Studies of a Mechanically Pumped Two-Phase Loop with a Pressure-Controlled Accumulator Under Pulsed Evaporator Heat Loads

Abstract

As avionics become more power dense, electronic device cooling has become a significant barrier to aircraft integration. A mechanically pumped two-phase loop (MPTL) is a thermal subsystem that enables near isothermal evaporator operation, which is desirable for electronics cooling. The goal of this study was to integrate an MPTL with a pressure-controlled accumulator and model a predictive control technique to demonstrate improvements for transient, isothermal evaporator operation for MPTLs under pulsed evaporator heat loads. The model predictive controller enables active control of MPTL compressible volume, which has not been demonstrated for pulsed evaporator heat loads. Experimental data were collected to validate a representative numerical model. A pressure-controlled accumulator was added to an MPTL to experimentally characterize the system thermodynamic response for three pulsed evaporator heat loads. Two statistical methods were used to assess the numerical model agreement with the experimental results. Under pulsed evaporator heat loads, the mean percent error agreed within 3.45% and the mean average percent error agreed within 0.74% for the three pulsed evaporator heat loads. Finally, a traditional proportional–integral (PI) controller and an advanced model predictive controller were developed and integrated into the validated numerical model. Both control methods were evaluated for an expanded set of evaporator heat load profiles to analyze transient behavior. For evaporator heat profiles with high heat transfer rates, the model predictive controller can maintain a target ±2 K refrigerant temperature at the evaporator exit throughout the evaporator heat load duration, whereas the PI-controlled MPTL cannot. Through this work, active control of a pressure-controlled accumulator within an MPTL is shown to improve refrigerant isothermal (±2 K) operation when compared to a traditional control technique.

1. Introduction

Aircraft electrical power requirements have increased exponentially over the first hundred years of powered flight to support on-board technological advancements including flight computers, actuators, power generation and distribution systems, directed energy, and propulsion systems [1]. This trend is projected to continue with the development of future high-power and power-dense electronic components [2]. Next-generation electronic systems, such as radar, are expected to generate a heat flux greater than 1000 ${\displaystyle \frac{\mathrm{W}}{\mathrm{c}{\mathrm{m}}^2}}$ in waste heat [3], which must be removed from an aircraft to ensure safe and continuous operation. Traditional aircraft thermal subsystems and heat sinks are approaching the limit of their ability to adequately transport and dissipate ever-increasing heat loads. To address this shortcoming in cooling capability, novel aircraft thermal management subsystem concepts, such as two-phase solutions, have been investigated and characterized to provide constant surface temperature heat rejection from a platform [4].

A mechanically pumped two-phase loop (MPTL) is a thermal subsystem that leverages refrigerant phase change to provide a near-isothermal evaporator temperature for cooling electronics [5]. MPTLs have been fielded in terrestrial applications [6], microgravity flow boiling experiments [7], and spacecraft thermal control systems [8]. In addition, they have been studied for high-heat-flux aircraft avionics applications [9]. MPTLs provide many benefits compared to single-phase pumped loop systems, including a lower required mass flow rate to maintain heat source surface temperature, a low-pressure differential throughout the cycle, a narrow temperature operation range, an ability to operate with physically distributed heat loads (i.e., multiple evaporators), and stable operation over a broad heat load range [10,11,12]. Unfortunately, MPTLs have known issues with evaporator start-up and shut-down [10,13], which may lead to pressure oscillations [14] and could result in hardware damage [15] through evaporator dryout or partial dryout [16].

These known issues led to the study of MPTL system dynamics with numerical and experimental methods by several researchers. A numerical model was developed and experimental data used to chart an increase in gas- and liquid-phase pressure when a 100 W step evaporator heat load was applied to an MPTL. This heat load decreased the refrigerant mass flow rate through the system as refrigerant mass was transferred to the accumulator throughout the heat load duration [17]. Refrigerant mass exchange between the accumulator and the rest of the MPTL system has been identified by other research groups as influencing system dynamics [18]. A research group described the importance of the accumulator in transient operation by equating it to the system ‘brain’ to manage temperature or pressure. This group found that temperature overshoot, pressure fluctuation, and evaporator temperature change are related to accumulator mass exchange [5]. The impact of the accumulator volume on the transient response time was evaluated and it was found that a large-volume accumulator is less sensitive to changes in evaporator input power than a small accumulator [19]. An analytical model was developed to describe the accumulator volume’s influence on MPTL system performance. This model showed that the accumulator volume has a large effect on boiling temperatures and evaporator exit vapor quality [20]. Another study demonstrated that pressure oscillations within the accumulator are determined by the evaporator heating load, accumulator volume, and initial thermodynamic state of the loop and accumulator [21]. Studies have been performed to evaluate the impact of accumulator volume [20] and the location of the compressible volume provided by an accumulator within an MPTL system [22]. Compressible volume exists in components where refrigerant vapor is present, such as an accumulator. The compressible volume allows a system to respond to internal pressure changes without transmitting the pressure change through the remainder of the system. Compressible volume has been passively implemented through the presence of a surge (expansion) tank to dampen flow instabilities [3]. In summary, transient evaporator events such as start-up and shut down are proven to impact MPTL system pressure, temperature, and refrigerant mass flow rate due to mass exchange between the compressible volume accumulator and the remainder of the system. Thus, there is a need to further explore the integration of a compressible volume accumulator within an MPTL. It is imperative to address highly transient evaporator heat loads to maintain the evaporator setpoint temperature.

