A Device to Emulate Diffusion,
Thermal Conductivity, and
Momentum Transport
Using Water Flow
Harvey Blanck
JCE October 2005
Department of Chemistry
Austin Peay State University
Clarksville, Tennessee
Steady State
Planar Diffusion
Fick’s First Law of Diffusion
J   D(dc / dx)
Fick’s Second Law of Diffusion
c / t  D( c / x )
2
2
Plane or Point Source
Plane or Point Source Solution
to
Fick’s Second Law
c0 x
2
c
exp(

x
/
4
Dt
)
1/ 2
2Dt 
Gaussian Curves for
Dt = 0.1, 0.3, and 1.0
Free Plane or Point Source
Diffusion
Half Gaussian Curve
c0 x
2
c
exp(

x
/
4
Dt
)
1/ 2
2Dt 
c0 x
2
c
exp(

x
/
4
Dt
)
1/ 2
Dt 
Free Step Boundary
Planar or Point Source
Diffusion
Full 1-erf Curve
Step-Boundary
Step-Function Solution
to
Fick’s Second Law
c0
2
c  [1  1/ 2 
2
 0
x / 2 ( Dt )1 / 2
exp(  y )dy ]
2
Step-Function Solution
to
Fick’s Second Law
c0
c  [1  erf ( z )]
2
Gaussian Curves for
Dt = 0.1, 0.3, and 1.0
Error Function Curves for
Dt = 0.1, 0.3, and 1.0
Constant Source
Step Boundary
Planar or Point Source
Diffusion
Lower Half 1-erf Curve
Error Function Curves for
Dt = 0.1, 0.3, and 1.0
c0
c  [1  erf ( z )]
2
c  c0 [1  erf ( z )]
Constant Exit
Step Boundary
Planar or Point Source
Diffusion
Upper Half 1-erf Curve
Error Function Curves for
Dt = 0.1, 0.3, and 1.0
c0
c  [1  erf ( z )]
2
c  c0 [1  erf ( z )]
Error Function Curves for
Dt = 0.1, 0.3, and 1.0
Confined
Step Boundary
Planar or Point Source
Diffusion
Steady State
Planar
Momentum Transport
Newton’s Law of Viscosity
dv x
J  
dz
To show momentum transport in a liquid
between two parallel plates do a 90 deg CCW
rotation of figure and a horizontal flip.
Momentum Transport
in liquid between parallel plates.
(Bottom plate moving.)
Momentum Transport
in liquid between parallel plates.
(Bottom plate moving.)
Steady State Momentum Transport
in liquid between parallel plates.
(Bottom plate moving.)
Newton’s Law of Viscosity
dv x
J  
dz
Summary: This device rapidly emulates diffusion and
thermal conductivity. It emulates all the diffusion
coefficient determination methods found in JCE.
Using Spreadsheet Transfer
Equations to Emulate Diffusion
and Thermal Conductivity
Harvey Blanck
JCE 2009
Department of Chemistry
Austin Peay State University
Clarksville, Tennessee
Fick’s First Law
J   D(dc / dx)
Fick’s Second Law
c / t  D( c / x )
2
2
Steady State
Planar Diffusion
Spreadsheet Emulation
Spreadsheet Formula for all
Diffusion Calculations
• B8 = B7+(A7-B7)*0.05-(B7-C7)*0.05
• explanation of the three terms:
• (1) Amount that was initially present in cell B.
• (2) Amount input from cell to the left (cell A)
calculated from the height (pressure) difference
times a flow proportionality constant.
• (3) Amount output to cell on the right (cell C)
calculated as in second term.
Spreadsheet For First Law
Steady State
• Boundary conditions:
(1) first cell always 100 (cell 1)
(2) exit always zero (‘cell’ 17)
• Transfer equation:
B8 = B7+(A7-B7)*0.05-(B7-C7)*0.05.
