Validated dynamic model of an organic Rankine cycle (ORC) for waste heat recovery in a diesel truck

Accepted Manuscript Validated dynamic model of an organic Rankine cycle (ORC) for waste heat recovery in a diesel truck Wolfgang R. Huster, Yannic Vaupel, Adel Mhamdi, Alexander Mitsos PII:

S0360-5442(18)30466-3

DOI:

10.1016/j.energy.2018.03.058

Reference:

EGY 12516

To appear in:

Energy

Received Date: 11 October 2017 Revised Date:

7 February 2018

Accepted Date: 9 March 2018

Please cite this article as: Huster WR, Vaupel Y, Mhamdi A, Mitsos A, Validated dynamic model of an organic Rankine cycle (ORC) for waste heat recovery in a diesel truck, Energy (2018), doi: 10.1016/ j.energy.2018.03.058. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Fundamental Equation of State l1

l2

wall SH

Tsat

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l0

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Transient Heat Source

Experimental Data

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wall

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Cooling Water

Experimental Data

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Dynamic Model

Validated Model

Parameter Estimation

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Validated Dynamic Model of an Organic Rankine Cycle (ORC) for Waste Heat Recovery in a Diesel Truck Wolfgang R. Hustera,1 , Yannic Vaupela,1 , Adel Mhamdia , Alexander Mitsosb,a,∗

Aachen University - Process Systems Engineering (AVT.SVT), Forckenbeckstr. 51, 52074 Aachen, Germany b JARA-ENERGY, Templergraben 55, 52056 Aachen, Germany

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a RWTH

Abstract

Recovering waste heat from the exhaust gas of diesel trucks using an organic Rankine cycle (ORC) is considered a promising option to reduce emissions and fuel consumption. As the system operates under highly transient boundary conditions, the development of a predictive model accounting for its dynamic behav-

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ior is of high relevance. In this contribution, such a model is presented. Evaporator and condenser are modeled using a dynamic moving boundary model,

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while pump and turbine are represented with pseudo-steady state models, as dynamics are assumed to be negligible here. A fundamental thermodynamic equation of state based on the free Helmholtz energy is implemented for the

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working fluid ethanol. As unknown parameters, e.g., in heat transfer, occur, a parameter estimation is conducted with the use of measurement data. The

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model is then validated using different experimental data. It is able to capture the system dynamics well both qualitatively and quantitatively, e.g., the mean relative temperature deviations are all smaller than 4 %. Keywords: organic Rankine cycle, waste heat recovery, dynamic modeling, moving boundary approach, parameter estimation, model validation

∗ Corresponding author: Alexander Mitsos. Tel.: +49 241 80 94704; fax: +49 241 80 92326. E-mail address: [email protected] 1 Both authors contributed equally to this publication.

Preprint submitted to Elsevier

March 10, 2018

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1. Introduction According to the U.S. Energy Information Administration [1], diesel fuel consumption accounted for 36 % of the transportation energy consumption worldwide in 2012. To reduce this consumption, hence cut specific emissions, which 5

are being regulated by the EURO norm within Europe [2], the idea of recov-

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ering heat from otherwise lost exhaust gas energy has become more and more attractive. This thermal energy can be used in a power cycle to regain power

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that can be transmitted to the powertrain.

The exhaust gas of diesel trucks, with temperatures typically ranging from 300 500 ◦ C [3], can be efficiently utilized as an energy source employing an organic Rankine cycle (ORC). An extensive review on waste heat recovery (WHR) in vehicles with internal combustion engines, published in 2013, is given in [4]. Comprehensive research has been carried out in the field of ORCs, as they can also be utilized for recovering other mid- or even low-temperature energy sources

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such as geothermal brine [5] and solar thermal energy [6].

In this contribution, a dynamic model of an ORC for WHR in a diesel truck is

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proposed that is suitable for optimization. To obtain a model that is valid over a broad range of operating conditions, model parameters are estimated using transient data obtained from a test rig. To enable parameter estimation via

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dynamic optimization, the thermodynamic formulation necessary for the calculation of the WF properties [7] is directly implemented in the model. Further, we introduce generic heat transfer correlations that can be parameterized with-

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out detailed knowledge about the heat exchanger geometry. This approach of estimating parameters with transient data on an ORC system for waste heat recovery has so far not been reported in literature and promises to provide a more accurate prediction of transients. In order to demonstrate the effectiveness of the approach, the model is validated with a simulation using input data

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from the World Harmonized Transient Cycle (WHTC), which exhibits strongly

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transient exhaust gas conditions.

Within the last decades, most of the research effort concentrated on optimal design of ORC systems for steady-state operation [8], including structural de35

cisions or optimal choice of the working fluid (WF) [9–11], even extending to

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the choice of WF mixtures [12–14]. Occasionally, dynamic models were used to

account for part-load conditions [15]. For the use within diesel trucks, the variations in exhaust mass flow and temperature due to changing driving conditions

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have to be considered. These occur as heavy duty diesel trucks are operated in long haul transport as well as urban traffic. Hence, the system is subject to significant fluctuation in its heat source, and therefore the dynamic system behavior is of high relevance. In [16], it was found that during transient operation less than half of the design point efficiency could be achieved. Thus, the focus of this contribution is on dynamic modeling of the ORC. As most authors do rarely validate their model on such a wide range of working conditions or

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do not report their errors in detail, this publication highlights the need for a

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parameter estimation and validation on transient measurement data, including error indicators.

When building a dynamic model, it has to be decided which dynamic effects

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have to be considered and which can be assumed to be quasi-stationary on the

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timescale of interest. In accordance with the findings in literature, we use dynamic models for the heat exchangers, as the most significant dynamics of an ORC are observed there [17], while the pump and the turbine are represented

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with pseudo-steady state models.

Dynamic modeling of heat exchangers has received significant attention, especially with regard to control system design [18]. The two most common approaches are either discretization using the finite volume method or the im-

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plementation of a moving boundary model. While historically a trade-off between more accurate finite volume models and computationally less expensive 3

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moving boundary models was assumed to exist, recent publications have questioned this notion [18–20]. It was found that, with a sufficient number of finite

volumes, both approaches can achieve similar accuracies. Under this constraint, 65

a smaller computational effort for the moving boundary approach is required, which explains its popularity in model-based control applications [21]. Further-

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more, often linearized models derived from either type are employed [22–25] or physically motivated simplifications are used [26]. As the computational effort can be crucial in dynamic optimization, the moving boundary approach is

chosen within this contribution. In contrast, the finite volume method offers

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higher flexibility to adapt the approach to different heat exchanger types and geometries [19].

An extensive description of moving boundary models can be found in [27], where 75

many important aspects regarding the implementation are discussed. In con-

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trast to discretization with the finite volume method, the moving boundary approach in its initial formulation can not handle the appearance or disappear-

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ance of phases that is associated with startup and shutdown of the system and can also occur during operation. This particular shortcoming and attempts to overcome it have been subject of several publications [28–34].

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While many publications dealing with details of dynamic heat exchanger model-

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ing are available, relatively few contributions report full ORC models for WHR. In the majority of publications on full ORC models heat exchangers are modeled

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with the finite volume method [35–38]. In [39], both heat exchanger modeling approaches are presented. In [38] an ORC model for a heavy duty diesel truck implemented in Simulink is presented. Most of these publications obtain thermodynamic properties of the WF from look-up tables or databases connected through an interface. Another option is presented in [35], where polynomials

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that are valid in certain intervals are fitted to the data.

