Duhamel's Theorem, part 2

As an additional example of Duhamel’s theorem that we talked about in the last post, we’ll use the solution to a sudden change in temperature at the surface of a semi-infinite solid to examine the temperature in a thin layer in the wall of the cylinder of an internal combustion engine.

Leveraging Solutions with Duhamel's Theorem

There are many transient heat transfer problems that can be solved analytically for a simple boundary condition, but would be difficult or impossible to solve with a more complicated, time-dependent boundary condition. In many cases, if the original solution can be formulated, possibly through parameterization, as the response to a unit step change on the boundary, Duhamel’s theorem can be used to extend the simple solution to a more complex case.

The Standard Atmosphere

Most people don’t need to worry too much about the temperature and pressure of the atmosphere, as long as there is some of it there to breathe.  However, pilots, airplane designers, meteorologists, and many other people need to know how the temperature and pressure typically change with altitude.  Of course, on any given day at a particular location, conditions might be different than the “typical” condition.  Still, the typical conditions are useful enough that there are several definitions of the Standard Atmosphere describing conditions clear out into what is usually considered outer space.

Fortuitous Properties of Cold Water

As cold water starts to get close to the freezing point, it behaves differently than most other substances in that it stops getting denser and actually gets less dense as it gets colder.  In this post, we’ll look at the variation of density with temperature, and talk about why it is lucky for all life on earth that it behaves this way.

Specific Heat Ratio

The specific heat ratio, k, is defined as the ratio of the constant pressure specific heat to the constant volume specific heat.  It depends weakly (over small temperature ranges) on temperature.  In this post, we’ll look at the effect of that temperature dependence.  This figure shows the value of the specific heat ratio of air as a function of temperature.

Compression Heating of a Gas

An ideal gas, like air, or helium, will tend to heat up when it is compressed.  Unless measures are taken to cool the gas during the compression process, this can lead to a pretty large temperature increase for relatively modest pressure increases.  If we assume that no cooling takes place (adiabatic) it is easy to calculate the minimum temperature to which the gas is heated from the compression process alone.

Quenching in a Finite Bath

In an earlier post we looked at the transient 1 dimensional temperature profiles and

temperature gradients across a piece of metal being quenched in a tank large enough that the temperature of the quench fluid never changed.  In this post, we’ll look at a different case: a piece of steel small enough that the temperature is uniform across the piece which is being quenched in a bath small enough that the temperature of the quench fluid goes up as the steel cools down.  We’ll assume that the quench fluid is well-stirred so that it can also be characterized by single temperature.

An Alternate View of Condensation

Usually when we think of condensation, we think of the droplets of water that form on a cold glass on a humid day, or perhaps we think of dew making the grass wet on a summer morning, or maybe the fog that forms on the bathroom mirror after you have a hot shower.  In those familiar cases of condensation, the water vapor is mixed in with dry air in a very dilute mixture.  For example, at 75 deg F, 60% relative humidity, the water vapor comprises only about 1.1% of the total moist air, by mass.  Even at 75 deg F, 100% relative humidity, (at which point the vapor is about to start condensing to liquid) the water vapor is only about 1.8% of the mixture.  So, we’re accustomed to condensation of water from a very dilute mixture of water vapor in air.  In this blog, we’ll consider the condensation of pure water vapor, and see some surprising forces.

Thermal Gradients from Quenching

Many heat treating operations involve a quench—that is, an immersion in a fluid at a lower temperature in order to achieve a rapid cooling rate. It is commonly used for hardening in ferrous metals. Quench fluids include air, water, oils, and many others. From a heat transfer standpoint, quenching of a hot metal has a lot of interesting aspects: determining the heat transfer coefficient (possibly with phase change) at the surface, calculating transient temperature profiles inside the material (with implications for thermal stresses and metallurgical properties), effects of thermal transport properties (possibly time-dependent, or spatially non-uniform) on the heat transfer, and others. In this post, we’ll discuss transient temperature profiles and temperature gradients induced by quenching a one-dimensional (wide enough and long enough that the main effects are controlled by the thickness) piece of tool steel. 

Isothermal and Isoflux Boundary Conditions

In analytical conduction problems, two commonly used boundary conditions are isothermal, meaning that the temperature at the boundary is fixed, and isoflux, which means that the heat flux at the boundary is fixed. The two are mutually exclusive (you can’t specify both the heat flux and the temperature on a single boundary) and lead to very different thermal behaviors inside the body. In this post, we’ll look at the effects of these two boundary conditions.

Laser Melting of a Plastic

A high-intensity laser beam that deposits energy over a millisecond time scale into a mostly transparent plastic is an ideal candidate for analysis using the equations of thermal explosions.