Time-varying Boundary Conditions Research Articles

On the analytical solution of single particle models and semi-analytical solution of P2D model for lithium-ion batteries

The Doyle–Fuller–Newman (DFN) model serves as a fundamental electrochemical modeling framework widely used in the field of lithium-ion batteries, especially in its pseudo-two-dimensional (P2D) formulation. Despite its widespread use, the DFN model imposes significant computational demands, especially for tasks such as cell characterization and state estimation in electric vehicles’ battery management systems (BMS). To mitigate these computational challenges, researchers have developed several reduced-order models. In this study, we tackle the associated computational efforts starting from the governing DFN equations. The analytical solutions for the transport equations in the electrode and the electrolyte are derived, considering generic time-varying boundary conditions under certain assumptions. The obtained exact solutions are then used to develop an analytical model for a single particle model with electrolyte dynamics (SPME) which provides higher predictive accuracy compared to a single particle model (SPM), especially in high C-rate scenarios, which are critical in many applications, including electric vehicles. Furthermore, the exact solution of lithium diffusion in active material particles is incorporated into the standard P2D model. This integration leads to a semi-analytical variant of the P2D model (SAP2D). A remarkable advantage of these analytical solutions is the significantly lower computational cost compared to corresponding numerical approaches. Spatial discretization becomes unnecessary, and the solutions can be obtained by a few explicit function evaluations. Moreover, when the time-varying boundary conditions are known, the need for time stepping is also eliminated.

Unsteady Multiphase Simulation of Oleo-Pneumatic Shock Absorber Flow

The internal flow in oleo-pneumatic shock absorbers is a complex multiphysics problem combining the interaction between highly unsteady turbulent flow and multiphase mixing, among other effects. The aim is to present a validated simulation methodology that facilitates shock absorber performance prediction by capturing the dominant internal flow physics. This is achieved by simulating a drop test of approximately 1 tonne with an initial contact vertical speed of 2.7 m/s, corresponding to a light jet. The flow field solver is ANSYS Fluent, using an unsteady two-dimensional axisymmetric multiphase setup with a time-varying inlet velocity boundary condition corresponding to the stroke rate of the shock absorber piston. The stroke rate is calculated using a two-equation dynamic system model of the shock absorber under the applied loading. The simulation is validated against experimental measurements of the total force on the shock absorber during the stroke, in addition to standard physical checks. The flow field analysis focuses on multiphase mixing and its influence on the turbulent free shear layer and recirculating flow. A mixing index approach is suggested to facilitate systematically quantifying the mixing process and identifying the distinct stages of the interaction. It is found that gas–oil interaction has a significant impact on the flow development in the shock absorber’s upper chamber, where strong mixing leads to a periodic stream of small gas bubbles being fed into the jet’s shear layer from larger bubbles in recirculation zones, most notably in the corner between the orifice plate and outer shock absorber wall.

Diffraction Problem with Time-Varying Boundary Conditions

Diffraction Problem with Time-Varying Boundary Conditions

Analysis of Multilayer Cylindrical Thermal Conduction With a Time-Varying Convective Boundary Condition

Abstract Heat transfer in a multilayer body plays a key role in design and optimization of several engineering systems. While the analysis of simple multilayer problems is quite straightforward, realistic scenarios such as time-dependent boundary conditions result in significant complications in analysis. This work presents thermal analysis of a one-dimensional heat-generating multilayer cylinder with time-varying convective heat transfer at the boundary. Such a scenario may occur in applications such as nuclear reactors, jet impingement cooling, turbine blade heat transfer, as well as casting and related manufacturing processes. Analysis is presented for both annular and solid cylinders. A derivation for the temperature distribution is carried out, using a shifting function to split the time-dependent boundary condition into two parts, followed by appropriate mathematical substitution. For particular special cases, the analytical results derived here are shown to reduce exactly to results from past work. Good agreement of the theoretical results with numerical simulations is also demonstrated. Thermal response to various realistic time-dependent boundary conditions is analyzed. This work contributes towards the design of realistic multilayer problems and may facilitate the optimization of engineering systems where multilayer thermal conduction plays a key role.

