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