This contribution extends the findings in literature by providing a dynamic 4

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parameter estimation procedure. To allow for this, we embed the fundamental equation of state in the model. Further, we introduce flexible heat transfer cor95

relations that do not require precise knowledge of the heat exchanger geometry. Finally, we present an extensive parameter estimation and model validation us-

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ing experimental data representative of street traffic conditions.

The remainder of this article is structured as follows: In Sec. 2, the topol100

ogy of the considered system is introduced, followed by a description of the

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implemented models in Sec. 3. The parameter estimation is described in detail in Sec. 4 and its results are discussed in Sec. 5, together with a validation using input data of different experiments. Finally, conclusions are drawn in Sec. 6, together with an outlook on potential future research directions.

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2. System Topology

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A schematic representation of the considered system is depicted in Fig. 1. The liquid WF, ethanol in this contribution, is pressurized by a pump 4 → 1 ,

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preheated, evaporated and superheated in an evaporator 1 → 2 , where the exhaust gas from a heavy-duty diesel truck engine serves as the heat source, expanded in the turbine to transfer mechanical power to a generator 2 → 3 and

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finally completely condensed (and subcooled) in a condenser 3 → 4 , in which the cooling capacity is provided by a dedicated cooling water cycle (CW). The

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input variables to the system are: exhaust gas mass flow and inlet temperature m ˙ exh , Texh,in , CW mass flow and inlet temperature m ˙ CW , TCW,in , WF mass

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flow m ˙ W F and turbine speed nturb .

The WF leaves the evaporator as superheated vapor (nominal operation mode), or as subcooled liquid or two-phase fluid, depending on the amount of heat transferred from the exhaust gas to the system. In the latter cases, the turbine

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bypass valve is opened in order to avoid droplet erosion in the turbine during expansion, thus no power is generated. Also, in case of insufficient cooling capac-

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ity, the amount of heat transferred into the system can be reduced by opening the exhaust gas bypass valve. In this contribution, only the nominal transient

operation mode is considered where the WF is not expanded into the two-phase 125

region and all available exhaust gas is passed through the evaporator. A liquid storage tank is installed in the system in order to be able to control the total

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WF volume in the cycle. The valve to the storage tank can be opened during

startup/shutdown which maintains the condenser pressure at ambient pressure

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3. Dynamic ORC Model

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and can be closed when the desired amount of WF in the cycle is reached.

In the following, all implemented dynamic models representing the ORC system are described. The model is implemented in the equation-based modeling environment gPROMS [40], which, in addition to providing numerical solvers for DAE-systems, offers the possibility to perform parameter estimation and solve dynamic optimization problems.

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3.1. Heat Exchanger Model using the Moving Boundary Approach The heat exchangers (evaporator, condenser) are modeled using the moving

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boundary approach where, instead of discretizing in the spatial domain along a grid with fixed cell volumes, the fluid volume is discretized in zones according to 140

the fluid’s aggregate state. Given an evaporator, in which subcooled fluid is fed

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and superheated vapor exits (vice versa for the condenser), the model consists of three zones. This case is illustrated in Fig. 2.

The quantities at the zone interfaces using the indexes 0 and 2 are in fact

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saturation quantities in the presence of all zones. However, to maintain generality, we use the notation where two additional differential quantities, the specific enthalpies h0 and h2 , are added to the model, as compared to the most com-

mon formulation [27]. This allows the model to be easily extendable to simulate non-nominal conditions, where not all zones are present and h0 or h2 might

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be the outlet enthalpy [32]. For each zone, a lumped mass and energy balance

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is formulated. The differential quantities characterizing the WF are pressure p and enthalpy h at the zone boundaries. The heat exchanger wall is divided

according to the zone lengths of the WF and an energy balance is derived for

each wall element. During simulations, the zones move along the evaporator length. The secondary side is modeled quasi-stationary, as the exhaust gas or

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cooling water normally have lower residence times in the heat exchanger and no

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phase change occurs [27].

The heat exchanger is a counter-current plate heat exchanger with multiple 160

fluid channels. For simplicity, the channels are not modeled separately. Instead, it is assumed that only one rectangular channel per fluid and one single separating wall exists. For this assumption to be valid, the flow conditions in the separate channels must all be similar. This simplification of the geometry is

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expected to contribute to the prediction error. 165

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In order to derive the moving boundary model equations, a number of assumptions are made, namely

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• averaged thermodynamic state variables for each zone (enthalpy, density), • spatially uniform pressure over the complete heat exchanger length (momentum balance can be omitted),

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• constant cross sectional fluid flow area, • no axial heat conduction in WF or wall, • homogeneous equilibrium two-phase flow, and • quasi-stationary behavior on the secondary sides (exhaust, CW).

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These assumptions are typically made for the implementation of the moving boundary approach in literature ([18, 27, 30]) and are applied when deriving the following equations. 7

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3.1.1. Governing Differential Equations For the sake of brevity, the equations are presented in the final form that is

implemented in the model, rather than discussing their derivation. A detailed

derivation of all of the following equations in this section can be found in [32].

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All variables and abbreviations used are defined in the nomenclature.

For control volumes with single-phase flow, we get the following mass (1) and

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energy (2) balances   dza d¯ ρ d (zb − za ) dzb A (za − zb ) + ρa A + ρ¯ − ρb A =m ˙ a−m ˙ b, dt dt dt dt

(1)

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  ¯ d¯ ρ d (zb − za ) dp dh ¯ ¯ A (zb − za ) ρ¯ + (zb − za ) h + ρ¯h − A (zb − za ) dt dt dt dt
dza dzb +ρa ha A − ρb h b A =m ˙ a ha − m ˙ b hb + bW F αW F (zb − za ) Tw − T¯ , dt dt (2) where A is the cross-sectional area of the fluid channel and z is the longitudinal

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coordinate. ρ, T and m ˙ are density, temperature and mass flow of the WF, where the subscripts a and b indicate quantities of the left-hand and right-hand boundary of the zones and the overline indicates averaged quantities. t is the

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time, bW F the width of the fluid channel and αW F is the heat transfer coefficient from WF to the wall. The last term on the right hand side of (2) is the heat

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flow from the wall into the WF. As ρ¯ and ¯h are algebraic quantities, their time dependence is accounted for by constructing their total differential w.r.t. the differential quantities (3) and (4) and inserting them into (1) and (2), in order to obtain formulations, where only actual differential quantities appear in time derivatives.