Influence of rapidly time-varying boundary conditions on the control of two-dimensional elastic pulses in laminates

Influence of rapidly time-varying boundary conditions on the control of two-dimensional elastic pulses in laminates

Numerical modeling for predicting service life of reinforced concrete structures exposed to chloride

In recent decades, there has been a growing interest in predicting the service life of reinforced concrete structures vulnerable to chloride-induced corrosion. The service life is divided into initiation and propagation phases, considering the de-passivation of the reinforcement's protective film as a threshold for active corrosion. However, it is observed that the corrosion process is gradual, requiring considerable time before significant damage occurs. This research presents a comprehensive numerical model that incorporates both passive and active corrosion contributions to predict the service life. The model utilizes a finite difference approach to simulate transportation processes, corrosion current, and rust growth. Time-varying boundary conditions, including relative humidity, temperature, and surface chloride concentration, are considered in solving the partial differential equation governing the transportation processes. The incorporation of both passive and active corrosion contributions, as well as the consideration of time-varying boundary conditions, enhances the accuracy of the predictions. Sensitivity analysis is conducted to evaluate the effects of corrosion rate, cross-section loss rate, and service life, considering variations in relative humidity, temperature, water-to-cement ratio, and cover depth. The results highlight the significant impact of relative humidity and water-to-cement ratio on corrosion rate and cross-section loss. Changing the water-to-cement ratio from 0.5 to 0.4 was observed to double the overall service life of an identical structure. Additionally, cover depth was identified as a crucial factor affecting the service life of the structure. Changing the cover depth from 40 mm to 80 mm enhanced service life by three and a half times in the present case study. Temperature fluctuations had a relatively minor effect within acceptable limits, allowing the use of average temperatures for service life estimation, particularly in coastal regions. The comprehensive model presented can provide aid in informed decision-making for corrosion prevention and mitigation strategies in practice.

AREHS: effects of changing boundary conditions on the development of hydrogeological systems: numerical long-term modelling considering thermal–hydraulic–mechanical (–chemical) coupled effects

Abstract. The objective of the AREHS project, funded by BASE (FKZ 4719F10402), is to model the effects of changing external boundary conditions on the hydrogeologically relevant parameters and effects (e.g. hydraulic permeability, porosity, migration pathways, fluid availability, hydraulic gradients) of a generic geological repository in Germany in all three potential host rocks (clay, salt, and crystalline rocks) in its hydrogeological setting. Special attention is paid to the cyclic mechanical loading due to glaciation events. This results in stress changes (M – mechanical processes) as well as induced temperature effects (T – thermal processes) due to permafrost and warm periods. Since the thermal, hydraulic (H – hydraulic processes), and mechanical processes are strongly coupled, they have to be covered by state-of-the-art coupled thermal–hydraulic–mechanical (THM) (C) modelling. The presentation consists of a (a) presentation of important findings for all three host rock types and a (b) presentation of the overall workflow. Complex, time-varying boundary conditions have been formulated for modelling glacial cycles. Investigating the effects of these boundary conditions with THM simulation reveals a variety of coupling effects. To conduct the complex modelling workflow for claystone and salt rock effectively, an automated workflow was developed and tested. It handles the transformation from a simulator-independent geological model to a numerical model specific to the simulator. This contains a suitable finite-element mesh and a parameterization of the fully coupled or optionally isolated thermal–hydraulic–mechanical processes, which are implemented and executed in the OpenGeoSys simulator. Simulation results are presented for selected physical quantities at characteristic local and temporal points with respect to the position of a migrating glacier. The investigation results are reproducible through full automation, container deployment, and a continuous integration process running on a GitLab instance (see Fig. 1a). To simulate the response of a fractured crystalline rock mass to THM impacts, the explicit integration of the fracture network in numerical models is necessary. The workflow couples the distinct element method (DEM) software “3DEC” and the discrete fracture network (DFN) software “DFN.Lab” automatically (Fig. 1b).

