2.1. Steam Cycle
Based on the academic literature, the steam condenses in a turbine down to a pressure between 0.03 and 0.08 bar [
13]; thus, the condenser was chosen to operate at a saturated fluid temperature of 39 °C and pressure of 0.07 bar. To analyze a range of heat exchanger configurations and calculate efficiencies, the pressure that the feed pump produces is taken at 8 different values between 10 and 70 bar. As the pump would likely only slightly increase the temperature of the saturated water (due to inefficiencies), 40 °C is considered as the boiler water inlet temperature. Considering the mass flow of the exhaust gases in comparison with the study conducted by Muench et al. [
14], it was concluded that a superheated vapor temperature of 400 °C is achievable. Following their method, the analysis of the heat exchanger is conducted by splitting it into four distinct regions shown in
Figure 5, as follows: economizer, low-quality evaporator, high-quality evaporator and superheater. The characteristic parameters of the heat transfer are separately calculated for each section due to their different properties, in order to determine the necessary total heat transfer area.
The water/steam circuit starts at the inlet of the economizer and ends at the outlet of the superheater. As the water enters the low- and high-quality boiler, it can be observed that the temperature does not increase until the inlet of the superheater. Phase change takes place throughout this stage, and the energy required to transform the whole mass flow into steam is given by the latent heat of the vaporization of water. The split between the LQ and HQ boiler is taken at the point where the vapor fraction reaches 0.7, where heat transfer turns from being predominantly dictated by nucleate boiling to being driven by mist flow.
On the gas side of the heat exchanger, it is recommended to keep the temperatures above the sulfuric acid dew point, in order to avoid serious corrosion problems. Since the standard marine diesel fuel that is used for this type of gas turbine can contain a maximum sulfur content of 1% [
15], that limit is taken as 135 °C. The location with the highest risk of reaching this dew point is the water inlet of the economizer, where both the water and the exhaust gases are at their lowest temperatures. There, the parameter that needs to be monitored is the outside wall temperature of the water tubes. This temperature tends to be significantly lower than that of the gas itself due to the high thermal conductivity of the cold water. It is a complex function of flow geometries, fluid thermal parameters and temperatures T1 and T8, but for the purposes of this paper, we proposed a minimum value for T8 of 250 °C to eliminate the possibility of sulfur dioxide formation. The discussed fluid properties are centralized in
Table 2.
The required water mass flow can be easily determined by using the heat flux of the exhaust gas, given by its mass flow (
), specific heat capacity (
) and temperature difference through the boiler (
):
It is assumed there are no losses to the environment and the heat flux is fully absorbed by the water/steam. The required mass flow (
) can be calculated using a similar formula, while also considering that phase change occurs, dictated by the specific heat of vaporization (
) and the temperature at which vaporization starts (
), and considering the average heat capacities of the water (
) and steam (
) in the economizer and superheater, respectively:
For every heat exchanger configuration analyzed, is should be verified that the boiler can operate under the specified conditions: the temperature difference needs to be above zero at the “pinch point” at the outlet of the economizer, to ensure heat flow between the two fluids.
The power output of the steam turbine is obtained using the Rankine cycle characteristic to the calculated properties. Isentropic expansion is initially assumed for the calculation. The inlet enthalpy (
) and the theoretical outlet enthalpy (
) can thus be determined based on the pressure and temperature at those points. By correcting the theoretical enthalpy drop (
) with the internal efficiency factor
, which has the usual value of 0.8 for steam turbines, the real internal enthalpy drop (H′) is obtained. This can now be used to calculate the real power extracted by the turbine while taking into account mechanical efficiency
caused by small frictional losses:
The real enthalpy of the steam at the outlet of the turbine is determined as follows:
After leaving the turbine, the steam needs to be condensed, in order to be recycled through the pump, boiler and the turbine again. As previously mentioned, in real Rankine recuperation cycles, the turbine will partly condense the steam through expansion. The resulting mixture of liquid and gas is characterized by the vapor quality x, defined using the enthalpies of saturated steam (
) and of vaporization (
) as follows:
As a result, the heat exchanger needs to condense the remaining steam, down to the saturated liquid point, which requires energy:
Assuming a mass flow of cooling sea water, with a fixed tube inlet temperature (
), its outlet temperature (
) can also be calculated. The shell-side wet steam mixture will remain at a constant temperature
throughout this whole process (
Figure 6). To accommodate the regulations regarding the level of thermal pollution, the outlet temperature of the cooling sea water needs to be limited to 5 °C higher than the intake water temperature [
16,
17]. According to Tsimplis et al. [
18], the average temperature of the Black Sea is around 13.5 °C, which gives a maximum outlet temperature of the cooling water of 18.5 °C. These temperatures set the intake and discharge conditions for the cooling side of the condenser.