¯ d¯ ρ ∂ ρ¯ dp ∂ ρ¯ dh = + ¯ dt ∂p dt ∂ h dt d¯h 1 = dt 2



dha dhb + dt dt

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(3)

(4)

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For the two-phase zone, the mass (5) and energy (6) balances are   d¯ d (zb − za ) γ ′′ A (¯ γ ρ′′ + (1 − γ¯) ρ′ ) + (zb − za ) (ρ − ρ′ ) dt dt dzb ∂ρ′′ dp ∂ρ′ dp  dza + (1 − γ¯ ) + ρa A − ρb A =m ˙ a−m ˙ b, +¯ γ ∂p dt ∂p dt dt dt

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 d (z − z )  d¯ γ ′′ ′′ b a A (¯ γ ρ′′ h′′ + (1 − γ¯) ρ′ h′ ) + (zb − za ) (ρ h − ρ′ h′ ) dt dt dρ′′ dp dρ′ dp dh′′ dp dh′ dp  +¯ γ h′′ + (1 − γ¯ ) h′ + γ¯ ρ′′ + (1 − γ¯ ) ρ′ dp dt dp dt dp dt dp dt dp dza dzb −A (zb − za ) + Aρa ha − Aρb hb dt dt dt
¯ =m ˙ a ha − m ˙ b hb + bW F αW F (zb − za ) Tw − T ,

(5)

(6)

where γ¯ is the average void fraction calculated with (7) and the superscripts and

′′

′

indicate quantities at liquid and vapor saturation respectively. The time

derivative of the average void fraction can be expressed by constructing the total

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differential w.r.t. the differential quantities (8).   ρ′ ′ ′′ γ¯ = (h − h ) ρ + ρ h2 − h0 0 2 2 (h0 − h2 ) (ρ′ − ρ′′ )    ′′ ′′ ρ (h − h0 ) ′ ′′ + (h − h ) ln ρ′ (h2 − h′ )

(8)

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d¯ γ ∂¯ γ dh0 ∂¯ γ dh2 ∂¯ γ dp = + + dt ∂h0 dt ∂h2 dt ∂p dt

(7)

Under nominal operating conditions, the heat exchanger model consists of two single-phase control volumes and one two-phase control volume, which are combined as can be seen in Fig. 2. The mapping of the zone boundary positions to the zone lengths li is given in Tab. 2. The energy balance for each wall zone (9) reads,  
dza,i
dzb,i dTwi li + Tw,Bi,i−1 − Twi + Twi − Tw,Bi,i+1 dt dt dt
= Q˙ exhi − bW F αW F,i li Twi − T¯i − αamb pevap li (Twi − Tamb ) ,

Aw ρw cpw

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(9)

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where Aw , ρw and cpw are the wall cross-sectional area, density and heat capacity. Twi is the temperature of the respective wall zone and Tw,Bi,i−1 and Tw,Bi,i+1 are the wall temperatures at the left- and right-hand boundary of the

zone, which are calculated using a length-weighted average, as suggested in [30]. Q˙ exhi is the amount of heat transfered from the exhaust gas to the wall, bW F

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is the WF channel width and αi the heat transfer coefficient for the WF in the

respective zone. A term accounting for heat loss from the exchanger wall to the environment is introduced, in which αamb is the heat transfer coefficient, pevap

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the heat exchanger perimeter and Tamb the ambient temperature.

3.1.2. Thermodynamic Model of the Working Fluid 190

We implemented a suitable thermodynamic formulation for the WF ethanol in form of a fundamental equation of state based on the free Helmholtz energy can be found in publications of Penoncello et. al. [7, 41]. This formulation uses

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temperature and density as key variables. A major advantage of this formu-

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lation is the availability of analytical derivatives for all important derivatives required for the moving boundary model. The aforementioned publications use

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extensive experimental data sets for validation.

The general fundamental equation of state uses the dimensionless Helmholtz energy α for the description of the fluid state [7]. It is the sum of an ideal term, α0 , and a term accounting for non-ideal behavior, α ¯ , which are both calculated

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only as a function of τ and δ, for which the equations are given in [7]. τ and δ are the dimensionless temperature and density, nondimensionalized by their respective critical values, marked by subscript c. (Tc = 514.71 K, pc = 62.68 · 105 Pa,

ρc = 273.186 kg/m3 ). With α0 and α ¯ and their first and second derivatives with

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respect to τ and δ, the calculation of all important thermodynamic variables (pressure, enthalpy, entropy, heat capacity, etc.) is possible. The analytical expressions for the derivatives can be found in [42]. Saturation properties are calculated from auxiliary equations which can be found in [7]. 10

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For a given pressure and temperature, the thermodynamic formulation could possibly find a solution being in the non desired thermodynamic state, as this

combination of characterizing variables is ambiguous. This is why it is necessary for the implementation of this formulation to force the density of each thermo-

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dynamic evaluation to be in the appropriate region (liquid zone: ρ > ρc , vapor

zone: ρ < ρc ), when the density is not known a priory. The two-phase zone therefore has to be characterized by a combination of the two saturation points

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and vapor fraction.

3.1.3. Exhaust Temperature Calculations

By analytical integration of the quasi-stationary energy balance on the exhaust side from interface i + 1 to interface i assuming static one dimensional flow [31], as depicted in Fig. 3, the temperature at the end of one element can

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be calculated as in (10) and the heat transfered to the wall as in (11). Both the exhaust heat capacity cp,exhi and the heat transfer coefficient αexhi are assumed

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constant over one element. m ˙ exh , Texh and bexh are mass flow, temperature and width of the exhaust channel.

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αexhi bexh Texhi = Twi + Texhi+1 − Twi exp − li , m ˙ exh cp,exhi
Q˙ exhi = m ˙ exh cp,exhi Texhi+1 − Texhi

i ∈ [0, 2]

(10) (11)

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3.1.4. Heat Transfer Correlations The dynamic behavior of the heat exchanger model is significantly influ-

enced by the chosen heat transfer correlations. In literature, different choices, ranging from fixed heat transfer coefficients [37] to Nusselt correlations [43], can be found. While fixing the heat transfer coefficients reduces computational effort, it can be expected to deteriorate the model accuracy. Nusselt correlations, which are based on similarity theory, promise high accuracy, but require detailed knowledge about the heat exchanger geometry. As a compromise, we

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decided to introduce generic correlations for the calculation of all heat transfer correlations, where the influences of mass flow and temperature on the heat

transfer coefficient can be captured without detailed information on geometry. An additional advantage of this approach is that finding adequate bounds on the parameters to be estimated is rather simple. A similar approach has been

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used in [35]. For all three sections of the evaporator, we choose the same generic

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functional approach, as given in (12), for the WF heat transfer coefficients.   ci,1,evap  ci,2,evap  m ˙ m ˙ αi,W F,evap = αi,0,evap · (1 − ki ) · + ki · + di,evap m ˙0 m ˙0 (12)

The approach assumes the heat transfer coefficient to depend on the WF mass flow rate. It is scaled using a maximum mass flow rate m ˙ 0 to ensure the scaled value ranges between zero and one. αi,0 and di in (12) represent a base coefficient and an additional factor. While the exponent ci,1 is bounded between zero and

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one, the exponent ci,2 is bounded to be greater than one. To allow for the dominance of one exponent within a region of low or high mass flow, the logistic

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function ki is used as an activation function, as shown in (13):

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ki =

1   ˙ i 1 + exp m−e m ˙ 0 ·fi

(13)

The parameters ei and fi are used to determine the mass flow rate at which the logistic function switches and how fast this switch occurs. While this functional

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approach offers great flexibility and a wide range of behaviors can be captured, six parameters per zone are added to the parameter estimation problem.

For the heat transfer in the condenser and the exhaust gas heat transfer coefficient, the comparable, yet slightly different correlations (14) and (15) are used, where the subscripts cond and exh represent the values of the WF in the condenser and the exhaust gas.  ci,1,cond  c m ˙ WF Ti,W F,in i,2,cond αi,W F,cond = αi,0,cond · · + di,cond m ˙ W F,0 TW F,0 12

(14)

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m ˙ exh m ˙ exh,0

cexh,1  c Ti,exh,in exh,2 · + dexh Texh,0

(15)

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αi,exh = αexh,0 ·



In addition to the mass flow rate, they also depend on the respective temperature TW F/exh,0 . Again, the temperature is normalized by the maximum permissible

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temperature in the heat exchanger.

