BetaCAP diluters are fully complying with their transfer function : the user inputs the wanted resulting flow, the dilution ratio and the gas composition. The gas divider puts those data in an equations system and get back the capillaries partition and the pressures to be applied at capillaries ends (span gas side and diluting gas side). The cited transfer function includes the logic reproducing the diluter performances and the physics of the flows in capillaries

Such a control type can be only applied thanks to the electronic pressures control

## WHICH IS THE STARTING POINT?

It’s the User request, the user can ask for:

- A target concentration for a component to be measured. In this case the standard target (dilution ratio) is immediately found.
- A dilution ratio (in this case the target is immediately available)

Where C_{Dil.}is the requested concentration after dilution, CBs is the concentration od the measurand component in the span gas bottle, CBz the concentration of the measurand component in the zero gas bottle (generally CBz=0). In both cases a) and b) the starting point is the same

## THE TRANSFER FUNCTION OF THE DILUTER

The tranfer functionis composed by three contributions that puts into mathematic form the three conditions that must be accomplished in the divider:

- The K
_{Dil.}formula in terms of capillaries partition (N) and flows through single capillaries Q(TG1) and Q(TG0).

b. The condition of having set the total output flow to a setpoint specified by the user

c. The Poiseuille formula, (modified to consider the fact that dilution occurs at capillaries outlet and gases are compressibles.

d. The system of the three above equations is calculated in one time to get the solution (N, dP1, dP0), but before that it’s better to prevent consequences due to special conditions :

- If the viscosities of TG1 and TG0 are very different (it’s possible to reach factors near to 3) the calculation of dP1 and dP0 lead to very different values, and this is not desirable, for various reasons. To avoid this, the more viscous gas is allowed to cross more capillaries and this reduces the applied pressure. The more suitable capillaries partition (N calculation) is found setting dP1 = dP0 in the equations system :

- Normally, for metrological applications it’s made the metrological certification which provides measuring of the flows deviation setting one at a time all of the main dilutions. Overlapping the deviation for each of the capillary groups involved by the requested “N”, it’s easy to calculate how much real dilution Kdil.r is deviated respect the target dilution. It’s then activated a corrected KDil.c applying to the target KDil.t a correction equal and opposite to the deviation Kdil.c= Kdil.t-Kdil.r.

It can be interesting to see the values of dP1 and dP0 resulting from the above calculations.The two trends shown below indicate how the dilution range can be fully covered. In this case the left side figure is showing a situation with the two viscosities equal and the right side picture is showing a case with η1=0,5*η0.

In the second case, N is growing slower to compensate the higher flow due to lower viscosity

In the charts we see two lines : one bleu is the ratio P1/P0 and one brown is relevant to capillaries partition : from the left (high dilution) with N small to the right low dilutions with N high.

It’s good to remember that N=Ni (intero) e copre I valori per i quali Ni-0,5 < Nd < Ni+0,5

It’s useful to observe what’s happening in the various points of the horizontal segment in the brown step (note that dilution grows linearly moving from left to right) :

- The left side shows points where N is higher than required by the dilution point : in this area dilution is regular when P1<P0
- The middle area where N is near to what is required by the dilution point : in this area dilution is regular when P1 and P0 are nearly equal
- The righy area where N is lower than is required by the dilution point : in this area dilution is regular when P1>P0.

Each front of step (when N changes to N+1), there is a sudden change from c) to a) : the pressures ratio do revert (P1/P0 changes from higher than one to lower than one) and N is incremented : with all this bailamme the Dilution is changed just a very small amount (step is nearly vertical). It’s the right point in which to check the calculation accuracy. We made those tests, not requiring a special instrumentation, and got the demonstration it runs very well.