**1. Introduction**

Agustín de Betancourt y Molina was one of the fathers of engineering in the period of the Spanish Enlightenment [1]. This article aims to analyse from the point of view of engineering one of the most controversial inventions of his career, the double-acting steam engine, the first steam engine of its kind to reach the European continent. This invention has already been the object of a detailed study from the graphic engineering point of view, which has allowed us to obtain a reliable 3D CAD model [2] from which the present investigation has been carried out and which led to the discovery of the concept of energy symmetry with which Betancourt designed this invention. Its novelty and scientific interest lies in the fact that from the point of view of industrial archaeology and the study of technical historical heritage there is no worldwide study on this invention, which marked a historic milestone in the design of the steam engines of the Industrial Revolution (1760–1840). This underscores the utility and originality of this research.

Until the end of the eighteenth century, the steam engine known in Europe was that of the English inventor Thomas Newcomen, a simple steam engine that worked thanks to pressure differences in the two chambers of a cylinder. This invention worked from the cooling of the water vapor inside the cylinder, which produced a vacuum. The upper face of the piston, open to the atmosphere, pushed it downward causing it to return to its initial lower position. This movement of the piston operated a rocker arm that impelled other rotating elements through a connecting rod-crank mechanism [3]. Subsequently, James Watt (mechanical engineer and Scottish inventor) worked from 1765 on the Newcomen steam engine, introducing a new element (the condenser) that would triple the performance compared to its predecessor. This simple element allowed advantage to be taken not only of the vacuum produced by the water vapor when condensing but also its expansion, decreasing in this way the amount of water vapor necessary to produce the movement of the piston. He also introduced other elements to increase performance such as the planetary gear system that facilitated the movement of the inertia flywheel, among others.

In 1782 the patent of James Watt reached perfection, when the Scottish engineer adapted the superior part of the cylinder so that the admission of the steam could be realized as much below as above the piston allowing the push of the steam on both its faces.

He also improved the rocker arm designed by Thomas Newcomen. Initially, this rocker arm was attached to the cylinder by means of a chain and therefore only transmitted the movement when the piston moved in the downward direction in the cylinder [3]. However, the double-acting steam engine was attached to the rocker arm by means of a lever that connected the piston shaft to the end of the rocker arm. By means of this connection, the upward stroke of the piston was also exploited but it presented the difficulty of adapting a certain movement of oscillation in the axle of the piston since, while the rocker arm described a circular movement, the axle of the piston moved vertically. Despite the joints (Watt's extended mechanism consisting of an articulated parallelogram) designed to eliminate this movement and ensure an effective rectilinear guidance of the piston, the engine in use presented a significant maladjustment [3].

In September 1785, Betancourt began his second stay in Paris, where he returned with a double commission: to supervise the group of Spanish pensioners and to obtain plans and documents for the Royal Cabinet of Machines of Buen Retiro. During those years, he travelled through different factories and French ports taking notes and making known in Spain a large number of ideas and new techniques with which to stimulate the country's industrial progress. It is during this period that he became aware of the existence of James Watt's steam engine and the enormous progress it represented compared to that of Newcomen [4].

The Spanish engineer, interested in the news about the steam engine, obtained an interview in November 1788 with its inventors James Watt and Matthew Boulton, to be told about their patent. They showed him their factories of buttons and plated silver but none of their steam engines. Even so, he managed to visit the Albion Mills which were being built near the Blackfriars bridge. This installation consisted of three steam engines, one of which was completed [5].

So on December 16, 1789, he presented to the Academy of Sciences of Paris a double-acting steam engine based on the one designed by Watt but improved where it included the theoretical study of the extended mechanism of Watt, as well as solving for the first time in the history of the mechanisms a problem of synthesis of generation of trajectories with three points of precision [6].

As a result of his studies on the steam engine he wrote a memoir on the expansive force of water vapor, which obtained the approval of the Paris Academy of Sciences in September of that same year [7].

There are also two studies on the impact of the Spanish engineer's work on steam engines at the time [8,9] but this invention has never been analysed from the engineering point of view, which highlights the originality and convenience of the present investigation.

The ultimate goal of this study is to perform a static analysis [10] of the double-acting steam engine by the finite-element method [11] under real operating conditions in order to determine whether it was properly sized and would function properly.

#### **2. Materials and Methods**

The starting material was only the information available on the website of the Betancourt Project of the Canary Orotava Foundation for the History of Science [12]. Here the information related to the invention is shown, as well as the letter written to his brother José on March 6, 1789 in which he gives news of his steam engine and the report of the Academy of Sciences of Paris on the examination of the invention, signed by Jean-Charles Borda, Mathurin-Jacques Brisson and Gaspard Monge [5].

On the other hand, there are two Betancourt works directly related to this invention. On the one hand in 1790 Betancourt wrote his 'Mémoire sur la force expansive de la vapeur de l'eau' [7], which was one of the first treatises on applied thermodynamics in which the results of the experiments carried out with the double-acting steam machine were shown; and on the other hand, the 'Explication d'une machine destinée à curer les ports de mer' (1808) [13], in which he proposes the design of a mechanical dredger installed on a ship and whose mechanism is propelled by its double- acting steam engine. In this second case, both in the drawings and in the memoir, Betancourt explains the mechanism of the double-acting steam engine, although in less detail.

