Dimensionless Numbers in Transport Phenomena: Heat Transfer, Fluid Mechanics, and Mass Transfer
Reynolds Number (Re): Inertia Versus Viscosity
The Reynolds number is one of the most important dimensionless numbers in fluid mechanics. It describes the relative importance of inertial forces and viscous forces in a flowing fluid.
\[ Re=\frac{\rho VL}{\mu}=\frac{VL}{\nu} \]
where:
- \(\rho\) is the fluid density (kg/m3).
- \(V\) is the characteristic fluid velocity (m/s).
- \(L\) is the characteristic length (m).
- \(\mu\) is the dynamic viscosity (Pa·s).
- \(\nu=\mu/\rho\) is the kinematic viscosity (m2/s).
A high Reynolds number indicates that inertial effects are strong relative to viscous effects. Such flows are more susceptible to instabilities and turbulence. A low Reynolds number indicates that viscous effects are relatively important, and the flow is often smooth and orderly.
For flow in a circular pipe, the Reynolds number is commonly defined using the pipe diameter:
\[ Re_D=\frac{\rho VD}{\mu} \]
For fully developed flow in a straight circular pipe, flow is generally considered laminar below approximately \(Re_D=2300\), transitional between approximately 2300 and 4000, and turbulent above approximately 4000. These are practical guidelines rather than universal boundaries; disturbances, geometry, and inlet conditions can affect the transition.
The Reynolds number is also important in heat and mass transfer because fluid motion influences the thickness of thermal and concentration boundary layers. Consequently, Reynolds number frequently appears in correlations for the Nusselt and Sherwood numbers.
Prandtl Number (Pr): Momentum Diffusion Versus Thermal Diffusion
The Prandtl number compares momentum diffusivity with thermal diffusivity:
\[ Pr=\frac{\nu}{\alpha}=\frac{\mu c_p}{k} \]
where:
- \(\nu\) is the kinematic viscosity.
- \(\alpha\) is the thermal diffusivity.
- \(c_p\) is the specific heat capacity at constant pressure.
- \(k\) is the thermal conductivity.
Thermal diffusivity is defined as:
\[ \alpha=\frac{k}{\rho c_p} \]
It measures how rapidly temperature disturbances spread through a material relative to its capacity to store thermal energy.
The Prandtl number indicates the relative rates of momentum and thermal diffusion.
- When \(Pr<1\), thermal diffusion is relatively rapid compared with momentum diffusion. The thermal boundary layer may be thicker than the velocity boundary layer.
- When \(Pr>1\), momentum diffusion is relatively rapid compared with thermal diffusion. The thermal boundary layer may be thinner than the velocity boundary layer.
Air near room temperature has a Prandtl number of approximately 0.7, while liquid water commonly has a Prandtl number of several units, depending on temperature. Liquid metals often have very low Prandtl numbers because their thermal diffusivity is high.
The Prandtl number is especially useful in forced convection because it helps describe how fluid properties influence heat transfer. It commonly appears in empirical correlations such as:
\[ Nu=CRe^mPr^n \]
The constants \(C\), \(m\), and \(n\) depend on the geometry, flow regime, and correlation being used.
Nusselt Number (Nu): Convection Versus Conduction
The Nusselt number characterizes convective heat transfer relative to heat conduction across a characteristic length in the fluid.
\[ Nu=\frac{hL}{k_f} \]
where:
- \(h\) is the convective heat transfer coefficient (W/m2·K).
- \(L\) is the characteristic length (m).
- \(k_f\) is the thermal conductivity of the fluid (W/m·K).
The Nusselt number is a dimensionless measure of convection relative to conduction across a fluid layer of characteristic thickness \(L\).
A Nusselt number near unity is associated with heat transfer comparable in scale to conduction across the characteristic length. A substantially larger Nusselt number generally indicates enhanced heat transfer associated with fluid motion. The exact interpretation depends on the geometry and reference conduction problem.
The Nusselt number is widely used to estimate heat transfer coefficients. Once a suitable correlation gives \(Nu\), the heat transfer coefficient can be calculated from:
\[ h=\frac{Nu\,k_f}{L} \]
This is valuable in the design of heat exchangers, cooling systems, reactors, electronic thermal management systems, and industrial process equipment.
The Nusselt number should be distinguished from the Biot number. Although their formulas can look identical, they describe different physical situations. The Nusselt number generally characterizes convection relative to conduction in the fluid, whereas the Biot number compares internal conduction in a solid with convection at its surface.
Biot Number (Bi): Internal Conduction Versus Surface Convection
The Biot number is particularly important in transient heat transfer involving solid objects.
\[ Bi=\frac{hL_c}{k_s} \]
where:
- \(h\) is the convective heat transfer coefficient at the solid surface.
- \(L_c=V_s/A_s\) is the characteristic length of the solid, defined using its volume \(V_s\) and exposed surface area \(A_s\).
- \(k_s\) is the thermal conductivity of the solid.
The Biot number compares the resistance to heat conduction within a solid with the resistance to convection between the solid surface and the surrounding fluid.
A small Biot number indicates that internal conduction is relatively effective compared with surface convection. The temperature within the solid therefore tends to remain relatively uniform.
When \(Bi<0.1\), the lumped-capacitance method is often a reasonable approximation for transient heating or cooling, provided that the other assumptions of the method are satisfied.
Under this approximation, the solid can be treated as having a single temperature that changes with time, rather than a temperature that varies significantly with position.
For larger Biot numbers, internal temperature gradients become more important, and a spatially resolved conduction model may be required.
For example, a thin metal component may have a relatively uniform internal temperature during cooling, whereas a thick concrete structure may develop substantial temperature differences between its surface and interior.
The Biot number is therefore useful for deciding whether a simplified thermal model is sufficient or whether the heat conduction equation must be solved throughout the solid.
Fourier Number (Fo): The Extent of Transient Heat Diffusion
The Fourier number describes the importance of elapsed time relative to the characteristic timescale of thermal diffusion.
\[ Fo=\frac{\alpha t}{L_c^2} \]
where:
- \(\alpha\) is the thermal diffusivity.
- \(t\) is the elapsed time.
- \(L_c\) is the characteristic length.
The Fourier number is dimensionless because thermal diffusivity has units of m2/s.
A useful way to understand the Fourier number is to compare the approximate thermal diffusion distance, proportional to \(\sqrt{\alpha t}\), with the characteristic length of the system:
\[ Fo\sim\left(\frac{\sqrt{\alpha t}}{L_c}\right)^2 \]
- When \(Fo\ll1\), relatively little time has elapsed for heat to diffuse across the characteristic length.
- When \(Fo\) is comparable to unity, thermal diffusion has had sufficient time to influence a substantial portion of the characteristic length.
- A large Fourier number indicates that thermal diffusion has proceeded for a relatively long time compared with the characteristic diffusion timescale.
The Fourier number is widely used in transient conduction problems, including cooling of food products, heating of metal components, thermal treatment of construction materials, and temperature evolution in concrete.
Unlike the Biot number, which compares internal conduction with surface convection, the Fourier number explicitly includes time. The two numbers are often used together in transient heat transfer analysis.
Grashof Number (Gr): Buoyancy Versus Viscous Forces
The Grashof number is important in natural convection, where fluid motion is generated by density differences caused by temperature variations rather than primarily by an external pump or fan.
For thermal natural convection, it is defined as:
\[ Gr=\frac{g\beta\Delta T L^3}{\nu^2} \]
where:
- \(g\) is gravitational acceleration (m/s2).
- \(\beta\) is the volumetric thermal expansion coefficient (1/K).
- \(\Delta T\) is the characteristic temperature difference (K).
- \(L\) is the characteristic length (m).
- \(\nu\) is the kinematic viscosity (m2/s).
When a fluid is heated, its density often decreases. In a gravitational field, this density difference produces buoyancy forces that can drive fluid motion.
The Grashof number compares the importance of buoyancy-driven motion with viscous resistance.
- A small Grashof number indicates that buoyancy-driven flow is relatively weak.
- A large Grashof number indicates that buoyancy can drive stronger fluid motion relative to viscous resistance.
The characteristic length has a cubic dependence:
\[ Gr\propto L^3 \]
Consequently, increasing the size of a system can strongly increase the importance of buoyancy, even if the temperature difference and fluid properties remain unchanged.
The Grashof number plays a role in natural-convection correlations in much the same way that the Reynolds number plays a role in forced-convection correlations. In natural convection, the characteristic velocity is generated by buoyancy itself, so the Reynolds number alone does not adequately describe the driving mechanism.
Rayleigh Number (Ra): The Overall Strength of Natural Convection
The Rayleigh number combines the Grashof and Prandtl numbers:
\[ Ra=GrPr \]
Substituting their definitions gives:
\[ Ra=\frac{g\beta\Delta T L^3}{\nu\alpha} \]
The Rayleigh number incorporates buoyancy, viscous diffusion, and thermal diffusion into a single dimensionless group. It is one of the principal parameters used to characterize natural convection.