Analysis of transient laminar forced convective heat transfer in parallel-plate channels with time-varying thermal boundary conditions of the inlet and wall

The problem of transient laminar forced convective heat transfer in parallel-plate channels with time-varying thermal boundary conditions of the inlet and wall is thoroughly investigated in this study. By employing the methodologies of generalized integral transform technique (GITT) and Duhamel's theorem, we successfully derive an analytical solution for this problem. To validate the accuracy of our analytical findings, a comparison is made with the corresponding numerical solution, demonstrating a remarkable level of agreement between the two approaches. In order to obtain the general patterns of heat transfer, we employ a decoupling technique to separate the thermal response in the case of harmonic temperature boundary conditions of the inlet and wall. This approach enables us to uncover the intricate relationship between the weight coefficient of heat input and the two-dimensional spatial location of the channel, as well as the dimensionless angular frequency of heat input. Furthermore, we present the corresponding dimensionless cut-off angular frequency of heat input at various locations within the channel. To gain further insights, we conduct an in-depth analysis of the effects of integration time or distance on the dimensionless optimum angular frequency corresponding to the dimensionless maximum average heat flux.

A new method for interpreting well-to-well interference tests and quantifying the magnitude of production impact: theory and applications in a multi-basin case study

Interference tests are used in shale reservoirs to evaluate the strength of connectivity between wells. The results inform engineering decisions about well spacing. In this paper, we propose a new procedure for interpreting interference tests. We fit the initial interference response with the solution to the 1D diffusivity equation at an offset observation point. It is advantageous to use the initial interference response, rather than the subsequent trend, because the initial response is less affected by nonlinearities, time-varying boundary conditions, and uncertainties about flow geometry and flow regime. From the curve fit, we estimate the hydraulic diffusivity and the conductivity of the fractures connecting the wells. For engineering purposes, it would be useful to quantify the impact of interference on well production. Thus, we seek a relationship between the ‘degree of production interference’ (DPI) and an appropriate dimensionless quantity that can be derived from the estimate of fracture conductivity. Using simulations run under a wide range of conditions, we find that the classical definition for dimensionless fracture conductivity does not achieve a consistent prediction of DPI. This occurs because the dimensionless fracture conductivity was derived assuming radial flow geometry, but the dominant flow geometry during shale production is linear. We also find that the Chow Pressure Group metric does not yield consistently accurate predictions of DPI. As an alternative, we derive a dimensionless quantity similar to the classical ‘dimensionless fracture conductivity,’ but which is derived for linear, not radial, flow geometry. Using this approach, we calculate a ‘dimensionless interference length’ that collapses all cases onto a single curve that predicts DPI as a function of fracture conductivity, well spacing, and formation properties. We conclude by applying the new method to field cases from the Anadarko and Delaware Basins.

Physics-guided identification of Euler–Bernoulli beam PDE model from full-field displacement response with SimultaNeous basis function Approximation and Parameter Estimation (SNAPE)

Full-field measurements of the continuous spatiotemporal response of the physical processes such as structural vibration or fluid flow generate large datasets. In many scientific fields, such continuous spatiotemporal dynamic models are represented by partial differential equations (PDEs). In the past, attempts have been made to identify the PDE models from the measured response by inferring its parameters by the use of either regression or deep learning-based techniques. But the previously presented regression-based methods fail to estimate the parameters of the higher-order PDE models in the presence of moderate noise. Likewise, the deep learning-based methods lack the much-needed property of repeatability and robustness in the identification of PDE models from the measured response. The authors introduced the method of SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE) in a recent paper which addresses such drawbacks by fitting basis functions to the measured response and simultaneously infer the parameters of the PDE model. In this paper the theory and formulation of SNAPE is presented to perform physics-guided identification of the Euler–Bernoulli beam PDE model which is widely applied in the modeling of large scale infrastructures to nanoscale structures.The domain knowledge of the physics is used as a constraint in the formulation of the optimization framework. The alternating direction method of multipliers (ADMM) algorithm is used to simultaneously optimize the loss function over the parameter space of the PDE model and coefficient space of the basis functions. The proposed method not only infers the parameters but also estimates a continuous function that approximates the solution to the PDE model. The efficacy of the method is both numerically and experimentally validated on noise corrupted full-field vibration response. The method neither requires the knowledge of the initial or boundary conditions of the beam nor comprises of model discretization error as in the case of finite element model updating. SNAPE demonstrates its applicability on various homogeneous and time-varying nonhomogeneous boundary conditions.

