One way to visualize Gibbs free energy is to think of it as a square function of temperature and depth. Since the mean energy level is the same for all temperature depths, we can think of the slope of Gibbs free energy as the area under the curve. The square function is not a special case. In the presence of many different temperature depths, a similar expression would look like this:

With a constant depth of 0 ° (or a constant temperature of 0 ° C), the square function would have a slope of 0.3. Then a rise from depth 0 would produce a temperature rise of 1, for a cumulative slope of 2.7.

The only time we would go much higher, above 0 ° C, is when we’re at rest. The constant temperature is still greater than zero. If we’re in a state of rest, we can get away with running higher, which would produce more energy. As we mentioned, for a temperature fall, we always want to give up less energy than we get from the start position.

On the other hand, we don’t get as much heat back for running at zero as we do for running at a higher temperature. For example, assuming a pressure of 100 Pa, a thermodynamic gradient of 0 ° C at 20 ° and 5 ° at 10 ° should yield a total heat transfer of 25 Watts, or one joule per kg of body weight. When we consider the energy transferred for 20 ° C and a pressure of 60 Pa, it drops to 14 Watts (one joule per kg) per joule. At 10 ° C we get a joule per kilogram, again about one joule per kg of body weight. The remaining energy is given up simply as heat.

Energy transferred to the environment by running a given distance is given by

where H is the amount of heat energy transferred per unit of work done. However, we can’t know if the energy we got for running was the energy we’re actually getting, by taking the heat energy as a square function of temperature and depth. That would produce a completely different expression.

Here, we have to figure out something much simpler. What is the temperature at which the energy needed for the next run falls short of the energy at work? The answer depends only on the total work done, including the initial work performed, the next work done, any further work done to recover the energy lost the first time, and the energy expended on all those subsequent

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