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Supplementary material
S1 - ORC off-design performance evaluation
The ORC off-design performance estimation is based on the model described in Baldasso et al. [1].
The turbine isentropic efficiency at off-design conditions was computed using a relationship derived
from Schobeiri [2] (see Equation 1), and it was assumed that the relationship between the mass flow
rate and the pressure was governed by the Stodola equation [3] (see Equation 2).
ηexp ,is
ηexp ,is ,des=2√ Δhexp , is, des
Δhexp ,is−
Δhexp , is, des
Δhexp ,is
(1)
CT=m√T¿
√ P¿2−Pout
2
(2)
where Δh is the specific enthalpy drop, and the subscripts is, des, in, out refer to isentropic, design,
inlet and outlet conditions, respectively. CT is the Stodola constant and ṁ, P and T represent the
working fluid mass flow rate, pressure and temperature, respectively.
The off-design efficiency of the electrical generator was estimated with the method described in
Haglind and Elmegaard [4], as indicated in Eq. 3, while the pump performance curve was derived as
described by Veres [5], as shown in Eq. 4:
ηgen=Lηgen .des
Lηgen ,des+( 1−ηgen ,des ) [ (1−F cu )+ Fcu L2 ](3)
ηpump=η pump, des[0.86387+0.3096 VV des
−0.14086( VV des
)2
−0.029265( VV des )
3] (4)
where L represents the generator load relative to its design power output, F cu is the copper loss
fraction (set to 0.235 [4]), and V the working fluid volumetric flow rate at the pump inlet.
The performance of the boiler (subdivided into pre-heater, evaporator and super-heater) and the
condenser (subdivided into de-superheater and condenser) were estimated by scaling their UA values
(product of the overall heat transfer coefficient U, with the heat transfer area, A) according to the flow
rates:
UA=UAdes( mmdes )
n (5)
The exponent n was set to 0.80 or 0.60 depending on whether the fluid which dominated the heat
transfer process was flowing inside or outside the tube banks [6]. Table S1 shows the fluid that was
assumed to be dominating the heat transfer process and the selected exponent n for the various heat
exchangers. The design values for the UA values (UAdes ¿and mass flow rates (mdes ¿were computed
for each optimized ORC design. The seawater mass flow rate was kept constant during off-design
operation (its design value was computed during the ORC design procedure), while the pump
rotational speed was adjusted to keep a fixed superheating degree at the turbine inlet.
S2 - Exhaust recovery heat exchanger design procedure
The design of the exhaust recovery heat exchanger was carried out using a thermodynamic model
described in Baldasso et al. [1]. The heat transfer coefficient and pressure losses on the exhaust gases
side were estimated using the correlations proposed by ESCOA [7], as they were specifically derived
for WHR applications. The performance of the fins was accounted for by implementing the empirical
correlation from Weierman [8], which accounts for the non-uniformity of the heat transfer along the
fins. The heat transfer on the thermal oil side was estimated by using the Gnielisnki correlation [9].
When designing the exhaust recovery heat exchanger, a minimum velocity of 20 m/s and a maximum
pressure drop of 0.025 bar were imposed for the gases. The first constraint ensures the minimization
of the risk of soot fires [10]. The second makes sure that the additional backpressure has a negligible
impact on the performance of the main engine. According the manufacturer’s product specifications,
the engine should be operated with a maximum backpressure of 0.04 bar [11], of which 0.015 bar
were assumed to be allocated for the other components constituting the exhaust system [10]. The
properties of the thermal oil and the exhaust gases were computed at the respective average
temperatures in the heat exchanger. The exhaust gases were assumed to behave as air at 1 bar, while
the properties of the thermal oil were retrieved from the manufacturer’s website [12]. The off-design
performance of the heat recovery heat exchanger was estimated using Eq. 5, taking the exhaust gases
as the fluid dominating the heat transfer process, and n = 0.6.
S3 - Stratified tank model
The behavior of the stratified tank, which was considered to have a cylindrical shape and a volume
Vtank, was estimated by means of a 1D model. The tank was described as the sum of N horizontal slices
(layers) of equal height, each behaving as a fully mixed tank. During operation, the thermal oil flows
between adjacent slices, either from the top to the bottom or vice versa, respectively, during
discharging and charging phases. The energy balance on each individual slice reads as follows:
mslice c p ,f
dT i
dti=Qflow+Qcnd+Qloss
(6)
where mslice , c p ,f , and T i represent the amount of thermal fluid mass contained in the slice, its specific
heat capacity at constant pressure, and temperature, respectively. The symbols Qflow, Qcnd, and Qloss
represent the heat transfer rates exchanged due to forced convection of the storage fluid through the
slice, conduction between subsequent slices, and heat losses to the environment, respectively.