Various accumulator technologies have been developed to improve MPTL evaporator temperature control. Pressure-controlled accumulators [7] and temperature-controlled accumulators [8] have been used in spacecraft applications by NASA. Temperature-controlled accumulators add or remove heat from the accumulator to maintain the system saturation pressure. To achieve a desired saturation pressure, pressure-controlled accumulators modulate the system volume. Pressure-controlled accumulators have been reported to respond faster than temperature-controlled accumulators [19], which is important for pulsed heat load applications. The fielded spacecraft accumulator technologies target evaporators with low magnitudes of heat input, at 10–100 W [23]. Accumulator technologies require further investigation to assess the integration feasibility of an accumulator into an MPTL for future high-performance aircraft thermal subsystems [2].

The MPTL transient performance is also impacted by the selected control scheme. Throughout the aerospace industry, proportional–integral (PI) controllers are commonly used to regulate flow rate and temperature. PI controllers are effective in systems with linear behavior and where response rates are slow [23]. However, they may not be suitable for systems that have highly nonlinear behavior or rapid dynamics, such as with rapidly pulsed evaporator heat loads. A more computationally intensive, and potentially improved, control strategy is model predictive control (MPC). MPC uses a representative mathematical plant model to predict future states by calculating the optimal control decision over a time horizon to minimize an objective function cost [24,25]. The performance impact that a control scheme has on an MPTL has been previously evaluated. Li et al. [26] compared the ability of an MPTL to track an evaporator setpoint temperature with a PI controller and MPC. The model predictive control more closely tracked the evaporator setpoint temperature and reduced the average transient time by at least four times relative to a PI controller. The system evaluated used a fixed-volume receiver and did not actively control compressible volume to influence transient performance. Another research group compared the MPTL response with two different control schemes: an active disturbance rejection controller (ADRC) and a PI controller [27]. The ADRC was able to improve temperature tracing and regulation relative to the PI controller. The evaluated MPTL systems were fixed-volume for both research groups. In another study seeking to improve the MPTL control methodology, the refrigerant flow rate was recommended as a process variable to include in controller design. The modulation of the refrigerant flow rate allowed the average wall temperature to be reduced by 0.74 °C relative to a fixed refrigerant flow rate [28]. In summary, previous research has demonstrated MPTL performance improvements from MPC compared to PI control techniques. However, these MPTL architectures did not include components to modulate the system volume, such as a pressure-controlled accumulators. The combination of MPC and a pressure-controlled accumulator within an MPTL has the potential to improve evaporator isothermal transient performance under pulsed, high-heat-flux electronics cooling. However, this combination of technologies has not previously been evaluated.

The present study explored the transient performance resulting from the integration of an MPTL, a pressure-controlled accumulator, and an advanced MPC scheme relative to a PI- controlled MPTL with an uncontrolled accumulator volume. The study collected experimental data to validate the thermal and flow performance of a numerical model for three-step evaporator heat loads. The validated model was used to quantify transient behavior and compare the two control schemes under multiple applied evaporator heat load profiles, to maintain a near-constant refrigerant setpoint temperature at the evaporator outlet. As individual technologies, MPC and controlled accumulators have been proven to improve temperature setpoint tracking for MPTLs. However, the combination of these two technologies has not previously been evaluated.

2. Experimental Description

Experiments were performed to evaluate compressible volume’s impact on MPTLs’ transient response under pulsed evaporator heat load applications. The experiments were conducted with constant setpoints to verify the thermodynamic and hydraulic performance of a numerical model. Through evaluation using a step input, system characterization was performed and experimental time constants for each control effector were determined [29] and incorporated into the numerical model. The numerical model was then validated and used to evaluate the control scheme’s impact on an expanded set of evaporator heat inputs, more representative of an operational system.

2.1. Experimental Hardware

The hardware for this experimental setup operated on the same principles as described in the literature [30]. Figure 1 shows the schematic of the MPTL architecture, which included a recuperator, a 100 W steady-state preheater, and a pressure-controlled accumulator downstream from the evaporator.

The MPTL working fluid was R-134a. The system was charged with refrigerant according to an accepted correlation, resulting in 1.5 kg of refrigerant [10]. A magnetically coupled, positive-displacement gear pump (Tuthill Series D, 0.38 ${\displaystyle \frac{\mathrm{m}\mathrm{L}}{\mathrm{m}\mathrm{i}\mathrm{n}}}$) was controlled via a variable-frequency drive to vary the refrigerant mass flow rate. The refrigerant passed through the first (i.e., cold) side of the recuperator to decrease the subcooling at the preheater inlet. The refrigerant flowed through an inline preheater (OMEGA AHP-3741, Ronkonkoma, NY, USA), which was controlled via a DC power supply to input a steady 100 W electrical heat load. The refrigerant then entered a cold plate (Lytron C-CP20G01, Woburn, MA, USA) with a DC-powered ceramic heater (ULTRAMIC CER-1-01-00374, St. Louis, MO, USA) mounted on the cold plate. The accumulators were bladder-style accumulators manufactured from material compatible with R-134a (Buna-N rubber). Refrigerant then passed through the second (i.e., hot) side of the recuperator. Finally, the refrigerant flowed through the condenser and then to the receiver. The refrigerant system used $3/8″$ diameter copper tubes, and the system volume was minimized during fabrication. To prevent ambient heat loss, the inline heater, cold plate, ceramic heaters, and copper tubes were insulated. Mounting brackets thermally isolated the hardware from the lab table with low-thermal-conductivity ($0.288{\displaystyle \frac{\mathrm{W}}{\mathrm{m}\mathrm{K}}}$) G-10 Garolite insulation pads.

