The first step before the actual optimization is to define the objective function that will be maximized. In this study, the comparison of the heat exchangers was based on the net power output of the power plant per unit of the heat exchanger surface area. The net power output of an OTEC system depends on the gross power output that can be produced by the harvested heat and the different losses that occur in the system.
In OTEC, the heat from the sea is harvested using plate heat exchangers. A plate heat exchanger consists of a stack of plates in which the thermal energy of a relatively hot fluid is transferred to a colder one, with each fluid flowing on a different side of a plate. Therefore, the harvested heat depends on the convection heat transfer coefficient of both fluids and the heat transfer conductivity of the plate through which the heat transfer occurs. In addition, deposit layers of different natures are observed on the heat exchanger plate as the OTEC plant operates. This is called fouling and is caused by elements; organic matter; or living organisms present in the seawater. Such layers add thermal resistance to the plate, and thus decrease the heat transfer performance of the exchanger.
Part of the gross power output generated by the power plant is used for water ducting. In addition, losses happen in the heat transfer process itself and in the form of pressure drop that will be countered using pumps, decreasing the net power output of the system. The pressure drop happens in the pipes in which each fluid circulates and in the heat exchangers.
In this study, an optimization was realized for an OTEC system based on a Carnot and a Rankine cycles. For both cycles, the following assumptions were considered:
Moreover, although it has a non-negligible impact on the net power output of an OTEC power plant, the pressure drop that occurs in the piping system is not accounted for in the present paper. Indeed, it mainly depends on the length of the pipes and therefore, it has no impact on the choice of a heat exchanger.
2.1. Carnot Cycle: Concept and Equations
In this section, the conceptual OTEC system in
Figure 1a is assumed to be working on a Carnot cycle, which is described in
Figure 1b. A working fluid is heated by the warm seawater in a first heat exchanger in which it evaporates. The resulting vapor is then used to operate a turbine before being condensed by the deep seawater through a second heat exchanger. Finally, the working fluid is pumped into the evaporator for another cycle. This is an ideal cycle using isothermal heat exchange as well as isentropic compression and expansion processes, although they cannot be achieved in a practical power plant.
As this study focuses on the cycle performance, the pumping power required for water ducting is not considered. Thus, the net power output is equal to the difference between the gross power generated by the heat engine,
Wgross, and the required pumping power to counter the seawater pressure drop in the heat exchanger,
WP. The net power output,
Wnet, of the Carnot cycle is therefore given by Equation (1) [
7]:
The power generated by the cycle is given by the balance between the heat added to the system (
Qe) and the heat taken from the system (
Qc) Equation (2) [
25]:
where
ε is the heat exchanger efficiency, which is equal to the ratio of the actual heat exchange that occurs in the heat exchanger and the maximum theoretical heat exchange that could occur for an infinite counterflow heat exchanger without losses.
C is the product of the mass flow rate,
m, and the specific heat capacity,
cp, of the seawater. Subscript
ws stands for warm source and
cs stands for cold source.
Twsi,
Tcsi,
Twfe, and
Twfc are the temperature of the warm seawater at the inlet of the evaporator, the temperature of the cold seawater at the inlet of the condenser, the temperature of the working fluid in the evaporator, and the temperature of the working fluid in the condenser, respectively.
In the case of an OTEC power plant, as a phase change occurs in both heat exchangers, the efficiency,
ε, can be written as Equations (3) and (4) [
25]:
where NTU is the number of transfer units and is defined in Equation (5):
where
U is the overall heat transfer coefficient and
A is the heat transfer area of the heat exchanger.
The Lagrange multiplier method is applied, similar to Ibrahim et al., to find the conditions on the temperatures that maximize the power output of the heat engine, as can be seen from Equations (6) to (9) [
26]. The entropy balance written in Equation (6) is used as the constraint function, as it is equal to 0 when there are no irreversibilities:
The Lagrange multiplier,
Λ, is then introduced and verifies Equation (7):
It leads, using Equations (2), (6), and (7), to the following relationship at the maximum point.
