray, nuclear, or other sources that may be a function of time and/or position. Fig. 2 examples the heat conduction driven by the initial temperature distribution, the internal source and the both, respectively, the readers are referred to30 for more practical examples.  

 

Fig. 2 Heat conduction in oven: (a) heat conduction driven by the initial temperature distribution after switching off the power, (b) heat conduction driven by the internal source when the power is first switched on, (c) heat conduction driven by the initial temperature distribution and the internal source after re-switching on the power.

Heat conduction initiated by the initial temperature distribution.  For the heat conduction driven by the initial temperature distribution, f ( x, y, t ) = 0, and Eq. (1) reduces into

                                            

Applying generalized Fourier extension to the solution of (2) yields,3
                             
Because
                                               b
                                              
                                                 
Eq. (2) becomes
                                 
                                                  
Let , thus 
                                                     
Therefore, one has, because a_mn≠0,
                             4
The solution of Eq. (4) reads3, 
                           
Define constantA_mn=a_mn c_mn, then substituting Eq. (5) into Eq. (3) yields the solution of Eq. (2)
                     
By using the initial condition and the orthogonality of function group  , one arrives at
                             
                           
where
                                          
Therefore, the temperature distribution T_φ (x,y,t) due to initial temperature distribution φ(x,y)can be expressed as 
                                     
where A = ab is the area of domain D. 
The total entropy in the rectangle is, by using the first ds equation from the first and second law of thermodynamics,^{1, 2, 6}
                                                       
                                                                        
Note that Sφ(t) is also the total entropy generation for the case of adiabatic boundary conditions with vanished thermal entropy flux.1, 6
By applying Eq. (11) and noting that ω_mn^2>0 for m+n≠0, we obtain

 
                
                      
By using Eqs. (8) and (9),
                         
which is the average of the initial temperature distribution φ(x, y) over the rectangle. Therefore,
                       
The limit of Sφ(t) is thus bounded as time trends to infinity and equals to system total entropy with the temperature being the average of the initial temperature distribution over the whole rectangle. Note that,

                        

                                               

Applying the principle of entropy increase    leads to, 

           

As  , one also has
                    
or
                                  
This yields the integral inequality 

                                             
                                                                           
Note that 
                                                               
and
                                                                                       
Applying  leads to 
                                                               
that is
                                                                    
which is the two-dimensional extension over the rectangle of well-known Arithmetic-mean---geometric-mean inequality^{32, 33} and can be proven mathematically as following as well.
           By the third law of thermodynamics, φ(x,y)=T_φ (x,y,0)>0 with T_φ  being the temperature during the heat conduction driven exclusively by the initial temperature distribution φ(x,y). Divide D into N area elements ∆σ_i,i=1,2,…,N, of equal area. For ∀(ξ_i,η_i )∈∆σ_i, one has N positive numbers:
                                                      φ(ξ_i,η_i ),i=1,2,…,N. 
Therefore,
                                                                   
and
                                                                 
or
                                                                  
which can be written as
                                                              
Note thatA/N=Δx_i Δy_i，one has
                                                
By the definition of double integral, 
                                                          
and
                                      
Therefore, one arrives at Eq. (18),
                                                                   
While Eq. (18) is developed from the 2nd law of thermodynamics with φ(x,y) being the initial temperature distribution, it is actually valid for any positive function φ(x,y). Consider the total multiplicity Ω of a macrostate, the number of microstates that compose the macrostate, in the Boltzmann entropy S=k"lnΩ"  with k being the Boltzmann’s constant.34-36  Eq. (18) leads to,
                                                            
                                                        
Therefore, the average of entropy over any domain can never be larger than the entropy calculated with the average of multiplicity over the domain.

 

2.1 Heat conduction driven by the internal source.

For the heat conduction driven exclusively by the internal source, φ(x,y) = 0 and Eq. (1) reduces
                                          
                                                                                  
Note that 3, 
                                                                 
where δ stands for the δ function. The solution of Eq. (23) is thus, by the superposition principle,3 
                                                                 
in which U(x,y,t,τ)is the solution of 
                                                        
or
                                                             
By replacing t and φ(x,y) in Eq. (10) with t - τ and f(x,y, τ), respectively, the solution of Eq. (25) reads, 
                                                 
Thus,
                                         
where Tf stands for the temperature for heat conduction driven exclusively by the internal heat source and can also be rewritten as,3 
                                                     
where
                                            
is the Green function3. 
The total entropy in the rectangle at time instant t is, which is also the entropy generation up to time instant t,^{1, 2, 6}
                                             

                                                                               
By applying Eq.(31)
                                       
                                                              
which can be unbounded for some internal heat sources [see the proof in the part regarding S_f (+∞), Eq. (43)].    
Also,
                                                            
. By applying the rule of taking derivatives of integral with respect to its parameters, one has 
                                                
where 

                         

                                                                  

                                                                         
One thus has
                                                      
By the second law of thermodynamics, 
                                                                                             .
One has thus, by noting that ρC_V>0,
                                                  