For the chilled water loop, a liquid-to-liquid recirculating cooler (PolyScience, Niles, IL, USA) was controlled to deliver chilled water to the condenser (Bell & Gossett BP-400-20LP, Cheektowaga, NY, USA) at the desired temperature. The refrigerant saturation temperature was chosen to minimize ambient heat transfer with the lab environment (291–293 K). On the accumulator pressure-control side, facility nitrogen was used to pressurize the system. For constant-setpoint experiments, the pressure regulator maintained the system pressure at 0.66 MPa. System control and data collection were achieved using LabView 2018. Ceramic heaters that input electrical current at a desired voltage were controlled via a DC power supply (MagnaPower 6010A, Flemington, NJ, USA). Local pressure and temperature measurements were recorded at the inlet and outlet of each component in Figure 1 at a sampling rate of 10 Hz. Additional instrument characteristics are listed in

Table 1. Measurement description and experimental uncertainty.

Measurement Device	Model Number	Calibrated Range	Unit	Uncertainty
Volumetric flow meter	McMillian S-112-6H	$100–1000$	${\displaystyle \frac{\mathrm{m}\mathrm{L}}{\mathrm{m}\mathrm{i}\mathrm{n}}}$	$\pm$0.5%
T-type thermocouple	OMEGA TMQSS-125G	$277–366$	$\mathrm{K}$	$\pm$0.4 K
Pressure transducer	OMEGA PX319-300AI	$0.1–1.37$	MPa	$\pm$1.0%

. Prior to data collection, each measurement device was calibrated.

Each component was cleaned with an industry-standard solvent, RX11-flush, to remove oil, debris, or other contaminates prior to installation. Before the system was charged with refrigerant, 500 μm of Mercury vacuum (67 Pa) was maintained within the system to ensure no leaks were present and to remove water vapor. Initial tests were completed to ensure the commanded electrical power was transferred into the refrigerant, with negligible heat loss to the environment. Between each applied evaporator heat load, the system returned to an initial state where the evaporator inlet temperature was $295\pm 0.5$ K.

2.2. Evaporator Heat Load Profile

Heat was added to the experimental hardware in two physically separate locations. The first evaporator, ‘pre-heater’ in Figure 1, represents a generic, constant 100 W steady-state aircraft load. The second evaporator, ‘Cold Plate’ in Figure 1, had various pulsed-transient heat load profiles applied, shown in Figure 2A,B. The evaporator heat load profiles shown in Figure 2A were used to generate experimental data for model validation. A single heat load pulse was determined sufficient to represent the system response, and three step input power levels were evaluated: 300 W, 600 W, and 1200 W.

The three evaporator heat load profiles shown in Figure 2B were used to evaluate how variation in evaporator heat pulse magnitude, duration, and duty cycle impacted the MPTL control response. Each evaporator heat load profile had 50 kJ of heat input, which was applied over various pulses, intervals between pulses, and pulse durations. With respect to an aircraft, heat loads on this evaporator could represent waste heat from an electrical device operated in different duty cycles.

3. Numerical Model Description and Validation

A transient numerical model was created in the physical modeling software MathWorks Simscape™ R2023a, to represent system behavior. Simscape domain libraries provide non-directional blocks where the equation formulation does not explicitly specify inlet and outlet ports. These blocks can be parameterized to represent physical components and track energy and mass flow throughout a system [31]. Thermo-fluid properties are calculated in Simscape through the domain-specific ‘Fluid Properties’ function. Figure 3 is the Simscape representation of the experimental hardware shown in Figure 1.

A brief formulation of the underlying equations can be described for the two-phase pipe component (i.e., Pipe (2P)). In the MPTL model shown in Figure 3, each subsystem denoted ‘PIPE_#’ contains a Pipe (2P) component. The assumptions of the Pipe (2P) component include fully developed flow, negligible influence of gravity, and no fluid inertia effects [32]. An additional assumption was imposed through the modeling strategy to eliminate heat transfer to the environment. These assumptions resulted in the governing equations below.

Equation (1) below is the energy equation [32].

$$m{\dot{u}}_I+\left({\dot{m}}_i+{\dot{m}}_o\right)u_I={\varphi }_i+{\varphi }_o+Q_H$$

(1)

In Equation (1), $m{\dot{u}}_I$ represents the time derivative of the fluid internal energy within the pipe, ${\dot{m}}_i$ and ${\dot{m}}_o$, are the mass flow rates into and out of the pipe, $u_I$ is the fluid internal energy, ${\varphi }_i$ and ${\varphi }_o$ are the energy flow rates into and out of the pipe, and $Q_H$ is the heat transfer rate into the pipe through the pipe wall. The Gnielinski correlation was used to determine the Nusselt number for single-phase turbulent flow through a pipe [33]. The Gnielinski correlation is valid for smooth tubes over the ranges $3000\le R{e}_D\le 5\times {10}^6$ and $0.5\le Pr\le 2000$. The Cavallini and Zecchin correlation was used to determine the Nusselt number for a two-phase mixture within the pipe [32]. The Cavallini and Zecchin correlation is a semi-empirical correlation, which has been demonstrated to show better accuracy compared to many other correlations [34].

Equation (2) is the mass equation [32].

$$\left[{\left({\displaystyle \frac{\mathsf{\partial }\rho }{\mathsf{\partial }P}}\right)}_u{\dot{P}}_I+{\left({\displaystyle \frac{\mathsf{\partial }\rho }{\mathsf{\partial }u}}\right)}_P{\dot{u}}_I\right]V={\dot{m}}_i+{\dot{m}}_o+{ϵ}_M$$

(2)

In Equation (2), $\rho$ is the fluid density, $P_I$ is the pipe internal pressure, $V$ is the fluid volume in the pipe, ${\dot{m}}_i$and ${\dot{m}}_o$ are the mass flow rates into and out pf the pipe, and ${ϵ}_M$ is a density smoothing correction term. This formulation accounts for compressible flow effects in the partial derivative terms. The density smoothing term, ${ϵ}_M$, is used to reduce numerical interpolation errors at phase boundaries for fluid properties.

