Components present in the cycles are modeled using various techniques and are discussed in more detail below. Turbines and compressors are analyzed using isentropic efficiencies. Counter-flow heat exchangers (HX) are modeled using the effectiveness-NTU method while simplified “black box” heat exchangers that use a simple energy balance for state point calculations are used in lieu of more detailed component models where data are available. The lead-cooled fast reactor is assumed to be a black box heat exchanger because of the constant heat input and state points on the sCO inlet and outlet are provided. The molten salt loop for the CSP is modeled with necessary components including hot and cold TES, receiver, pumps, and counter-flow heat exchangers.
2.1.2. Black Box and Counter-Flow Heat Exchangers
Black box heat exchangers are simplified heat exchangers which have no approach temperature or pinch point and are modeled as a perfect heat transfer into or out of the cycle. These heat exchangers use an energy balance with mass flow inlet energy, heat input or output, and mass flow outlet energy. The energy balance equation used for all black box heat exchangers is Equation (
1).
In this equation, the energy input to the system is on the left hand side with multiplied by being energy from the mass flow while is heat transfer directly into, positive, or out of, negative, the flow from an outside source. The right hand side of the equation is heat leaving the black box heat exchanger with and enthalpy of . Black box energy balances are used in three situations, the receiver, LFR heat exchanger, and pre-cooler heat exchanger. These heat exchangers are not exhaustively modeled because the state points on the inlet and outlet are defined by design parameters.
Counter-flow heat exchangers are modeled with two fluids flowing in opposite directions exchanging heat from the hot side to the cold side. The smallest temperature difference of the hot and cold flows on either the low or high end of the heat exchanger is defined as the approach temperature of the counter-flow heat exchanger and a calculation is performed to identify whether the hot end of cold end is limiting A diagram showing a simplified counter-flow heat exchanger is illustrated in
Figure 1.
Additional assumptions of the counter-flow heat exchanger model are: no heat loss to the surroundings, no pressure drops across the heat exchangers, and no fouling resistances. In
Figure 1, the subscript ‘out’ denotes where the streams are leaving, ‘in’ denotes the entering streams, ‘c’ and ‘h’ signify cold and hot streams respectively,
is the total heat transfer from the hot to cold stream, and
is the heat transfer to the surroundings.
Counter-flow heat exchanger calculations require two known state points, fluid properties, mass flow rate of hot and cold side, and a specified approach temperature. In the modeled cases, the approach temperature is set to value of
C, based off prior model development of sCO
Brayton cycle heat exchangers [
22]. The fluid libraries referenced are built into Engineering Equation Solver (EES) for Carbon Dioxide and Salt (60% NaNO
40% KNO
) [
23,
24].
To analyze the counter-flow heat exchanger, a side is chosen, usually the high side, to start the calculations. The approach temperature is initially subtracted from the hot stream on the high-temperature side to find the missing cold temperature according to Equation (
2).
where T
is the cold stream outlet temperature and T
is the hot stream inlet temperature. Knowing the two state points allows for the enthalpy out to be found using correlations from the fluid property libraries. This enthalpy then allows for the heat transfer of the heat exchanger to be found with Equation (
3).
where
is the total heat transfer rate from the hot stream to the cold stream,
is the mass flow rate of the cold stream,
is the enthalpy at the outlet of the cold side, and
is the inlet of the cold side. The known heat transfer of the counter-flow heat exchanger can then solve for the enthalpy out of the hot stream, h
. This is accomplished with Equation (
4).
Knowing the hot stream enthalpy out allows for all states to be set on the outlets and inlets of the counter-flow heat exchanger. The temperature difference of the low side is then checked to ensure that it is larger than the approach temperature, defined at C. If the temperature difference on the low side is smaller than the approach temperature, the same computations are carried with the low side as the starting point.
Knowing the state points on all inlets and outlets of the counter-flow heat exchanger allows for the heat exchanger performance metrics to be calculated. Performance metrics include effectiveness (
), capacitance (CR), conductivity (UA), and number of transfer units (NTU). for heat exchangers. Effectiveness is the ratio of the actual heat transfer rate to the maximum heat transfer rate, a perfect heat exchanger has an effectiveness of one with no approach temperature. Assuming the approach temperature is on the high side, the maximum heat transfer rate,
is found with the maximum enthalpy. Maximum enthalpy of the cold stream is found with correlations by setting the temperature to T
with same pressure on the cold outlet. Using the maximum enthalpy,
, the maximum heat transfer rate is calculated using Equation (
5).
Calculating the maximum heat transfer rate allows for effectiveness to be calculated using the ratio in Equation (
6).
All of the prior equations are carried out in a built-in function within EES. EES is an iterative solver, and therefore if there is a feasible solution, the functions can take any of the four state points around the heat exchanger and converge on a solution.
After the effectiveness is solved for, capacitance ratio is necessary. The capacitance ratio is defined as the average minimum capacitance rate,
, over the average maximum capacitance rate,
. Average capacitance rates for the hot and cold streams are found by multiplying the addition of the specific heat at the inlet and outlet of the stream by the mass flow and dividing by two as shown in Equation (
7).
where
is the average capacitance rate across the hot or cold stream and
and
is the specific heat at the inlet and outlet respectively. Specific heat is found using library correlations, with the average capacitance rate assumed to be constant during the analysis. Once both average capacitances are calculated for the hot and cold streams, one has a larger value,
, and one has a smaller value,
. These maximum and minimum values are used to find the capacitance ratio,
, in Equation (
8).
