This week the professor finally reached the content that actually relates to the course’s name. He introduced first-order linear equations. It seemed such an abstract idea to me after I finished the lecture, but with further research and study, I found that such an equation is highly applicable in the real world. For more detail about first order linear equations click here
this picture shows the general solution formula for all first order linear equations (picture from http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx, which also explains the process very well)
One of the applications of first order linear equations is the conduction-diffusion model, which is the model built according to Newton’s law of cooling. Such a model can be used in situations including heat conduction and diffusion of certain substances through two environments. The conduction-diffusion model is used to calculate how fast temperature or concentration changes within a chamber due to the conduction of heat or diffusion of certain substances from the outer environment through a membrane. The formula for such model is by nature a first order linear equation which has an important characteristic: it is solvable. By using a technique to solve the equation, a direct relationship between the time and temperature/concentration can be reached. For further information about Newton’s law of cooling click here
What surprised me was how perfectly the variables in math and physics matched each other. There are all kinds of factors that can influence the diffusion rate, such as the concentrations within and outside the membrane and the speed with which the membrane is capable of allowing substances to pass. Every single variable fits perfectly into the equation, as if the method were designed for the physical problem.
Many people may think that heat conduction and diffusion sound unimportant to rocket science and high-level biology, but many important ideas in both fields are established on the above model and the solution reached by the process for solving first-order linear equations.
Osmosis is an important concept in biology. It is the spontaneous movement of water through the cellular membranes due to the difference between the concentrations of solutions both within and without the cells. The cellular membranes are semi-permeable membranes so they only allow substances like water to flow through. Due to the difference between concentrations appearing, water flows through the membrane and causes the cell either to burst or to shrink. If the difference of concentration is too large and too much water flows into the cells, cytolysis (meaning cell death due to the burst caused by osmosis) will occur (for further information please click here. Cytolysis happens in mammalsas a result of stroke. With the death of large amount of blood cells, hemolysis would happen, and their capacity of delivering oxygen would decrease vastly, resulting in further consequences. The diffusion model is able to calculate very accurately how fast water can flow through the cell and how long would it take generally to cause the massive death of cells, which gives a time range for doctors and vets to rescue either human or other mammals from it.
picture above shows how osmosis causes cellular death (picture from http://en.wikipedia.org/wiki/Cytolysis)
The lives of astronauts depend upon the conduction model. Thermal insulation materials are used on space shuttles and manned spacecraft, but in the real world, nothing can insulate heat perfectly. Even the thermal insulation materials used by NASA still conduct heat. The spacecraft experiences extremely high temperature (as high as 1600 degrees Celsius) caused by the friction between the ship and atmosphere both during the launch and return. Without insulation, the astronauts may very well literally fry to death in the time of landing. In fact, that is what happened to Laika, the Soviet space dog; It died because the malfunction of the heat insulation board. The conduction model and its first order linear solution can be used to accurately calculate the direct relationship between time and temperature within the spacecraft, and predict the overall temperature changes astronauts may need to experience, and under what circumstances such heat conduction can cause any kind of malfunction of the space shuttle.
A picture of Laika (picture from http://www.fromquarkstoquasars.com/laika-the-first-earthling-in-space/, an article that briefly introduces the fate and the mission of Laika)
In fact, I was very scared when I felt I understood nothing in the professor’s lecture, because it was extremely abstract and theoretical. After the study of such model, however, the practical uses of first order linear equations appear quite intriguing to me.