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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. - PowerPoint PPT Presentation
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A Device to Emulate Diffusion, Thermal Conductivity, and
Momentum Transport Using Water Flow
Harvey BlanckJCE October 2005
Department of Chemistry
Austin Peay State University
Clarksville, Tennessee
Steady StatePlanar Diffusion
Fick’s First Law of Diffusion
)/( dxdcDJ
Fick’s Second Law of Diffusion
)/(/ 22 xcDtc
Plane or Point Source
Plane or Point Source Solution to
Fick’s Second Law
)4/exp(
22
2/10 DtxDt
xcc
Gaussian Curves for Dt = 0.1, 0.3, and 1.0
Free Plane or Point Source Diffusion
Half Gaussian Curve
)4/exp(
22
2/10 DtxDt
xcc
)4/exp( 2
2/10 Dtx
Dt
xcc
Free Step Boundary Planar or Point Source
Diffusion
Full 1-erf Curve
Step-Boundary
Step-Function Solution to
Fick’s Second Law
])exp(2
1[2
2/1)(2/
0
22/1
0 Dtx
dyyc
c
Step-Function Solution to
Fick’s Second Law
)](1[20 zerf
cc
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
)](1[0 zerfcc
)](1[20 zerf
cc
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
)](1[0 zerfcc
)](1[20 zerf
cc
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
dz
dvJ x
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
dz
dvJ x
Summary: This device rapidly emulates diffusion and thermal conductivity. It emulates all the diffusion coefficient determination methods found in JCE.
Using Spreadsheet TransferEquations to Emulate Diffusion
and Thermal Conductivity
Harvey BlanckJCE 2009
Department of Chemistry
Austin Peay State University
Clarksville, Tennessee
Fick’s First Law
)/( dxdcDJ
Fick’s Second Law
)/(/ 22 xcDtc
Steady StatePlanar 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 TransferEquations to Emulate
Diffusion and Thermal Conductivity in Cylindrical and Spherical
Systems
Cylindrical Diffusion and Thermal Conductivity
Fick’s First Law--a second look--
)/( dxdcDJ
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 = 2rh for a cylinder
and A = 4r2 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)*2r1h - (B4-C4)* 2r2h]*0.02/ [hr2
2 - hr12]
• 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)*4r12
- (B4-C4)* 4 r22]*0.2 /
[(4r23/3) - (4r1
3/3)]
• B5 = B4+[(A4-B4)*r12
- (B4-C4)*r22]*0.2 /
(r23 - r1
3)/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 Coolingwith 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: [email protected]
• Prandtl-Glauert Condensation around an F-18