The heat flow rates were computed as shown in Eqs. 7-9:
Qflow=mslice cp , f (T i−1−T i+1 ) (7)
Qcnd=λeff Acnd
H slice(T i+1+T i−1−2T i )
(8)
Qloss=−UAtank
N slice(T i−T amb)
(9)
where mslice is the mass flow rate through the slices, which was assumed not to vary among the slices
[13]. Conduction was assumed to be taking place both through the fluid in the inner subsection of the
tank and the ring of the tank walls. The symbol Acnd represents the surface area through which heat is
exchanged by conduction, while λeffwas computed as the weighted average of storage fluid and wall
thermal conductivities. Finally, H slice, N slice, and U tank represent the slice height, the number of slices
used to discretize the storage tank, and the heat transfer coefficient between the tank and the
environment, respectively.
The UA value for the tank, UA tank, was computed as the sum of the UA value of the top, the bottom
and the sides of the tank. These were computed taking into account the thermal resistances of the
storage tank wall (Rwall), the insulation layer (Rinsu), and the external air (Rconv). The impact of the
thermal resistance between the thermal oil and the storage wall was assumed to be negligible. Due to
the injection of the liquid, mixing effects are commonly taking place around the inlet area in the tank.
The impact of this phenomenon on the performance of the tank was taken into account by assuming
that the first K slices of the tank behaved as a single fully mixed tank. The number of slices K
constituting the fully mixed inlet section of the tank was fixed through a mixing factor, rmixing. The
same approach was previously utilized by Andersen et al. [14] and Furbo et al. [15] to investigate the
impact of mixing during hot water draw-offs on the performance of small solar domestic hot water
systems. Both studies reported the suitability of this simplification to represent the effect of mixing
phenomena in stratified tanks. More advanced methods to treat this phenomenon were described in
previous studies, including CFD simulations [16] and quasi-1D models [17]. It was decided to use the
simplified approach, because the objective of this paper was to evaluate the feasibility of the proposed
concept and not to derive a detailed model of the storage tank.
Following the approach presented in Klein et al. [18], the temperature profile through the slices was
checked at every time step and, if a temperature inversion was found, the two layers were mixed to
their average temperature. This procedure enables avoiding physically inconsistent conditions when
the tank is not subject to external circulation. The tank model was solved with a maximum time step
Δt = mslice/ṁslice, as a way to avoid numerical diffusion [19].
The described modelling framework is generally used for water tanks. In these cases, the water
density can be assumed to be constant [13]. Similarly, the assumption of constant fluid density was
previously successfully used to describe the behavior of the thermal oil stratified tank presented in the
work of Mawire [20]. However, considering the temperature ranges investigated in this study, the
thermal oil density was found to vary up to 16 %.
As a way to ensure safe operation of the tank during all working conditions, it was decided to design
the tank taking into account the density of the thermal oil at its highest operating temperature. The
density of the oil was considered to be constant at this value, to ensure the validity of the mass
conservation equations. This assumption led to solutions where both the tank size and the mass flow
rates between slices are overestimated. In practice, due to the variation of the density of thermal oil, a
portion of the tank would remain empty and the tank pressure would be kept constant by installing an
expansion vessel.
The oil thermal conductivity and specific heat capacity, on the contrary, were considered to be
temperature-dependent properties and were therefore individually calculated for each discretization
slice.
S4 - Two-tank storage model
For the case of the two-tank system, both thermal storage units were modelled as fully mixed and
were assumed to have the same dimensions and insulation levels. The energy balance of the fluid
inside a tank reads as follows:
d ( M c p, f T )dt
=mc p , f (T¿−T out )−UAtank ( T−Tamb )(10)
where M is the total mass of oil inside the tank, and m is the flow rate of mass entering/exiting
to/from the tank.
The energy balance was solved by assuming that the density (ρ f) and the specific heat capacity at
constant pressure (c p , f ) of the thermal oil were constant during a time step. The variation of the
volume of the fluid in each time step was computed as the ratio of the fluid flow rate to its density:
dVdt
= mρf
(11)
The temperature variation in each time step was computed according to the derivation shown in Eqs.
