modeling radiant heating and cooling systems: integration with a whole-building simulation program
TRANSCRIPT
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Energy and Buildings 37 (2005) 389–397
Modeling radiant heating and cooling systems: integration with
a whole-building simulation program
R.K. Strand*, K.T. Baumgartner
School of Architecture, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL 61820, USA
Received 12 May 2004; received in revised form 13 July 2004; accepted 30 July 2004
Abstract
One of the primary goals of the radiant heating and cooling community for the past two decades has been the fair and accurate comparison
of radiant space conditioning systems with conventional forced air systems. The desire of the community was to give architects and engineers
the ability to compare the energy consumption of different system types while requiring the systems to produce equivalent thermal
environments. Unfortunately, radiant systems, though relatively simple in concept, are relatively complex from a modeling standpoint. A
rigorous model must include fundamental laws of thermodynamics and properly account for heat transfer due to radiation, convection, and
conduction. In addition, the model must do this in a generic way within a program that is capable of modeling buildings of any size and type
and their associated HVAC systems. Such challenges are not easily overcome and require careful attention to detail as well as a solid
foundation from which to build. This paper summarizes a model which addresses these concerns and provides a radiant system model within a
popular building energy simulation program.
# 2004 Elsevier B.V. All rights reserved.
Keywords: Simulation programs; HVAC systems; Heating and cooling systems
1. Introduction
Determining the energy efficiency of a building before it
is even built requires sophisticated science. While simula-
tion has the potential to reduce and has been shown to be
successful at reducing energy consumption, energy costs,
and the impact of buildings on the environment, the process
through which a simulation program has to go to simulate a
building is inherently complex. Though computer technol-
ogy certainly helps architects and engineers have better
access to improved predictions of energy efficiency, the fact
remains that simulation programs are, by their nature,
complex and detail oriented.
This is evident in the processes that take place within the
building envelope as well as other parts of the building. The
building envelope transmits heat to and from the inside and
outside environments that are experienced by a building.
Modeling a building properly requires that a simulation
* Corresponding author.
E-mail address: [email protected] (R.K. Strand).
0378-7788/$ – see front matter # 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.enbuild.2004.07.009
program adequately addresses fundamental heat transfer
processes such as conduction through the solid materials in
the building envelope, convection between the envelope and
the surrounding air, and radiation between the envelope and
the inside and outside environments. None of these
processes are simple linear functions, and most are
dependent not only on the material properties of the
envelope but also the inside and outside environmental
conditions. While some programs are capable of handling
most heat transfer effects for general building types and
materials, there are also challenges to modeling the way a
building is heated or cooled. Most modern buildings have
some form of HVAC system which has the function of
maintaining thermally comfortable conditions inside the
building regardless of the conditions on the outside of the
building. One of the more common methods of maintaining
thermal conditions within a building is through the use of
conventional forced air systems. These systems aim to
deliver conditioned air to a space to maintain a particular air
temperature since air temperature is one of the main factors
that determines thermal comfort. While in the case of most
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397390
conventional forced air systems many of the system
interactions have been well-studied and can be modeled,
there are a wide variety of different system types and
applications. To be able to evaluate how well and efficiently
different system types will condition a building requires a
program that is capable of modeling various components as
well as various reasonable configurations of those compo-
nents.
Radiant heating and cooling systems pose additional
problems to simulation programs. The idea of using
radiation to heat and cool environments is not new and is
becoming more commonplace in Asia and Europe and to a
lesser extent in North America [8]. Radiant systems aim to
provide conditioning directly to space occupants by radiant
exchange between ‘‘conditioned’’ surfaces within the space
rather than indirectly via transferring conditioned air to the
space which then provides comfort to the occupants via
convection.
While the concept of a radiant system seems relatively
simple—directly transfer heat to or from people—there are
significant challenges to properly modeling a radiant system
and incorporating such a model into a whole-building energy
simulation program. A radiant system, by definition, is both
an HVAC system as well as a surface within the space. This
means that it is part of the calculation to determine how
much heating or cooling is needed as well as the system that
actually responds to those space conditions. Since some
radiant systems have thermal mass associated with them,
controls and transient behavior of the system are critical
components of a realistic model. Moreover, since radiant
systems condition in a significantly different way in
comparison to standard conventional forced air systems,
the system model must be able to evaluate thermal comfort
in the space and also account for the possibility of con-
densation when the system is used for cooling. This high
level of integration must then also account for a generic
simulation environment that is established to ensure the
ability to simulate buildings of many different shapes,
orientations, sizes, etc.
