Determination of the Dimensionless Schmidt Number
for Gases and Liquids using Matlab Functions

Schmidt Number:  Sc = mu/rho/DAB = viscosity/density/diffusivity

The Schmidt number, often abbreviated Sc, is a dimensionless quantity with important applications to transport phenomena.  According to Bird, Stewart, and Lightfoot, "The Schmidt number is the ratio of momentum diffusivity to mass diffusivity and represents the relative ease of molecular momentum and mass transfer.  It is analogous to the Prandtl number, which represents the ratio of the momentum diffusivity to the thermal
diffusivity" (BSL 600).  The Schmidt number becomes important in calculations of binary mass transfer between phases, as demonstrated in Chapter 22 of our text.

While the CENG 402 class already has a Matlab program, prcalc, to calculate the Prandtl number, no analogous program existed for the purpose of determining the Schmidt number -- until now.  In April of 2002 we
unveil a nifty feat of engineering: not one, but two Matlab programs which can be used to easily determine the Schmidt number.  The first, sccalc, calculates Sc for low density binary gas mixtures, while scliqcalc determines Sc for dilute binary mixtures of several common liquids.  Both functions require the prior use of start301 to obtain the necessary data for the desired compounds.  The contents of the two Matlab m-files may be found below.
Also included are examples of their use and applications.

Matlab Program to Calculate Schmidt Number of Binary Gas Mixtures at Low Density

The Schmidt number appears most frequently in correlations used to calculate mass transfer coefficients, which
can then be used to calculate such quantities as rate of evaporation.  For instance, example 22.3-1 on
page 682 of the text (BSL) gives the following problem:

"A spherical drop of water, 0.05 cm in diameter, is falling at a velocity of 215 cm/sec through dry, still air at 1
atm pressure with no internal circulation.  Estimate the instantaneous rate of evaporation from the drop, when the drop surface is at T0 = 70 degrees F and the air (far from the drop) is at Tinf = 140 degrees F.  The vapor
pressure of water at 70 degrees F is 0.0247 atm.  Assume quasi-steady state conditions."

The mass transfer coefficient can be found from the Sherwood number, but to calculate that, we need the
Schmidt number!  What if the problem didn't give the density, viscosity, and diffusivity?  Okay, so this
particular example does, but life is not always so simple.  That's where sccalc comes to the rescue.  Here's
how it works:
______________________________________________________________________________

function sc = sccalc(p,T,zs,index)

% function sc = sccalc(p,T,zs,index)
%
% Calculates the Schmidt number for low density, binary gas mixtures, using the Matlab functions
% mixmu and dcalc to calculate the viscosity and diffusivity of the mixture, respectively.
%
% mixmu uses equations 1.4-15 and 1.4-16 in BS&L, 2nd ed to give viscosity mu in kg/m/sec
% dcalc uses the Chapman-Enskog formula to give diffusivity DAB in m^2/sec
% Density rho is calculated using the ideal gas law with a weighted average of the molecular weights
% of the two compounds.  Units are kg/m^3.
%
% Argument List:
%   p        [=] total pressure in atm
%   T       [=] temperature in units of Tdeg, as specified using start301
%   zs       [=] vector of mol fractions for the compounds
%   index [=] indices of the compounds from start301.  If no
%             index vector is given, compounds 1 and 2 will be used.
% Returns:
%   sc      [=] Schmidt number (unitless)
%
% Example:  >> start301 (specify new session, mass & energy balances, ceng301 database,
%                        temperature in Kelvin, 2 compounds: chlorine and air)
%                  >> Sc = sccalc(1,297,[0.25 0.75],[1 2])
%
%                       DAB = 1.2015e-05
%                       mu =  1.6407e-05
%                       rho = 1.6183
%                       Sc =  0.8439
%
% Created April 2002 by Danielle Dunn and Jessica Dunn

global mw lenjones Tdeg

if nargin==3
index = [1 2];
end

pkPa = p*1.0133*10^2;              % dcalc requires pressure in kPa
DAB = dcalc(pkPa,T,index)       % DAB in units of m^2/sec
mu = mixmu(zs,T,index)          % mu in units of kg/m/sec

T = at(T);                                    % converting temperature to K

j = index(1);
k = index(2);

mixmw = zs(1)*mw(j) + zs(2)*mw(k);

R = 0.08206;
rho = p*mixmw/R/T                 % rho is in kg/m^3

sc = mu/rho/DAB;
____________________________________________________________________________________

So now let's see what this program can do!  Back to example 22.3-1:  Using a pressure of 1 atm, the
film temperature of 105 degrees F, and the mole fraction of water in the water/air mixture (given in the
problem as 0.0124), the Matlab program gives:

>> Sc = sccalc(1,105,[0.0124 1-0.0124])
DAB = 2.2951e-05  m^2/sec
mu =   1.8973e-05  kg/m/sec
rho =    1.1194  kg/m^3
Sc =   0.7385

From here, the Sherwood number, the mass transfer coefficient, and finally the rate of evaporation can be
found as demonstrated in the example problem in the text.