12 – 15.
ρ f cp ,fd (V T )
dt=m cp , f (T ¿−T out )−UA tank (T−T amb )
(12)
d (V T )dt
=T dVdt
+V dTdt
(13)
V dTdt
=m (T ¿−T out )
ρf−
UA tank (T−T amb )ρf c p , f
−T dVdt
(14)
dTdt
= 1V ( m (T ¿−Tout )
ρf−
UA tank (T−T amb )ρf c p , f
−T mρf
) (15)
S5 - Economic estimations
For the ORC unit, three scenarios were considered for the total capital cost: 2,000 $/kW, 3,000 $/kW
and 4,000 $/kW. For comparison, Quoilin et al. [21] suggested a total specific investment of around
3,000 $/kW for WHR ORC units of this size. The purchasing cost of the exhaust recovery heat
exchanger C ¿¿ and of the storage tank (CTank¿ were estimated using the correlations indicated as
follows, and presented in Smith et al. [22]:
CTank=11,500( V tank
5 )0.53 (16)
C¿=156,000( Abare
200 )0.89
¿(17)
where V tankand Abare represent the volume of the tank, and the bare surface area of the heat exchanger
tubes, respectively. A factor of three was used to account for indirect, engineering and working capital
costs [22] and the estimated total costs were upgraded to $2015 by using the chemical engineering plant
cost index (CEPCI).
The cost for the insulation material used for the storage tank was assumed to be included in the total
cost factor, as this generally accounts for around 10 % of the cost of the storage tank [23]. The price
of the thermal oil was retrieved from the manufacturer and set to 8 $/kg. For the batteries, the total
capital cost was assumed to be in the range of 600 $/kWh to 900 $/kWh, which is the total cost for a
marine battery system by 2020 according to DNV GL [24]. The average price of 750 $/kWh was used
in the current investigation, and the impact of this assumption was detailed through sensitivity
analyses. The annual O&M costs were assumed equal to 1.5 % of the total investment cost for both
systems.
For the LNG, three price scenarios were considered: 6 $/mmBTU, 12 $/mmBTU and 18 $/mmBTU,
with 12 $/mmBTU being the average cost of European and Japanese LNG in the years 2008 – 2016
[25].
S6 - Models validation
The ORC numerical model was previously validated for both mixtures and pure working fluids in
Andreasen et al. [26]. The validation procedure, based on previous studies in literature [27–30],
showed that the model is suitable to predict the cycle first and second law efficiency with a maximum
relative deviation of 3.3 %.
The part load model was checked against the operational data acquired through the PilotORC project
[31]. The operational data included variations in the ORC heat source and heat sink mass flow rates.
The numerical model was previously proved to be suitable to predict the ORC power output, pressure
levels, and mass flow rate with an accuracy within 5 % [31].
The exhaust recovery heat exchanger model was previously described and validated in Baldasso et al.
[1]. The sizing procedure of the heat exchanger was verified against an example from Verei Deutscher
Ingenieure [32], showing relative deviations of 0.66 % and 0.75 %, for the estimated heat transfer
coefficient and heat transfer area, respectively. For the gas side, the calculation of the heat transfer
coefficient was checked against the examples provided in Weierman [8], while the pressure drop
calculations were validated against the results presented in Mon [33]. The gas-side pressure drop
validation was based on the results of the model for 13 tube bundle geometries and 51 working
conditions and showed that, for 90 % of the working conditions, the predicted pressure drop agreed
with the results from Mon [33], with a relative deviation within 20 %. The maximum deviation was
40 %.
The performance of the developed stratified tank model was validated against the experimental results
presented by Oliveski et al. [34] and Zachar et al. [35]. Both experiments investigated the charging of
a small cylindrical water tank initially at ambient temperature. The validation procedure was carried
out by first calibrating the tank inlet mixing factor.
Figure S1 depicts the validation results against the data from Oliveski et al. [34]. After tuning the tank
inlet mixing factor to a value of 5 %, the numerical model is able to predict the temperature
distribution of the water in the tank as a function of the tank height and after different times with an
average relative deviation of 2.8 %. The maximum deviation is 18 % and occurs at t = 100 min. As
can be appreciated in the plot, the deviation between the predicted temperature and the experimental
values increases with time. This suggests an inaccurate estimation of the heat losses to the
environment in the model. However, this could be justified by the fact that the properties of the
insulation material used in the experiment were not described in the paper, and therefore they had to
be estimated.