It is also important that the resulting simulation not be so
‘‘radiant-centric’’ that it cannot model more conventional
systems or combined load-sharing systems that are a hybrid
combination of radiant and forced air systems. One of the
most significant barriers to the specification of radiant
systems in buildings is the inability to fairly compare the
radiant and conventional systems to each other or to
combinations of the two. This is really the most pressing
need of the radiant community today—to be able to simulate
various system types and make energy and/or cost
comparisons based on reliable simulation data.
The remainder of this paper focuses on the incorporation
of radiant system models into a whole-building energy
analysis program sponsored by the US Department of
Energy. It will begin with a brief description of the
capabilities of the program that are important for the proper
modeling of radiant systems. It will then delve into the
mathematical background of the algorithm and controls of
two distinct radiant system types: flow controlled systems
and temperature controlled (constant flow) systems.
2. Modeling radiant systems
As was briefly mentioned in the preceding section, a
program that will accurately model radiant systems and
compare them to other system types based on equivalent
thermal (comfort) environments needs to be able to analyze
a building from fundamental heat transfer and thermo-
dynamic perspectives, the HVAC system, and the inter-
connections between the two. In addition, to be successful, it
needs to be general enough to handle any building type,
shape, or size and do so in a relatively efficient manner.
The program [3] chosen for implementing the radiant
system model has incorporated all of the features needed to
successful simulate buildings and conventional systems. In
addition, it has features that make it an ideal application for the
integration of a radiant system model. From a heat transfer
perspective, it uses a fundamental heat balance approach [7]
that applies the First Law of Thermodynamics to three key
interfaces: the inside surfaces for walls, roofs, floors, etc.; the
outside surfaces for walls, roofs, floors, etc.; and the air
contained within the various zones in the building. The inside
and outside heat balances are done for individual surfaces
within a building and incorporate fundamental approaches for
handling convection, long wavelength radiation, and short
wavelength radiation and their impact on each surface.
Moreover, heat transfer through each surface is modeled using
a solution to the transient conduction equation known as
conduction transfer functions (CTFs).
Another key feature of the program is the integrated
nature of the simulation [13]. In the program, the impact of
the HVAC system on various spaces is calculated based on
the real capacity of the system. This means that when a
system is not sized large enough to meet all of the heating or
cooling needs for the spaces it is conditioning, the program
will simulate the deviation from the normal control band as
it would occur in a real building. This type of simulation
algorithm is particularly important when dealing with a
radiant system simulation since the radiant system is both a
surface within the building and an HVAC system. Many
more details about the program [4] and the initial
implementation of the radiant system model [10,11] and
thermal comfort models [6] in the program are available in
the literature. The main focus of this paper is two-fold: the
interconnection between the building and HVAC portions of
the simulation and how this connection is affected by the
type of controls used with the radiant system.
There are two main categories of radiant system controls
[1]: variable flow/fixed temperature systems and fixed flow/
variable temperature systems. The variable flow/fixed
temperature system has a central loop and pump that
supplies water at a particular temperature to all of the
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397 391
Fig. 1. Variable flow low temperature radiant system controls.
Fig. 2. Variable temperature low temperature radiant system controls.
systems connected to the loop. Valves control how much
water flows through each portion of the system based on
some criteria relating back to conditions within the space.
The fixed flow/variable temperature system has a central
loop and pump but also has local pumps associated with each
portion of the system. The local pumps maintain a constant
flow rate through that portion of the system while valves
control the amount of recirculation in the local loops. A
central pump maintains flow so that there is enough fluid to
inject to each local loop to maintain the proper temperatures
at each local loop. While some systems have several local
loops or perhaps use slight variations on these main control
strategies, most systems can be categorized into one of these
two classes.
