Have you ever noticed how blowing on a hot beverage, say a cup of coffee, cools it down
much faster than if you just left it on the table?
What you are unconsciously doing is employing forced convection to cool your coffee faster.
Let’s take a deeper look at what forced convection is all about.
Convection is the mode of heat transfer where energy is transferred to or from a cold or
hot body by fluid motion.
Back to the coffee example, the air you blow extracts heat from the coffee and convects
it away from the mug.
Based on how the fluid flow is created, convection can be classified into Natural or Forced convection.
In forced convection, the fluid flow is generated by external means i.e. by a fan or because
of an object’s motion.
Cooling of electronic components, localized space heaters are a few examples of forced
convection.
In natural convection, a fluid flow is created because of buoyancy forces, caused by temperature
gradients, due to the energy transfer between the body and the surroundings.
Cooling of a hot coffee left on a table, heating a room through heaters are examples of heat
transfer via natural convection.
In this course, our focus will be on forced convection.
Consider a hot plate at a temperature of Tw.
If the incoming fluid is at temperature T∞ and the heat transfer coefficient is ‘h’,
according to Newton’s law of cooling, the convection heat flux can be calculated using
the following equation.
At the surface of the plate, there is no fluid motion due to the no-slip condition and therefore,
there will be only conduction from the plate into the fluid, which is calculated using
the Fourier’s law as shown here.
Neglecting radiation, these two fluxes must be equal at the wall and as a result, we get
the following flux balance.
Here, q’’ is local heat flux and therefore, ‘h’ is generally referred to as a local
heat transfer coefficient.
In general, heat transfer is a function of space and hence the heat transfer coefficient
will vary over the surface.
For engineering estimates, the averaged value of heat transfer coefficient for a surface
can be defined as shown here.
Subsequently, the total heat transfer rate is defined by the following relation.
Note that ‘h’ is always a positive number and therefore the sign of ‘q’ will depend
on the temperature difference Tw and T∞.
The essence of any convection heat transfer problem is determining the local and averaged
heat transfer coefficients, which can subsequently be used to calculate the local flux or the
total heat transfer rate.
Let’s go back to the heat flux balance at the wall.
If we now non-dimensionalize this equation using reference length (L) and reference temperature
(T∞), the heat flux balance can be rewritten as shown here.
Note that Tw, T∞, h and Kf are assumed to be constant.
The circled term here is generally referred to as the Nusselt number (Nu).
It is a non-dimensional measure of convective heat transfer relative to the heat transfer
that would occur by conduction into the fluid.
If the heat transfer coefficient used to define the Nusselt number is local, then the Nusselt
number is also local.
To obtain the average Nusselt number, one can instead use the average heat transfer
coefficient.
Experiments have shown that the Nusselt number correlates with the Reynolds number, which
represents the ratio of inertial to viscous forces in a fluid flow, and the Prandtl Number
which represents the ratio of viscous diffusion of momentum to thermal diffusion in the fluid.
To determine the heat transfer coefficient, there are two ways.
Through a series of experiments where key geometric and thermal parameters are varied,
and fluxes are measured carefully.
The second way is through simulations wherein the governing equations are solved using CFD techniques
for specific geometries and flow/thermal boundary conditions.
This is akin to doing an “experiment” except now we are simulating the flow field
numerically.
However, it is impossible to perform experiments or simulations for every conceivable geometry
and flow condition.
To address this issue, engineers make use of similarity methods and dimensional groups
to develop correlations for common geometries and flow configurations.
There are two types of similarities: Geometrics and Dynamic.
Geometric similarity means that two bodies are geometrically similar if they are perfectly
scaled versions of each other.
Consider the following two objects.
If they are geometrically similar, then it is required that the ratio of corresponding
lengths of the two objects i.e L1/L2, W1/W2 etc. are the same.
Dynamic similarity requires that, for geometrically similar bodies, the flow patterns are also
similar i.e. the velocities and velocity gradients, fluid forces, and streamlines all scale with
the geometry.
For heat transfer problems, it is also required that the temperatures, temperature gradients,
and heat fluxes scale with the geometry.
For the case of forced convection, two geometrically similar systems are dynamically similar if
their Nusselt, Reynolds, and Prandtl Numbers are the same.
Is it possible to formulate a relationship between these dimensionless numbers?
In order to determine that, multiple experiments can be setup at varying Reynolds and Prandtl
numbers, to calculate the corresponding Nusselt numbers.
And as it turns out, researchers have found that for a fixed Prandtl number i.e. for a
given fluid, an empirical correlation of the following form can be written to calculate
the global Nusselt number.
On a log-log scale, this correlation looks like a straight line as shown here.
The values of ‘C’, ‘m’, and ‘n’ are dependent on the geometry of the object
and the flow type, and are oftentimes independent of the fluid itself.
That was a quick brief on the basics of forced convection.