Figure S1. Temperature profiles along the tank height for different times. Markers represent the
experimental values from Oliveski et al. [34], while the simulation results are indicated as follows:
solid line (10 min), dashed line (40 min), dotted line (70 min), and dash-dot line (100 min).
Figure S2 depicts the comparison of the model predictions with the experimental results of Zachar et
al. [35]. In this case, the optimal inlet mixing fraction was found to be 3 %, and the relative deviation
between predicted and experimental values for the water temperature in the tank is 3.2 %. The
maximum deviation between simulated and experimental temperatures is 17 %. The tank investigated
by Zachar et al. [35] was not insulated and, as indicated by the plot, the simulation model is able to
match the final temperature at the various tank heights (except for T10, whose final temperature is
underestimated by 8.4 %). The deviations that appear during the initial phase of the temperature ramp
ups can be attributed to the simplified approach used to take into account the mixing phenomena
taking place due to the flow of water entering the tank.
The performance comparison of the results of the numerical model and the experimental datasets from
Oliveski et al. [34] and Zachar et al. [35] indicates that the numerical model, despite its simplicity, is
suitable for characterizing the temperature distribution inside a thermocline storage, and therefore it
was considered to be sufficiently accurate for the scope of the present work.
Figure S2. Temperature evolution as a function of time captured by thermocouples (T1 – T3 – T6
– T10) positioned at 0.8 m, 0.71 m, 0.60 m, and 0.42 m from the bottom of the tank. Markers
represent the experimental values from Zachar et al. [35], while the simulation results are indicated
as follows: solid line (T1), dashed line (T3), dotted line (T6), and dash-dot line (T10).
Tables
Table S1. Fluids which were assumed to be dominating the heat transfer process and the exponent n
for the various heat exchangers.
Heat exchanger Fluid dominating heat
transfer process
n
WHR boiler (super-heater) ORC fluid 0.8
WHR boiler (two-phase) Thermal oil 0.6
WHR boiler (pre-heater) Thermal oil 0.6
Condenser (two-phase) Seawater 0.8
Condenser (de-superheater) ORC fluid 0.6
Table S2. Fixed parameters in the overall optimization procedure.
Fixed parameter ValueEngine (load = 85 %)Exhaust gases flow rate [kg/s] 11.4Exhaust gases temperature [°C] 365.4System fixed parametersTank type [-] Stratified/ two tanksNumber tanks discretization layers 300Oil maximum temperature [°C] 260 - 340ORC design power [kW] 1000ORC fixed parametersMaximum evaporation pressure [bar] 30 Minimum condensation pressure [bar] 0.045 Maximum working fluid reduced pressure [-] 0.8Minimum boiler pinch point temperature [°C] 20Minimum condenser pinch point temperature [°C] 5Minimum superheating degree at turbine inlet [°C] 5Seawater temperature at condenser inlet [°C] 15Seawater temperature increase [°C] 5Seawater pump head [bar] 2Seawater pump isentropic efficiency [%] 0.7Pump isentropic efficiency [%] 0.7Expander isentropic efficiency [%] 0.85Gearbox efficiency [%] 0.98Electrical generator efficiency [%] 0.98
Exhaust recovery heat exchangerLayout Staggered (equilateral triangle)Material Carbon steelCarbon steel thermal conductivity [W/ m K] 48Tube thickness [mm] 2.0Maximum gas-side back-pressure [bar] 0.025Minimum gas velocity [m/s] 20 [10]Storage tankHeight to diameter ratio [-] 2rmixing [%] 5einsu [m] 0.1λinsu [W/m K] 0.03λwall [W/m K] 17αside [W/m2 K] 7.69αtop [W/m2 K] 10αbot [W/m2 K] 5.88Mixing fraction [%] 5
Table S3. Optimized variables in the overall optimization procedure.
