Draft:Scale Analysis of External Natural Convection

Scale Analysis of External Natural Convection

External natural convection is a heat transfer process where fluid motion is driven by buoyancy forces resulting from temperature differences within the fluid. This phenomenon occurs when a fluid near a heated surface becomes less dense and rises, while cooler fluid moves in to replace it, creating a continuous circulation pattern. In contrast to forced convection, external natural convection does not depend on mechanical devices like fans or pumps to drive fluid motion.

Laminar Boundary Layer equations

Diagram showing thermal ($\delta _{T}$) and hydrodynamic ($\delta$) boundary layers along a heated vertical wall. It illustrates the temperature (𝑇) and velocity (𝑢) profiles of the fluid, highlighting the distinct scales for thermal and hydrodynamic effects

The laminar boundary layer is a key concept in fluid dynamics and heat transfer, describing the region of fluid flow near a solid surface where viscous forces dominate. This boundary layer is crucial in understanding heat transfer from a heated wall to the surrounding fluid, especially during laminar flow, where the fluid moves in smooth, parallel layers, free of turbulence.

The complete Navier–Stokes equations for the steady constant-property two-dimensional flow for the buoyancy driven flow system:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ (Continuity Equation)

$\rho \left(u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=-{\frac {\partial P}{\partial x}}+\mu \nabla ^{2}u$

(Momentum equation in x-direction)

$\rho \left(u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)=-{\frac {\partial P}{\partial y}}+\mu \nabla ^{2}v-\rho g$

(Momentum equation in y-direction)

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha \nabla ^{2}T$ (Energy equation)

where:

• $u$ and $v$ are the velocity components in the $x$- and $y$-directions, respectively,
• $\mu$ is the dynamic viscosity,
• $P$ is the pressure,
• $g$ is the acceleration due to gravity,
• $\rho$ is the density,
• $T$ is the temperature at any given point in the fluid, and
• $\alpha$ is the thermal diffusivity

These equations, with appropriate boundary conditions, allow for the determination of the velocity and temperature fields within the boundary layer.

The body force term, −ρg, in the vertical momentum equation can be simplified by considering the boundary layer region, where $x\sim \delta _{T}$, $y\sim H$, and $\delta _{T}\ll H$, where $H$ is the characteristic length, and $\delta _{T}$ is the thermal boundary layer thickness.

As a result, only the ${\frac {\partial ^{2}}{\partial x^{2}}}$ term remains relevant in the $\nabla ^{2}$ (del operator). The transversal(horizontal) momentum equation reduces to indicate that the pressure in the boundary layer only depends on the longitudinal(vertical) position.

The simplified equation for pressure becomes:

${\frac {\partial P}{\partial y}}={\frac {dP}{dy}}={\frac {dP_{\infty }}{dy}}$

The boundary layer equations for momentum and energy are as follows:

Momentum equation:

$\rho \left(u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)=-{\frac {dP_{\infty }}{dy}}+\mu {\frac {\partial ^{2}v}{\partial x^{2}}}-\rho g$

Energy equation:

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha {\frac {\partial ^{2}T}{\partial x^{2}}}$

Additionally, by considering that the pressure gradient ${\frac {dP_{\infty }}{dy}}$ is determined by the hydrostatic pressure distribution in the reservoir fluid of density $\rho _{\infty }$, with ${\frac {dP_{\infty }}{dy}}=-\rho _{\infty }g$, the momentum equation simplifies to,

$\rho \left(u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)=\mu {\frac {\partial ^{2}v}{\partial x^{2}}}+(\rho _{\infty }-\rho )g$

Boussinesq Approximation

In many practical cases, especially in natural convection, the Boussinesq approximation is applied to simplify the analysis of buoyancy effects in the momentum equation. This approximation assumes that variations in density are negligible except in the buoyancy term . The linearized form of density variation with temperature is:

$\rho \approx \rho _{\infty }\left[1-\beta \left(T-T_{\infty }\right)+\cdots \right]$

where β is the volume expansion coefficient at constant pressure,

$\beta =-{\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial T}}\right)_{P}$

This allows the momentum equation to be written as:

$u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=\nu {\frac {\partial ^{2}v}{\partial x^{2}}}+g\beta (T_{0}-T_{\infty })$

where, $\nu$ is the kinematic viscosity

This equation captures the balance between inertial forces, viscous effects, and buoyancy forces due to temperature differences.

Boundary Conditions

For the analysis of laminar boundary layers, appropriate boundary conditions must be applied.

Typically, for an impermeable, no-slip, isothermal wall, the velocity and temperature at the wall are given by:

$u=v=0\quad {\text{and}}\quad T=T_{0}\quad {\text{at}}\quad x=0$

At large distances from the wall, the flow reaches an undisturbed state, where the velocity and temperature approach those of the free stream:

$v=0\quad {\text{and}}\quad T=T_{\infty }\quad {\text{as}}\quad x\to \infty$

Scale Analysis

In external natural convection, scale analysis is performed to understand the balance between various forces—such as inertia, buoyancy, and friction—and the energy transfer mechanisms, including convection and conduction.

Scale Analysis of boundary layer equations

• The scale analysis domain can be described as $x\sim \delta _{T}$ and $y\sim H$, for thermal boundary layer region
• In this region, the heating effect influences fluid behavior, causing noticeable changes.
• At steady-state, the heat conducted from the wall into the fluid is transported upward by the fluid, forming an enthalpy stream.

