Volume 3 solar thermal systems components and applications 3 11 – modeling and simulation of passive and active solar thermal systems

3.11 Modeling and Simulation of Passive and Active Solar Thermal
Systems
A Athienitis, Concordia University, Montreal, QC, Canada
SA Kalogirou, Cyprus University of Technology, Limassol, Cyprus
L Candanedo, Dublin Institute of Technology, Dublin, Ireland
© 2012 Elsevier Ltd. All rights reserved.

3.11.1
3.11.2
3.11.2.1
3.11.2.1.1
3.11.2.1.2
3.11.2.1.3
3.11.3
3.11.3.1
3.11.3.2
3.11.3.3
3.11.3.3.1
3.11.3.3.2
3.11.3.3.3
3.11.3.4
3.11.4
3.11.4.1
3.11.4.1.1
3.11.4.2
3.11.5
3.11.6
3.11.6.1
3.11.6.1.1
3.11.6.1.2
3.11.6.1.3

3.11.6.2
3.11.6.2.1
3.11.6.2.2
3.11.6.3
3.11.6.4
3.11.7
3.11.7.1
3.11.7.2
3.11.7.3
3.11.7.3.1
3.11.7.3.2
3.11.8
3.11.8.1
3.11.8.2
3.11.9
3.11.9.1
3.11.9.2
3.11.9.3
3.11.10
References

Introduction
Passive Solar Design Techniques and Systems
Direct-Gain Modeling
Transient heat conduction and steady-periodic (frequency domain) solution
Building transient response analysis
Simplified analytical direct-gain room model and solution (passive)
PV/T Systems and Building-Integrated Photovoltaic/Thermal (BIPV/T) Systems
Integration of Solar Technologies into the Building Envelope and BIPV/T
A Simplified Open-Loop PV/T Model

Transient and Steady-State Models for Open-Loop Air-Based BIPV/T Systems
Air temperature variation within the control volume
Radiative heat transfer
Inlet air temperature effects
Heat Removal Factor and Thermal Efficiency for Open-Loop BIPV/T Systems
Near-Optimal Design of Low-Energy Solar Homes
Envelope and Passive Solar Design
HVAC and renewable energy systems
Overview of the Design of Two Net-Zero Energy Solar Homes
Active Solar Systems
The f-Chart Method
Performance and Design of Liquid-Based Solar Heating Systems
Storage capacity correction
Collector flow rate correction
Load heat exchanger size correction
Performance and Design of Air-Based Solar Heating Systems
Pebble-bed storage size correction
Airflow rate correction
Performance and Design of Solar Service Water Systems
General Remarks
Utilizability Method
Hourly Utilizability
Daily Utilizability
Design of Active Systems with Utilizability Methods
Hourly utilizability
Daily utilizability
The Φ
¯, f-Chart Method
Storage Tank Losses Correction
Heat Exchanger Correction

Modeling and Simulation of Solar Energy Systems
The F-Chart Program
The TRNSYS Simulation Program
WATSUN Simulation Program
Limitations of Simulations

Nomenclature
A collector exposed area, m2; collector area, m2
Ac cross-sectional area, m2
Acv area of the control volume, m2

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b conditions at bulk temperature
Cmin minimum capacitance of the two fluid streams in
the heat exchanger, W °C−1
cp specific heat capacity of the air, J kg−1 K−1

doi:10.1016/B978-0-08-087872-0.00311-5

357


358

Components

d tube diameter, m
Dh hydraulic diameter of the cavity, m
Ep electric power, W
f friction factor

