Here is the mathematical explaination for what is occuring in the sprinkler.

The change in length of a material due to thermal expansion in a unidirectional expansion is given by

dL = a(dT)L

where dL is the change in length, a is the coefficient of thermal expansion (which is different for every material), dT is the change in temperature, and L is the original length of the material. In the situation described for this project, we are dealing with a strip that is made of two metals welded together to form what is called a bimetallic strip. When heated the strip will expand, but the metal that has the smaller value for its coefficient of thermal expansion, we will call this metal A, will not expand at the same rate as the other metal. This will cause the strip to bend in towards the metal A side. The other metal, metal B, will be proportionally larger, by this rule:

dA = aA(DT)L

dB = aB(DT)L

Solving for L and combining we get

dA/aA = dB/aB

or

dB = dA(aB /aA)

As metal B expands (faster than metal A), the strip curves upwards. Because the strip is fixed at one end, it only curves on the non constrained edge. The amount the strip will curve, theta, is dependent on the temperature, the thickness of the strip, t, and the overall length. The radius of curvature, r, is defined as the distance from the strip to the center of curvature, m.

(L) (1+ dB-dA) = (theta) (r+.5t)

theta = [(L)(1+dB-dA)] / (r+.5t)

When theta becomes large enough, the sprinkler is activated.As the fire is put out, the air cools and the sprinkler cycles off. If fire spreads to that parts of the building again, the sprinkler will cycle back on.