equation of continuity ii. summary of equations of change
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Equation of Continuity II
Summary of Equations of Change
( πππ‘ππππππππππ πππ πππ‘ππ‘π¦ )=(πππ‘ πππ‘ππππππππ‘πππ
ππ πππ‘ππ‘π¦ )+( πππ‘ πππ‘ππππππππ’ππ‘πππππ πππ‘ππ‘π¦ )
Summary of Equations of Change
Summary of Equations of Change
π =ππΉ+πThe momentum molecular flux,
*
molecular stresses = pressure + viscous stresses
Summary of Equations of Change
π=(hπππ‘ π‘ππππ ππππ‘ππ¦ πππππ’ππ‘πππ)+( hπππ‘ π‘ππππ ππππ‘ππ¦ hπππ πππππ’π ππππ ππππππ )
The energy molecular flux is the partial molar enthalpy of species Ξ±
Summary of Equations of Change
Recall: the combined energy flux vector e
Combined Energy Flux Vector
Convective Energy FluxHeat Rate from Molecular Motion
Work Rate from Molecular Motion
Combined Energy Flux Vector:
π=( 12 ππ£2+π οΏ½ΜοΏ½)π+ [π βπ ]+π
We introduce something new to replace q:
Combined Energy Flux Vector
Combined Energy Flux Vector:
We introduce something new to replace q:
π =ππΉ+πRecall the molecular stress tensor:When dotted with v: [π βπ ]=ππ+[π βπ ]
Substituting into e:
π=( 12 ππ£2+π οΏ½ΜοΏ½)π+ππ+[π βπ ]+π
Summary of Equations of Change
Recall: Substituting the equation for q into e
Summary of Equations of Change
Recall: Substituting the equation for q into e
partial molar
per unit mass
Summary of Equations of Change
Recall: Substituting the equation for q into e
Summary of Equations of Change
Summary of Equations of Change
Dp
Dt
vg
ππΆππ·ππ·π‘
=β (π» βπ)β (π βπ» π )
ππ·π€ π΄
π·π‘=(π» β π π·π΄π΅π»π€π΄ )+π π΄
Simultaneous Heat and Mass Transfer
Example 1.
Hot condensable vapor, A, diffusing through a stagnant film of non-condensable gas, B, to a cold surface at y=0, where A condenses
Find:
Simultaneous Heat and Mass Transfer
Assumptions:
1. Steady-state2. Ideal gas behavior3. Total c is constant4. Uniform pressure5. Physical properties are
constant, evaluated at mean T and x.
6. Neglect radiative heat transfer
Simultaneous Heat and Mass Transfer
πππ΄ππ‘
+(π» βππ΄π£π )β (π» βππ· π΄π΅π»π₯π΄ )=π π΄
Equations of Change:
Continuity (A)
πππ΄ππ‘
=β (π» βππΌ )+π π΄
ππ π΄ π¦
ππ¦=0
Simultaneous Heat and Mass Transfer
Equations of Change:
Energyπππ‘π(οΏ½ΜοΏ½+1
2π£2)=β (π» βπ )+(ππ βπ )
πππ¦ππ¦
=0
* Both NAy and ey are constant throughout the film
Simultaneous Heat and Mass Transfer
To determine the mole fraction profile:
Recall: The molar flux for diffusion of A through stagnant B
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
π π΄=βππ·π΄π΅
ππ₯π΄ππ§
+π₯π΄(π π΄+ππ΅)
Since B is stagnant,
π π΄=βππ· π΄π΅
(1βπ₯π΄)ππ₯π΄ππ§
Simultaneous Heat and Mass Transfer
To determine the mole fraction profile:
Recall: The molar flux for diffusion of A through stagnant B
π π΄ π¦=βππ· π΄π΅
(1β π₯π΄)ππ₯π΄π π¦
Recall: Integration of the above equation
Concentration Profiles
I. Diffusion Through a Stagnant Gas Filmβ ln (1β π₯π΄ )=πΆ1π§+πΆ2
Let C1 = -ln K1 and C2 = -ln K2,
1βπ₯π΄=πΎ 1π§πΎ 2
B.C.