For calculating the heat transfer coefficient of the CW in the condenser, we employ a simple approach, where the CW mass flow m ˙ CW is the independent

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variable, i.e., αCW = αCW,0 ·



m ˙ CW m ˙ CW,0

cCW

+ dCW .

(16)

The mass flow rate is once again normalized by the maximal CW mass flow. 3.2. De Laval’s Nozzle

An important variable that strongly influences the pressure in the evapora-

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tor and therefore most characteristics of the WF is its mass flow leaving the evaporator m ˙ evap,out . Its value can be determined by the implementation of

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a model for a de Laval’s nozzle, which is placed within the turbine to convert thermal energy into kinetic energy. The nozzle is modeled similar to an orifice, thus can also be used for the turbine bypass. Herein, the calculation of the mass

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flow depends on the state of the entering fluid and the ratio between the inlet

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and outlet pressure.

The implemented model is based on the description in the VDI W¨armeatlas [44] and the previous work of Leung [45], assuming homogeneous flow equilibrium conditions. With the simplification of neglecting friction on the wall, m ˙ evap,out is calculated from (17) - (19), in which Cd is the discharge coefficient, pHP the

high pressure level, νin the specific inlet volume, ψ the discharge function and ALaval the smallest cross-sectional area of the Laval nozzle. r pHP m ˙ evap,out = Cd 2 · · ψ · ALaval νin

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(17)

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r     (1 − ηs ) + ω · ηs · ln ηηs − (ω − 1) · (ηs − η)   ψ= ω · ηηs − 1 + 1

(18)

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  ω2 − 2 · ω + 1 3 ηc 2 0= · ηs − 2 · (ω − 1) · ηc + ω · ηs · ln + · ω · ηs − 1 2 · ηs · ω ηs 2

(19)

The calculation of ψ in (18) depends on whether the flow is in choked state (reaching sonic velocity at ALaval ). This can be verified by calculating the

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critical flow pressure pc,Laval and comparing it to the low pressure level pLP . In (18), the nondimensionalized pressures ηc =

pc pHP

and η =

p1 pHP

are used. ηs

is set to one [44]. Using this formulation, Cd is the only unknown parameter that can be determined in the parameter estimation. 230

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3.3. Models for Pump, Turbine, Piping and Fluid Tank

The pump is modeled assuming a fixed isentropic and mechanical efficiency

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(ηis,pump , ηmech,pump ), according to (20). Within the model, both efficiencies are set to 0.9. Changes of either efficiency would have a minor impact on the

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system model, as the pump power Ppump is far smaller than that produced by the turbine Pturb and the effect of ηis,pump on the WF outlet temperature is negligible. hout,is is the enthalpy of the WF in the case of an isentropic

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compression/expansion.

Ppump =

1 ηmech,pump

·m ˙ WF ·

hout,is − hin ηis,pump

(20)

For the turbine, (21) is used to calculate the power output Pturb . Pturb = ηmech,turb · m ˙ W F · ηis,turb · (hin − hout,is )

(21)

For both the isentropic and mechanical efficiency (ηis,turb , ηmech,turb ), a set of data points of the turbine efficiencies is available. Using this data, polynomial fits are created. The isentropic efficiency is a function of pressure ratio between

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high and low pressure and turbine speed. For this, a polynomial function of third order with respect to pressure ratio respectively fifth order to turbine

speed is chosen. The mechanical efficiency, in contrast, is a function of turbine

speed n (second order polynomial) and torque M (fifth order polynomial). Both

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polynomial functions reflect the measured data accurately, as shown in Tab. 3.

When comparing outlet and corresponding inlet temperatures of two sequential units in the ORC, it becomes obvious that it is necessary to account for

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heat losses in between. For this, between two temperature sensors, pipe segments are placed in the model. As the heat transfer in these pipe segments stat does not take place instantly, the stationary outlet temperature TW F,out is not

immediately reached. In order to account for this behavior within the pipes, a first order system with the time-constant τpipe is used, i.e.,
dTW F,out 1 stat = · TW F,out − TW F,out . dt τpipe,i

(22)

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stat The stationary outlet temperature of the WF TW F,out is computed from (23),

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similar to (10). It is a function of ambient temperature Tamb , inlet temperature TW F,in and heat capacity flow m ˙ W F · cp,W F , assumed to be constant over the pipe segment. Furthermore, the heat transfer to the pipe wall is described by

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the factor αpipe · Apipe .

= Tamb + (TW F,in − Tamb ) · exp



(αpipe · Apipe )i m ˙ W F · cp,W F



(23)

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stat TW F,out

Both τpipe and (αpipe · Apipe ) depend on the respective pipe segment and fluid state, thus are not known a priori. Therefore, they have to be estimated within the parameter estimation described in Sec. 4.

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We did not implement a dynamic model including storage terms for the tank since no sensor signals for validation are available. Instead, we introduced an orifice, representing the connection to the tank, using (17) - (19). The tank is assumed to operate at atmospheric level. With an adequate cross-sectional

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area, it is possible to calculate the mass flow leaving the condenser in order to 15

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keep the pressure level constant. 4. Parameter Estimation

The models of the units in the ORC described in Sec. 3 contain a large number of unknown parameters. These include parameters for the heat trans-

fer correlations in the heat exchangers and geometry parameters. As discussed

present results of parameter estimation.

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in Sec. 1, few publications in the field of dynamic heat exchanger modeling

In the rare cases where a parameter

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estimation is carried out, it is generally done with a steady-state model. This yields a nonlinear programming problem (NLP), which can be solved to minimize some least-squares criterion [46]. In [38], Nusselt correlations are extended with multipliers, which are separately optimized for various steady-state points using Particle Swarm Optimization. In [47], coefficients for heat transfer correlations, that are very similar to the ones introduced in Sec. 3.1.1, are estimated. However, the estimation is performed with a steady state model and only base

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coefficients, that correspond to α0 , are estimated. Parameter estimation based on data measured at steady-state points does not necessarily improve capturing

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the system’s transient behavior.

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To estimate the parameters using data measured under transient conditions means that the dynamic model has to be embedded in the parameter estimation problem, which results in a dynamic optimization problem [48]. The parameter

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estimation utility in gPROMS allows to perform dynamic parameter estimation using a Maximum Likelihood formulation by minimizing the objective function (24a) subject to the dynamic model (24b) - (24c), where data of j measured

variables in i experiments, taken at k time points, are considered. ξ are the

differential and algebraic state variables, u are the input values recorded in the respective experiments, θ is the vector of parameters to be estimated and x are 2 the differential state variables. The sensor variances σijk are assumed to have

constant values of 2 K for temperature and 0.1 bar for pressure sensors. As we used a constant variances model, changes in the variance values correspond to

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a different weighting in the objective function between pressure and temperatures. Delays in the response of the sensors are neglected. Eqs. (24a) to (24c) are reported in [40].

θ

s.t.

i=1 j=1 k=1

 2  ˜ijk − ξijk ξ     2 σijk

  ˙ ξ, u, θ 0 = f ξ, x (t = 0) = x0

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min

Mij Vi NX NE N X X

(24a)

(24b) (24c)

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To solve the parameter estimation problem, the initial values for the differential quantities (24c), indicated by subscript 0, have to be specified. However, the vector of differential variables contains quantities such as the wall temperatures and zone lengths that cannot be measured. To circumvent this issue, we assume the system to be in steady-state at the beginning of the experiment. x0 = xstat (u0 , θ)

(25)

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The steady-state of the system xstat is a function of the parameters to be es-

245

We derive the steady-state by simulation by

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timated as indicated in (25).

simulating from an initial state with fixed inputs for 5000 seconds before the first measurement is recorded. Depending on the chosen set of parameters,

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the simulation still can fail during the transient from the initial value to the steady-state associated with the parameter set, which might, in consequence,

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cut off some feasible solutions. Another issue associated with the method is 250

that dynamic optimization problems typically have multiple minima, while we only employ a local solver. Hence, the quality of the solution depends on the chosen initial values of the parameters.