A small Rayleigh number generally indicates that buoyancy-driven convection is weak relative to diffusive effects. A large Rayleigh number indicates that natural convection can become increasingly important.
The Rayleigh number is especially useful in studying heated vertical plates, natural circulation in enclosures, heat transfer around buildings, cooling of electronic components, and thermal convection in fluids.
For example, when a warm surface is exposed to cooler surrounding air, the air near the surface becomes warmer and less dense. The resulting buoyancy can cause the air to rise and induce circulation. The Rayleigh number helps characterize the resulting heat transfer behavior.
For an idealized horizontal fluid layer heated from below, the onset of convection under classical conditions occurs at a critical Rayleigh number of approximately 1708. This value applies to a specific configuration and set of boundary conditions; it should not be treated as a universal threshold for every natural-convection problem.
Natural-convection heat transfer is often represented by a correlation of the form:
\[ Nu=f(Ra,Pr) \]
For many common geometries and applicable flow regimes, correlations can sometimes be simplified to:
\[ Nu=C\,Ra^n \]
The constants depend on the geometry and the relevant range of Rayleigh numbers. The Rayleigh number describes the driving conditions for natural convection, while the Nusselt number describes the resulting dimensionless heat transfer.
Schmidt Number (Sc): Momentum Diffusion Versus Mass Diffusion
The Schmidt number is the mass-transfer analogue of the Prandtl number.
\[ Sc=\frac{\nu}{D_{AB}} \]
where:
- \(\nu\) is the kinematic viscosity of the fluid.
- \(D_{AB}\) is the molecular diffusivity of species A in species B.
The Schmidt number compares momentum diffusivity with molecular mass diffusivity.
- When \(Sc<1\), molecular mass diffusion is relatively rapid compared with momentum diffusion.
- When \(Sc>1\), momentum diffusion is relatively rapid compared with molecular mass diffusion.
The Schmidt number is particularly important in chemical engineering because mass transfer often involves a chemical species moving through a fluid while the fluid itself is in motion.
Examples include the evaporation of a liquid into air, dissolution of a solid into a flowing liquid, absorption of gases into liquids, and transport of reactants in chemical reactors.
The Schmidt number influences the relationship between the velocity boundary layer and the concentration boundary layer. Its effect is analogous to the way the Prandtl number influences the relationship between velocity and thermal boundary layers.
This similarity is a key reason that heat-transfer and mass-transfer correlations often have comparable mathematical forms.
Sherwood Number (Sh): Convective Mass Transfer Versus Molecular Diffusion
The Sherwood number is the mass-transfer analogue of the Nusselt number.
\[ Sh=\frac{k_cL}{D_{AB}} \]
where:
- \(k_c\) is the convective mass-transfer coefficient (m/s).
- \(L\) is the characteristic length (m).
- \(D_{AB}\) is the molecular diffusivity (m2/s).
The Sherwood number characterizes mass transfer relative to molecular diffusion across a characteristic length. A higher Sherwood number generally indicates stronger convective mass transfer relative to the selected diffusion reference.
The mass-transfer coefficient can be calculated from:
\[ k_c=\frac{ShD_{AB}}{L} \]
This coefficient is useful in estimating species-transfer rates in absorption towers, packed columns, drying operations, dissolution processes, catalytic systems, and membrane-related applications.
The Sherwood number often appears in correlations of the form:
\[ Sh=CRe^mSc^n \]
Such correlations allow engineers to estimate mass-transfer coefficients from fluid properties, flow conditions, and geometry.
The analogy between the Sherwood and Nusselt numbers is especially useful when a heat-transfer correlation is known for a particular flow configuration and an appropriate mass-transfer correlation is needed. The analogy can guide the mathematical form of the model, although the coefficients and assumptions must be validated for the specific system.
Stanton Number (St): Transfer Relative to the Flowing Fluid's Transport Capacity
The Stanton number characterizes convective heat transfer relative to the heat-capacity transport associated with the flowing fluid.
\[ St=\frac{h}{\rho Vc_p} \]
where:
- \(h\) is the convective heat transfer coefficient.
- \(\rho\) is the fluid density.
- \(V\) is the characteristic velocity.
- \(c_p\) is the specific heat capacity at constant pressure.
The Stanton number compares heat transferred from a surface with the heat that the flowing fluid can carry through the system per unit temperature change.
