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Abstract
In U.S. district heating (DH) systems, steam is the most common heat transport medium. Industry demand for new advanced
modeling capabilities of complete steam DH systems is increasing; however, the existing models for water/steam thermodynamics
are too slow for large system simulations because of computationally expensive algebraic loops that require the solution to nonlinear
systems of equations. For practical applications, this work presents a novel split-medium approach that implements numerically
efficient liquid water models alongside various water/steam models, breaking costly algebraic loops by decoupling mass and energy
balance equations. New component models for steam DH systems are also presented. We implemented the models in the equation
based Modelica language and evaluated accuracy and computing speed across multiple scales: from fundamental thermodynamic
properties to complete districts featuring 10 to 200 buildings. Compared to district models with the IF97 water/steam model
and equipment models from the Modelica Standard Library, the new implementation improves the scaling rate for large districts
from cubic to quadratic with negligible compromise to accuracy. For an annual simulation with 180 buildings, this translates to a
computing time reduction from 33 to 1-1.5 hours. These results are critically important for industry practitioners to simulate steam
DH systems at large scales.
Keywords: Steam, District Heating, Computing Speed, Modeling, Simulation, Modelica
1. Introduction
1.1. Motivation
District heating (DH) can effectively reduce CO2emissions
and enable communities to leverage economizes of scale ben-
efits [1]. The global DH market is large, with an estimated
80,000 systems in operation that distribute hot water and steam
through almost 600,000 km of distribution pipes [2]. In the
United States, steam is the most common medium for DH, rep-
resenting 97% of all installations [3]. Beyond heating build-
ings alone, steam DH provides beneficial waste-heat recovery
opportunities when coupled with power plants (e.g., combined
heat and power) or other industrial systems (e.g., wastewater
treatment, metal refineries, etc.). While the conversion of steam
DH systems (first generation) to hot water DH systems (sec-
ond generation and later) is an important mechanism to realize
deep carbon savings [2], high-quality energy such as steam will
likely still be present in future energy grids due to its mutu-
alistic benefits with power generation and high-heat industrial
processes such as those listed in [4].
Modeling and simulation can improve the design and oper-
ation of DH systems while also aiding the transition of legacy
∗Corresponding author.
Email address: wangda.zuo@psu.edu (Wangda Zuo)
steam-based systems to newer technologies. While many soft-
ware tools can simulate individual buildings, the demand for
district-scale simulations are rapidly increasing. This work
is part of a larger collaborative project [5] involving the UR-
BANopt software development kit [6] and the open-source
Modelica Buildings Library (MBL) [7] to create district-scale
software modeling tools for energy optimization, grid respon-
siveness, and waste-heat recovery. Because of the prevalence of
steam in DH, combined heat and power (CHP) systems, and as
waste-heat from industrial processes, it is imperative that these
modeling tools are capable of simulating steam systems. In par-
ticular, steam DH simulation times need to be relatively fast
for industry to adopt the computing tools in practice. To ac-
complish this task, the first step is to implement a modeling
approach for water and steam thermodynamics that is fast and
accurate for DH simulations at large scales.
1.2. Literature Review
Table 1 summarizes several water/steam models that can be
considered typical examples of analytical models in scientific
and industrial practices. The International Association for the
Properties of Water and Steam (IAPWS) develops formulations
for thermodynamic properties of water and steam for various
applications. Based on the Helmholtz free energy, the IAPWS-
95 model is a single-set of equations with density ρand temper-
ature Tas independent variables; the model covers the largest
Journal: Energy Accepted 7 May 2022, DOI: 10.1016/j.energy.2022.124227
K. Hinkelman, S. Anbarasu, M. Wetter, A. Gautier, W. Zuo. 2022. "A Fast and Accurate Modeling
Approach for Water and Steam Thermodynamics with Practical Applications in District Heating System
Simulation," Energy, 254(A), p. 124227, https://doi.org/10.1016/j.energy.2022.124227.
A Fast and Accurate Modeling Approach for Water and Steam Thermodynamics with
Practical Applications in District Heating System Simulation
Kathryn Hinkelmana, Saranya Anbarasub, Michael Wetterc, Antoine Gautierc, Wangda Zuoa,d,∗
aArchitectural Engineering, Pennsylvania State University, University Park, 16802, PA, USA
bCivil, Environmental and Architectural Engineering, University of Colorado, Boulder, 80309, CO, USA
cLawrence Berkeley National Laboratory, Berkeley, 94720, CA, USA
dNational Renewable Energy Laboratory, Golden, 80401, CO, USA
Nomenclature
Abbreviations
c condensate (generic)
CD constant density liquid water
CHP combined heat and power
CVRMSE coefficient of variation of the root mean square
error
DH district heating
IAPWS international association for the properties of water
and steam
IAPWS95 scientific formulation for water/steam (Appendix
A)
IF97 industrial formulation for water/steam (Appendix B)
MBL Modelica Buildings Library
MSL Modelica Standard Library
RMSE root mean square error
S simplified steam model (Appendix C)
s steam (generic)
SBTL spline-based table look-up
TDD temperature dependent density liquid water
Accents
¯ mean
˙ flow rate
ˆ approximate
Parameters
δsmall number for regularization
S0pump performance data
aregression coefficient
Fnumber of fluid ports
iindex
kconstant coefficient
Nnumber in series
nnumber of operating points
pscaling power
Subscripts
0 nominal
1 inlet port
2 outlet port
a atmospheric
cpu central processing unit
f fuel
fg vaporization
fl flow
h hydraulic
l loss
m motor
max maximum
sat saturated state
t turbulent transition point
Variables
χvapor quality
∆pchange in pressure
ηefficiency
ρdensity
cpisobaric specific heat
cvisochoric specific heat
hspecific enthalpy
mmass
Pelectrical power
ppressure
Qheat
rrotational speed
sspecific entropy
Ttemperature
ttime
Uinternal energy
Vvolume
yload ratio
range of conditions and is intended for scientific applications
where accuracy is of utmost importance [8]. The IAPWS also
released their industrial formulation (IF97) for water and steam
that is based on the Gibbs free energy [9]; this model contains
multiple equations corresponding to different regions of water
and steam thermodynamics, and includes forward and back-
ward equations with various combinations of independent vari-
ables: (p,T), ( p,h), and (p,s), where pis pressure, his specific
enthlapy, and sis specific entropy. With this formulation, the
computing time is typically faster than the IAPWS-95 model
and produces negligibly small accuracy sacrifices. The IF97
model is by far the most widely adopted medium model for
2
Table 1: Representative analytical models for calculating water and steam thermodynamic properties.
Name Applicable Range(s) Regions1Intended Freely Ref.
p (MPa) T (◦C) Application Available
IAPWS-95 0–1000 −22–1000 1–6 Scientific Yes [8]
IF97 0–100 0–800 1–6 All Industrial Yes [9]
800–2000 0–10 5 High-temperature
Polynomial Fits with Splines 0.08–100 0–800 1–6 All Industrial No [10]
Polynomial Fits 0.01–100 0–800 1–6 All Industrial No [11]
Power Series with Log Transformations 0–21 0–370 2,4 All Industrial Yes [12]
MBL Constant Density (CD)20–4 –1–130 1 Building HVAC Yes [7]
MBL Temperature Dependent Density (TDD)30–4 –1–373 1 Building HVAC Yes [7]
1Region numbers correspond to: 1 subcooled liquid region, 2 saturated water line, 3 wet steam region, 4 dry saturated
steam line, 5 superheated vapor region, and 6 supercritical region. See Figure 2 for graphical representation.
2Referred as TemperatureDependentDensity in the MBL [7].
3Referred as Water in the MBL [7].
simulation applications [13, 14, 15].
Largely based on the IF97 model, other researchers have de-
veloped approximation models for water and steam in order
to further improve the computing speed. Åberg [10] fit fifth
order polynomials to the IF97 models with spline functions
to smooth the phase change transitions. Hofmann et al. [11]
implemented linear and quadratic approximation functions for
pressure-based sub-regions with interpolation methods for in-
between states. Further, some models cover a subset of phase
regions, such as the saturated water and steam line functions by
Affandi et al. [12] and the incompressible liquid water models
in the MBL [7]. In addition, there are other interpolation and
table-based methods for reducing computing time and smooth-
ing phase-change transitions. These include spline-based table
look-up (SBTL) [16], look-up table interpolation [17], bi-cubic
spline interpolation [18], bi-quadratic spline interpolation [19],
and tabular Taylor series expansion [20, 21]. However, these
methods are data intensive and are often most suitable for com-
putational fluid dynamics and other applications that require a
fine discretization of property evaluations, such as a pressurized
water reactor in nuclear systems. This is in contrast to thermo-
fluid modeling for DH applications, where the scalability from
small to large districts is a primary concern. Thus, table-based
methods were excluded from Table 1.
For nuclear and power plant applications, there are numer-
ous computational codes available that meet the requirements
of fast and accurate two-phase flow modeling for those do-
mains. To evaluate thermodynamic properties, most of these
codes store in memory the IAPWS-95 or IF97 models as look-
up tables and evaluate properties over an indexed region us-
ing either fitting functions (e.g., polynomials, splines) or in-
terpolation methods. Further, they often extend and modify
the IAPWS base models to include non-equilibrium thermody-
namics. For example, RELAP [22] implements the previously-
mentioned SBTL method by Kunick and Kretzschmar [16] with
the tabulated IAPWS-95 model. Similarly, TRACE [23] uses a
tabulated IAPWS-95 dataset and applies polynomial fits (de-
fault) or interpolation, while CATHARE [24] uses the REF-
PROP Library by NIST, which is similarly based on “high ac-
curacy Helmholtz energy equations of state” [25] (just as the
IAPWS-95). ATHLET-SC [26] and APROS [27] implement
the IF97 model with table look-up methods. Targeting nuclear
and power plant systems, these codes require and provide fine
discretization for property evaluations, implementing finite dif-
ference, finite volume, and finite element methods. Further,
they often aim to simulate failure conditions such as a small
break loss-of-coolant accident, which require accurate evalua-
tion of fast transients at extreme temperature/pressure condi-
tions. For these applications, the table-based methods are ef-
fective both in terms of accuracy and computational speed, as
demonstrated by Zhong et al. [28] in their comparative eval-
uation between TRACE and RELAP (table-based IAPWS-95)
and the IF97 models implemented analytically.
