2.2.1. Pre-Processing

CAE analysis of the Betancourt double-acting steam engine involves high computational requirements since it consists of a large number of parts subjected to various types of stress. For this reason, it is essential to simplify the model that facilitates the analysis (Figure 4). In a similar manner, the mechanism works in several positions and the valves change noticeably the stresses of the different parts of the mechanism according to whether they are open or closed. Finally, for the static analysis of the engine the two positions in which the valves work have been chosen, since a priori one cannot see which of them is going to have greater stresses.

**Figure 4.** Simplified model of the double-acting steam engine for static analysis.

In the first place, it has been decided to remove the water boiler and the brick building that houses it since it is not a structural element of the mechanism and its stresses do not affect the rest of the assembly. However, the pipes that enter and leave the building have been taken into account and it is planned to restrict their movements in order to simulate what we would have if this construction existed.

Secondly, all the elements acting as a foundation have been removed, namely the brick supports of the rocker arm and the inertia flywheel. These elements have been replaced by actions that simulate their behaviour. Specifically, the brick supports have been eliminated and on the contrary, the axes supports that allow the movement of the rocker arm and the inertia flywheel are acted upon. Both supports are defined as articulated supports so that the lower faces do not have any degree of freedom but the axis of the supported element is free to rotate.

Thirdly, the last element that is excluded from the simulation is the water tank where the condenser is submerged and the water that cools it, since both elements do not contribute anything structurally to the mechanism and, on the other hand, water cannot be studied in the static analysis. This element is important when explaining the stresses to which the pipes are subjected due to the pressure and temperature differences that it facilitates but its influence does not go any further.

To conclude this section, it can be said that two different positions have been taken for the study based on the opening of the valves of the steam boxes. Valves B and C, as already explained, allow the passage of water vapor at high pressure and temperature to the upper part of the cylinder chamber or to the lower part. In the first position, when valve B is open and valve C is closed, water vapor passes

into the PP pipe and enters the lower part of the cylinder pushing the lower face of the piston and, therefore, causing the rocker arm to rise. In the second position, valve C is open and B closed so that water vapor enters directly into the upper part of the cylinder, pushing the upper face of the piston and therefore lowering the rocker arm. The pressures and the vacuum generated in each of them will be considered when preparing their study.

#### 2.2.2. Assignment of Materials

The next stage is to assign material to each of the elements that make up the assembly. However, the original documents of the invention do not specify the materials, although they do show different parts of the machine made of wood, metal or brick.

From the descriptions of other steam engines of the time, it is known what materials were used in those (e.g., the Watt and Boulton steam engine has been widely studied) and according to those specifications and the simple materials that Betancourt would have access to they have been specified. The materials chosen from the library of materials provided by Autodesk Inventor Professional software have been oak, cast iron for metal parts and brick for structural elements. It should be noted that since the structural elements have been excluded from the analysis for the reasons cited in the previous section, the description of the properties of the brick will be omitted.

Autodesk Inventor Professional establishes specific physical characteristics for each material such as thermal, mechanical, elasticity and breaking properties, among others. Cast iron has an isotropic behaviour and its main physical properties are: Young's modulus (120,500 MPa), Poisson's coefficient (0.30), density (7150 kg/m3) and breaking stress (758 MPa). On the other hand, oak wood has physical properties that depend on the direction in which the elements are studied, since it is an orthotropic material. The most favourable conditions occur when the material works in the direction of the grain since, in the other two orthogonal directions, the physical properties are more limited. For this reason, it is important that in the wooden parts the main axes are always those of the direction of the grain. Thus its main physical properties would be: Young's modulus (9300 MPa), Poisson coefficient (0.0001), density (760 kg/m3) and breaking stress (46.6 MPa).

#### 2.2.3. Boundary Conditions

Once the materials have been assigned and those elements that cannot be subjected to static analysis eliminated from the simulation, the next stage is to define the boundary conditions of the elements that have a support function. Each support can work in a certain way according to the degrees of freedom that define its mobility, so that the definition of this mobility will affect the static analysis of the complete assembly. The supports can be embedded, articulated, mobile or roller. Thus, the software studies the boundary conditions based on the freedom of movement of each component of the support.

