Quick cooling facilitates rapid CWF formation; however, thermal stress can cause the coalesced product to crack into pieces. This paper predicts the thermal stress that develops at different stages of cooling, which is critical to achieve optimization of cooling cycles and has the potential to form a more durable and, if desired, tempered product and speed product turnaround. While this discussion is limited to the behavior of the axial stress, similar arguments can also be applied to circumferential stress.
The CWF is envisioned as being made up of a series of concentric annuli Heat is lost through the outer surface of the initially liquid cylinder, so annuli solidify from the outer annuli inwards. The vertical lengths of all annuli are the same at solidification; however, when the second inner annulus solidifies, the outermost becomes shorter due to its thermal contraction. This process repeats as the annuli progressively solidify inward resulting in a dome shape on the top end of the cylinder. The height of the dome increases with higher cooling rates.
The difference between the length of any annulus when it solidifies and the length of the annulus next to it is termed the “annulus length deficit” because this is the length that would be removed if the dome were removed from the cylinder. The length deficit causes a stress, (called here the “set-in stress”) when the temperature becomes uniform. A temperature deficit is related to this length deficit through the coefficient of thermal expansion, being greatest in the center and decreasing to zero at the outer radius.
The temperature deficit is used to calculate the set-in stress. If the cylinder is at a uniform temperature (Tu), the set-in stress is determined using the thermal stress from the temperature profile, which is obtained by subtracting the temperature deficit profile from Tu. The deficit profile is essentially the inverse of the temperature profile that exists when the last portion (center) of the cylinder reaches solidification.
Outside the average temperature circle, the set-in stress is in compression, while in the inside region, it is in tension. This is the opposite of the thermal cooling stress due to the temperature distribution. The total stress is the sum of the thermal cooling stress and the set-in stress. The total stress is zero until the temperature reaches solidification, and then it increases as it cools. The total stress reaches a maximum when the solid reaches room temperature, and the set-in stress is all that remains.
This is the opposite behavior of that exhibited by a high-temperature solid that is annealed so that the set-in stress is near zero and then cooled. In this instance, the total stress is only the thermal cooling stress and reaches a maximum near the initiation of cooling after the temperature profile is developed. The magnitude of the maximum thermal cooling stress will be approximately the same as the set-in stress in the previous case but will have tension in the outer region and compression in the inner region.
If the maximum tension stress is above the tension stress limit, failure will occur in either case but in a much different way. In the annealed solid, it will start in the outer region near the initiation of cooling. In the unannealed case, failure will occur much later by cracking in the inner region as the cylinder reaches room temperature.
The results from the model developed here show that the most desirable cooling procedure would be to chill through solidification at a rate that produces a set-in stress that is less than the failure tension limit. After solidification, cooling can be increased to a value that causes tension in the outer region but keeps it below the tension-failure limit. Set-in stress will aid in keeping the tension in the outer region less than the limit during the transient. When room temperature is finally reached, the stress in the outer region will be compressive because only the set-in stress will remain. This residual compressive stress will add the benefit of making the CWF more resilient to damage from stresses caused while moving the CWF to storage. It will also resist external stresses such as impacts much in the way tempered glass does.