Derivation of the Pressure Retarded Osmosis (PRO) Mass Transport

(1)

Derivation of the Pressure Retarded Osmosis (PRO) Mass Transport

Equations

Korea University, July 2012

Fall Semester 2012 ACE 644

Water Treatment Process

(2)

Ideal Membrane and Hydrodynamics:

NO Concentration Polarizations

J _w

Feed Solution

Pressurized Draw Solution

c

_D,b

c

_F,b

By definition:

   

w m b

J  A      P A     P

A: Membrane Intrinsic

Water Permeability [L t

^‐1

P

^‐1

]

P: Applied Hydraulic Pressure Difference

Again, assume the van’t Hoff relation to be valid:

c c nR Tg



   



 

w m

b g

w

g

J

J A c nR T P A c nR T P

   

(3)

Ideal Semi‐Permeable Membrane and Hydrodynamics

By definition:

B: Membrane Intrinsic

Solute Permeability [L t

^‐1

]

J _s

   

s m b

J B c B c

      

   

w m b

J  A      P A     P

J _w

Feed Solution

c

_D,b

c

_F,b

(4)

Real World

J _w

Feed Solution

c

_D,b

z

J _s

c

_F,b

c_D,m c_F,m

c_F,s

x

t_s t_a δ

Therefore:

   

s m b

J B c B c

      

δ: thickness of dECP[L]

t_a: thickness of membrane active layer [L]

t_s: thickness of membrane support layer [L]

,

, ,

,

, ,

Concentrative (feed) External Concentration Polariza

Dilutive (draw) ECP Concentrative Internal Conce

tio ( ECP)

n ( P)

n

EC

D

F b F

D

F s m

m F

s

b d

f

c c



 



tration Polarization (ICP)

) (

)

( P A P

A

J

_w

  

_m

    

_b

 

(5)

Internal Concentration Polarization

, , Concentrative Internal

Concentration Polarization (ICP)

F s F m

c  c 

• As water permeates across the membrane, the feed solutes are selectively retained by the semi‐

permeable active layer and build up within the support layer

• Diffusion works to restore c to c_F,s but

porous support acts as unstirred boundary layer

• J_s across the porous support is the sum of the diffusive component, driven by Δc, and the convective component, arising from J_w:

( ) ( )

s

s w

J D dc z J c z

   dz 

s

/

D  D  

J _w

Feed Solution

c

_D,b

z

J _s

c

_F,b

c

_D,m c_F,m

c

_F,s

x

t_s t_a δ

(6)

ICP: Maths

( ) ( )

s

s w

J D dc z J c z

   dz 

F s F m

c c 



^{D m}^, ^{F m}^,



^s ^{dc z}^{( )} ^w ^{( )}

B c c D J c z

  dz 

Integrate with boundary layers:

, ,

0

_{F s}

s F m

z c c

z t c c

  

    



 

, , exp ^{w s} , , exp ^{w s} 1

F m F s D m F m

w

J t B J t

c c c c

D J D

 

 

 

   

       

Substitute J_sat steady state:

Plus algebra:

J _w

z

J _s

c_F,m

c

_F,s

t_s

(7)

ICP: Final Equation

 

, , exp ^{w s} , , exp ^{w s} 1

F m F s D m F m

w

J t B J t

c c c c

D J D

 

 

 

   

       

 

, , ,

,

exp exp 1

w w

F s D m F m

w

w s w

F s

w

J S B J S

c c c

D J D

J S J J S

c D J D

 

   

       

 

   

      

F , m F , m

c c

S: structural parameter of the support layer: [L]

D/S = 1/K: mass transfer coefficient of the support layer [L t^‐1]

s / S  t 

c_F,m is the sum of:

1. The effect of concentrative ICP described by the first term

2. The gain in feed salt concentration at the

membrane interface due to the reverse permeation of draw solution, J_s, in the second term

c

_D,m

J _w

z

J _s

c_F,m

c

_F,s

t_s

F s F m

c c 

(8)

Dilutive (Draw) External Concentration Polarization

, , Dilutive (draw) External

Concentration Polarization ( ECP)

D m D b

f

c  c 

• J_w dilutes c_D in the boundary layer

• Diffusion works to restore c to c_D,b but is limited by mass transfer

• Similar to ICP, the salt flux within this dECP boundary layer comprises diffusive and convective components

( ) ( )

s w

J D dc x J c x

   dx 

J _w

Feed Solution

c

_D,b

z

J _s

c

_F,b

c_D,m

c

_F,m

c

_F,s

x

t_s t_a δ

(9)

dECP: Maths

( ) ( )

s w

J D dc x J c x

   dx 



^{D m}^, ^{F m}^,



^{dc x}^{( )} ^w ^{( )}

B c c D J c x

  dx 

Substitute J_sat steady state:

Integrate with boundary layers:

, ,

0

_{D m}

D b

x c c

x  c c

  

    



 

, , exp ^w , , 1 exp ^w

D m D b D m F m

D w D

J B J

c c c c

k J k

 

   

       

    

Plus algebra:

, , Dilutive (Draw) External

D m D b

d

c c 

J _w

c

_D,b

J _s

c_D,m

c

_F,m

x

t_a δ k_D  D /

(10)

dECP: Final Equation

 

, , ,

,

exp 1 exp

w w

D b D m F m

D w D

w s w

D b

D w D

J B J

c c c

k J k

J J J

c k J k

 

   

       

    

 

   

      

    

D , D , m

c m

c

k_D: mass transfer coefficient of the draw boundary layer [L t^‐1]

c_D,m is the sum of:

1. The effect of dilutive ECP described by the first term 2. The loss in draw salt concentration at the membrane

interface due to the reverse permeation of draw solution, J_s, in the second term

, , Dilutive (Draw) External

D m D b

d

c c 

J _w

J _s

c_D,m

c

_F,m

x

t_a δ

c

_D,b

(11)

Concentrative (Feed) External Concentration Polarization

, , Concetrative (feed) ECP ( ECP)

F b F s f

c  c 

• Solute built up in support layer tries to diffuse out to the bulk feed solution, but is limited by mass transfer at the boundary layer.

However,

• Therefore, we assume fECP to be negligible and

1 / K  D S /  k

_F

,

c

F b



F , s

c

k_F: mass transfer coefficient of the feed boundary layer [L t^‐1]

J _w

Feed Solution

c

_D,b

z

J _s

c

_F,b

c

_D,m

c

_F,m

c_F,s

x

t_s t_a δ

(12)

Combining Everything

,

, ,

,

exp

e ex

1 e p x x p

p

^w

D b

D m F m

w

D

w D w

F b

w

c J S

D J

J k

J S

D c

c c

B

J k

 

  

  

 

 



  

 



 

 

 



 





    

 

, ,

exp

exp exp

exp

w

D m F m D b

D

w D b

w

w F

s w

c c c J S

D J S

J J

J k

J k D

  c

  

 

 

 

   

 

  

 



 

  

  

  

  

Alternatively:

J _w

Feed Solution

z

J _s

c

_F,b

c_D,m c_F,m

x

t_s t_a δ

c

_D,b

(13)

 

^, ^,

exp exp

1 exp exp

w w

D b F b

D

s m

w w

w D

J J S

c c

k D

J B c B

J S J

B

J D k

       

       

   

                               

 

^, ^,

exp exp

1 exp exp

w w

D b F b

D

w m

w w

w D

J J S

k D

J A A P

J S J

B

J D k

 



       

       

   

                                 

Final Equations for Water Flux, J _w , and

Reverse Salt Flux, J _s

Derivation of the Pressure Retarded Osmosis (PRO) Mass Transport