As (dS_f (t))⁄(dt≥0), one also has 
                                                                
Note that

                                             

 

The following inequality is thus obtained:
                                                             
                                                                    
Also,

                                

                                 

                                             

                                   

As  , one has,
                                
Let’s now examine S_f (∞)in details. By Eq.(24),
                                                        
where U(x,y,t,τ) is the solution of (26) and reads [Eq. (27)],
                                       
As
                                       
Therefore, for sufficiently large t0,
                                         
and
                         
Consider now the source with a positive average over the rectangle so that for any time instant τ, there exists a positive value ε such that
                                             
or
                                     
Thus,
                                         
As ε＞0, εt0 can be sufficiently large for sufficiently large t0. Note also that lnx always increases with x. Therefore, with Eq. (42),
                                 
This is 

                             

Heat conduction driven by the initial temperature distribution and the internal source.  The temperature field subjected to the effect of both initial temperature distribution and the internal source is, by the superimposition principle, 
                                  
With Eqs. (10) and (28), we have

              

                                                 

where
                                   
The total entropy in the rectangle at time instant t is, which is also the entropy generation up to time instant t,^{1, 2, 6}

                          

                                      

Also,

                                 

                                       

Note that
 

                 

                                           

By Eqs. (13) and (34), one has
                                                            
                                           
By the second law of thermodynamics that requires dS_φf (t)/dt≥0, one has
                       

note that 

                                   

                                 

By applying
                                             
one obtains

                                         

                                         

When t_1=0,t_2=+∞, 

                         

                                     

                               

thus one has
                           
Consider now the source with a positive average over the rectangle so that for any time instant τ, there exists a positive value ε such that
                                           
By using the results in 2.1 and 2.2 and for sufficiently large t,

                                              

                                           

                                                   

As ε＞0, εt can be sufficiently large for sufficiently large t. Note also that lnx always increases with x. Eq. (54) thus leads to
                                                  
that is 
                                                 

3. Discussion
As time tends to infinity, the entropy generation from the heat conduction in adiabatic rectangles has a bounded limit [Eq. (12)] when the heat conduction is initiated by the initial temperature distribution. This is not valid anymore [Eqs. (43) and (55)] when the heat conduction includes an internal source with a positive average over the rectangle. As the entropy generation degrades the energy quality,7 the unbounded entropy generation will cause serious problems. We must thus constrain the use and of technologies that depend on positive average heat generation, such as electric-resistance heating, nuclear fission and exothermic chemical reactions.

Table 1 Integral inequalities developed in present work.
Heat conduction    Second law of thermodynamics    Inequalities

By initial temperature distribution
           
Table 1 lists the nine integral inequalities from applying the second law of thermodynamics to the heat conduction in rectangles. Despite of the similar form, they differ from their 1D counterparts in literature6 due to different temperature fields. One of these inequalities [Eq. (18)] has also been proven mathematically. For their fundamental nature (from the three fundamental laws: the second law of thermodynamics, the first law of thermodynamics and the Fourier law of heat conduction via the heat-conduction equation), they are important both for studying heat-conduction equations and for assessing accuracy of solutions from analytical, numerical and experimental approaches. Application of Eq. (18) to the total multiplicity Ω of a macrostate, the number of microstates that compose the macrostate in the Boltzmann entropy, has also led to Eqs. (21) and (22) showing that the entropy average over any domain can never be larger than the entropy calculated with the average multiplicity over the domain. Table 2 lists   and  with a=1m, b=2m for four typical, and further demonstrates the correctness of Eq. (18).Equations (18), (21) and (22) are classical and have been acquired by the other approach in the literature. The present work develops a new approach, the second-law approach, to obtain them, thus building their relation to the second law of thermodynamics. The second-law approach works also for developing new mathematical inequalities.
Table 2 Comparison between and   ).

 

4. Concluding remarks

As the time tends to infinity,the entropy generation during the heat conduction in adiabatic rectangles evolves into a finite value when the heat conduction comes from the initial temperature distribution, but infinity when the conduction contains any heat source of positive average over the rectangle. Therefore, it is critical to use as less technologies as possible that could yield positive average heat generation and to develop their replacing technologies, for preventing unbounded entropy generation and thus exergy or energy/quality destruction. 
  The nine inequalities are obtained by applying the second law of thermodynamics to the heat conduction in rectangles: eight being new and the other one being classical. They are capable for examining correctness and accuracy of various studies of heat conduction processes. An application of one of these nine inequalities to the total multiplicity of a macrostate in the Boltzmann entropy concludes that the entropy average over any domain can never be larger than the entropy calculated with the average multiplicity over the domain.