Fig. 4 shows the ORC topology including all relevant sensors and introduced

255

pipe segments required for the parameter estimation and following model validation. Five pipe segments are introduced according to the formulation presented in Sec. 3.2. The geometry of the evaporator and condenser have many uncertainties such as the fluid flow areas, heat exchange areas, wall thicknesses etc., 17

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260

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therefore several parameters have to be estimated here. Another important parameter is the discharge coefficient for the Laval nozzle in the turbine. Additionally, all coefficients for the heat transfer correlations introduced in (12) to

(16) have to be estimated. As no sensor data of the ambient conditions during the measurement is available, the sink term with a constant heat transfer

265

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coefficient multiplied with the perimeter is to be estimated, together with an assumed ambient temperature.

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The tables summarizing all estimated parameters together with their respective lower and upper bounds are given in Tab. 4, 6 and 7.

5. Parameter Estimation Results and Model Validation 270

This section consists of three parts: In Sec. 5.1, the pipe model used to connect the respective models representing the ORC components is validated.

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The results of the parameter estimation for the main ORC model components are presented in Sec. 5.2, followed by a validation of the full cycle model in

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Sec. 5.3 using measurement data from an experiment not incorporated in the previous parameter estimation. For confidentiality reasons, all signals are scaled.

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5.1. Pipe Model Parameter Estimation The parameter estimation for the pipe models is carried out with data from

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the temperature sensors depicted in Fig. 4. The data collected from the sensor before the respective pipe segments is the input to (22) and (23), while the data

280

from the sensor behind the respective element is the desired output. For all segments, measurements from a number of experiments are taken into account. Fig. 5 shows trajectories for one of the experiments using the obtained parameters for the pipe segment between evaporator and turbine, with the WF outlet temperature of the evaporator as a model input. It can be seen that the model

285

is able to sufficiently capture the dynamics of the temperature at the turbine inlet. For all other pipe segments, results of similar quality are achieved by the

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parameter estimation. Therefore, all pipe models are used together with their respective values for τ and α · A within the validation of the ORC in the next section. All values are given in Tab. 4. 290

5.2. Parameter Estimation Results of the ORC model

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The parameter estimation for the remaining parts of the ORC model, namely

the evaporator and condenser, is carried out using transient measurement data from a test rig described in Sec. 2.

As the storage tank valve after the con-

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denser is open throughout the experiments, the low pressure of the ORC is nearly constant at around 1 bar. Therefore, the low pressure level is not taken into account for the objective function of this parameter estimation problem. For the measurements taken for the parameter estimation, the turbine in the test rig was bypassed using the expansion valve. As a result, the superheated vapor is assumed to undergo an isenthalpic expansion to the low pressure level and no further detailed model considering the geometry of the valve is imple-

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mented. To account for losses in the bypass, an additional pipe segment is

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introduced.

The input variables for one of the experiments included in the parameter esti-

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mation are depicted in Fig. 6. In this particular experiment, the response on steps in the WF mass flow is examined. Step changes in the mass flow to three different constant levels are applied, each of the steps corresponding to about

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10 % change in mass flow. The simulation results obtained after the parameter estimation using the method described in Sec. 4, are shown together with the measurement data in Fig. 7. The simulated WF outlet temperatures after the evaporator and condenser are shown, as well as the outlet temperatures of the CW and exhaust gas after the respective heat exchangers. Furthermore, the high pressure level is used for comparison. Although there is a bias between the simulated and measured data, the transient behavior caused by time-varying inputs is captured by the model, as the plots of the measured values show similar dynamics, e.g., in comparing the high pressure level to the WF mass flow. 19

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To provide quantitative observations, four statistic indicators are introduced in (26) to (29): the maximal absolute deviation between the simulated result

and measurement ∆ξmax , the maximal relative deviation ∆ξmax,rel , the abso-

lute deviation averaged over all samples ∆ξ¯ and the relative mean deviation

∆ξmax,rel

ξ˜k − ξk

i2

!

,

k ∈ [0, ktot ]

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∆ξmax = max

r h

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∆ξ¯rel .

 rh i2  ˜k − ξk ξ   , = max    ξ˜k

k=0

! rh i2 ˜ ξk − ξk ktot

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∆ξ¯ =

kP tot

∆ξ¯rel =

k ∈ [0, ktot ]

kP tot

k=0

q

[ξ˜k −ξk ] ξ˜k

ktot

2

!

(26)

(27)

(28)

(29)

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The values of these indicators for the presented case are summarized in Tab. 8. In general, the deviations are small with a maximum relative error of less than

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4 % and a relative averaged error of less than 2 % for the temperatures. For the high pressure level, the mean relative error of 3 % supports the impression

295

obtained from inspection of the plot in Fig. 7a, where the dynamics are captured appropriately.

The high pressure level of the system can be adequately predicted for steadystate points. It is predominantly influenced by the WF inlet mass flow into

300

the evaporator as the steps in the inlet mass flow rate are qualitatively similar compared to the high pressure level. In the measurement data, after each step in the mass flow, the pressure shows a notable overshoot before it reaches a 20

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steady level. This overshoot is not appropriately captured by the model. The reason for the phenomenon is unclear and subject of ongoing investigation. 305

The remaining simulated output variables associated with the evaporator model

also exhibit an acceptable representation of the measurements. The maximum

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deviation of the simulated WF evaporator outlet temperature, which is of ma-

jor interest for the operation of the system, is 10.2 K and the mean deviation 310

is 3.7 K. The maximum and mean deviations of the exhaust gas outlet tem-

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perature are 4.6 K and 1.0 K respectively. In the condenser, a constant offset between the simulated outlet temperatures and the respective measurements is observed. This can be attributed to uncertainties in the geometry, heat transfer coefficients and heat capacity of the CW. 315

The resulting values of all parameters characterizing evaporator and condenser

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are given in Tab. 6 and 7. Quantitative measures of errors for direct comparison are rarely provided in literature. When comparing the errors found in this

320

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study to [39], where the deviation of the WF outlet temperature in the evaporator is at around 8 K, the absolute average error in our publication with 3.9 K for the measured value seems to be smaller. In general, a fair comparison of

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model accuracy is hardly possible as the system topology and scenarios vary from publication to publication. For example, in [38], comparable mean errors

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of ∆ξ¯pHP ,rel = 2.2 % and ∆ξ¯Tturb,in = 8 K are reported, but this system includes 325

a second evaporator in parallel, which also applies to [49, 50]. While [49] lists the errors for steady-state validation, the averaged errors in the dynamic validation are not provided. Compared to the errors given in [50] (∆ξ¯pHP ,rel = 12.4 %,

∆ξ¯hturb,in,rel = 5.4 %), our proposed approach seems to be more accurate. In contrast, the relative errors reported in [19] are comparable small (< 5 % when

330

using the MB approach). It has to be noted that the system is a vapor compression cycle operating at lower temperature and pressure levels.

