Dimensionless Numbers #1

Dimensionless Groups in Chemical Engineering

Dimensionless Groups or numbers are relationships with no units of measurement and are often used in chemical engineering. 

There are numerous dimensionless numbers used by chemical engineers and this blog entry discusses the more common ones. The equations below are all in metric units, however, if you use consistent units, the dimensionless numbers remain unchanged.

Reynolds Number (Re)

This is arguably the most commonly used dimensionless group in chemical engineering.  It gives a measure of the ratio of inertial and viscous forces in fluid flow and is often used to determine if the flow is either laminar or turbulent:

• In laminar flow, viscous forces dominate. The flow paths are smooth, streamline and constant.
• In turbulent flow, inertial forces dominate. The flow regime is unstable, generating eddies and vortices.

The Reynolds Number can be calculated using the following equation:

Where:

• ρ = Fluid density (kg/m³)
• u = Fluid velocity (m/s)
• L = Characteristic dimension (m)
• μ = Dynamic viscosity (Pa.s)

When calculating Reynolds Number, the units used are not important but MUST be consistent. For pipes or channels with circular cross-section, the characteristic dimension can be taken as the pipe diameter.

For flow through pipes, Reynolds Number below 2000 indicates laminar flow while a Reynolds Number above 4000 indicates turbulent flow. Reynold Numbers between 2000 and 4000 indicate transitional flow i.e., rapidly changing between laminar and turbulent flow.

Prandtl Number (Pr)

This is the ratio of kinematic viscosity to the thermal diffusivity. Prandtl Number can also be defined as the ratio of momentum and thermal diffusivities. It tells us how fast the thermal diffusion occurs as compared to momentum diffusion in fluids. It is used in many calculations involving heat transfer in flowing fluids, as it gives a measure of the relative thickness of the thermal and momentum boundary layers.  It can be calculated using the following equation:

Where:

• C_P = Fluid Specific Heat Capacity (J/kg.K)
• μ = Dynamic viscosity (Pa/s)
• k = Thermal conductivity (W/m.K)
• ν = Momentum or Kinematic diffusivity (m²/s)
• α = Thermal diffusivity (m²/s)

It should be noted that the Prandtl number is dependent on the fluid’s physical properties alone and hence it is often found in physical properties.  

For many gases (with the notable exception of hydrogen), the Prandtl number has a value of 0.6 to 0.8 over a wide range of conditions. If the Pr << 1, then thermal diffusivity dominates and when Pr >> 1, then momentum diffusivity dominates

Nusselt Number (Nu)

This is the ratio of convective to conductive heat transfer in a fluid over a given length, L:

Where:

• h = Heat Transfer Coefficient (W/m².K)
• L = Characteristic length (m)
• k = Thermal conductivity (W/m.K)

For heat transfer in pipes, the characteristic length is the pipe diameter.

When Nu ≈ 1 then convection and conduction are about equal.  Typically, this occurs in laminar conditions.  As the Nusselt number becomes larger, convective heat transfer becomes relatively more important – this occurs as the flow becomes more turbulent.  

The mass transfer equivalent of the Nusselt number is the Sherwood Number discussed next.

Sherwood Number (Sh)

This is a measure of the ratio of convective and diffusive mass transfer in a fluid.  It is analogous to the Nusselt Number in heat transfer and summarized as shown:

Where:

• h_D = Mass Transfer Coefficient (m/s)
• L = Characteristic length (m)
• k = Molecular Diffusivity (m²/s)

Froude Number (Fr)

It is a measure of the ratio of the inertial and gravitational forces and can be expressed as:     

Where:

• v = Velocity (m/s)
• g = Acceleration due to gravity (m/s²)
• L = Characteristic length (m)

It is often used to analyze fluid flow problems on a free surface.  For example, in agitated vessels, Fr governs the formation of free surface vortices:

Grashof Number (Gr)

This is a ratio of the buoyancy and viscous forces. It is used to calculate heat transfer in natural convection where the fluid velocity depends on buoyancy.  It can be expressed as:

Where:

• β = Volumetric coefficient of thermal expansion (1/K)
• g = Acceleration due to gravity (m/s²)
• ΔT = Temperature difference (K)
• L = Characteristic length (m)
• ρ = Fluid Density (kg/m³)
• μ = Dynamic Viscosity (Pa.s)

Mach Number (Ma)

This is the ratio of the fluid velocity to the velocity of sound in that medium.  It can be expressed as:

Where:

• u = Fluid velocity (m/s)
• a = Speed of sound in fluid medium (m/s)

In Chemical Engineering, the Mach Number is commonly used in calculations that involve high velocity gas flow.

Schmidt Number (Sc)

This is the ratio of kinematic viscosity to the diffusivity and it characterizes mass transfer in a flowing fluid. It can be expressed as:

Where:

• μ = Dynamic viscosity (Pa/s)
• ρ = Fluid Density (kg/m³)
• D = Diffusivity (m²/s)
  
  

Biot Number (Bi)

It is the ratio of the internal thermal resistance of a solid to the boundary layer species transfer resistance. It can also be defined as the ratio of internal conductive resistance to the external convective resistance. It is represented as shown:

Where:

• h = Convective heat transfer coefficient (W/m².K)

• L_C = Characteristic Length (m)

• k = Thermal conductivity (W/m.K)

Bi < 0.1 indicates the applicability of the lumped heat analysis.

Biot Number helps analyze the interaction between conduction in a solid and convection at its surface. Smaller Bi number values signify that conduction is dominating the heat transfer mechanism while larger Bi number values signify that convection is dominating the heat transfer process.

