Lecture 30, March 19, 2004 • Quest Evaluations Monday please fill them out • Mr. Caners apologizes (to all 11 of you), the

quizzes will be available Monday • ‘An Album of Fluid Motion’, Milton Van Dyke ($20

from amazon.ca) Natural or Free Convection (Ch9) Why do we say that heat rises? Recall Fourier’s Law, which tells us that conduction heat transfer occurs against the gradient of temperature.

Is the up direction always cooler? The flight of a hot air balloon is often referred to as ‘lighter than air flight’. The balloon flies because the gas inside the balloon is lighter than cooler air around the balloon. This gives us insight into why we say that heat rises. Hot air happens to be less dense than cooler air, and hence is subject to buoyancy forces exactly as is a ship in water. This is not true of all fluids. Water near its triple point actually gets less dense as the temperature

decreases, and this is why lakes freeze from the top down and are able to sustain life through the winter. Free convection comes about due to fluid motion caused by temperature changes in the fluid which cause a density change and hence buoyancy forces. The resulting velocity fields are usually of much lower magnitude than the velocity fields imposed in forced convection, but the heat transfer can be significantly enhanced compared to the case of no fluid motion (conduction only). Applications of Natural Convection • Cooling of electronic components. • Heat transfer from refrigerator coils • Heat transfer from power lines • Power Generation (Solar Chimneys) • Window Gaps • We can see natural convection when we look at a

road surface in the summer sun. Imagine a vertical plate which is heated No slip zero velocity on the plate surface Buoyancy forces case heated fluid to rise (assuming density decreases with temperature) The pressure remains atmospheric cooler air mist come in and replace the heat air which has ‘risen’

What would happen if the plate were cooler than the air? This still will occur (under the right conditions), but the flow will be from the top of the plate towards the bottom of the plate. What drives natural (free) convection? Bouyancy forces due to changes in density. These density changes can be described by the volumetric thermal expansion coefficient, beta,

For small changes, we can look at differences rather than at gradients.

For an ideal gas, it is very simple to determine beta,

Where T must be expressed in Kelvin. We now want to determine how the fluid is going to react to changes in density, or to buoyancy forces. We are clearly interested in conservation of

momentum for the fluid, or the Navier-Stokes Equations. All we have to do is add the buoyancy force to this equation and we are in business. The big assumption that is used to simplify matters greatly, is that density changes are only important in the buoyancy force term, and the density can otherwise be considered constant. This is called the Boussinesq approximation.

Now, the momentum equation is coupled to the energy equation and they must be solved together. This is quite different from our previous work where we solved the momentum equation (Blasius equation) and then used the velocity field to solve the temperature field after the fact. Note: The Navier-Stokes equations are non-linear and there is always the possibility of multiple solutions. This can be a real challenge in natural convection problems.