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5.3. Validation of the Model In order to validate the model, we performed simulations with inputs taken from an experiment not considered in the parameter estimation. For this ex335

periment, parts of the World Harmonized Transient Cycle (WHTC) were taken as inputs, thus representing realistic data for the exhaust mass flow and inlet

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temperature in street traffic. The experiment was also executed with an open

tank valve thus constant low pressure level. In contrast to the case used for the parameter estimation, the WF was expanded using the turbine. Hence, the turbine model described in Sec. 3.2 is used.

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340

All model inputs are shown in Fig. 8. The exhaust gas mass flow exhibits strong fluctuations, as expected under real-world driving conditions, with the lowest value being 70 % smaller than the highest. However, the variations in the 345

exhaust gas temperature are small. The CW mass flow and inlet temperature

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are almost constant with the lowest input value being less than 5 % than the highest value. The change in WF inlet mass flow and turbine speed is high with

350

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values of 59 and 14 %.

Again, the simulated output variables of the evaporator are in good agreement

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with the measurements and the relative deviations are of the same magnitude as those in the previous section, as can be seen from Tab. 8. It has to be noted

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that the simulated WF evaporator outlet temperature does not completely capture the dynamics measured in the system. The maximum deviation is three

355

times higher than the deviation averaged over all sampling points. For the turbine outlet and condenser, the simulated values capture the dynamics of the measured temperatures properly, but deviate by an almost constant offset. Accordingly, the difference between the maximum absolute deviation ∆ξmax and

the mean absolute deviation ∆ξ¯ is comparatively small. 360

The constant offset in the turbine outlet temperature can be attributed to the turbine model. As the turbine map is created with a broader range of measure22

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ment data with respect to the pressure ratio, small deviations can occur for the simulated turbine speed. With more measurement data for the fit, this offset 365

could be reduced. Between the evaporator and bypass valve and between the bypass valve and the turbine, one pipe segment was introduced for experiments with an active turbine to account for heat loss in both sections. Therefore, the

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heat loss before the turbine is captured by the model and does not contribute to this error. Another reason for the error could be a pressure drop along the 370

piping and condenser, which would result in a higher low pressure level after

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the turbine. As this pressure drop can not be taken into account with the pipe model described in Sec. 3.2, this potential error is not further evaluated here. With a lower turbine outlet temperature, the remaining offset of ∆ξ¯ = 1.3 K in 375

the WF condenser outlet temperature could be further reduced. The offset in the CW temperature may be a result of inaccuracies in the CW heat capacity

D

or the employed heat transfer correlations.

380

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It is also worth noting that the system appears not to be at steady state at the beginning of the simulation. An offset in the first 50 s is present in Fig. 9b and Fig. 9d. Therefore, the steady state assumption at the beginning does not

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seem to be completely justified here, but has to be used due to the lack of

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another appropriate procedure to initialize the simulation.

6. Conclusions and Outlook

385

In this manuscript, a dynamic model representing an ORC for waste heat re-

covery in a diesel truck is presented. As the dominant transient phenomena can be observed in the heat exchangers, dynamic models are implemented for the evaporator and condenser. More precisely, the moving boundary approach, one of two prominent approaches in dynamic heat exchanger modeling, is chosen.

390

The pump and expander are described with pseudo-steady state models. To account for heat losses between components, a simple pipe model is chosen. Since

23

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the ORC model is to be used for dynamic optimization, a fundamental thermodynamic equation based on the free Helmholtz energy is directly embedded.

This holds the advantage that the sensitivities with respect to the optimization 395

variables can be obtained directly by integrating the sensitivity system. If a

thermodynamic property database was used instead, as is commonly done in

SC

literature, the sensitivities would have to be obtained by finite difference per-

turbation, which tends to be time-consuming and inaccurate. To validate the model, rather than estimating unknown parameters based on steady-state measurements, transient measurement data are used and the arising dynamic op-

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400

timization problem is solved in gPROMS. Here, simple functional approaches, instead of more complex Nusselt correlations for the calculation of the heat transfer coefficients, are used.

405

The dynamics in the heat exchangers are mostly dominated by the pressure

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level, as many thermodynamic variables, such as saturation temperature and enthalpy of vaporization, are pressure-dependent. As, for example, the outlet

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mass flow of the evaporator strongly influences the high pressure level, the system dynamics are strongly connected and the initialization of the system is a 410

challenging task. Furthermore, the use of accurate heat transfer correlations in

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all zones of the moving boundary model is crucial for the implementation a predictive dynamic model. The approach presented in this contribution increases

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the parameter estimation effort, but allows for a high flexibility.

415

The results of a validation study show that the ORC model is able to capture the relevant dynamics qualitatively as well as quantitatively, with a mean relative error smaller than 4 % for all measured temperatures for a transient measurement data set. Note that the highest error, located at the turbine outlet, has to be mostly attributed to the turbine model. The mean relative error for

420

both evaporator outlet temperatures is smaller than 1 %. For the high pressure level, a mean relative deviation of 5.5 % arises.

This model accuracy could

not have been obtained without the parameter estimation applied to transient 24

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425

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data.

Future work further includes the extension of the model in order to handle

operational modes that are associated with the (dis-)appearance of zones as, for example, startup and shutdown of the ORC. As the governing equations are

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implemented in a general way, the model can be extended to these cases with

reasonable effort. Also, dynamic optimization of the model in order to deter430

mine optimal operation in the presence of highly transient disturbances, as it is

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the case in automotive WHR, can be conducted.

Acknowledgments

The work leading to this contribution was funded by the Federal Ministry for Economic Affairs and Energy (BMWi) according to a resolution passed by the German Federal Parliament under grant number 19U14010C.

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System topology of the ORC test rig including the exhaust pipe (grey, dashed), the WF cycle (gr Application of the moving boundary approach to the counter-current evaporator in the nominal c Illustration of the qualitative temperature distribution assumed for the exhaust gas in the evapor ORC topology for parameter estimation including pipe segments within the WF cycle. Temperat Result of pipe model between evaporator outlet and turbine inlet, showing the measured tempera Trajectories of all input variables within a step-response experiment used for the parameter estim Simulated trajectories of relevant thermodynamic variables of the ORC in comparison to their co Trajectories of all controlling inputs within the measurement used for the model validation. 40 Simulated trajectories of relevant thermodynamic variables of the ORC in comparison to their co

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1 2 3 4 5 6 7 8 9

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List of Figures

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Exhaust Texh,in, mexh 1

2

Working fluid

nturb

SC

meth

M AN U

4

3

Cooling water

Tcw,in, mcw

AC C

EP

TE

D

Fig. 1. System topology of the ORC test rig including the exhaust pipe (grey, dashed), the WF cycle (green, solid), and cooling water (blue, dash-dot). Circled numbers mark the states of the working fluid. Disturbances and control inputs, which are recorded during experiments, are labeled within arrows.

33

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Fig. 2. Application of the moving boundary approach to the counter-current evaporator in the nominal case. Numbers represent the discretized zones (subcooled, two-phase, superheated) respectively their boundaries.

34

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EP

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D

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Fig. 3. Illustration of the qualitative temperature distribution assumed for the exhaust gas in the evaporator.

35

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Pipe segments

T

Exhaust

T

T

p

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T

Temperature sensors T

SC

T

T

M AN U

T

T

T

T

Cooling water

AC C

EP

TE

D

Fig. 4. ORC topology for parameter estimation including pipe segments within the WF cycle. Temperature and pressure sensor positions are labeled in circles. The pipe segment in the bypass path is also accounting for losses in the expansion through the valve.

