Diffusivity: A Fundamental Transport Property
Diffusivity (D) is a critical transport property that characterizes the mobility of a component within a medium. Mathematically, it is defined as the ratio of the molar flux (J) to the corresponding concentration gradient (∇C), expressed in units of m²/s.
D = J / ∇C
The diffusivity of a component is a function of various thermodynamic and kinetic parameters, including temperature (T), pressure (P), and the nature and concentration of other components in the system.
Temperature and Pressure Dependence
The temperature and pressure dependence of diffusivity varies between gases and liquids. For gases at atmospheric pressure, the diffusivity is typically less than 1 cm²/s, whereas for liquids, it is on the order of 1 × 10⁻⁵ cm²/s.
The temperature and pressure dependence of gas diffusivity can be described by the following relation:
D ∝ T¹.⁵ / P
In contrast, the diffusivity of liquids varies linearly with temperature:
D ∝ T
Estimation of Liquid Diffusivity
The Wilke-Chang equation is a commonly used correlation for estimating the diffusivity of liquids:
D = (7.4 × 10⁻⁸ × (χ × M)⁰.⁵ × T) / (μ × V₀²⁸)
where χ is the association factor, M is the molecular weight, μ is the viscosity, and V₀ is the molar volume.
This correlation provides a useful estimate of liquid diffusivity, which is essential for designing various engineering processes, including mass transfer operations, chemical reactions, and separation processes.
The Diffusivity Framework
Diffusivity (D) measures the rate at which molecules or energy diffuse through a medium. This framework applies across various engineering disciplines:
- Heat and Mass Transfer: Diffusivity affects transfer rates, influencing design and efficiency.
- Fluid Dynamics: Diffusivity influences fluid behavior, impacting flow patterns and pressure drops.
- Biological Engineering: Diffusivity plays a crucial role in biological systems, informing biomedical device design and tissue engineering.
Key Principles and Relationships
- Fick's Laws: Diffusivity relates to concentration gradients and molecular diffusion rates.
- Dimensionless Numbers: Numbers like Schmidt and Prandtl provide a framework for comparing transport processes.
Several research gaps persist, hindering the development of innovative technologies and applications. One of the primary challenges is scalability, as diffusivity measurements are often limited to small scales, making it difficult to translate these findings to larger systems. This scalability issue is particularly pronounced in complex systems, such as porous media, biological tissues, and nanomaterials, where the intricate interactions between components and the surrounding environment are not yet fully understood.
Furthermore, non-equilibrium systems, characterized by high gradients or rapid changes, pose another significant research gap. In these systems, the traditional assumptions of equilibrium and steady-state conditions no longer apply, and new theoretical frameworks and experimental techniques are needed to accurately capture the dynamics of diffusivity. By addressing these research gaps, scientists and engineers can unlock new insights into the behavior of complex systems, ultimately leading to breakthroughs in fields such as energy storage, biomedical engineering, and materials science.
The fascinating realm of aeronautics and space exploration relies heavily on the intricate principles of diffusivity. This fundamental concept plays a critical role in designing and optimizing various systems that enable spacecraft to venture into the vastness of space. One of the most significant applications of diffusivity is in the design of heat shields, which protect spacecraft from the intense heat generated during re-entry into the Earth's atmosphere. By carefully manipulating the diffusivity of materials, engineers can create heat shields that efficiently dissipate heat, ensuring the safety of both the spacecraft and its occupants.
Diffusivity also has a profound impact on the performance of propulsion systems, including rocket engines and ion thrusters. The rate at which fuel and oxidizer diffuse through the combustion chamber influences the efficiency and thrust of the engine. Furthermore, diffusivity plays a crucial role in life support systems, such as air and water recycling, which rely on efficient mass transfer processes to sustain life during extended space missions. In addition, space suits, which must maintain a safe internal environment while protecting against extreme external conditions, are designed with diffusivity in mind. The careful balance of diffusivity in these systems ensures that astronauts can survive and thrive in the harsh environment of space. By continuing to advance our understanding of diffusivity, scientists and engineers can develop innovative technologies that will propel humanity further into the cosmos.
