There are a number of physical parameters required to model heat flow. This is particularly so for fluids -- liquids and gases. A lot of this data is readily available on the web. However the data is not often in a convenient format. There is also the complication that many of the physical properties vary with both temperature and pressure. This complicates modelling in systems where both the pressure and temperature are input variables.

This page documents the modelling of the physical properties used in calculating thermal heat flows through gases and solids used in our projects. This includes: hydrogen, argon, borosilicate and quartz. As a gift to the research community we are providing some of our modeled data here.

A lot of data is available on the National Institute of Standards and Technology (NIST). This data is available in a number of formats and locations. For fluids the NIST web book facility provides all the data in a convenient web enquiry system that will output the data requested in a number of tabular and graphical formats. The NIST site is Thermophysical Properties of Fluid Systems.

The problem with the NIST web book is that you can only choose a single parameter (temperature of pressure) at a time to output the data against... we need data for both temperature and pressure. Furthermore we would prefer that out parameter data were generated by a single formula with both temperate and pressure as input variables. We have output a great deal of the data one parameter at a time and then modeled surface fits to that data to generate simple formuala that fit the data very well.

These formulas are embedded in our thermal calculator page.

We have also modeled the thermal conductivity data for fused quartz and borosilicate glass from data in the NIST database.

Data and analysis provided by Dr Mark Snoswell, CTO Chava. All inquiries should be directed to

Copyright Chava 2012. No copying permitted – refer all links back to this web page.

No comment -- This information is not meant to imply the existence or results of any Chava project and no correspondence will be entered into regarding commercial projects.

No Warranty – This information is provided as is. There is no warranty, real or implied, as to the accuracy or fitness of the information provided for any use.

The following formula were fitted to NIST data for the thermodynamic properties of Hydrogen between 0.1 - 3 bar and 300 - 1000 oK. Modelling was done with the assistance of Lab Fit software. Formula were selected were the simplest and lowest order that did not compromise the fit to the data.

The Formula are expressed in terms of Pressure in bar (p) and temperature in oC (t).

● Thermal Conductivity of Hydrogen W/moK

Red. ChiSq. = 0.163745E-5

● Density of Hydrogen kg/m3

Red. ChiSq. = 0.750017E-10

● Specific Heat of HydrogenJ/goK

Red. ChiSq. = 0.128133E-2

● Dynamic Viscosity of HydrogenuPa*S

Red. ChiSq. = 0.109154E-4

● κ = 0.48314E-3 * t + 0.65446E-4 * p + 0.039557

● ρ = 24.1264 * (t / p)0.99926

● cp = (0.10203E6 * p) / (0.71258E4 + 0.31261E-3 * t2)

● μ = 0.18563 * (1.00001p) * t0.67955{/AC}

The Nusselt number is the ratio of convective to conductive heat transfer across (normal to) the boundary. The Nusselt number is dependent on the physical arrangement as well as fluid properties. It can be theoretically calculated for various different physical configurations or derived from experimental data. For heating wires we are interested in the Nusselt number fo cylinders -- both horizontal and vertical. The formulation of the Nusselt number if often very complicated as we can see here.

Fortunately Anita Eisakhani et all reported data for 0.5mm dia wire recently in their article NATURAL CONVECTION HEAT TRANSFER MODELLING OF SHAPE MEMORY ALLOY WIRE which they presented at the SMART MATERIALS, STRUCTURES & NDT in AEROSPACE Conference in Canada in November 2011.

Eisakhani et all measured heat transfer for 0.5mm wire for a range of angles between horizontal (0 deg) and vertical (90deg).

From their data they determined a simple formula that was a far better fit to the experimental data than previous formulations. Their formula and parameter table:

NuAngle | A | C | n |

0 | 0.190 | 0.820 | 0.170 |

15 | 0.180 | 0.815 | 0.162 |

30 | 0.169 | 0.797 | 0.154 |

45 | 0.159 | 0.767 | 0.146 |

60 | 0.148 | 0.724 | 0.139 |

75 | 0.138 | 0.670 | 0.131 |

90 | 0.127 | 0.603 | 0.123 |

average | 0.159 | 0.742 | 0.146 |

We have two possible configurations for a heating coil that we have modeled in our thermal calculator: Vertical axis coil and a horizontal acis coil.

For a vertical axis coil all the wire is effectively horizontal giving the following formula for the Nusselt number.

NuD = 0.19 + 0.82 (RaD)0.17

For a horizontal axis coil we must use an average of all angles to make and average estimate for the Nusselt number.

NuD = 0.159 + 0.742 (RaD)0.146

The parameters are accurately modeled with angle (a) as an input:

A = -0.0007 * a + 0.1902 | R^{2}=0.9999 |

C = -3E-5 * a^{2} + 4E-5 * a + 0.8202 |
R^{2}=1 |

n = -0.0005 * a + 0.1698 | R^{2}=0.9997 |

The Nusselt number is the ratio of convective to conductive heat transfer across (normal to) the boundary.

Recently (2009) Ş. Özgür Atayılmaz and İsmail Teke have looked at the predictive accuracy of various empirically derived equations for predicting the Nusselt number for a horizontal cylinder in free air. In their article Experimental and numerical study of the natural convection from a heated horizontal cylinder which appears in the International Communications in Heat and Mass Transfer 36 (2009) 731-738 they report that the best equation for the Nusselt number for our conditions was that reported by V.T. Morgan, The overall convective heat transfer from smooth circular cylinders, Advances in Heat Transfer 11 (1975) 199–211.

With an error of ~ 2.5% the following equation for the Nusselet number correctly predicts the heat transfer for a horizontal cylinder in free air:

Nu = 0.85 * Ra0.188

Thermal conductivity of these glasses varies quite a lot with temperature. This is the normal behavior of solids -- with the conductivity rising with temperature. The rise in thermal conductivity with temperature can be quite dramatic over the working range of a solid and needs to be properly modeled to get the most accurate thermal flow predictions.

Using data published by NIST the following relationships have been determined for the thermal conductivity of fused quartz and borosilicate glasses. The temperature parameter (t) is in celcius.

Thermal conductivity (k) of borosilicate glass....

Red.ChiSq. = 0.182308E-5

Thermal conductivity (k) of fised quatrz glass....

Red.ChiSq. = 0.4E-5

Using data published by NIST the following relationships have been determined for the thermal conductivity of fused quartz and borosilicate glasses. The temperature parameter (t) is in celcius.

κ = t / (440.22 + 2.2208 * t)+0.92585

κ = 4W-9 * t3 - 3E-6 * t2 + 0.0019 * t + 1.2796

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