The water and steam medium models reviewed above have
been adopted in steam-based simulation studies spanning mul-
tiple domains and system boundary scopes. Numerous studies
focus on steam plants. For example, Beiron et al. [29] simu-
lated with Modelica waste-fired CHP plants using Modelon’s
Thermal Power Library [30] (its Water Polynomial model uses
IF97 as a base). Huang et al. [31] simulated combined cooling,
heating and power systems during off-design conditions and
adopted the IF97 model. Zhang et al. [32] similarly used the
IF97 model while simulating integrated solar combined cycle
systems. Several steam plant simulation studies do not declare
which medium model they adopted for steam and water ther-
modynamics, but still ought to be mentioned. These simulation
studies include ultra-critical biomass steam power plants [33],
new industrial waste heat recovery systems with seasonal ther-
mal storage [34], new waste heat DH systems with CHP [35],
and optimization of coal-fired CHP to decouple heat and power
modes [36]. In addition to Modelon’s Thermal Power Li-
brary [30], there are numerous Modelica libraries suitable for
steam plants, such as ClaRa+[37] and ThermoSysPro [38].
Contrary to studies focusing on steam plants, others have
focused on DH networks. For example, commercial software
packages such as Termis by Schneider Electric [39] can help
3
with design and planning of steam DH piping networks but does
not include plant models. Termis uses the ASME steam tables
for property evaluation, and the condensate return temperature
from buildings must be explicitly defined by the user for steam
systems. In literature, some studies focused on steam piping
networks as well. For example, Wang et al. [40] modeled steam
piping networks dynamically in Modelica for industrial appli-
cations. Similarly, Wang et al. [41] considered drainage loss
in steam pipeline network models for recovery purposes, while
Jie et al. [42] optimized pressure drop in steam piping networks
from environmental perspectives. Among all of these exam-
ples, thermodynamic states at source nodes are user-defined in-
puts (i.e., the plant supply or building condensate return), either
via historic measured data or real-time measurements through
Supervisory control and data acquisition systems, and the sim-
ulations solve for states at sink nodes. Lastly, numerous studies
and tools focus on buildings connected to DH networks. While
a comprehensive review of these building-level works is out
of the scope of this paper, many established simulation codes
can simulate buildings fed by steam piping, including Energy-
Plus [43].
1.3. Contributions
As exemplified by the literature review, steam system simu-
lation codes and literature to date have focused on one of three
system boundaries: (1) the plant, (2) the distribution network,
or (3) the building. However, there are increasing demands for
complete steam DH simulations. These three scope boundaries
as well as the fourth emerging system boundary for complete
districts are depicted in Figure 1. It is important to note that lit-
erature on modeling and simulation of complete hot water DH
systems is abundant [44, 45, 46, 47] but out of the scope of this
paper. The primary contribution of this work is to enable simu-
lations of complete steam DH systems that are sufficiently fast
and accurate for practical adoption by industry professionals for
annual simulations.
While existing simulation codes are effective within their
scope boundaries, new simulation challenges arise with the
need for complete steam DH simulations. First, complete dis-
trict simulations (system boundary #4) contains multiple paral-
lel closed loops in the thermo-fluid system that grow linearly
with the number of buildings N. These thermo-fluid loops can
create coupled systems of nonlinear equations. Solving these
coupled systems of nonlinear equations require the application
of iterative nonlinear solvers, which can be computationally
costly. System boundaries 1 through 3 do not have this prob-
lem, since they are either open loops (boundary #2) or con-
tain a smaller set of algebraic loops within the plant (boundary
#1) or building (boundary #3). High-order building models can
have this same challenge when used with implicit ordinary dif-
ferential equation solvers, but reduced-order building models
are frequently used for district-scale analysis to meet compu-
tational needs. Second, steam heating systems involve phase
changes that typically cross the two-phase wet steam region.
With this, discontinuities in the thermodynamic functions at the
phase change boundary can cause chattering [18] and at times
simulation failure. Further, the IF97 model contains separate
Plant
System Boundary #1
s
c
System Boundary #2
Distribution Network
System Boundary #3
Building
(Emerging) System Boundary #4
Complete District
Building
Figure 1: Scope boundaries for steam district heating simulation with steam (s)
and condensate (c) connections.
forward and backward equations that are not analytical inverses,
i.e., h(p,T(p,h∗)) ,h∗for some p,h∗∈ <. This can cause se-
rious error accumulation in lengthy calculations [20] and can be
an additional source of chattering [48]. Yet on the other hand,
the IAPWS-95 model contains only a single set of equations
in terms of (ρ, T); thus, costly iterative methods are required to
solve thermodynamic properties if other inputs are known, such
as (p,T) or ( p,h).
To address these gaps and limitations, we propose a new wa-
ter and steam modeling approach with sufficient accuracy and
computing speed for large-scale thermo-fluid system modeling
and simulation in practical industry applications. First and fore-
most, we replace the mathematical formulations for the sub-
cooled liquid region with a numerically efficient model while
retaining the commonly-adopted IF97 formulations of thermo-
dynamic properties in the other regions. To our best knowl-
edge, this split medium approach is atypical and has not yet
been tested in the context of system simulation of steam heating
and industrial processes. Our hypothesis is that by splitting the
water/steam medium into different models, we can break the al-
gebraic loops of steam DH systems, therefore improving com-
puting speed with negligible sacrifice to accuracy. Secondly,
we also replace commonly-called thermodynamic functions for
the superheated vapor region with polynomial approximations
in a reduced p-Trange. While this reduced range does not ap-
ply to all steam systems, its intention is to evaluate the effects of
IF97’s inconsistent forward-backward equations and high-order
mathematical models in the superheated vapor region. When
suitable for the intended application, these secondary simplifi-
cations may further reduce computing time and numerical chal-
lenges for industry practitioners. In addition, we develop sev-
eral new component, equipment, and system models that allow
numerically efficient simulations of DH systems with our novel
approach.
1.4. Paper Organization
The rest of this paper is organized as follows. In section 2,
we present the models adopted through this work, including our
split-medium implementations and several component models.
In section 3, we present our approach for evaluating the water
4
and steam implementation across several scales: at thermody-
namic property scale, component scale, and district scale. The
results and discussion for each of these three levels of evalua-
tion are then presented in section 4 and section 5, respectively.
Lastly, final conclusions are provided in section 6.
2. Modeling Approach
To meet the objective of fast and accurate steam DH mod-
eling for practical applications, we propose new models from
mediums to components that can be assembled into complete
district energy systems. Component models leverage the novel
split-medium approach when phase-change is present (e.g.,
steam boilers, heat exchanges). It is worth noting that the Mod-
elica Standard Library (MSL) and the MBL decouple balance
equations and media model equations; thus the same balance
equations and components can adopt various medium models,
regardless of whether the media uses (T,p) or (ρ, T) as inde-
pendent variables, or whether the media is incompressible or
compressible. Thus, the various models in subsection 2.1 can
be used in components and systems models presented through-
out this paper.
2.1. Medium Models
The medium modeling objective is to evaluate the accuracy
and computing speed of split-medium models with respect to
the standard implementations of IF97 and IAPWS-95. All six
regions for water/steam thermodynamics, depicted in Figure 2,
are covered by the IF97 and IAPWS-95 formulations. As previ-
ously mentioned, the IF97 model is most commonly adopted; as
such, it was readily available through the open-source MSL [49]
as their Standard Water model. The IAPWS-95 model was pre-
viously implemented in Modelica by M´
arquez et al. [50], but
their implementation was not compatible with the MSL and
therefore could not be used directly. For comparative evalua-
tion, we implemented the IAPWS-95 model in a way that fol-
lows the medium definition in the MSL [51]. This new im-
plementation allows us to compare the IAPWS-95 model with
the IF97 and our novel split-medium approaches across multi-
ple scales. Our IAPWS-95 implementation was validated with
the computer program verification test values given in the stan-
dard [8]. For reference, the IAPWS-95 equations and the IF97
equations for the superheated vapor region that we implemented
in this work are included in Appendix A and Appendix B, re-
spectively.
Shown in Table 2, our novel split-medium models cover the
same regions as the IF97 and IAPWS-95 models, except the su-
percritical region 6 in two of the three cases, and reduced pres-
sures in regions 1 and 5. First, case IF97+TDD pairs the IF97
model with the temperature dependent density water model
(TDD) from the MBL. This TDD model assumes an incom-
pressible fluid with, as the name implies, density as a function
of temperature. In addition, the TDD model assumes by de-
fault a constant value of 4184 J/kg-K for both isobaric cpand
isochoric cvspecific heats, which corresponds to 20◦C. Sec-
ond, case S+TDD further replaces the superheated vapor re-
gion 5 with the simplified steam model S. Here, S refers to the
0
50
100
150
200
250
300
350
400
450
500
0 500 1000 1500 2000 2500 3000 3500
Temperature ( °C)
Specific Enthalpy (kJ/kg)
Supercri�cal (Region 6)
Wet Steam (Region 3)
Subcooled
Liquid
(Region 1)
Superheated
Vapor
(Region 5)
Saturated Water Line
(Region 2)
Saturated Vapor Line
(Region 4)
Figure 2: Temperature-enthalpy diagram for water and steam with adopted re-
gion definitions.
complete simplified steam formulation, which includes thermo-
dynamic property formulations from the IF97 model with the
equations for h(·) and s(·) replaced by approximation functions
ˆ
h(·) and ˆs(·), which are given in Appendix C. Further, the wet
steam region 3 is covered through the balance equations with
equilibrium assumptions, as presented in section 2.3.3. Third,
case S+CD is the same as S+TDD except we replace the TDD
water model with a constant density water model from the MBL
(CD), which is referred as Water in the library [7]. This CD
model also assumes a constant value for cpand cv. Because the
TDD and CD models both assume an incompressible liquid,
they are less accurate for high pressure conditions. As such,
the supercritical region 6 is also excluded from the S+TDD and
S+CD cases, which is typically not required for steam heating
applications. These three split medium cases in addition to the
two commonly-used cases, referred as IF97 and IAPWS95, are
evaluated herein.
2.2. Balance Equations
Several component models can optionally select dynamic or
steady state balances for mass and energy dynamics. Since
these equations are fundamental to all of the following com-
ponent models, they are presented here at the onset. Assuming
a generic control volume with ˙mi(·) being the mass flow rate
through fluid port iand F∈Fbeing the number of fluid ports,
the steady state mass mass balance equation used during time
step tis
0=
F
X
i=1
˙mi(t).(1)
Similarly, the dynamic mass balance is
dm(t)
dt =
F
X
i=1
˙mi(t).(2)
5
Table 2: Split-medium model implementations by region, with the entire wa-
ter/steam model covered by two models delineated at the saturated water line.
Abbreviations IF97 is the IAPWS IF97 formulation (Appendix B), S is the
simplified steam model (Appendix C), TDD is referred as TemperatureDepen-
dentDensity in the MBL [7], and CD is referred as Water in the MBL [7]
.