Transient simulation of reversible pump turbine during pump mode's starting up

A pumped storage power station is an electric power system that utilizes low-peak electricity to pump water into a high-altitude reservoir and releases water to generate electricity during high-peak electricity demand. The pump mode's starting up impacts the operational stability and lifespan of the reversible pump turbine (RPT). Especially, the pressure pulsation signal during the pump mode's starting up is a typical non-stationary signal in RPT, which is an important basis for judging the operation status of the unit. In this study, the time-varying boundary conditions are used to calculate the pressure fluctuation and flow pattern evolution for a prototype RPT. This paper analyzed and compared the time-frequency characteristics of pressure fluctuations by continuous wavelet transform (CWT) and by variational mode decomposition (VMD). And the internal flow of RPT was also visualized using entropy generation theory. In this study, the effect of the penalty coefficient on the VMD algorithm is investigated in the form of a signal-noise ratio (SNR). With the increase of the penalty coefficient, the SNR of the signal gradually decreases and tends to be stable. Several modal components with strong physical significance and center frequency separation are obtained. Influenced by the boundary conditions that change with time, the pressure fluctuation presents a five-stage trend of “descending - ascending - descending - ascending - descending”. There are three main types of high-flow energy dissipation (FED) areas during this process of the RPT. The research and analysis in this paper will provide a good reference for the stable and safe operation of RPT and pumped storage power stations.

The onset of turbulence in decelerating diverging channel flows

This work investigates the stability and transition to turbulence in a diverging channel subjected to a time-varying trapezoidal-shaped inflow boundary condition. Numerical simulations are performed for different deceleration rates and Reynolds numbers while maintaining a constant acceleration rate. The flow transition begins with two-dimensional primary instability with the formation of inflectional velocity profiles, followed by local separation and the emergence of an array of shear layer vortices. We divide simulation cases systematically into three categories based on the onset of secondary instability and the generation of streamwise vorticity. At low and medium Reynolds numbers (type I), the spanwise vortex rolls formed by inflectional instability remain two-dimensional and diffuse at the channel centre without exhibiting further instabilities. At high Reynolds numbers and deceleration rates (type II), the rolled shear layer exhibits secondary instability during the zero mean inflow phase, followed by local incipient turbulent structure formation. The streamwise vorticity that develops over the shear layer structures causes oscillations with a spanwise wavelength similar to those associated with the elliptic instability in a counter-rotating vortex pair. Using the Lamb–Oseen approximation of vortices in conjunction with the dynamic mode decomposition algorithm of the three-dimensional flow field, we captured successfully the characteristics of the secondary instability. The third category (type III) is characterized by periodic unsteady separation, secondary instability, and merging of shear layer vortices, which occurs when Reynolds numbers are high and deceleration rates are low.

Numerical Study of Flow Control on Simplified High-Lift Configurations

Enhancement in the aerodynamic performance of wings and airfoils is very notable when Active Flow Control (AFC) is applied to Short Take-off and Landing aircraft (STOL). The present numerical study shows the application of steady, pulsed and synthetic tangential jets applied to the plain flap shoulder of a modified NASA Trapezoidal Wing. Pulsed jets are modeled by sinusoidal and square waveforms while synthetic jets are modeled only by pure sine waveform. The freestream airflow conditions are Mach number equal to 0.2 and Reynolds number equal to 4.3 million based on the mean aerodynamic chord. The presented simulations are two-dimensional and based on RANS for steady jet cases and URANS for pulsed and synthetic cases, compiled with the open-source suite SU2 and adapted for time varying boundary conditions. Numerical results for modified configurations based on the same baseline wing profile considering different leading edges, jet slot height, flap position, blowing mass flow, type and frequency of the jets are presented. Curves of pressure coefficient distribution revealed a substantial influence upstream of the AFC, around the slat and main element. The jet slot height analysis showed that the lift gain is also influenced by the slot size due to the change of the local flow velocity considering the same blowing momentum coefficient. Regarding the jet frequency, no significant differences on the lift coefficients were found between the reduced frequencies F+ equal to 1 and 2. Results of aerodynamic loads showed an improved lift coefficient in relation to the baseline airfoil when pulsed and steady jets are employed. Pulsed jets under square waveform were effective even at high deflected flap condition at 50°, with a significant gain in the lift coefficient of 36%, in relation to the uncontrolled case, combined with a drag reduction of 20%, and a decrease in mass flow up to 49% in relation to the steady jet for the same lift gain. Although sine and square waveform results presented similar improvements for lift, the drag is around 15% higher for the former. When compared with the steady jet case, the mass flow reduction is 36% for the sinewave. Synthetic jets with zero-net-mass-flux proved superior to the baseline conventional multi-element airfoil only with deployed flap at 50°, where a modest lift improvement of 5% was observed.