Fe emissivity factor, 1/(1/ε2 + 1/ε3 − 1)
Fplate,insu view factor between plate and insulation
FR collector heat removal factor
G total incident solar radiation, W m−2
g gravitational acceleration, m s−2
Gr Grashof number, g β(Tw – Tbulk)Dh3/υ2
Grb Grashof number, (%b – ρ)d3g/%b2
Grq Grashof number based on heat flux qw,
ðgβqw Dh 4 Þ=ðv2 kÞ
Grx Grashof number based on inlet distance,
x3gβ(Tw – Tbulk)/υ2
h hour angle in degrees at the midpoint of each hour,
degrees
ho, hw exterior/wind convective heat transfer coefficient,
W m−2 K−1
hr cavity radiative heat transfer coefficient, W m−2 K−1
hro exterior radiative heat transfer coefficient, W m−2 K−1
hss sunset hour angle, degrees
h΄ss sunset hour angle on the tilted surface, degrees
H t monthly average daily total radiation on the tilted
collector surface, MJ m−2
hcb,hct convective heat transfer coefficient in cavity,
W m−2 K−1
hci convective heat transfer coefficient in attic,
W m−2 K−1
It total radiation incident on the collector surface per unit
area, kJ m−2
k thermal conductivity, W m−1 K−1
L length of the channel, m; total heating load during the
integration period; monthly heating load or demand, MJ;

local latitude, degrees
Ls solar energy supplied to the load, GJ
Lu useful load, GJ
m average mass flow rate, kg s−1
M actual mass of storage capacity, kg
_ a actual collector flow rate per square meter of collector
m
area, l s−1 m−2
Mb,a actual pebble storage capacity per square meter of
collector area, m3 m−2
Mb,s standard storage capacity per square meter of
collector area, 0.25 m3 m−2
_ s standard collector flow rate per square meter of
m
collector area, l s−1 m−2
Mw,a actual storage capacity per square meter of collector
area, l m−2
Mw,s standard storage capacity per square meter of
collector area, 75 l m−2
N number of days in a month
NDR diffusivity ratio
Nu Nusselt number, hDh k−1
Pelect electrical power per unit area, W m−2
Pe Peclet number, RePr
Pr Prandlt number (v/α)
qrad radiative heat exchange between cavity surfaces per
unit area, W m−2

qrec heat recovered in the control volume per unit area,
W m−2

qw heat flux on the wall, W m−2
Qincv convective heat transfer rate in control volume, W
Qradcv radiative heat transfer rate in control volume, W
Ra Rayleigh number, GrPr = g%2cp β(Tw – Tbulk)Dh3/(μk) = g
β(Tw – Tbulk)Dh3/υα
RB monthly mean beam radiation tilt factor
Re Reynolds number, %VDh/μ
Rex Reynolds number based on inlet distance, %Vx/μ
Rinsu insulation R-value, m2K W−1
Rmix combined thermal resistance, m2K W−1
Rplywood plywood layer R-value, m2K W−1
Rs ratio of standard storage heat capacity per unit of
collector area of 350 kJ m−2 °C−1 to actual storage capacity
RTefzel Tefzel R-value, m2K W−1
t time from midnight, h
Ta ambient temperature, °C
T a monthly average ambient temperature, °C
Tb air bulk temperature in the control volume, °C
Tdp dew point temperature, °C
Ti inlet collector fluid temperature, °C
Tinlet inlet air temperature, °C
Tinsu interior side temperature of the insulation, °C, or in
K for radiative heat transfer computation
Tm mains water temperature, °C
To, Ta exterior air temperature, °C
Toutlet outlet air temperature, °C
Tplate interior side temperature of the metal sheet, °C, or
in K for radiative heat transfer computation
TPVMID temperature of the photovoltaic (PV) module at
midpoint, °C

TPVTOP temperature of the PV module at its external
surface, °C
T s monthly average storage tank temperature, °C
Tsky sky temperature, K
Tw minimum acceptable hot water temperature, °C
U wetted perimeter, m
UL energy loss coefficient, kJ m−2−K−1
(UA)L building loss coefficient and area product used in
degree-day space heating load model, W K−1
V average air velocity in the channel, m s−1
Vw average wind velocity, m s−1
W air moisture content, kgv (kga)−1
WPV width of the control volume, m
x distance from inlet of flow channel, m

