Diffusion Through a Stagnant Gas Film
Short Description
Download Diffusion Through a Stagnant Gas Film...
Description
Diffusion through a stagnant stagnant gas film
Table of Contents ACKNOWLEDGEMENT ................................................................................................................................... 2 DIFFUSION THROUGH A STAGNANT GAS FILM............................................................................................. 3 SHELL MASS BALANCES; BOUNDARY CONDITIONS: ..................................................................................... 3 Boundary Condition: ..................................................................................................................................... 8 Average Concentration ............................................................................................................................... 11 Applications: ............................................................................................................................................... 15 References: ................................................................................................................................................. 16
Transport Phenomena
Page 1
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film ACKNOWLEDGEMENT ACKNOWLEDGEMENT We dedicate our effort to our teacher (Engr. Kamran Sibtain), who guided us and helped us in the preparation of this report. Because we think that without his guidance it was almost impossible to create this. Secondly we dedicate this effort to our parents as they pray for our success and we did not able to do that without their prayers.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film DIFFUSION THROUGH A STAGNANT GAS FILM The procedure for the shell mass balance is as follows:
A mass balance is made over a thin shell perpendicular to the direction of mass transport, and this shell balance leads to a first-order differential equation, which may be solved to get the mass flux distribution.
Into this expression we insert the relation between mass flux and concentration gradient, which results in a second-order differential equation for the concentration profile. The integration constants that appear in the resulting expression are determined by the boundary conditions on the concentration and/or mass flux at the bounding surfaces.
We know several kinds of mass fluxes are in common use. A For simplicity, we shall in this chapter use the combined flux N A-that is, the number of moles of A that go through a unit area in unit time, the unit area being fixed in space. We shall relate the molar flux to the concentration gradient for the z-component is:
SHELL MASS BALANCES; BOUNDARY CONDITIONS: The diffusion problems in this chapter are solved by making mass balances for one or more chemical species over a thin shell of solid or fluid. Having selected an appropriate system, the law of conservation of mass of species A in a binary system is written over the volume of the shell in the form:
(1)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film overall motion of the fluid (i.e., by convection), both of these being included in N A . In addition, species A may be produced or consumed by homogeneous chemical reactions. After a balance is made on a shell of finite thickness by means of Eq.1, we then let the thickness become infinitesimally small. As a result of this process a differential equation for the mass (or molar) flux is generated. If, into this equation, we substitute the expression for the mass (or molar) flux in terms of the concentration gradient, we get a differential equation for the concentration. When this differential equation has been integrated, constants of integration appear, and these have to be determined by the use of boundary conditions. The boundary conditions are very similar to those used in heat conduction:
The concentration at a surface can be specified; for example, x example, x A = x A0 .
The mass flux at a surface can be specified; for example, N Az = N A0 . If the ratio N Bz /N Az is known, this is equivalent to giving the th e concentration gradient.
If diffusion is occurring in a solid, it may happen that at the solid surface substance A is lost to a surrounding stream according to the relation:
in which N AO is the molar flux at the surface,
C AO
is the surface concentration, c Ab is the
concentration in the bulk fluid stream, and the proportionality constant k c is a "mass transfer coefficient."Equation.1 is analogous to "Newton's law of cooling".
The rate of chemical reaction at the surface can be specified. For example, if substance A disappears at a surface by a first-order chemical reaction, then N then N AO = k cA0 . That is, the rate of disappearance at a surface is proportional to the surface concentration, the proportionality constant k being a first-order chemical rate coefficient. 1
Let us now analyze the diffusion system shown in Fig.1 in which liquid A is evaporating into gas B. We imagine there is some device that maintains the liquid level at z =z1 . Right at the liquid-gas A, expressed as mole interface, the gas-phase concentration of A,
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film
Gas stream of A and B Z=Z 2
X B 1.0
X B2 B2
X A2
Z 2
N A│Z+∆Z ∆ Z
X A+X B=1
X B
X A
Z
N A│Z
Z 1 ------------ Z=Z 1 ------------
1.0 liquid A
X B1 B1
X A1
0
X A
-----------Fig.1: Steady-state diffusion of A through stagnant B with the liquid vapor
interface maintained at a
fixed position. The graph shows how the concentration profiles deviate from straight lines because of the convective contribution to the mass flux.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film fraction, is xAl. This is taken to be the gas-phase concentration of A corresponding to equilibrium1 with vap
the liquid at the interface. That is, xAl is the vapor pressure of A divided by the total pressure, p A /p, provided that A and B form an ideal gas mixture and that the solubility of gas B in liquid A is negligible.
A stream of gas mixture A-B of concentration mole fraction of A at
X A2 A2
X A2 A2
flows slowly past the top of the tube, to maintain the
for z= z2. The entire system is kept at constant temperature and pressure. Gases
A and B are assumed to be ideal.
We know that there will be a net flow of gas upward from the gas-liquid interface, and that the gas velocity at the cylinder wall will be smaller than that in the center of the tube. To simplify the problem, we neglect this effect and assume that there is no dependence of the z-component of the velocity on the radial coordinate.
When this evaporating system attains a steady state, there is a net motion of A away from the interface and the species B is stationary. Hence the molar flux of A is given by Eq. given below with NBz = 0. Solving for NAz, we get:
(2)
A steady-state mass balance (in molar units) over an increment Az of the column states that the amount of A entering at plane z equals the amount of A leaving at plane z +Δz:
[Rate of mass in] – [ Rate of mass] + [Rate of production or accumulation]=0 S. N AZ │Z ––– S. N AZ │Z +∆ Z + 0 = 0
(3)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film Now, dividing by equation (3) by S.∆Z
S[ NAZ│Z Taking limit as ∆Z
–––
NAZ│Z+∆Z ]/S( ∆ Z ) = 0
0
(d – (d
N AZ /dZ)= O
(4)
Now putting the value of N AZ in equ (4).
