A Device to Emulate Diffusion, Thermal Conductivity, and Momentum Transport Using Water Flow

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: blanckh@apsu.edu

• Prandtl-Glauert Condensation around an F-18