Greek Letters
α solar absorptivity; thermal diffusivity (=k/%cp), m2 s−1
β thermal expansion coefficient, 1/T or PV module
temperature coefficient, (%K−1)
βmp maximum power point PV module temperature
coefficient, %K−1
δ declination
ΔtL number of seconds during a month the load is
required, s
Δx length of the control volume, m
εL effectiveness of the load heat exchanger


Modeling and Simulation of Passive and Active Solar Thermal Systems

ε1, ε2, ε3 long-wave emissivities
ηe electrical efficiency
ηPV electrical efficiency of the PV module
ηSTC electrical efficiency at standard test conditions
θ incidence angle, degrees
μ dynamic or absolute viscosity, kg m−1s−1
υ kinematic viscosity, m2 s−1

359

ρ average air density 1/(Tw – Tb) ∫∫TTwb ρdT, kg m−3
% air density, kg m−3
σ Stefan–Boltzmann constant, W m−2 K−4
(τα) effective transmittance–absorptance product
(τα) monthly average value of (τα)
φ tilt angle, degrees

3.11.1 Introduction
There are two principal categories of building solar heating and cooling systems: passive and active. Passive systems integrate into
the structure of the building technologies that admit, absorb, store, and release solar energy, thereby reducing the need for electricity
use to transport fluids. In contrast, active systems also include fans and pumps controlled to move air and heat transfer fluids,
respectively, for space heating and/or cooling and domestic hot water (DHW) heating.
Current international trends, which are expected to continue, will increasingly rely on a combination of active and passive solar
systems as enabling technologies for net-zero energy solar buildings (NZESBs) – solar buildings that produce as much energy as they
consume over a year. Similarly, hybrid systems – active/passive and thermal/electric – will gain popularity, such as the photovoltaic/
thermal (PV/T) systems that are described later in this chapter.
This section presents approaches that are used for modeling and simulating both passive and active solar systems. First, techniques
are discussed for modeling direct gains, analyzing transient responses of buildings, and developing simplified analytical thermal
models of direct-gain rooms. Next, methods are presented for the thermal analysis of hybrid PV/T collectors and building-integrated
photovoltaic (BIPV) systems. Then, to conclude the section, an overview of the design of two net-zero energy houses is described.

In the second part of the chapter, various design methods are presented that include the simplified f-chart method, which is
suitable for both solar heating and solar cooling of buildings, as well as for domestic water heating systems, utilizability Φ, and the
Φ; f-chart methods. Subsequently, various packages for advanced modeling and simulation of active systems are presented.
Finally, it should be noted that the components and subsystems discussed in other chapters of this volume may be combined to
create a wide variety of building solar heating and cooling systems.

