3.3 Natural convection and mixed convection

Natural convection happens when temperature differences make fluids move on their own. It's all about buoyancy forces and how they balance with fluid friction. We'll look at how this works on different surfaces and what factors affect it.

Mixed convection combines natural and forced fluid movement. We'll see how to tell which type is stronger and how they work together. This matters for real-world cooling systems and heat transfer applications.

Natural Convection Heat Transfer

Mechanisms and Governing Principles

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• Natural convection is driven by buoyancy forces resulting from density variations due to temperature gradients in the fluid
• The buoyancy force is proportional to the density difference and the volume of the fluid displaced, as described by Archimedes' principle
• The fluid motion in natural convection is governed by the balance between buoyancy forces and viscous forces, characterized by the Grashof number (Gr)
  • The Grashof number represents the ratio of buoyancy forces to viscous forces and is defined as $Gr = (gβ(Ts - T∞)L³)/ν²$, where $g$ is the gravitational acceleration, $β$ is the thermal expansion coefficient, $Ts$ is the surface temperature, $T∞$ is the ambient temperature, $L$ is the characteristic length, and $ν$ is the kinematic viscosity

Dimensionless Parameters and Flow Regimes

• The Rayleigh number (Ra) is another dimensionless parameter used to characterize natural convection, defined as the product of the Grashof number and the Prandtl number (Pr), $Ra = Gr × Pr$
  • The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity and is defined as $Pr = ν/α$, where $α$ is the thermal diffusivity
• The flow regime in natural convection can be laminar or turbulent, depending on the value of the Rayleigh number
  • The critical Rayleigh number for the transition from laminar to turbulent flow depends on the geometry and boundary conditions (vertical plates, horizontal plates)

Natural Convection over Surfaces

Vertical Surfaces

• For vertical surfaces, the heat transfer coefficient in laminar natural convection can be determined using the empirical correlation: $Nu = C(Ra)^n$, where $Nu$ is the Nusselt number, $C$ and $n$ are constants that depend on the Rayleigh number range and the surface geometry
  • For laminar flow ($10⁴ < Ra < 10⁹$), the correlation is $Nu = 0.59(Ra)^{1/4}$
  • For turbulent flow ($10⁹ < Ra < 10¹³$), the correlation is $Nu = 0.10(Ra)^{1/3}$

Horizontal Surfaces

• For horizontal surfaces, the heat transfer coefficient depends on whether the surface is facing upward (heated surface facing upward or cooled surface facing downward) or downward (heated surface facing downward or cooled surface facing upward)
  • For the upper surface of a heated plate or the lower surface of a cooled plate, the correlation is $Nu = 0.54(Ra)^{1/4}$ for $10⁴ < Ra < 10⁷$ and $Nu = 0.15(Ra)^{1/3}$ for $10⁷ < Ra < 10¹¹$
  • For the lower surface of a heated plate or the upper surface of a cooled plate, the correlation is $Nu = 0.27(Ra)^{1/4}$ for $10⁵ < Ra < 10¹⁰$
• The Nusselt number is defined as $Nu = hL/k$, where $h$ is the heat transfer coefficient, $L$ is the characteristic length, and $k$ is the thermal conductivity of the fluid
• The heat transfer rate in natural convection can be calculated using Newton's law of cooling: $Q = hA(Ts - T∞)$, where $A$ is the surface area

Fluid Properties and Geometry Influence

Fluid Properties

• The fluid properties that affect natural convection include density, viscosity, thermal conductivity, specific heat capacity, and thermal expansion coefficient
• Fluids with high thermal conductivity (liquid metals) and low viscosity generally enhance natural convection heat transfer
• The thermal expansion coefficient determines the density variation with temperature and directly influences the buoyancy force
  • Fluids with higher thermal expansion coefficients (gases) exhibit stronger natural convection
• The Prandtl number of the fluid affects the relative thickness of the velocity and thermal boundary layers
  • Fluids with lower Prandtl numbers (liquid metals) have thicker thermal boundary layers compared to the velocity boundary layer, enhancing convective heat transfer

Geometry and Surface Characteristics

• The geometry of the surface and the surrounding environment influence the flow patterns and heat transfer in natural convection
  • The characteristic length in the Rayleigh number depends on the geometry (height for vertical plates, area-to-perimeter ratio for horizontal plates)
  • The presence of nearby surfaces or enclosures can restrict or enhance the fluid motion and affect the heat transfer rate
• Roughness of the surface can promote turbulence and increase the heat transfer coefficient in natural convection
  • Rough surfaces (sandpaper) enhance heat transfer compared to smooth surfaces (polished metal)

Mixed Convection vs Dominant Mode

Mixed Convection

• Mixed convection occurs when both natural convection and forced convection are present and significantly contribute to the heat transfer process
• The relative importance of natural convection and forced convection can be assessed using the Richardson number (Ri), defined as $Ri = Gr/Re²$, where $Gr$ is the Grashof number and $Re$ is the Reynolds number
  • When $Ri << 1$, forced convection dominates, and the effect of natural convection is negligible
  • When $Ri >> 1$, natural convection dominates, and the effect of forced convection is negligible
  • When $Ri ≈ 1$, both natural and forced convection are significant, and mixed convection occurs
• In mixed convection, the flow and heat transfer characteristics depend on the direction of the buoyancy force relative to the forced flow
  • Assisting flow occurs when the buoyancy force acts in the same direction as the forced flow, enhancing heat transfer (heated vertical plate with upward forced flow)
  • Opposing flow occurs when the buoyancy force acts in the opposite direction to the forced flow, potentially reducing heat transfer (heated vertical plate with downward forced flow)

Dominant Heat Transfer Mode

• The heat transfer coefficient in mixed convection can be estimated using correlations that account for both natural and forced convection effects, such as the Churchill-Usagi correlation: $Nu_m = (Nu_n^p + Nu_f^p)^{1/p}$, where $Nu_m$ is the mixed convection Nusselt number, $Nu_n$ is the natural convection Nusselt number, $Nu_f$ is the forced convection Nusselt number, and $p$ is an empirical exponent
• In some cases, such as in heat exchangers or electronic cooling applications, mixed convection can be intentionally exploited to enhance heat transfer performance
  • Designing heat sinks with fins that promote both natural and forced convection can improve cooling efficiency in electronic devices