---

References
    Y. A. Cengel and M. A. Boles, 2011, Thermodynamics: An Engineering Approach, 7th ed, McGraw-Hill, New York.
    A. Bejan, 2016, Advanced Engineering Thermodynamics, 4th ed, Wiley, Newark.
    L. Q. Wang, X. S. Zhou and X. H. Wei, 2008, Heat Conduction: Mathematical Models and Analytical Solutions, Springer-Verlag, Heidelberg.
    A. Bejan, 1993, Heat Transfer, John Wiley & Sons, New York.
    L. Q. Wang, Int. J. Heat Mass Tran., 1994, 37, 2627-2634.
    X. W, Tian, X. Lai, P. A. Zhu and L. Q. Wang, P. Roy. Soc. A, 2016, 472, 20160362.
    L. Q. Wang, Int. J. Heat Mass Tran., 1998, 41, 1869-1871.
    X. W. Tian and L. Q. Wang, Int. J. Heat Mass Tran., 2018, 121, 1137-1145.
    M. Lange, M. Roeb, C. Sattler and R. Pitz-Paal, Entropy, 2016, 18 (1), e18010024.
    X. J. Zhu, X. Du, Y. Q. Ding and Q. G. Qiu, Int. J. Heat Mass Tran., 2017, 114, 20-30.
    Y. Ji, H. C.Zhang, X. Yang and L. Shi, Entropy, 2017, 19 (3), 108.
    E. Guelpa and V. Verda, Entropy, 2017, 19 (8), 433.
    Z. X. Li and Z. Y. Guo, 2011, Advances in Transport Phenomena2010 (Ed. By L. Q. Wang), 1-91, Springer-Verlag, Heidelberg.
    M. T. Xu, J. F. Guo and X. F. Li, 2014, Advances in Transport Phenomena2011 (Ed. By L. Q. Wang), 63-167, Springer-Verlag, Heidelberg.
    D. L. Goodstein, 2015, Thermal Physics: Energy and Entropy, Cambridge University Press, Cambridge.
    G. F. Naterer and J. A. Camberos, 2008, Entropy-based Design and Analysis of Fluids Engineering Systems, CRC Press, Boca Raton. 
    A. Bejan, 1996, Entropy Generation Minimization: the Method of Thermodynamic Optimization of Finite-Size Systems and Finite-time Processes, CRC Press, New York.    
    M. M. Rashid and M. A. Abbas, Entropy, 2017, 19 (8), 414.
    H. Wang, D. Lin, X. R. Su and X. Yuan, Entropy, 2017, 19 (7), 324.
    T. Choudhary, S. Sanjay, Energy, 2017, 134, 1013-1028. 
    T. Hayat, S. Nawaz, A. Alsaedi and M. Rafiq, Entropy, 2016,18 (10), 355.
    I. Dinçer and M. A.Rosen, 2015, Exergy Analysis of Heating, Refrigerating and Air Conditioning: Methods and Applications, Elsevier, Amsterdam. 
    V. M. Brodyanbsky, M. V. Sorin and P. Le Goff, 1994, The Efficiency of Industrial Processes : Exergy Analysis and Optimization, Elsevier, Amsterdam.
    D. H. Yang, 1986, Exergy and Energy-order Analyses, Science Press, Beijing.
    N. Freidoonimehr, M. M. Rashidi, S. Abelman and G. Lorenzini, Entropy, 2016, 18 (5), 131.
    C. E. Damian-Ascencio, A. Saldana-Robles, A. Hernandez-Guerrero and S. Cano-Andrade, Energy, 2017, 133, 306-316. 
    M. A. Sheremet, H. F. Oztop, I. Pop and N. Abu-Hamdeh, Entropy, 2016, 18 (1), e18010009.
    V. P. Singh, 2015, Entropy Theory in Hydrologic Science and Engineering, McGraw-Hill, New York.
    Q. Jia, M. B. Muhammad, A. A. Munawwar, M. R. Mohammad and E. A. Mohamed, Entropy, 2016, 18, e18040123. 
    H. A. Muhammad, F. Navid, N. Foad and M. R. Mohammad, Adv. Pow. Technol., 2017, 26, 542-552.
    Y. A. Cengel, 2007, Heat and Mass Transfer: An Engineering Approach, 3rd ed, McGraw-Hill, New York.
    E. F. Beckenbach and R. Bellman, 1965, Inequalities, Springer-Verlag, Berlin.
    G. H. Hardy, J. E. Littlewood and G. P´olya, 1934, Inequalities, Cambridge University Press, Cambridge.
    D. S. Lemons, 2013, A Student’s Guide to Entropy, Cambridge University Press, Cambridge.
    I. Ford, 2013, Statistical Physics: An Entropic Approach, Wiley, Hoboken.
    T. Downarowicz, 2011, Entropy in Dynamical Systems, Cambridge University Press, New York.

Acknowledgements. 
The financial support from the Research Grants Council of Hong Kong (GRF 17237316,17211115 and 17207914) and the University of Hong Kong (URC 201411159074 and 201311159187) isgratefully acknowledged. The work is also supported in part by the Zhejiang Provincial, Hangzhou Municipal and Lin'an County Governments.

Author Contributions
X.T and L.W. conceived the project. X.T. and L.W. designed the project. X.T. performed mathematical derivation. X.T. and L. W. analyzed the results. X.T. and L.W. wrote the manuscript. L.W. supervised the study. 

Additional Information
Competing financial interests: The authors declare no financial and non-financial competing interests.