The classic radial-conduction problem: enter the pipe size, insulation material and thickness, and the temperatures, and get the heat loss per meter, the outer surface touch temperature, and the savings versus the bare pipe.
The diagram is labeled with the same symbols as the input fields below.
At t = 0 the pipe goes hot. A real 10-shell radial transient shows the wave diffusing outward through the insulation while q' climbs to exactly the steady value the calculator reports.
This preview solves a handful of lumped nodes. The NovaThermal engine behind ThermalResults.com (coming soon) runs the same physics on tens of thousands of nodes - full transients with phase change, radiation, fluid loops, and Monte-Carlo design envelopes, GPU-accelerated at 400× real-solver speed - and hands you review-ready margin reports.
This is the textbook cylindrical-insulation problem every heat-transfer course teaches, and it stays the workhorse of plant energy audits because pipes are long: 30 W per meter across a few hundred meters of steam line is a real utility bill. The logarithm in the conduction term is the cylindrical geometry talking - each added centimeter of insulation buys less than the one before it, which is why an ECONOMIC thickness exists rather than a physical optimum.
The famous wrinkle is the critical radius, r = k/h. On small-diameter lines - refrigeration suction lines, tubing, wires - adding a thin layer of insulation INCREASES the outer surface area faster than it adds resistance, and the heat loss goes UP until the radius passes k/h. For mineral wool in still air that is only a few millimeters, so it rarely bites on process pipe; for a thin refrigerant line with high-k elastomeric foam it is a genuine design trap this calculator flags.
The two numbers plant engineers actually act on are here: the surface touch temperature (60 °C is the common personnel-protection threshold; insulate or clad past it) and the percentage saving versus bare pipe, which routinely lands above 85% for the first sensible thickness. What a closed form cannot do: multi-layer systems with temperature-dependent k, wet or degraded insulation, freeze-protection transients on stagnant lines, or condensation control on chilled service - those are simulation territory.
r = k/h: below this outer radius, adding insulation increases surface area faster than it adds conduction resistance, so heat loss RISES. With k = 0.04 W/mK and h = 10 W/m2K it is just 4 mm - irrelevant for process pipe, but a real effect on small tubes and wires.
60 C is the widely used personnel-protection limit for metal jacketing (ASTM C1055-class guidance: burns in about 5 seconds above it). This calculator reports the jacket temperature so you can add thickness until it clears the limit.
The first sensible thickness typically cuts loss by 85-95% versus bare hot pipe. Beyond that, returns diminish logarithmically - the economic thickness balances the next centimeter's material cost against the energy it saves over the plant's life.