CSTR Design

 

Short Notes #2

Isothermal Process Vs. Adiabatic Process

Isothermal and adiabatic processes are important concepts with thermodynamics and chemical engineering students should grasps these concepts, Below are some basic differences between the two processes:

Isothermal Process	Adiabatic Process
It is a thermodynamic process that occurs under a constant temperature	It is a thermodynamic process that occurs without any heat transfer between a system and its surroundings
The temperature is constant	The temperature can change
Heat transfer can be observed	No heat transfer
Work done is due to the change in the net heat content of the system	Work done is due to the change in its internal energy

Absorption vs Adsorption

Both are some of the most important mass transfer processes used in chemical and process industries and are called sorption process. A Sorption Process is a physical or a chemical process by which one substance becomes attached to another substance. Another sorption process is Ion Exchange process.

Absorption is a process where components from a gas phase transfer into a liquid phase when the gas phase and liquid phase are brought into contact. On the other hand. 

Adsorption is a process where components from a gas phase or a liquid phase are attached to the surface of a solid phase when the gas phase or the liquid phase is brought in contact to the solid phase. Adsorption is a surface phenomenon.

Absorption	Adsorption
The substance penetrates the surface	It is a surface phenomenon
Occurs at a uniform rate	Initially the rate increases then decreases
Unaffected by temperature	Affected by temperature
It is the same throughout the material	Concentration on the surface of absorbent is different from that in the bulk
It is an endothermic process	It is an endothermic process

Reactor Sizing

Space time can also be determined using the following graph:

One of the most common numerical methods used is the Simpson's One-Third Rule

Simpson’s One-Third Rule is one of the most common numerical methods. It uses three data points. One numerical method for evaluating integrals include:

Trapezoidal Rule (uses two data points)

Five-Point Quadrature Formula

Introduction to Mass Transfer

Mass transfer is the movement of particles or molecules from one point to another under the influence or a concentration gradient i.e., from an area of high concentration to an area of low concentration. This movement of particles occurs due to two mechanisms: Molecular diffusion and mass convention.

Molecular Diffusion

This refers to the movement of individual molecules through a group of molecules without the aid of a bulk fluid flow like stirring. Imagine a few molecules of a liquid A placed in a particular location within a larger group of molecules of liquid B. As the molecules undergo movement through Brownian Motion, the molecules of A will eventually be distributed into various locations within liquid B as shown below:

The rate of transfer of molecules of A through molecules of B from one location to another is through molecular diffusion. This is mathematically described using Fick's Law. This is simplified into rate of transfer of species A by molecular diffusion in the x-direction as shown in the molecular diffusion equation below:

 

Where:

Ṅ_A,x = diffusion transfer rate of species A (moles transferred per unit time e.g., gmol/s) across area A in the x direction between locations 1 and 2.

A = cross sectional area across which the diffusion occurs (perpendicular to the x direction)

D_AB = the binary diffusivity coefficient of species A in species B with units of area per time (e.g., m²/s)

c_A = concentration of species A (e.g., gmol/mol)

 

The figure below shows the molecular diffusion of a molecule in the x direction through a cross sectional area, A.

 

The transfer rate can be simplified in terms of a driving force (Δc_A), which tends to produce the diffusion, and a resistance (Δx/D_ABA), which tends to oppose the diffusion. The binary diffusivity, D_AB, describes the ease with which a molecule of species A moves through the molecules of species B. Thus, when D_AB is large and transfer occurs rapidly, the resistance is small and when D_AB is small and transfer occurs slowly, the resistance is large. D_AB is not a constant and it varies with the physical conditions like temperature of the system. The temperature affects the motion of the molecules. Thus, the factors affecting D_AB depend on the properties of the molecules of A and B and are listed below:

• molecular size – this determines the distances and spaces between the molecules
• molecular shape – includes the presence of long chains that can tangle
• molecular charge – affects the attractive or repulsive forces between the molecules.

 

Mass Convection

This is the method by which velocities and flow help the molecules of different types mix. In the figure below we can see a small current of fluid moving in the direction represented by the dark arrow which carries the molecules in its path to the new locations. Even though the molecules undergo molecular diffusion, the convection dominates as a transfer mechanism.

Mass convection distributes molecules faster than molecular diffusion alone.

A frequent application of mass transfer in chemical engineering processes is the transfer of materials across a phase boundary (interface) where two phases (solid, liquid and/or gas) meet. In the design of process that have mass transfer across phase boundaries, chemical engineers include mass convection flow to increase that transfer. In mass transfer by convection at phase boundaries, the net effect is the transport perpendicular to the phase boundary as shown below, where the net transfer direction is indicated by the large arrow. In this figure, the transfer mode of species A in Phase I is not specified in order to focus on Phase II.

The following equation for mass convection quantitatively describes the transport to/from phase boundaries via convection:

where:

c_A1 = concentration of species A at the starting location (1) of transfer (at the phase boundary)

c_A2 = concentration of species A at the ending location (2) of transfer (at the bulk of fluid)

Ṅ_A = convection transfer rate of species A through area A from locations 1 to 2

h_m = mass transfer coefficient, which accounts for the effects of diffusion and fluid motion (unit length per time)

A = cross sectional area through which the transfer occurs

In the equation of mass convection shown above, the driving force is present in the concentration difference just as in the molecular diffusion equation. For mass convection, the resistance is 1/h_mA. Thus, if h_m is large, the resistance is small and transfer occurs rapidly and when h_m is small, the resistance is large and transfer occurs slowly.