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Evaporation and condensation of droplets in the presence - Welcome

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Evaporation and condensation of droplets in the presence - Welcome Powered By Docstoc
					Evaporation and condensation
of droplets in the presence of
 inert admixtures containing
         soluble gas


     B. Krasovitov, T. Elperin and A. Fominykh



           Department of Mechanical Engineering
 The Pearlstone Center for Aeronautical Engineering Studies
            Ben-Gurion University of the Negev
           P.O.B. 653, Beer Sheva 84105, ISRAEL
Outline of the presentation


   Motivation and goals

   Fundamentals

   Description of the model

   Results and discussion

   Conclusions


                              Ben-Gurion University of the Negev
Gas absorption by droplets
Spray tower                                                Scavenging of
absorbers     • SO2 absorption of
                                                           air pollutions
                boiler flue gas
              • HF absorption in the
                aluminum industry
              • In-cloud scavenging
                of polluted gases
                (SO2, CO2, CO, NOx,
                NH3)


Spray
scrubbers

                                           Air
                                             Soluble gas
                       Single Droplet




                                        Ben-Gurion University of the Negev
Gas absorption by stagnant droplets:
Scientific background
   Dispersed-phase controlled isothermal absorption model (Newman
   A. B., 1931)

               C t        6  1       n 2 π 2 Dl t 
                       1  2  2 exp                 
               Csat        π n 1 n         R2        

   Particle Sherwood number:
                                   
                                   exp n 2 2 
                         2 2    n 1
                  Sh p         
                          3
                                
                                       1
                                         2
                                              
                                           exp  n 2 2   
                                n 1 n

                  Dl t
   where     
                  R2


                                                     Ben-Gurion University of the Negev
Gas absorption by stagnant droplets:
Scientific background

   Gas absorption in the presence of inert admixtures (see e.g., Plocker
   U.J., Schmidt-Traub H., 1972)

   Effect of vapor condensation at the surface of stagnant droplets on the
   rate of mass transfer during gas absorption by growing droplets:
        uniform temperature distribution in both phases was assumed (see
        e.g., Karamchandani, P., Ray, A. K. and Das, N., 1984);
        liquid-phase controlled mass transfer during absorption was
        investigated when the system consisted of liquid droplet, its vapor
        and soluble gas (see e.g., Ray A. K., Huckaby J. L. and Shah T.,
        1987, 1989);

   Simultaneous heat and mass transfer during droplet evaporation or
   growth:
       model of physical absorption (Elperin et al., 2005);
       model taking into account subsequent dissociation reaction (Elperin
       et. al, 2007).


                                                Ben-Gurion University of the Negev
Absorption equilibria

                                Air
                                        SO2
                                                        is the species in dissolved state
                                                Henry’s Law:
                                                Aqueous phase sulfur dioxide/water
                                                chemical equilibria
Droplet
                         Gas-liquid interface



          = molecule of soluble gas
          = pollutant captured in solution




          Total dissolved sulfur in
          solution in oxidation state 4:


                                                      Ben-Gurion University of the Negev
Absorption equilibria: Aqueous phase sulfur
dioxide/water chemical equilibria

  The equilibrium constants (Maahs, 1982):

                              1376.1                                      853
           [ SO2  H 2O]             4.521           [H     ][ HSO3 ]           4.74
    KH                   10   T          ;   K1                      10  T
               pSO2                                    [ SO2  H 2O]                        (1)

                                                                         6717 
                         621.9                                                  1 1 
     [ H ][ SO3  ]
                                                            
              2               9.278                                                
K2          
                     10  T           ;   Kw  H                   14
                                                       OH  1.008 10 e        T 298 
        [ HSO3 ]

 Langmuir‟s Electro neutrality principle (1920):         Ci zi  0
  Electro neutrality equation:


                                     H    HSO3  2SO32 
  Huckaby & Ray (1989),
  Walcek et al. (1984):                                                                     (2)

                                                       Ben-Gurion University of the Negev
Absorption equilibria: Aqueous phase sulfur
dioxide/water chemical equilibria

Eqs. (1) – (2) yield the following equation for concentration of ions :

                                                                                        (3)


Using (1) – (3) we obtain:
      pH is a measure of the acidity
      or alkalinity of a solution .                                                     (4)


Eq. (4) yields the following expression for the effective Henry's constant:

                                                       Figure 1. Equilibrium dissolved (5)
                                                       S(IV) as a function of pH, gas-phase
                                                       partial pressure of SO2 and pressure
                                                       (Seinfeld, 1986).