Mass Transfer Coefficient: A Critical Parameter in Diffusion Processes
The mass transfer coefficient (k) is a fundamental parameter in diffusion processes, quantifying the rate at which a species transfers from one phase to another. Mathematically, k is defined as the ratio of molar flux to concentration difference, expressed as:
N = k × (C1 - C2)
where N is the molar flux, k is the mass transfer coefficient, and C1 and C2 are the concentrations of the species at the interface and in the bulk phase, respectively.
The units of k depend on the choice of concentration units. When mole/volume units are used, k has units of cm/s, whereas when mole fraction units are employed, k has units of mole/cm² s, as mole fractions are dimensionless.
Advanced Mass Transfer Concepts
In equimolar counter-diffusion, where species A and B diffuse in opposite directions, the ratio of mass flux for diffusion of A to the mass flux through non-diffusing B is greater than one. This phenomenon is attributed to the differences in diffusivities and concentrations between the two species.
The mass transfer coefficients for gas (kg) and liquid (ky) phases are related through the following equation:
kG/P = kY/P2
This relationship highlights the dependence of mass transfer coefficients on the phase and total pressure.
Film Theory and Boundary Layer Theory
According to the film theory, the mass transfer coefficient (kl) is related to diffusivity (D) through the following proportionality:
kl µ D
Boundary layer theory predicts a slightly different relationship:
kl α D^0.67
This discrepancy arises from the differences in assumptions and simplifications between the two theories.
Mass Transfer in Dilute Mixtures
When a solute A is present in a dilute mixture with solvent B, the term PB,M tends to the total pressure P. This simplification enables the use of approximate methods for calculating mass transfer coefficients and fluxes.
By understanding these advanced mass transfer concepts, engineers and researchers can design and optimize various industrial processes, including chemical reactors, separation units, and environmental remediation systems.
Relationship between Diffusivity and Mass Transfer Coefficient
The mass transfer coefficient (k) is related to molecular diffusivity (D) by:
k = D / δ
where δ is the thickness of the stagnant layer (film theory)
Theories for Calculating Mass Transfer Coefficient
- Film Theory: Steady-state model, assumes a stagnant film at the interface
- Boundary Layer Theory: Describes mass transfer in terms of boundary layer thickness and concentration gradient
- Penetration Theory: Assumes a finite exposure time for the interface, used for liquid-liquid and gas-liquid systems
- Surface Renewal Theory: Accounts for the renewal of the interface due to turbulence or other disturbances
- Surface Stretch Theory: Combines elements of film and surface renewal theories
- Combination of Film and Surface Renewal Theory: Hybrid approach, combining the strengths of both theories
These theories provide a framework for understanding and calculating mass transfer coefficients, essential for designing and optimizing various engineering processes.
Dimensionless
Groups in Mass Transfer
|
Dimensionless Group |
Definition |
Physical Significance |
Typical Values |
|
Schmidt Number (Sc) |
μ/ρDv |
Momentum diffusivity / Mass diffusivity |
Gases: ~1, Liquids: ~0.1 |
|
Sherwood Number (Sh) |
kcD/Dv |
Convective flux / Diffusive flux |
Evaporation from spherical naphthalene ball:
2 |
|
Stanton Number (St) |
Sh /(Re.Sc) |
Mass transfer coefficient / Convective
velocity |
- |
The analogy between Heat and Mass Transfer
|
Heat Transfer |
Mass Transfer |
|
Prandtl Number (Pr) |
Schmidt Number (Sc) |
|
Nusselt Number (Nu) |
Sherwood Number (Sh) |
|
Reynolds Analogy |
St = f / 2 |
Note:
- μ: Dynamic viscosity
- ρ: Density
- Dv: Diffusion coefficient
- kc: Mass transfer coefficient
- D: Diameter
- Re: Reynolds number
- f: Friction factor
Mass transfer coefficients can be predicted using various theories, including:
- Danckwerts Surface Renewal Theory: kl' = (DAB * S)^0.5, where S is the fraction of surface renewed per unit time.