3.11.2 Passive Solar Design Techniques and Systems
Passive solar technologies do not use fans or pumps in the collection and usage of solar heat. Instead, these technologies use the
natural modes of heat transfer to distribute the thermal energy of solar gains among different spaces. When applied to buildings, this
generally refers to passive energy flows among rooms and envelope, such as the redistribution of absorbed direct solar gains or night
cooling [1]. Buildings that use primarily these technologies to reduce heating and/or cooling energy consumption are called ‘passive
solar buildings’ (i.e., a building that uses solar gains to reduce heating and possibly cooling energy consumption based on natural
energy flows – radiation, conduction, and natural convection). The major driving forces for thermal energy transfer in a passive solar
building are long-wave thermal radiation exchanges and natural convection, that is, buoyancy [1].
Passive technologies are integrated within the building and may include:
1. ‘Near-equatorial facing windows’ with high solar transmittance and high thermal resistance. These properties maximize the
amount of direct solar gains into the living space, while reducing envelope heat losses and gains in the heating and cooling
seasons, respectively. Skylights are often employed for daylighting in office buildings and in sunspaces (solaria).
2. Building-integrated thermal storage. Thermal storage, which is commonly referred to as thermal mass, may consist of sensible heat
storage materials, such as concrete or brick, or phase-change materials. Two design options are ‘isolated thermal storage’ passively
coupled to a fenestration system or solarium/sunspace and ‘collector-storage walls’. A collector-storage wall – known as a Trombe
wall – consists of thermal mass that is placed directly in front of the glazing; however, this system has not gained much acceptance
since it limits the views to the outdoor environment. Direct-gain systems are the most common implementation of thermal storage.
3. Airtight insulated opaque envelope. Such an envelope reduces heat transfer to and from the outdoor environment, but must be
chosen to be appropriate for the local climate. In most climates, this energy efficiency aspect is an essential part of the passive
design. A solar technology that may be employed in conjunction with opaque envelopes is transparent insulation combined with
thermal mass to store solar gains in a wall so as to turn it into an energy-positive element.
4. Daylighting technologies and advanced solar control systems. These technologies provide passive daylight transmission. They include
electrochromic and thermochromic coatings, motorized shading (internal, external) that may be automatically controlled, and
fixed shading devices, particularly for daylighting applications in the workplace. Newer technologies, such as transparent

photovoltaics (PV) panels, can also generate electricity.


360

Components

Passive solar heating systems are generally divided into two categories: direct gain and indirect gain. Four common types of passive
solar systems are shown in Figure 1.
Direct-gain systems have two essential components: near-equatorial facing windows that transmit incident solar radiation
and thermal mass distributed in the interior surfaces of the room to store much of that radiation. Since the direct-gain zone
of a building collects, stores, and releases thermal energy from the sun, it is not only technologically simple but also one of
the most thermodynamically efficient solar systems – it is essentially a live-in solar collector, in which thermal comfort must
be satisfied and very often visual comfort as well with glare reduction measures. Although technologically simple, these
systems require proper integration with the active (heating, ventilation, and air-conditioning (HVAC)) systems to achieve

(a)

DIRECT
GAIN

qsolar

qsolar

TROMBE
WALL

mass

FRESH AIR IN

Shading

device

(roller blind)


HOT AIR

EXHAUSTED


TIM

element

Glass sheet

ENVELOPE SECTION
Gypsum board


FRESH AIR

Concrete

wall

Air gap

TRANSPARENT
INSULATION
(b)

Shading

device

(roller blind)


AIRFLOW COLLECTOR
WINDOW

AIR PASSING

THROUGH WALL

TO TRANSFER

HEAT TO ROOM


TIM

element

Glass sheet
Gypsum board

Concrete


Air gap

Figure 1 (a) Common types of passive systems. (b) Transparent insulation and an option for air circulation in a wall accelerate heat release.


Modeling and Simulation of Passive and Active Solar Thermal Systems

361

high performance. In the case of the workspace such as offices integration with design and operation of the lighting system
is also essential.
Thermal storage is essential in direct-gain systems since it performs two important roles: storing much of the absorbed direct
gains for slow release and maintaining satisfactory thermal comfort conditions by limiting the rise in maximum operative (effective)
room temperature [2].
The key design choices for such a system are type, quantity, and position of thermal mass, as well as the choice of window area
and type. To satisfy thermal comfort requirements, the ratio of peak solar heat gains to thermal mass should not exceed the
maximum room temperature swing; this can be determined using dynamic thermal analysis.
In indirect-gain systems, the thermal storage mass is separated from the main building envelope. Such systems include Trombe
wall (i.e., collector-storage wall) systems, transparent insulation systems, and air heating systems (i.e., airflow windows and solar
collectors) (Figure 1). Various controlled devices may be employed such as motorized reflective shades and controlled inlet/outlet
dampers to control transmission of solar heat gains and the rate of their release from the thermal storage layers (Figure 1(b)).

