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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 83 "Section 17.5: Diffusion i
nto a Falling Liquid Film: Forced Convection Mass Transfer" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "Az:=W*dx; Ax:=W*dz; " }{TEXT -1 65 "Az and Ax are c
ross-sectional areas.  W is the width of the film." }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#AzG*&%\"WG\"\"\"%#dxGF'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#AxG*&%\"WG\"\"\"%#dzGF'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 58 "eq:=(NAz(x,z)-NAz(x,z+dz))*Az + (NAx(x,z)-NAx(x+dx,
z))*Ax;" }{TEXT -1 140 "This is the mass balance on component A and it
 is equal to zero.   cA changes both with x and with z. This equation \+
should be equal to zero." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,&*(
,&-%$NAzG6$%\"xG%\"zG\"\"\"-F)6$F+,&F,F-%#dzGF-!\"\"F-%\"WGF-%#dxGF-F-
*(,&-%$NAxGF*F--F86$,&F+F-F4F-F,F2F-F3\"\"\"F1F-F-" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 30 "eq2:=limit(eq/(W*dx*dz),dx=0);" }{TEXT -1 
39 " The limit is taken as dx goes to zero." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G,$*&,(-%$NAzG6$%\"xG%\"zG!\"\"-F)6$F+,&F,\"\"\"%
#dzGF1F1*&F2F1--&%\"DG6#F16#%$NAxGF*F1F1\"\"\"F2!\"\"F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pde:=-limit(eq2,dz=0)=0; " }{TEXT 
-1 127 "The limit of the previous equation is taken as dz goes to zero
.  The resulting equation is equal to zero and matches Eqn 17.5-3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG/,&--&%\"DG6#\"\"#6#%$NAzG6$%\"
xG%\"zG\"\"\"--&F*6#F26#%$NAxGF/F2\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "NAz:=(x,z)->vz(x)*cA(x,z);" }{TEXT -1 429 "A moves in
 the z direction due to the flow of the film.  In the z direction, the
 convective term dominates and the diffusive term is negligible.  Thus
 Eqn 17.0-1 reduces to xA(NAz+NBz), but this can be rewritten as cA*vz
 as long as vz is taken to be the molar average velocity.  See Table 1
6.1-3.  the assumption that the molar average velocity and the mass av
erage velocity are identical is another approximation in the problem.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$NAzGR6$%\"xG%\"zG6\"6$%)operato
rG%&arrowGF)*&-%#vzG6#9$\"\"\"-%#cAG6$F19%F2F)F)F)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "NAx:=(x,z)->-DAB*D[1](cA)(x,z);" }{TEXT -1 
143 "A moves in the x-direction due to diffusion and the convective te
rm is negligible.  Thus Eqn 17.0-1 reduces to -DAB dcA/dx for the x di
rection." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$NAxGR6$%\"xG%\"zG6\"6$%
)operatorG%&arrowGF),$*&%$DABG\"\"\"--&%\"DG6#F06#%#cAG6$9$9%F0!\"\"F)
F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "pde;" }{TEXT -1 22 "
This equals eq. 17.5-6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%#vzG6
#%\"xG\"\"\"--&%\"DG6#\"\"#6#%#cAG6$F)%\"zGF*F**&%$DABGF*--&F.6$F*F*F1
F3F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Pdsolve does n
ot solve the pde as we would like.  Therefore we will enter the books \+
solution and see if we can work backwards to make it mathc our pde." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "cA:=(x,z)->cA0*(1-erf(x/sq
rt(4*DAB*z/vmax)));" }{TEXT -1 37 "The book's solution form Eqn 17.5-1
5." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cAGR6$%\"xG%\"zG6\"6$%)operat
orG%&arrowGF)*&%$cA0G\"\"\",&F/F/-%$erfG6#*&9$\"\"\"-%%sqrtG6#,$*&*&%$
DABGF/9%F/F6%%vmaxG!\"\"\"\"%F@!\"\"F/F)F)F)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "vz:=x->vmax;" }{TEXT -1 171 "Substance A only pe
netrates a short distance into the film and thus for most parts, subst
ance A regards the velocity of the film to equal vmax.  ????????other \+
part 17.5-7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vzGR6#%\"xG6\"6$%)o
peratorG%&arrowGF(%%vmaxGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "D[2](NAz)(x,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--&%\"DG6#