Regardless of which type of system (variable temperature
or variable flow) is being simulated, both systems are
essentially a heat exchanger between the primary/secondary
equipment loop and the building elements (floors, walls,
ceilings, etc.). The primary/secondary side of the model is
simply a fluid loop that receives conditioned water from a
boiler, chiller, heat pump, etc. and then sends unconditioned
water back to the primary equipment. The building element
portion of the model connects to the heat balance of the
surfaces via a modification of the conduction transfer function
(CTF) formulation that accepts heat sources and sinks [9]. This
link is critical because the two sides of the radiant system
are simulated in very different areas of the program code. The
next two sections discuss the details behind this integration
and the mathematical derivation that was necessary to make
the model generic enough to both fit in the program structure
and model as many systems as possible and to make the model
efficient by avoiding unnecessary iterations that can cause
excessive program runtimes.
2.1. Simulation control overview
The controls for variable flow low temperature radiant
systems within the simulation program are fairly simple
though there is some flexibility through the use of schedules.
The program user is allowed to define a setpoint temperature
as well as a throttling range through which the system varies
the flow rate of water (or current) to the system from zero to
the user defined maximum flow rate. The flow rate is varied
linearly with the flow reaching 50% of the maximum when
the controlling temperature reaches the setpoint tempera-
ture. Setpoint temperatures can be varied on an hourly basis
throughout the year if desired. The controlling temperature
can be the mean air temperature, the mean radiant tem-
perature, the operative temperature of the zone, the outdoor
dry-bulb temperature, or the outdoor wet-bulb temperature,
and this choice is also left to the user’s discretion. Since
flow rate is varied, there is no explicit control on the inlet
water temperature or mixing to achieve some inlet water
temperature in the variable flow system. However, the user
does have the ability to specify on an hourly basis through a
schedule the temperature of the water that would be supplied
to the variable flow radiant system. Graphical descriptions of
the controls for the low temperature radiant system model in
the program are shown in Fig. 1 for a variable flow radiant
system.
In the constant flow-variable temperature systems, the
controls are also considered piecewise linear functions, but
in this case the user selects both the control temperatures and
the water temperatures via schedules. This offers greater
flexibility for defining how the radiant system operates
though it may not model every situation. Fig. 2 shows how
the ‘‘desired’’ inlet water temperature is controlled based on
user schedules. The user has the ability to specify the high
and low water and control temperature schedules for heating
and cooling separately (a total of eight temperature
schedules). Note that this inlet temperature is a ‘‘desired’’
inlet temperature in that there is no guarantee that the system
will provide water to the system at that temperature. The
model includes a local loop that attempts to meet this
demand temperature through mixing and recirculation.
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397392
Fig. 3. Variable temperature low temperature radiant system component
details.
The constant flow (variable temperature) low temperature
radiant system model is actually a combination of mixing
valves, a pump (constant speed, but the maximum flow can
be modified by a schedule), and the radiant system (surface,
panel, or group of surfaces/panels). This is connected to the
main loop through the standard inlet connections as shown
in Fig. 3. The system controls determine the desired inlet
temperature and system flow rate while loop controls
determine the flow rate and temperature of the loop. Note
that pump heat also factors into the model through a simple
constant speed pump model and user input.
Control of the variable flow system is considerably easier
because it simply accepts whatever temperature water is
available and varies the flow rate based on the controls
defined by Fig. 1. There is also no recirculation to account
for nor is there a local pump that adds heat to the loop.
Control of the variable temperature systems is more
challenging because of the desire to control temperature.
There are four possible conditions (separate for heating
and cooling) for the variable temperature radiant system.
First, if the loop has adequate temperature and flow to meet
system requests, then the model sets the radiant system inlet
temperature and controls to the desired values based on the
controls and simulates. This is the best condition and
recirculation and bypass amounts are adjusted accordingly
based on radiant system outlet temperatures. Second, if the
loop temperature is adequate but the loop flow rate is less
than the radiant system flow rate, we may or may not be able
to meet the desired inlet temperature since recirculation
might lower the temperature below the desired temperature.
In this second case, the model first simulates the radiant
system with the desired conditions and then resimulates it to
solve for the actual inlet temperature (see later in this
section) if it cannot achieve the desired inlet temperature.