3.11.2.1

Direct-Gain Modeling

The primary objective in the design of a direct-gain solar building or thermal zone is to achieve high savings in energy consumption
through optimal utilization of passive solar gains, while preventing frequent room overheating above the acceptable comfort limit.
During the thermal analysis stage of a solar building, it is necessary to determine heating loads and room temperature
fluctuations either for design days or with given typical annual weather data. For sizing equipment and components, it is desirable
to evaluate the building response under extreme weather conditions for many design options, each time changing only a few of the
building parameters, until an optimum or acceptable response is obtained. For a solar building that includes direct gain as its main
solar energy utilization mechanism, it is also essential to study the free (passive) response of the building as it enables the designer
to determine the relation between room temperature fluctuation and storage of passive solar gains.
There are two main steps in creating a mathematical model that describes the heat transfer processes in a solar building. First, the
thermal exchanges must be modeled as accurately as is practical; while a high level of precision is desired, too much complexity can
limit the model’s usefulness in analysis and design. Second, an appropriate method of solution must be chosen to determine the
room temperature and auxiliary energy loads. The type of solution may be numerical or analytical, as long as the variables of interest
can be determined. As an optional third step, a method of analyzing the system without simulation can be developed.
The degree of detail and model resolution required during the analysis of a building depends on the stage of the design. For the
early stages of design, a steady-state or an approximate dynamic model is often adequate. However, more detail is required for a
preliminary design, taking into account all objectives of thermal design and the specific characteristics of the system considered.
Modeling the radiant heat exchanges of the zone interior is more important with direct-gain than with indirect-gain systems and
generally requires more modeling detail. In designing direct-gain buildings (i.e., a building with at least one direct-gain room), a key
objective is to store energy in the walls during the daytime for release at night without having uncomfortable temperature swings.
A basic characteristic of passive solar building is the strong convective and conductive coupling between adjacent thermal zones.
This coupling is very important between equatorial-facing direct-gain rooms that receive a significant amount of solar radiation
transmitted through large windows and adjacent rooms that receive very little solar radiation. For example, heat transfer by natural
convection through a doorway connecting a warm direct-gain room or a solarium and a cool north-facing room can be an effective
way of heating the cool room.
The design of direct-gain buildings can be separated into two phases. First involves the determination of room temperature swings
on relatively clear days during the heating season (assuming no active or passive cooling) in order to decide how much storage mass to
include so as to ensure that overheating does not occur frequently. Second, to determine the optimum amount of insulation and
window area and type, the net increase in the mean (daily or monthly) room temperature above the ambient temperature due to the

solar gains is calculated, or auxiliary heating loads are computed until the desired energy savings are achieved.
Periodic conditions are usually assumed (explicitly or implicitly) in dynamic building thermal analysis and load calculations.
Heating or cooling load, that is, the auxiliary heat energy input/removal required to maintain comfort conditions, is usually calculated
for a design day. The peak heating load is used to size heating equipment and the peak cooling load to size cooling equipment.
The following three types of approximations are commonly introduced in mathematical models to facilitate the representation
of building thermal behavior:
1. Linearization of heat transfer coefficients. Convective and radiative heat transfers are nonlinear processes, and the respective heat
transfer coefficients are usually linearized so that equations to derive system energy balance can be solved by direct linear
algebraic techniques and possibly represented by a linear thermal network.
2. Spatial and/or temporal discretization. The equation describing transient heat conduction is a parabolic, diffusion-type partial
differential equation. Thus, when finite-difference methods are used, a conducting medium with significant thermal capacity
such as concrete or brick must be discretized into a number of regions, commonly known as control volumes, which may be
modeled by lumped network elements (thermal resistances and capacitances). Also, time domain discretization is required in
which an appropriate time step is employed. In response factor methods, only time discretization is necessary. For frequency
domain analysis, none of these approximations are required; in periodic models however, the number of harmonics employed
must be kept within reasonable limits.