Equation (3) is the momentum equation [32].

$$P_i-P_I={\displaystyle \frac{{\dot{m}}_i}{S}}\left|{\displaystyle \frac{{\dot{m}}_i}{S}}\left({\nu }_I-{\nu }_i\right)\right|+F_{visc,i}$$

(3)

In Equation (3), $P_i$ is pressure, ${\dot{m}}_i$ is the mass flow rate, $S$ is the cross-sectional area, ${\nu }_I$ is the fluid specific volume inside the pipe, and $F_{visc,i}$ represents the effects of viscous friction forces. The viscous friction force depends on the flow regime and the Darcy friction factor, estimated by the explicit Haaland equation [33]. The Haaland equation avoids the need for iteration, as required by use of a Moody diagram and, thus, is convenient for numerical calculations. The pipe component was parameterized with measured values, such as length and diameter. A similar process was followed for each component shown in the Figure 3 MPTL representation.

A variable-step Simulink solver (daessc) with default tolerances was selected for model convergence [31]. There were no noted issues with model convergence. The initial conditions were manually assigned to match the experiment data five seconds before the pulse, to prevent numerical initialization errors from impacting the results. To evaluate the control scheme’s impact on transient performance, a PI controller and an MPC controller were designed and implemented in the numerical model.

3.1. Control Design

The numerical model was used to create state-space representations of the dynamic system to develop the PI control and MPC logic. While the two control methods contained different effectors, a similar process was used to generate each state-space representation. The state-space representations were reduced to improve computational speed and stability. The process is described in more detail for each controller below.

3.1.1. Proportional–Integral (PI) Controller Design and Model Linearization

A single-input, single-output (SISO) feedback PI controller was designed to serve as a baseline control architecture. In this controller, the evaporator exit pressure ($P_2$) was the measured variable (MV) and the condenser water-side volumetric flow rate was the process variable (PV). As the refrigerant evaporator exit pressure increases, the condensing water-side valve will open to transfer heat from the MPTL to the water-side reservoir.

The representative Simscape model was linearized to develop a PI controller. To linearize the system, the Simscape numerical model was evaluated with a constant condenser water-side volumetric flow rate to arrive at a steady-state condition. Equation (4) shows the output vector, $y$, and derivative of the state equation, $\dot{x}$, which were developed after the physical Simscape equations were linearized. From the state-space representation, shown in Equation (4), pole-zero pairs were canceled within the MATLAB default tolerance to remove unobservable and uncontrollable states [29]. Then, unstable parts of the state-space representation were decomposed into a stable state-space representation, as shown in Equation (5) [35]. In Equation (5), $A,B,C$and $D$ are the state-space representation. $A_1$ are the unstable modes, and $A_2$ are the stable modes.

A pole map was generated to compare the original state-space representation with the reduced state-space representation. The reduced state-space representation contained many high-frequency, oscillatory poles and roots and fewer low-frequency poles and roots. The reduced state-space linear representations of the system gains, $K_p=-1.79$and $K_i=-0.0876$, were determined to achieve the fastest stable response.

$$\begin{gathered} \dot{x}=Ax+BU \\ y=Cx+DU \end{gathered}$$

(4)

$$T_{L}A{T}_R=\left(\begin{array}{cc}{A}_1& 0\\ 0& A_2\end{array}\right),T_{L}B=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right),C{T}_R=\left(\begin{array}{cc}{C}_1& C_2\end{array}\right)$$

(5)

3.1.2. Model Predictive Controller (MPC) Design and Model Linearization

A multiple-input, multiple-output (MIMO) MPC was designed with three manipulated variables (MVs), one measured disturbance (MD), and two measured outputs (MOs). These manipulated variables were selected according to previous research, as refrigerant pump voltage (refrigerant volumetric flow rate), accumulator-nitrogen pressure regulator voltage (refrigerant saturation pressure), and condenser water-side throttle valve current (condenser water-side volumetric flow rate). The measured disturbance was the pulsed evaporator heat load, denoted ${\dot{Q}}_{load}$. The measured outputs were the evaporator outlet pressure ($P_2$) and pressure-controlled accumulator displacement ($V_{acc}$). The MPC quadratic program objective function is

$$J\left(z_k\right)=J_y\left(z_k\right)+J_u\left(z_k\right)+J_{\mathsf{\Delta }u}\left(z_k\right)+J_{ϵ}\left(z_k\right)$$

(6)

In Equation (6), $z_k$ is the quadratic program decision at each timestep, $J_y\left(z_k\right)$ is the output reference tracker term, $J_u\left(z_k\right)$ is the manipulated variable tracking term, $J_{\mathsf{\Delta }u}\left(z_k\right)$ is the manipulated variable move suppression to eliminate large changes in manipulated variables, and $J_{ϵ}\left(z_k\right)$ is the term to prevent constraint violation. The output reference tracker term $J_y\left(z_k\right)$ is shown in Equation (7). This term evaluates each plant output variable $n_y,$ at each prediction step $p$, determines the difference between the reference value $r_j\left(k+i|k\right)$ and predicted value $y_j\left(k+i|k\right)$, and applies a weight and scale term ${\displaystyle \frac{w_{i,j}^y}{s_j^y}}$. Each of the terms shown in Equation (6) has a similar form to the output reference tracker term $J_y\left(z_k\right)$ shown in Equation (7). Additional information about the optimization solver is provided by MathWorks [25].

$$J_y\left(z_k\right)=\sum _{j=1}^{n_y}\sum _{i=1}^p{\left\{{\displaystyle \frac{w_{i,j}^y}{s_j^y}}\left[r_j\left(k+i|k\right)-y_j\left(k+i|k\right)\right]\right\}}^2$$

(7)

A multiple-input, multiple-output (MIMO) MPC was designed following a similar linearization and state-space reduction process as that performed with the PI controller. The MPC was designed with three manipulated variables (MVs), two measured outputs (MOs), and one measured disturbance (MD). The variables captured by the MPC were selected from reference MPTL control methodologies, recommendations from prior studies [28], and parameters enabled by the incorporation of the pressure-controlled accumulator. The numerical model was evaluated with a constant condenser water-side volumetric flow rate, refrigerant volumetric flow rate, and accumulator pressure to arrive at a steady-state condition for linearization. The linearized state-space representation was decomposed and reduced with the same methods as the PI controller. The reduced state-space representation contained many high-frequency, oscillatory poles and roots and fewer low-frequency poles and roots.

