Question

What is the physical significance of the Sherwood number? How is it defined? To what dimensionless number does it correspond in heat transfer? What does a Sherwood number of 1 indicate for a plane fluid layer?

Short answer

Answer: The Sherwood number (Sh) is a dimensionless quantity representing the ratio of the total mass transfer rate to the mass transfer rate by purely molecular diffusion, allowing us to determine the effectiveness of mass transfer mechanisms in fluid flow situations. It is defined as \( Sh = \frac{(k_c L)}{D} \), where \(k_c\) is the mass transfer coefficient, \(L\) is the characteristic length, and \(D\) is the molecular diffusivity. The counterpart in heat transfer is the Nusselt number (Nu), which represents the ratio of convective heat transfer to conductive heat transfer. A Sherwood number of 1 in a plane fluid layer indicates that mass transfer occurs predominantly due to molecular diffusion, rather than convection, typically when fluid velocities are very low or the fluid is stagnant.

Step-by-step solution

Physical Significance of Sherwood Number

The Sherwood number (Sh) is a dimensionless quantity used in mass transfer operations, like diffusion and convection in fluid flow. It represents the ratio of the total mass transfer rate in a system to the mass transfer rate by purely molecular diffusion. In other words, it compares the actual mass transfer occurring at an interface with that of pure diffusion, allowing us to determine the effectiveness of mass transfer mechanisms in various situations. #Step 2: Definition of Sherwood number#

Definition of Sherwood Number

The Sherwood number is defined as the ratio of convective mass transfer to diffusive mass transfer. Mathematically, it can be written as: \[ Sh = \frac{(k_c L)}{D} \] where \(Sh\) is the Sherwood number, \(k_c\) is the mass transfer coefficient, \(L\) is the characteristic length, and \(D\) is the molecular diffusivity. #Step 3: Counterpart in heat transfer#

Counterpart in Heat Transfer

In heat transfer, the Nusselt number (Nu) is the dimensionless number analogous to the Sherwood number. The Nusselt number represents the ratio of convective heat transfer to conductive heat transfer and is defined as: \[ Nu = \frac{(h L)}{k} \] where \(Nu\) is the Nusselt number, \(h\) is the heat transfer coefficient, \(L\) is the characteristic length, and \(k\) is the thermal conductivity of the fluid. #Step 4: Sherwood number of 1 in a plane fluid layer#

Sherwood Number of 1 in a Plane Fluid Layer

A Sherwood number of 1 indicates that the convective mass transfer is negligible compared to molecular diffusion. When the Sherwood number is 1 in a plane fluid layer, it means that mass transfer occurs predominantly due to molecular diffusion, rather than due to convection. This situation typically occurs when fluid velocities are very low or the fluid is stagnant.

Key concepts

Dimensionless Quantities in Mass Transfer

Dimensionless quantities play a crucial role in the study of mass transfer processes by providing a way to analyze systems without the direct need for units, thus making comparisons and scaling much easier. Let's delve into this concept with an example that's often encountered in textbooks: the Sherwood number (Sh).

Sherwood number is a dimensionless quantity that illustrates the relative significance of convective mass transfer over molecular diffusion. It is the mass transfer analog to other dimensionless numbers like the Reynolds number for flow regimes or the Nusselt number for heat transfer. These dimensionless parameters allow engineers and scientists to predict mass transfer rates in different systems and under various conditions. By determining the Sherwood number, students can assess how effective a mass transfer process is, whether in a laboratory setting or an industrial application.

Mass Transfer Coefficient

The mass transfer coefficient, denoted by the symbol \(k_c\), is a measure indicating the rate at which mass is transferred per unit area of a phase boundary. This parameter is discovered when diving into both natural and forced convection scenarios within mass transfer operations.

Imagine a scenario where a solute moves between two phases, such as from a gas into a liquid. The mass transfer coefficient quantifies how readily the solute crosses the interface. A higher coefficient means faster transfer and is associated with more turbulent or active systems like stirred tanks or packed columns. This coefficient is a pivotal part of the Sherwood number's definition, as seen in the textbook exercise, and is essentially the convective mass transfer term in the ratio.

Molecular Diffusivity

Molecular diffusivity, symbolized by \(D\), is another key term in our mass transfer exploration. It quantifies the rate at which molecules disperse or spread out by purely molecular effects, often driven by a concentration gradient. For students, it's important to comprehend that molecular diffusivity is akin to the 'spreading' rate of a drop of ink in water.

This property is inherent to specific substances and is affected by factors like temperature and the medium they're in. In the context of the Sherwood number, it represents the rate of mass transfer if it was only due to diffusion, i.e., without any external forces or convection involved.

Nusselt Number

The Nusselt number (Nu) is the thermal counterpart to the Sherwood number and offers a way to quantify the convective heat transfer relative to conductive heat transfer within a fluid. It's a dimensionless number that includes the heat transfer coefficient \(h\) and the thermal conductivity \(k\) of the fluid.

In the equation \[ Nu = \frac{(h L)}{k} \], \(L\) is the characteristic length, which can vary depending on the physical situation described. For instance, it could represent the diameter of a pipe in a flow scenario. A greater Nusselt number implies a stronger influence of convective heat transfer, highlighting an analogy between thermal and mass transfer phenomena.

Convective Mass Transfer

Convective mass transfer involves the movement of a solute between phases, driven not solely by molecular diffusion but also by the bulk motion of the fluid. This mechanism plays a significant role in many industrial and environmental processes, like the absorption of pollutants in the air or the extraction of valuable components in a chemical reactor.

When studying convective mass transfer, it is vital to understand that it can either enhance or diminish the overall mass transfer rate depending on the system's dynamics. This type of mass transfer is captured in the numerator of the Sherwood number equation alongside the mass transfer coefficient, emphasizing its contribution relative to diffusion. For a plane fluid layer, a Sherwood number of 1, as explained in the solution, suggests that molecular diffusion dominates over convective effects.