From here, it is possible to express the maximum power output of the heat engine, as in [
7,
19,
26]:
In the case of OTEC, as heat exchangers are the most expensive components [
16], the goal of optimization is to maximize the net power output per unit of heat exchanger surface area. Therefore, the objective function is defined as the net power output of the heat engine, assuming it operates at the optimized temperature ratio shown in Equation (8) [
7], divided by the total surface area of both the evaporator and condenser. In addition, heat exchangers, for which the calculation is made, do not have the same surface area. Thus, given that the surface area can be changed by the addition or subtraction of plates, the net power output per unit of surface area,
, allows a more efficient comparison. It is expressed in Equation (10), which is deduced from Equations (1) and (9):
where
As is the heat exchanger surface area calculated as
, with
L and
Wi representing the length and width of the plate, respectively.
As differs from
A as it does not take into account any plate surface patterns, such as herringbone.
WP is defined as in [
7]:
where
D is the equivalent diameter of the heat exchanger,
L is the length of a plate and the Reynolds number, Re, is defined as:
with
ρw the seawater density,
v the seawater mean velocity, and
µw the seawater dynamic viscosity.
f is the friction factor and is defined as:
where
τ is the wall shear stress of the heat exchanger. However, in this study, an experimental correlation of the friction factor is used. It is assumed that the friction factor can be written as:
where
β and
ξ are two constants that depend on the heat exchanger properties.
UA, from Equation (5), is defined as follows:
where
αw is the seawater convective heat transfer coefficient,
αwf is the working fluid heat transfer coefficient,
t is the plate thickness,
Rf is the resistance due to fouling, and
λp is the thermal conductivity of the plate.
As specified at the beginning of
Section 2, fouling is neglected and
αwf is assumed to be much greater than
αw. Therefore:
where
B is a constant and
αw is calculated from the Nusselt number (Nu), defined as follows, with
λw being the water thermal conductivity and
D the equivalent diameter:
Furthermore, the following assumption is taken:
with
d,
γ, and
n being constant coefficients for the Nusselt correlation. The Prandtl number, Pr, is defined as:
leading to:
In addition, using Prandtl and Reynolds numbers definitions from Equations (12) and (19),
C can be written as:
Then, replacing NTU and
C in Equation (10) by their respective expressions described in Equations (22) and (21), the following equation of the net power output per unit of heat exchanger surface area,
wnet, is deduced:
It has been possible here to express the net power output of the heat engine as a function of the properties of the seawater side only, which will facilitate the calculations.
As the Carnot cycle is ideal, it does not reflect the actual net power output of an OTEC system. It is possible to introduce a FTT irreversibility factor, as done by Ibrahim et al. [
26]; however, this coefficient regroups different sources of internal irreversibilities, which makes it difficult to assess, and is therefore not usable for practical applications like the OTEC power plant. A better way is to proceed with a cycle that can be used for OTEC, such as the Rankine cycle. A comparison must be carried out to see if the use of a different cycle will have an impact on the choice of a heat exchanger.
2.2. Rankine Cycle: Concept and Equations
For the Rankine cycle, described in
Figure 2a,b, the net power output can be given by Equation (24). It was assumed that the working fluid at the outlet of the heat exchangers was in a saturated state. This cycle differs from the Carnot cycle, as it can be used practically in power plants. In this case, heat exchange is not isothermal, and a change in entropy occurs in both compression and expansion processes.
As for the Carnot cycle, the objective function is the net power output of the heat engine divided by the total heat exchanger surface area:
where
mwf is the mass flow rate of the working fluid.
h1,
h2,
h3, and
h4 are the enthalpy values of the corresponding points in
Figure 2 and with:
h’1 and
h’3 are the enthalpy values of the corresponding points in the case of an isentropic process, and
ηT and
ηp are the turbine and pump efficiencies, respectively. For the calculations, the turbine efficiency is equal to 0.85 and the pump efficiency is equal to 0.8.
Enthalpy values can be calculated from temperatures
T2 and
T4 at Points 2 and 4 using the software REFPROP [
27].
T2 and
T4 can be evaluated from Equation (26):
h2 and
h4 are calculated using the respective temperature and because the fluid is at a saturated state. It is possible to calculate
h3 using
T’3 =
T4,
s2 =
s’3, and Equation (26). For Point 1,
s4 =
s’1 and
P2 =
P’1;
P is the pressure of the fluid and
s is its specific entropy. NTU is calculated as detailed for the Carnot cycle in Equations (15)–(22). Details for the calculations to obtain Equation (26) are given in
Appendix A.