Each of the manipulated variables and measured outputs was assigned a hard constraint, according to hardware limitations, which the MPC controller objective function could not violate. Objective function weights were selected to modulate the system as desired and improve controller stability. Manipulated variable hard constraints and objective function weights are shown in

Table 2. MPC objective function weight and hard constraints.

Variable	Variable Type	Objective Function Weight	Hard Constraint	Unit
Refrigerant Mass Flow Rate, ${\dot{m}}_{ref}$	MV	0.01	$0.015\le MV\le 0.050$	${\displaystyle \frac{\mathrm{k}\mathrm{g}}{\mathrm{s}}}$
Accumulator Pressure, $P_{N_2}$	MV	0.01	$0.520\le MV\le 0.800$	$\mathrm{M}\mathrm{P}\mathrm{a}$
Condenser Water Mass Flow Rate, ${\dot{m}}_{wat}$	MV	0.001	$0.001\le MV\le 0.178$	${\displaystyle \frac{\mathrm{k}\mathrm{g}}{\mathrm{s}}}$
Evaporator Exit Pressure, $P_{o,2}$	MO	10	None	$\mathrm{M}\mathrm{P}\mathrm{a}$
Accumulator Displacement, $V_{acc}$	MO	10	None	${\mathrm{m}}^3$
Evaporator Heat Input, ${\dot{Q}}_{in}$	MD	-	-	W

.

System dynamics dictated the sampling period $t_s$ to be 0.1 s, prediction horizon $p_H$ to be 60 s, and the control horizon $c_H$ to be 12 s. The controller performance was verified in terms of closed-loop internal stability, closed-loop nominal stability, steady-state gains, and constraints.

3.2. Statistical Metrics

To compare the experimental results with the numerical model for validation, a 25-s measurement window, $W$, was defined. The measurement window was defined as a five-second pre-evaporator load pulse ($0<t<5$), five-second evaporator load pulse ($5<t<10$), and then fifteen seconds after the evaporator load pulse response ($10<t<25$).

Over this measurement window, two metrics are defined: maximum percent error (MPE) and mean average percent error (MAPE). MPE reports the maximum percent error over the measurement window to determine the condition with the maximum difference between the experimental data and predictions. MAPE reports the average percent error over the measurement window to provide an estimate for transient model assessment. For a function $y\left(t\right)$, with measured values $y{\left(t\right)}_{exp}$ and predicted values $y{\left(t\right)}_{pred}$, MPE and MAPE are useful and defined by Equations (8) and (9).

$$MPE=\max \left({\displaystyle \frac{\left|y{\left(t\right)}_{exp}-y{\left(t\right)}_{pred}\right|}{y{\left(t\right)}_{exp}}}\times 100\right)$$

(8)

$$MAPE={\displaystyle \frac{\sum \left(\frac{\left|y{\left(t\right)}_{exp}-y{\left(t\right)}_{pred}\right|}{y{\left(t\right)}_{exp}}\times 100\right)}{W}}$$

(9)

Furthermore, in applications where multiple evaporator load pulses can be applied, it is useful to define a metric for which the system returns to a steady state and the evaporator load can be applied a second time. This parameter was chosen to be a 3% settling time, which is the time it takes for the system to reach $\pm 3\%$ of the initial condition after an evaporator pulse load has been applied [36].

To evaluate the impact of an evaporator heat pulse, the pressure and temperature increases are calculated across the duration of the pulse. Equation (10) shows the pressure rise at the evaporator outlet for the pulse duration.

$$P_{rise}={P_{evap}}_{o,t_{start}}-{P_{evap}}_{o,t_{end}}$$

(10)

3.3. Model Validation

Previous work has demonstrated that the system transient response is influenced by the total system volume and system charge [17]. Prior to experimental data collection, the MPTL volume was measured. A nitrogen tank with a known initial pressure and volume was expanded into the MPTL. The measured final pressure was measured, and the final volume was calculated by the Ideal Gas Law. The experimentally measured volume was within 0.85% of the model-represented volume.

The purpose of the collected experimental data was to describe the impact a passive accumulator has on an MPTL transient response under a pulsed evaporator heat load. A brief description of pressure and temperature response is provided below. MPE and MAPE were used as metrics to validate the numerical model response, and the 1200 W evaporator step heat input was used for discussion. Within the MPTL cycle, system transient performance was strongly influenced by the pressure at each cycle point. Since an MPTL cycle is nearly isobaric, increased pressure at one cycle point will increase pressure throughout the system. As slightly subcooled refrigerant flowed through the evaporator and a 1200 W heat load was applied, $t_{start}$, the subcooled liquid changed phase to a two-phase liquid–vapor mixture. The two-phase liquid–vapor mixture had a lower density (107 ${\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}$) than the subcooled liquid (1225 ${\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}$), which increased the system volume through the accumulator and increased system pressure. Upon termination of the evaporator heat load,$t_{end}$, heat continued to be removed from the cycle via the condenser and the MPTL pressure returned to the initial state. This behavior is presented in Figure 4A. Over the five-second, 1200 W pulsed evaporator load, the evaporator outlet pressure increased by 0.043 MPa. After the evaporator pulse was removed, the MPTL system required 6.8 seconds to reach a 3% settling time. The pressure-controlled accumulator had a constant setpoint of 0.66 MPa. As the evaporator outlet pressure increased, the pressure-controlled accumulator increased the system volume to maintain a system 0.66 MPa saturation pressure. Figure 4A shows the pressure response measured in the experiments, as captured by the numerical model. The predicted pressure response has a similar slope and maximum pressure when compared to the experimental data. Deviations between the model and experimental data are within the experimental measurement uncertainty ($\pm$0.13 MPa) and can be attributed to minor differences in the model initial conditions and heat rejection via the condenser.

