cryoheatflow 1.0.0

Cryogenic thermal calculations library for cryostat design

CryoHeatFlow

A Python package for cryogenic thermal analysis and heat transfer calculations. This package provides functions for calculating thermal conductivity, thermal power transfer, thermal boundary conductance, and multilayer insulation effectiveness.

Installation

pip install cryoheatflow

Table of Contents

• Quick Start
  • Calculate Thermal Conductivity
  • Calculate Thermal Power Transfer
  • Calculate Temperature Rise
  • Multilayer Insulation Analysis
  • Plotting Thermal Conductivity Curves
• Available Materials
• Data Sources

Features

• Thermal conductivity calculations for various materials at cryogenic temperatures
• Thermal power transfer through conductors and insulators
• Thermal boundary conductance across joints and interfaces
• Multilayer insulation effectiveness calculations
• Area calculations for various cross-sectional geometries (coax, AWG wire, etc)

Quick Start

Calculate Thermal Conductivity

import cryoheatflow

# Get thermal conductivity for stainless steel at 10K
k_conductivity_function = cryoheatflow.conductivity.k_ss
T = 10  # Temperature in Kelvin
result = k_conductivity_function(T)
print(f'Thermal conductivity = {result} W/m*K')

Calculate Thermal Power Transfer

Let's say you wanted to connect a stainless-steel microwave coax line from a 40K stage to a 4K stage. The coax has a diameter of 0.085" (so-called "085" coax), and is 30mm long. How much heat would be transferred?

import cryoheatflow

# Select stainless steel as the material
k = cryoheatflow.conductivity.k_ss
area = cryoheatflow.area.coax_085  # 0.085" outer-diameter coax
length = 30e-3  # 30 mm
T1 = 40  # 40 K 
T2 = 4   # 4 K

P, G, R = cryoheatflow.calculate_thermal_transfer(k, area, length, T1, T2)
print(f'Power transmission = {P*1e3:0.3f} mW')
print(f'Thermal conductance = {G:0.6f} W/K')
print(f'Thermal resistance = {R:0.3f} K/W')

giving us

Power transmission = 4.844 mW
Thermal conductance = 0.000135 W/K
Thermal resistance = 7432.015 K/W

Calculate thermal boundary conductance

Now let's say you want to anchor a 1.5x1.5 cm^2 sample to your 4K stage, and you put grease between the sample and the 4K stage. Your sample is going to generate 2 mW of heat load and going to warm up a little. What temperature is your sample going to be at?

First, we calculate the thermal boundary conductance (in watts per kelvin), and/or its inverse quantity, the thermal boundary resistance:

import cryoheatflow

# Calculate thermal boundary conductance across a solder joint
T = 4  # Temperature in Kelvin
area_m2 = 15e-3 * 15e-3  # 15 mm x 15 mm contact area

h = cryoheatflow.conductivity.h_grease(T=T, area=area_m2)
print(f'Thermal conductance = {h:0.3f} W/K')
print(f'Thermal resistance = {1/h:0.3f} K/W')

This gives us Thermal resistance R = 38.384 K/W. We can then estimate the temperature by the simple relation

(temperature increase) = (thermal resistance) x (heating power)

Giving us a temperature increase of ~76.8 mK.

Calculate Temperature Rise

If you have a thermal conductor with a known heat load applied at one end and the other end anchored at a known temperature, you can calculate the temperature rise at the hot end.

For example, assume you have a 4mm thick x 2mm wide x 100mm long strip of aluminum 6061-T6 that's attached to a 40K coldhead at one end. If the other end of the strip has 250 mW of heat load applied to it, what will the temperature be at the hot end?

import cryoheatflow

k = cryoheatflow.conductivity.k_al6061
area = 4e-3 * 2e-3  # 4 mm x 2 mm
length = 100e-3  # 100 mm
T1 = 40  # 40 K (cold end temperature)
heat_load = 0.25  # 250 mW

