12.2 Dimensionless numbers in transport phenomena

Dimensionless numbers are crucial in transport phenomena, helping us understand and predict fluid behavior. They simplify complex systems by relating different forces and properties, making it easier to analyze and design processes.

These numbers, like Reynolds and Nusselt, are essential for characterizing flow regimes and heat transfer. They allow engineers to develop correlations, scale up processes, and optimize designs in various applications, from heat exchangers to mass transfer operations.

Dimensionless Numbers in Transport Phenomena

Key Dimensionless Numbers

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• Dimensionless numbers are ratios of forces, energies, or rates that characterize the relative importance of different physical phenomena in a system
• The Reynolds number ($Re$) represents the ratio of inertial forces to viscous forces in a fluid flow, determining the flow regime (laminar, transitional, or turbulent)
  • $Re = \frac{\rho v L}{\mu}$, where $\rho$ is density, $v$ is velocity, $L$ is characteristic length, and $\mu$ is dynamic viscosity
  • Example: Flow in a pipe with $Re < 2300$ is typically laminar, while $Re > 4000$ is typically turbulent
• The Nusselt number ($Nu$) represents the ratio of convective heat transfer to conductive heat transfer, quantifying the enhancement of heat transfer due to convection
  • $Nu = \frac{hL}{k}$, where $h$ is the convective heat transfer coefficient, $L$ is the characteristic length, and $k$ is the thermal conductivity
  • Example: A high $Nu$ indicates effective convective heat transfer in a heat exchanger
• The Prandtl number ($Pr$) represents the ratio of momentum diffusivity to thermal diffusivity, characterizing the relative thickness of the velocity and thermal boundary layers
  • $Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$, where $\nu$ is kinematic viscosity, $\alpha$ is thermal diffusivity, $c_p$ is specific heat, $\mu$ is dynamic viscosity, and $k$ is thermal conductivity
  • Example: Air has a $Pr \approx 0.7$, while water has a $Pr \approx 7$ at room temperature
• The Schmidt number ($Sc$) represents the ratio of momentum diffusivity to mass diffusivity, characterizing the relative thickness of the velocity and concentration boundary layers
  • $Sc = \frac{\nu}{D}$, where $\nu$ is kinematic viscosity and $D$ is mass diffusivity
  • Example: For gases, $Sc$ is typically around 1, while for liquids, $Sc$ can range from 100 to 1000
• The Sherwood number ($Sh$) represents the ratio of convective mass transfer to diffusive mass transfer, quantifying the enhancement of mass transfer due to convection
  • $Sh = \frac{k_c L}{D}$, where $k_c$ is the convective mass transfer coefficient, $L$ is the characteristic length, and $D$ is the mass diffusivity
  • Example: A high $Sh$ indicates effective convective mass transfer in a gas absorption process

Fluid Properties and Dimensionless Numbers

• Fluid properties play a crucial role in determining the values of dimensionless numbers and the behavior of transport phenomena
• The Prandtl number is a fluid property that depends on the specific heat, dynamic viscosity, and thermal conductivity of the fluid
  • Fluids with high $Pr$ (e.g., oils) have a thicker thermal boundary layer relative to the velocity boundary layer
  • Fluids with low $Pr$ (e.g., liquid metals) have a thicker velocity boundary layer relative to the thermal boundary layer
• The Schmidt number is a fluid property that depends on the kinematic viscosity and mass diffusivity of the fluid
  • Fluids with high $Sc$ (e.g., liquids) have a thicker concentration boundary layer relative to the velocity boundary layer
  • Fluids with low $Sc$ (e.g., gases) have a thicker velocity boundary layer relative to the concentration boundary layer
• The values of $Pr$ and $Sc$ affect the heat and mass transfer rates, respectively, and the development of the corresponding boundary layers
  • Example: In a heat exchanger, a fluid with a high $Pr$ will have a thinner thermal boundary layer and a higher heat transfer rate compared to a fluid with a low $Pr$
  • Example: In a gas absorption process, a fluid with a high $Sc$ will have a thinner concentration boundary layer and a higher mass transfer rate compared to a fluid with a low $Sc$

