Entropy equation

From Thermal-FluidsPedia

(Difference between revisions)
Jump to: navigation, search
Line 4: Line 4:
  \end{array}</math> from [[Integral entropy equation]] can be rewritten as volume integrals:
  \end{array}</math> from [[Integral entropy equation]] can be rewritten as volume integrals:
-
  <center><math>\int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA}  = \int_V {\nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s)dV}      \qquad \qquad(1) </math></center>(2.109)
+
  <center><math>\int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA}  = \int_V {\nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s)dV}      \qquad \qquad(1) </math></center>
 
 
-
<center><math>\int_A {\frac{{{\mathbf{q''}}}}{T} \cdot {\mathbf{n}}dA}  = \int_V {\nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right)dV}    \qquad \qquad(2) </math></center>(2.110)
+
<center><math>\int_A {\frac{{{\mathbf{q''}}}}{T} \cdot {\mathbf{n}}dA}  = \int_V {\nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right)dV}    \qquad \qquad(2) </math></center>
Substituting eqs. (1) and (2) into eq. <math>\begin{array}{l}
Substituting eqs. (1) and (2) into eq. <math>\begin{array}{l}
Line 13: Line 13:
  \end{array}</math> and considering eq. <math>\frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV</math> from [[Continuity equation]] yields
  \end{array}</math> and considering eq. <math>\frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV</math> from [[Continuity equation]] yields
-
  <center><math>\int_V {\left[ {\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T}} \right]dV}  = \int_V {{{\dot s'''}_{gen}}dV}  \ge 0    \qquad \qquad(3) </math></center>(2.111)
+
  <center><math>\int_V {\left[ {\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T}} \right]dV}  = \int_V {{{\dot s'''}_{gen}}dV}  \ge 0    \qquad \qquad(3) </math></center>
In order for eq. (3) to be true for any arbitrary control volume, the integrand in eq. (3) should always be positive, i.e.,
In order for eq. (3) to be true for any arbitrary control volume, the integrand in eq. (3) should always be positive, i.e.,
-
  <center><math>\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0    \qquad \qquad(4) </math></center>(2.112)
+
  <center><math>\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0    \qquad \qquad(4) </math></center>
Equation (4) can be rewritten as  
Equation (4) can be rewritten as  
 
 
-
<center><math>s\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}})} \right] + \rho \left[ {\frac{{\partial s}}{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla s} \right] + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0    \qquad \qquad(5) </math></center>(2.113)
+
<center><math>s\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}})} \right] + \rho \left[ {\frac{{\partial s}}{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla s} \right] + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0    \qquad \qquad(5) </math></center>
Considering the continuity equation, eq. <math>\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho {{\mathbf{V}}_{rel}} = 0</math> from [[Conservation of mass species equation]], and definition of the substantial derivative, eq. <math>\frac{D}{{Dt}} \equiv \frac{\partial }{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla</math> from [[Conservation of mass species equation]], eq. (5) can be reduced to  
Considering the continuity equation, eq. <math>\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho {{\mathbf{V}}_{rel}} = 0</math> from [[Conservation of mass species equation]], and definition of the substantial derivative, eq. <math>\frac{D}{{Dt}} \equiv \frac{\partial }{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla</math> from [[Conservation of mass species equation]], eq. (5) can be reduced to  
Line 31: Line 31:
For a multicomponent system without internal heat generation (<math>q''' = 0</math>), [[#References|Curtiss and Bird (1999; 2001)]] obtained the entropy flux vector and the entropy generation as  
For a multicomponent system without internal heat generation (<math>q''' = 0</math>), [[#References|Curtiss and Bird (1999; 2001)]] obtained the entropy flux vector and the entropy generation as  
   