In the first place, the elements that have no freedom of movement are defined, these being the surfaces that are screwed into the brick, such as the supports of the different axles (Figure 5a), as well as those that are geometrically inserted into the brick wall (Figure 5b).

Second, the articulations or elements that rotate freely are defined (Figure 6). These components cannot move longitudinally but can rotate, so they have a lesser degree of freedom. Among them are the hinges of the counterweights, the pulleys and the shaft fastenings.

Finally, it would be necessary to define those surfaces of the support that have freedom in only one direction but in the double-acting steam engine there are none.

On the other hand, the boundary conditions do not depend on the two extreme positions that have been proposed for the static analysis but the contacts between the parts change depending on the position to be studied. Autodesk Inventor Professional automatically detects existing contacts between contiguous surfaces as long as the surface is not excessively complex.

**Figure 5.** Embedded supports: (**a**) fixed elements (**b**) inertia flywheel defined as a built-in element to simulate stop situation.

**Figure 6.** Rotary supports.

2.2.4. Forces Applied

The next step before performing the simulation is to define the forces acting on the invention, since defining each of them correctly is key to quantifying the stresses that affect the steam engine.

The first of the actions that affects the mechanism is gravity. Autodesk Inventor Professional allows the user to define severity using any direction and magnitude. For modelled engine, it is defined as a generic vector of intensity 9.81 m/s2 in the direction of the Z axis and in the negative direction. When defined in a generic way, the software represents the gravity vector applied at the centre of gravity of the engine (Figure 7).

**Figure 7.** Gravitational force applied at the centre of gravity of the engine.

The second action that is going to be characterized is the force that is exerted on the face of the piston of the steam cylinder (Figure 8) and for this the treatise of the English engineer Thomas Tredgold [18] was used, which proposed a method for the calculation of the effect of the strength of the water vapor, as well as its losses and useful work.

**Figure 8.** Force transmitted by the piston: (**a**) downward movement and (**b**) upward movement.

According to Tredgold, the quantifiable losses in the steam engine would be:


Thus, the sum of all losses is 0.368 and therefore the useful energy would be 0.632.

On the other hand, the steam strength in the boiler was generally 900 mm Hg, the temperature of the non-condensed steam 50 ◦C and its force 100 mm Hg. So it will be necessary to:

$$0.900 \times 0.632 \text{ - } 100 = 468.8 \text{ mm Hg} = 0.6373 \text{ kg/cm}^2 \tag{1}$$

Since the pressure on the piston is 0.6373 kg/cm<sup>2</sup> and its area of 2827.43 cm2, a force will act on it of:

$$F\_{\text{piston}} = 0.6373 \text{ kg/cm}^2 \times 2827.43 \text{ cm}^2 \times 9.81 \text{ m/s}^2 = 17,677 \text{ kN} \tag{2}$$

Moreover, the force on the piston will be located on the lower face when valve B is open and on the upper face when valve C is open.

On the other hand, the pressure received by the face of the piston is the same as that received by the pipes directly connected to the boiler (Figure 9). Thus, depending on which valves are open or closed, the pressure in those pipes can be characterized.

**Figure 9.** Pipes with steam at high pressure and temperature: (**a**) affecting the upper face of the piston and (**b**) affecting the lower face of the piston.

When valve B is open, the pipe coming from the boiler, the upper valve box, the PP pipe, the lower valve box, the entrance to the steam cylinder and the lower part of it have a pressure of 0.6373 kg/cm2.