Significance of Key Dimensionless Numbers

Flow Regime Characterization

• The Reynolds number is used to characterize flow regimes in various fluid flow situations
  • Laminar flow occurs at low $Re$, typically $Re < 2300$ in pipes, characterized by smooth, parallel streamlines and minimal mixing
  • Turbulent flow occurs at high $Re$, typically $Re > 4000$ in pipes, characterized by chaotic, fluctuating motion and enhanced mixing
  • Transitional flow occurs between laminar and turbulent regimes, typically $2300 < Re < 4000$ in pipes, exhibiting intermittent turbulent bursts and increased mixing
• The flow regime affects the pressure drop, heat transfer, and mass transfer in fluid systems
  • Example: In laminar flow, the pressure drop is proportional to the fluid velocity, while in turbulent flow, the pressure drop is proportional to the square of the fluid velocity
  • Example: Turbulent flow enhances heat and mass transfer due to increased mixing and the presence of eddies, leading to higher $Nu$ and $Sh$ compared to laminar flow

Transport Mechanism Characterization

• The Nusselt number is used to characterize the relative importance of convective and conductive heat transfer
  • A high $Nu$ indicates that convective heat transfer dominates over conductive heat transfer, leading to more effective heat exchange
  • A low $Nu$ indicates that conductive heat transfer is significant compared to convective heat transfer, limiting the heat exchange effectiveness
• The Sherwood number is used to characterize the relative importance of convective and diffusive mass transfer
  • A high $Sh$ indicates that convective mass transfer dominates over diffusive mass transfer, leading to more effective mass exchange
  • A low $Sh$ indicates that diffusive mass transfer is significant compared to convective mass transfer, limiting the mass exchange effectiveness
• The Prandtl and Schmidt numbers are used to characterize the relative thickness of the velocity, thermal, and concentration boundary layers
  • The relative thickness of these boundary layers affects the heat and mass transfer rates and the development of temperature and concentration profiles
  • Example: In a heat exchanger, a fluid with a high $Pr$ will have a thinner thermal boundary layer and a steeper temperature gradient near the wall compared to a fluid with a low $Pr$
  • Example: In a gas absorption process, a fluid with a high $Sc$ will have a thinner concentration boundary layer and a steeper concentration gradient near the interface compared to a fluid with a low $Sc$

Role of Dimensionless Numbers in Transport

Empirical Correlations

• Dimensionless numbers are used to develop empirical correlations for heat and mass transfer coefficients in various transport processes
• These correlations relate the dimensionless numbers to the transport coefficients, allowing for the prediction of heat and mass transfer rates in different systems
• The Dittus-Boelter correlation is a widely used empirical correlation for the Nusselt number in turbulent pipe flow
  • $Nu = 0.023 Re^{0.8} Pr^n$, where $n = 0.4$ for heating and $n = 0.3$ for cooling
  • This correlation is valid for $0.7 \leq Pr \leq 160$, $Re \geq 10,000$, and $L/D \geq 10$, where $L$ is the pipe length and $D$ is the pipe diameter
• Similar correlations exist for other transport processes, such as the Chilton-Colburn analogy for mass transfer in turbulent flow
  • $Sh = 0.023 Re^{0.8} Sc^{1/3}$, valid for $0.6 \leq Sc \leq 2500$ and $Re \geq 10,000$
• These correlations enable engineers to estimate transport coefficients and design efficient heat and mass transfer equipment

Dimensional Analysis and Scaling

• Dimensionless numbers are used to perform dimensional analysis and scale-up of transport processes
• Dimensional analysis involves combining the relevant variables into dimensionless groups, reducing the number of independent variables and simplifying the problem
• The Buckingham Pi theorem states that a physical problem with $n$ variables and $m$ fundamental dimensions can be reduced to a relationship between $n-m$ dimensionless groups
  • Example: In convective heat transfer, the relevant variables are heat transfer coefficient ($h$), thermal conductivity ($k$), characteristic length ($L$), density ($\rho$), velocity ($v$), and dynamic viscosity ($\mu$)
  • Applying the Buckingham Pi theorem yields the relationship $Nu = f(Re, Pr)$, where $f$ is a function determined experimentally or theoretically
• Scaling laws derived from dimensional analysis ensure that the key dimensionless numbers remain similar when moving from laboratory-scale to industrial-scale systems
  • Example: In a scale-up of a heat exchanger, maintaining the same $Re$ and $Pr$ ensures that the flow regime and heat transfer characteristics are preserved
  • This allows for the prediction of the performance of large-scale systems based on small-scale experiments or simulations