Third, if the loop flow is greater than the radiant flow but the
temperature of the loop is not adequate, then there is no
amount of mixing that will solve this problem. All of the
radiant flow comes from the loop and the loop temperature
(after pump heat addition) becomes the radiant system inlet
regardless of the temperature controls. Finally, if both the
temperature and the flow of the loop are inadequate, then the
model simply solves for the actual radiant system inlet
temperature and does not try to meet the controls (merely
tries to get as close as physically possible given the loop
conditions).
Note that Fig. 3 shows that multiple surfaces can be
attached to a particular local loop. This option is available
for both the variable flow and variable temperature hydronic
radiant systems in the program. This is intended to account
for potential situations where a local loop serving a
particular zone has multiple active surfaces within that
zone. For example, both the floor and ceiling of a space may
serve as part of the radiant system, several different ceiling
surfaces may also be linked together, etc.
2.2. Modeling the connection between the heat balance
and HVAC system
As was mentioned earlier, the radiant system itself is a
heat exchanger with a ‘‘stationary’’ fluid (i.e., the building
element) assumed to be at a single temperature on one side
and a water loop on the other. The stationary fluid side
relates to the heat balance portion of the program and must
be integrated with the surface heat balances. The water loop
side is part of the primary plant simulation. A heat exchanger
formulation merges the two sides as described below.
As shown by [9], the standard conduction transfer
function formulation of heat conduction through a surface
can be adapted to include the effect of an internal heat
source/sink. This leads to in the following equation which is
used for the inside surface heat balance:
q00i;t ¼XM
m¼1
XmTi;t�mþ1 �XM
m¼1
YmTo;t�mþ1
þXk
m¼1
Fmq00i;t�m þXM
m¼1
Wmqsource;t�mþ1 (1)
A similar equation can be established for the outside surface
heat balance:
q00o;t ¼XM
m¼1
YmTi;t�mþ1 �XM
m¼1
ZmTo;t�mþ1
þXk
m¼1
Fmq00o;t�m þXM
m¼1
Vmqsource;t�mþ1 (2)
Additionally, we can characterize the temperature at the
source/sink location with a similar equation:
Ts;t ¼XM
m¼1
xmTi;t�mþ1 �XM
m¼1
ymTo;t�mþ1
þXk
m¼1
fmTs;t�m þXM
m¼1
wmqsource;t�mþ1 (3)
In this series of equations, X, x, Y, y, Z, z, F, and f are
conduction transfer function (CTF) coefficients that are
calculated once at the beginning of a simulation for each
unique surface construction. The V, W, and w terms are heat
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397 393
source transfer function (QTF) coefficients that allow the
evaluation of the response of a building element to an
embedded heat source or sink. Coefficients, temperatures,
and fluxes at time ‘‘t’’ are for the current moment in time
while terms at other times are history terms—values
from past time steps which are known as a result of the
simulation.
Eqs. (1)–(3) serve as a starting point for the actual heat
balance solution. These equations alone, however, are not
enough information to solve for what is happening in the
radiant system at a given point in time. The heat source/
sink term is not known directly for a radiant system since
this relates back to the inlet water temperature, the water
flow rate, and the temperature of the radiant surface. The
actual heat transferred between the building element and
the water loop is related to the temperature of the building
element at the source location as well as the water system
inlet and outlet temperatures. In the model, it is assumed
that the inlet temperature to the slab (defined by a user
schedule and/or controls and the primary equipment
simulation) and the mass flow rate (determined by the
control algorithm and/or user input) are known and that the
remaining parameters must be calculated. However, the
heat balance equations require the heat transferred to the
building element from the water loop in order to calculate
the heat transferred from the element to the building
environment.
Even though systems defined by this model can vary
somewhat, the same characteristic link between the system
variables exist. For modeling purposes, the overall water/
slab system can be thought of as a heat exchanger. While
in principle there are two alternative heat exchanger
methodologies, it is more convenient to use the effective-
ness-NTU method in this case. Several assumptions will be
incorporated into the heat exchanger analysis. It is assumed
that the building element that contains the water loop is
stationary and that its temperature along the length of the
tubing is constant. The latter part of this assumption stems
from assumptions made in both the one- and two-
dimensional heat source transfer function derivations. In
either case, the source was added at a single node that
was characterized by a single temperature. For consistency,
this assumption must be made again in the heat exchanger
analysis. Another assumption for the model is that the
fluid in the tubing is water. Additionally, it is assumed
that the thermal properties of the water do not vary
significantly over the length of the tubing. Finally, the
temperature at the inside surface of the water tubing is
assumed to be equal to the temperature at the source
location. In other words, it is assumed that the water tubing
itself has no appreciable effect on the heat transfer process
being modeled.
Using these assumptions and the effectiveness-NTU heat
exchanger algorithm, several equations can be defined which
establish the relationship between the heat source and the
water temperatures. First, a heat balance on the water loop
results in:
q ¼ ðmcpÞwaterðTwi � TwoÞ (4)
where q is the energy transferred between the water
loop and the building element, m the mass flow rate of
the water, cp the specific heat of the water, Twi the
inlet water temperature, and Two the outlet water tempera-
ture.
The maximum amount of heat transfer that is possible
according to the Second Law of Thermodynamics is:
qmax ¼ ðmcpÞwaterðTwi � TsÞ (5)
where qmax is the maximum amount of energy transfer that is
possible and Ts the temperature at the source location.
The effectiveness of the heat exchanger, e, is defined as
the ratio of the actual energy transfer to the maximum
possible, or:
e� q
qmax(6)
For a heat exchanger where one fluid is stationary, the
effectiveness can be related to NTU, the number of transfer
units, by the following equation [5]:
e ¼ 1 � e�NTU (7)
where NTU is defined by:
NTU� UA
ðmcpÞwater
(8)
Since the water tubes were assumed to have no effect on the
heat transfer process, the only term present in the overall
heat transfer coefficient, UA, is a convection term. Thus, the
equation for UA is:
UA ¼ hðpDLÞ (9)
where h is the convection coefficient, D the interior tube
diameter, and L the total length of the tube.
The convection coefficient can be obtained from internal
flow correlations that relate the Nusselt dimensionless
number to other flow properties. For laminar flow in a tube of
constant surface temperature, the Nusselt number is defined
by:
NuD ¼ hD
k(10)
where k is the thermal conductivity of the water.
For turbulent internal flow, the Colburn equation can be
used to define the Nusselt number:
NuD ¼ hD
k¼ 0:023 Re
4=5D Pr1=3 (11)
where Pr is the Prandtl number of water and ReD the
Reynolds number which is defined by:
ReD ¼ 4m
pmD(12)
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397394
The parameter m is the absolute viscosity of water. For
internal pipe flow, the flow is assumed to be turbulent for
ReD � 2300.
Knowledge of the flow conditions allows Eqs. (6)–(12) to
be calculated. This fixes the value for e which can then be
used to relate Eqs. (4) and (5) to each other. The controls and
the plant define the water mass flow rate and the inlet water
temperature, leaving two equations (Eqs. (4) and (5)) and
three unknowns. The third equation that can be used in
conjunction with Eqs. (4) and (5) is Eq. (3), which is the CTF/
QTF equation for the temperature at the source location.
Unfortunately, it has two additional unknowns—the inside
and outside surface temperatures. This requires the addition
of modified forms of Eqs. (1) and (2) which adds two more
equations without the introduction of any further unknowns.
Eq. (1) is the standard conduction transfer function
formula for the inside surface of a building element with an
embedded source/sink of heat. In the actual simulation
program, the surface flux on the left hand side of the
equation is replaced with a surface heat balance:
surface heat balance ¼XM
m¼1
XmTi;t�mþ1 �XM
m¼1
YmTo;t�mþ1
þXk
m¼1
Fmq00i;t�m þXM
m¼1
Wmqsource;t�mþ1
(13)
The surface heat balance includes terms for incident solar
energy, radiation heat transfer from internal heat sources
such as lights and electrical equipment, radiation between
surfaces using Hottel’s Gray Interchange concept, and con-
vection to the surrounding air. A similar equation can be
derived from Eq. (2) for the outside surface heat balance and
includes thermal forces such as solar radiation, convection to
the outside air, long wavelength radiant exchange with the
external environment, etc. It should be noted that the internal
radiation heat balance is dependent on conditions at the
other surfaces within the space. Thus, an iterative loop is
required to provide a more accurate estimate of the radiative
exchange within the building interior. Thus, the terms of the
heat balance on the left hand side of the equation have been
set with the only unknown quantity being Ti, the inside
surface temperature at the current time step. On the right
hand side of Eq. (13), most of the terms are already defined
since they depend on known values from previous time steps
(temperature, flux, and source histories). The only terms
which are not defined are the inside surface temperature (Ti),
outside surface temperature (To), and internal heat source/
sink (qsource) of the current time step. Similar arguments can
be made for reformulation of the outside heat balance
equation and the base form of the heat source/sink location
temperature equation (Eq. (3))—the only undefined terms
are the inside surface temperature (Ti), outside surface
temperature (To), and internal heat source/sink (qsource) of
the current time step.