                                                   Ben-Gurion University of the Negev
Gas absorption by stagnant droplet:
Description of the model
         Governing equations
  1. gaseous phase r > R (t)
               r2
                    2
                     
                   t r
                           r  vr  0          (6)                       Droplet
                                                                                                  Far field

                                                                                                          Gaseous phase
                                       2 Y j 
r 2 
    t
                  
       Y j    vr r Y j    D j r  (7)
                 r
                         2
                                 
                                r      r 
                                                                               Z
                                                                                                 d
                                                                                                               mL
                                                                                                                
                                                                                        ds
          c pTe 
r   2
            t
                         
                             
                             r
                                              
                                                       T 
                                 vr r 2c pTe   ke r 2 e  (8)
                                               r      r               mA
                                                                                         q                      Gas-
                                                                                                      R          liquid
                                                                                                                 interface
    2. liquid phase 0 < r < R (t)                                                                               Y
                  T  L        2 T  L  
               r 2
                            L r 
                                                            (9)                     j
                   t          r    r     
                                                                     X

                                      2 YA 
             2 
                                  
                                                ( L)
                         L
           r       LYA    L DL r
                                 
                                                            (10)                              j  1,..., K  1,
                                              r 
                                                                         In Eqs. (7)
               t             r                    
                                       S IV  M S (IV )                      
                                                                                         K
                             L                                                             Y
                                                                                         j 1 j
                                                                                                   1;
                         YA        
                                                                  Ben-Gurion University of the Negev
Gas absorption by stagnant droplet:
Description of the model


 anelastic approximation:

  v 2 c 2  1 Eq. 6            v   0.                              (11)

  In spherical coordinates Eq. (11) reads:
                                            2
                                           r
                                               
                                              r  vr  0           (12)

 The radial flow velocity can be obtained by integrating equation (12):

                             vr r 2  const                              (13)

 subsonic flow velocities (low Mach number approximation, M << 1)

                              p ~  v2
                                           K  Yj 
                        p  p  Rg Te                                (14)
                                             M 
                                        j 1    j



                                                   Ben-Gurion University of the Negev
Description of the model
Stefan velocity and droplet vaporization rate
  The continuity condition for the radial flux of the absorbate at the droplet
  surface reads (Elperin et al. 2005, 2007):
                                YA                    
                                                     YAL 
      j A r  R  YA vs  DA              DL  L                       (15)
                                 r r  R             r r  R
                                                                  
  Other non-soluble components of the inert admixtures are not absorbed
  in the liquid
                  J j  4R 2 j j  0,  j  1, j  A                  (16)

  Taking into account Eq. (16) and using anelastic approximation (Eq.12)
  we can obtain the expression for Stefan velocity:
                                    
                        DL  L YAL                    D1 Y1
                vs                                                                   (17)
                        1  Y1  r   r  R
                                                     1  Y1  r r  R


   where subscript “1” denotes water vapor species

                                                            Ben-Gurion University of the Negev
Description of the model
Stefan velocity and droplet vaporization rate

   The material balance at the gas-liquid interface yields:

                        d mL
                         dt
                                               
                              4 R 2  s vR, t   R
                                                                                   (18)

   Then assuming  L   we obtain the following expressions for the
   rate of change of droplet's radius (Elperin et al. 2005, 2007):

                        DL YAL 
                                                       D1      Y1
                    R                                                              (19)
                       1  Y1  r   r  R
                                                    L 1  Y1  r r  R
                                                                        




                                                           Ben-Gurion University of the Negev
Description of the model
Stefan velocity and droplet vaporization rate

                         
             DL  L YAL                         D1 Y1
     vs                                  
             1  Y1  r        r  R
                                               1  Y1  r r  R
                                                                


          DL YAL 
                                          ρ D 1 Y1
      R                          
         1  Y1  r     r R
                                      ρ L 1  Y1  r r  R
                                                                




                                                                          D1 Y1
                                                                vs  
   In the case when all of the inert                                   1  Y1  r r  R  
  admixtures are not absorbed in
  liquid the expressions for Stefan
                                                                          D1 Y1
  velocity and rate of change of                                R
  droplet radius read                                                   L 1  Y1  r r  R
                                                                                            



                                                                    Ben-Gurion University of the Negev
Description of the model
Stefan velocity and droplet vaporization rate