Like the pressure response, the evaporator outlet temperature increased while the evaporator pulse was applied, and then it decreased once the evaporator load was removed. Figure 4B shows the evaporator outlet temperature transient response under a 1200 W evaporator pulse. The evaporator outlet refrigerant temperature increased by 2.5 K over the evaporator load pulse duration. In addition, the model underpredicted the maximum temperature relative to the experimental results. However, the difference between the measured and calculated maximum temperatures was relatively small, at roughly 1 K. Figure 4B shows that the numerical model captured the general shape of the measured temperature response.

The predicted refrigerant mass flow rate agreed within 3% of the experimentally measured mass flow rate across the evaporator pulse duration. The numerical model captured a decrease in refrigerant mass flow rate upon evaporator heat load application, in accordance with a previous study [17]. However, the refrigerant volumetric flow meter was placed upstream of the recuperator in a subcooled liquid refrigerant region, and therefore did not capture these dynamics.

Table 3. Measured evaporator outlet refrigerant pressure and temperature from numerical model.

Load [W]	Pressure Rise [MPa]	Temperature Rise [K]
300	0.0122	1.4
600	0.0246	1.8
1200	0.043	2.5

shows the measured evaporator outlet refrigerant pressure and temperature for each of the three evaporator heat pulses in Figure 2A. As the magnitude of heat input increased, the associated pressure and temperature rise increased.

Table 4. Comparison between measured and predicted pressures and temperature at each component inlet and outlet (1200 W).

Cycle Point	Description	MPE	MAPE	MPE	MAPE
		Pressure		Temperature
-	-	[%]	[%]	[%]	[%]
$1,i$	Pump Inlet	1.61	0.40	0.70	0.10
$1,o$	Pump Outlet	5.81	3.57	1.54	1.10
$2,i$	Recuperator, Cold-Side Inlet	-	-	1.27	1.10
$2,o$	Recuperator, Cold-Side Outlet	-	-	0.94	0.66
$3,i$	Evaporator Inlet	1.20	0.02	0.49	0.28
$3,o$	Evaporator Outlet	1.53	0.33	0.70	0.35
$4,i$	Recuperator, Hot-Side Inlet	-	-	0.81	0.31
$4,o$	Recuperator, Hot-Side Outlet	-	-	1.02	0.35
$5,i$	Condenser Inlet	1.73	0.16	0.70	0.38
$5,o$	Condenser Outlet	1.42	0.53	0.70	0.88
-	Receiver	1.77	0.22	-	-
-	Mean 1200 W	2.15	0.41	0.89	0.38
-	Mean 300 W	3.17	0.72	1.24	0.61
-	Mean 600 W	3.45	0.74	1.18	0.58

compares the measured and predicted pressures and temperatures at each cycle point inlet and outlet for the 1200 W heating pulse. The cycle points correspond to those in Figure 3. Relatively small values of the MPE and MAPE indicate that the model predicts the measured pressure reasonably well. In addition, the MPE and MAPE were averaged across all cycle points to create a single value to represent the model fit for each of the three pulse load values, shown in Table 4. With this metric, the MPE agrees within 3.45% and the MAPE agrees within $0.74\%$. This indicates that the model predicted a reasonable transient pressure and temperature performance under multiple pulsed evaporator loads relative to the experimental data.

4. Results and Discussion

The numerical model validated with experimental data was used to evaluate three additional evaporator heat load profiles, shown in Figure 2B, for both PI- and MPC-control schemes. The PI- and MPC-controlled responses are shown in the same plot to highlight differences from the selected control action and the effect on thermodynamic parameters. The MPTL thermodynamic responses ($P_2$and $T_2$) at the evaporator exit and control responses (${\dot{m}}_{wat}$ and $V_{acc}$) are shown in Figure 5, Figure 6 and Figure 7. In all cases, the numerical model was simulated for 100 s to arrive at a steady operating condition before evaporator heat load profiles were applied. A comparison of the control responses highlights the importance of actively controlling compressible volume within the system to maintain an evaporator setpoint temperature. This section concludes with a comparison of temperature rise between the two control architectures.

4.1. Profile 1: Ten 1000 W Amplitude Square Waves with a 5 s Pulse Width and 30 s Period (50 kJ Total)

Through the Profile 1 duration, the MPC-controlled system maintained a $293\pm 2\mathrm{K}$ isothermal temperature condition, whereas the PI-controlled system could not maintain this temperature condition. This is shown in Figure 5B.