36

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0.84 0.83 0.82

T ∗ [-]

0.81

evap,out,meas turb,in,sim turb,in,meas

0.8

D

0.79

0.77 0.76

TE

0.78

100

EP

0

200

300

400

500

600

t [s]

AC C

Fig. 5. Result of pipe model between evaporator outlet and turbine inlet, showing the measured temperatures of the evaporator outlet and turbine inlet together with the simulated turbine inlet temperature.

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0.985

0.16 0.159 0.158

0.98

0.156 0.155 0.154

SC

T ∗ [-]

m˙ ∗ [-]

0.157

0.975

0.153 0.152 0.151 500

1000

1500

2000

t [s]

2500

3000

0

500

1000

1500

2000

2500

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t [s]

(a) Exhaust mass flow 0.565

(b) Exhaust inlet temperature

0.5435

0.56

0.543

0.555

0.5425

T ∗ [-]

m˙ ∗ [-]

M AN U

0.97

0

0.55

0.542

0.5415

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0.535 500

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0

D

0.545

0.541

0.5405 0

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1000

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(d) CW inlet temperature

(c) CW mass flow

EP

0.021

500

t [s]

t [s]

0.02

m˙ ∗ [-]

0.019

AC C

0.018

0.017

0.016

0

500

1000

1500

2000

2500

3000

t [s]

(e) Evaporator WF inlet mass flow

Fig. 6. Trajectories of all input variables within a step-response experiment used for the parameter estimation.

38

0.39

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0.86

meas sim

0.38 0.84 0.37

∆ξmax,rel = 2.0 %, ∆ξ¯rel = 0.8 % 0.82

0.34 0.33

0.8

0.78

0.32

∆ξmax,rel

= 7.7 %, ∆ξ¯rel = 3.2 %

0.31

0.76

meas sim

0.3

0.74 500

1000

1500

2000

t [s]

2500

3000

0

500

1000

1500

M AN U

0

SC

0.35

T ∗ [-]

p∗ [-]

0.36

2000

2500

3000

t [s]

(a) Evaporator WF outlet pressure

(b) Evaporator WF outlet temperature 0.547

0.71

0.705

meas sim

0.546

∆ξmax,rel = 1.1 %, ∆ξ¯rel = 0.2 %

0.545

T ∗ [-]

T ∗ [-]

0.7

0.695

0.544 0.543 0.542

∆ξmax,rel = 0.6 %, ∆ξ¯rel = 0.3 %

D

0.69 0.541

meas sim

0.685 0

500

1000

1500

2000

2500

(c) Exhaust outlet temperature 0.576

0

500

1000

1500

2000

2500

3000

t [s]

(d) Condenser WF outlet temperature

meas sim

0.574

EP

0.572 0.57

T ∗ [-]

0.54

3000

TE

t [s]

0.568

∆ξmax,rel = 2.0 %, ∆ξ¯rel = 1.7 %

0.566

AC C

0.564 0.562

0.56

0.558

0

500

1000

1500

2000

2500

3000

t [s]

(e) CW outlet temperature

Fig. 7. Simulated trajectories of relevant thermodynamic variables of the ORC in comparison to their corresponding measured trajectories in a measurement used for the parameter estimation, including maximal and mean relative errors.

39

0.35

0.98

0.3

0.975

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0.97

0.2

0.965

SC

T ∗ [-]

m˙ ∗ [-]

0.25

0.96

0.15 0.955

0.1

0.95

0

100

200

300

400

500

0

600

100

200

300

400

500

600

t [s]

t [s]

(a) Exhaust mass flow

(b) Exhaust inlet temperature

0.554

0.925

0.552

0.92

0.55

T ∗ [-]

0.915

m˙ ∗ [-]

M AN U

0.945

0.05

0.91

0.905

0.548

0.9

0.895 100

200

300

400

500

TE

0

D

0.546

0.544

0.542

600

0

100

200

t [s]

(c) CW mass flow

m˙ ∗ [-]

0.025

AC C

0.02

500

600

(d) CW inlet temperature 0.92 0.9

n∗ [-]

0.03

400

0.94

EP

0.035

300

t [s]

0.88 0.86 0.84

0.015

0.82

0.01

0

100

200

300

400

500

0.8

600

0

t [s]

100

200

300

400

500

600

t [s]

(e) Evaporator WF inlet mass flow

(f) Turbine speed

Fig. 8. Trajectories of all controlling inputs within the measurement used for the model validation.

40

0.55

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0.835 meas sim

0.83

0.5

0.825 0.82 0.815 0.81

0.4 0.805

0.795

∆ξmax,rel = 9.7 %, ∆ξ¯rel = 5.5 %

0.79

0.3 100

200

300

400

500

0

600

100

200

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400

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M AN U

0

meas sim

∆ξmax,rel = 2.7 %, ∆ξ¯rel = 0.8 %

0.8

0.35

SC

p∗ [-]

T ∗ [-]

0.45

t [s]

t [s]

(a) Evaporator WF outlet pressure 0.68

(b) Evaporator WF outlet temperature 0.72

meas sim

0.67

0.715

0.66

T ∗ [-]

T ∗ [-]

0.71

0.65

0.705

0.64

∆ξmax,rel = 4.8 %, ∆ξ¯rel = 3.2 % 0.62 0

100

200

300

400

500

meas sim

0

600

100

200

300

400

500

(d) Exhaust outlet temperature 0.575

meas sim

0.552

meas sim

EP

∆ξmax,rel = 2.7 %, ∆ξ¯rel = 0.4 %

AC C

0.548

0.57

T ∗ [-]

T ∗ [-]

0.55

600

t [s]

(c) Turbine WF outlet temperature 0.554

∆ξmax,rel = 1.2 %, ∆ξ¯rel = 0.4 %

0.695

TE

t [s]

D

0.7

0.63

0.565

0.56

0.546

∆ξmax,rel = 1.4 %, ∆ξ¯rel = 1.1 %

0.544

0

100

200

0.555 300

400

500

600

0

t [s]

100

200

300

400

500

600

t [s]

(e) Condenser WF outlet temperature

(f) CW outlet temperature

Fig. 9. Simulated trajectories of relevant thermodynamic variables of the ORC in comparison to their corresponding measured trajectories within the model validation, including maximal and mean relative errors.

41

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Mapping of zone boundary positions and zone lengths [27] . . . . 44 Summary of the data fit for the isentropic and mechanical efficiency of the turbine, including vali Overview of all estimated parameters for the pipe and turbine models, including bounds and fina Summary of statistic error indicator values for relevant variables resulting from a measurement u Overview of all estimated parameters for the evaporator model, including bounds and final values Overview of all estimated parameters for the condenser model, including bounds and final values. Summary of statistic error indicator values for relevant variables resulting from the measurement

AC C

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610

2 3 4 5 6 7 8

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List of Tables

42

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Nomenclature Latin

ρ density [kg/m3 ]

m ˙ mass flow [kg/s] Q˙ heat flow [kW]

σ 2 variance [-]

b width [m]

θ model parameter

SC

A area [m ]

τ nondimensionalized temperature [-], time constant [s]

2

c parameter for heat exchange cor- ξ state variable Superscript relations [-] ′

saturated liquid

M AN U

Cd discharge coefficient [-] cp specific heat capacity [kJ/(kg K)]

