Dimensionless Numbers in Heat Transfer

Why This Matters

Dimensionless numbers are the universal language of heat transfer analysis—they let engineers compare systems of vastly different scales and predict thermal behavior without solving the full governing equations every time. You're being tested on your ability to recognize what physical phenomena each number captures and when to apply each one. Whether you're analyzing a heat exchanger, predicting flow transition, or determining if a simple lumped analysis will work, these numbers are your diagnostic tools.

Don't fall into the trap of memorizing definitions in isolation. The real exam skill is understanding which forces or resistances each number compares, how numbers relate to each other (like how Rayleigh combines Grashof and Prandtl), and what the magnitude tells you about system behavior. Master the physical intuition behind each ratio, and you'll handle any problem they throw at you.

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Flow Regime and Momentum Transport

These numbers characterize how fluid moves and whether inertial or viscous effects dominate—the foundation for all convective heat transfer analysis.

Reynolds Number (Re)

• Ratio of inertial to viscous forces—defined as $Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}$ where $L$ is characteristic length
• Flow regime indicator: laminar ($Re < 2000$), transitional, or turbulent ($Re > 4000$) for internal pipe flow
• Critical for convection correlations—nearly every forced convection Nusselt correlation includes $Re$ as a primary variable

Prandtl Number (Pr)

• Ratio of momentum diffusivity to thermal diffusivity—$Pr = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}$, a fluid property only
• Boundary layer relationship: determines whether velocity or thermal boundary layer is thicker ($Pr > 1$ means thinner thermal layer)
• Fluid characterization: liquid metals have $Pr \ll 1$, oils have $Pr \gg 1$, gases typically $Pr \approx 0.7$

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Convective Heat Transfer Performance

These numbers quantify how effectively heat moves from surfaces to fluids—the outputs you're solving for in most convection problems.

Nusselt Number (Nu)

• Ratio of convective to conductive heat transfer—$Nu = \frac{hL}{k}$, essentially a dimensionless heat transfer coefficient
• Higher values mean stronger convection—turbulent flow, enhanced surfaces, and favorable geometry all increase $Nu$
• The target variable in most convection problems; correlations express $Nu = f(Re, Pr)$ or $Nu = f(Ra)$

Stanton Number (St)

• Ratio of heat transferred to fluid thermal capacity—$St = \frac{h}{\rho V c_p} = \frac{Nu}{Re \cdot Pr}$
• Heat exchanger efficiency metric—directly relates wall heat flux to bulk fluid energy transport
• Connects momentum and heat transfer through Reynolds analogy: $St \approx \frac{C_f}{2}$ for $Pr \approx 1$

Peclet Number (Pe)

• Ratio of advective to diffusive transport—$Pe = Re \cdot Pr = \frac{VL}{\alpha}$
• Large $Pe$ means convection dominates—thermal diffusion is negligible compared to bulk fluid motion
• Simplifies analysis: when $Pe \gg 1$, axial conduction terms can often be dropped from energy equations

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Natural Convection and Buoyancy Effects

When temperature differences drive fluid motion through density variations, buoyancy forces replace imposed velocity as the driving mechanism.

Grashof Number (Gr)

• Ratio of buoyancy to viscous forces—$Gr = \frac{g \beta \Delta T L^3}{\nu^2}$, the natural convection analog to $Re^2$
• Drives natural convection strength—larger temperature differences and longer surfaces increase $Gr$
• Flow stability indicator: critical $Gr$ values determine transition from laminar to turbulent natural convection

Rayleigh Number (Ra)

• Product of Grashof and Prandtl—$Ra = Gr \cdot Pr = \frac{g \beta \Delta T L^3}{\nu \alpha}$
• Convection onset criterion: $Ra > Ra_{critical}$ (often ~1708 for horizontal layers) triggers convective motion
• Primary correlation variable for natural convection; most $Nu$ correlations use $Ra$ directly rather than $Gr$ alone

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Transient Conduction Analysis

These numbers govern time-dependent heat transfer in solids—essential for heating/cooling process design and determining appropriate solution methods.

Biot Number (Bi)

• Ratio of internal to external thermal resistance—$Bi = \frac{hL_c}{k_s}$ where $L_c$ is characteristic length of the solid
• Lumped capacitance criterion: if $Bi < 0.1$, internal temperature gradients are negligible and simple exponential solutions apply
• Physical meaning: small $Bi$ means the solid conducts heat much faster than the surface can transfer it to the fluid

Fourier Number (Fo)

• Dimensionless time for conduction—$Fo = \frac{\alpha t}{L^2}$, ratio of heat conduction rate to thermal storage rate
• Measures thermal penetration—larger $Fo$ means heat has diffused further into the solid
• Appears in all transient solutions—Heisler charts, semi-infinite solid solutions, and numerical schemes all use $Fo$

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Special Applications

These numbers address specific phenomena that become important under particular conditions—high-speed flows and combined transport mechanisms.

Eckert Number (Ec)

• Ratio of kinetic energy to enthalpy difference—$Ec = \frac{V^2}{c_p \Delta T}$
• Viscous dissipation indicator—when $Ec$ is significant, frictional heating affects temperature profiles
• Critical for high-speed flows—compressible aerodynamics, high-velocity lubricant films, and polymer processing

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Quick Reference Table

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Self-Check Questions

1. You're analyzing natural convection from a vertical plate. Which two dimensionless numbers would appear in your Nusselt correlation, and how are they related to each other?

2. A small steel sphere is quenched in oil. What dimensionless number determines whether you can use lumped capacitance analysis, and what threshold value must it satisfy?

3. Compare and contrast Reynolds number and Grashof number: what role does each play in forced vs. natural convection, and what physical forces does each ratio represent?

4. If an FRQ asks you to evaluate heat exchanger performance and compare different flow velocities, which dimensionless number directly relates heat transfer to the fluid's thermal capacity rate?

5. Two fluids have the same Reynolds number in identical tubes, but one is liquid metal ($Pr = 0.01$) and one is oil ($Pr = 1000$). How would their thermal boundary layers differ, and which would have the higher Nusselt number for the same $Re$?