Region Split-medium approaches
IF97+TDD S+TDD S+CD
Model 1
1 TDD TDD CD
Model 2
2 IF97 IF97 IF97
3 IF97 Control volume1Control volume1
4 IF97 IF97 IF97
5 IF97 IF97+ˆ
h(·)+ˆs(·)2IF97+ˆ
h(·)+ˆs(·)2
6 IF97 N/A N/A
1The control volume is presented in section 2.3.3.
2Approximation equations for the enthalpy (Equation C.5)
and entropy (Equation C.8) are given in Appendix C.
The steady state energy balance is
0=
F
X
i=1
˙mi(t)hi(t)+˙
Q(t),(3)
where hi(·) is the specific enthalpy for fluid connector iand ˙
Q(·)
is the heat flow rate entering the volume. Meanwhile, the dy-
namic energy balance is
dU(t)
dt =
F
X
i=1
˙mi(t)hi(t)+˙
Q(t),(4)
where U(·) is the energy stored in the volume. Because a steady
state energy balance coupled with a dynamic mass balance can
lead to inconsistent equations, all models require the energy and
mass balances to be the same type. For the specific evaluation
models included in this paper, the mass and energy balances
assigned to each component model as well as the initial condi-
tions are included in section 3.
2.3. Component Models
This section presents component models that are fundamen-
tal to this work. While a complete description of all models
available for steam DH systems is out of the scope, interested
readers can find more information in the open-access MBL [7]
and MSL. For example, the check valve, feedwater tank, the
PID control block, and various thermodynamic sensors (see
Figure 7) are publicly available in the MBL, while the table in-
put used for the building heat load profile as well as the various
mathematical operation blocks (see Figure 8) can be found in
the MSL. All component models that involve water/steam phase
change and the two-phase region were implemented specifically
for the split-medium approach with this work; however, the fun-
damental mathematics apply for both the split medium and tra-
ditional single medium approaches. Lastly, some of these com-
ponents are existing models in the MBL (pump, pressure drop),
while the control volume, boiler, and steam trap are new.
2.3.1. Pump
In steam DH systems, pumps supply feedwater at the plant
and at times return condensate to the plant from buildings. Be-
cause pumps notably contribute to the fluid dynamics and elec-
tricity consumption, we include the model details here. We im-
plement the pump model from the MBL that uses performance
curves to compute pressure rise ∆p, electrical power draw P,
and efficiency ηas functions of volumetric flow rate ˙
Vand ro-
tational speed r. The pump model is consistent with the affinity
laws ∆p∝r2and ˙
V∝r. To ensure that solutions to the differ-
ential algebraic system of equations posed by the thermo-fluid
model can be computed robustly and efficiently by Newton-
based solvers, the pump is formulated in such a way that the
resulting equations of the fluid flow network has a unique solu-
tion in each operable region and is differentiable in all inputs.
While complete details for the pump formulation are available
in [52], the fundamental formulation when the pump operates
far from the origin is as follows. Let δ=0.05 be a small
number that is below the typical normalized pump speed and
S0
n={(˙
Vi,∆pi)}n
i=1be the user-supplied performance data at
full speed r=1, with ˙
Vi≥0 and ∆pi≥0 for all i∈ {1,...,n}.
Here, nrepresents the total number of operating points. For
conditions r> δ (i.e., far from the origin), the affinity laws
are satisfied while the maximum volumetric flow rate ˙
Vmax and
maximum pressure change ∆pmax are linearly extrapolated as
˙
Vmax =˙
Vn−˙
Vn−˙
Vn−1
∆pn−∆pn−1
∆pnand (5)
∆pmax = ∆ p1−∆p2−∆p1
˙
V2−˙
V1
˙
V1.(6)
The pump performance curve for r> δ is defined as
∆p+(r,˙
V)=−∆ˆp(˙
V)+r2h ˙
V
r,S0
n!,(7)
with the curve end points represented by ∆p+(1,˙
Vmax )=0 and
∆p+(1,0) = ∆pma x, while h(·,S0
n) is a cubic Hermite spline that
maps ˙
Vto ∆p, and ∆ˆp(˙
V) approximate the flow resistance of the
pump, for reasons of numerical robustness, by a linear function
as
∆ˆp(˙
V)=˙
V∆pmax
˙
Vmax
δ2
10.(8)
In addition to conditions r> δ, Wetter [52] also defines formu-
lations for near origin (r< δ/2) and composite (r∈[δ/2, δ])
conditions to complete the pump model. In this case study, we
assume a constant hydraulic efficiency ηh=70% and a constant
electric motor efficiency ηm=70%. Total efficiency ηis then
η=ηhηm,(9)
6
and the electrical power draw is computed as
P=Wf l
η,(10)
where Wf l is the flow work defined per the first law as
Wf l =|˙
V∆p|.(11)
In this model, ˙
Vis calculated from ˙mwith the density defined
at the inlet port ρ1as ˙
V=˙m/ρ1.
2.3.2. Pressure Drop
To account for pressure drop in pipes and other components,
we use a flow resistance model from the MBL that has a fixed
flow coefficient, named Pressure Drop in the library. To decou-
ple the energy and mass balance equations, pressure drop ∆pis
a function of the mass flow rate, rather than the volumetric flow
rate. This model computes ∆pas
∆p=sign( ˙m)˙m
k2
,(12)
where ˙mis the mass flow rate and kis a constant flow coeffi-
cient calculated from the nominal mass flow rate ˙m0and nomi-
nal pressure drop ∆p0as
k=˙m0
p∆p0
.(13)
With the inverse ˙m=kp∆palso implemented in the library,
this model replaces the square root with a differentiable func-
tion with a finite slope for conditions ˙m< δt˙m0, where δtis
the fractional mass flow rate where the transition to turbulent
flow occurs (set to 0.3 by default but adjustable by the user).
Further information about the regularization near the origin and
the basic flow models is available in the MBL [7].
2.3.3. Control Volume
The new control volume model can represent either evapora-
tion or condensation processes with the liquid and vapor sub-
components in equilibrium. This model is designed to assign
each of the two medium formulations of the split-medium ap-
proach (Table 2) at the inlet and outlet ports. For an evapora-
tion process, the subcooled liquid water medium is assigned to
the inlet, while the composite water/steam medium is assigned
to the outlet. The opposite is true for the condensation pro-
cess. The mathematical formulation for the evaporation pro-
cess is consistent with the existing drum boiler implemented in
the MSL [53]. Both the evaporation and condensation control
volumes have the following assumptions:
1. The fluid within the volume is wet steam (region 3);
2. Liquid and vapor subcomponents are at equilibrium; and
3. Fluid is discharged from the volume as either saturated liq-
uid or saturated vapor.
It should be noted that these assumptions restrict the possible
use cases (e.g., the volume cannot model superheated or sub-
cooled fluids). An additional limitation is that any sensible
heat losses/gains downstream of the control volume must be
included in a separate component. While this additional step
can increase model development time, from our experience, the
time saved by avoiding common numerical challenges greatly
outweighs the former inconvenience.
The fundamental equations are as follows. Let subscripts 1
and 2 represent the inlet and outlet ports, respectively. The fluid
mass min the volume is calculated as
m=ρ1V1+ρ2V2,(14)
where ρis density and Vis volume.
The total internal energy Uis
U=ρ1V1h1+ρ2V2h2−pV,(15)
where his specific enthalpy, pis pressure, and V=V1+V2is
the total volume of the fluid.
More specifically, since the volume contains a saturated mix-
ture, h1and h2are the specific enthalpies of saturated vapor and
saturated liquid for a condensation process. These assignments
are reversed for an evaporation process. As a saturated mixture
at equilibrium, the vapor quality χis defined as
χ=
m1/m,if condensation,
m2/m,if evaporation, (16)
which is used to calculate thermodynamic properties of the wet
steam two-phase mixture. For example, the specific enthalpy of
wet steam is
h=m1h1+m2h2
m.(17)
As previously mentioned, the control volume is configured
to allow both steady state and dynamic mass and energy bal-
ances. The steady mass, dynamic mass, and dynamic energy
balances are consistent with Equation 1, Equation 2, and Equa-
tion 4, respectively, with F=2. However, the steady energy
balance for a wet steam control volume at equilibrium has an
additional constraint. Because the discharging fluid is con-
strained at pressure p=psat with T=Tsat (psat) and saturated
enthalpy h=h2(psat), where subscript sat is the saturated state,
if the mass and energy balances are steady, then prescribing the
heat flow into the volume ˙
Qover-constrains the problem. Thus,
Equation 3 is not included in this model; instead, ˙
Qis directly
proportional to the mass flow rate and is calculated as
˙
Q=˙m1hf g ,(18)
where hf g =h2−h1is the enthalpy of vaporization. Thus, when
used as a steady state model, this heat must be removed from
the system in which this volume is used.
2.3.4. Boiler
Figure 3 shows the schematic model view of the boiler that
discharges saturated steam and has an efficiency curve defined
by a polynomial. The rate of heat transferred to the water
7
500*mDry
QWat_flow
G=UA
Dry Heat
Capacity
Overall
Thermal
Conductance
Pressure
Drop
Heat Port
Inlet
Port
Outlet
Port
Control Volume
(Evaporation)
Liquid
Volume
Load Ratio
Prescribed Heat Flow
Rate into Water
s
c
Figure 3: Modelica diagram for the boiler with an efficiency curve defined by
a polynomial. Components in the green shaded region (including the heat port
for the control volume, but not the volume itself) are conditionally removed if
the boiler is configured with steady state mass and energy balances.
medium ˙
Qis
˙
Q=y˙
Q0η
η0,(19)
where y∈[0,1] is the load ratio, ˙
Q0is the nominal heat capac-
ity, ηis the total efficiency at the current operating point, and
η0is the total efficiency at y=1 and boiler output temperature
T=T0, where T0is the nominal temperature. With efficiency
η=˙
Q/˙
Qfand ˙
Qfrepresenting the rate of heat released by the
fuel combustion, the three polynomial options to compute ηare
η=a1,(20)
η=a1+a2y+a3y2+... +anyn−1,and (21)
η=a1+a2y+a3y2+(a4+a5y+a6y2)T,(22)
where a1through anare regression coefficients.
Similar to the control volume, the boiler model can have
steady or dynamic mass and energy balances. If the boiler
is configured in steady state, then several components (high-
lighted in green in Figure 3) are conditionally removed to main-
tain a consistent set of equations. The reason is the same as the
control volume, where ˙
Q=f( ˙m); therefore, if the mass and en-
ergy balances are steady, then prescribing the heat flow into the
fluid over-constrains the problem, and thus they are removed.