Huckaby and Ray (1989)

 H 2O g  DL   2  H 2O g    L   g    R 2 t  d R  H 2O g 
               2  r                          
     t       DLr YrA
                          r         g           r 2 dt            r
                        L
                                                  
            1  Y1  r
                       r R

       Tg   g    Tg    L   g      R 2 t  d R  Tg
           2                           
                       2
                 r
       t  r  r   r   g
                            
                                            r 2 d t r
                                           

                                                                 for r  Rt 




                                                    Ben-Gurion University of the Negev
Description of the model
Initial and boundary conditions
   The initial conditions for the system of equations (6)–(10) read:
                                                                           L
    At t = 0, 0  r  R0 :              T  L   T0 L            Y AL  YA,0
                                                                                                  (20)
    At t = 0,       r  R0 :            Y j  Y j ,0 r            Te  Te,0 r 
   At the droplet surface:
                                      Y j
                                Dj                    Y j v s                                  (21)
                                      r     r  R
                                           Y                            
                                                                       YAL 
                            YA v s  D A  A                 DL  L                            (22)
                                            r    r  R
                                                                        r      r  R

        T                        dR      T  L 
                                                                                  
                                                                                YAL 
      ke e                 L Lv     kL                            La  L DL                   (23)
        r      r  R
                                  dt       r              r  R
                                                                                 r
                                                                                         r  R

                                       Te R  T  L                                              (24)
                                                              R

                                                                         Ben-Gurion University of the Negev
Description of the model
Initial and boundary conditions
  The equilibrium between solvable gaseous and dissolved in liquid species
  reads:
                       C  S IV   H *
                             A               p  S ( IV ) A
                                                                     (25)

  where
                                              K1   K1K 2                           (26)
                          *
                                     K H 1            
                                                
                        H S ( IV)                       2
                                          
                                              H   H   


  Huckaby & Ray (1989):

       4 K 2 S IV  3   K1  12K 2  SO2  g  K H S IV  2  2SO2  g K H 
       6 K 2  K1   2 K1K 2 S IV  K H SO2  g   K H SO2  g 
       K H SO2  g 2 K1  4K 2   K1K 22  K H SO2  g 4K1K 2  K12  0
          2



                                                             Ben-Gurion University of the Negev
Description of the model
Vapor concentration at the droplet surface and Henry’s constant
The vapor concentration (1-st species) at the droplet surface is the function
of temperature Ts(t) and can be determined as follows:

                 1, s p1, s Ts  M 1
 Y1, s Ts                            (27)
                            pM


The functional dependence of the Henry's
law constant vs. temperature reads:

       K T   H  1 1 
     ln H 0                             (28)
       K H T  RG  T T0 
                         

                                                   Figure 2. Henry‟s constant vs. temperature


                                                   Ben-Gurion University of the Negev
Description of the model
Initial and boundary conditions

  In the center of the droplet symmetry conditions yields:


                YAL 
                  
                                        T  L 
                                 0                       0                   (27)
                 r                      r        r 0
                          r 0


  At t  0 and r   the „soft‟ boundary conditions at
  infinity are imposed:

                    Y j                    Te
                                   0                       0                 (28)
                    r      r 
                                            r      r 




                                                          Ben-Gurion University of the Negev
Method of numerical solution
  Spatial coordinate transformation:
           r
  x  1       ,       for 0  r  Rt ;
         Rt 
       1 r         
  w             1, for r  Rt ;
        Rt  
  The gas-liquid interface is located at x  w  0;
  w  0, 1 x  0, 1  Coordinates x and w can be treated identically in
  numerical calculations;
  Time variable transformation:   DLt R0 ;
                                         2


  The system of nonlinear parabolic partial differential equations (6)–(10)
  was solved using the method of lines;
  The mesh points are spaced adaptively using the following formula:
                                  n
                          i  1
                    xi             i  1,, N  1
                          N 
                                               Ben-Gurion University of the Negev
Results and discussion
                            
                   YA L   YA,L0
                          
                   YA,Ls  YA,L0
Average concentration of absorbed
CO2 in the droplet:
            1
YA L  
 
                YAL  r  r 2 sin  dr d dj
                 
           Vd 
 Analytical solution in the case of
 aqueous-phase controlled diffusion in a
 stagnant non-evaporating droplet:

                                          
                   