2.2. Hot Air Cycle
In the case of the hot air—exhaust gas heat exchanger—it is similarly assumed that the air can only be heated up to 400 °C. In this case, however, the compressor is responsible for part of the air temperature increase. For example, for a compression ratio of 4 and assuming room temperature air at the inlet, the air leaves the compressor at 180 °C. That means that the heat exchanger only needs to provide a temperature increase of around 220 °C. The lower inside the heat exchanger and the lower of the air allow for a higher mass flow to be heated and circulated through the auxiliary turbine system.
To determine the output of the power turbine in the air cycle, it is first necessary to establish the thermodynamic cycle corresponding to the specific parameters. It is first assumed that the air at the outlet of the gas generator turbine (with pressure
and temperature
) is iso-entropically expanded through the power turbine down to atmospheric pressure (
) according to the following:
The outlet temperature
is used to find the real enthalpy difference extracted by the turbine, by considering again the internal efficiency factor
:
The mass flow of the hot air and the power output of the turbine were determined in the same way as for the steam turbine. This analytical analysis was performed for a range of pressure ratings between 10 and 70 bar for steam and 2.25 and 4 bar for air. The upper limit of 4 bar for air was chosen because, for higher compression ratios, a two-stage compressor is needed and that implies higher costs and a more complex geometry for the entire assembly.
2.3. Heat Exchanger Dimensioning
The next objective is to determine the approximate weight and size of the equipment that would be needed to convert each gas turbine to a COGES configuration. To that extent, the ε-NTU method described by Shah and Sekulić [
19] is used to first calculate the surface areas necessary for heat transfer.
Starting with the boiler, the energy transferred between the fluids is calculated separately in each four-section split (see
Figure 5). For the economizer and superheater, that energy is the product of the tube-side mass flow (
), temperature difference and the average specific heat capacity of the water or steam, according to Equation (1). Meanwhile, for the vaporization process, the low- and high-quality boiler energies can be calculated, respectively, as
where each fraction is given by the percentage of steam evaporated. The effectiveness of heat transfer (ε) of each section is then calculated as follows:
is the lower value from the two rate capacities of the fluids (
) and
is the difference between the maximum temperature of the hot fluid and the minimum temperature of the cold fluid. The rate capacity ratio (CR) is defined as follows:
In the LQ and HQ boilers, the specific heat capacity of the wet steam is considered infinite, since its temperature does not change; therefore, CR = 0.
The number of transfer units (NTU) is a nondimensional parameter that can be defined based on ε, CR and the heat exchanger flow arrangement. The formula for the NTU is based on a crossflow heat exchanger (single pass), for which the fluid with the higher rate capacity is mixed, while the other is unmixed:
For the boiler section, where the NTU is calculated at the limit CR
0, the square bracket becomes 1 − ε. It should be remarked that the chosen boiler configuration is not necessarily a single pass; however, formulas for more complex geometries were tried, and similar results were obtained for the NTU. Once this number is calculated, the value for the heat transfer area (A) is obtained from the following definition:
where U is the overall heat transfer coefficient. For each section, the value of the area depends on whether the outside value (
) or the inside value (
) of the coefficient is used. In this paper, the outside areas of the tubes (
) are calculated using the shell-side value of the heat transfer coefficients,
. These values were initially chosen based on empirical data of heat exchangers utilizing the same fluids as this paper, with values between 50 and 100 W/m
2K for the boiler sections and 1000 W/m
2K for the condenser. More accurate values for
were subsequently calculated and utilized, based on the fluid temperatures and boiler areas obtained with the initial values, the method for which is explained in
Appendix A. The total transfer area of the boiler is calculated as the sum of the areas for each section. The heat transfer area for the condenser was calculated in a similar manner, though it was taken as only one heat exchanging section.
