Steam Thermodynamic Model

A separate model was created for the case of operation with steam. This model required the following parameters; supply steam pressure, supply steam temperature, supply steam quality, exhaust back pressure, operating speed, cylinder stroke volume, CS [%], CR [%] and EP [%]. The P-V diagram was determined in a similar manner to the air model. However, it required the use of steam tables to take into account the change in steam properties during the engine cycle.

Steam properties at point 1 were determined using the inlet steam pressure, temperature and quality. Again, the volume at point 1 was set equal to the specified 𝑉𝑐𝑠.

The steam properties were assumed to stay constant during the steam admission from point 1 to 2. The volume at point 2 was determined using Equation (3.1). The specific volume at point 2, 𝑣2, was obtained using steam tables. From this, the mass of steam in the cylinder at point 2, 𝑚2, could be calculated using

The expansion process from point 2 to 3 was assumed to be an isentropic process therefore 𝑠3 = 𝑠2. The volume in the cylinder at point 3, 𝑉3, was calculated using Equation (3.3). The mass of steam in the cylinder at point 3, 𝑚3, was set equal to 𝑚₂ assuming no leakage of steam past the piston. The specific volume at point 3 was then calculated by

The enthalpy, ℎ3, internal energy, 𝑢3, and pressure, 𝑃3, at point 3 were obtained from steam tables using 𝑠3 and 𝑣3. Point 3 represented the conditions in the cylinder just before the exhaust port was exposed. The exhaust process from point 3 to 4 is shown in Figure 3.2. Here 𝑚e and ℎe represent the mass and enthalpy of the steam exhausted to the atmosphere from the cylinder.

𝑚₂ =𝑉2

𝑣₂ (3.12)

𝑣₃ = 𝑉3

Figure 3.2: Exhaust process schematic

The mass and properties of steam left in the cylinder at point 4 after exhaustion were calculated using the conservation of energy equation for an open system (Cengel & Boles, 2011).

The steam inside the cylinder was modelled as the system. Here 𝐸mass,in and 𝐸_mass,out represent the energy transfer associated with mass flow into and out of the system respectively. The terms 𝑄 and 𝑊 represent the heat transfer and work transfer respectively.

Equation (3.14) was reduced to

for the case of negligible potential and kinetic energy changes in the system. It was assumed that no work was performed during this process therefore 𝑊𝑖𝑛 and 𝑊𝑜𝑢𝑡 are set to zero. The process was assumed to be adiabatic such that 𝑄𝑖𝑛= 0 and 𝑄_𝑜𝑢𝑡 = 0. There is no mass flow into the system therefore 𝐸𝑚𝑎𝑠𝑠,𝑖𝑛 = 0. Here the subscript 𝑒 terms refer to the properties of steam leaving the system.

The mass leaving the cylinder, 𝑚e, is equal to 𝑚3− 𝑚4. Equation (3.15) was further reduced to

Δ𝐸_system= 𝑄_in− 𝑄_out+ 𝑊_in − 𝑊_out + 𝐸_mass,in− 𝐸_mass,out (3.14)

𝑚₄𝑢₄− 𝑚₃𝑢₃ = −𝑚_e(𝑢⏞ _e + 𝑃_b 𝑣_e ℎ𝑒

) (3.15)

Rearranging gave

which required iterative solving for 𝑚4. The exhaust steam enthalpy, ℎe, was taken to be the average between ℎ3 and ℎ4. The steam remaining at point 4 was assumed to be in the wet steam region at atmospheric pressure. The mass of steam at point 4 is given by Equation (3.18).

Here 𝜌4 is the density at point 4 and 𝑉4 = 𝑉3.

The enthalpy of the steam at point 4, ℎ₄, was acquired from the saturated steam tables using atmospheric pressure and 𝜌₄. The internal energy, 𝑢₄, was then acquired from the saturated steam tables using atmospheric pressure and the previously acquired ℎ4. The mass calculated in Equation (3.18), the enthalpy, ℎ4, and the internal energy, 𝑢4, were then substituted into Equation (3.17).

Since 𝜌4 was unknown, the above process was performed for density values from 𝜌4 = 𝜌liquid to 𝜌4 = 𝜌vapor of saturated steam at 101.325 kPa. The density and correlating enthalpy and internal energy which resulted in the minimum difference between the LHS and RHS of Equation (3.17) was taken to be the solution.

The compression stage from point 4 to 5 was assumed to be an isentropic process therefore 𝑠5 = 𝑠4. At point 5 the piston is back at TDC such that 𝑉5 = 𝑉1. Assuming no leakage during compression, the specific volume at point 5 could be calculated using

The enthalpy, internal energy and pressure at point 5 were obtained from the steam tables using the known entropy, 𝑠5, and the calculated specific volume, 𝑣5. A constant volume pressure rise was assumed to occur from point 5 to 1 as the inlet valve opened. The steam consumption was determined using Equation (3.11).

The work performed in a single engine cycle was calculated by determining the work output during each stage and summating them. The work during steam admission was calculated using Equation (3.20). This simplified to Equation (3.21) with the cylinder pressure 𝑃 assumed constant during steam admission.

𝑚₄ = 𝑚3ℎe+𝑚4𝑢4−𝑚3𝑢3

ℎe (3.17)

𝑚₄ = 𝜌₄𝑉₄ (3.18)

𝑣₅ = 𝑉5

The work output during the expansion stage 2-3 was obtained by simplifying Equation (3.14) with the assumption that no leakage occurs from the cylinder during the expansion stage such that 𝑚3= 𝑚2. This gave

for the case of negligible changes in kinetic and potential energy. The process was assumed to be adiabatic such that 𝑄in = 0 and 𝑄out= 0 . The process was modelled as a closed system therefore 𝐸mass,in and 𝐸mass,out are zero. Here 𝛥𝑈 is the internal energy change from point 2-3. The work performed during the expansion process, 𝑊2−3, was determined from

with 𝑢₂ and 𝑢₃ being the previously obtained specific internal energies at point 2 and 3 respectively. Again, it was assumed that no work is performed during the process from point 3 to 4 while the exhaust port is exposed. The work performed during the compression stage, 𝑊4−5, was determined using

This results in a negative value as work is required to compress the remaining steam from point 4-5. The constant volume pressure rise from point 5-1 was assumed to perform no work. The total work performed during a single engine cycle, 𝑊cycle, was then calculated using

The power output, 𝑃out, could then be calculated using

with 𝑓 being the frequency of engine cycles per second. 𝑊₁₋₂ = ∫ 𝑃 𝑉2 𝑉1 𝑑𝑉 _(3.20) 𝑊1−2 = 𝑃1(𝑉2− 𝑉1) (3.21) 𝑄_in− 𝑄_out ⏞ 0 + 𝑊₂₋₃ = 𝛥𝑈 + 𝛥𝐾𝐸 + 𝛥𝑃𝐸⏞ 0 (3.22) 𝑊₂₋₃= 𝑚₃(𝑢₂− 𝑢₃) (3.23) 𝑊₄₋₅ = 𝑚₄(𝑢₄− 𝑢₅) (3.24) Wcycle= Δ𝑊1−2+ Δ𝑊2−3+ Δ𝑊4−5 (3.25) 𝑃_out= 𝑊_cycle× 𝑓 (3.26)

4 Design Process

The following chapter presents the design process for the conversion of an existing ICE into a prototype reciprocating steam engine with variable valve duration.