It can also be expressed in terms of the Nusselt, Reynolds, and Prandtl numbers:
\[ St=\frac{Nu}{RePr} \]
To see this relationship, recall that:
\[ Nu=\frac{hL}{k_f},\qquad Re=\frac{VL}{\nu},\qquad Pr=\frac{\nu}{\alpha} \]
and:
\[ \alpha=\frac{k_f}{\rho c_p} \]
Substitution gives:
\[ \frac{Nu}{RePr} =\frac{h}{\rho Vc_p} =St \]
This relationship makes the Stanton number useful for connecting heat-transfer coefficients with flow conditions. It is frequently used in turbulent-flow heat transfer and in the analysis of thermal boundary layers.
For mass transfer, an analogous Stanton number can be defined as:
\[ St_m=\frac{k_c}{V} \]
Under the same characteristic-length conventions, it can be written as:
\[ St_m=\frac{Sh}{ReSc} \]
The thermal and mass-transfer versions should be distinguished because their definitions involve different transport coefficients and physical quantities.
Lewis Number (Le): Thermal Diffusion Versus Mass Diffusion
The Lewis number compares thermal diffusivity with molecular mass diffusivity:
\[ Le=\frac{\alpha}{D_{AB}} \]
It can also be expressed in terms of the Schmidt and Prandtl numbers:
\[ Le=\frac{Sc}{Pr} \]
Because:
\[ \frac{Sc}{Pr} =\frac{\nu/D_{AB}}{\nu/\alpha} =\frac{\alpha}{D_{AB}} =Le \]
The Lewis number describes the relative rates at which heat and a chemical species diffuse through a fluid.
- When \(Le\approx1\), thermal and mass diffusivities are comparable.
- When \(Le>1\), thermal diffusion is faster than molecular mass diffusion.
- When \(Le<1\), molecular mass diffusion is faster than thermal diffusion.
The Lewis number is particularly important in systems where heat and mass transfer occur simultaneously, including humidification, drying, evaporation, combustion, and gas-liquid contact operations.
For example, during evaporation, energy must be transferred to provide the latent heat of vaporization, while vapor must simultaneously diffuse or be transported away from the liquid surface. The Lewis number helps characterize the relative importance of thermal and species diffusion.
The Heat–Mass Transfer Analogy
One of the most important concepts in transport phenomena is the mathematical similarity between momentum, heat, and mass transfer.
The governing equations for momentum, thermal energy, and species concentration often contain analogous diffusion and convection terms. As a result, many dimensionless numbers appear in corresponding pairs.
Prandtl and Schmidt Numbers
The Prandtl number is:
\[ Pr=\frac{\nu}{\alpha} \]
The Schmidt number is:
\[ Sc=\frac{\nu}{D_{AB}} \]
Both compare momentum diffusivity with another transport mechanism. The Prandtl number uses thermal diffusivity, while the Schmidt number uses molecular mass diffusivity. Thus, the two numbers play analogous roles in heat and mass transfer.
Nusselt and Sherwood Numbers
The Nusselt number is:
\[ Nu=\frac{hL}{k_f} \]
The Sherwood number is:
\[ Sh=\frac{k_cL}{D_{AB}} \]
Both describe convective transfer relative to a diffusion-based reference. Their mathematical correspondence is:
\[ Nu\longleftrightarrow Sh \]
\[ h\longleftrightarrow k_c \]
\[ k_f\longleftrightarrow D_{AB} \]
The thermal conductivity \(k_f\) describes heat conduction in the fluid, whereas the molecular diffusivity \(D_{AB}\) describes species diffusion.
Fourier Number and Mass-Diffusion Timescales
The Fourier number describes transient thermal diffusion:
\[ Fo=\frac{\alpha t}{L^2} \]
An analogous dimensionless mass-diffusion time is:
\[ Fo_m=\frac{D_{AB}t}{L^2} \]
The second quantity is often called a mass-transfer Fourier number or a dimensionless diffusion time. The terminology is less standardized than that of the thermal Fourier number, but it serves the same mathematical purpose.
Both quantities compare elapsed time with a characteristic diffusion timescale.
Lewis Number as the Bridge
The Lewis number connects thermal and mass diffusivities:
\[ Le=\frac{\alpha}{D_{AB}}=\frac{Sc}{Pr} \]
When the Lewis number is near unity, heat and mass diffusion may occur on comparable timescales. This is useful when modeling processes such as drying, humidification, evaporation, and combustion, where heat and species transport are coupled.
However, equal diffusivity scales alone do not guarantee identical heat- and mass-transfer behavior. Boundary conditions, reaction kinetics, phase changes, and property variations can still produce important differences.
A Unified Framework for Understanding the Eleven Numbers
The eleven dimensionless numbers can be organized by the physical question each one answers.
| Dimensionless Number | Primary Physical Meaning | Typical Application |
|---|---|---|
| Reynolds (Re) | Inertia relative to viscosity | Flow regime and forced convection |
| Prandtl (Pr) | Momentum diffusivity relative to thermal diffusivity | Thermal boundary layers |
| Nusselt (Nu) | Convective heat transfer relative to a conductive reference | Estimating heat-transfer coefficients |
| Biot (Bi) | Internal solid conduction relative to surface convection | Lumped-capacitance validity |
| Fourier (Fo) | Elapsed time relative to thermal diffusion time | Transient heat conduction |
| Grashof (Gr) | Buoyancy relative to viscous effects | Natural convection |
| Rayleigh (Ra) | Combined buoyancy and thermal-diffusion effects | Natural-convection strength |
| Schmidt (Sc) | Momentum diffusivity relative to mass diffusivity | Concentration boundary layers |
| Sherwood (Sh) | Convective mass transfer relative to a diffusion reference | Estimating mass-transfer coefficients |
| Stanton (St) | Convective heat transfer relative to fluid heat-capacity transport | Boundary-layer and turbulent heat transfer |
| Lewis (Le) | Thermal diffusivity relative to mass diffusivity | Coupled heat and mass transfer |
- The first group describes the relative importance of transport mechanisms. Reynolds, Prandtl, Schmidt, Grashof, and Rayleigh numbers help characterize flow and the mechanisms that drive it.
- The second group characterizes transfer rates. Nusselt and Sherwood numbers quantify dimensionless heat and mass transfer, while the Stanton number relates transfer coefficients to the transport capacity of the flowing fluid.
- The third group characterizes internal or transient diffusion. Biot and Fourier numbers are especially useful in transient heat transfer, while the Lewis number connects thermal and species diffusion.
Practical Applications in Chemical Engineering
These dimensionless numbers are useful because engineering systems rarely involve only one transport mechanism.
Heat Exchangers
In a heat exchanger, the Reynolds number characterizes the flow, the Prandtl number describes the relative rates of momentum and thermal diffusion, and the Nusselt number helps estimate the convective heat transfer coefficient. The Stanton number can then relate that coefficient to the heat-capacity transport of the flowing fluid.
Heating and Cooling of Solids
When a solid undergoes heating or cooling, the Biot number helps determine whether internal temperature gradients can be neglected, while the Fourier number indicates how much time has elapsed relative to thermal diffusion across the object.
Natural Convection
Near a heated surface, the Grashof number describes the relative importance of buoyancy and viscosity, while the Rayleigh number incorporates thermal diffusion and helps determine the natural-convection regime. The Nusselt number then provides a way to estimate heat transfer.
Evaporation and Gas Absorption
In evaporation or gas absorption, the Reynolds number describes the flow, the Schmidt number characterizes momentum diffusion relative to species diffusion, and the Sherwood number helps estimate the mass-transfer coefficient. If heat and mass transfer occur simultaneously, the Lewis number indicates whether their diffusion timescales are comparable.
These examples show that dimensionless numbers are not isolated mathematical definitions. They form a connected framework for translating physical properties and operating conditions into practical engineering predictions.
The Biot, Fourier, Grashof, Nusselt, Prandtl, Rayleigh, Reynolds, Schmidt, Sherwood, Stanton, and Lewis numbers provide a unified language for analyzing transport phenomena.
The Reynolds number compares inertia with viscosity. The Prandtl and Schmidt numbers compare momentum diffusion with thermal and mass diffusion, respectively. The Nusselt and Sherwood numbers characterize convective heat and mass transfer. The Biot and Fourier numbers are essential for analyzing internal temperature gradients and transient thermal diffusion. The Grashof and Rayleigh numbers characterize buoyancy-driven natural convection, while the Stanton number relates convective transfer to the transport capacity of a flowing fluid. The Lewis number links heat and mass diffusion.
The most useful relationships to remember are:
- \(Ra=GrPr\)
- \(Le=\alpha/D_{AB}=Sc/Pr\)
- \(St=Nu/(RePr)\)
- \(Nu\longleftrightarrow Sh\)
- \(Pr\longleftrightarrow Sc\)
The broader lesson is that dimensionless analysis reveals which physical mechanisms dominate a system and provides a foundation for scaling experiments, developing correlations, and designing engineering equipment. Once these relationships are understood, many apparently different problems in fluid mechanics, heat transfer, and mass transfer become variations of the same underlying transport principles.
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