Conversely, dynamic balances enable the heat flow rate into the
control volume to be calculated based on the heat transfer from
the fuel and through the boiler’s enclosure with the external en-
vironment.
2.3.5. Steam Trap
A required component of steam heating systems, steam traps
effectively ensure that only liquid condensate leaves compo-
nents (e.g., steam heat exchanger), while any flashed steam is
returned to a liquid state before discharge. This prevents the
loss of steam while protecting pipes for water from damage by
hot and high pressure steam vapor. In this model, we assume
steady state mass and energy balances. The steam trap rep-
resents an isenthalpic thermodynamic process that transforms
liquid water from an upstream high pressure (state 1) to atmo-
spheric pressure (state 2a), followed by an isobaric condensa-
tion process to return flashed steam to a saturated liquid (state
2). The heat loss in the trap ˙
Qlis
˙
Ql=˙m(h2a−h2),(23)
where ˙mis the steady state mass flow rate, and h2aand h2are
the specific enthalpies at states 2a and 2, respectively.
3. Evaluation Approach
To evaluate the accuracy and computing speed across mul-
tiple spatial scales, Modelica models are developed at thermo-
dynamic property scale (simple function evaluations), compo-
nent scale (a control volume), and district scale (complete heat-
ing districts of several sizes). Several medium model imple-
mentations are used within the component models for the com-
ponent and district scale evaluations. Two cases involve the
IF97 model, IF97(MBL) and IF97(MSL); one case involves the
IAPWS-95 model, IAPWS95(MSL); and three cases use our
split-medium approach, IF97+TDD, S+TDD, and S+CD. With
the two IF97 cases, models from two separate libraries are eval-
uated: the MBL and MSL, while the IAPWS95 model is only
evaluated with the MSL. Even though the models from both the
MBL and MSL libraries can be applied for steam DH appli-
cations, their original design intentions differ, and correspond-
ingly, they are based on different assumptions. The IF97(MBL)
and IAPWS95(MSL) cases are only used in the component-
scale evaluation. The remaining four cases are common to both.
Further details regarding the model setup for each simulation
case are provided in the following sections.
Several new component and system models for DH modeling
were developed for these case studies, as presented in section 2.
These models are in the process of being refined and open-
source released in the MBL. Beyond newly developed models,
all components used are existing in the MSL v3.2.3 and MBL
v7.0.0.
3.1. Thermodynamic Property Scale
The accuracy and computing speed of several thermody-
namic property functions are evaluated with respect to the IF97
and IAPWS-95 models. These include the subcooled liquid wa-
ter models from the MBL (CD and TDD) as well as the su-
perheated vapor approximation functions for specific enthalpy
(Equation C.5) and entropy (Equation C.8), which are inte-
grated into model S. The objective was to evaluate solely the
thermodynamic properties separate from higher-level effects
from components, equipment, and systems. For accuracy, we
calculated the absolute and percent differences between the nu-
merically improved functions (CD, TDD, S, and IF97) and the
IAPWS95 model across their respective p-Tranges.
To evaluate computing speed, we also present the comput-
ing time for thermodynamic property evaluations using differ-
ent formulations – IF97, IAPWS95, CD, TDD, S – and different
independent variables – (p,T), (p,h), and (ρ, T). As previously
mentioned, the IF97 and the numerically efficient liquid water
8
Q_flow
G=10 W/K
T=300 °C
K
s
V=V
Fixed
Temperature Thermal
Conductor
Heat Flow
Rate Sensor
Replaceable
Volume
Specific Entropy
Sensor
Boundary with
Reference Pressure
(1) IF97(MBL)
(2) IF97(MSL)
(3) IAPWS95(MSL)
(4) IF97+TDD
(5) S+TDD
(6) S+CD
}
Figure 4: Diagram of Modelica model for control volume evaluations.
and steam vapor models contain both forward and backward
equations for various properties, while the IAPWS95 formula-
tion strictly has independent variables of (ρ, T). Thus, the se-
lection of independent variables can significantly impact com-
puting time.
3.2. Component Scale
Because of the discontinuities at the phase-change barrier,
numerical solvers have shown to experience difficulties solv-
ing thermo-fluid problems with two-phase flow. Chattering is
one example of a well-known issue that has been demonstrated
previously with a Modelica-based simulation of a boiler pipe
model featuring the IF97 medium [18]. For this case, we use
a control volume (section 2.3.3) to evaluate the performance of
the split-medium implementations compared to the IF97 and
IAPWS95 models with control volume models in the MSL and
MBL. This component-scale experiment was selected to isolate
some of the common numerical challenges of thermo-fluid sys-
tem modeling involving phase change that may not appear at
smaller scales while being harder to diagnose at larger scales.
Shown in Figure 4, the Modelica-based evaluation features
a water control volume of 0.1 m3that is exposed to a constant
temperature boundary of 300◦C via a thermal conductor with a
constant thermal conductance of 10 W/K. The volume is config-
ured with dynamic energy and mass balance equations, and the
initial condition p(t0) is set at the reference pressure of 200 kPa
due to the connection with the Boundary component. The con-
trol volume is replaceable in order to allow six different medi-
ums to be simulated with the same experimental setup.
In the component-scale evaluation, three baseline cases are
included – IF97(MBL), IF97(MSL), and IAPWS95(MSL) –
and three cases from Table 2 are included – IF97+TDD,
S+TDD, and S+CD. First, the MBL case with the IF97 model –
IF97(MBL) – includes the control volume designed for single-
phase fluid, referred as Mixing Volume in the MBL Fluid pack-
age. Second, the MSL cases with the IF97 and IAPWS-95
models – IF97(MSL) and IAPWS95(MSL) – implement a two-
phase equilibrium boiler model [53], referred as Equilibrium
Drum Boiler in the MSL Fluid package. This model was de-
signed for water-steam phase change and is mathematically
equivalent to the control volume presented in section 2.3.3. The
remaining three split-medium cases all use the new control vol-
ume. The other model components in Figure 3 (e.g., thermal
conductor, sensors, pressure-temperature boundary) are freely
available in the MBL. For accuracy, we evaluated the Root
Mean Square Error (RMSE) and Coefficient of Variation of the
Root Mean Square Error (CVRMSE) using
RMS E =rP(yi−ˆyi)2
Nand (24)
CV RMS E =RMS E
¯y,(25)
where yiis the individual reference data generated by the IF97
model, ˆyiis the corresponding evaluation data predicted by the
new model, ¯yis the mean of the reference dataset, and Nis the
total number of data points.
3.3. District Scale
The objective of the district scale evaluation was to assess the
accuracy and numerical performance of the new split-medium
approach in a typical DH system design across districts of sev-
eral sizes. An overarching description of the selected system is
presented next, followed by the Modelica implementation.
3.3.1. System Description
Figure 5 depicts the schematic diagram for the DH evalua-
tion. This DH system is broken into three subsystems: a central
plant, the distribution network, and building end users. The
central plant features a feedwater tank, feedwater pump, and
a single steam boiler. Saturated steam is discharged from the
boiler at 300 kPa. The fuel load ratio for the boiler is controlled
to maintain the boiler discharge pressure, while the feedwater
pump speed is controlled to maintain the liquid water level in
the boiler. While there are several mechanical and control de-
signs seen in central plants [54], this configuration was selected
because it represents real-world control dynamics while repre-
senting one of the more simple designs. For the distribution
network, we assume there are no mass nor energy losses in the
steam supply pipe, while the condensate return pipes have fixed
pressure drops without any heat transfer. Although heat and
mass losses in steam supply pipes are not negligible in real-
world systems, there is no Modelica model available for a sat-
urated steam pipe with drip-leg, to our knowledge. While out
of the scope of work for this paper, we are currently developing
a model for this purpose that will be made public in the fu-
ture, which features a steam pipe (heat losses, mass losses, flow
resistance, transport delays) with a drip-leg (condensate recol-
lected). Lastly, the buildings contain a steam heat exchanger
(modeled as a condensation control volume), a steam trap, and
a condensate return pump. For demonstration purposes, we ap-
ply the same variable heating load profile to all buildings, which
can be seen in Figure 8 of section 4. The mass and energy bal-
ances for all components in the district scale simulation as well
as the initial conditions are summarized in Table 3.
3.3.2. Modelica Implementation
To evaluate the steam medium implementation across a va-
riety of district sizes, a vector-style DH system model was de-
veloped in Modelica (Figure 6). This top-level Modelica di-
agram has a clear one-to-one relationship with the subsystems
9
Pump
speed
Measured
pressure
Part load ratio
Steam
Boiler
Condensate
Return Pump
Steam Supply Pipe
(lossless)
Condensate Return Pipe
(pressure drop)
Condensate Return Pump
(Ideal mass flow control)
Trap Trap
Heat
Exchanger
Heat
Exchanger
Feedwater Tank
Vent
Feedwater
Pump
s
c
Measured
liquid volume
Liquid volume
setpoint
Building N
Building 1
Central Plant
Δp
Δp
Δp
Distribution Network
Pressure setpoint
Figure 5: Schematic diagram of DH steam loop with a central plant, distribution network, and Nnumber of interconnected buildings.
Table 3: Initial conditions as well as mass and energy balances for the district scale evaluation model, where Nis the total number of buildings and niis the number
of buildings connected to pipe segment i.
Subsystem Component Initial Conditions Balance
p(Pa) T(◦C) ˙m(kg/s) Equations
Plant
Feedwater tank 101325 20 (7.38 ×10−3)NDynamic
Feedwater pump 101325 20 (7.38 ×10−3)NDynamic
Check valve 101325 20 (7.38 ×10−3)NSteady
Boiler 300000 133.5 (7.38 ×10−3)NDynamic
Distribution network Supply pipes (lossless) 300000 133.5 (7.38 ×10−3)niSteady
Return pipes (pressure drop) 101325 100 (7.38 ×10−3)niSteady
Building
Heat exchanger 300000 133.5 7.38 ×10−3Steady
Steam trap 300000 133.5 7.38 ×10−3Steady
Condensate return pump 101325 100 7.38 ×10−3Dynamic
depicted in Figure 5. A parameter Nrepresenting the total num-
ber of buildings can be adjusted to represent districts of multiple
sizes. The boiler’s rated heating capacity is scaled by Nin order
to adjust for the variable heating capacity of the entire system.
This DH system is simulated for two days.
The plant model diagram is shown in Figure 7. In the central
plant, the feedwater pump and boiler both have dynamic energy
and mass balances, and PI controllers are used to maintain the
water level and pressure setpoints. A check valve was also in-
cluded in the central plant model in order to prevent unintended
reverse flow. Both the thermal conductance and heat capacity of
the boiler drum metal and insulation are included in the model.