For the PI-controlled system, upon heat load application, the evaporator exit quality increased from the initial subcooled condition to a two-phase mixture condition with exit quality, $x_2\cong 0.24$. As with the step pulse, the refrigerant pressure increased as vapor was generated, shown in Figure 5A. The vapor increased the MPTL pressure, $P_2$, above the constant accumulator nitrogen-side pressure setpoint ($P_{N_2}=0.578$ MPa). The refrigerant-side pressure, $P_2$, increase above the nitrogen-side pressure, and ${P_N}_2$ expanded the flexible bladder within the accumulator and displaced nitrogen from the accumulator. The initial refrigerant-side accumulator displacement was $V_{acc}=1.0\times {10}^{-4}{\mathrm{m}}^3$. After the first evaporator heat load pulse, the accumulator was fully expanded and filled with refrigerant vapor ($V_{acc}=9.5\times {10}^{-4}{\mathrm{m}}^3$). For the first square wave heat load within Profile 1, the accumulator expansion enabled the PI-controlled MPTL to maintain saturation pressure and, thus, maintain the evaporator temperature within $293\pm 2\mathrm{K}$. However, subsequent heat pulses in Profile 1 were applied with the accumulator fully expanded. Additional generated vapor contributed to a saturation pressure increase to 0.705 MPa and, therefore, an increased saturation temperature to 300 K. Both the pressure rise and accompanying temperature rise resulting from the Profile 1 heat load occurred on the same timescale as the heat input (modeled as $\tau <0.1\mathrm{s}$). As the MPTL pressure increased above the controller setpoint, the water flow rate through the condenser increased, ${\dot{m}}_{wat}$, as shown in Figure 5D. The time constant associated with this valve is 3.5 seconds Thus, the valve did not respond adequately to maintain the saturation pressure throughout the duration of this transient profile. After the fourth square wave, the PI-controlled responses are similar.

Figure 5B also shows that the MPC-controlled MPTL response can maintain the 293 $\pm 2\mathrm{K}$ isothermal temperature condition for the Profile 1 duration. In addition to the water-side condenser flow rate, the MPC includes the refrigerant pump speed (i.e., refrigerant flow rate) and the nitrogen-side pressure regulator (i.e., accumulator pressure) as effectors. Like the PI-controlled scheme, as the heat load was applied, vapor was generated. However, the MPC increased the refrigerant mass flow rate from $0.015{\displaystyle \frac{\mathrm{k}\mathrm{g}}{\mathrm{s}}}$ to the maximum allowable flow rate of $0.05{\displaystyle \frac{\mathrm{k}\mathrm{g}}{\mathrm{s}}}$. The increase in refrigerant mass flow rate decreased the evaporator exit quality to a maximum of $x_2=0.10$, and thus generated less vapor than the PI-controlled system. The accumulator expanded to accommodate the generated vapor for the first square wave, $V_{acc}={6.9\times {10}^{-4} \mathrm{m}}^3$, shown in Figure 5C. After the first square wave heat input, the MPC increased nitrogen pressure ($P_{N_2}$) from 0.571 MPa to 0.594 MPa, which was above the refrigerant saturation pressure (0.571 MPa). As the nitrogen pressure was greater than the refrigerant saturation pressure, the flexible bladder displaced refrigerant from the accumulator with nitrogen. This decreased the accumulator refrigerant-side volume, shown in Figure 5C, to the near the initial condition ($V_{acc}={1.4\times {10}^{-4}\mathrm{m}}^3$). With MPC, the accumulator displacement is a measured output, which has a weighted penalty for setpoint deviation. The combination of these control actions reset the system to the initial conditions before a sequential heat input was applied and the system was able to continue for each of the evaporator pulses of Profile 1. After Profile 1 was completed, the refrigerant mass flow rate, ${\dot{m}}_{ref}$, and water flow rate through the condenser, ${\dot{m}}_{wat},$ decreased.

The addition of refrigerant pump speed and nitrogen-side accumulator pressure as effectors allowed the MPC-controlled system to generate less vapor than the PI-controlled system by increasing the refrigerant mass flow rate. Additionally, the MPC-controlled system manipulated the accumulator compressible volume to maintain a $293\pm 2$ K isothermal temperature condition for Profile 1. The PI-controlled MPTL did not adjust the refrigerant mass flow rate or modulate the compressible accumulator volume beyond the first square wave heat input, as these parameters are not process variables. As a result, the PI-controlled system did not maintain the $293\pm 2$ K isothermal temperature for Profile 1.

4.2. Profile 2: One 178 W Amplitude Square Wave with a 281 s Pulse Duration (50 kJ Total)

Profile 2 demonstrates that the system transient response is dependent upon the applied evaporator heat load. As shown in Figure 6, when 50 kJ of heat is inputted to the system as a long-duration (281 s) square wave with a low amplitude (178 W) compared to Profile 1 (1000 W amplitude), the saturation temperature and pressure responses are similar between the two control schemes. Both control schemes operated within the 293 $\pm 2$K temperature limit.

For the PI-controlled MPTL, the lower-amplitude square wave (178 W) decreased the rate of accumulator nitrogen displacement as compared to Profile 1. The lower heat input allowed additional time for the PI controller to actuate the condensing water throttle valve, shown in Figure 6D, to remove heat from the system. Upon heat load application, the refrigerant-side accumulator volume increased, but the condensing water flow rate responded on a similar timescale. The accumulator was fully expanded between 25 s and 70 s as the condensing water flow rate increased, but the system maintained a saturation temperature within 293 $\pm 2$K.

The MPC-controlled system maintained the saturation pressure (Figure 6A), and therefore temperature (Figure 6B), with the minimal accumulator actuation. Accumulator nitrogen-side pressure, $P_{N_2}$, did not require actuation. Instead, the MPC increased the refrigerant mass flow rate and condensing water flow rate to maintain the saturation pressure and temperature, according to the MPC objective function weights.

This profile demonstrated that process and manipulated variables are actuated differently under a long-duration square wave with a low-magnitude evaporator load, as compared to a highly transient evaporator heat load, such as Profile 1. For low-magnitude, long-duration square wave heat inputs, the thermodynamic responses ($P_2$ and $T_2$) for both control methods were similar.

4.3. Profile 3: Ten 1000 W Amplitude Square Waves with a 5 s Pulse Width and 30 s Period (50 kJ Total), Grouped into Two Duty Cycles, Spaced 80 s Apart

Profile 3 demonstrates the effect of the heat input grouped into two distinct cycles. As in Profile 1, Figure 7B shows that the MPC-controlled MPTL is able to maintain the $293\pm 2\mathrm{K}$ evaporator temperature condition throughout the profile duration, whereas the PI-controlled MPTL cannot.