One can summarize the situation by hiding some of the
complexity for the purpose of deriving equations to solve the
system of equations at hand. The three basic equations for
the temperature at the inside and outside surface as well as at
the location of the heat source/sink can be reduced to:
Tinside ¼ Ca þ CbToutside þ Ccq00 (14)
Toutside ¼ Cd þ CeTinside þ Cfq00 (15)
Tsource ¼ Cg þ Chq00 þ CiTinside þ CjToutside (16)
where Tinside is the current temperature at the inside surface,
Toutside the current temperature at the outside surface, Tsource
the current temperature within the radiant system at the
location of the source/sink, q00 the heat source/sink from the
radiant system water loop, Ca is all of the other terms in the
inside heat balance (solar, LW exchange, conduction history
terms, etc.), Cb the current cross CTF term, Cc the QTF
inside term for the current heat source/sink, Cd is all of the
other terms in the outside heat balance (solar, LW exchange,
conduction history terms, etc.), Ce the current cross CTF
term, Cf the QTF outside term for the current heat source/
sink, Cg the summation of all temperature and source history
terms at the source/sink location, Ch the QTF term at the
source/sink location for the current heat source/sink, Ci the
CTF inside term for the current inside surface temperature,
Cj the CTF outside term for the current outside surface
temperature.
Eqs. (14) and (15) above can each be solved to remove the
other surface temperature. Then, substituting these new
equations for Tinside and Toutside as a function of C and q00 into
the equation for Tsource and simplifying, one obtains the
following equation:
Tsource ¼ Ck þ C1q00 (17)
Ck ¼ Cg þCiðCa þ CbCdÞ þ CjðCd þ CeCaÞ
1 � CeCb(18)
Cl ¼ Ch þCiðCc þ CbCfÞ þ CjðCf þ CeCcÞ
1 � CeCb(19)
Combining this with the heat exchanger analysis as shown
above, we eventually arrive at the following equation to
relate the flux to the slab to the water inlet temperature and
mass flow rate:
q00 ¼ Twater;in � Ck
ðCl=AÞ þ ð1=eðmcpÞwaterÞ(20)
which includes all of the inside and outside heat balance
terms (‘‘hidden’’ in the Ck and Cl coefficients). Once the flux
to the slab is known, the remaining terms of interest (outlet
water temperature, inside and outside surface temperatures,
etc.) can be calculated using the relatively simpler equations
shown above.
The above development is valid for both the variable flow
radiant system and for variable temperature radiant systems
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397 395
where the inlet temperature is known (based on controls).
However, due to loop conditions and the presence of
recirculation, it is not always possible to know the inlet
temperature to the variable temperature radiant system
without actually simulating it. For cases where the main loop
temperature and flow rates are adequate, one can assume an
inlet temperature and then determine the recirculation
needed after the fact. Yet, when the main loop flow or
temperature is not adequate to meet the needs of the local
loop as determined by the system controls, one is faced with
a ‘‘chicken-and-egg’’ situation. One needs to know the inlet
temperature in order to solve for the system output, but one
needs the system output in order to solve for the inlet
temperature. As a result, the model must either iterate using
a guess for inlet temperature as a starting point or perform
more mathematics to arrive at the inlet temperature as a
function of system characteristics. This implementation
chose to avoid iteration and solved for the inlet temperature
as shown below.