The boiler is assumed to have constant efficiency with η=90%.
The building model diagram is shown in Figure 8. Through
the Tabulated Heating Load data reader, the generic variable
heat flow rate ˙
Qis input directly. A condensate return pump
Central Plant
s
c
Buildings [:]
Distribution Network
Figure 6: Top level diagram of Modelica model for the DH system.
prescribes the mass flow rate, set to ˙m=˙
Q/(h1−h2), where h1
and h2are the measured inflowing and outflowing specific en-
10
M
y
PI k=1/VBoiWatSet
PI
k=1/pSteSet
p
k=1
s s
Pump
Controller
Feedwater
Tank
Inflowing Port
(Liquid Water)
Feedwater
Pump
Check
Valve
Boiler
Controller
Normalized
Liquid Volume
Normalized
Pressure
Boiler
Outflowing Port
(Steam Vapor)
Pump Power
Fuel Heat Flow Rate
s
c
Figure 7: Diagram of Modelica model for the central plant.
thalpy values, respectively. The Heat Exchanger Volume model
is an instance of the control volume (subsubsection 2.3.3), con-
figured as a condensation process with steady state mass and
energy balances. Lastly, the Steam Trap represents a steady
isenthalpic process where liquid condensate is discharged at at-
mospheric pressure, as described in section 2.
I
k=1
M
m_flow
s
s
+
+1
-1
u1 / u2
Tabulated
Heating Load
Heat Exchanger Volume
(Steady State)
Steam
Trap
Condensate
Return Pump
Mass Flow Rate
Integral
Heat Flow
Rate
Heating
Energy
Inflowing Port
(Steam Vapor)
Outflowing Port
(Liquid Water)
h
h
1
2
1
2
h2
h
1-
s
c
Figure 8: Diagram of Modelica model for the interconnected building.
Four medium model configurations were included in this
evaluation: IF97(MSL), S+CD, S+TDD, and IF97+TDD.
We were unable to resolve the numerical challenges of the
IAPWS95 model for the complete steam DH simulations, par-
ticularly for large district sizes. Thus, this model was not in-
cluded in the district scale evaluation. These challenges are
primarily due to (1) the lack of backward equations to represent
thermodynamic functions in terms of variables other than (ρ, T)
and (2) the highly nonlinear thermodynamic property functions
that require precise starting values with implicit solvers. How-
ever, this is not to say that the IAPWS95 model cannot be used
for complete DH simulations, but only that the numerical hur-
dles are significantly greater than the other implementations.
3.4. Simulation Settings
All simulations ran in Dymola 2021 on a Windows 10 work-
station with a Intel®Xeon®3.60GHz CPU and 32.0GB of
RAM. The DASSL solver was selected for all case studies, after
preliminary tests demonstrated its lower computing time com-
pared to other numerical solvers. The simulation tolerance was
set to 10−6for all cases. For the fast thermodynamic and com-
ponent scale evaluations, computing times presented are aver-
age values across 10 repeated simulation runs.
4. Results
This section presents the results from simulations across the
three scales: thermodynamic properties, component, and dis-
trict scales. First the accuracy of the model will be presented,
followed by the computing speed.
4.1. Thermodynamic Property Scale
Following the methodology prescribed in subsection 3.1, the
results for model accuracy and computing speed at the ther-
modynamic property scale are evaluated. Because IAPWS95
and IF97 have previously been evaluated over their entire p-T
range [8, 9], these results focus on the numerically efficient wa-
ter models (CD, TDD) in the subcooled liquid region 1 and the
simplified steam model (S) with polynomial approximations for
commonly called functions in the superheated vapor region 5.
4.1.1. Accuracy
For the subcooled liquid water region, the accuracy of the
CD and TDD water models are evaluated with respect to IF97
and IAPWS95 at several sub-critical pressure states. The re-
sults for 0.1 MPa, 2 MPa, and 4 MPa are shown in Fig-
ure 9. The IF97 model produces the highest accuracy relative to
IAPWS95, as expected, with relative differences ranging from
3.13 ×10−4kJ/kg-K (0.140%) for sto −5.18 ×10−3kJ/kg-K
(−0.109%) for cpfor pressures 0.1 to 4 MPa. Because the func-
tions for h,s, and cpare the same for CD and TDD models,
they produced the same results. Errors in the hand scalcula-
tions were generally low, with the highest differences relative
to IAPWS95 for both CD and TDD cases being −4.06 kJ/kg
(−7.18%) and −5.99 ×10−2kJ/kg-K (−2.24%), respectively.
Calculating cpwith both CD and TDD models produces the
largest difference relative to IAPWS95 of −0.560 kJ/kg-K
(−11.8%), which occurs near the saturated liquid line at 4 MPa.
Lastly for ρ, the TDD case produced better accuracy than the
CD case, as expected. The largest errors for the CD and TDD
cases also occurred near the saturated liquid line at 4 MPa, with
differences relative to IAPWS95 of 17.8×105kg/m3(21.8%)
and 4.42 ×104kg/m3(5.41%), respectively. Based on these
results, we recommend the CD water model to be used in its
original design range indicated in Table 2. The TDD model is
suitable for a larger range of p-Tconditions compared to CD,
but this is still a subset of the IF97 range as currently designed.
11
Figure 9: Percent difference of specific enthlapy h, specific entropy s, density ρ, and isobaric specific heat cpwith respect to IAPWS95 values over the range of
subcooled temperatures in region 1.
For the new polynomial approximations in the superheated
vapor regions (model S), property evaluations for specific en-
thalpy and entropy produced acceptable accuracy across the en-
tire reduced p-Trange. Figure 10 presents the percent differ-
ences in hand scalculations for both S and IF97 models rela-
tive to IAPWS95. For h, the largest difference between S and
IAPWS95 models was −2.42 kJ/kg (−0.090%), which occurred
near the saturated vapor line at low pconditions. The reason for
this is that the nonlinearities in h(p,T) are higher along the sat-
urated vapor line than other regions. For s, the largest difference
between S and IAPWS95 was 0.0470 kJ/kg-K (0.691%), which
occurred at high pressure-temperature conditions. The shape
of the residuals indicate that the errors are relatively consistent
with respect to temperature, but the quadratic fit in pressure un-
derestimates sslightly in the middle of the range while it over-
estimates sat the high and low limits. Based on these results,
the polynomial approximations in model S produce sufficient
accuracy in the reduced p-Trange. This reduced range is suit-
able for many steam DH applications, but not all. For steam
vapor applications involving higher pressure and temperature
states, the IF97 model can still be used with our split-medium
approach.
4.1.2. Computing Speed
Table 4 presents the computing speed results for thermody-
namic property evaluations in the subcooled liquid Region 1,
where h,s,ρ, and cpare evaluated over the p-Trange shown in
Figure 9. The IAPWS95 calculations were significantly slower
than all other models, with time savings achieved from 79-92%
with IF97, TDD, and CD models. With inputs of (p,T), the
TDD and CD models were 24% faster than IF97, while they
were both 12% faster with inputs (p,h). With inputs of (ρ, T),
the IAPWS95 computing times were 25% and 54% faster than
IAPWS95 with inputs of (p,T) and ( p,h), respectively. This
Figure 10: Percent difference of specific enthlapy hand specific entropy swith
respect to IAPWS95 values over a range of superheated temperatures.
result follows expectations, since inputs of (ρ, T) induce no
nonlinear systems of equations for IAPWS95, while (p,T) and
(p,h) induce one nonlinear system with 1 and 2 iteration vari-
ables, respectively. However, even when the IAPWS95’s de-
sign independent variables (ρ, T) are used, IF97 was still sig-
nificantly faster than IAPWS95 (79% time savings).
Further, we evaluated the computing speed for individual
thermodynamic property functions ˆ
h(p,T) (Equation C.5) and
ˆs(p,T) (Equation C.8) and their backward functions with re-
spect to those from the IF97 and IAPWS95 formulations. The
results in Table 5 show that the new polynomial approximations
12
Table 4: Computing times for thermodynamic property evaluations in sub-
cooled liquid region 1 with time savings evaluated with respect to IAPWS95.
Results are averages over 10 simulation runs.
Medium Input Computing Time Savings
Model Variables Time (s) (%)
IAPWS95
(p,T)
0.101 –
IF97 0.017 84
TDD 0.013 87
CD 0.013 87
IAPWS95
(p,h)
0.166 –
IF97 0.015 91
TDD 0.013 92
CD 0.013 92
IAPWS95
(ρ, T)
0.076 –
IF97 0.016 79
TDD N/A1N/A1
CD N/A1N/A1
1Properties cannot be calculated from (ρ, T) for an
incompressible fluid.
Table 5: Computing times for thermodynamic property evaluations in super-
heated vapor region 5 with time savings evaluated with respect to IAPWS95.
Results are averages over 10 simulation runs.
Medium Equation Computing Time Savings
Model Time (s) (%)
IAPWS95 h(p,T), A.9 0.100 –
IF97 h(p,T), B.4 0.017 83
Sˆ
h(p,T), C.5 0.014 86
IAPWS95 s(p,T), A.10 0.101 –
IF97 s(p,T), B.7 0.017 83
S ˆs(p,T), C.8 0.014 86
and the IF97 formulations reduce the computing time by 83%-
86% relative to IAPWS95. Relative to IF97, ˆ
h(p,T) and ˆs(p,T)
were 17% and 14% faster, respectively. Even for these simple
thermodynamic property evaluations, the coupled systems of
nonlinear equations differed between the IAPWS95, IF97, and
S model cases. Both the IAPWS95 and IF97 models produced
a nonlinear system of equations that contains a single time-
varying variable, while the S formulations did not have any
nonlinear systems. In addition, the IAPWS95 model produced
a numerical Jacobian, which can be computationally expensive
compared to symbolic processing. These factors in the problem
formulation can partially explain the computing time results.
While the approximation functions ˆ
hand ˆsproduced savings in
computing time at the thermodynamic property scale, the sav-
ings are often compounded for larger thermo-fluid system mod-
els that involve nonlinear systems of equations. This will be
evaluated with the district scale models. However, the comput-
ing time savings and calculation accuracies demonstrate how
fundamental improvements in steam property modeling can be
achieved through function replacement.
4.2. Component Scale
Following the methodology prescribed in subsection 3.2, the
results for model accuracy and computing speed at the compo-
nent scale evaluation are as follows.
