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Derivation of the differential equation of entropy generation rate

t.me/exergy_destruction

 q    q    +   dy  dx  T  y y  T  y 

q    dy  T x

    ρvs + y ( ρvs ) dy  dx  

 q    q   T  + x  T  dx  dy  x    x

y

  ρs ( )  t  dxdy

ρusdy y, v

x

x, u

q    dx T y

ρvsdx

S g = −

Q S −  ms +  ms + T t i e t.me/exergy_destruction

   ρus + ρus dx ( )   dy x

 q    q    q    q q q   S gdxdy =   +   dx  dy +   +   dy  dx −   dy −   dx  T x  T y  T  y y  T  y   T  x x  T  x        +  ρus + ( ρus ) dx  dy +  ρvs + ( ρvs ) dy  dx − ρusdy − ρvsdx x y      ( ρs ) + dxdy t

 q S g =  x x  T

   qy +   y  T

    + ρus + ρvs + ( ) ( ) ( ρs )  y t  x

1 qx qx T 1 q y q y T ρ u s ρ v s s ρ  Sg = − 2 + − 2 + us + ρs + ρu + vs + ρs + ρv + ρ + s T x T x T y T y x x x y y y t t t.me/exergy_destruction

 s  u v   1  qx q y  1  T T  s s   ρ ρ ρ S g =  + + qy  − 2  qx  + ρ  + u + v  + s  + u + v + ρ  +  T  x y  T  x y  x y   t x y  t  x y  

S g =

1 1 Ds 1 1 Ds  Dρ  .q − 2 q.T + ρ +s + ρ.V  → S g = .q − 2 q.T + ρ T T Dt T T Dt  Dt 

1 ρ P Ds ρ De P Dρ Tds = de + Pd   → ρds = de − dρ → ρ = − T ρT Dt T Dt ρT Dt  ρ

t.me/exergy_destruction

S g =

1 1 ρ De P Dρ .q − 2 q.T + − T T T Dt ρT Dt ρ

S g =

De = −.q − P ( .V ) + μΦ Dt

1 1 1 P μ P Dρ .q − 2 q.T − .q − ( .V ) + Φ − T T T T T ρT Dt

1 μ P  Dρ  S g = − 2 q.T + Φ − + ρ  . V ( )  T T ρT  Dt 

t.me/exergy_destruction

1 μ S g = − 2 q.T + Φ T T

1 μ q =− K T K μ 2 S g = − 2 q.T + Φ → S g = 2 ( T ) + Φ T T isotropic T T 2 2 2 2 2       u v   K  T   T  μ   u   v  S g = 2    +  2   +    +  +    + T  x   y   T   x   y    y x    

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