MTO lecture chapter 1

8/13/2019 MTO lecture chapter 1

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Chapter One

Review of Diffusion and Mass Transfer Theories1. Introduction:When a system contains two or more components whose concentrations vary from point to point,

there is a natural tendency for mass to be transferred, minimizing the concentration differences

within the system and moving it towards equilibrium.

(a) Mass Transfer:Transport of one component from a region of high concentration to a region of low concentration.

(b) Dail !ife "#a$ples: Numerous of our daily activities involve mass transfer phenomena such as:

1. iffusion of perfume through air.

!. iffusion of coffee smell through air.

". issolution and diffusion of sugar throughout a cup of coffee.

#. $aundry hanging under the sun during a breezy day.

(c) Industrial "#a$ples:%ass Transfer plays an important role in many industrial processes. & group of operations for

separating the components of mi'tures is based on the transfer of material from one homogeneous

1



phase to another. These methods (i.e. %ass Transfer )perations* include +as &bsorption,

istillation, umidification, $iquid -'traction, &dsorption, and %embrane eparations.

%ote:The driving force for mass transfer in these operations is a concentration gradient.

&. Diffusivit:

(a) 'ics !aw:/lu' of %ass transfer by molecular diffusion is proportional to concentration gradient.

na α X C a∂∂ na = - D X

C a∂∂

Where na: /lu' of %ass transfer,sec.

!cm

mole g

D: iffusivity coefficient and its function in temperature and pressure,sec

!cm

X

C a

∂

∂: 0oncentration gradient,

cm

cmmole g "

%otes:

1. /ic23s law proportionality factor 45 is 2nown as the diffusion coefficient or diffusivity. 6tsfundamental concept is identical to the other transport properties as 2inematic viscosity (78* and

thermal diffusivity 9.

!. The mass diffusivity is usually reported in units of cm !s or m!s. The diffusion coefficient

depends upon the pressure and temperature.

!



". -'perimental values for the diffusivities of gases, liquids, and solids are tabulated.

#. ue to the molecules mobility, the diffusivities are generally higher for gases ( ;.<=1; >< to

1.;=1;>< m!s* than for liquids ( 1;>1; to 1;>? m!s* which are higher than the values reported for

solids ( 1;>1# to 1;>1; m!s*.

<. 6n the absence of e'perimental data, semi>theoretical e'pressions have been developed which give

appro'imated values, which sometimes as valid as e'perimental values due to the difficulties

encountered in their measurements.

'or ideal *ases:

ince PV = nRT P = RT V

n P = C RT where C

V

n=

0 @ RT

P na = - (D/RT)(∂P A / ∂x)

(b) Calculation of Diffusivit coefficient +D, fro$ other phsical properties:(i) -ases:

-illiland Correlation:

B M M V V P

T D

A B A

AB

11

*(

;;#".;!"1"1

<.1

++

=

sec

!cm

T : A P : atm V A, V B:

gmole

cm"

M A, M B: molecular weight

"#a$ple:

"



0alculate the mass diffusivity of mi'ture 0)! B )! at ; o0 and 1 atm using +illiland 0orrelation.

tudy the effect of temperature and pressure as:

(a* 6ncrease the temperature from ; to 1;;o

0 in 1;o

0 increment at constant pressure.(b* 6ncrease the pressure from 1 to 1; atm in 1 atm increment at constant temperature.

When diffusion 4 D15 is 2nown at T 1 B P 1, D!" can be calculated at T ! B P ! from:<.1

1

!

!

11!

=

T

T

P

P D D

irschfelder et al. Correlation:

irschfelder et al. (1?#?* used the $ennard>Cones potential to evaluate the influence ofintermolecular forces, presented an equation for the diffusion coefficient for gas pairs of non>polar

and non>reacting molecules as:

&D @ (;.;;!EE T1.<* (F %&D;.< G&D

! H*

Where %&D @ ! (1 %& I 1 %D*>1

&D : diffusion coefficient, cm!s %& B %D : molecular weight of & B D

T : temperature, A F : pressure, bar

G&D : collision diameter, &o H : diffusion collision integral, dimensionless

H is a function of temperature and intermolecular potential field for & and D molecules. H is

usually tabulated as a function of T=.