To obtain the 3D CAD model of the steam engine [2], both the six sheets and the 34-page memory were used, explaining the invention and its operation, which appear in the original Betancourt file [5].

Once the 3D CAD model was obtained the methodology followed for the static analysis object of the present investigation was the same as that used in the study of other Betancourt inventions [14–17], giving it a substantial degree of credibility.

#### *2.1. Operation of the Double-Acting Steam Engine*

Although the 3D CAD model of the double-acting steam engine and its operation are perfectly described in the previously referenced publication [2], it has been considered convenient to briefly summarize it in order to facilitate the reader's understanding, due to the high number of components and the complexity of the invention. This explanation is based on two plans along with an indication of the elements that compose the invention (Figures 1 and 2).

Figure 1 represents an isometric view of the set where it can be seen first, a brick building that houses the boiler (18) of the steam engine. This building is not large and fits the models of coal boilers of the time. Secondly, there is a large rocker arm whose balancing axis is supported by two square section columns (the previous one omitted to better see the rest of the elements). Finally, to the left and right of these columns there are two well differentiated parts: on the right, a handle-crank mechanism that moves an inertia flywheel (5) of large dimensions and on the left, the hydro-pneumatic circuit composed of a series of components that regulate its movement.

Through the metal doors of the brick building there is access to a room where the water located in a large boiler over a concave space is heated. In this space coal is burned, causing a thermal plume in the lower zone of the boiler so that the water in the boiler reaches a temperature of 100 ◦C, transforming it into steam and it leaves the boiler building through an upper pipe A (20).

The steam at high temperature reaches a steam box FF (22) that functions as a double-pass valve, that is, one position allows entry to the upper area of the steam cylinder (23) and the other position directs the steam to the lower steam box PQ (16). Thus, the valve system makes it possible to direct the water vapor to the upper or lower face of the piston (27). Valves B and C (44) are correlated to valves D and E (37), so that when the right valve of the upper steam box is open the left valve is closed and in those of the lower steam box the opposite occurs, the right valve closes and the left valve opens.

**Figure 1.** Isometric view of the double-acting steam engine.

**Figure 2.** Detail of the plan of the regulating mechanism of the double-acting steam engine.

Figure 3 shows the path of the water vapor in the pipes depending on the action of the valves. Figure 3a shows the water vapor at high pressure and temperature impacting on the upper face of the piston of the steam cylinder and causing it to fall, while in Figure 3b the opposite occurs, the water vapor at high pressure and temperature affects the lower face of the piston causing upward movement of the same.

**Figure 3.** Movement of water vapor in the hydropneumatic system: (**a**) downward movement of the piston; (**b**) upward movement of the piston.

The incidence of water vapor is not the only cause of the movement of the piston of the steam cylinder. A vacuum is also generated in the condenser (35), creating a remarkable pressure difference which favours its movement. This can be seen in Figure 3b. When the piston is reaching its highest point, water vapor at a lower temperature is dislodged from the cylinder. At that moment valve Y (13) opens, allowing the momentary entry of cold water. This drop in temperature, together with the increase of the space where the steam is located (since the piston of the air pump is rising), produces the condensation of part of the water vapor. Thus, when condensing this steam the pressure decreases locally, becoming lower than the atmospheric pressure and, consequently, both the piston of the steam cylinder and that of the air pump (29) descend.

The piston of the air pump has two small valves, E' and F' (28) that allow the upward passage of the water vapor and prevent its return, facilitating also the evacuation of non-condensed water vapor. Also, a last pipe communicates the pump with the atmosphere, presenting at its end a condensation hood so that the water condensed in that hood returns to the boiler for its reuse through a return pipe M (19).

The movement of the piston of the steam cylinder and the downward movement of the piston of the air pump produce the movement of the rocker arm (2). This is connected to an inertia flywheel by a handle-crank mechanism. On one side of the rocker arm the pistons of the steam cylinder and the air pump are connected and at the opposite end there is the connecting rod-crank mechanism. The connecting rod (4) is connected by means of a joint to a satellite gear (9) and this gear is engaged with a planetary gear (6) solidly connected to the inertia flywheel (5). Finally, the inertia flywheel can move any mechanism.

Finally, it should be noted that there is a water pump, propelled by the same movement of the rocker arm, which serves to flood a space (Figure 3) where are submerged both the air pump and the lower pipe that connects the steam cylinder to the air pump. Thus, this pump maintains the water at a constant level and serves as a water reservoir for use in the condenser.

#### *2.2. Computer-Aided Engineering (CAE)*

Based on the 3D CAD model obtained in previous studies [2], reliable results can be obtained in the computer-aided engineering phase, allowing the static analysis to be carried out correctly using Autodesk Inventor Professional software (release 2016, Autodesk, Inc., San Rafael, CA, USA).