- Penetration Theory: kL,av = 2 (DAB / π t)^0.5.
Effective diffusivity in porous solids is influenced by porosity (ε) and tortuosity (τ). For a gaseous solute with mass diffusivity DAB, the effective diffusivity Deff is given by Deff = DAB * ε / τ. For example, if DAB = 0.5 cm²/s, ε = 0.5, and τ = 2, then Deff = 0.125 cm²/s.
Knudsen Diffusion
Knudsen diffusion occurs when the mean free path is much greater than the pore diameter. In this regime, molecule-pore wall collisions dominate, and Knudsen diffusivity (DK) is independent of total pressure. DK increases with the square root of temperature and inversely with the square root of molecular weight, typically ranging from 10⁻¹ to 10⁻⁴ cm²/s.
Permeability
Permeability (P) is defined as the product of solubility (S) and diffusivity (D): P = S * D. This parameter characterizes the ability of a species to pass through a material.
Approximate Diffusivities of gases at standard atmospheric pressure, 101.325 KPa:
s.no
|
System
|
Temperature, 0C
|
Diffusivity,m2/s X 10-5
|
Reference
|
1
|
H2 - CH4
|
0
|
6.25
|
Chapman,s.&T.G. Cowling
|
2
|
O2 - N2
|
0
|
1.81
|
,,
|
3
|
CO – O2
|
0
|
1.85
|
,,
|
4
|
CO2 – O2
|
0
|
1.39
|
,,
|
5
|
Air – NH3
|
0
|
1.98
|
Wintergeist
|
6
|
Air – H2O
|
25.9
59.0
|
2.58
3.05
|
Gilliland
|
7
|
Air – C2H5OH
|
25.9
|
1.02
|
Gilliland
|
8
|
Air – n-Butanol
|
25.9
59.0
|
0.87
1.04
|
International critical table
|
9
|
Air – Ethyl Acetate
|
25.9
59.0
|
0.87
1.06
|
Gilliland
|
10
|
Air – Aniline
|
25.9
59.0
|
0.74
0.90
|
Gilliland
|
11
|
Air – Chlorobenzene
|
25.9
59.0
|
0.74
0.90
|
Gilliland
|
12
|
Air - Toluene
|
25.9
59.0
|
0.86
0.92
|
Gilliland
|
Approximate Diffusivities of Liquids at 1 atm, pressure:
s.no
|
System
|
Temperature,
0C
|
Solute con:
Kmole/m3
|
Diffusivity,
m2/s X 10-9
| |
solute
|
solvent
| ||||
1
|
Cl2
|
Water
|
16
|
0.12
|
1.26
|
2
|
HCl
|
Water
|
0
,,
10
,,
16
|
9
2
9
2.5
0.5
|
2.7
1.8
3.3
2.5
2.44
|
3
|
NH3
|
Water
|
5
15
|
3.5
1.0
|
1.24
1.77
|
4
|
CO2
|
Water
|
10
20
|
0
0
|
1.26
1.21
|
5
|
NaCl
|
Water
|
18
,,
,,
,,
,,
|
0.05
0.2
1.0
3.0
5.4
|
1.24
1.36
1.54
1.28
0.82
|
6
|
Methanol
|
Water
|
15
|
0
|
0.91
0.96
|
7
|
Acetic acid
|
Water
|
12.5
,,
18
|
1.0
0.01
1.0
|
0.50
0.83
0.90
|
8
|
Ethanol
|
Water
|
10
,,
16
|
3.75
0.05
2.0
|
0.50
0.83
0.90
|
9
|
n-butanol
|
Water
|
15
|
0
|
0.77
|
10
|
Co2
|
Ethanol
|
17
|
0
|
3.2
|
11
|
chloroform
|
ethanol
|
20
|
2.0
|
1.25
|