The mass of the boiler is calculated as the sum of three components: the pipes, the pipe fins and the shell. The arrangement and sizes of the tubes and fins of the heat exchanger must be chosen, in order to calculate the component masses. The configuration CF-9.05-3/4 J(c) from Kays and London [
20] was selected, as its dimensions are fitting for the counter-crossflow needed for the boiler, with large enough spacing between the pipes to allow for a high mass flow of hot gas. The metal thickness of the pipes was chosen based on industrial steel pipe schedule ratings, such that the boiler would withstand the chosen inlet pressure, given the diameter of the pipe. These thicknesses range from 2.11 mm for a 10-bar inlet to 2.77 mm for 70 bar [
21]. The heat exchanger configuration chosen specifies the ratio of the transfer area of the fins to the total one. This allows us to calculate the pipe transfer area and fin transfer area separately. Multiplying these values with the thickness of the pipes and of the fins (also given in the configuration), respectively, and further by a chosen density of steel, the inside mass of the boiler is calculated. Assuming that the shell and other components of the boiler would make up a third of its total weight, the total mass can be obtained for any chosen pressure. While the configuration of the condenser would, in reality, differ slightly from the one for the boiler, it was assumed that their masses would be proportional to the transfer area. Moreover, the boiler inlet water pressure value has little impact on the condenser transfer area, and thus a fixed condenser mass was determined as suitably accurate for any steam cycle, for a chosen sea water temperature.
A possible installation location for the heat exchanger (explained further in the Conclusions section) and its shape are shown in
Figure 7. Assuming a square cross-section of size W × W and a length L, the next step is to determine these lengths.
Before choosing the desired configuration for the boiler, another design aspect to consider is the backpressure added to the exhaust gases. This pressure is a result of losses created by friction, blockages or sudden bends that the gas turbine needs to overcome in order to discharge gases at atmospheric pressure. High backpressure reduces the extraction of energy by the turbine and can cause overheating, mechanical failures and overall reduced gas turbine efficiency [
23]. Thus, losses induced by the boiler should be minimized. For the purposes of this paper, a pressure drop lower than 5% is considered acceptable and produces no significant effect on the energy extracted. The percentage pressure drop for the exhaust gas (Δp) can be calculated using the formula given in Kays and London [
20]:
where G is the mass flow divided by the free-flow area, v is the specific volume at the inlet (
in), outlet (
out) and the mean value (
m), σ is the ratio of free flow to frontal area,
is the mean hydraulic radius of the flow, L is the flow length through the boiler and f is the friction factor. The latter is obtained by using the average Reynolds number of the flow in the specific graph for the heat exchanger configuration, available for CF-9.05-3/4 in reference [
14]. The only parameters in Equation (15) that vary with boiler size are G, dependent on the W
W area, and L. The latter value is tied to W, since the total boiler volume is fixed, determined by the ratio of total transfer area to total volume given by the heat exchanger configuration. As a result, the pressure drop can be calculated for a range of W values, aiming to keep it below 5%.
The lower bound for W was found by making sure that the free-flow area for the exhaust gas should not be reduced throughout the exhaust pipe—the heat exchanger configuration of the gas turbine—which would otherwise create significant backpressure. As these calculations are based on an ST40M turbine, the exhaust pipe diameter is predetermined, which sets a minimum free-flow area for the gas in the boiler, and thus a minimum size of W. It should be noted that, while large values of W give smaller pressure drops, these also result in small L values, which would alter the initial cross-counterflow assumption for the heat exchanger by modifying its length–width proportions. Thus, a balanced W value was selected and presented in the Results section.
For a W value greater than the diameter of the exhaust pipe, the flow path should be expanded using a diffuser, similar to the one illustrated in
Figure 8. This is an important feature of the system, as angles 2θ larger than 15° result in flow separation (and thus large pressure losses) [
24], but low angles increase the length L needed for the diffuser. Since both its inlet and outlet dimensions are known based on the exhaust and the boiler, the length of the diffuser can be calculated.