Determining the volume fraction in 2-phase composites and bodies using time varying applied fields

A body Θ containing two phases, which may form a periodic composite with microstructure much smaller that the body, or which may have structure on a length scale comparable to the body, is subjected to slowly time varying boundary conditions that would produce an approximate uniform field in Θ were it filled with homogeneous material. Here slowly time varying means that the wavelengths and attenuation lengths of waves at the frequencies associated with the time variation are much larger than the size of Θ, so that we can make a quasistatic approximation. At least one of the two phase does not have an instantaneous response but rather depends on fields at prior times. The fields may be those associated with electricity, magnetism, fluid flow in porous media, or antiplane elasticity. We find, subject to these approximations, that the time variation of the boundary conditions can be designed so boundary measurements at a specific time t=t0 exactly yield the volume fractions of the phases, independent of the detailed geometric configuration of the phases. Moreover, for specially tailored time variations, the volume fraction can be exactly determined from measurements at any time t, not just at the specific time t=t0. We also show how time varying boundary conditions, not oscillating at the single frequency ω0, can be designed to exactly retrieve the response at ω0.

Static load identification of nonlinear structure based on internal force estimation exemplified by underground straight-wall arch structure

Due to the complexity of the interaction between surrounding rock and support structure, underground structures have complex and time-varying boundary conditions that is difficult to determine in advance. It is necessary to perform damage/load identification of underground structures in real time, but there is a lack of relevant studies. Therefore, a load inversion method of nonlinear structure based on internal force estimation is proposed in this paper. First, a static damage experiment of straight-wall arch model was conducted to analyze its crack development pattern and damage mode. Then, interpolation and integration method (IIM) was used to calculate the internal force of the section through numerous strain data, and its estimation accuracy was verified by experiment and simulation. Finally, a new method of load identification, IVW-LAD (Incremental Variable Weighted - Least Absolute Deviation), using internal forces as optimization variables was proposed and validated. In summary, the findings of this paper can provide a reference for health diagnosis and load identification of nonlinear underground structures.

Dynamic characteristics of a running away pump-turbine with large head variation: 1D + 3D coupled simulation

The dynamic characteristics of running away pump turbines (PTs) with a large head variable amplitude have not been understood thus far, primarily because of two difficulties in simulation and analysis. The first is how to provide accurate time-varying boundary conditions for transient simulation of the turbine runaway process (TRP). The other is how to determine the specific appearance time of each frequency component of the complex pressure fluctuations. This study presented a one- and three-dimensional (1D-3D) coupled approach considering waterway dynamics to provide accurate unsteady boundary conditions for the transient flow simulation of a PT with a large head variable amplitude. The short-time Fourier transformation (STFT) approach was adopted to analyse the time-frequency characteristics of the transient pressures and impeller forces. The study found that the fluctuations of pressures and impeller forces during the TRP of the PT with a large head variable amplitude contained two exclusive fluctuation frequency components. The former was approximately three times the rated rotational frequency of the impeller. The later was a series of integer fold transient rotational frequencies of the impeller, which was irrelevant to the rotor-stator interactions. The findings have important value for controlling the pressure fluctuations during the TRP of PTs.

Investigation of Electromagnetic Sub-Modeling Procedure for the Breeding Blanket System

The outcome of the electromagnetic (EM) analyses carried out during the DEMO pre-conceptual phase demonstrated that EM loads are relevant for the structural assessment of the breeding blanket (BB) and, in particular, for the definition of the boundary conditions at the attachment system with the vacuum vessel. However, within the scope of the previous campaign, the results obtained using simplified models only give a rough estimation of the EM loads inside the BB structure. This kind of data has been considered suitable for a preliminary assessment of the BB segments, but it is not considered representative as input for structural analysis in which a detailed BB internal structure (that considers cooling channels, thin plates, etc.) is analyzed. Indeed, mesh dimensions and computational time usually limit EM models that simulate a whole DEMO sector. In many cases, these constraints lead to a strong homogenization of the BB structure, not allowing the calculation of the EM loads on the internal structure with high precision. To overcome such limitations, an EM sub-modeling procedure was investigated using ANSYS EMAG. The sub-modeling feasibility is studied using the rigid boundary condition method. This method consists of running a global “coarse” mesh, including all the conducting structures that can have some impact on the component under investigation and inputting the obtained results on the detailed sub-model of the structure of interest as time-varying boundary conditions. The procedure was tested on the BB internal structure, taking as reference a DEMO 2017 baseline sector and the helium cooled pebble bed (HCPB) concept with its complex internal structure made by pins. The obtained results show that the method is also reliable in the presence of non-linear magnetic behaviour. The methodology is proposed for application in future BB system assessments.