T2, thermal_conductance, thermal_resistance = cryoheatflow.calculate_temperature_rise(k, area, length, T1, heat_load)
print(f'Temperature at cold end = {T1:0.3f} K')
print(f'Temperature at hot end = {T2:0.3f} K')
print(f'Thermal conductance = {thermal_conductance:0.3f} W/K')
print(f'Thermal resistance = {thermal_resistance:0.3f} K/W')

giving us

Temperature at cold end = 40.000 K
Temperature at hot end = 83.680 K
Thermal conductance = 0.006 W/K
Thermal resistance = 174.719 K/W

Multilayer Insulation Analysis

from cryoheatflow import solve_multilayer_insulation
from cryoheatflow.emissivity import Al_polished, Al_oxidized, mylar

# Calculate effectiveness of multilayer insulation
T1 = 4   # Cold side temperature (K)
T2 = 85  # Warm side temperature (K)
N = 2    # Number of mylar layers
emissivity1 = Al_oxidized    # Emissivity of the first layer (e.g. 300K walls)
emissivity_mylar = mylar     # Emissivity of the multilayer mylar layers
emissivity2 = Al_polished    # Emissivity of the last layer (e.g. 40K walls)
area = (20e-2)**2           # Area in m^2

layer_temps, qdot = solve_multilayer_insulation(T1, T2, N, emissivity1, emissivity_mylar, emissivity2, area)
print(f'Layer temperatures: {layer_temps}')
print(f'Thermal power: {abs(qdot)} W')

Plotting Thermal Conductivity Curves

Plotting code here: https://github.com/amccaugh/cryoheatflow/blob/main/plot_thermal_conductivities.py

Available Materials

Thermal Conductivity Functions

The package provides thermal conductivity functions for various materials from the NIST cryogenic thermal conductivity reference:

• k_ss - Stainless steel (316/314/304L)
• k_cuni - 70-30 CuNi cupronickel
• k_al6061 - Aluminum 6061-T6
• k_al6063 - Aluminum 6063-T5
• k_al1100 - Aluminum 1100
• k_brass - Brass (UNS C26000)
• k_becu - Beryllium copper
• k_cu_rrr50 - Copper (RRR=50, typically ETP or OFHC)
• k_cu_rrr100 - Copper (RRR=100)
• k_g10 - Fiberglass-epoxy (G-10)
• k_nylon - Nylon (polyamide)

Thermal Boundary Conductance Functions

• h_grease - Thermal conductance of grease for given contact area
• h_solder_pb_sn - Thermal conductance of standard lead-tin (PbSn) solder for given contact area

Emissivity Values

• Al_polished - Polished aluminum (ε = 0.03)
• Al_oxidized - Oxidized aluminum (ε = 0.3)
• Cu_polished - Polished copper (ε = 0.02)
• Cu_oxidized - Oxidized copper (ε = 0.6)
• brass_polished - Polished brass (ε = 0.03)
• brass_oxidized - Oxidized brass (ε = 0.6)
• stainless - Stainless steel (ε = 0.07)
• mylar - Mylar (ε = 0.05)

Area Calculations

The package includes functions for calculating cross-sectional areas:

• tube_area(diameter, wall_thickness) - Annular cross-section area
• cylinder_area(diameter) - Circular cross-section area
• wire_gauge_area(awg) - Wire cross-section area based on AWG gauge
• coax_141, coax_085, coax_047, coax_034 - Predefined coaxial cable areas

Data Sources

Thermal conductivity data is sourced from the NIST Cryogenics Materials Database: https://trc.nist.gov/cryogenics/materials/materialproperties.htm

Emissivity and thermal boundary conductance values are from Ekin, J. (2006), Experimental Techniques for Low-Temperature Measurements, Oxford University Press, Oxford, UK.

Requirements

• Python >= 3
• NumPy
• SciPy
• Matplotlib (for plotting examples)

Acknowledgements

This package was developed by Adam McCaughan. Special thanks to the wider cryogenic-science community for the invaluable data used in this package. If you use this package in your work, please consider citing the relevant sources and acknowledging the authors.