For an ideal gas mixture the equation of state is p = cRT, so that at constant temperature and pressure c must be a constant. Furthermore, for gases is very nearly independent of the composition. Therefore, c DAB be moved to the left of the derivative operator to get:
This is a second-order differential equation for the concentration profile expressed as mole fraction of A. Integration with respect to z gives:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film Boundary Condition:
i. At Z=Z 1 ii.
ln – ln
(1
– ln
(1
At
–––
–––
X A= X A1
;
Z=Z 2
;
X A=X A2
X A1 )= C 1. 1 Z . 1 + C 2
(a)
X A2 )= C 1. 1 Z . 2+ C 2
(b)
Now subtract equation (b) from (a)
ln – ln
(1
–––
X A1 ) + Ln (1
Ln{ (1
–––
X A2 )/( 1
C 1=ln { (1
–––
–––
X A2 )/( 1
–––
X A2 )= C 1 (Z 1 – Z 2 )
X A1 )}= C 1 (Z 1 – Z 2 )
–––
X A1 )}/ (Z 1 – Z 2 )
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film Now putting the value of C1 and C2 in equ (5)
ln – ln
(1
–––
X A )= ln { (1 – ln
ln – ln
ln – ln
(1
–––
–––
{ (1
(1
X A ) + ln (1
X A2 )/( 1 –––
–––
–––
ln{ (1
–––
X A1 )/ (1
–––
X A2 )/( 1
X A1 )} Z / . (Z 1 – Z 2 ) –––
X A1 )} Z 1 / . (Z 1 – Z 2 )
X A1 )
X A1 ) =
ln { (1 – ln
–––
{ (1
–––
X A2 )/( 1
–––
X A1 )} Z / . (Z 1 – Z 2 )
–––
X A2 )/( 1
–––
X A1 )} Z 1 / . (Z 1 – Z 2 )
X A )}=ln{(1
–––
X A2 )/( 1
–––
X A1 )}
Z / (Z1 – Z 2)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film Multipling both side by ( – – )
ln{ – ln{
(1
–––
X A1 )/ (1
–––
X A )}= ln{(1 – ln{(1 + ln { (1
ln{ (1
–––
X A )/ (1
–––
ln{(1 – ln{(1
–––
X A )/ (1
ln { (1
–––
–––
–––
X A1 )}
–––
X A2 )/( 1
–––
X A1 )}
–––
X A2 )/( 1
–––
X A1 )}
Z / (Z1 – Z 2)
Z1 / (Z1 – Z 2)
X A1 )}= ln { (1
ln{ (1
X A2 )/( 1
–––
–––
X A2 )/( 1
–––
X A1 )}
Z1 / (Z1 – Z 2)
Z / (Z1 – Z 2)
X A1 )}=
X A2 )/( 1
[{ Z1 [{ Z1 / / (Z1 – Z 2 )– { Z / (Z1
–––
X A1 )}
– Z 2)}]
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film (1
–––
X A )/ (1
–––
{ (1
X A1 )=
–––
X A2 )/( 1
–––
X A1 )}
{ (Z1 – Z )/ (Z1 – Z 2 )}
(6)
OR also can be written as,
(X B /X B1 B1 ) = (X B2 B2 /X B1 B1 )
{ (Z1 – Z )/ (Z1 – Z 2 )}
(7)
The equation (6) and (7) represent the concentration profile for diffusion through a stagnant medium. The profiles for gas B are obtained by using x B = 1-x A. The concentration profiles are shown in Fig.1. It can be seen there that the slope dxA/dz is not constant although N is; this could have been A0 anticipated from Eq.1.
Average Concentration Concentration Once the concentration profiles are known, we can get average values and mass fluxes at surfaces. For example, the average concentration of B B in the region between z1 , and z2, is obtained as follows:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film X B avg /X B1 B1 =
/ ∫ ∫
Now using equation (7), we get,
( ) Now introducing reduced length,
ξ= As
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film
|
1 0
Now by simplifying, We get
X B avg
OR Can be written as,
(8)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film The rate of mass transfer at the liquid-gas interface-that is, the rate of evaporation- may be obtained from Eq.1 as follows:
(10)
By combining Eqs. 9 and 10 we get finally
This expression gives the evaporation rate in terms of the characteristic driving force x A1 – xA2.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film Applications:
Diffusivity or diffusion co-efficient of a gas can be measured by diffusion through stagnant gas film. The apparatus used for this is called “Arnold Diffusion Cell”. Now we consider the case of diffusion through a gas-liquid interface Ammonia, NH,, is being selectively removed from an air-NH, mixture by absorption into water. In this steady-state process, ammonia is transferred by molecular diffusion through a stagnant gas layer 5 mm thick and then through stagnant water layer 0.1 mm thick. The concentration of ammonia at the outer boundary of the gas layer is 3.42 mol% and the concentration at the lower boundary of the water layer is essentially zero.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Diffusion through a stagnant stagnant gas film References:
Wikipedia.org
Transport Phenomena by R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot
PRINCIPLES AND MODERN APPLICATIONS OF MASS TRANSFER OPERATIONS by Jaime Benitez
View more...
Comments
