Flux Weighted Temperature
In my two previous posts (“Why does load matter?” & “Projecting Future Load – Compound Growth”), I used the phrase, “Flux Weighted Temperature.” Since we coined that term at Novinium, I have an obligation to define it. In mass transfer, flux is the mass that flows through a unit area per unit time. In the illustration nearby, the arrows show the flux of fluid through the cable radius as it exudes out of the insulation shield into the adjacent soil. Exudation flux is the flux at the insulation shield outside diameter1 and is a one-way affair. Once fluid hits the infinite dilution of the soil, it is quickly transported away and metabolized by the flora and fauna in the soil. The cable manufacturers call this exudation flux “sweat-out.” Certain volatile components may sweat-out of the insulation shortly after the cable is manufactured, leaving beads of fluid on the surface. Within the field of rejuvenation, sweat-out is not a good metaphor, because the exudation flux is so very small, there is no obvious indication that it is occurring at all.
Flux is a non-linear function (Tech translation: That means it’s complicated!) of the dynamic temperature profile across the radius of the cable, the geometry of the cable, the materials of which the various layers of a cable are made, and the dynamic chemistry of the rejuvenation fluid. At Novinium we use a tool, which enjoys two U.S. patents, 7,643,977 and 7,848,912, to provide accurate estimates of the flux. We call this tool, MFlux. I introduced MFlux in an August 3, 2010 post, “40-year Life.”
The alternative to a flux weighted temperature would be a time weighted temperature. Assume for example that a cable spent 12 hours at a radially uniform 40°C and 12 hours at 20°C, the time weighted temperature would be 30°C. Time weighting would lead you grossly astray, because the exudation flux, while the cable is at 40°C, is almost an order of magnitude greater than the flux at 20°C. This is because permetion rates change by about a factor of 3 for each 10°C. If we apply flux weighting to this example the flux weighted temperature would be 39°C [(40°C x 12 hrs x 10 flux-weight + 30°C x 12 hrs x 1 FW) ∕ (12 hrs x 10 FW + 12 hrs x 1 flux-weight)].
It is fortunate that cables do not have a radially uniform temperature profile as in the example of the previous paragraph. If they did, treatment fluids would exude much faster than they actually do. In fact, for all direct-buried cables with non-zero loads it is always warmer in the inner portions of the cable than the surrounding soil. Since the solubility in insulation and shield polymers of fluids in general, and rejuvenation fluids in particular, is greater at higher temperature, the exudation flux is greatly reduced. This is so, because the exudation flux is proportional to the difference in chemical potential or the fugacity2 of the fluid in the polymer. The fugacity gradient mitigates the inside-out concentration gradient. Let’s take that in pieces – last piece first. The concentration of the rejuvenation fluid in the soil around the cable is zero and has some value greater than zero inside the cable. The concentration gradient provides a second law3 driving force for fluid exudation. A healthy radial temperature profile reduces the chemical potential in the inner portions of the cable, but the exudation flux must still always be greater than zero.
Only Novinium can calculate the flux weighted temperature. Only Novinium can use that knowledge to tailor the fluid delivery and fluid formulation to the requirements of an individual cable as described by U.S. Patent 7,611,748. Only from Novinium can you learn so much from a frog!
Resting on my flux weighted belly,
1If there were a jacket present, the exudation flux would be at the outside diameter of the jacket.
2To learn more about fugacity in cables, check out “Molecular Thermodynamics of Water in Direct-buried Power Cables,” from the Nov/Dec issue of IEEE Electrical Insulation Magazine.
3The second law of thermodynamics provides that entropy of a system always increases. In common language, systems become more disordered with time – if only it were not so. The concentration of rejuvenation fluid within a cable is an ordered state, compared to the disordered state of infinite dilution outside of the cable.