**1. Introduction**

Noninvasive methods used to characterize human and animal respiration require advanced techniques and are costly, such as: computed tomography and densitometry [1–6], electrical impedance tomography [7,8], magnetic resonance imaging [9–11], contrast radiology [12], image ultrasound [13,14], ultrasonic sensors [15–17], pulse oscillometry [18,19], electrostatic methods [20], closed circuits with inert gases [21], impedance pneumography and plethysmography [22,23] and electronic noses [24].

One of the procedures used in plethysmography is spirometry, which uses physical concepts to study the air going in and out of the lungs, characterizing human breathing [25,26]. The technique is used to evaluate pulmonary function [1,18], chronic obstructive pulmonary disease [1,27], cystic fibrosis [2], smokers [4], air pollution [28–30], hyperinflation [31,32], exposure of particulates such as nanotubes and nanofibers [33] and airway resistance [34], among others.

To perform spirometry, a spirometer is used, which can be: (i) of volume (sealed in water, piston, and bellows [35]); (ii) of flow (differential pressure or pneumotacometers [36], thermistors, Pitot and turbinometers [35]); or (iii) portable [37] of volume or flow [25]. The respiratory volume sensors used in these types of equipment have a high cost and some of them perform indirect measurements, discarding their application in medical simulators or dummies, as in the case of Hamiltonian sensors coupled to differential pressure sensors [38], flow mass sensors [39] and airflow meter [40] rotary or vibratory beam and shell flow meters [41].

Cardiopulmonary resuscitation (CPR) is a recurring practice in medical urgencies and emergencies. CPR is characterized by a set of maneuvers performed in an attempt to reanimate the victim of cardiac and/or respiratory arrest, to restore the heart and lung to normal functions while maintaining the oxygenation of the brain. These procedures provide continuous improvement in the quality of healthcare professional skills by using automated dummies for teaching.

The pioneers in automating CPR dummies [42] used the Resusci Anne® manikin. The authors developed compression and ventilation sensors using digital logic and normally open contacts. The monitoring of the system was done using a display panel or indicator lamps, which showed indications of "insufficient", "acceptable", and "above acceptable". Currently, identical models are still widely used [43].

In the present study, we propose measuring the volume of air supplied to the lungs in rescue ventilation during CPR training using a rotor-type flow sensor with propellers. Also, we develop a theoretical model to make it equivalent to spirometric models. This brings more realism to the dummies and introduces advantages to possible debriefings after various simulations [44].

This work is an extension of [44], (doi:10.3390/ecsa-5-05724), where only the idea was put forward to assess the feasibility of the application. Additionally, in the present article, we perform the validation of the adapted water flow sensor to measure airflow considering concepts of fluid mechanics. Also, we apply spirometric concepts to the results, defining a theoretical model for the curves obtained.

### **2. Mathematical Modeling of Propeller Type Flow Sensors**

The analysis of the behavior of any material contained in a finite region of space, control volume, solves many problems involving fluid mechanics [45]. The Reynolds transport theorem ensures that the time rate of change of mass within a system is equal to the sum of the time rate of mass within the control volume (*CV*) and the net flux of mass through the control surface (*CS*), that is,

$$\frac{DM\_{\rm sys}}{Dt} = \frac{\partial}{\partial t} \int\_{CV} \rho dV + \int\_{CS} \rho v \cdot \hbar dS\_{\prime} \tag{1}$$

where *Msys* is the mass of the system (kg), ρ is the specific mass of the fluid (kg/m3), *V* is the control volume (m3), and *v* is the velocity vector perpendicular to the differential area *dS* (m/s).

Using the principle of mass conservation for a system, the material derivative of the mass of the system is

$$M\_{\rm sys} = \int\_{\rm sys} \rho dV\_{\prime} \tag{2}$$

therefore,

$$\frac{DM\_{\rm sys}}{Dt} = \, \, 0. \tag{3}$$

At permanent regime, the properties at any point in the system remain constant over time, so

$$\frac{\partial}{\partial t} \int\_{CV} \rho dV = 0.\tag{4}$$

Applying (3) and (4) to (1) and adding up all the differential contributions that exist on the control surface, we obtain the net flux of mass in the control volume, that is,

$$\int\_{CS} \rho v \cdot \hbar dS = \sum \dot{m}\_{\bullet} - \sum \dot{m}\_{l} = 0. \tag{5}$$

Considering that the input of the sensor in question has the same characteristics of the output, that is, *SI* = *SO*, and applying Equation (3) to its control volume, we conclude that:

$$
\dot{m}\_I = \dot{m}\_{O^\*} \tag{6}
$$

A widely used expression for mass flow assessment . *m* (kg/s), in a section of the control surface with area *<sup>S</sup>* (m2), is .

$$
\dot{m} = \rho \mathbb{Q} = \rho Sv,\tag{7}
$$

where *Q* is the volume flow (m3/s), and *v* is the velocity vector perpendicular to area *S* (m/s).

We can adequately analyze many mechanical fluid problems considering a fixed and undeformable control volume. In addition, considering a uniform distribution of the specific mass of the fluid in each flow section (of the compressible flows) allows specific mass variations to occur only from one section to another.

Substituting (7) into (6), we obtain

$$v\_l = \frac{\rho\_O}{\rho\_I} v\_O. \tag{8}$$

An ideal gas can be characterized by having a large number of molecules, considered as spherical beads with a mass greater than zero and negligible individual volume when compared to the volume containing them [46]. Thus, the evident macroscopic properties of an ideal gas are consequences mainly of the independent movement of the molecules as a whole.

In various conditions the gases deviate from ideality, being characterized as a real gas, which is constituted by particles endowed with chaotic movement, and subjected to the forces of attraction of long-distance and forces of repulsion at a short distance. It is important to know the specific mass range in which an ideal gas equation describes the behavior of one real gas with adequate accuracy. It is important also to know how much the behavior of a real gas can deviate from the ideal gas at a given pressure and temperature. This information originates from the compressibility factor *Z*. When it is an ideal gas (*Z* = 1), the distance from *Z* of the unit is a measure of the behavior deviation of the actual gas from that predicted by the ideal gas equation [47]

$$PV = nRT\_\prime \tag{9}$$

where *P* is the absolute pressure of the gas (Pa), *V* is the volume occupied by the gas (m3), *n* is the number of moles of the gas (mols), *R* is the ideal gas constant (8.31 J/(mol·K)) and *T* is the temperature (K).

If the temperature ranges from 250 K up to 400 K, and at a pressure of 101,325 Pa, atmospheric air (compressible fluid) approaches to an ideal gas with acceptable accuracy for the system of this work [48]. If the pressure and temperature differences are small, generally less than 10%, air can be considered incompressible. Therefore, we can write Equation (8) as:

$$
v\_I = \upsilon\_O.\tag{10}$$

### **3. Materials and Methods**