Spreadsheet initial condition
Spreadsheet row 50
Spreadsheet row 100
Spreadsheet row 150
Spreadsheet row 200
Spreadsheet row 250
Spreadsheet row 300
Spreadsheet row 350
Spreadsheet row 400
Spreadsheet row 450
Spreadsheet row 500
Spreadsheet row 550
Spreadsheet row 600
Spreadsheet row 650
Spreadsheet row 700
Spreadsheet row 750
Spreadsheet row 800
Spreadsheet row 850
Spreadsheet row 2500
Spreadsheet initial condition
Spreadsheet row 50
Spreadsheet row 100
Spreadsheet row 150
Spreadsheet row 200
Spreadsheet row 250
Spreadsheet row 300
Spreadsheet row 350
Spreadsheet row 400
Spreadsheet row 450
Spreadsheet row 500
Spreadsheet row 550
Spreadsheet row 600
Spreadsheet row 650
Spreadsheet row 700
Spreadsheet row 750
Spreadsheet row 800
Spreadsheet row 850
Spreadsheet row 2500
Linear Curve Fit to
Spreadsheet Data
Free Plane or Point Source
Diffusion
Spreadsheet Emulation
Gaussian Curve
Plane or Point Source
Gaussian Curves for
Dt = 0.1, 0.3, and 1.0
Spreadsheet for
Gaussian Diffusion
• Boundary conditions for model:
Set leftmost and rightmost cell to always read 0 which means
there will be no flow from these cells to adjacent cells.
Note: The center cell has no input but there is no need to
alter the transfer equation. The transfer equation will have
two negative values i.e. two outputs – one left and one right.
• Transfer equation for all other cells (e.g.):
B8 = B7+(A7-B7)*0.05-(B7-C7)*0.05
Spreadsheet for
Gaussian Diffusion
-2
-1
0
1
2
0.0000
100.0000
100.0000
100.0000
0.0000
6.0000
94.0000
100.0000
94.0000
6.0000
10.9200
89.0800
99.2800
89.0800
10.9200
14.9760
85.0024
98.0560
85.0024
14.9760
18.3373
81.5840
96.4896
81.5840
18.3373
Spreadsheet initial condition
Spreadsheet row 20
Spreadsheet row 50
Full Gaussian
Step-Boundary
Error Function Curves for
Dt = 0.1, 0.3, and 1.0
Spreadsheet for Full
Step Boundary Diffusion
• Boundary conditions for 16 cell model:
(1) first cell has no input so transfer equation
for it is (e.g.): A8=A7-(A7-B7)*0.05
(2) exit always zero (‘cell’ 17)
• Transfer equation for all other cells (e.g.):
B8 = B7+(A7-B7)*0.05-(B7-C7)*0.05
Spreadsheet for Full
Step-Boundary Diffusion: initial
Spreadsheet for Full
Step-Boundary Diffusion: row 30
Spreadsheet for Full
Step-Boundary Diffusion: row 60
Spreadsheet row 60 with
1- erf curve superimposed
Spreadsheet row 60 with
1- erf curve superimposed
Step boundary full 1-erf curve
Confined Diffusion
or Thermal
Conductivity
Spreadsheet for Confined Diffusion
or Thermal Conductivity
• Boundary conditions:
(1) first cell has no input so transfer equation
for it is (e.g.): A8 = A7-(A7-B7)*0.05
(2) last cell has no output so transfer equation
for it is (e.g.): Q8 =Q7+(P7-Q7)*0.05
• Transfer equation for all other cells is (e.g.):
B8 = B7+(A7-B7)*0.05-(B7-C7)*0.05
Copper Sulfate Diffusion
Spreadsheet initial condition
Spreadsheet row 10
Spreadsheet row 100
Spreadsheet row 200
Spreadsheet row 300
Spreadsheet row 400
Spreadsheet row 500
Spreadsheet row 600
Spreadsheet row 700
Spreadsheet row 800
Spreadsheet row 900
Spreadsheet row 2500
Spreadsheet row 5000
Thermal energy transport from a hot central
section to uniform temperature throughout
Spreadsheet initial condition
Spreadsheet row 10
Spreadsheet row 50
Spreadsheet row 100
Spreadsheet row 200
Spreadsheet row 500
Spreadsheet row 1000
Spreadsheet row 1500
Spreadsheet row 2500
Spreadsheet row 5000
Thermal energy transport from a hot central
section to uniform temperature throughout
MODEL
SUMMARY
•
Rapidly emulates planar diffusion and
thermal conductivity and is a useful
classroom demonstration device.