Applications of Dimensionless Numbers in Transport Problems

Heat Exchangers

• In heat exchangers, dimensionless numbers are used to calculate the convective heat transfer coefficient and determine the overall heat transfer rate and effectiveness
• The Nusselt number is used to calculate the convective heat transfer coefficient ($h$) from the thermal conductivity ($k$) and the characteristic length ($L$)
  • $h = \frac{Nu \cdot k}{L}$
  • Example: In a shell-and-tube heat exchanger, the Nusselt number for the tube-side flow can be estimated using the Dittus-Boelter correlation, allowing for the calculation of the tube-side convective heat transfer coefficient
• The overall heat transfer coefficient ($U$) depends on the convective heat transfer coefficients of the hot and cold fluids ($h_h$ and $h_c$), the thermal conductivity of the heat exchanger wall ($k_w$), and the wall thickness ($t$)
  • $\frac{1}{U} = \frac{1}{h_h} + \frac{t}{k_w} + \frac{1}{h_c}$
  • Example: In a plate heat exchanger, the overall heat transfer coefficient can be determined by calculating the convective heat transfer coefficients for the hot and cold fluids using appropriate Nusselt number correlations
• The effectiveness ($\varepsilon$) of a heat exchanger is a function of the number of transfer units ($NTU$) and the heat capacity ratio ($C_r$)
  • $NTU = \frac{UA}{C_{min}}$, where $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, and $C_{min}$ is the smaller of the hot and cold fluid heat capacity rates
  • $C_r = \frac{C_{min}}{C_{max}}$, where $C_{max}$ is the larger of the hot and cold fluid heat capacity rates
  • Example: For a counter-flow heat exchanger, the effectiveness is given by $\varepsilon = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r \exp[-NTU(1 - C_r)]}$

Pipe Flows

• In pipe flows, dimensionless numbers are used to determine the flow regime, friction factor, pressure drop, and pumping power requirements
• The Reynolds number is used to determine the flow regime in pipes
  • $Re = \frac{\rho v D}{\mu}$, where $\rho$ is the fluid density, $v$ is the average velocity, $D$ is the pipe diameter, and $\mu$ is the dynamic viscosity
  • Example: For water flowing in a pipe with a diameter of 0.05 m at a velocity of 1 m/s, the Reynolds number is approximately 50,000, indicating turbulent flow
• The friction factor ($f$) depends on the Reynolds number and the relative pipe roughness ($\varepsilon/D$), where $\varepsilon$ is the absolute pipe roughness
  • For laminar flow, $f = \frac{64}{Re}$
  • For turbulent flow, the friction factor can be estimated using the Colebrook equation or the Moody diagram
  • Example: For a smooth pipe with $Re = 100,000$, the friction factor is approximately 0.018, which can be determined using the Moody diagram or the Colebrook equation
• The pressure drop ($\Delta P$) in a pipe is a function of the friction factor, fluid density, average velocity, pipe length ($L$), and pipe diameter
  • $\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$
  • Example: For a 100 m long, 0.05 m diameter pipe with water flowing at 1 m/s and a friction factor of 0.018, the pressure drop is approximately 36 kPa
• The pumping power ($P$) required to overcome the pressure drop is given by $P = \Delta P \cdot Q$, where $Q$ is the volumetric flow rate
  • Example: For the previous example, with a flow rate of 0.002 m³/s, the pumping power is approximately 72 W

Convective Mass Transfer

• In convective mass transfer, dimensionless numbers are used to calculate the convective mass transfer coefficient and determine the overall mass transfer rate and effectiveness
• The Sherwood number is used to calculate the convective mass transfer coefficient ($k_c$) from the mass diffusivity ($D$) and the characteristic length ($L$)
  • $k_c = \frac{Sh \cdot D}{L}$
  • Example: In a gas absorption process, the Sherwood number for the gas-liquid interface can be estimated using the Chilton-Colburn analogy, allowing for the calculation of the convective mass transfer coefficient
• The overall mass transfer coefficient ($K$) depends on the convective mass transfer coefficients of the phases involved ($k_1$ and $k_2$) and the equilibrium distribution coefficient ($m$)
  • $\frac{1}{K} = \frac{1}{k_1} + \frac{1}{mk_2}$
  • Example: In a gas absorption process with a soluble gas, the overall mass transfer coefficient can be determined by calculating the convective mass transfer coefficients for the gas and liquid phases using appropriate Sherwood number correlations
• The effectiveness of a mass transfer process can be characterized by the number of transfer units ($NTU_m$) and the height of a transfer unit ($HTU$)
  • $NTU_m = \frac{KA}{Q}$, where $K$ is the overall mass transfer coefficient, $A$ is the mass transfer area, and $Q$ is the volumetric flow rate
  • $HTU = \frac{L}{NTU_m}$, where $L$ is the height of the mass transfer equipment
  • Example: In a packed bed absorption column, the height of the column can be determined by calculating the number of transfer units and the height of a transfer unit based on the overall mass transfer coefficient and the flow rates of the gas and liquid phases