The equation below represents a balance between longitudinal convection and transverse conduction:

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha {\frac {\partial ^{2}T}{\partial x^{2}}}$

This can be approximated as:

$u{\frac {\Delta T}{\delta _{T}}},\,v{\frac {\Delta T}{H}}\sim \alpha {\frac {\Delta T}{\delta _{T}^{2}}}$

Here the left-hand side represents the convective heat transfer components in the x and y directions, and the right-hand side represents the conductive heat transfer in the x direction. Also, $\Delta T=T_{0}-T_{\infty }$ is the scale of the variable $T-T_{\infty }$.

From the principle of mass conservation in the same layer:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ (Continuity Equation)

The velocities are related by the approximation:

${\frac {u}{\delta _{T}}}\sim {\frac {v}{H}}$

This indicates that the two convection terms in equation are of the order of ${\frac {v\Delta T}{H}}$.

The energy balance involves two characteristic scales:

${\frac {v\Delta T}{H}}\sim \alpha {\frac {\Delta T}{\delta _{T}^{2}}}$

which results in:

$v\sim {\frac {\alpha H}{\delta _{T}^{2}}}$

Determination of Thermal Boundary Layer Thickness

The thermal boundary layer thickness, $\delta _{T}$, remains undetermined. To resolve this, we examine the vertical momentum equation, considering the interaction among three forces:

$u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=\nu {\frac {\partial ^{2}v}{\partial x^{2}}}+g\beta (T-T_{\infty })$

These forces can be approximated as follows:

• Inertia: ${\frac {uv}{\delta _{T}}},\,{\frac {v^{2}}{H}}$

• Friction: ${\frac {\nu v}{\delta _{T}^{2}}}$
• Buoyancy: $g\beta \Delta T$

Dividing the terms by $g\beta \Delta T$ and applying $v\sim {\frac {\alpha H}{\delta _{T}^{2}}}$, the resulting expressions are:

• Inertia: $\left({\frac {H}{\delta _{T}}}\right)^{4}{\text{Ra}}_{H}^{-1}{\text{Pr}}^{-1}$

• Friction: $\left({\frac {H}{\delta _{T}}}\right)^{4}{\text{Ra}}_{H}^{-1}$
• Buoyancy: $1$

where:

• $H$ is characteristic length,
• $\delta _{T}$ is the thermal boundary layer thickness.
• $Ra_{H}$ is the Rayleigh number,
• $Pr$ is the Prandtl number.

For High-Pr Fluids

For fluids with a high Prandtl number (Pr >> 1), the thermal boundary layer is generally thin compared to the velocity boundary layer. This difference arises due to the fluid's low thermal diffusivity relative to its momentum diffusivity. This characteristic influences specific scaling relations within the thermal boundary layer.

In these cases, the balance between frictional and buoyant forces yields a thermal boundary layer thickness, $\delta _{T}$, which scales as:

$\delta _{T}\sim H\,\mathrm {Ra_{H}} ^{-1/4}$

The characteristic velocity ($v$) within the boundary layer is determined by the thermal diffusivity ($\alpha$) and the thermal boundary layer thickness, yielding:

$v\sim {\frac {\alpha H}{\delta _{T}^{2}}}\sim {\frac {\alpha }{H}}\,\mathrm {Ra_{H}} ^{1/2}$

The convective heat transfer coefficient ($h$) is inversely proportional to $\delta _{T}$. Consequently, the Nusselt number ($\mathrm {Nu}$), defined as $\mathrm {Nu} ={\frac {hH}{k}}$, where $k$ is the thermal conductivity, scales as:

$\mathrm {Nu} \sim \mathrm {Ra_{H}} ^{1/4}$

These scaling relations for the thermal boundary layer thickness, velocity, and Nusselt number have been validated by experimental studies, confirming their applicability in high-Prandtl number fluid convection.

Momentum Conservation in Viscous Layer

The movement of the fluid is not confined to a thermal boundary layer of thickness $\delta _{T}$. The heated $\delta _{T}$ layer can viscously entrain an adjacent layer of unheated fluid, with the outer layer having a thickness $\delta$, where $\delta$ >> $\delta _{T}$.

In this case, the momentum conservation within the boundary layer of thickness $\delta$ is considered. Since the outer layer of fluid remains isothermal, there is no influence from buoyancy forces. The $\delta$ layer is driven by viscous forces from the thinner $\delta _{T}$ layer, while being opposed by its own inertia. This results in a balance between inertia and friction in the layer with thickness $\delta$:

$v{\frac {v}{H}}\sim \nu {\frac {v}{\delta ^{2}}}$

The vertical velocity scale $v$ is determined by the driving mechanism, specifically the $\delta _{T}$ layer. By eliminating $v$ from the following two equations, the system can be simplified:

$v{\frac {v}{H}}\sim \nu {\frac {v}{\delta ^{2}}}$ and $v\sim {\frac {\alpha }{H}}\,\mathrm {Ra_{H}} ^{1/2}$

Thus, characteristic length scale for thermal convection can be expressed as:

$\delta \sim H\cdot Ra_{H}^{-1/4}\cdot Pr^{1/2}$

where, $\delta$ is the characteristic length scale.

This relationship indicates how the characteristic length scale is influenced by the Rayleigh and Prandtl numbers in a convection system.

The ratio of the thickness of the outer layer to the inner layer is given by:${\frac {\delta }{\delta _{T}}}\sim {\text{Pr}}^{1/2}>1$

This relationship indicates that the characteristic length scale is greater than the thermal boundary layer thickness when the square root of the Prandtl number is greater than one.

Flow Characteristics of the Boundary Layer with two length scales on a heated vertical plate for High Prandtl fluids

In conclusion, the higher the Prandtl number, the thicker the layer of unheated fluid driven upward by the heated layer. This fundamental difference between forced convection boundary layers and natural convection boundary layers is illustrated by the velocity profile, which is described by two length scales ($\delta _{T}$ and $\delta$) instead of a single length scale ($\delta$) as in forced convection.