4.2.1. Accuracy
Figure 11 depicts the temperature, density, and mass results
of the control volume evaluation cases for the model shown
in Figure 4. In this simulation, the control volume fluid is
heated via a temperature boundary that is at 300◦C. With the
IF97(MBL) case, undesirable erratic behavior is clear. At ini-
tialization, the liquid in the volume is at 40◦C. For the first
28 minutes, the temperature gradually increases. However,
when water starts to boil, the fluid temperature oscillates be-
tween the saturation temperature and the boundary tempera-
ture of 300◦C. Mass and density are also oscillating in the
IF97(MBL) case, as the fluid switches back and forth between
one-phase (liquid or vapor) and two-phase states (liquid-vapor
mixture). The volume in the MBL case was designed for single
phase fluid (air, water, etc.), and functions correctly in those in-
stances. However, it exhibits chattering at phase change when
used with the two-phase IF97 medium and the numerical solver
DASSL. It is important to note that the MBL behavior seen
here is a numerical problem, not a physics problem. Indeed,
this numerical problem can be avoided if an explicit, fixed-
time step method is employed (i.e., Euler) rather than an im-
plicit, variable-time step method (i.e., DASSL). With Euler, the
IF97(MBL) case performs physically correct and avoids the nu-
merical chattering issue. Further, the IF97(MBL) volume can
simulate not only a saturated fluid, but supercooled and super-
heated fluids as well, which may be of interest for some use
cases. However, implicit, variable-time step methods are often
used for large thermo-fluid system simulations because of their
ability to deal with stiffsystems. Thus, the Euler results were
excluded from this paper.
The later five cases – IAPWS95(MSL), IF97(MSL),
IF97+TDD, S+TDD, and S+CD – perform with the correct
physics. Because the MSL and new control volumes were de-
signed for boiling processes with the fluid at a saturated state,
each of these three cases initialize at the saturation tempera-
ture. Throughout the constant pressure boiling process, the liq-
uid volume gradually decreases as more of the water is con-
verted from liquid into vapor, while the temperature of the fluid
is held constant. With the simulation entirely within the wet
steam region 3, the density of liquid water and steam vapor are
similarly maintained.
With the split-medium implementations, some minor accu-
racy errors are introduced. Since this work targets normal oper-
ating conditions, the RMSE and CVRMSE are presented in Ta-
ble 6 for control volumes with liquid-to-vapor ratios within the
ratio of 1 to 9 which are common in real operations. Calculation
errors between IF97(MSL) and IAPWS95(MSL) were negligi-
ble to none, as expected. Relative to the IAPWS95(MSL) case,
the IF97+TDD and S+TDD cases generally produced lower er-
rors than the S+CD case. This was to be expected, because
the liquid water with constant density is being applied outside
13
Figure 11: Evolution of (a) fluid temperature and (b) fluid density, including
both the liquid and vapor components, and (c) fluid mass in the control volume
through the boiling process.
of its design temperature range, causing higher than normal er-
rors to be introduced. However, with that said the S+CW case
did produce less than 1% errors in terms of CVRMSE for most
property evaluations. The largest errors were introduced in the
S+CW case for ρ1and m, with CVRMSE values of 5.9% and
7.6% respectively, while CVRMSE for IF97+TDD and S+TDD
were less than 0.4% for all property evaluations. However, be-
cause boiling a volume of water completely is not a typical
normal-operation scenario for DH applications, all of the split-
medium models can be deemed acceptable.
4.2.2. Computing Speed
Table 7 presents the computing speed results for the control
volume component evaluations. The computing speed for the
IF97(MBL) case was significantly slower than all other cases,
indicative of the chattering problem. Across the remaining five
cases, the IAPWS95(MSL) case had the slowest computing
time. Relative to IAPWS95(MSL), time savings of 18%, 25%,
37%, and 35% were achieved for the IF97(MSL), IF97+TDD,
S+TDD, and S+CD cases, respectively. These trends are con-
sistent with expectations, as computing times generally reduced
as additional simplifications to the medium model implementa-
tions were made.
While the overall computing times for this small simulation
case are notably small, the structure of the differential alge-
braic system of equations indicates the likelihood of comput-
ing speed differences for larger thermo-fluid system models.
Except for IF97(MBL), the other five cases contain six con-
tinuous time state variables; however, the nonlinear systems of
equations varies among these five cases. The IAPWS95(MSL)
and IF97(MSL) cases each contain a nonlinear system with one
iteration variable: dV1/dt. Conversely, the new split-medium
implementations (IF97+TDD, S+TDD, and S+CD) contain no
nonlinear systems. While the nonlinear systems are small for
this component-scale evaluation, their sizes and quantities will
grow as DH system models increase in the number of build-
ings. From experiences, the time required to iteratively solve
the nonlinear systems largely contribute to the total computing
time. Thus, this likely will have an impact on computing time
for DH applications, which will be quantitatively tested in the
next section.
4.3. District Scale
Following the methodology prescribed in subsection 3.3, the
results for model accuracy and numerical performance at the
district scale evaluation are as follows. To evaluate impacts with
respect to the number of buildings N, simulations are repeated
with 1 to 10 buildings.
4.3.1. Accuracy
Figure 12 depicts the high accuracy achieved between the
four cases for heat flow rate at each building, the fuel con-
sumption rate at the boiler, and electric power at the feedwa-
ter pump. In Figure 12(a), the measured heat flow rate at the
building for each of the four cases followed the input data file
with minimal deviation (RMSE of 4 W and CVRMSE of 0.03%
for all cases with respect to the input data). Given that the in-
put to the building model was the data table and the conden-
sate return pumps controlled the mass flow rate ideally, this
was expected. In Figure 12(b) and (c), there are slight devi-
ations in the boiler’s fuel consumption rate and pump power
across the four cases. For the boiler’s fuel consumption rate, the
IF97+TDD and S+TDD cases both produced RMSE of 5.3 kW
(CVRMSE of 2.6%) compared to IF97(MSL), while the S+CD
case produced RMSE of 7.1 kW (CVRMSE of 3.5%) compared
to IF97(MSL). For feedwater pumping power, the S+CD case
also produced slightly higher errors than the other two cases
with CVRMSE of 4.2%, compared to 1.5% for IF97+TDD and
S+TDD. These differences were primarily caused by deviations
in the density at the pump inlet port ρ1, which impacts pumping
power as presented in section 2.3.1. For the feedwater pump in-
let, the density RMSE and CVRMSE for IF97+TDD, S+TDD,
and S+CD were 0.52 kg/m3(0.1%), 0.52 kg/m3(0.1%), and
37 kg/m3(4%), respectively. Over the two-day simulation pe-
riod, the total boiler fuel consumption was within 1% error of
the IF97(MSL) for all three split-medium cases. For the feed-
water pump energy consumption, the IF97+TDD and S+TDD
cases were within 0.3% of IF97(MSL), while the S+CD case
produced 4% error.
As seen in Table 8 and Table 9, the accuracy of thermody-
namic property evaluations at building 1 had insignificant dif-
ferences across all cases. Here, the building 1 results are used as
14
Table 6: RMSE and CVRMSE of control volume evaluations relative to the IAPWS95(MSL) case.
Property RMSE CVRMSE (%)
S+CD S+TDD IF97+TDD IF97(MSL) Units S+CD S+TDD IF97+TDD IF97(MSL)
ρ158.6 1.9 1.9 <0.001 kg/cm35.9 0.2 0.2 <0.001
ρ2<0.001 <0.001 <0.001 <0.001 kg/cm3<0.001 <0.001 <0.001 <0.001
h1.5 1.5 1.5 0.005 kJ/kg 0.06 0.06 0.06 <0.001
u2.4 2.3 2.3 0.002 kJ/kg 0.4 0.4 0.4 <0.001
s0.005 0.005 0.004 <0.001 kJ/kg-K 0.07 0.07 0.05 <0.001
T<0.001 <0.001 <0.001 <0.001 K <0.001 <0.001 <0.001 <0.001
m0.6 0.02 0.02 <0.001 kg 7.6 0.3 0.3 <0.001
Table 7: Computing times for the boiler component evaluation with time sav-
ings relative to IAPWS95(MSL). Results are averages over 10 simulation runs.
Case Computing Time (s) Time Savings (%)
IAPWS95(MSL) 0.020 –
IF97(MSL) 0.016 18
IF97(MBL) 14.3 None
IF97+TDD 0.015 25
S+TDD 0.012 37
S+CD 0.013 35
Table 8: RMSE of building 1 thermodynamic properties from DH evaluations
relative to IF97(MSL). Subscript 1 is for the inlet and 2 is for the outlet.
Property S+CD S+TDD IF97+TDD Units
T10.007 0.007 0.001 K
p17.49 8.58 5.79 Pa
h11.74 1.74 0.001 kJ/kg
s10.007 0.007 0.0 kJ/kg-K
T20.017 0.017 0.017 K
p27.49 8.58 5.79 Pa
h22.79 2.79 2.79 kJ/kg
s20.007 0.007 0.007 kJ/kg-K
proxies for all simulation cases, since the accuracy did not sig-
nificantly change building to building, with respect to N, and
at different points throughout the system. Subscripts 1 and 2
refer to the inlet and outlet ports of the building model, where
the inlet state is steam vapor and the outlet state is liquid wa-
ter condensate. As expected, the errors generally increased in-
versely to the steam/water model complexity. In this experi-
ment, the IF97+TDD case produced the lowest errors compared
to the IF97(MSL) case, followed by the S+TDD case. While
the S+CD case produced some of the higher errors, all errors
were within acceptable ranges. For example, the CVRMSE
values are within 1% for all property evaluations. With the
split-medium approaches, these DH simulations were able to
produce acceptable accuracy for all thermodynamic property
states.
Table 9: CVRMSE (%) of building 1 thermodynamic properties from DH eval-
uations relative to IF97(MSL). Subscript 1 is for the inlet and 2 is for the outlet.
Property S+CD S+TDD IF97+TDD
T10.002 0.002 0.0
p10.002 0.003 0.002
h10.064 0.064 0.0
s10.11 0.11 0.0
T20.004 0.004 0.004
p20.002 0.003 0.002
h20.50 0.50 0.50
s20.40 0.40 0.40
4.3.2. Computing Speed
Table 10 presents the structure of the translated model for
each of the four cases. The number of continuous time states
did not vary across the four cases. However importantly, the
split-medium approaches reduced the size of linear and nonlin-
ear systems present in the district models. For the IF97(MSL)
case, both the number of linear systems and the dimension of
the largest nonlinear system increase with the N. Conversely,
the linear and nonlinear systems do not change with respect to
Nfor the three split-medium cases. Because solving nonlin-
ear systems of equations are computationally expensive, these
results are highly advantageous for simulations of large DH
systems, especially when considering computing time require-
ments for industry applications.