T= @ ( 2 T J&D*

#



Where 2 : Doltzmann constant (1."K=1;>1E ergA*

J&D : energy of molecular interaction for & and D erg

& very accurate appro'imation of H can be obtained from (Neufield et al., 1?L!* as:

H @ (a T= b* I (c e'p d T=* I (e e'p f T=* I (g e'p h T=*

Where a @ 1.;E;# b @ ;.1<E1 c @ ;.1?"; d @ ;.#LE#

e @ 1.;"<? f @ 1.<!?? g @ 1.LE#L h @ ".K?#1

/or a binary system composed of non>polar molecular pairs:

G&D @ (G& I GD* ! J&D @ (J& JD *;.<

<



E



(ii) !i/uids: A: solute B: solvent

0toes "inestein correlation:&D @ (?.?E = 1;>1E T* (7D M&

;."""* for V A ;.< m" 2 mol

2ile Chan*:

&D @ O11L." = 1;>1K (P %D*;.< TQ (7D M&;.E* for V A R ;.< m" 2 mol

Where

D AB: diffusivity of 4 A5 in very dilute solution in solvent B, m!sec

M B: %olecular weight of solvent T : temperature, A

# B: solvent viscosity, 2g m . s V A: solute molal volume, m"2mol

P : solvent association factor (!) is !.!E, 0") is 1.?, 0!<) is 1.<, etcS*

L



K



(c) 0tead state one3di$ensional diffusion:

n& @ > ( d 0& d '*

4. Mechanis$ of Mass Transfer:

There are two types:

(a* %olecular iffusion (b* 0onvective (as in mi'ing*

When a mi'ture of gases or liquids is contained such that there concentration gradient of one ormore of the constituents across the system, there will be a mass transfer on a microscopic level as a

result of diffusion from regions of high concentration to regions of low concentration. %ass is also

transported from one place to another due to convection in the flow system. This type of mass

transfer occurs on a macroscopic level.

(a) Molecular Diffusion:(i* -quimole > counter iffusion

(ii* iffusion of 4&5 in non> diffusing of 4D5

(i) "/ui$ole counter Diffusion of +5, and +6,:

?

A B

1 mole of 4&5

1 mole of 4D5



B A nn −=$x

$C Dn A

A −= )ne direction

$x

$P

RT

D

n A AB

A −= and $x

$P

RT

D

n

B BA

−=B

since

B A P P P += ;=+$x

$P

$x

$P B A

$x

$P

$x

$P B A −=

Then$x

$P

RT

Dn A BA=B B A nn −=

$x

$P

RT

D

$x

$P

RT

D A BA A AB −=− D D D BA AB ==

Conclusion:

iffusivity does not depend on concentration.

$x

$P

RT

Dn

A

A −= ∫ ∫ −=

!

1

!

1

A A $P RT

D$xn ( ) ( )

1!1! A A A P P RT

D x xn −−=−

−

−=

−

−=

1!1!

!1!1

x x

C C D

x x

P P

RT

Dn

A A A A

A

"#a$ple:

& large tan2 filled with a mi'ture of gases &BD at 1 atm and "; o0 is connected to another

large tan2 filled with different concentration at 1 atm and "; o0. The connection between the

tan2s is ! inch 6.. and 1! ft long. 0alculate the steady state mass transfer rate of gas &

1;



through the tube when the concentration of & in one tan2 is ?; mole and in the other is <

mole . Ta2e the diffusivity equals to ;.#!< cm!sec.