               6      1
     1          n 2 exp  4 2n 2Fo
               2 n1                           Figure 3. Comparison of the numerical results with
                                                the experimental data (Taniguchi & Asano, 1992)
                            DL t                and analytical solution (Elperin et al 2005).
                    Fo 
                            4R 2

                                                       Ben-Gurion University of the Negev
Results and discussion




Average concentration of the
absorbed SO2 in the droplet:

 relative absorbate concentration
 is determined as follows:


                                       Figure 4. Dependence of average aqueous
                                       sulfur Dependence concentration vs. time
                                    Figure 5.dioxide molarof dimensionless average
                                    aqueous SO2 concentration vs. time for various
                                       for various values of relative humidity
                                       (Elperin of evaporating
                                    initial sizes et al. 2005). droplet R0
                                    (Elperin et al. 2005).


                                        Ben-Gurion University of the Negev
Results and discussion

                                                 Figure 8. Droplet surface temperature vs. time:
                                                      Figure 7. Effect of Stefan flow and heat of
                                                 1 – model taking into account the equilibrium
                                                      absorption on droplet surface temperature
                                                 dissociation reactions; 2 – model of physical
                                                      (Elperin et al. 2005).
                                                 absorption (Elperin et al., 2007).




Figure 6. Droplet surface temperature vs. time
(Elperin et al., 2007).




                                                          Ben-Gurion University of the Negev
Results and discussion
                                              Figure 9b. Temporal evolution of surface
                                              temperature for a water droplet evaporating in
                                              N2/NH3/H2O gaseous mixture (Elperin et al.,
                                              2007).




 Figure 9a. Droplet surface temperature vs.
 time (Elperin et al., 2007).

                                                  Ben-Gurion University of the Negev
Results and discussion

                                               Figure 10b. Droplet surface temperature vs.
                                               time, [SO2(g)]0 = 10-6 mole/m3 (Huckaby and
                                               Ray, 1989).




 Figure 10a. Droplet surface temperature vs.
 time, YA = 0.01 (Huckaby and Ray, 1987).

                                                   Ben-Gurion University of the Negev
Results and discussion
                                                  Figure 12. Temporal evolution of surface
                                                  temperature for a water droplet evaporating in
                                                  N2/SO2/H2O gaseous mixture (Elperin et al.,
                                                  2005).




 Figure 11. Temporal evolution of the surface
 temperature for a water droplet condensation
 in N2/CO2/H2O gaseous mixture (Elperin et al.,
 2005).
                                                      Ben-Gurion University of the Negev
Results and discussion
                                              Figure 14. Average concentration of aqueous
                                              sulfur species and their sum vs. time, RH =
                                              101% (Elperin et al., 2007).




Figure 13. Average concentration of aqueous
sulfur species and their sum vs. time, RH =
70% (Elperin et al., 2007).


                                                     Ben-Gurion University of the Negev
Results and discussion: the interrelation between
heat and mass transport
                                              Absorption during
                                             droplet evaporation


                     Diffusion of              Thermal effect of         Reactions of
                      absorbate                  absorption              dissociation

                   Decreases Stefan
                       velocity

                 Decreases vapor flux

               Increases droplet surface
                     temperature



     Decreases effective              Increases
      Henry’s constant                vapor flux

     Decreases absorbate        Decreases droplet
            flux               surface temperature

                                Increases effective
       Increases Stefan
                                 Henry’s constant
            velocity
                               Increases absorbate
                                       flux
                                                            Ben-Gurion University of the Negev
Conclusion

   The obtained results show, that the heat and mass transfer rates in water
   droplet-air-water vapor system at short times are considerably enhanced
   under the effects of Stefan flow, heat of absorption and dissociation reactions
   within the droplet.
   It was shown that nonlinearity of the dependence of droplet surface
   temperature vs. time stems from the interaction of different phenomena.
   Numerical analysis showed that in the case of small concentrations of
   absorbate in a gaseous phase the effects of Stefan flow and heat of absorption
   on the droplet surface temperature can be neglected.
   The developed model allows to calculate the value of pH vs. time for both
   evaporating and growing droplets. The performed calculations showed that
   the dependence of pH increase with the increasing relative humidity (RH).
   The performed analysis of gas absorption by liquid droplets accompanied by
   droplets evaporation and vapor condensation on the surface of liquid droplets
   can be used in calculations of scavenging of hazardous gases in atmosphere
   by rain, atmospheric cloud evolution, and in design calculations of gas-liquid
   contactors.


                                                  Ben-Gurion University of the Negev

				
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