   
-
<center><math>{\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right)    \qquad \qquad(7) </math></center>(2.115)
+
<center><math>{\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right)    \qquad \qquad(7) </math></center>
-
   <center><math>T{\dot s'''_{gen}} =  - \left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \cdot \nabla \ln T - \sum\limits_{i = 1}^N {\left( {{{\mathbf{J}}_i} \cdot \frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i}} \right)}  - {\mathbf{\tau }}:\nabla {\mathbf{V}} - \sum\limits_{i = 1}^N {\frac{{{{\bar g}_i}}}{{{M_i}}}{{\dot m'''}_i}}    \qquad \qquad(8) </math></center>(2.116)
+
   <center><math>T{\dot s'''_{gen}} =  - \left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \cdot \nabla \ln T - \sum\limits_{i = 1}^N {\left( {{{\mathbf{J}}_i} \cdot \frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i}} \right)}  - {\mathbf{\tau }}:\nabla {\mathbf{V}} - \sum\limits_{i = 1}^N {\frac{{{{\bar g}_i}}}{{{M_i}}}{{\dot m'''}_i}}    \qquad \qquad(8) </math></center>
where <math>{d_i}</math> is the diffusional driving force [see eq. (2) – (6)], <math>{\bar g_i}</math> is partial molar Gibbs free energy, <math>{\mathbf{q''}}</math> is total heat flux, obtained by eq. <math>{\mathbf{q''}} =  - k\nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}}  + c{R_u}T\sum\limits_{i = 1}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{{{D}_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)</math> from [[Energy equation]], including conduction, interdiffusional convection, and Dufour effect.
where <math>{d_i}</math> is the diffusional driving force [see eq. (2) – (6)], <math>{\bar g_i}</math> is partial molar Gibbs free energy, <math>{\mathbf{q''}}</math> is total heat flux, obtained by eq. <math>{\mathbf{q''}} =  - k\nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}}  + c{R_u}T\sum\limits_{i = 1}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{{{D}_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)</math> from [[Energy equation]], including conduction, interdiffusional convection, and Dufour effect.

Revision as of 02:01, 19 November 2009

To obtain the differential form of the second law of thermodynamics, the surface integrals in eq. \begin{array}{l}
 \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} }  \\ 
  + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA}  - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ 
 \end{array} from Integral entropy equation can be rewritten as volume integrals:

\int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA}  = \int_V {\nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s)dV}      \qquad \qquad(1)
\int_A {\frac{{{\mathbf{q''}}}}{T} \cdot {\mathbf{n}}dA}  = \int_V {\nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right)dV}    \qquad \qquad(2)

Substituting eqs. (1) and (2) into eq. \begin{array}{l}
 \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} }  \\ 
  + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA}  - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ 
 \end{array} and considering eq. \frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV from Continuity equation yields

\int_V {\left[ {\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T}} \right]dV}  = \int_V {{{\dot s'''}_{gen}}dV}  \ge 0     \qquad \qquad(3)

In order for eq. (3) to be true for any arbitrary control volume, the integrand in eq. (3) should always be positive, i.e.,

\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0     \qquad \qquad(4)

Equation (4) can be rewritten as

s\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot (\rho {{\mathbf{V}}_{rel}})} \right] + \rho \left[ {\frac{{\partial s}}{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla s} \right] + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0    \qquad \qquad(5)

Considering the continuity equation, eq. \frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho {{\mathbf{V}}_{rel}} = 0 from Conservation of mass species equation, and definition of the substantial derivative, eq. \frac{D}{{Dt}} \equiv \frac{\partial }{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla from Conservation of mass species equation, eq. (5) can be reduced to

\rho \frac{{Ds}}{{Dt}} + \nabla  \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0     \qquad \qquad(6)

where the three terms on the left-hand side represent rate of change of entropy per unit volume, rate of change of entropy per unit volume by heat transfer and internal heat generation, respectively. Equation (6) means that the entropy generation per unit volume must not be negative at any time or location.

For a multicomponent system without internal heat generation (q''' = 0), Curtiss and Bird (1999; 2001) obtained the entropy flux vector and the entropy generation as

{\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right)     \qquad \qquad(7)
T{\dot s'''_{gen}} =  - \left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \cdot \nabla \ln T - \sum\limits_{i = 1}^N {\left( {{{\mathbf{J}}_i} \cdot \frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i}} \right)}  - {\mathbf{\tau }}:\nabla {\mathbf{V}} - \sum\limits_{i = 1}^N {\frac{{{{\bar g}_i}}}{{{M_i}}}{{\dot m'''}_i}}     \qquad \qquad(8)

where di is the diffusional driving force [see eq. (2) – (6)], {\bar g_i} is partial molar Gibbs free energy, {\mathbf{q''}} is total heat flux, obtained by eq. {\mathbf{q''}} =  - k\nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}}  + c{R_u}T\sum\limits_{i = 1}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{{{D}_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right) from Energy equation, including conduction, interdiffusional convection, and Dufour effect.

References

Curtiss, C.F., and Bird, R.B., 1999, “Multicomponent Diffusion,” Industrial and Engineering Chemistry Research, Vol. 38, pp. 2115-2522.

Curtiss, C.F., and Bird, R.B., 2001, “Errata,” Industrial and Engineering Chemistry Research, Vol. 40, p. 1791.

Further Reading

External Links