Analytical solution to heat transfer in stationary wet granular mixtures with time-varying boundary conditions

Analytical solution to heat transfer in stationary wet granular mixtures with time-varying boundary conditions

Mathematical modeling of the process of vacuum drying of corn cobs

Purpose. To determine the effect of vacuum on the process of drying corn seeds on the cob using mathematical modeling. Methods. Differential calculus methods were used, in particular, methods for solving differential diffusion equations with time-varying boundary conditions, as well as methods of differential and analytical geometry. The mathematical model was analyzed using graphs constructed by the computer program MathCad 13 from the solutions of differential equations and methods of technical analysis. Results. On the basis of the differential diffusion equation with variable boundary conditions in time, the processes of moisture removal from corn seeds in the cob were described. These processes are described by analytical dependencies that describe the kinetics of drying corn seeds and the entire cob, while taking into account diffusion processes both inside the seed and in the cob core. They are used to construct graphs that describe the dynamics of the moisture field inside a single seed and all the seeds along the cob. They show that the part of the seed that is outside the cob takes the longest to dry. The graphs also show the unevenness of drying along the cob: the ends of the cob dry the fastest, and the middle dries the slowest. Conclusions. The vacuum drying of corn cobs consists of 3 processes that take place in parallel: evaporation of moisture from the end surface of the cob, diffusion of moisture to the outer surface of the cob, and diffusion of moisture from the seed to the cob. The outer part of the seed takes the longest to dry, resulting in a moisture gradient that moves toward the core. Reducing the pressure inside the drying chamber reduces the time for moisture equalization, both over the volume of the seed and along the length of the cob, and increases the drying speed of the cob. To increase the uniformity of drying, the distribution of the heating temperature of the seeds along the length of the cob should correspond to a parabolic relationship with the highest temperature in the middle of the cob length, or it is necessary to divide the cob into parts to reduce the length of the moisture path. Keywords: seeds, moisture diffusion, drying time, moisture evaporation, vacuum, humidity, corn cob, cob, drying kinetics, drying chamber.

Experimental and numerical analysis of variable volume ratio as additional optimization parameter in organic Rankine cycle expanders

• Variable volume ratio piston expander has been experimentally characterized. • Semi-empirical ORC model quantifies the effects of a variable volume ratio expander. • Good selection of built-in volume ratio can eliminate need for variable valve timing. • Variable valve timing extends the working conditions of a volumetric expander. In this paper, the use of a variable volume ratio expander for a 10 kWe organic Rankine cycle (ORC) to be integrated in a Carnot battery has been studied. The reciprocating piston expander used in this work has a built-in mechanism which allows changing the inlet valve timing of the expander and hence the expander internal volume - and pressure ratio. The experimental results of a 10 kWe ORC prototype are presented. A total of 58 steady state points has been collected by varying the heat source and cold sink temperature, the refrigerant mass flow rate, the expander speed and the expander inlet valve timing. The experimental results for the ORC are presented in terms of electrical power output and cycle efficiency and for the expander the isentropic and volumetric efficiencies are determined for different valve timings. Although the experimental results clearly show the effect of changing the internal volume ratio on the expander performance we here present a combined experimental-numerical approach which is necessary to evaluate the performance under varying external conditions. Therefore, an empirical quasi-steady state model of the ORC has been calibrated based on the experimental results. Subsequently, in a numerical analysis, the use of various valve timings has been evaluated under time-varying boundary conditions. A maximal difference of 14.2 % in the net power production has been observed between the most optimal and least optimal valve timing. However, for the presented numerical study the use of variable valve timing only increased the net power production maximally 1 % compared to the optimal fixed timing. Altogether, this study demonstrates the benefit of variable volume ratio reciprocating piston expanders for ORCs under varying external conditions.