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What is the direction of temperature gradient, when we study about thermoelectric power. And what is exactly meant by up and down the thermal gradient?

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The general formula for the Seebeck effect (the thermoelectric effect for power generation) is: $$V=S\;\nabla T$$ $S$ is the Seebeck coefficient. The temperature gradient $\nabla T$ causes majority-charge-carrier drift from hot to cold end (due to increased charge-carrier random motion, corresponding to charge-carriers "filling more space" when hot). This means that

  • in an n-type thermoelectric material where the charge-carriers are electrons, electrons will flee from the hot end, corresponding to positive charge moving towards the hot end (current direction from cold to hot), while
  • in a p-type material with charge-carriers being positive "holes", positive charges flee from the hot end (current direction from hot to cold).

The potential difference that the above charge motion corresponds to is called the Seebeck voltage $V$. The different types of material have different signs of $S$, and you must put together one of each type in a series connection to establish a circuit with steady current. Couple more in series and you add up the voltage and thus power output. See a good illustration of a possible minimal circuit here.

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  • $\begingroup$ Strictly speaking, there's nothing preventing a material from having both a negative Seebeck coefficient and a positive one, depending on the temperature the material is submitted to. Example: Pt. $\endgroup$ – thermomagnetic condensed boson Jul 25 '18 at 19:47
  • $\begingroup$ Also your sentence "The different types of material have different signs of S, and you must put together one of each type in a series connection to establish a circuit with steady current." is not really true in the sense that it is not a must. One does not need to use different materials such that they have opposite Seebeck coefficients. Sure, it helps in terms of power generated, but it is by no mean a necessity. But I guess you already know that :) I've learned that from your own question at $\endgroup$ – thermomagnetic condensed boson Jul 25 '18 at 19:51
  • $\begingroup$ physics.stackexchange.com/questions/150295/… $\endgroup$ – thermomagnetic condensed boson Jul 25 '18 at 19:51
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    $\begingroup$ It is not essential for the answer, but, in the first formula, there should be E (field strength) on the left side, not V (voltage). $\endgroup$ – V.F. Jul 25 '18 at 23:08
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According to Wikipedia, the gradient is a vector that "points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction".

The gradient is applicable to various physical properties, like temperature, pressure or electric potential. A simple example is the slope of an inclined plane, shown below, where the function is the height of the plane.

enter image description here

Here the gradient is shown as the red vector, which points in the direction of the maximum slope or maximum rate of height growth. The blue vector is also pointing in the direction of the height growth, but it is not a gradient, because it is not pointing in the direction of the maximum slope.

Going up and down the gradient in this setup means going up and down the slope along the red vector.

If applied to the temperature distribution in space, the gradient would point in the direction of the fastest temperature change (or the steepest slope of the temperature curve as a function of space coordinates) and its value will be equal to the rate of that change.

In a thermoelectric generator, the electric field at any point is proportional to the temperature gradient at that point and it points in the same direction as the gradient or the opposite depending on the sign of charges, negative or positive, respectively.

In this context, the phrase "up and down the gradient" could be used in a sentence like "As we move up and down the temperature gradient, the potential increases or decreases".

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