The scaling results in Figure 13 clearly depict the comput-
ing speed benefits of the three split-medium approaches com-
pared to the standard IF97(MSL) case. For large N, the com-
puting time is expected to scale as tcpu =k Np, where kis some
constant and pis the order of the scaling. On a log-log plot,
this curve becomes log(tcpu )=log(k)+plog N. Making a
data fit for N≥100, which is where we see the expected lin-
ear behavior reproduced in the data, we obtain for IF97(MSL)
p=3.3 (i.e., cubic scaling), while for IF97+TDD, S+TDD,
and S+CD, pis 2.1, 2.3, and 2.1, respectively (i.e., quadratic
scaling). Thus, the main computational advantages were not
from the superheated steam simplifications (S), but from the
numerically-efficient liquid water models (TDD or CD) with
the split-medium approaches. This results was achieved by
decoupling the mass and energy balances through the split-
15
Table 10: Translation results with respect to the number of buildings Nin the DH system model. Results given are after Dymola’s built-in model manipulation.
NContinuous Linear Systems Nonlinear Systems
Time States IF97(MSL) IF97+TDD S+TDD S+CD IF97(MSL) IF97+TDD S+TDD S+CD
10 71 {2,2,...,2(11)} {2} {2} {2} {11,4} {3} {3} {1}
20 131 {2,2,...,2(21)} {2} {2} {2} {21,4} {3} {3} {1}
30 191 {2,2,...,2(31)} {2} {2} {2} {31,4} {3} {3} {1}
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
n6n+11 {2,2,...,2(n+1)} {2} {2} {2} {n+1,4} {3} {3} {1}
Figure 12: Accuracy evaluation for district scale results including (a) heat flow
rate at each building compared to the input data file, (b) boiler fuel consumption
rate, and (c) feedwater pump power. Input data refers to the prescribed heat flow
rate at each building and thus only appears in (a).
medium approach, which eliminated costly nonlinear systems
of equations. As an example, the computing time for an annual
simulation with 180 buildings will improve from 33 hours with
IF97(MSL) to 1-1.5 hours with the split-medium implementa-
tions. These results are critically important to enable industry
to simulate complete steam DH systems at large scales.
5. Discussion
To our knowledge, this is the first modeling and simulation
setup for complete steam DH systems featuring a plant, a dis-
tribution network, interconnected buildings with closed fluid
Figure 13: Computing time in seconds, presented on a log-log scale (base 10),
with respect to (a) the total number of interconnected buildings Nand (b) the
number of continuous time states. Linear fits indicating scaling rates are shown
for N∈[100,200].
loops and feedback control, which can also be coupled to build-
ing models of various levels of detail or to electrical system
models which are part of the MBL [7, 55, 56]. As complete
steam DH systems present new computational challenges, par-
ticularly at large scale, we focus on the modeling of water/steam
thermodynamics, which affects both accuracy and computing
speed across scales. We propose a novel split-medium approach
that couples numerically efficient liquid water models with var-
ious water/steam models and evaluate performance across ther-
modynamic property, component, and district scales. As a first
of its kind for steam DH systems, there are several important
limitations to consider and opportunities for improvement.
First, at the thermodynamic property scale, the accuracy of
16
the TDD and CD liquid water models from the MBL were eval-
uated with respect to IF97 and IAPWS-95. This study extended
these models beyond their original design intention, thus reveal-
ing that they are most suitable for pressures less than 4 MPa.
This upper pressure limit was most restricted by the constant
cpassumption, which produced errors as high as 11.8% under
high temperature and pressure conditions. If the TDD model
were to be improved in the future, changing the cpfunction
from constant to a temperature-dependent polynomial function
likely can increase the upper pressure limit with minimal im-
pacts to numerical performance. This hypothesis can be eval-
uated in the future. Further, this study focused on analytical
models for thermodynamic properties rather than table-based
methods. It is possible that the table-based methods can outper-
form analytical models for steam DH systems. This has yet to
be studied and can be evaluated in the future.
Second, the component scale evaluation revealed that errors
in internal energy and mass calculations increase as mass goes
to zero. This is an atypical scenario in normal operation, be-
cause boilers are controlled to maintain a water level setpoint.
However, it is important to recognize modeling sensitivity at
low mass quantities and mass flow rates, particularly with large,
complex, DH system models where problems can be difficult
to diagnose. One solution to this problem can be to include
smoothing functions that account for regularization at low mass
conditions to avoid diverging extensive property evaluations.
Third, the district scale evaluation importantly revealed how
the split-medium approaches can reduce the computing time
from cubic to quadratic with negligible compromise to accu-
racy. While additional time savings with the simplified steam
vapor model (S) and liquid water model (CD) are possible, the
most significant benefits can be achieved with the IF97+TDD
case. This finding is critical for practical applications of steam
DH simulations in industry, particularly since the IF97+TDD
case had the largest p-Tapplicable range of all split-medium
approaches.
However, as a first of its kind demonstration, simplifying as-
sumptions were made in the complete steam DH model that
present opportunities for improvement in the future. Most no-
tably, only the pressure drop in the condensate return pipes of
the distribution network were included, while the steam sup-
ply pipes were lossless. To our knowledge, a model for a sat-
urated steam pipe with a drip-leg that can accommodate both
pressure and heat losses is not available. Indeed, the Termis
software [39] cannot model saturated steam pipes nor conden-
sation in the distribution network. Although this lossless as-
sumption was appropriate based on this study’s objectives, it is
unrealistic for real-world case studies, since heat and pressure
losses are not negligible in both the steam supply and conden-
sate return pipes, and mass losses are not negligible in steam
pipes. With that said, we are currently developing a model for
a steam supply pipe (saturated or superheated) with a drip-leg
for condensate return, which will be made publicly available
in the future. For saturated or superheated steam pipes with
condensation, the steam supply with drip-leg model instantiates
the saturated control volume (section 2.3.3) developed herein.
Meanwhile, for superheated steam supply pipes without con-
densation, several pipe models are readily available (e.g., the
MBL plug flow pipe).
In addition, the studied system at district scale included
steam traps that vent to atmospheric pressure, which decreased
the pressures of subcooled liquid water. While this is a common
system design, some systems implement high pressure steam
traps or have flash tanks that can increase the back pressure
downstream of the steam trap; based on the thermodynamic
scale results herein, these higher pressure states for liquid con-
densate have lower accuracy with the TDD and CD models.
This can correspond to lower accuracy in pumping power and
boiler fuel consumption rate. With the IF97+TDD case as cur-
rently designed, the split-medium approach is currently limited
to 0-4 MPa for subcooled liquid condensate, while the full 0-
100 MPa are retained for all other water/steam phases. Lastly, a
single plant configuration was included in this study, represent-
ing one of the more simple designs. Future work will include
more advanced plant configurations and controls.
6. Conclusion
With district-scale simulations growing in importance as so-
cieties strive to meet energy and climate targets, this work en-
ables the modeling and simulation of complete steam-based
DH systems that, for all intents and purposes, was not previ-
ously possible for industry applications. Based on the medium
models considered, the IF97+TDD performed the best in terms
of both accuracy and computing speed for complete steam DH
simulations at large scales, reducing the computing time scal-
ing rate from cubic to quadratic while maintaining accuracies
within 3.5% CVRMSE across energy, power, and fuel con-
sumption rate calculations. The significant computing time sav-
ings were achieved by replacing the subcooled liquid water re-
gion of the IF97 model with a numerically efficient model from
the MBL, as the medium model in the MBL allows Modelica
translators to formulate a differential equation for the system
model in which mass and energy balance are decoupled. This
avoids costly nonlinear systems of equations. These results are
critically important to enable industry practitioners to simulate
complete steam DH systems at large scales. While there is room
for improvement in the models from thermodynamic properties
through complete systems, the split-medium approach can help
aid the transition of legacy DH systems to newer sustainable
designs, while providing a pathway for practical district-scale
analysis and optimization in the many steam heating applica-
tions that are likely to remain.
7. Acknowledgements
This research was supported in part by an appointment with
IBUILD Graduate Student Research Program sponsored by
the U.S. Department of Energy (DOE), Office of Energy Effi-
ciency and Renewable Energy, and Building Technologies Of-
fice. This program is managed by Oak Ridge National Labora-
tory (ORNL). This program is administered by the Oak Ridge
Institute for Science and Education (ORISE) for the DOE.
17
ORISE is managed by ORAU under DOE contract number
DESC0014664. All opinions expressed in this paper are the
author’s and do not necessarily reflect the policies and views of
DOE, ORNL, ORAU, or ORISE. In addition, this material is
based upon work supported by the DOE’s Office of Energy Ef-
ficiency and Renewable Energy under the Advanced Manufac-
turing Office, award number DE-EE0009139, and the Building
Technologies Office, contract number DE-AC02-05CH11231.
Further, this work emerged from the IBPSA Project 1, an in-
ternational project conducted under the umbrella of the Interna-
tional Building Performance Simulation Association (IBPSA).
Project 1 will develop and demonstrate a BIM/GIS and Mod-
elica Framework for building and community energy system
design and operation.
8. Disclaimer
This report was prepared as an account of work sponsored
by an agency of the United States Government. Neither the
United States Government nor any agency thereof, nor any of
their employees, makes any warranty, express or implied, or as-
sumes any legal liability or responsibility for the accuracy, com-
pleteness, or usefulness of any information, apparatus, product,
or process disclosed, or represents that its use would not in-
fringe privately owned rights. Reference herein to any specific
commercial product, process, or service by trade name, trade-
mark, manufacturer, or otherwise does not necessarily consti-
tute or imply its endorsement, recommendation, or favoring
by the United States Government or any agency thereof. The
views and opinions of authors expressed herein do not neces-
sarily state or reflect those of the United States Government or
any agency thereof.
Appendix A. IAPWS-95
The primary innovations of this work include the split-
medium approach and numerically efficient component mod-
els. We then simulated complete steam DH systems at large
scales, which to our knowledge, is novel among existing lit-
erature. Consistent with [8, 57], we provide additional details
regarding the mathematical formulations of water/steam used
in this work in appendices. In this work, we implement the
IAPWS-95 formulation [8] in Modelica for comparative eval-
uation. This formulation includes a single-set of equations ex-
plicit in the Helmholtz free energy fwith density ρand tem-
perature Tas independent variables. The fundamental equation
expresses the dimensionless form of the Helmholtz free energy
φin terms of the ideal-gas part φ◦and residual part φras
φ(δ, τ)=f(ρ, T)
RT =φ◦(δ, τ)+φr(δ, τ),(A.1)
where the reduced density δ=ρ/ρcand the reduced tempera-
ture τ=Tc/T, with subscript cindicating critical points. For
reference, the critical density and temperature for water are
322 kg/m3and 647.096 K, respectively.
