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Natural Convection
Storyboard
Natural convection is triggered by gravity. Lower temperature zones, in which the mass has contracted and is therefore greater, tends to fall by displacing a mass of higher temperature which, when dilated, is less dense and is therefore lighter.
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Convection speed
Equation
The average velocity of a turbulent flow in convection can be modeled as a function of the lift force generated by the density variation due to heat, using the equation:
 $ v =\displaystyle\frac{ g }{ \eta }( \rho_b - \rho_m ) h ^2$ |
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Different viscosities
Description
Viscosity has a profound effect on the behavior of a fluid, as can be seen in the following three examples:
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Grashof number
Equation
The Grashof number describes the instability of a convection flow and is related to the Reynolds number for a velocity of the order of
| $ v =\displaystyle\frac{ g }{ \eta }( \rho_b - \rho_m ) h ^2$ |
Its expression is
| $ Gr =\displaystyle\frac{ \rho ^2 g \alpha }{ \eta ^2}( T_b - T_t ) h ^3$ |
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Mixing ratio of water vapor with air
Equation
The mixing ratio of water vapor with air is defined as the ratio of the masses of each component present in a volume:
$\displaystyle\frac{M_v}{M_a}=\displaystyle\frac{n_vM_{mol,v}}{n_aM_{mol,a}}=\displaystyle\frac{p_v}{p_a}\displaystyle\frac{M_{mol,v}}{M_{mol,a}}\sim 0.01$
Where $M_v$ and $M_a$ are the masses of water vapor and air respectively, $n_v$ and $n_a$ are the moles of water vapor and air, $M_{mol,v}$ and $M_{mol,a}$ are the molar masses of water vapor and air, $p_v$ and $p_a$ are the relative pressures of water vapor and air, and $r$ is the mixing ratio. Therefore, we have
| $ r =\displaystyle\frac{ M_v }{ M_a }$ |
In the specific case of water vapor in air, the mixing ratio is proportional to the relative pressures, which can be quantified using the vapor pressure of water $p_v\sim 1500 Pa$ and the air pressure $p_a\sim 10^5 Pa$. By applying the ideal gas equation and the definition of molar mass, it can be determined that the mixing ratio is approximately $0.01$. This means that the amount of water vapor compared to air is low under normal conditions.
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Reynold Number
Equation
The key criterion for determining whether a medium is laminar or turbulent is the Reynolds number, which compares the energy associated with inertia to that associated with viscosity. The former depends on the liquid density ($\rho_w$), maximum Speed ($v_{max}$), and the typical Dimension of the System ($R$), while the latter depends on the viscosity ($\eta$), defining it as:
| $ Re =\displaystyle\frac{ \rho R v }{ \eta }$ |
The inertia of a medium can be understood as proportional to the density of kinetic energy, given by:
$\displaystyle\frac{\rho_w}{2}v^2$
where the liquid density ($\rho_w$) and the mean Speed of Fluid ($v$) are.
If we consider the viscose force ($F_v$) as:
$F_v=S\eta\displaystyle\frac{v}{R}$
where the section or Area ($S$), the viscosity ($\eta$), the mean Speed of Fluid ($v$), and the typical Dimension of the System ($R$) are properties of the medium.
Let's recall that energy equals the viscose force ($F_v$) multiplied by the distance traveled ($l$). The density of energy lost due to viscosity will be equal to the force multiplied by the distance divided by the volume $S l$:
$\displaystyle\frac{F_vl}{Sl}=S\eta\displaystyle\frac{v}{R}\displaystyle\frac{l}{Sl}=\eta\displaystyle\frac{v}{R}$
Therefore, the relationship between the density of kinetic energy and the density of viscous energy is equal to a dimensionless number known as the number of Reynold ($Re$). If the number of Reynold ($Re$) is several orders of magnitude greater than one, inertia dominates over viscous force and the flow becomes turbulent. On the other hand, if the number of Reynold ($Re$) is small, viscous force dominates and the flow is laminar.
| $ Re =\displaystyle\frac{ \rho R v }{ \eta }$ |
The original article in which Osborne Reynolds introduces the number bearing his name is:
"An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels," authored by Osborne Reynolds, published in Philosophical Transactions of the Royal Society of London, Vol. 174, pp. 935-982 (1883).
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Turbulence generated by a Cigarette
Description
A cigarette has a glowing tip that heats the air in its surroundings. Additionally, the expelled smoke allows us to visualize the movement of the air. The heating leads to an expansion of the air, resulting in a decrease in density and, consequently, generates a lift force. As a result, the smoke starts to rise in a laminar fashion, forming the typical lines that are seen.
During this process, the gas begins to cool down, losing lift force, and certain regions start ascending more slowly, obstructing the upward movement of the air. This obstruction creates turbulence, and the same regions that ascend slower begin to rotate, forming part of the vortices observed in that area.
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