The book gives a Schmidt number of 0.58.  Notice that the density and viscosity values given by Matlab
both compare very favorably to those given in the text on page 683.  The difference is in the calculation of
DAB.  The text uses equation 17.2-1, based on kinetic theory and corresponding states, while sccalc
references dcalc, which uses the Chapman-Enskog equation.  Equation 17.2-1 is more accurate than Chapman-Enskog for polar compounds, such as water in this case, but the opposite is true of nonpolar compounds.  The Chapman-Enskog equation is also less limited in the variety of systems to which it can be applied.

Now consider a mixture of chlorine and air, two nonpolar substances.  Problem 17A.6 involves the calculation
of the Schmidt number for various mole fractions of chlorine in air.

Solution of BSL Problem 17A.6 (c): Calculation of
Schmidt Number for Chlorine in Air at 1 atm and 297 K

 Mole fraction of Cl2 0 0.25 0.5 0.75 1 Sc from sccalc 1.2828 0.8439 0.61 0.4684 0.3752 Sc from BSL 1.27 0.832 0.602 0.463 0.372
The values in this table indicate that the results from our program compare very well to those given by
Bird, Stewart, and Lightfoot for this nonpolar system.

It might also be informative to compare the results of our program to the experimental values provided in Table 17.1-1 of BSL:

Experimental and Sccalc Limiting Schmidt Numbers at 1 atm

 Sc from text Sc from Matlab Sc from text Sc from Matlab Gas Pair A-B Temp (K) xA->1 xA->1 xB->1 xB->1 H2-CH4 298.2 1.95 1.5207 0.23 0.2309 H2-N2 273.2 1.40 1.4426 0.19 0.1996 H2O-N2 308 0.58 0.6840 0.62 0.7398

Note that most of the sccalc values compare quite well to those from the textbook, except for the water-nitrogen pair.  The difference again may be attributed to the fact that water is polar.

Matlab Program to Calculate Schmidt Number of Dilute, Binary Liquid Mixtures
Well, we had created our sccalc program for calculating Sc for low density gas mixtures.  Still, something
seemed missing.  What could it be?  Then it hit us: liquids!  Liquids have Schmidt numbers, too!  How
might we determine those?

In an effort to do so, we assembled our team of expert students.  Rebecca Sisson and Eric Lee provided the
liquid viscosity component, while Garrick Malone and Stanley Chan held up their end of the bargain with a
liquid diffusivity algorithm.  In the end, our collaborative efforts led to a brand new program, fondly known
as scliqcalc, which may be used to determine the Schmidt number for dilute, binary liquid mixtures.  This
program will only work for the following compounds:

H2O    water
C2H5C2H5O    diethyl ether
C6H6    benzene
Br2    bromine
C2H5OH    ethanol
H2SO4    sulfuric acid
CH2CH2    acetylene
HCl    hydrogen chloride
NH3    ammonia
CHCl3    chloroform
CH3COOH    acetic acid
CH2OHCH2OH    ethylene glycol
C3H6    propylene
CH3COCH3    acetone
C3H8    propane
C6H5CH3    toluene
C6H5OH    phenol
CCl4    carbon tetrachloride

The secret recipe is given below:

________________________________________________________________________________

function scliq = scliqcalc(T,zs,solvent,index)

% function scliq = scliqcalc(T,zs,solvent,index)
%
% Calculates the Schmidt number for dilute binary mixtures of
% liquids, using the following:
%
% liqmucalc --  calculates the viscosity of the solvent (the compound
%               with higher mole fraction) as an approximation to
%               the viscosity of the mixture.  It is assumed that the dilute
%               solute contributes negligibly to the viscosity of
%               the mixture.
% ldiff code -- a modified version of the ldiff program created by
%               Garrick Malone and Stanley Chan is included in our code.
%               This program uses the empirical Wilke-Chang equation, eq. 17.4-8 to
%               calculate the diffusivity for small concentrations
%               of solute A in solvent B.
%
%  The density used in calculating the Schmidt number is the density of the
%  solvent at or near room temperature (it is assumed that the
%  dilute solute has negligible effect on density of the mixture).
%
% Argument List:
%
%   T      [=] temperature in units of Tdeg, as specified using start301
%   zs     [=] vector of mole fractions for the two compounds
%   solvent[=] association parameter of solvent liquid species as follows
%              1 - 'water'
%              2- 'methanol'
%              3- 'ethanol'
%              4- 'propanol'
%              5- 'other'  (other unassociated solvents)
%   index  [=] indices of the compounds from start301.  If no
%              index vector is given, compounds 1 and 2 will be
%              used.
%
% Returns:
%   sc     [=] Schmidt number (unitless)
%
% Note: If the program returns NaN for the Schmidt number, this
% means that the start301 database does not include the necessary
% data for the compounds you chose.
%
%                  Created April 2002 by Danielle Dunn and Jessica Dunn

global mw lenjones Tdeg lrho

if nargin==3
index = [1 2];
end

if zs(1)>zs(2)
k = index(1);               % k is the index of the solvent
j = index(2);                % j is the index of the solute
else
k = index(2);
j = index(1);
end