In a similar manner, when valve C is open the upper valve box, the upper entrance of the steam cylinder and the upper part thereof are subjected to a pressure of 0.6373 kg/cm2. The pressure that comes directly from the boiler is called the upper steam pressure. The calculation of the pressure in the lower pipe section with steam at low pressure and temperature (Figure 10) and in the air pump is somewhat more complicated. The air with water vapor that leaves the cylinder does so at a pressure of 0.6373 kg/cm2 but when it reaches the CC' pipe the water vapor is mixed with a small amount of the water in the tank that is at a lower temperature and higher pressure, causing an immediate condensation of part of the water vapor and therefore producing the vacuum.

**Figure 10.** Lower pipe with steam at low pressure and temperature: (**a**) subjected to atmospheric pressure and (**b**) subjected to a pressure below atmospheric pressure (by condensation).

Similarly, the pressure outside the pipeline is close to the atmospheric pressure since the pipe is submerged at 0.75 m, so when it is submerged at this depth the external pressure will be:

$$P = 1000 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 0.75 = 7357.5 \text{ Pa} \tag{3}$$

This pressure is equivalent to 0.0725 atm which, added to the atmospheric pressure, results in an external pressure of 1.0725 atm. On the other hand, the vacuum pressure starts at 0.2960 atm, so the difference in pressure will be that required by the submerged water pump and piping, that is:

$$1.0725 - 0.2960 = 0.7765 \text{ atm}, \text{ equivalent to } 0.8023 \text{ kg/cm}^2 = 78,678.8 \text{ Pa} \tag{4}$$

Finally, the piston of the air pump moves due to the pressure difference between the pipe and atmospheric pressure, so it will be necessary to:

$$11 - 0.2960 = 0.704 \text{ atm}, \text{ equivalent to } 0.727 \text{ kg/cm}^2 = 71.294.3 \text{ Pa} \tag{5}$$

## 2.2.5. Meshing

Discretization is the last stage before executing the stress analysis of the invention and its object is to obtain a mesh that realistically fits its geometry. As a rule, the greater the density of the mesh, the better it adjusts to its geometry. However, smaller elements need a smaller mesh size than larger ones, although this rule admits some exceptions in the case of complex geometries. Similarly, in places where a specific force is applied it is advisable to establish a higher density mesh, because if the geometry of these points is distorted the results suffer important alterations.

The software used (Autodesk Inventor Professional) presents the option of obtaining the automatic meshing of the part by adjusting some variables in a simple way (Figure 11). By default, this software establishes a mesh formed by tetrahedrons whose average size is 10% of the length of the element, a minimum size of the tetrahedron of 20% of the average size, a maximum variation between tetrahedrons of 1.5 and a maximum angle of rotation of 60◦. In the case of the present invention these values are acceptable, although a mesh of higher density will be necessary in the chain links and smaller elements.

**Figure 11.** Automatic mesh obtained from the double-acting steam engine.

In order to modify the automatic discretization it is necessary to refine the mesh on some surfaces, directly indicating the size of the tetrahedron side. This is the procedure with small elements such as valves, valve axles and screws (Figure 12). On the other hand, the software presents a serious drawback with the chain links that appear in some elements of the steam engine since these links, defined individually by sweeping a circular sector on a closed curve, have a toric geometry, meaning that the assembly of these links is carried out by defining a contact between a point of the surface of the upper link and another of the lower one.

**Figure 12.** Refinement of the mesh in areas of complex geometry.

In a similar way, the density of the mesh in the zones of contact between links is very large and the contact is established as a single point, so that all the stress is applied to an infinitesimal surface unit, obtaining enormous pressures at these points and therefore distorting the stress results. In order to control and reduce this error at these points the manual mesh control allows the operator to take a mesh density lower than the one established by default, achieving stresses more in line with the real capacity of the chain.

In terms of finite-element analysis with Autodesk Inventor Professional, a convergence criterion has been established since this analysis has been carried out by iterative methods and for a maximum number of ten cycles. Thus, the software compares the result with that of the previous cycle and if that result varies more than 5%, reiterates. However, if the difference is less than 5% the analysis is stopped, adopting the result as definitive. In this study, taking into account computational resources, the analysis used a mesh size of 1,924,288 elements and 3,353,725 nodes.
