# Verification of Valve2022

In order to verify the Valve2022, a benchmark model is designed in which the Valve2022 has a prescribed proximal and distal pressure and area.

## Code Verification of the State Variable \(q\)

The Valve2022 module solves the differential equation:

This benchmark setup places the Valve2022 module in a setup with prescribed constant proximal and distal pressures. The proximal pressure is a stepfunction, for which the pressure is \(p=2000Pa\) for \(0.1<t<0.4\) and \(p=0Pa\) elswhere. The distal pressure is constant at \(p=1000Pa\). This results in a \(\Delta p = \pm 1000Pa\).

This benchmark problem has 3 phases. Phase 1 has a negative pressure drop with a positive decreasing flow. Phase 2 has a negative pressure drop with a negative stabalizing flow. Phase 3 has a positive pressure drop with a positive increasing to stabalizing flow. After phase 3, phase 1 is repeated.

Two different valve states occur in these phases, namely open and closed valve. This results into 3 different analytical solutions for each of the phases.

To analytically solve the equation, we rewrite the equation into

In this equation, \(L_{total}\) is given by

in which \(A_{eff}=A_{open}\) if \(\Delta p<0\) and \(A_{eff}=A_{leak}\) if \(\Delta p>=0\). Furthermore, \(\alpha\) is given by

As a result of the \(|q|\), there are two analytical solutions. One for positive flow and one for negative flow. Additionally, the constant \(\alpha\) changes when

### Phase 1: Positive pressuredrop with positive decreasing flow

with \(c_1\) such that \(q(0) = q_0\). Notably in this analytical solution is that if \(\Delta p\) is negative, \(\alpha\) is also negative. Consequently, must be \(A_{open} < A_{prox}\). The limit of these solution goes to \(q=-\infty\), however, it will go to phase 2 when the sign of \(q\) changes negative.

### Phase 2: Positive pressuredrop with negative flow

with \(c_2\) such that \(q(t_2) = 0\). This phase is stable, with an asympote at \(q=-\sqrt{-\frac{\Delta p}{\alpha_{leak}}}\).

### Phase 3: Negative pressuredrop

with \(c_3\) such that \(q(t_3) = 0\). This phase is stable, with an asympote at \(q=\sqrt{\frac{\Delta p}{\alpha_{open}}}\)

### Verification

(`Source code`

, `png`

, `hires.png`

, `pdf`

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