a device to emulate diffusion, thermal conductivity, and momentum transport using water flow
DESCRIPTION
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 PresentationTRANSCRIPT
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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
![Page 2: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/2.jpg)
Steady StatePlanar Diffusion
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Fick’s First Law of Diffusion
)/( dxdcDJ
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Fick’s Second Law of Diffusion
)/(/ 22 xcDtc
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Plane or Point Source
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Plane or Point Source Solution to
Fick’s Second Law
)4/exp(
22
2/10 DtxDt
xcc
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Gaussian Curves for Dt = 0.1, 0.3, and 1.0
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Free Plane or Point Source Diffusion
Half Gaussian Curve
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)4/exp(
22
2/10 DtxDt
xcc
)4/exp( 2
2/10 Dtx
Dt
xcc
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Free Step Boundary Planar or Point Source
Diffusion
Full 1-erf Curve
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Step-Boundary
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Step-Function Solution to
Fick’s Second Law
])exp(2
1[2
2/1)(2/
0
22/1
0 Dtx
dyyc
c
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Step-Function Solution to
Fick’s Second Law
)](1[20 zerf
cc
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Gaussian Curves for Dt = 0.1, 0.3, and 1.0
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Error Function Curves for Dt = 0.1, 0.3, and 1.0
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Constant Source Step Boundary
Planar or Point Source Diffusion
Lower Half 1-erf Curve
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Error Function Curves for Dt = 0.1, 0.3, and 1.0
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)](1[0 zerfcc
)](1[20 zerf
cc
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Constant Exit Step Boundary
Planar or Point Source Diffusion
Upper Half 1-erf Curve
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Error Function Curves for Dt = 0.1, 0.3, and 1.0
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)](1[0 zerfcc
)](1[20 zerf
cc
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Error Function Curves for Dt = 0.1, 0.3, and 1.0
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Confined Step Boundary
Planar or Point Source Diffusion
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Steady State Planar
Momentum Transport
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Newton’s Law of Viscosity
dz
dvJ x
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To show momentum transport in a liquid between two parallel plates do a 90 deg CCW rotation of figure and a horizontal flip.
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Momentum Transport in liquid between parallel plates. (Bottom
plate moving.)
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Momentum Transport in liquid between parallel plates. (Bottom
plate moving.)
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Steady State Momentum Transport in liquid between parallel plates. (Bottom
plate moving.)
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Newton’s Law of Viscosity
dz
dvJ x
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Summary: This device rapidly emulates diffusion and thermal conductivity. It emulates all the diffusion coefficient determination methods found in JCE.
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Using Spreadsheet TransferEquations to Emulate Diffusion
and Thermal Conductivity
Harvey BlanckJCE 2009
Department of Chemistry
Austin Peay State University
Clarksville, Tennessee
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Fick’s First Law
)/( dxdcDJ
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Fick’s Second Law
)/(/ 22 xcDtc
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Steady StatePlanar Diffusion
Spreadsheet Emulation
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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.
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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.
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Spreadsheet initial condition
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Spreadsheet row 50
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Spreadsheet row 100
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Spreadsheet row 150
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Spreadsheet row 200
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Spreadsheet row 250
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Spreadsheet row 300
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Spreadsheet row 350
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Spreadsheet row 400
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Spreadsheet row 450
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Spreadsheet row 500
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Spreadsheet row 550
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Spreadsheet row 600
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Spreadsheet row 650
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Spreadsheet row 700
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Spreadsheet row 750
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Spreadsheet row 800
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Spreadsheet row 850
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Spreadsheet row 2500
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Spreadsheet initial condition
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Spreadsheet row 50
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Spreadsheet row 100
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Spreadsheet row 150
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Spreadsheet row 200
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Spreadsheet row 250
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Spreadsheet row 300
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Spreadsheet row 350
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Spreadsheet row 400
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Spreadsheet row 450
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Spreadsheet row 500
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Spreadsheet row 550
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Spreadsheet row 600
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Spreadsheet row 650
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Spreadsheet row 700
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Spreadsheet row 750
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Spreadsheet row 800
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Spreadsheet row 850
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Spreadsheet row 2500
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Linear Curve Fit to Spreadsheet Data
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Free Plane or Point Source Diffusion
Spreadsheet Emulation
Gaussian Curve
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Plane or Point Source
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Gaussian Curves for Dt = 0.1, 0.3, and 1.0
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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
![Page 123: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/123.jpg)
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
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Spreadsheet initial condition
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Spreadsheet row 20
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Spreadsheet row 50
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Full Gaussian
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Step-Boundary
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Error Function Curves for Dt = 0.1, 0.3, and 1.0
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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
![Page 139: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/139.jpg)
Spreadsheet for Full Step-Boundary Diffusion: initial
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Spreadsheet for Full Step-Boundary Diffusion: row 30
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Spreadsheet for Full Step-Boundary Diffusion: row 60
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Spreadsheet row 60 with 1- erf curve superimposed
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Spreadsheet row 60 with 1- erf curve superimposed
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Step boundary full 1-erf curve
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Confined Diffusion or Thermal
Conductivity
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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
![Page 148: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/148.jpg)
Copper Sulfate Diffusion
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Spreadsheet initial condition
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Spreadsheet row 10
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Spreadsheet row 100
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Spreadsheet row 200
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Spreadsheet row 300
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Spreadsheet row 400
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Spreadsheet row 500
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Spreadsheet row 600
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Spreadsheet row 700
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Spreadsheet row 800
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Spreadsheet row 900
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Spreadsheet row 2500
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Spreadsheet row 5000
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Thermal energy transport from a hot central section to uniform temperature throughout
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Spreadsheet initial condition
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Spreadsheet row 10
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Spreadsheet row 50
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Spreadsheet row 100
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Spreadsheet row 200
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Spreadsheet row 500
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Spreadsheet row 1000
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Spreadsheet row 1500
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Spreadsheet row 2500
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Spreadsheet row 5000
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Thermal energy transport from a hot central section to uniform temperature throughout
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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.
![Page 175: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/175.jpg)
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.
![Page 176: A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow](https://reader035.vdocuments.mx/reader035/viewer/2022070418/56815734550346895dc4d3a4/html5/thumbnails/176.jpg)
Using Spreadsheet TransferEquations to Emulate
Diffusion and Thermal Conductivity in Cylindrical and Spherical
Systems
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Cylindrical Diffusion and Thermal Conductivity
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Fick’s First Law--a second look--
)/( dxdcDJ
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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
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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)
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Temperature profile with four inches of insulation surrounding a one inch radius pipe containing a hot liquid.
(Each cell is 0.05 inches thick.)
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Theoretical Cylindrical Thermal Conductivity Temperature Distribution Equation
T = T2 + ΔT[ ln (r/r2) / ln (r1/r2) ]
T = A ln r + B
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Temperature profile for row 10,000
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Diffusion of a fixed amount originating as a cylinder
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Curve fit for row 80 and row 300
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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)
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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.)
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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
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Spherical Steady State Thermal Conductivity
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Diffusion of a fixed amount originating as a central sphere
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Temperature Profile for Sphere Coolingwith constant temperature surroundings
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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.
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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.
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Information: www.apsu.edu/blanckh
Web search for: “Harvey Blanck”
e-mail: [email protected]
• Prandtl-Glauert Condensation around an F-18
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