The form of the ideal-gas part φ◦is
φ◦=ln δ+n◦
1+n◦
2τ+n◦
3ln τ+
8
X
i=4
n◦
iln 1−exp (−γ◦
iτ),(A.2)
where each of the 8 coefficients for n◦
iand exponents γ◦
iare
tabulated in Table 1 of [8]. Similarly, the form of the residual
part φris
φr=
7
X
i=1
niδdiτti+
51
X
i=8
niδdiτtiexp (−δci)(A.3)
+
54
X
i=52
niδdiτtiexp −αi(δ−εi)2−βi(τ−γi)2
+
56
X
i=55
ni∆biδψ,
with ∆ = θ2+Bh(δ−1)2iai,(A.4)
θ=(1 −τ)+Aih(δ−1)2i1
2βi,and (A.5)
ψ=exp −Ci(δ−1)2−Di(τ−1)2,(A.6)
where each of the coefficients and parameters for ci,di,ti,αi,
βi,γi,εi,Ai,Ci, and Diare tabulated in Table 2 of [8].
Based on the ideal-gas and residual parts of the dimension-
less Helmholtz free energy and their derivatives, thermody-
namic properties are calculated. Each of the following deriva-
tives are tabulated in Table 4 (ideal-gas part) and Table 5 (resid-
ual part) of [8]. By definition, pressure p=ρ2(∂f/∂ρ)T, and
the IAPWS-95 relation is
p(δ, τ)
ρRT =1+δ"∂φr
∂δ #τ
.(A.7)
Internal energy is defined in terms of the Helmholtz free energy
as u=f−T(∂f/∂T)ρ, and the relation to the dimensionless
form is u(δ, τ)
RT =τ "∂φ◦
∂τ #δ
+"∂φ◦
∂τ #δ!.(A.8)
Similarly, specific enthalpy, h=f−T(∂f/∂T)ρ+ρ(∂f/∂ρ)T,
and specific entropy, s=−(∂f/∂T)ρ, are calculated in IAPWS-
95 with
h(δ, τ)
RT =1+τ "∂φ◦
∂τ #δ
+"∂φ◦
∂τ #δ!+δ"∂φr
∂δ #τ
,(A.9)
and
s(δ, τ)
R=τ "∂φ◦
∂τ #δ
+"∂φ◦
∂τ #δ!−φ◦−φr.(A.10)
Formulations for other thermodynamic properties in terms of
the dimensionless Helmholtz free energy are in Table 3 of [8].
These include the isochoric heat capacity cv, isobaric heat ca-
pacity cp, and the Maxwell criterion for the phase-equilibrium
conditions, among others.
18
Appendix B. IF97 Region 2 Equations
Complete details for the IF97 formulation of water and steam
are available in [9, 57]. However, to provide a reference point
for the functions we replaced with the superheated steam ap-
proximations (Appendix C), relevant formulations for specific
enthalpy and entropy are reproduced here. In the IF97 formula-
tion for Region 2 (superheated vapor, referred as region 5 in this
work), basic equations for Gibbs free energy gare expressed in
a dimensionless form
γ(π, τ)=g(p,T)
RT =γ◦(π, τ)+γr(π, τ),(B.1)
where γ◦represents the ideal-gas part and γrrepresents the
residual part. This equation is formulated in terms of the spe-
cific gas constant of ordinary water R=0.461526 kJ/(kg ·K),
the reduced pressure π=p/p∗, and the inverse reduced tem-
perature τ=T∗/T, where superscript ∗indicates the reducing
quantity. For these equations, p∗=1 MPa and T∗=540 K.
The form of the ideal-gas part γ◦is
γ◦(π, τ)=ln π+
9
X
i=1
n◦
iτJ◦
i,(B.2)
where each of the 9 coefficients for n◦
iand J◦
iare tabulated in
Table 10 of [57]. Similarly, the form of the residual part γris
given as
γr(π, τ)=
43
X
i=1
niπIi(τ−0.5)Ji,(B.3)
where each of the 43 coefficients for niand exponents Iiand Ji
are tabulated in Table 11 of [57].
In terms of Gibbs free energy, specific enthalpy h=g−
T(∂g/∂T)p. Thus, the dimensionless relation in terms of the
ideal-gas and residual parts can be expressed as
h(π, τ)
RT =τ "∂γ◦
∂τ #π
+"∂γr
∂τ #π!,(B.4)
where the partial derivatives of γ◦and γrwith respect to τat
constant πare
"∂γ◦
∂τ #π
=
9
X
i=1
n◦
iJ◦
iτJ◦
i−1and (B.5)
"∂γr
∂τ #π
=
43
X
i=1
niπIiJi(τ−0.5)Ji−1.(B.6)
In terms of Gibbs free energy, specific entropy s=
−(∂g/∂T)p, and the IF97 relation is
s(π, τ)
R=τ "∂γ◦
∂τ #π
+"∂γr
∂τ #π!−(γ◦+γr).(B.7)
This completes the forward equations for h(p,T) and s(p,T)
in the IF97 formulation, Region 2. In total, h(p,T) includes
5 equations, 52 coefficients, and 95 exponents, while s(p,T)
includes 7 equations, 104 coefficients, and 190 exponents. In
addition to the forward equations, separate backward equations
are similarly implemented. However, Region 2 is further di-
vided into three subregions 2a, 2b, and 2c (see [57] for sub-
region divisions). Thus, three T(p,h) and three T(p,s) cover
Region 2, but only subregion 2a is applicable for the reduced
p-Trange specified for DH applications in Table 1. The dimen-
sionless backward equation T(p,h) for subregion 2a is
T(p,h)
T∗=
34
X
i=1
niπIi(η−2.1)Ji,(B.8)
where η=h/h∗,h∗=2000 kJ/kg, and each of the 34 coeffi-
cients for niand exponents Iiand Jiare tabulated in Table 20 of
[57].
Lastly, the backward function T(p,s) for subregion 2a is
T(p,s)
T∗=
46
X
i=1
niπIi(σ−2)Ji,(B.9)
where σ=s/s∗,s∗=2 kJ/(kg ·k), and each of the 46 coeffi-
cients for niand exponents Iiand Jiare tabulated in Table 25
of [57]. In total, T(p,h) includes 34 coefficients and 68 expo-
nents, while T(p,s) includes 46 coefficients and 92 exponents.
Modelica implementations of the above forward and backward
equations for specific enthalpy and entropy along with the other
thermodynamic properties are used for comparative evaluation
in this study.
Appendix C. Superheated Steam Approximations
To evaluate the potential numerical benefits of reduced-order
polynomial approximations for superheated steam thermody-
namics, we replaced commonly called thermodynamic func-
tions in the formulation of medium model S. Because this en-
deavor was a secondary evaluation beyond the split-medium ap-
proach, we include the mathematical details for these approxi-
mations in the appendix. In energy and exergy analysis of steam
DH systems, specific enthalpy hand specific entropy sare fre-
quently called. Thus, we replace these two functions with in-
vertible polynomial approximations such that h(p,T(p,h∗)) =
h∗for any p,h∗∈ <. Polynomial coefficients are determined
via a linear least squares method for two-dimensional polyno-
mial surface fits. The goal of this is not only to reduce the com-
putational load through lower-order functions, but also to avoid
numerical challenges due to inconsistent forward and backward
equations in the IF97 formulation. First, to minimize the sensi-
tivity to round offerrors and alleviate numerical problems, we
improve the conditioning of input variables pressure and tem-
perature by centering and scaling the inputs to standard-normal
distributions using
p=p−pmean
psd
and (C.1)
T=T−Tmean
Tsd
,(C.2)
19
where pand Tare the normalized pressure and temperature,
respectively; and subscripts mean and sd signify the mean and
standard deviation, respectively, of the selected input data set.
A dense input grid was developed at standard increments of
0.1 K and 1 kPa from the IF97 formulation. Several invert-
ible polynomials were evaluated before selecting (1) the lowest
order fits and (2) an acceptable reduced p-Trange with accept-
able accuracy and suitability for industrial applications. Based
on this process, the following pressure and temperature ranges
were selected for this reduced range implementation
0.1≤p≤0.55 MPa and (C.3)
100 ≤T≤160 ◦C.(C.4)
Table C.11 presents the normalization parameters used that per-
tain to the selected p-Tinput grid for the polynomial approxi-
mations.
Table C.11: Normalization parameters for polynomial approximations.
Parameter Value Units
pmean 1.235e+06 Pa
psd 8.326e+05 Pa
Tmean 6.776e+02 K
Tsd 1.611e+02 K
For specific enthalpy, we selected a linear approximation
ˆ
h(p,T)≈h(p,T), with h(p,T) given in Equation B.4, as
ˆ
h(p,T)=a1+a2p+a3T,(C.5)
where his specific enthalpy and aiare regression coefficients.
The corresponding backward equation is formulated directly
from Equation C.5 by solving for Tas
T(p,h)=c1+c2p+c3h,(C.6)
where
c=[c1,c2,c3]="−a1Tsd
a3Tmean
,−a2Tsd
a3,Tsd
a3#.(C.7)
Table C.12 lists the regression coefficients for Equation C.5
found using this least squares approach.
Table C.12: Regression coefficients for specific enthalpy h(p,T) and T(p,h).
Parameter Value Units
a13.272e+06 kJ/kg
a2-1.838e+04 kJ/kg-Pa
a33.504e+05 kJ/kg-K
Similar to specific enthalpy, we selected an invertible poly-
nomial approximation for specific entropy ˆs(p,T)≈s(p,T),
with s(p,T) given in Equation B.7. Using a function that is
quadratic in pressure and linear in temperature, specific entropy
can be approximated as
ˆs(p,T)=d1+d2p+d3T+p(d4p+d5T),(C.8)
where sis specific entropy and diare regression coefficients.
While higher order models could produce better fits, Equa-
tion C.8 was selected because its accuracy was sufficient while
still meeting our goal of having invertible functions. Solving for
Tdirectly from Equation C.8, we write the backward function
as
T(p,s)=(s−d1−p(d2+d4p))Tsd
d3+d5p+Tmean.(C.9)
Regression coefficients for Equation C.8 are listed in Ta-
ble C.13.
Table C.13: Regression coefficients for specific entropy s(p,T) and T(p,s).
Parameter Value Units
d17530 kJ/kg-K
d2-636.9 kJ/kg-K-Pa
d3532.8 kJ/kg-K2
d4159.8 kJ/kg-K-Pa2
d54.130 kJ/kg-K2-Pa
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