Assuming constant average capacitance rate is suitable for most engineering purposes, especially when there is uncertainty associated with other design parameters [
25]. To justify the assumption of constant average capacitance rate, two graphs for the LTR and HTR are plotted. To ensure that there is no internal pinch point, the temperatures of the hot and cold streams as a function of dimensionless length in the LTR and HTR are shown in
Figure 2b,d respectively. Additionally, to confirm approximate linearity of specific heats at differential steps throughout the counter-flow heat exchanger, the specific heat as a function of dimensionless length of the hot and cold streams in the LTR and HTR are plotted in
Figure 2a,c respectively.
The calculations used to discretize the counter-flow heat exchangers into a sub-heat exchanger model, shown in
Figure 2, is from Dyreby [
26].
Figure 2 is constructed using the most extreme temperature values experienced by the recuperators—the cold inlet and hot outlet are the lowest and highest modeled temperature values, respectively. As demonstrated, the capacitance ratio determines the approach temperature instead of any possible internal pinch point within the recuperator. All pinch points recorded are on the high-temperature end.
2.1.3. Lead-Cooled Fast Reactor
Lead-cooled fast reactors use energy from a controlled nuclear reaction to heat molten lead. This lead is used to cool the core and transfer heat into the sCO
Brayton power cycle [
27,
28]. The lead-cooled fast reactor is assumed to be a black box heat transfer and is labeled in the cycle models LFR HX. The inlet, outlet and heat transfer rates are provided by our industry partner, Westinghouse, making the black box simplification viable. The energy balance for the black box assumption can be seen in Equation (
9).
where the left hand side,
,
, and
, is the energy into the flow and the right hand side,
and
, is the energy brought out from the flow of sCO
. The amount of energy transferred into the cycle,
, is set at 950 MW, and outlet temperature of the sCO
from the LFR HX is set at a value of
C. The outlet temperature of the LFR is specified because of high-temperature material limits on the LFR lead side. The low-temperature side is allowed to vary over a range of values with some considerations. The lead flow velocity is limited by the erosion of the fuel, the slower the lead flow velocity reduces fuel erosion and therefore leads to a more desirable compact design. At constant lead velocity (and hence mass flow rate), reducing sCO
inlet temperature allows for a higher coolant temperature increase in the LFR core and hence a higher thermal power output LFR sCO
inlet temperature has a lower bound of 340
C before the lead begins to freeze, which is operationally unacceptable. When the inlet temperature of sCO
is increased the temperature difference across the LFR is decreased leading to an increase in power conversion cycle thermodynamic efficiency but a reduction in LFR power below 950 MW. There is a compromise between high LFR efficiency and LFR power, and therefore a temperature of 400
C for the sCO
inlet temperature is the optimal value provided by Westinghouse.
2.1.4. Concentrating Solar Power Cycle
The CSP salt cycle modeled in this paper is composed of hot and cold thermal energy storage, pumps, receiver, sCO
-to-salt counter-flow heat exchanger (C2S), and CSP counter-flow heat exchanger (CSP HX). The diagram for this CSP salt loop is shown in
Figure 3.
The CSP salt cycle uses 60% sodium nitrate, NaNO
, and 40% potassium nitrate, KNO
, ‘solar salt’ as the heat transfer fluid. Solar salt stored in the hot TES can be dispatched on demand through the CSP HX when grid demand increases and held when grid demand is low. Current CSP salt cycles heat solar salt with receivers and store it in hot TES tanks at temperatures around 565
C. Future CSP salt cycles are hypothesized to have bulk hot TES temperatures of up to
C, but here the hot TES temperature is set at
C for all modeled cycles [
29] as this has been commercially proven. The cold TES temperature takes on three different values according to cycle configuration capabilities:
C,
C, and
C. In addition to the lower hot TES temperature, current CSP salt cycles lack a secondary option for charging the hot TES [
30]. The studied CSP salt cycle has two TES charging options: a receiver, which generates heat from a heliostat field, and C2S heat exchanger, which draws excess heat from the sCO
Brayton cycle. While the hot TES is charging, the receiver and LFR are storing heat for later use when grid demand increases. The hot TES storage is not dispensing salt for use in the CSP cycle while charging.
The C2S heat exchanger is active in the ‘charging’ cycle operating modes, when the focus is on heat storage for later use. Pump 1 is actively moving solar salt from cold TES to hot TES through the C2S heat exchanger extracting heat from the sCO Brayton cycle. Additionally, while the focus is on heat storage, and the heliostat field is inputting heat, pump 2 is actively transporting solar salt through the receiver to be stored in the hot TES.
‘Non-charging’ cycle operating modes are characterized by operations wherein the CSP salt cycle is discharging the hot TES, the C2S heat exchanger is not transferring heat, and the LFR is dispatching heat directly to generate electricity. When electrical generation is occurring and solar resource is available, the heat input in the CSP salt cycle is modelled through a black box energy balance across states S6-A and S1-A with a heat addition of 750 MW from the heliostat field. The hot TES solar salt is moved through Pump 3 and transfers heat into the sCO Brayton cycle through CSP HX to be converted into electricity. The cooled salt is stored in cold storage and moved through Pump 2, where the heat from the receiver is again transferred into the CSP cycle.
When grid demand for electrical power increases, a series of operating modes are activated. During the highest demand times, cycle operation focuses on maximum electrical generation. This is achieved through the C2S being turned off for direct electrical production from the LFR and the hot TES is discharging heat through the CSP HX for electrical production in the sCO Brayton cycle. As grid demand diminishes, CSP HX ramps down heat extraction until no power is being dispatched through the salt and the hot TES begins charging. During this process, the LFR gradually adds a larger fraction of heat input to the TES through C2S, supplementing the heat produced by the CSP which is also used to charge the TES. This process continues until no electrical production is occurring in the cycle and all heat is stored in TES for later use.