Cryogenic Design by the Numbers

About this book

Below about 100 K, the material properties an engineer learns to treat as constants stop being constant. A kilogram of copper that needs 389 J to warm by one kelvin at room temperature needs 58 J at 40 K and about a tenth of a joule at 4 K. Pure copper conducts heat several times better at 40 K than at room temperature, while stainless steel conducts less than a third as well. Aluminium shrinks by 0.4% on the way down, stainless steel by 0.3%, PTFE by 2% and Invar by almost nothing. Every one of those changes has to be designed for, and every watt of heat that reaches the cold end costs hundreds of watts at the wall socket.

This book is about designing apparatus that works in that world: test cryostats, cryocooled sensors, sample stages and the mechanical and electrical parts that hold and feed them. It starts with why the properties change, which the physics of phonons and electrons in solids explains, then follows a cold assembly through its cooldown, its contraction and the choice of its materials. Five chapters then count the heat that leaks in, through supports, straps and joints, radiation, wiring and current leads, and the last two set that heat against what a cryocooler can remove and look at the superconducting coils such apparatus is often built to test.

It is for mechanical, electrical and instrumentation engineers, technicians and students who know basic heat transfer and electrical circuits. Each chapter works the running case with real numbers, includes one plausible mistake and how it was caught, links the KEVOS® tool that performs the chapter’s calculation where one exists, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional cryostat. This is an illustration. An electrical equipment maker’s laboratory builds a test cryostat, CT-40, to characterise small high-temperature superconducting coils at 40 K and currents up to 100 A. It is cooled by a cryocooler, with no liquid cryogens.

• Vacuum vessel: 304 stainless steel, 400 mm inside diameter and 600 mm high, 1.005 m² of inside surface, at room temperature.
• Cryocooler: a single-stage Gifford–McMahon cold head whose fictional capacity curve gives 2.0 × (T − 25 K) watts up to 120 W, so 30 W at 40 K, driven by a water-cooled compressor drawing 7.0 kW.
• Cold mass: an OFHC copper cold plate 300 mm in diameter and 20 mm thick (12.64 kg), a test coil of 4.0 kg treated as copper, and a 6061 aluminium coil former of 2.0 kg: 18.64 kg in all, with 0.35 m² of outer surface wrapped in ten layers of multilayer insulation.
• Supports: three G-10 tubes, 25 mm outside diameter, 2 mm wall, 150 mm long, from the vessel’s top flange to the plate.
• Thermal link: a laminated copper strap of residual resistance ratio 50, 300 mm² in section and 100 mm long, with two bolted joints.
• Electrical: two copper current leads for 100 A, and 32 phosphor bronze instrument wires 0.127 mm in diameter and 1 m long.
• Target: the cold plate at or below 40 K with 100 A flowing, and a cooldown of no more than six hours.

The laboratory, its cryostat, the cryocooler’s curve and the coil are invented and do not describe KEVOS® work or any real product or business. Equipment data are of the kind makers publish, not the data of any product.

How to use the book

Read Chapter 1 first: it explains why the rest of the book integrates properties over temperature instead of using constants. Chapters 2 to 4 cover the cold mass itself: its heat, its contraction and its materials. Chapters 5 to 8 each take one path by which heat leaks in and can be read in any order. Chapter 9 assembles the heat budget and sets it against the cryocooler, and Chapter 10 turns to the coils CT-40 exists to test.

All solid properties come from NIST’s cryogenic material fits, integrated numerically over temperature. Figures are given to more digits than a measurement would justify, so that a reader can follow each step. The linked KEVOS® tools use constant properties; the chapters show how to give them the right mean values.

Source limits and professional responsibility

The relations in this book are the simplified forms used for first design and checking. Real heat loads depend on surface finish, workmanship, vacuum quality and contact pressure, and real apparatus is commissioned by measurement. Property data vary with alloy, temper and purity; NIST’s fits are typical values within their stated accuracy, and critical designs need data for the actual material.

Cryostats are vacuum vessels and sometimes pressure vessels, superconducting coils store energy that can be released suddenly, and cryogenic equipment can cause cold burns and oxygen deficiency. Their design and operation are subject to the relevant standards and to work health and safety law. This book is educational. It is not a design for any apparatus, and a competent person must approve real work.