′′

saturated vapor

nonideal d parameter for heat exchange cor- ¯ averaged, Helmholtz energy relations [kW/(m2 K)]

part

of

D

e parameter for heat exchange cor- ˜ measured value relations [kg/s] stat stationary f parameter for heat exchange corSubscript relations [-] 0 ideal part of Helmholtz energy, inih specific enthalpy [kJ/kg] tial condition, scaling value for heat exchange correlations k logistic function [-]

TE

l length [m] M torque [Nm]

a, b zone boundaries amb ambient

EP

N total number of measurement B zone boundary wall points [-] c critical n speed [1/min] cond condenser N E number of experiments [-]

CW cooling water

N M number of measurements [-]

AC C

evap evaporator N V number of measured variables [- exh exhaust ] HP high pressure level P power [kW] i, j, k counter variable p pressure [Pa] in inlet pevap perimeter of evaporator [m] is isentropic T temperature [K] liq liquid t time [s] LP low pressure level u model input max maximum x differential state variable mech mechanic z longitudinal coordinate [m] out outlet Greek rel relative α heat transfer coefficient [kW/(m2 scale scaling value K)] tot total δ nondimensionalized density [-] η efficiency, pressure [-]

nondimensionalized trans transition turb turbine

γ void fraction [-]

vap vapor 3

ν specific volume [m /kg]

w wall

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za

zb

dza dt

0 l0 l0 + l1

l0 l0 + l1 l

dl0 dt d(l0 +l1 ) dt

0

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0 1 2

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Tab. 2. Mapping of zone boundary positions and zone lengths [27]

44

dzb dt dl0 dt d(l0 +l1 ) dt

0

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pHP pLP

∈ [0, 40], nturb ∈ [0.636, 1.636] · nscale Mturb ∈ [0, 1] · Mscale , nturb ∈ [0.636, 1.636] · nscale

54 192

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ηis ηmech

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Tab. 3. Summary of the data fit for the isentropic and mechanical efficiency of the turbine, including validity range, size of data set and R2 -value of the fit. The subscript scale indicates a scaling value. Valid input range Data set R2

45

0.992 0.995

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(α · A)evap,turb [W/ K] (α · A)cond,pump [W/ K] (α · A)pump,evap [W/ K] (α · A)turb,cond [W/ K]

lower bound

upper bound

final value

1 · 10−4

100

9.24782

1 · 10

−4

100

1 · 10

−4

100

1 · 10

−4

0.001

τcond,pump [s]

0.001

τpump,evap [s]

0.001

τturb,cond [s]

0.001

AC C

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Tamb [K]

46

15.4042

100

1.64795

150

56.6023

150

14.8807

150

150

150

0.188313

M AN U

τevap,turb [s]

17.061

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parameter

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Tab. 4. Overview of all estimated parameters for the pipe and turbine models, including bounds and final values. Missing bounds indicate assumed values.

300

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7.7 2.0 0.6 2.0 1.1

0.22 5.7 1.1 3.7 1.0

SC

0.52 6.8 2.0 10.2 4.6

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pW F,evap,out TCW,cond,out TW F,cond,out TW F,evap,out Texh,evap,out

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Tab. 5. Summary of statistic error indicator values for relevant variables resulting from a measurement used for the parameter estimation. ∆ξmax [bar]/[K] ∆ξmax,rel [%] ∆ξ¯ [bar]/[K] ∆ξ¯rel [%]

47

3.2 1.7 0.3 0.8 0.2

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lower bound

upper bound

final value

0

1000

427.287

cliq,1 [−]

0

1

cliq,2 [−] 
 dliq W/ m2 K

1

10

-1000

bliq [−]

0.01

uliq [kg/s] 
 αtrans,0 W/ m2 K

1 · 10

btrans [−]

bvap [−]

cexh,1 [−]

AC C

cexh,2 [−] 
 dexh W/ m2 K

-2.3888

10

0.01

0.0499 1 · 10

0

1

1

10

5

4

0.005803 278.844

0.458011 8.83129

1 · 10

0.01

10

0.088959

0.0499

0.02759

0

1000

57.6655

0

1

0

1

10

2.46687

-1000

1000

17.2663

0.01

10

0.062251

0.0499

0.01665

0

1000

153.173

0

10

0.976345

−4

D

1 · 10

EP

uvap [kg/s] 
 αexh,0 W/ m2 K

1000

-1000

TE

cvap,2 [−] 
 dvap W/ m2 K

3.67548

M AN U

ctrans,2 [−] 
 dtrans W/ m2 K

cvap,1 [−]

−4

0

ctrans,1 [−]

utrans [kg/s] 
 αvap,0 W/ m2 K

0

SC

parameter 
 αliq,0 W/ m2 K

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Tab. 6. Overview of all estimated parameters for the evaporator model, including bounds and final values. Missing bounds indicate assumed values.

1 · 10

−4

158.358

0

10

1.21767

-1000

1000

-1.09746

m ˙ W F,0 [kg/s]

0.05

m ˙ exh,0 [kg/s]

0.4

Texh,0 [K] 
 αamb W/ m2 K

923 0

300

*depending on experiment conditions

48

*

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lower bound

upper bound

final value

0

1 · 104

93.7257

cliq,1 [−]

0

10

cliq,2 [−] 
 dliq W/ m2 K 
 αtrans,0 W/ m2 K

0

10

2.55708 2.86173

SC

parameter 
 αliq,0 W/ m2 K

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Tab. 7. Overview of all estimated parameters for the condenser model, including bounds and final values. Missing bounds indicate assumed values.

1000

6.69519

0

0

1 · 106

36411.7

ctrans,1 [−]

0

10

5.0614

ctrans,2 [−] 
 dtrans W/ m2 K 
 αvap,0 W/ m2 K

0

M AN U −1 · 10

hplate [m] hW F [m] bplate [m] ltot [m]

16491.6

300

86.1252

0

10

1.37435

0

10

0

1 · 10

4

0.963504 1862.48

100

5000

2722.54

1 · 10−5

350

108.122

10

0.503491

EP

Nplates [−]

6.02665

0

0

−4

3.74 · 10−4

1 · 10−5

3 · 10−4

1.85 · 10−4

0.2

0.4

0.283959

100

200

141.98

0.2

0.5

0.343441

1 · 10

−5

5 · 10

TE

cCW [−]

1 · 10

5

D

cvap,1 [−]

cvap,2 [−] 
 dvap W/ m2 K 
 αCW,0 W/ m2 K 
 dCW W/ m2 K

10

4

0.05

TW F,0 [K]

503

AC C

m ˙ W F,0 [kg/s] m ˙ CW,0 [kg/s]

2

49

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9.7 1.4 2.7 2.7 4.8 1.2

0.49 3.7 1.3 3.9 12.4 1.7

SC

0.95 4.9 2.2 13.5 18.3 4.9

AC C

EP

TE

D

M AN U

pW F,evap,out TCW,cond,out TW F,cond,out TW F,evap,out TW F,turb,out Texh,evap,out

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Tab. 8. Summary of statistic error indicator values for relevant variables resulting from the measurement used for the model validation. ∆ξmax [bar]/[K] ∆ξmax,rel [%] ∆ξ¯ [bar]/[K] ∆ξ¯rel [%]

50

5.5 1.1 0.4 0.8 3.2 0.4

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HIGHLIGHTS

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- A dynamic organic Rankine cycle (ORC) model based on a WHR test rig is presented - Thermodynamic properties are obtained from an embedded EOS, avoiding external calls - A parameter estimation is carried out using transient data from various experiments - The model is successfully validated using different data