A while ago we looked at some basics in Heat Transfer Basics – Part Zero.
Equations aren’t popular but a few were included.
As a recap, there are three main mechanisms of heat transfer:
In the climate system, conduction is generally negligible because gases and liquids like water don’t conduct heat well at all. (See note 2).
Convection is the transfer of heat by bulk motion of a fluid. Motion of fluids is very complex, which makes convection a difficult subject.
If the motion of the fluid arises from an external agent, for instance, a fan, a blower, the wind, or the motion of a heated object itself, which imparts the pressure to drive the flow, the process is termed forced convection.
If, on the other hand, no such externally induced flow exists and the flow arises “naturally” from the effect of a density difference, resulting from a temperature or concentration difference in a body force field such as gravity, the process is termed natural convection. The density difference gives rise to buoyancy forces due to which the flow is generated..
The main difference between natural and forced convection lies in the mechanism by which flow is generated.
From Heat Transfer Handbook: Volume 1, by Bejan & Kraus (2003).
The Boundary Layer
The first key to understanding heat transfer by convection is the boundary layer. A typical example is a fluid (e.g. air, water) forced over a flat plate:
This first graphic shows the velocity of the fluid. The parameter u∞ is the velocity (u) at infinity (∞) – or in layman’s terms, the velocity of the fluid “a long way” from the surface of the plate.
Another way to think about u∞ – it is the free flowing fluid velocity before the fluid comes into contact with the plate.
Take a look at the velocity profile:
At the plate the velocity is zero. This is because the fluid particles make contact with the surface. In the “next layer” the particles are slowed up by the boundary layer particles. As you go further and further out this effect of the stationary plate is more and more reduced, until finally there is no slowing down of the fluid.
The thick black curve, δ, is the boundary layer thickness. In practice this is usually taken to be the point where the velocity is 99% of its free flowing value. You can see that just at the point where the fluid starts to flow over the plate – the boundary layer is zero. Then the plate starts to slow the fluid down and so progressively the boundary layer thickens.
Here is the resulting temperature profile:
In this graphic T∞ is the temperature of the “free flowing fluid” and Ts is the temperature of the flat plat which (in this case) is higher than the free flowing fluid temperature. Therefore, heat will transfer from the plate to the fluid.
The thermal boundary layer, δt, is defined in a similar way to the velocity boundary layer, but using temperature instead.
How does heat transfer from the plate to the fluid? At the surface the velocity of the fluid is zero and so there is no fluid motion.
At the surface, energy transfer only takes place by conduction (note 1).
In some cases we also expect to see mass transfer – for example, air over a water surface where water evaporates and water vapor gets carried away. (But not with air over a steel plate).
So a concentration boundary layer develops.
Newton’s Law of Cooling
Many people have come across this equation:
q” = h(Ts – T∞)
where q” = heat flux in W/m², h is the convection coefficient, and the two temperatures were defined above
The problem is determining the value of h.
It depends on a number of fluid properties:
- thermal conductivity
- specific heat capacity
But also on:
- surface geometry
- flow conditions
The earlier examples showed laminar flow. However, turbulent flow often develops:
Flow in the turbulent region is chaotic and characterized by random, three-dimensional motion of relatively large parcels of fluid.
Check out this very short video showing the transition from laminar to turbulent flow.
What determines whether flow is laminar or turbulent and how does flow become turbulent?
The transition from laminar to turbulent flow is ultimately due to triggering mechanisms, such as the interaction of unsteady flow structures that develop naturally within the fluid or small disturbances that exist within many typical boundary layers. These disturbances may originate from fluctuations in the free stream, or they may be induced by surface roughness or minute surface vibrations
from Incropera & DeWitt (2007).
Imagine treacle (=molasses) flowing over a plate. It’s hard to picture the flow becoming turbulent. That’s because treacle is very viscous. Viscosity is a measure of how much resistance there is to different speeds within the fluid – how much “internal resistance”.
Now picture water moving very slowly over a plate. Again it’s hard to picture the flow becoming turbulent. The reason in this case is because inertial forces are low. Inertial force is the force applied on other parts of the fluid by virtue of the fluid motion.
The higher the inertial forces the more likely fluid flow is to become turbulent. The higher the viscosity of the fluid the less likely the fluid flow is to become turbulent – because this viscosity damps out the random motion.
The ratio between the two is the important parameter. This is known as the Reynolds number.
Re = ρu∞x / μ
where ρ = density, u∞ = free stream velocity, x is the distance from the leading edge of the surface and μ = dynamic viscosity
Once Re goes above around 5 x 105 (500,000) flow becomes turbulent.
For air at 15°C and sea level, ρ=1.2kg/m³ and μ=1.8 x 10-5 kg/m.s
Solving this equation for these conditions, gives a threshold value of u∞x > 7.5 for turbulence.. This means that if the wind speed (in m/s) x the length of surface over which the wind flows (in m) is greater than 7.5 we will get turbulent flow.
For example, a slow wind speed of 1 m/s (2.2 miles / hour) over 7.5 meters of surface will produce turbulent flow. When you consider the wind blowing over many miles of open ocean you can see that the air flow will almost always be turbulent.
The great physicist and Nobel Laureate Richard Feynman called turbulence the most important unsolved problem of classical physics.
In a nutshell, it’s a little tricky. So how do we determine convection coefficients?
Empirical Measurements & Dimensionless Ratios
Calculation of the convection heat transfer coefficient, h, in the equation we saw earlier can only be done empirically. This means measurement.
However, there are a whole suite of similarity parameters which allow results from one situation to be used in “similar circumstances”.
It’s not an easy subject to understand “intuitively” because the demonstration of these similarity parameters (e.g., Reynolds, Prandtl, Nusselt and Sherwood numbers) relies upon first seeing the differential equations governing fluid flow and heat & mass transfer – and then the transformation of these equations into a dimensionless form.
As the simplest example, the Reynolds number tells us when flow becomes turbulent regardless of whether we are considering air, water or treacle.
And a result for one geometry can be re-used in a different scenario with similar geometries.
Therefore, many tables and standard empirical equations exist for standard geometries – e.g. fluid flow over cylinders, banks of pipes.
Here are some results for air flow over a flat isothermal plate (isothermal = all at the same temperature) – calculated using empirically-derived equations:
Click for a larger view
The 1st graph shows that the critical Reynolds number of 5×105 is reached at 1.3m. The 2nd graph shows how the boundary layer grows under first laminar flow, then second under turbulent flow – see how it jumps up as turbulent flow starts. The 4th graph shows the local convection coefficient as a function of distance from the leading edge – as well as the average value across the 2m of flat plate.
Not much of a conclusion yet, but this article is already long enough. In the next article we will look at the experimental results of heat transfer from the ocean to the atmosphere.
Note 1 – Heat transfer by radiation might also take place depending on the materials in question.
Note 2 – Of course, as explained in the detailed section on convection, heat cannot be transferred across a boundary between a surface and a fluid by convection. Conduction is therefore important at the boundary between the earth’s surface and atmosphere.