Optimization variable Lower bound Upper boundORCTurbine inlet pressure [bar] 1 0.8 Pcrit
ORC superheating [°C] 5 40Condensation temperature [°C] 15 40Thermal oil outlet temperature [°C] 15 340Exhaust recovery heat exchangerGases outlet temperature [°C] 60 200Tube inner diameter [mm] 21.4 216Tube length [m] 0.6 7.16Fin height [mm] 6.4 31.8Fin thickness [mm] 0.9 4.2Fin spacing [mm] 3.6 25.6Transversal pitch [mm] 42.85 114.3
Table S4. Optimized ORC design variables as a function of the maximum oil temperature. The results
refer to the configurations whose overall performance is presented in Figure 3.
Optimized variable Maximum oil temperature [ ]℃260 270 280 290 300 310 320 330 340
Turbine inlet pressure [bar]
12.42 15.24 18.10 24.28 30.00 30.00 30.00 30.00 30.00
ORC superheating [°C] 5.00 5.00 5.00 5.02 5.00 5.00 11.86 11.86 21.47Condensation temperature [°C]
29.42 29.35 29.30 29.22 29.17 29.17 29.04 29.04 28.88
Thermal oil outlet 94.82 94.49 90.91 89.29 75.32 53.89 50.86 50.86 50.70
temperature [°C]
Table S5. Optimized exhaust recovery heat exchanger design variables as a function of the maximum
oil temperature. The results refer to the configurations whose overall performance is presented in
Figure 3.
Optimized variable
Maximum oil temperature [ ]℃260 270 280 290 300 310 320 330 340
Gases outlet temperature [°C]
120.25 124.97 124.19 126.32 123.43 116.39 121.34 132.37 150.52
Tube inner diameter [mm]
21.40 21.41 21.42 21.40 21.41 21.40 21.42 21.50 21.40
Tube length [m]
6.57 6.66 6.80 7.13 6.90 7.15 7.15 7.03 7.03
Fin height [mm]
6.40 6.41 6.41 6.41 6.41 6.42 6.40 6.40 6.40
Fin thickness [mm]
1.10 1.14 1.29 1.71 1.40 1.87 2.42 1.80 1.28
Fin spacing [mm]
3.60 3.60 3.67 3.71 3.67 3.77 5.06 4.40 3.60
Transversal pitch [mm]
44.50 44.23 44.26 44.24 44.22 44.25 44.22 44.30 44.20
Table S6. Optimized exhaust recovery heat exchanger design variables as a function of the oil flow
rate reduction factor. The results refer to the configurations whose overall performance is presented in
Figure 4.
Optimized variable
Oil flow rate relative to maximum attainable flow rate [%]80 82.5 85 87.5 90 92.5 95 97.5 100
Gases outlet temperature [°C]
116.32 111.79 124.10 129.80 129.50 135.39 142.02 148.48 157.56
Tube inner diameter [mm]
21.40 25.87 21.40 21.41 21.40 21.40 21.40 21.40 21.40
Tube length [m]
7.09 6.88 7.16 7.00 7.00 7.00 6.77 6.60 7.16
Fin height [mm]
6.43 6.43 6.40 6.40 6.41 6.41 6.40 6.41 6.40
Fin thickness [mm]
1.77 3.83 0.90 0.90 0.90 1.04 0.96 0.94 0.90
Fin spacing [mm]
3.73 4.94 3.71 3.84 3.81 3.60 3.60 3.60 3.76
Transversal pitch [mm]
44.28 48.76 44.32 44.21 44.24 44.23 44.21 44.22 44.64
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Nomenclature
Acronyms
CFD computational fluid dynamics
L load
Symbols
A area, m2
cp specific heat capacity at constant pressure, kJ/kg K
CT Stodola constant
e thickness, m
E electricity generation, $
F fuel expenditures, $
F cu copper loss fraction
H height, m
h specific enthalpy, kJ/kg K
M mass, kg
m mass, kg
ṁ mass flow rate, kg/s
N number
P pressure, bar / pitch, m
Q heat transfer rate, kW
R thermal resistance, m2 K/kW
T temperature, K
U overall heat transfer coefficient, kW/m2 K
V volumetric flow rate, m3/s
V volume, m3
Greek symbols
α heat transfer coefficient, kW/ m2 K
Δ difference
ρ density, kg/m3
η efficiency
λ thermal conductivity, kW/m K
Subscripts and superscripts
amb ambient
bot bottom
cnd conduction
conv convective
crit critical
des design
exp expander
f fluid
gear gearbox
gen generator
hex heat exchanger
in inlet
is isentropic
insu insulation
is isentropic
out outlet