Air Temperature Measurement — Page 39, Lesson 42

Let’s pick up right where we left off — we’ve got the accurate formula for converting TAT to SAT, and now I want to walk you through the full worked example, because this is where the theory becomes real. The accurate formula is: SAT equals TAT divided by (1 + 0.2 × Kr × M²), where M is the Mach Number. Now, this formula only works if the temperatures are quoted in degrees Absolute, °A, or Kelvin, K. That’s a hard requirement — if you plug Celsius in, the math falls apart. So let’s talk about the Absolute, or Kelvin, temperature scale. When Celsius designed his scale, he picked the boiling point of water as 100°C and the freezing point as 0°C. That seemed perfectly reasonable given the physics of his day. But here’s the problem: the Celsius scale changes sign at 0°C — the freezing point of water. Everything above zero is positive, everything below is negative. To measure the relative hotness of objects in a meaningful way, we need a baseline of absolute zero — the point of no heat at all. Absolute zero occurs at -273°C on the Celsius scale. So -273°C is 0°A, or 0 K, on the Kelvin scale. One kelvin, or one degree Absolute, represents exactly the same temperature change as one degree Celsius — the scales just start from different baselines. That means the freezing point of water is 273 K, the boiling point is 373 K, and so on. Now let’s run the worked example. Assume the indicated TAT — which is actually RAT, the ram air temperature — is -20°C. The Mach Number is 0.73, which is a typical long-range cruise speed for a B737. The Recovery Factor is 0.98, a typical value for a modern TAT probe. And Kr — that recovery factor constant — is determined by flight testing and published in the operating instructions for the aircraft. First, convert the TAT to Kelvin. -20°C is 253 K. Now substitute into the formula: SAT equals 253 divided by (1 + 0.2 × 0.98 × 0.73²). Let’s work the denominator: 0.73 squared is 0.5329. Multiply by 0.98 gives 0.5222. Multiply by 0.2 gives 0.1044. So the denominator is 1.1044. Now 253 divided by 1.1044 equals 229 K. And 229 K is -44°C on the Celsius scale. So the ram rise — the heating from compression at Mach 0.73 — has pushed the indicated temperature up by 24 degrees. The true static air temperature is -44°C, even though the probe reads -20°C. Now, let’s move to calibration. Because the atmosphere varies with temperature and pressure, we need a standard calibration for our instruments. The conditions used for calibration are those of the International Standard Atmosphere — the ISA. Here are the relevant assumptions. At Mean Sea Level: pressure is 1013.25 hectopascals, temperature is +15°C, and density is 1225 grams per cubic metre. From MSL up to 11 kilometres — that’s 36,090 feet — temperature falls at 6.5°C per kilometre, or 1.98°C per 1000 feet. From 11 km to 20 km — 65,617 feet — the temperature is constant at -56.5°C. And from 20 km to 32 km — 104,987 feet — temperature rises at 1°C per kilometre, or 0.3° per 1000 feet. With these assumptions, the pressure at any given level in the ISA can be calculated from the calibration formulae. From that, we can produce graphs or tables showing height in terms of pressure under standard conditions. These tables are used for the manufacturer’s calibration of the altimeter scale. Any discrepancies — if they fall within certain agreed tolerances — are listed over the operating height ranges as instrument errors. And note this important detail: the calibration is carried out with both increasing and decreasing readings, so that the amount of lag at calibration conditions can be determined. That lag is the instrument’s response delay — how long it takes to settle on the true value when conditions change. It’s a real, measurable quantity, and the calibration process is designed to capture it. So the full picture is: we measure TAT with the probe, we correct for ram rise using the recovery factor and Mach number, and we get SAT. And all of this is anchored to the ISA so that every instrument we fit is calibrated against the same standard atmosphere.