Draft:State analysis of internal natural convection

Scale Analysis of Internal natural convection

Internal natural convection is a process of fluid movement that occurs due to temperature differences within the fluid. When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks. This movement creates a circulation pattern, allowing for the transfer of heat within the fluid.

This phenomenon holds considerable importance across a spectrum of applications, encompassing heating systems, cooling mechanisms, and environmental processes. A comprehensive understanding of internal natural convection is imperative for engineers and scientists, as it enables the analysis and prediction of fluid behaviour under diverse conditions.

Introduction

Enclosures filled with fluid are fundamental components in various engineering and geophysical systems. The flow within the internal airspace of a double-pane window system differs significantly from the external natural convection boundary layer.

Natural convection phenomena in enclosures vary based on the geometry and orientation of the enclosure. Broadly, these phenomena are categorized into two primary classes:

•    Side-Heated Enclosures: This category includes applications such as solar collectors, double-wall insulations, and air circulation systems within buildings
•    Bottom-Heated Enclosures: These systems are commonly found in horizontally oriented thermal insulations, such as heat transfer through flat-roof attic spaces.

                 

                                 

Consider a two-dimensional rectangular enclosure with a height H and a horizontal length 𝐿. This enclosure is filled with a Newtonian fluid, such as air or water. In this scenario, we examine the transient behaviour of the fluid inside the cavity as the sidewalls are subjected to instantaneous temperature changes. The left and right walls are heated and cooled to temperatures + Δ 𝑇 / 2 and − Δ 𝑇 / 2 respectively, while the top and bottom walls (at 𝑦 = 0 and 𝑦 = 𝐻) remain insulated throughout the experiment. Initially, the fluid is in a state of thermal equilibrium (isothermal, 𝑇 = 0) and is motionless, with no fluid velocity ( u = 0 and 𝑣 = 0).

Governing Equations

The equations governing the internal natural convection can be derived from conservation of mass, momentum and energy equations. Here the assumptions taken to derive are that fluid is modelled to be as Boussinesq incompressible in other words the fluid density 𝜌 is assumed to be constant throughout the enclosure, except in the body force term of the vertical (y) momentum equation. In that term, the density is replaced by 𝜌 [ 1 − 𝛽 ( 𝑇 − 𝑇 ₀ )], where 𝛽 is the coefficient of thermal expansion, and 𝑇 ₀ is the reference temperature.

${\begin{aligned}{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}&=0\\{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial x}}+v\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)\\{\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial y}}+v\left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)-g\left[1-\beta \left(T-T_{0}\right)\right]\\{\frac {\partial T}{\partial t}}+u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}&=\alpha \left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}\right)\end{aligned}}$

Scale Analysis

Instead of solving the governing equations numerically, we utilize scale analysis to theoretically predict the flow and heat transfer patterns within the enclosure. Scale analysis helps in determining the dominant factors and characteristic behaviour of the system, providing insight into the types of flow regimes and thermal interactions that may develop.

Thermal Boundary layer

Immediately after 𝑡 = 0, the fluid adjacent to each sidewall is nearly motionless, with 𝑢 → 0 and 𝑣 → 0. At this moment, the thermal boundary layer thickness (𝛿 _T) is much smaller than the height H of the system (𝛿 _T<<<H). As a result, near the sidewall, the energy equation reflects a balance between thermal inertia and heat conduction normal to the wall.

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

In the governing equations, we assume that the fluid velocities 𝑢 and 𝑣 are initially zero, as the fluid is at rest. This simplification is valid because, near 𝑡 = 0 ⁺, the thermal boundary layer thickness 𝛿 _T is much smaller than the height of the enclosure. In other words, 𝑦 ∼ 𝐻 and 𝑥 ∼ 𝛿 _T, where 𝐻 is the height of the enclosure, leading to the conclusion that temperature variations along the vertical axis are much more gradual than along the horizontal axis.

From here, we get

$\delta _{T}\sim (\alpha t)^{1/2}$

This means that boundary layer thickness rises along with the heated wall.

Velocity Scale

To calculate this, the first step is to eliminate 𝑃 from the equations. To do this, we first take the partial derivative of equation with respect to 𝑦, and then the partial derivative of equation with respect to 𝑥. By subtracting these two resulting equations, we effectively eliminate the pressure term, simplifying the system for further analysis.

${\begin{aligned}{\frac {\partial }{\partial x}}&\left({\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)-{\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)\\&=v\left[{\frac {\partial }{\partial x}}\left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)-{\frac {\partial }{\partial y}}\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)\right]+g\beta {\frac {\partial T}{\partial x}}\end{aligned}}$

The resulting equation consists of three fundamental groups of terms: inertia terms on the left-hand side, along with four viscous diffusion terms and a buoyancy term on the right-hand side. It can be easily demonstrated that three dominant terms emerge from each of these fundamental groups.

${\begin{array}{ccc}{\text{ Inertia }}&{\text{ Friction }}&{\text{ Buoyancy }}\\{\frac {\partial ^{2}v}{\partial x\partial t}},&v{\frac {\partial ^{3}v}{\partial x^{3}}},&g\beta {\frac {\partial T}{\partial x}}\end{array}}$

Now performing the scale analysis, we get

${\frac {v}{\delta _{T}t}},\quad v{\frac {v}{\delta _{T}^{3}}}\sim {\frac {g\beta \Delta T}{\delta _{T}}}$

The driving force in the natural convection process within the enclosure is the buoyancy effect. A critical aspect of the analysis is determining whether this buoyancy effect is counteracted primarily by viscous friction or by inertial forces.

To simplify it more multiply by (𝛿 _T)³ on each side and doing a little bit of calculation we get-

${\begin{aligned}&{\text{ Inertia Friction Buoyancy }}\\&{\frac {1}{\operatorname {Pr} }},\quad 1\quad \sim \quad {\frac {g\beta \Delta T\delta _{T}^{2}}{vv}}\end{aligned}}$

Thus, for fluids with a Prandtl number of approximately 1 or higher (Pr>=1), the appropriate momentum balance immediately after 𝑡 = 0 ⁺ is between the buoyancy force and frictional forces.

$1\sim {\frac {g\beta \Delta T\delta _{T}^{2}}{vv}}$

We conclude that the initial vertical velocity scale is,

$v\sim {\frac {g\beta \Delta T\alpha t}{v}}$

This velocity scale is applicable to fluids such as water and oils (Pr >1 ) and is only partially applicable to gases (Pr ~ 1).

Graphical Analysis of δ_v and δ_T

In the analysis of natural convection within enclosures, the energy equation reveals that the heat conducted from the sidewall into the fluid layer 𝛿_T  is not only used to thicken the layer but is also carried away by the fluid as it rises with velocity 𝑣. This leads to a competition among three distinct effects in the energy equation:

${\begin{array}{lll}{\text{ Inertia }}&{\text{ Convection }}&{\text{ Conduction }}\\{\frac {\Delta T}{t}},&v{\frac {\Delta T}{H}}&\sim \alpha {\frac {\Delta T}{\delta _{T}^{2}}}\end{array}}$

As time 𝑡 increases, the convection effect becomes more significant, following the relationship 𝑣 ∼ 𝑡. Eventually, at a specific time 𝑡_f, the energy equation reaches a balance between the heat conducted from the wall and the enthalpy carried away by the rising buoyant layer.

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

From here, we get

$t_{f}\sim \left({\frac {vH}{g\beta \Delta T\alpha }}\right)^{1/2}$

and using previous results , we get ,

$\delta _{T,f}\sim \left(\alpha t_{f}\right)^{1/2}\sim H\mathrm {Ra} _{H}^{-1/4}$

where Ra_H is the Rayleigh number based on the enclosure height.

$\mathrm {Ra} _{H}={\frac {g\beta \Delta TH^{3}}{\alpha \nu }}$

In addition to thermal layers with thickness δ_T,f ​ , the sidewalls develop viscous (velocity) wall jets. The thickness of these jets, δv, is determined from the momentum balance for the region where the thickness 𝑥 ∼ 𝛿_v lies outside the thermal layer. In this region, the buoyancy effect is minor, resulting in a balance between inertia and viscous diffusion.

${\frac {v}{\delta _{v}t}}\sim v{\frac {v}{\delta _{v}^{3}}}$

So from here we have,

$\delta _{v}\sim (vt)^{1/2}\sim \operatorname {Pr} ^{1/2}\delta _{T}$

In the steady state, when 𝑡 > 𝑡_f, the fluid near each sidewall exhibits a two-layer structure:

·      A thermal boundary layer of thickness δ_T,f

·      A thicker wall jet represented as 𝛿 𝑣_f ∼ 𝑃r^1/2 δ_T,f

• 
  Development of two-layer structure near the warm wall.

References-

[1] Bejan, Adrian. Convection Heat Transfer. 4th ed., Wiley, 2013. ISBN: 9780470900376.