In addition to resolving the iteration concerns, one must
also account for the fact that there could be several potential
parts to a particular local loop as shown in Fig. 3. Each
of these individual surfaces may have a different outlet
temperature. Thus, all of the surfaces that are part of an
individual loop must be solved at the same time to obtain
both the combined outlet temperature for the local loop
and thus the inlet temperature based on the amount of
recirculation needed.
The previous equation combines with the following
equation which is valid for an individual surface in the
current local loop of a radiant system:
qj ¼ mjcpðTwater � Twater;out;jÞ (21)
where qj is the heat transfer to the jth surface in the radiant
system, mj the mass flow rate only to this surface, and
Twater,out,j the outlet temperature for the jth surface. Combin-
ing the previous two equations results in:
Twater;in � Twater;out;j ¼ejðTwater;in � Ck;jÞ
1 þ ððejmjcpCl;jÞ=AÞ (22)
where Ck,j and Cl,j are evaluated for each individual surface
as shown in Eqs. (18) and (19). If for each surface in the
radiant system, we let:
Cm;j ¼ej
1 þ ððejmjcpCl;jÞ=AÞ (23)
then the previous equations becomes the slightly less com-
plex:
Twater;in � Twater;out;j ¼ Cm;jðTwater;in � Ck;jÞ (24)
Rearranging to obtain the outlet temperature for the jth
surface:
Twater;out;j ¼ ð1 � Cm;jÞTwater;in þ Cm;jCk;j (25)
The overall outlet temperature from the radiant system is just
a simple mixing of all of the surface outlet temperatures
based on flow fraction and results in Twater,out as shown in:
Twater;out ¼X
j
½ðFlowFractionjÞTwater;out;j (26)
An energy balance on the mixing valve-pump group results
in the following equation that relates the radiant system inlet
temperature (Twater,in) to the loop inlet water temperature
(Tloop,in), the radiant system outlet temperature (Twater,out),
and the pump heat addition
Twater;in ¼ mloop
msystemTloop;in þ
mrecirc
msystemTwater;out
þ PumpHeat
msystemcp(27)
Plugging in the definition of Twater,out based on the summa-
tion of Twater,out,j equations results in:
Twater;in ¼ LoopTerm þ RecircTerm
TwiCoeff(28)
where
LoopTerm ¼mloop
msystemTloop;in þ
PumpHeat
msystemcp(29)
mrecircX
RecircTerm ¼msystem j
FlowFractionjTwater;out;j (30)
mrecirc
� �X
TwiCoeff ¼ 1 �msystem j
½FlowFractionjð1 � Cm;jÞ
(31)
Once the actual water inlet temperature is calculated with
this equation, it is then possible to calculate individual outlet
temperatures for each surface, the overall outlet tempera-
ture, and finally all of the necessary flow and loop quantities.
This procedure avoids iteration but is somewhat complex to
follow. However, this second mathematical process is only
needed for select cases of the variable temperature radiant
system when the inlet temperature is not known explicitly.
With the proper establishment of input data, such cases can
be avoided, though one cannot simply assume this for a
comprehensive energy simulation program.
The proceeding development shows several important
points. First, it is possible to model the connection between
the heat balance and the HVAC portions of the simulation in
a fundamental, generic way. Second, it is possible to handle
this connection without resorting to excessive iterations
despite the intimate link between the two sides of the radiant
system. In the simulation program, iteration does take place,
but the radiant system model as described here does not
result in additional iteration loops that could lead to long
execution times. Finally, the implementation works within
the current structure of the program—meaning that it does
not require special rules for this particular system and that it
automatically has access to any improvement made to either
the heat balance or HVAC portion of the simulation.
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397396
2.3. Avoiding undesirable operating conditions
Both the variable flow and variable temperature systems
have checks within their algorithms to avoid operating the
radiant system under what might be considered undesirable
conditions. One check is aimed at avoiding heat additions by
the system when cooling is required and heat removal by the
system when heating is required. While this likely will never
happen in most systems, such conditions might occur during
transitional seasons or changes in controls from heating to
cooling or cooling to heating. Thus, the system compares the
water inlet temperature with the current temperature of the
radiant system where the source/sink term will be applied.