For the PI control scheme, the first cycle (square waves 1–5) of square waves had an identical response to Profile 1. After the first heat input cycle was completed, the condensing water flow rate (Figure 7D) remained at the maximum value to remove system heat and decrease the accumulator refrigerant-side volume. However, after the first cycle (square waves 1–5), the MPTL was not able reject enough heat to fully re-compress the refrigerant within the accumulator to the initial condition. Therefore, as the second group of heat inputs (square waves 6–10) was applied, the accumulator refrigerant side was fully expanded. Subsequent heat input caused saturation temperature setpoint exceedance.

In the MPC control system, the accumulator refrigerant-side volume (Figure 7C) was reset to near the initial volume after each square wave heat input. As in Profile 1, nitrogen pressure, ${P_N}_2$, increased above the MPTL saturation pressure to displace and compress refrigerant vapor from the accumulator after each heat input. Since the accumulator refrigerant-side volume decreased after each heat input, grouping the heat input into two cycles did not negatively impact saturation temperature. Instead, the MPC decreased the refrigerant flow rate and water flow rate through the condenser in the period between grouped heat inputs. The water flow rate through the condenser is shown in Figure 7D.

4.4. Discussion

When the two control schemes were compared, access to the additional manipulated variables (refrigerant pump speed and accumulator nitrogen-side pressure) proved to be useful to the MPC-controlled system for transient performance improvement and temperature regulation, as compared to the PI-controlled system. The addition of refrigerant mass flow rate as an effector decreased the generated refrigerant vapor, which resulted in less saturation pressure rise throughout the heating profile. As MPTL system pressure increased from generated vapor, the MPC-controlled system was able to modulate accumulator refrigerant-side compressible volume, to maintain a given saturation pressure. In Profiles 1 and 3, multiple heat inputs caused the PI-controlled accumulator to fully expand the refrigerant compressible volume. In these scenarios, the MPTL was not able to transiently remove enough heat from the refrigerant, and saturation temperature increased beyond the 293 $\pm 2\mathrm{K}$ temperature condition. The results for each evaporator heat profile are summarized in

Table 5. Overall comparison between PI control and MPC schemes for the three evaluated evaporator heat input profiles.

Profile Number	Profile Description	Control Method	Maintained $\pm 2$ K Limit?	Maximum Saturation Temperature Rise, ${\mathit{T}}_2$ [K]
1	Ten 1000 W amplitude square waves with a 5 s pulse width and 30 s period (50 kJ total)	PI	No	7.1
		MPC	Yes	1.5
2	One 178 W amplitude square wave with a 281 s pulse duration (50 kJ total)	PI	Yes	1.4
		MPC	Yes	0.2
3	Ten 1000 W amplitude square waves with a 5 s pulse width and 30 s period (50 kJ total), grouped in two cycles, spaced 80 s apart	PI	No	7.0
		MPC	Yes	1.5

.

Table 5 shows that the MPC scheme was able to maintain a constant refrigerant temperature setpoint more accurately than the PI control scheme. In all cases, the MPC scheme resulted in a smaller temperature rise than the PI-controlled MPTL.

5. Summary and Conclusions

In this study, experimental methods were used to characterize a system response for an MPTL with a pressure-controlled accumulator evaluated for three different pulsed evaporator heat loads. The pressure-controlled accumulator contained a flexible bladder and was actuated by compressed nitrogen. The experimental results were compared with numerical simulation data for model validation. With the validated model, traditional PI control and an advanced MPC were developed. Both control strategies were numerically evaluated against three evaporator heat load profiles, which varied heat input magnitude, duration, cycles, and spacing to determine the impacts that the control scheme and compressible volume have on MPTL performance for isothermal ($293\pm 2\mathrm{K}$) evaporator operation.

In the current work, a representative, transient numerical model well-represented the pressure, temperature, and mass flow rate response for multiple cycle points and multiple step evaporator heat loads. Under pulsed evaporator heat loads, the MPE agreed within 3.45% and the MAPE agreed within $0.74\%$. The model-predicted refrigerant mass flow rate agreed within 3% of the experimentally measured mass flow rate across the evaporator pulse duration. The model was leveraged for control development.

The transient performance levels of a PI- and MPC-controlled MTPL were evaluated for three simulated evaporator heat input profiles. In all profiles, the MPC scheme produced less saturation pressure variability than the PI control scheme, which led to a more uniform saturation temperature across the evaporator load duration. As evaporator heat input transients increased (Profiles 1 and 3), the MPC-controlled MPTL was able to maintain temperature requirements, whereas the PI-controlled MPTL did not. However, for low-heat-flux applications (e.g., Profile 2), the two control schemes showed similar thermodynamic responses. The addition of the refrigerant mass flow rate and nitrogen-side pressure to control the accumulator compressible volume as effectors improved the MPC thermodynamic response compared to the PI-controlled response. These additional effectors enabled the MPC system to decrease the generated refrigerant liquid–vapor mixture upon evaporator heat input and provided actuation to re-compress the accumulator for heat rejection.

From this work, it can be concluded that the incorporation of a controlled compressible volume, via an accumulator, into an MPTL has the potential to provide near-isothermal ($293\pm 2\mathrm{K}$) evaporator operation throughout a pulsed heat load duration. Furthermore, the incorporation of advanced control algorithms and novel manipulated variables, such as nitrogen-side pressure, to actively control accumulator compressible volume offers performance advantages for pulsed evaporator heat loads. As future aerospace designs demand increased thermal capability, an MPTL with an accumulator should be evaluated, especially when design constraints require isothermal evaporator performance.