(ii) Diffusion of +5, in non3Diffusin* of +6,:

+as 4&5 dissolves in the liquid,

however 4D5 does not dissolve in the liquid.

n& @ (> U T* d F& d V

nD @ (> U T* d FD d V @ ( U T* d F& d V

ince (nD* for bul2 flow @ > (nD* for molecular diffusion

(nD* for bul2 flow @ (> U T* d F& d V

ince (n& bul2 flow nD bul2 flow* @ (F& FD*

(n&* bul2 flow @ (> UT* (d F&d V* (F&FD*

(n&*total @ n& I (n&* bul2 flow @ (> UT* (d F&d V* (1 I F&FD* @ (> UT* (d F&d V* (F&I FD* FD

(n&*total @ (UT* (F pDm* (F&1 F&!* (V! V1*

11

Liquid Gas

(& and D*

nD

&

D

n&

(&ID*

Dul2 flow



pDm @ (FD! FD1* $n (pD! pD1*

(n&*total @ (0 0Dm* (0&1 0&!* (V! V1*

Co$parison between "/ui$ole and 7ni3direction:

6tem -quimole Xni>direction

/lu'1!

!1

x x

P P

RT

D A A

−

−

maller 1!

!1

x x

P P

P

P

RT

D A A

BM −

−

$arger

Dul2 flow Not available &vailable

&pplication distillation &bsorption

-'traction

(b) Convective Mass Transfer:

This mode of transfer involves both the transport properties and the dynamic characteristics of theflowing fluid.

The controlling resistance to convective mass transfer is often a thin film of fluid.

1!



6n the turbulent region, particles of fluid no longer flow in the orderly manner found in the laminar

sub>layer. 6nstead, large portions of the fluid (called eddies* move rapidly from one position to

another. These eddies contribute considerably to the mass transfer process. Decause the eddy motionis rapid, mass transfer in the turbulent region is much more rapid than laminar conditions. This

situation can be modeled in terms of mass transfer coefficient.

( )$x

$C % Dn a

Da +−=

where

na : /lu' of mass transfer

D: iffusivity % D: eddy diffusivity and depend on degree of turbulence

( )!1

1!

aa

D

a C C x x

% Dn −

−+

= -quimole diffusion

( )!1

1!

aa

BM

T D

a C C C

C

x x

% Dn −

−+

= Xni>direction diffusion

the above equations can be written as: (!1

aaa C C & n −=

where

& : mass transfer coefficient,

.sec

cm

−+

=1!

x x

% D D

for -quimole diffusion

1"



BM

T D

C

C

x x

% D

−+

=1!

for Xni>direction diffusion

& is function in D: diffusivity, V : velocity, ': density, #: viscosity, and : linear dimension

8. Di$ensional 5nalsis:%ost practically useful mass transfer situations involve turbulent flow, and for these situations, it is

generally not possible to compute mass transfer coefficients from theoretical conditions. 6nstead, we

must rely on e'perimental data. The data are limited in scope w.r.t. circumstances and situations as

well as to range of fluid properties. Therefore, it is important to be able to e'tend their applicability

to conditions not covered e'perimentally. & very useful procedure is dimensional analysis.

6n dimensional analysis, the significant variables in a given situation are grouped into dimensionless

parameters which are less than the original variables. Dy combining the variables into a smaller

number of dimensionless parameters, the wor2 of e'perimental data is considerably reduced.

/rom dimensional analysis, we found that: h @ a Ue b cc

Where:

h: herwood number, 2 $ Ue: Ueynolds number, 8 v $7

c: chmidt number, 7 8

"#a$ple:

1#



0alculate the mass transfer coefficient for the absorption of )! from a dilute mi'ture with air in

wetted wall column using the given e'perimental data:

6.. @ !< mm Mgas @ !.! msec Tgas @ !?" A 7gas @ 1.LK (1;*><

2gm.sec8gas @ 1.!! 2gm" @ 1!.!(1;*>E m!sec h @ ;.;!" Ue;.K< c;.##

1<

MTO lecture chapter 1

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