If sending water to the radiant system will produce the
opposite of what is desired from a heat addition/removal
standpoint, the algorithm will interrupt flow to the radiant
system.
Another check is related to the potential for condensation
in a radiant cooling system. If moisture levels within the
space being cooled are high enough either due to indoor or
outdoor conditions (or both), the surface temperature of the
radiant system could drop below the dew-point temperature
of the space. If this occurs, condensation will take place.
This can be a potential problem in radiant cooling systems,
especially in many of the climates found in North America.
Currently, the program compares the dew-point temperature
of the space with the predicted surface temperature of the
radiant system based on what the normal controls dictate
should be happening within the radiant system. If the pre-
dicted surface temperature drops below the dew-point
temperature plus a small temperature tolerance, the
algorithm will stop flow to the radiant system.
2.4. Connecting to primary systems
One of the advantages to having access to the HVAC
portion of the program through the primary plant water loop
is that it allows the ability to try out different equipment and
various different ideas. One idea that has already been
addressed in the literature [12] is the possibility of hooking
the radiant system up to a water loop that is served only by an
evaporative condenser. Recent additions of ground and pond
loops make it possible to investigate the energy efficiency of
using temperate water in a radiant system and determine
whether or not such a system can produce thermally
comfortable conditions inside the building. The beauty of
such a simulation model is such alternative systems can be
tested for a variety of conditions and environments—even
ones that would not be thought of as climates where such
technology would be successful.
3. Concluding remarks
The model described in this paper is already available to
the general public and has been available for some time. It
incorporates all of the algorithm, control, modeling,
and other features described in the preceding sections.
While this is a significant advance toward the ultimate
goal of modeling all features of radiant systems and having
the ability to compare them fairly and accurately to
conventional forced air systems, there are still features that
needed to be added and are the focus of upcoming research
work on the program. The two issues that will likely be
addressed next are the incorporation of a more sophisti-
cated condensation limit for variable temperature radiant
systems and the modeling of ‘‘load-sharing’’ hybrid
systems. Currently, the condensation check is a fairly
blunt strategy that shuts the system down when it perceives
that condensation may occur. This is based on a comparison
between the dew-point temperature and the predicted
surface temperature if the radiant system is running. There
is a 1 8C tolerance that is presently used within the
program. This was chosen so that the system will shut off
when the surface temperature would be close but still
higher than the dew-point temperature. This limit may be
somewhat strict in that some systems that have the radiant
system connected to an evaporative condenser may
actually trigger the condensation limit. Further study is
needed here to determine if this tolerance is, in fact,
appropriate and whether this needs to be a parameter that
the user should be allowed to change. In addition, the
variable temperature systems do not need to interrupt flow
when condensation is predicted to occur. Since they have
the ability to increase the amount of recirculation, it would
be more appropriate to increase the water inlet temperature
first in an attempt to provide some amount of cooling
without causing condensation.
Concerns about condensation often result in the
specification of a ‘‘load-sharing’’ or hybrid system. In such
a system, a radiant system is combined with a conventional
forced air system to provide the conditioning needs of the
space. Generally, the radiant system provides most or all of
the sensible cooling while the air system handles the latent
cooling. While it currently is possible to model a space being
served by both a radiant and a conventional forced air system
within the program, their interaction is serial rather than
parallel. In the current structure of the program, one system
will run to capacity in an effort to meet the current space
conditioning needs of the zone. Then, additional systems
will attempt to meet any remaining load by operating in
series. True ‘‘load-sharing’’ systems are controlled differ-
ently and thus this type of system is the subject of a current
research project.
Despite the continuing work on the radiant model within
the simulation program, the current model has many
capabilities and has been the subject of recent validations
studies [2]. Many current systems can already be modeled
and compared to conventional forced air systems, and the
comparisons can be made on the more accurate basis of
equivalent thermal comfort rather than simply equal air
temperatures.
R.K. Strand, K.T. Baumgartner / Energy and Buildings 37 (2005) 389–397 397
Acknowledgment
The authors wish to thank the US Department of Energy’s
National Energy Technology Laboratory (NETL) for
funding the work under Grant DE-FC26-01NT41250 which
led to this paper.
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