\"\"#6#%$NAzG6$%\"xG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "diff(NAz(x,z),z);" }{TEXT -1 46 " Take the derivative of NAz with \+
respect to z." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**%$cA0G\"\"\"-%$
expG6#,$*&*&)%\"xG\"\"#\"\"\"%%vmaxGF'F1*&%$DABG\"\"\"%\"zG\"\"\"!\"\"
#!\"\"\"\"%F'F/F'F4F1F1*&-%%sqrtG6#%#PiGF1)*&*&F4F'F6F'F1F2F8#\"\"$F0F
1F8#F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(pde);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&--&%\"DG6#\"\"#6#%$NAzG6$%\"xG%
\"zG\"\"\"--&F(6#F06#%$NAxGF-F0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "simplify(diff(NAz(x,z),z)+diff(NAx(x,z),x));" }{TEXT 
-1 73 "The sum of these derivatives equals zero.  Thus Eqn 17.5-3 is s
atisfied.." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "limit(cA(x,z),z=0,right);" }{TEXT -1 113 
"BC 1 eq. 17.5-12  According to the figure, the concentration of A is \+
zero when z approaches zero from the right.." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%&limitG6%*&%$cA0G\"\"\",&F(F(-%$erfG6#,$*&%\"xG\"\"\"
*$-%%sqrtG6#*&*&%$DABGF(%\"zGF(F0%%vmaxG!\"\"F0F:#F(\"\"#!\"\"F(/F8\"
\"!%&rightG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "assume(DAB>0
,vmax>0,x>0,z>0);" }{TEXT -1 70 "Assuming these variables to be positi
ve helps to simplify the problem." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "limit(cA(x,z),z=0,right);" }{TEXT -1 72 "The limit is
 taken again after the variables are assumed to be positive." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "cA(0,z);" }{TEXT -1 63 "BC 2 eq. 17.5-13.  At x=0,  cA is at it
s initail concentration." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$cA0G" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(cA(x,z),x=infinity);
" }{TEXT -1 64 "BC 3 eq. 17.5-4.  At x= infinity, the concentration of
 A is zero" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 41 "This clears all th
e previous assumptions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
cA:=(x,z)->cA0*(1-erf(x/sqrt(4*DAB*z/vmax)));" }{TEXT -1 11 "Eqn 17.5-
15" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cAGR6$%\"xG%\"zG6\"6$%)operat
orG%&arrowGF)*&%$cA0G\"\"\",&F/F/-%$erfG6#*&9$\"\"\"-%%sqrtG6#,$*&*&%$
DABGF/9%F/F6%%vmaxG!\"\"\"\"%F@!\"\"F/F)F)F)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "NAx:=(x,z)->-DAB*D[1](cA)(x,z);" }{TEXT -1 92 "N
Ax is redefined, using the assumption that the convective term in Eqn \+
17.0-1 is negligible." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$NAxGR6$%\"
xG%\"zG6\"6$%)operatorG%&arrowGF),$*&%$DABG\"\"\"--&%\"DG6#F06#%#cAG6$
9$9%F0!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "NAx(0,z
);" }{TEXT -1 65 "The local mass flux at x=0 is found which resembles \+
Eqn. 17.5-16." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%$DABG\"\"\"%$cA0
G\"\"\"F&*&-%%sqrtG6#%#PiGF&-F+6#*&*&F%F(%\"zGF(F&%%vmaxG!\"\"F&F4" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "WA:=W*int(NAx(0,z),z=0...L)
;" }{TEXT -1 119 "The total moles of A transferred per unit time from \+
the gas to the liquid film is found.  This is based on Eqn 17.5-17." }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WAG,$*&**%\"WG\"\"\"-%%sqrtG6#*&*&
%\"LGF)%$DABGF)\"\"\"%%vmaxG!\"\"F1%$cA0GF)F2F)F1*$-F+6#%#PiGF1F3\"\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(%);" }{TEXT -1 
23 "Compare to Eqn. 17.5-17" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**%
\"WG\"\"\"%$cA0GF'%%vmaxGF'-%%sqrtG6#*&*(%#PiGF'%\"LGF'%$DABGF'\"\"\"F
)!\"\"F2F2F/F3\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simp
lify(%-W*L*cA0*sqrt(4*DAB*vmax/Pi/L),assume=positive);" }{TEXT -1 95 "
Subtract Eqn 17.5-17 and simplify to show that the previous equation a
nd Eqn 17.5-17 are equal." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
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