## Modeling and Analysis of Engine Performance

The Marnoch engine uses a pneumatic rotary actuator and a transmission system that converts mechanical energy provided through a flywheel to electricity in an electrical generator. Any cylinder configuration can be used and it is not restricted to a rotary actuator. The transmission can be a belt drive or direct drive, similar to systems used in wind turbines. The differential in pressure between the heated tank and the cooled tank drives the actuator. The size of the actuator depends on the size of the generator used. When the gas is highly compressed within the piston cylinder, the temperature differential needed to generate a sufficient pressure change is proportionately less than at a lower initial pressure. A prototype Mar-noch engine is shown in Fig. 14.2.

A thermodynamic model was developed to predict the performance and efficiency of the Marnoch system. Also, a control system was developed for automation of valves and flow exchange between the pressure vessels.

Analysis results are presented in Figs. 14.3 and 14.4 and Tables 14.1-14.5 of operating performance over a range of operating temperatures and time intervals. Based on the results, the performance of the device is promising and significant potential exists for higher power output at higher pressures within the pressure vessels.

Fig. 14.2 Photograph of Marnoch prototype engine.

A pressure differential is initially created between two pressure vessels, through a temperature difference. When the device begins operation, these vessels have the same initial condition. Heat exchangers within the vessels are used to transfer heat from an external source to the interior of the vessel. Depending on whether the system is heated or cooled, a thermal energy source or sink is connected accordingly. Once a pressure differential has been generated between the vessels, a specialized piston assembly is utilized to convert the pressure differential to mechanical energy.

The assembly consists of two chambers separated by the piston. The piston moves back and forth, thereby varying the sizes of the chambers. As the volume of the first chamber increases, that of the second decreases. The chamber of minimum volume is connected to the vessel with a higher pressure, while the lower pressure vessel is connected to the chamber with the larger volume. The pressure difference between the two chambers results in a net force on the piston, causing it to move. Once the piston has reached the end of the cylinder, a valve is activated. This yields a net mass flow from the high pressure to low pressure vessel, as the gas within the chamber of decreasing volume is transferred to the low pressure vessel.

 200 -1-1-1-1- 113 F 190 140 F ----- ' - 167 F 180 194 F ____—-- 1 170 160 1 1 [ 1 1

Time(s)

Time(s)

Plot of Pressure vs. Time in Tank 2

Plot of Pressure vs. Time in Tank 2

Fig. 14.3 Variations of pressures in tanks at varying source temperatures, for initial conditions in the tank of 69.8°F and 164.7 psia.

Plot of Pressure vs. Time in Tank 1

Plot of Pressure vs. Time in Tank 1

Fig. 14.4 Pressure in tank 1 at varying source temperatures, for initial conditions in the tank of 69.8"F and 200 psia.

Time(s)

Fig. 14.4 Pressure in tank 1 at varying source temperatures, for initial conditions in the tank of 69.8"F and 200 psia.

Table 14.1 System operating parameters (for case 2 with a 40 C temperature difference in Table 14.4).

Tank inner diameter (in.) Tank volume (US gallons) Tank operating pressure (psig) Tank design pressure (psig) Coil tube outer diameter (in.) Coil tube thickness (in.) Coil tube inner diameter (in.) Number of turns Pitch (in.)

Coil outer diameter (in.)

Coil length (in.)

Coil tube length (ft)

Percent of total tank volume in tubes

Total surface area of coil (ft2)

Inner surface area of tank (ft2)

23.5 431.83

3000 (205.08 atm absolute)

3500

0.109 1.282 160 1.5 22 210.00 921.53 19.59 361.89 117.92

An analytical model was developed to predict the power output and number of strokes, based on the initial conditions within the pressure vessels. This model is based on principles of conservation of mass and Bernoulli's equation (Cengel and Kern, 1950; Mulley, 2004; Turner, 2005). For a fixed time interval, it is assumed that each side of the system is at constant temperature and no losses occur during the pressure transfer. Using the Bernoulli equation, it can be shown that dm "dt~

In Eq. (14.1), k denotes the ratio of specific heats. Also, m, t, A, p, and p, respectively, denote mass, time, area, pressure, and density, and the subscripts u and d denote upstream and downstream, respectively. Using this equation, the mass flow rate from each tank into the cylinder can be determined. Based on the mass flow rates, numerical integration of the functions below lead to the predicted change of mass with time in each tank as follows:

Table 14.2 Data for heat exchangers (for case 2 with a 40 C temperature difference in Table 14.4).

Overall heat transfer coefficient, U (Btu/h ft2 F) 23.19

Table 14.3 System performance with a 80 C (144 F) temperature difference.

Temperature of heat source (oC/oF) 90/194

Temperature of cold sink (oC/oF) 10/50 48 gallon tank configuration (*4 tanks)

Overall heat transfer coefficient, U (Btu/h ft2 F) 9.88

Heat transfer rate, Q (Btu/h)*(per tank) 62,165

Operating pressure (psig) 300

Carnot efficiency (%) 22.03

Projected hourly output for 4-tank (2 pair) configuration, each tank having 48 8.06 gal. (kW h)

Projected daily output for 4-tank configuration, for 10 h/day operation (kW h) 80

Annual revenue for 10 h operation/day at \$110/MW h (\$) 3212.00 432 gallon tank configuration (*8 pair of tanks)

Overall heat transfer coefficient, U (Btu/h ft2 F) 9.46

Operating pressure (psig) 300

Carnot efficiency (%) 22.03

Projected hourly output for 8-pair tank configuration, each tank having 432 128 gal., at 50% efficiency (kW h)

Projected daily output for 8-pair tank configuration, for 10 h/day operation (kW h) 1277

Projected daily output for 8-tank configuration, operating 10 h/day high output 2171 and 14 h/day low output (kW h)

Projected daily output for 8-tank configuration, for 24 h/day steady operation (kW h) 3065 Revenue per day at \$110/MW h for 10 h/day high output and 14 h/day low output (\$) 238.84

Revenue per day at \$110/MW h for 24 h/day steady operation (\$) 337.19

Revenue per annum at \$110/MW h for 24 h steady operation (\$)_123,074

* The heat transfer rate is evaluated as Q = UAA.T.

These equations are based on the pistons and pressure vessels having the configuration shown in Fig. 14.1b. Based on the mass within each of the piston chambers, the pressures at points 3 and 4 are also calculated as a function of time. With this information, it can be shown that the boundary work produced by the system during a given power stroke can be calculated as follows: