Introduction & Context
Forced convection mass transfer involving a spherical geometry is a fundamental concept in chemical and process engineering. This calculation determines the mass transfer coefficient, which quantifies the rate at which a chemical species moves between the surface of a sphere and the surrounding fluid stream. This analysis is critical for designing equipment such as packed-bed reactors, spray dryers, and absorption towers, where the interaction between dispersed phases and continuous fluid phases dictates overall process efficiency and reaction kinetics.
Methodology & Formulas
The calculation follows a dimensionless analysis approach to determine the mass transfer coefficient (hm). The process begins by characterizing the flow regime and the fluid properties using the Reynolds number (Re) and the Schmidt number (Sc).
The Reynolds number represents the ratio of inertial forces to viscous forces, defined as:
\[ Re = \frac{V \cdot D}{\nu} \]
The Schmidt number represents the ratio of momentum diffusivity to mass diffusivity, defined as:
\[ Sc = \frac{\nu}{D_{ab}} \]
The Sherwood number (Sh), which represents the ratio of convective mass transfer to diffusive mass transfer, is calculated using the empirical correlation for flow over a sphere:
\[ Sh = 2 + 0.6 \cdot Re^{0.5} \cdot Sc^{1/3} \]
Finally, the mass transfer coefficient (hm) is derived from the Sherwood number and the molecular diffusivity:
\[ h_m = \frac{Sh \cdot D_{ab}}{D} \]
The validity of this empirical correlation is restricted to specific flow and transport regimes. The following table outlines the operational limits for the dimensionless numbers:
| Parameter |
Lower Bound |
Upper Bound |
| Reynolds Number (Re) |
3.5 |
80,000 |
| Schmidt Number (Sc) |
0.6 |
400 |
The Sherwood number (Sh) represents the ratio of convective mass transfer to the rate of diffusive mass transport. In the context of a sphere, it is dimensionless and serves as a measure of the efficiency of the mass transfer process at the surface. A higher Sherwood number indicates that convection is dominating the transport process, effectively thinning the concentration boundary layer around the sphere.
Worked Example: Forced Convection Mass Transfer from a Spherical Catalyst Pellet
In a chemical reactor design scenario, a spherical catalyst pellet with a diameter of 0.005 m is suspended in a flowing gas stream. To determine the rate of mass transfer, we must calculate the mass transfer coefficient under the given operating conditions.
Knowns:
- Diameter of sphere (Dm): 0.005 m
- Velocity of gas (Vmps): 2.0 m/s
- Temperature (Tcelsius): 25.0 degrees Celsius
- Kinematic viscosity (νm2s): 1.56e-05 m²/s
- Binary diffusion coefficient (Dab_m2s): 2.5e-05 m²/s
Step-by-Step Calculation:
- Calculate the Reynolds number (Re) to characterize the flow regime:
\[ Re = \frac{V \cdot D}{\nu} = \frac{2.0 \cdot 0.005}{1.56 \times 10^{-5}} = 641.026 \]
- Calculate the Schmidt number (Sc) to relate momentum and mass diffusivity:
\[ Sc = \frac{\nu}{D_{ab}} = \frac{1.56 \times 10^{-5}}{2.5 \times 10^{-5}} = 0.624 \]
- Determine the Sherwood number (Sh) using the Ranz-Marshall correlation for a sphere:
\[ Sh = 2 + 0.6 \cdot Re^{0.5} \cdot Sc^{0.333} \]
\[ Sh = 2 + 0.6 \cdot (641.026)^{0.5} \cdot (0.624)^{0.333} = 14.981 \]
- Calculate the mass transfer coefficient (hm):
\[ h_m = \frac{Sh \cdot D_{ab}}{D} = \frac{14.981 \cdot 2.5 \times 10^{-5}}{0.005} = 0.075 \text{ m/s} \]
Final Answer: The mass transfer coefficient for the spherical catalyst pellet is 0.075 m/s.