• It emulates all the diffusion coefficient
determination methods found in JCE.
SPREADSHEET
SUMMARY
• The transfer equations emulate model operation.
• Spreadsheet emulation lends itself well to use in
PowerPoint presentations concerning planar
diffusion and thermal conductivity behavior.
• Spreadsheet emulation is easily extended to more
cells.
Using Spreadsheet Transfer
Equations to Emulate
Diffusion and
Thermal Conductivity in
Cylindrical and Spherical
Systems
Cylindrical Diffusion and Thermal Conductivity
Fick’s First Law
--a second look--
J   D(dc / dx)
Fick’s and Fourier’s First Law
• J is the flux and has units of rate per area. It is only
constant for planar conditions.
• Although the flux is not constant for cylindrical and
spherical diffusion and thermal conductivity, the
rate is constant so the rate equations are:
rate = -DAdc/dr
and
rate = -kAdT/dr
where A = 2rh for a cylinder
and
A = 4r2 for a sphere
Cylindrical Diffusion and Thermal Conductivity
Spreadsheet Transfer Equations
• The change in concentration (or temperature) of a cell
depends on the amount in and out (which depends upon the
area of the cell wall) and the volume of the cell.
• B5 = B4+[(A4-B4)*2r1h - (B4-C4)* 2r2h]*0.02/
[hr22 - hr12]
• B5 = B4+((A4-B4)*A$3- (B4-C4)*B$3)*2*0.02/
(B$3^2-A$3^2)
Temperature profile with four inches of insulation
surrounding a one inch radius pipe containing a hot liquid.
(Each cell is 0.05 inches thick.)
Theoretical Cylindrical Thermal Conductivity
Temperature Distribution Equation
T = T2 + ΔT[ ln (r/r2) / ln (r1/r2) ]
T = A ln r + B
Temperature profile for row 10,000
Diffusion of a fixed amount originating as a cylinder
Curve fit for row 80 and row 300
Spherical diffusion and Thermal Conductivity
Spreadsheet Transfer Equations
• B5 = B4+[(A4-B4)*4r12 - (B4-C4)* 4 r22]*0.2 /
[(4r23/3) - (4r13/3)]
• B5 = B4+[(A4-B4)*r12 - (B4-C4)*r22]*0.2 /
(r23 - r13)/3
• B5 = B4+((A4-B4)*A$3^2-(B4-C4)*B$3^2)*3*0.2/
(B$3^3-A$3^3)
Spherical Thermal Conductivity
for a one inch radius center and four inches of insulation.
(Cell one is the outer edge of the central core which remains at
constant temperature.)
Theoretical Spherical Thermal Conductivity
Temperature Distribution Equation
T = T1 - ΔT[ (1- r1/r) / (1- r1/r2) ]
T = T1 - [ΔT / (1- r1/r2)][1- r1/r]
T = [ΔT r1/ (1- r1/r2)]/r + T1 - [ΔT / (1- r1/r2)]
T=A/ r +B
Spherical Steady State Thermal Conductivity
Diffusion of a fixed amount originating as a central sphere
Temperature Profile for Sphere Cooling
with constant temperature surroundings
Comments
• The plastic model emulates diffusion and
thermal conductivity in a variety of planar
systems.
• The spreadsheet transfer equation approach
for these planar systems produces the same
results as the plastic model.
• The spreadsheet transfer equation approach
appears to satisfactorily emulate transport
processes in cylindrical and spherical systems
to show the concentration and temperature
distribution changes with time.
References
• Blanck, H. F., J. Chem. Educ. 2005, 82, 1523
(October 2005)
(plastic model emulation)
• Blanck, H. F., J. Chem. Educ. 2009, 86, page ?
(May, June, or July 2009) (spreadsheet emulation)
• Incropera, F. ; DeWitt, D. Fundamentals of Heat and
Mass Transfer, 3rd ed.; John Wiley & Sons, 1990.
• Google “Harvey Blanck” to find my Web pages.
Information: www.apsu.edu/blanckh
Web search for:
“Harvey Blanck”
e-mail: blanckh@apsu.edu
• Prandtl-Glauert Condensation around an F-18