A= input('If you want to try to use the CENG301 data bank, type 1, otherwise type 2: ');

if A==1
VA=1/(lrho(j,2)/1000/mw(j));    % VA = molar volume of solute
else
VA=input('What is the molar volume of the solute in cm^3/g-mol: ');
end;

mu=liqmucalc(T,k);             % mu is in units of cp
T=at(T);
molwt=mw(k);
if solvent==1
psiB=2.26;
elseif solvent==2
psiB=1.9;
elseif solvent==3
psiB=1.5;
elseif solvent==4
psiB=1.2;
elseif solvent==5
psiB=1.0;
end;

DAB=(7.4E-8)*sqrt(psiB*molwt)*T/mu/(VA^.6)/(100^2)  % m^2/sec     Wilke-Chang equation

mu = mu*10^(-3)                % mu is now in kg/m/sec

rho = lrho(k)                  % kg/m^3

scliq = mu/rho/DAB;

______________________________________________________________________________

In order to test the accuracy of our scliqcalc program, we located data for viscosity, density, and diffusivity
for some common liquid pairs, using the CRC handbook*.  From this data, we calculated the Schmidt numbers
and compared them to those given by scliqcalc.  The results are shown below:

From start301:

Here are your compounds' formulae and names:
No. Formula   Name
----------------------------------------
1 H2O       water
2 C2H5OH    ethanol
3 C6H5CH3   toluene
4 CH3COCH3  acetone
----------------------------------------
>> Sc = scliqcalc(25,[0.001 0.999],3,[1 2])
If you want to try to use the CENG301 data bank, type 1, otherwise type 2: 1

DAB = 3.0324e-09  m^2/sec
mu    = 0.0010  kg/m/sec
rho    =   789  kg/m^3
Sc      =  434.8772  for solute water, solvent ethanol

From CRC*, the density of ethanol at 20 deg C is 0.7893 g/cm^3, viscosity of ethanol at 25 deg C is 1.074 mPa*sec, and the DAB of infinitely dilute water-ethanol solution is 1.24*10^-5 cm^2/sec.  This gives a Schmidt number of 1097.

>> Sc = scliqcalc(25,[0.001 0.999],5,[1 3])
If you want to try to use the CENG301 data bank, type 1, otherwise type 2: 1

DAB =  6.6204e-09   m^2/sec
mu    =   5.5031e-04   kg/m/sec
rho    =   866   kg/m^3
Sc     =   95.9847  for solute water, solvent toluene

From CRC, the density of toluene at 20 deg C is 0.8669 g/cm^3 and the viscosity at 25 deg C is 0.560 mPa*sec.  The DAB of an infinitely dilute water-toluene solution at 25 deg C is 6.19*10^-5 cm^2/sec.  This gives a Schmidt number of 104.4.

>> Sc = scliqcalc(25,[0.001 0.999],5,[1 4])
If you want to try to use the CENG301 data bank, type 1, otherwise type 2: 1

DAB =   9.5753e-09  m^2/sec
mu    =   3.0215e-04  kg/m/sec
rho    =   791   kg/m^3
Sc      =   39.8923  for solute water, solvent acetone

From CRC, the density of acetone at 20 deg C is 0.7899 g/cm^3 and the viscosity at 25 deg C is 0.306 mPa*sec.  The DAB of an infinitely dilute water-acetone solution at 25 deg C is 4.56*10^-5 cm^2/sec, which gives a Schmidt number of 84.95.

Notice that some results are more accurate than others.  This could be because of the psiB values used.  In the
case of the water-toluene system, the psiB value used is 1.0.  The textbook (BSL) says that this value is good for benzene, and toluene is a very similar compound chemically.  However, the value of 1.0 is also used for other compounds for which more accurate values are not available, as was the case with acetone.  This could explain why the results for acetone are not as good.  We should also note that most of the DAB values given by CRC "cannot be relied upon to better than 10%."  Our viscosity and density values appear to coincide very well with those given by the CRC handbook.

* Source of data for comparison: CRC Handbook of Chemistry and Physics, 75th edition. Editor-in-Chief David R. Lide.  Boca Raton: CRC Press, 1994.