## Heat Loss Rates from Multi-layer, Vacuum Insulated, Liquid Helium Flexible Transfer Hoses

**How does this study relate to heat transfer with liquid helium?**

One of the questions that comes up in a study like this is “You made the measurements of heat transfer to liquid nitrogen, what will be the heat transfer rate with liquid helium?” To be strictly correct the answer has to be that we will not be sure unless we measure the heat transfer rate to liquid helium experimentally, which we have not done. We can however usea theoretical model of the hose to attempt to address the question of heat transfer to other cryogenic liquids.

**Hose construction:**

Cryogenic hoses are typically constructed of multiple concentric layers of heat reflecting “radiation shields” separated by synthetic “spacer materials”. The rest of the volume between the inner and outer flex materials is evacuated to a low pressure to reduce conduction via the gas occupying the vacuum space. The geometry and fundamental phenomena involved are complicated, but a simple model where the insulation is considered to consist of concentric radiation shielding layers with layers of low conducting “space” in between is a reasonable approximation to the actual situation. The conducting space is a composite of the remaining gas in the hose and the spacer material which prevent thermal contact of the radiation shields. The exact geometry of how the spacers and shields are installed is also important.

**Heat transfer parameter estimation:**

In order to mathematically model this construction, two parameters must be estimated. The first parameter is the emissivity (absorptivity) of the shielding material surface(s). This value is typically on the order of 0.1 to 0.3, with lower being better (more reflecting). The conductivity of the space in between the shielding layers would be about 0.025 W/m-K for air at 1 atm (if the space was not evaculated) but is more typically on the order of 0.001 W/m-K (1.0 mW/m-K) or even lower when the air is properly evacuated and the spacer material is properly installed.

**Model:**

Once the parameters for the shielding and spacer have been estimated and the geometry (e.g. how many shield layers and how far apart they are) is set, the radial heat transfer can be modeled. Our approach is to consider that each shielding layer transfers heat by both radiation and conduction to the adjacent shielding layers and no others. Also we assume that the inner flex tube outer surface is at the normal boiling temperature of the liquid in the tube, and the outer flex tube inner surface is at 300 K (room temperature). (Refinements to these approximations can be added but do not add to the accuracy of the model.)

An energy balance is done on each layer by setting the net heat transfer to each layer equal to zero (20 layers = 20 energy balances). The sum of the net radiative heat transfer from the next most inner layer and to the next most outer layer and the net conductive heat transfer from the next most inner layer and to the next most outer layer is set equal to zero by guessing the layer temperatures and using Fourier’s Law and the Stefan-Boltzman equations as appropriate. When the net heat transfer to all the layers is the same for a set of guessed temperatures, the solution is achieved.

**Modeling heat transfer to fluids with different boiling temperatures**

To model heat transfer rates to other cryogenic fluids the only change in the input to the model is the temperature of the outer surface of the inner flex tube. Re-solving the equations for the new inner tube surface temperature allows one to estimate the radial heat transfer rate to the new fluid.

**Results**

Figure 1 shows the results of predicting the heat transfer rate using a modelof the heat transfer through a hose constructed like the Technifab hose. The colder the fluid being transported in the hose, the higher the heat transfer rate. The heat transfer to liquid helium (Tb = 4 K) is expected to be about 30% faster than it is to liquid nitrogen (Tb = 77 K). Heat transfer rates to liquid oxygen (Tb = 90.1 K) and liquid methane (Tb = 111.7 K) are appropriately lower.

Also plotted on this graph is the apparent conductivity one would observe if it was assumed that all of the heat transfer was by conduction.

**Figure 1.** Predicted heat loss and effective thermal conductivity for the Technifab hose with different cryogenic fluids. Liquid helium (cold boundary temperature = 4.1 K) is predicted to have a radial heat loss rate of about 1.6 W/m, still less than half of the heat loss rate of the next best host using LN2.

Rose-Hulman Institute of Technology

Terre Haute, Indiana

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