# Engineering

BIOE2397 Exam #3 magnetic levitation

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ANY TYPE OF COLLABORATION ON THIS ASSIGNMENT IS EXPLICITLY PROHIBITED

(This assignment is due on Tuesday, 11/20, 11:59pm) ABSTRACT:

Researchers at Harvard have developed an analytical system that uses magnetic levitation to measure densities of solids and water-immiscible organic liquids with accuracies ranging from ± 0.0002 g/cm3 to ± 0.02 g/cm3, depending on the type of experiment. The technique is compatible with most solids and water-insoluble organic liquids with densities of 0.8 – 3 g/cm3 and is applicable to samples with volumes of 1 pL – 1 mL; the samples can be either spherical or irregular in shape. The method implements a simple and inexpensive device that comprises two permanent NdFeB magnets positioned with like poles facing one another, with the axis between the poles aligned with the gravitational field, and a container filled with paramagnetic medium (e.g., MnCl2 dissolved in water) placed between these magnets. Density measurements are obtained by placing the sample into the container, and measuring the position of the sample relative to the bottom magnet. The balance of magnetic and gravitational forces determines the vertical position of the sample within the device; knowing this position makes it possible to calculate the density of the sample. Figure: Theory of magnetic levitation. (a) A 2D numerical simulation (COMSOL Multiphysics) of the magnetic field, B

 ,

between two identical permanent magnets (5 × 5 × 2.5 cm) separated by h =4.5 cm and arranged in an anti-Helmholtz configuration (i.e., with like poles facing each other). Arrows point in the direction of the field, and the grayscale contour plot indicates the magnitude of the field; darker shades correspond to stronger field. (b) The magnitude of the Z component of the magnetic field vector, zB , along the centerline (dotted line in panel (a)); 0B is the magnitude of the magnetic field in the center of the top surface of the bottom magnet in the configuration shown in panel (a). (c) Schematic illustration of the experimental setup; the Y axis points into the plane of the image and is not shown.

BIOE2397 Exam #3 magnetic levitation

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DESCRIPTION: In one experiment, the authors measured, 0t (s), the time it took a spherical polystyrene particle

of radius R (m) to reach its levitation height above the surface of the bottom magnet 0z (m) from its

initial position at the bottom of the cuvette, 31 10iz −= × (m), filled with aqueous 470 mM MnCl2 at room

temperature. The levitation height, 0z , is the point between the two magnets where the force of gravity and the magnetic force acting on the object balance each other exactly (equation 1):

( ) 2 0 2

0( )4 2 s m o

s m

g h hz B

ρ ρ µ χ χ −

= + −

(1)

In equation (1), 70 4 10µ π −= × (N A-2) is the permeability of free space, 0 0.375B = (N A-1 m-1) is the

magnitude of the magnetic field at the surface of the bottom magnet, 345 10h −= × (m) is the distance between the magnets, 9.8g = (m s-2) is the acceleration due to gravity, 0sχ ≈ is the bulk magnetic

susceptibility of the sample particle, 1049sρ = (kg m-3) is the density of the particle, 56.4214 10mχ −= ×

is the magnetic susceptibility of paramagnetic medium, and mρ is the density of paramagnetic medium.

The empirical dependence of mρ (kg m-3) on the concentration of the paramagnetic salt, 0.47c = (M), and temperature, 23T = (°C), is given by equation (2):

3/2 2 3/2 3/2 3/2 2 0 1 2( , )m m T c W W T W T Ac BcT CcT Dc Ec T Fc Tρ ρ= = + + + + + + + + (2)

In equation (2), 0 999.65W = , 1

1 2.0438 10W −= × , 22 6.1744 10W

−= − × , 210022.1 ×=A , 110966.4 −×=B , 210307.1 −×−=C , 3.659D = − , 110631.1 −×−=E , and 310774.4 −×=F are

empirical constants (the units for these constants are omitted to reduce confusion).

The position of the particle between the magnets, z (m), can be described using equation (3):

dz z dt

α β= + (3)

In equation (3), 2 2 0 2

0

( )8 9

s m B R h

χ χα µ η −

= , 2

2 0

0

( )2 2( ) 9

s m s m

R g B h

χ χβ ρ ρ η µ

 − = − − 

  , R is the radius of the

spherical polystyrene particle, and 48.9 10η −= × (kg/m/s) is the dynamic viscosity of the suspending medium at room temperature. Ordinary differential equation (3) can be solved analytically to yield an expression for 0t (equation 4) – note that the only remaining unknown in equation (4) is the radius of the particle:

00 1 ln

i

zt z

α β α α β

 + =  + 

(4)

BIOE2397 Exam #3 magnetic levitation

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Just to make sure that all of the above actually makes sense, Harvard researchers performed an experiment with many particles of different size in which they measured the dependence of 0t (in minutes) on the size of the particles R (in microns):

R ( 610−× m) 325 200 100 45 22.5 10

0t (min) 0.092 0.384 1.311 6.139 15.123 71.167

ASSIGNMENT:

(1) [5 points] Create a user-defined m-file function frho_m that calculates mρ using equation (2) from the concentration of the paramagnetic salt, c (M), and temperature, T (°C), as the arguments.

(2) [5 points] Plot the dependence of 0t (min) and R (µm) obtained experimentally (data from the table) on log-log scale.

(3) Obtain the dependence of 0t (min) on R (µm) numerically, and plot the numerical solutions on the same figure (log-log scale).

a. [10 points] Re-write equation (3) as 1dt dz zα β

= +

and solve it using an ODE solver (note

that 0iz z z≤ ≤ , ( ) 0it z = and ( )0 0t z t= ) for every R

b. [10 points] Re-write equation (3) as dzdt zα β

= +

and solve it by integrating the left-hand

side from 0 to 0t and the right-hand side from iz to 0z numerically (that is, 0

0 1

i

z

z

t dz zα β

= +∫ )

for every R

(4) [5 points] On the same figure, plot the analytical solution given by equation (4). Make the numerical and the analytical solutions look distinct on the graph (yes, all three of them).

(5) [10 points] Fit the experimental data with a power-law function my bx= . In a new figure, plot the data and the fitted curve. Find 0t (min) for R = 87 (µm) (using the power-law fit).

(6) [10 points] Interpolate the experimental data with the ‘cubic’ method. In a new figure, plot the data and the interpolant. Find 0t (min) for R = 70 (µm) (using cubic interpolation).

BIOE2397 Exam #3 magnetic levitation

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IMPORTANT:

Please do not forget to label the axes, to give your graph a title, and to generate a legend for the graph. Set the limits so that the combined graph looks good. Choose the colors, markers, and spacing between points on the plots carefully so that all individual plots are visible and distinct.

EXAM #3 REPORT FORMAT:

• Create a script ‘Lastname_Firstname_ExamThree.m’ m-file:

% Lastname, Firstname % Exam #3 % Date clear all close all clc disp(‘Problem 1 start:’) <solve problem, produce output of the result> disp(‘Problem 1 end’) disp(‘———————————————————–‘) disp(‘Problem 2 start:’) <solve problem, produce output of the result> disp(‘Problem 2 end’) <…and so on for all problems>

• TEST-RUN YOUR CODE BEFORE SUBMITTING!!!

• Compress your files in one .zip file and e-mail your it as an attachment to ALL of us: