When you blow over a cup of hot coffee to cool it down, you are applying external forcing
condition to induce the flow of cool air over the hot fluid. This is known as forced convection.
But, even if you don’t blow over the cup and let it sit on the table, it will slowly,
but surely, cool down. In such situations, where there is no external forcing condition,
convection heat transfer still occurs due to the convection currents that exist within
the ambient air owing to the density gradients. This is referred to as natural convection
and is the focus of this lesson.
Natural convection occurs due to a buoyancy force that acts on a fluid having density
gradients. These density gradients are usually due to temperature differences. In the case
of natural convection, the flow velocities are generally smaller than those associated
with forced convection and therefore the corresponding convection heat transfer rates are also smaller.
As a result, applications relying on natural convection are typically larger in size compared
to their forced convection counter parts. Natural convection plays an important role
in many industrial applications such as power generation, HVAC systems, electronics cooling.
Moreover, it is the driving force of many natural processes such as ocean and atmospheric
currents. Buoyancy induced flow in a fluid is driven
by an external body force when density gradients arise due to temperature differences. In most
cases, this body force is gravitational. However, it can also be centrifugal force, for e.g.
in the case of rotating machinery, or it can be Coriolis force, as in the case of oceanic
and atmospheric currents. Density of fluids depends on temperature and it generally decreases
with increasing temperature. This is because as the temperature increases, the fluid expands
and its volume increases. In this course we will focus on natural convection in which
density gradient is due to a temperature variations and the body force is gravitational.
Now let’s explore these natural convection currents in detail using the following example.
Consider the conditions shown here, the fluid is enclosed by two large horizontal plates
where the lower plate is at a higher temperature than the upper plate. As a result, the density
decreases in the direction of the gravitational force. If the temperature difference is such
that the buoyancy forces can overcome the viscous forces, fluid circulation would begin.
The heavier fluid will descend, being warmed in the process, while the lighter fluid will
rise and cool down as it moves. On the other hand, if the temperature of the upper plate
is higher, then the fluid density will not decrease in the direction of gravity and the
flow pattern will not set up. In this case heat will be transferred from the upper plate
to the lower plate but it will be via conduction and radiation.
Natural convection can occur in many different types of settings. For example, we can have
unbounded natural convection in which the buoyant currents move in the absence of any
bounding surface. Two common examples are plume formation over an immersed heated object
and buoyant jet from a heated discharge in a cooler fluid. A plume is formed from the
fluid rising from a heated object submerged in a quiescent fluid. The hot fluid rises
due to buoyancy forces and entrains more fluid from the quiescent region. The plume formed
from a hot immersion water heater is a good example of this type of natural convection
flow. The width of the plume increases with distance from the heated object, however,
the plume itself eventually dissipates due to viscous effects and reduction in the buoyancy
forces. Unlike plumes, buoyant jets have some initial velocity as shown. They arise when
a heated fluid is being discharged as a jet into a quiescent medium of lower temperature.
Due to the buoyant forces, the jet gains an extra vertical velocity component. A good
example of this type of natural convection flow is the flow setup arising from the discharge
of warm water from the condenser of a power plant into a reservoir of cooler water.
Another type of natural convection flow is when the flow is bounded by a surface. A canonical
example of such flows is the boundary layer development on a heated vertical plate. The
plate is fully immersed in an quiescent fluid having a lower free stream temperature than
the plate. As a result, the fluid close to the plate is warmer compared to the fluid
that is further away from the plate. This makes the fluid close to the plate less dense
and the buoyancy forces induce a free convection boundary layer in which the heated fluid rises
vertically, entraining fluid from the quiescent region. The resulting boundary layer is different
from the one we get in the case of forced convection. This is because, the velocity
is zero as y approaches infinity, as well as where 𝑦 is equal to zero. A natural
convection boundary layer will also develop if the plate is immersed in a hotter fluid.
However, the fluid motion in this case will be downwards as shown here.
Now that we have some understanding of natural convection, lets quickly look at its governing
equations. Consider the laminar boundary layer flow shown here. We assume the flow to be
two dimensional, steady with constant property conditions, and that the boundary layer approximations
are valid. With gravity acting in the negative x-direction, the x-momentum equation can be
written as this…. dp_infinity by dx is the free stream pressure gradient outside the
boundary layer, where u is equal to zero. Outside the boundary layer this equation reduces
to this expression. If we substitute this in the x-momentum equation, then we get the
following….here delta rho is the difference between the freestream and the local density.
This term represents the buoyancy force and if the density variations are only due to
temperature variations, then this term can be related to the volumetric thermal expansion
coefficient using this expression. The volumetric thermal expansion coefficient is a measure
of the amount by which the density changes in response to a change in temperature at
constant pressure. It can be approximated using the following equation. This simplification
is known as the Boussinesq approximation and substituting this in the momentum equation
we obtain the following equation for the x-momentum. The expansion coefficient beta can be determined
for ideal gases using the following expression. In case of liquids and non-ideal gases, beta
is obtained from property tables. In conclusion, these are the three governing
equations for natural convection flows. Since the buoyancy effects are confined to the momentum
equation, the mass and energy conservation equations are similar to those describing
forced convection. However, unlike forced convection, the solution to the momentum equation
is now dependent on the temperature and thus the solution of the energy equation. And that
is why this system of equations is strongly coupled and must be solved simultaneously.
With that we have come to the end of this lesson.