Problem 3 - Bioseparations Science and Engineering

Question

Isopycnic Sedimentation You wish to capture \(3 \mu \mathrm{m}\) particles in a linear density gradient having a density of \(1.12 \mathrm{~g} / \mathrm{cm}^{3}\) at the bottom and \(1.00\) at the top. You layer a thin particle suspension on the top of the \(6 \mathrm{~cm}\) column of fluid with a viscosity of \(1.0 \mathrm{cp}\) and allow particles to settle at \(1 \mathrm{~g}\). (a) How long must you wait for the particles you want (density \(=1.07 \mathrm{~g} / \mathrm{cm}^{3}\) ) to sediment to within \(0.1 \mathrm{~cm}\) of their isopycnic level? Is it possible to determine the time required for particles to sediment to exactly their isopycnic level? (b) If instead of \(1 \mathrm{~g}\) you use a centrifuge running at \(800 \mathrm{rpm}\), and the top of the fluid is \(5 \mathrm{~cm}\) from the center of rotation, how long must you centrifuge for the particles to move to within \(0.1 \mathrm{~cm}\) of their isopycnic level?

Step-by-Step Solution

Answer

The exact time to reach the isopycnic level cannot be determined because theoretically, it will take an infinite amount of time for the particles to reach this point. However, in terms of approximations, and under normal gravity (1 g), it would take a certain amount of time for the particles to sediment to within 0.1 cm of their isopycnic level. Changes to this time are introduced with the application of a centrifuge.

Step 1: Derive time for particles to settle (Part a)

First, we need to calculate the time it would take for the particles to settle. To do that, we use Stoke's Law, which gives the terminal settling velocity of an individual spherical particle for a given fluid medium and gravitational force. The formula to use is \(v_s = \frac{2}{9} \cdot \frac{(density_{particle} - density_{fluid}) \cdot g \cdot r^2}{\mu}\), where \(density_{particle}\) = 1.07 g/cm³, \(density_{fluid}\) = 1.12 g/cm³, g = 1 g, r = 1.5 μm = 0.0015 cm, and µ = 1.0 cp = 1.0 g/(cm·s). The negative value of velocity means that the particles are settling down. The time to reach within 0.1 cm of the isopycnic position can then be approximated by dividing the travel distance of 0.1 cm by the absolute velocity.

Step 2: Discuss ability to determine time to reach isopycnic level

As the particle approaches its isopycnic level, the density gradient causes the settling velocity to approach zero, i.e., the particle slows down as it near its isopycnic point. As the velocity approaches zero, the time to reach the isopycnic level approaches infinity. Thus, it is not possible to determine an exact time for the particles to reach their isopycnic level because theoretically, they will take an infinite amount of time to settle exactly to this point.

Step 3: Derive time for particles to sediment in centrifuge (Part b)

In the scenario of a centrifuge, we need to calculate the time it would take for the particles to move to within 0.1 cm of their isopycnic level. In this case, the velocity can be determined by using the principle of centrifugation: \(v_s = (density_{particle} - density_{fluid}) \cdot r \cdot \omega^2 / \mu\), where \(density_{particle}\) = 1.07 g/cm³, \(density_{fluid}\) = 1.00 g/cm³, r = 5 cm, and ω = 800 rpm = 83.775 rad/s. µ = 1.0 cp = 1.0 g/(cm·s). The actual radius used for the calculation is the average radius, which is the distance from the center of rotation to the middle of the distance the particles travel. The time to reach within 0.1 cm of the isopycnic point can be approximated by dividing the travel distance of 0.1 cm by the absolute velocity.

Key Concepts

Density Gradient

In isopycnic sedimentation, a density gradient plays a crucial role. This gradient is a gradual change in density within a column of fluid, from a lower value at the top to a higher one at the bottom. In the provided example, the fluid column has a density gradient ranging from 1.00 to 1.12 g/cm³. This setup allows particles to reach a stage where their density matches the density of the surrounding fluid, known as their isopycnic level. An effective density gradient ensures that particles of different densities separate effectively based on where they find buoyancy equilibrium.

An important aspect of the density gradient is its influence on particle motion. As particles move through this gradient, they experience varying buoyant forces depending on their location in the gradient. Initially, denser particles are more affected and settle faster until they reach their corresponding isopycnic level. Meanwhile, less dense particles move slower, eventually finding their own equilibrium points. Utilizing a density gradient is essential in separating particles by density, making it integral in techniques like centrifugation, which benefits from the accelerated sedimentation in a rotating system.

• Different densities separate within the fluid column due to the gradient.
• Particles settle at layers where their density equals the fluid's density.
• Density gradient enhances separation efficiency and accuracy.

Stoke's Law

Stoke's Law provides us with a way to calculate the settling velocity of small spherical particles under the influence of gravity through a fluid. This was particularly useful in the original problem to determine how long particles take to reach their isopycnic level. The law states that the terminal velocity, denoted as \(v_s\), is given by the formula:\[ v_s = \frac{2}{9} \cdot \frac{(density_{particle} - density_{fluid}) \cdot g \cdot r^2}{\mu} \]
Where:

• \(density_{particle}\) and \(density_{fluid}\) are the densities of the particle and the fluid, respectively.
• \(g\) represents the gravitational acceleration.
• \(r\) is the radius of the particle.
• \(\mu\) is the dynamic viscosity of the fluid.

In the context of isopycnic sedimentation, as particles settle, their density difference with the fluid dictates their velocity. However, as they approach their isopycnic point, the density difference decreases, which in turn slows their settling velocity dramatically. Ultimately, according to Stoke's Law, the time to reach the exact isopycnic level is essentially infinite because the velocity becomes insignificantly small as the particle nears this point. Understanding Stoke's Law is vital for predicting sedimentation behavior in fluids by accounting for particle and fluid characteristics.

Centrifugation

Centrifugation is a powerful technique used to accelerate the sedimentation of particles using a rotating centrifugal force rather than relying solely on gravity. This force effectively increases the apparent gravitational field allowing particles to settle much faster than they would otherwise. In the exercise scenario, centrifugation was applied by spinning the sample at 800 rpm to achieve a faster separation of particles to their isopycnic level.

During centrifugation, the applied force can be expressed with the formula for velocity in centrifugation:\[ v_s = (density_{particle} - density_{fluid}) \cdot r \cdot \omega^2 / \mu \]
Where:

• \(density_{particle}\) and \(density_{fluid}\) are the particle and fluid densities.
• \(r\) denotes the radial distance from the center of rotation.
• \(\omega\) is the rotational speed in radians per second.
• \(\mu\) is the fluid's viscosity.

By manipulating these parameters, particles can be made to reach their isopycnic level much more quickly. In practical applications, centrifugation allows precise adjustments through speed and time settings, aiding in the efficient separation of particles based on density gradients. Despite its utility, reaching the exact isopycnic layer remains challenging because as the difference in density equilibrates, the induced velocity decreases, complicating the precise separation.