at z = z1, xA = xA1
at z = z2, xA = xA2 ( 1βπ₯π΄1β π₯π΄1 )=( 1βπ₯π΄21βπ₯π΄1 )π§β π§ 1π§ 2β π§1
Simultaneous Heat and Mass Transfer
To determine the mole fraction profile:
Recall: The molar flux for diffusion of A through stagnant B
π π΄ π¦=βππ· π΄π΅
(1β π₯π΄)ππ₯π΄π π¦
( 1βπ₯π΄1β π₯π΄ 0 )=( 1βπ₯π΄Ξ΄1βπ₯π΄0 )π¦πΏ
Using the appropriate B.C.s
at y = 0, xA = xA0
at y = Ξ΄, xA = xAΞ΄
Simultaneous Heat and Mass Transfer
To determine the mole fraction profile:
( 1βπ₯π΄1β π₯π΄ 0 )=( 1βπ₯π΄Ξ΄1βπ₯π΄0 )π¦πΏπ π΄ π¦=β
ππ· π΄π΅
(1β π₯π΄)ππ₯π΄π π¦
Evaluating NAy from the equations aboveNote that:
π(1βπ₯ΒΏΒΏ π΄)
ππ¦=β
ππ₯π΄π π¦
ΒΏ π π΄ π¦=ππ· π΄π΅
(1βπ₯π΄)π
(1βπ₯ΒΏΒΏ π΄)ππ¦
ΒΏ
Simultaneous Heat and Mass Transfer
π(1βπ₯ΒΏΒΏ π΄)
ππ¦=β
ππ₯π΄π π¦
ΒΏ π π΄ π¦=ππ· π΄π΅
(1βπ₯π΄)π
(1βπ₯ΒΏΒΏ π΄)ππ¦
ΒΏ
π π΄ π¦ π¦=ππ·π΄π΅ ln1βπ₯π΄1βπ₯π΄0
( 1βπ₯π΄1β π₯π΄ 0 )=( 1βπ₯π΄Ξ΄1βπ₯π΄0 )π¦πΏ
BUTβ¦]
π π΄ π¦β«0
π¦
ππ¦=ππ·π΄π΅
(1βπ₯π΄)β«π₯ π΄ 0
π₯ π΄
π(1βπ₯ΒΏΒΏ π΄)ΒΏ
Simultaneous Heat and Mass Transfer
π π΄ π¦=ππ· π΄π΅
πΏln (1βπ₯π΄Ξ΄1βπ₯π΄0 )
]
Simultaneous Heat and Mass Transfer
π π΄ π¦=ππ· π΄π΅
πΏln (1βπ₯π΄Ξ΄1βπ₯π΄0 )
Rearranging and combining
( 1βπ₯π΄1β π₯π΄ 0 )=( 1βπ₯π΄Ξ΄1βπ₯π΄0 )π¦πΏ
π π΄ π¦
ππ·π΄π΅
= 1πΏln (1βπ₯π΄Ξ΄1βπ₯π΄0 ) ln ( 1βπ₯π΄1βπ₯π΄0 )= π¦πΏ ln(
1βπ₯π΄Ξ΄1βπ₯π΄0 )
ln ( 1βπ₯π΄1βπ₯π΄0 )=π¦ππ΄ π¦
ππ· π΄π΅
Simultaneous Heat and Mass Transfer
@ y = y, xA = xA
ln ( 1βπ₯π΄1βπ₯π΄0 )=π¦ππ΄ π¦
ππ· π΄π΅
( 1βπ₯π΄1β π₯π΄ 0 )=exp[(π π΄ π¦
ππ·π΄π΅)ΒΏπ¦ ]ΒΏ
1β( 1βπ₯π΄1βπ₯π΄0 )=1βexp [(π π΄ π¦
ππ· π΄π΅)ΒΏ π¦ ]ΒΏ
( π₯π΄βπ₯π΄01βπ₯π΄0 )=1β exp[( ππ΄ π¦
ππ· π΄π΅)ΒΏ π¦ ]ΒΏ
Simultaneous Heat and Mass Transfer
@ y = y, xA = xA
( π₯π΄βπ₯π΄01βπ₯π΄0 )=1β exp[( ππ΄ π¦
ππ· π΄π΅)ΒΏ π¦ ]ΒΏ
@ y = Ξ΄, xA = xAΞ΄
( π₯π΄πΏβπ₯π΄01β π₯π΄ 0 )=1β exp[( π π΄ π¦
ππ·π΄π΅)ΒΏπΏ]ΒΏ
Taking the ratios of the two equations
Simultaneous Heat and Mass Transfer
To determine the temperature profile:
Note:
where the enthalpy of mixing is often neglected for gases at low to moderate pressures
Simultaneous Heat and Mass Transfer
To determine the temperature profile:
πππ¦ππ¦
=0
βπ π2πππ¦2
+ππ΄π¦~πΆππ΄
ππππ¦
=0
The general solution is
π=π1+π2 exp(π 2π¦ ) where
Simultaneous Heat and Mass Transfer
where
At y = 0, T = T0 π 0=πΆ1+πΆ2
At y = Ξ΄, T = TΞ΄
π=π1+π2 exp(π 2π¦ )
π πΏ=π1+π2 exp(π2πΏ)
Subtracting the two equations
π 0βπ πΏ=π2 ΒΏ
π2=π 0βπ πΏ
1β exp (π2πΏΒΏ)ΒΏ
Simultaneous Heat and Mass Transfer
π1=π 0(1β exp (π 2πΏ ))β(π 0βπ πΏ)
1βexp (π 2πΏ)
π2=π 0βπ πΏ
1β exp (π2πΏΒΏ)ΒΏSince π 0=πΆ1+πΆ2
π=π1+π2 exp(π 2π¦ )
π=π 0(1β exp (π 2πΏ ))β(π 0βπ πΏ)
1βexp (π 2πΏ)+
π0βπ πΏ
1βexp (π2πΏΒΏ)exp (π2 π¦ )ΒΏ
Simultaneous Heat and Mass Transfer
π=π 0(1β exp (π 2πΏ ))β(π 0βπ πΏ)
1βexp (π 2πΏ)+
π0βπ πΏ
1βexp (π2πΏΒΏ)exp (π2 π¦ )ΒΏ
π (1β exp (π 2πΏ ))=π 0 (1βexp (π2πΏ ))β (π 0βπ πΏ )+(π 0βπ πΏ)(exp (π 2π¦ ))
(π βπ 0 ) (1βexp (π2πΏ ) )= (π πΏβπ 0 )(1βexp (π2 π¦ ))
where (π βπ 0 )(π πΏβπ 0 )
=1β exp (π 2 π¦ )1βexp (π2πΏ )
Simultaneous Heat and Mass Transfer
If we did not consider mass transfer
ππ¦=βπππππ¦πππ¦ππ¦
=0βπ π2πππ¦2
=0
βπππππ¦
=π1
ππππ¦
=π1βπ
π=π1π¦βπ
+π2
@ π¦=0 ,π=π 0
π2=π 0
@ π¦=πΏ ,π=π πΏ
π πΏ=π1πΏβπ
+π0
π1=(π0βπ πΏ ) (ππΏ )=βπ ππππ¦
Simultaneous Heat and Mass Transfer
With mass transfer
βkππππ¦
β£@ π¦=0=π0βπ πΏ
1β exp[(ππ΄π¦ πππ΄π )πΏ ] (
π π΄π¦ πππ΄π )βπ
Simultaneous Heat and Mass Transfer
Comparison of the energy flux with & without the presence of mass transfer:
Rate of heat transfer is directly affected by simultaneous mass transferBUT mass flux is not directly affected by simultaneous heat transfer
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