Problem 13 - Bioseparations Science and Engineering

Question

Estimation of Flow Rate for Centrifugation of Yeast Cells A maximum flow rate of 50 liters/min was achieved for the centrifugation of bacteria cells in a tubular centrifuge. The cells were \(2.0 \mu \mathrm{m}\) in diameter and had a density of \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\). The medium had a density of \(1.01 \mathrm{~g} / \mathrm{cm}^{3}\) and viscosity of \(1.2 \mathrm{cp}\). It is desired to centrifuge yeast cells in this same centrifuge. The yeast cells have a diameter of \(5.0 \mu \mathrm{m}\) and a density of \(1.10 \mathrm{~g} / \mathrm{cm}^{3}\). The medium has a density of \(1.02\) \(\mathrm{g} / \mathrm{cm}^{3}\) and a viscosity of \(1.3 \mathrm{cp}\). Estimate the maximum flow rate that can be used to centrifuge the yeast cells.

Step-by-Step Solution

Answer

The maximum flow rate for centrifugation of yeast cells is approximately 216.45 liters/minute.

Step 1: Use Stokes' Law

The settling velocity of the particles in a centrifuge can be defined using Stokes' Law as \(V = \frac{d^2 (p-p_0) gw}{18μ}\). This can be rearranged to a ratio form representing bacteria to yeast cell as \(\frac{V_b}{V_y} = \frac{d_b^2 (p_b - p_{0b})}{d_y^2 (p_y - p_{0y})}\). Here, \(V\) is the settling velocity, \(d\) is the diameter of the cell, \(p\) is the density of the cell, \(p_0\) is the density of the medium, \(g\) is the acceleration due to gravity, and \(w\) is the angular velocity.

Step 2: Substitute Given Values

Substitute the values of the given quantities into the formula from Step 1. This gives a ratio of \(\frac{V_b}{V_y} = \frac{(2 \times 10^{-4})^2 (1.08 - 1.01)}{(5 \times 10^{-4})^2 (1.10 - 1.02)} = 0.231.\)

Step 3: Find the Flow Rate for Yeast Cells

As the ratio of velocities is equivalent to the ratio of flow rates, the flow rate for yeast cells can be calculated by rearranging the formula \(V_y = \frac{V_b}{0.231}\). Substituting \(V_b = 50\) liters/min, we find \(V_y = 50/0.231 = 216.45\) liters/min, which is the desired maximum flow rate for yeast cells.

Key Concepts

Stokes' Law

In the centrifugation process, one fundamental principle used to calculate the settling velocity of particles is Stokes' Law. This law determines how fast a particle settles in a fluid under the influence of gravity or centrifugal force. The law is mathematically represented as:

\[ V = \frac{d^2 (p-p_0) g w}{18μ} \]

The variables in this equation signify the following: \( V \) is the settling velocity, \( d \) is the particle's diameter, \( p \) and \( p_0 \) are the densities of the particle and fluid respectively, \( g \) is the acceleration due to gravity, and \( w \) is the angular velocity. Viscosity of the medium is denoted by \( μ \).
The essence of Stokes' Law is that it illustrates how the settling velocity increases with the square of the particle's diameter and the density difference between the particle and fluid, whereas it is inversely proportional to the viscosity of the fluid. For example, larger or denser particles will settle faster in a less viscous fluid.

Cell Density in Centrifugation

When separating cells like bacteria or yeast by centrifugation, the density of the cells plays a critical role. The cell’s density, \( p \), compared to the density of the medium, \( p_0 \), determines the buoyant force experienced by the cells. A higher cell density means that the cell will experience a greater gravitational pull as opposed to buoyancy, leading to a higher settling velocity under centrifugation.

Considering the scenario with bacteria and yeast cells, the bacteria have a density of \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\), and the yeast cells have a slightly higher density of \(1.10 \mathrm{~g} / \mathrm{cm}^{3}\). Due to this minute difference in density, yeast cells will settle slightly faster than bacteria cells if all other conditions are equal. However, we must also consider other factors like cell size and the medium's properties to accurately estimate flow rates for centrifugation.

Settling Velocity

Settling velocity is the rate at which a particle moves through the medium under the influence of centrifugal force in a centrifuge. From Stokes' Law, we know settling velocity depends on several factors, including the size and density of the particles as well as the density and viscosity of the medium.

In the given example, the diameter of the yeast cells is \(5.0 \mu \mathrm{m}\) as opposed to the \(2.0 \mu \mathrm{m}\) of the bacteria, meaning that the yeast cells are larger and thus have a higher settling velocity. We use this information along with the cells' density to calculate the rate at which they'll sediment. Specifically, the larger diameter of the yeast cells, when squared in Stokes' equation, greatly increases the settling velocity compared to that of the smaller bacteria cells.

Viscosity Impact on Centrifugation

Viscosity, denoted as \( μ \), is a measure of a fluid’s resistance to flow. In centrifugation, the viscosity of the fluid medium can greatly affect the settling velocity of particles. According to Stokes' Law, an increase in the medium's viscosity will result in a decrease in settling velocity.

This is an important consideration when centrifuging different types of cells. In our exercise, the viscosity of the medium for yeast cells is slightly higher (\(1.3 \mathrm{cp}\)) compared to that of the bacteria (\(1.2 \mathrm{cp}\)). This increased viscosity means that, all else being equal, yeast cells will sedate at a slightly lower velocity due to the higher resistance posed by the medium. However, because the yeast cells are substantially larger, their increased settling velocity more than compensates for the higher viscosity, as reflected in the calculated flow rates.