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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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