362

Components

3. Approximations for reduction in model complexity – selecting model resolution. These approximations are employed in order to reduce
the required data input and the number of simultaneous equations to be solved or to enable the derivation of closed-form
analytical solutions. They are by far the most important approximations. Examples include combining radiative and convective
heat transfer coefficients (so-called film coefficients commonly employed in building energy analysis), assuming that many
surfaces are at the same temperature, or considering certain heat exchanges as negligible.
A major aspect of the modeling process considers heat conduction in the building envelope. In most cases relating to heating or
cooling load estimations, energy savings calculations, and thermal comfort analysis, it is generally accepted that one-dimensional
heat conduction may be assumed. Thermal bridges such as those present around corners and at the structure are generally accounted

for in calculating the effective thermal resistance of building envelope elements. However, the thermal storage process may usually
be adequately modeled as a one-dimensional process for insulated buildings.
Direct-gain zone modeling entails certain important requirements in addition to those involved in traditional building
modeling. In particular, there is an increased need to deal with thermal comfort requirements and a need to allow the room
temperature to fluctuate in order to enable storage of direct solar gains in building-integrated exposed thermal mass.
Calculation of peak heating and cooling loads – a major aspect of heating and cooling equipment sizing – needs to take into
account the building thermal storage capacity and dynamic variation of solar radiation and outdoor temperature in order to avoid
oversizing of HVAC systems. For most mild temperate climates, a heat pump will provide an efficient auxiliary heating and cooling
system. Well-insulated buildings with effective shading systems and natural ventilation will have a reduced need for auxiliary cooling.
Similarly, appropriate sizing of the fenestration systems facing the equator will meet most heating requirements on sunny days.
Frequency domain analysis techniques with complex variables are usually employed for steady-periodic analysis of multilayered
walls and zones. They provide a convenient mean for periodic analysis, in which parameters like magnitude and phase angle of
room temperatures and heat flows are obtained.
Generally, materials with significant thermal storage capacity must be modeled, particularly room interior layers. The thermal
properties of major thermal storage materials and a few other materials (for comparison) are given in Table 1. Generally, thermal
mass has high thermal capacitance but low thermal resistance. For example, a 1 m2 concrete block that is 10 cm thick can store
(for 1 °C temperature rise)
Q ¼ cp ρ Vol ΔT ¼ 840 J kg − 1 ˚C Â 2200 kg m − 3 Â 0:10 Â1 m2 Â 1˚C
Q ¼ 184 800 J
By contrast, its thermal resistance is negligible (0.1/1.73 = 0.058 K W−1). The properties of concrete can vary considerably with
density and moisture content.

3.11.2.1.1

Transient heat conduction and steady-periodic (frequency domain) solution

The equation describing heat conduction is a parabolic, diffusion-type partial differential equation. Thus, the use of finite-difference
methods requires the discretization of a conducting medium with significant thermal capacity into a number of regions which are
modeled by lumped elements. Also, time domain discretization is required in which an appropriate time step is employed. In
response factor methods, only time discretization is necessary.

Table 1

Properties of thermal mass and other building materials [3, 4]

Material
Heavyweight concrete
Clay tile
Gypsum
Gas-entrained concrete
Water
Plasterboard
Expanded polystyrene
Timber
Softwood
Hardwood
Plywood
Chipboard
Common brick (full)
Stone
Granite
Limestone
Sandstone
Marble
Screed finish (lightweight)

Mass density
(kg m−3)

Thermal conductivity
(W m−1 K−1)

Specific heat
(J kg−1 K−1)

2243
1121
1602
400
1000
840
25

1.73
0.57
0.73
0.14
0.58
0.16
0.035

840
840
840
1000
4200
950
1400

630
630

530
800
1922

0.13
0.15
0.14
0.15
0.727

1360
1250
1214
1286
840

2600
2180
2000
2500
1200

2.50
1.59
1.30
2.00
0.41

900
720

712
802
840


Modeling and Simulation of Passive and Active Solar Thermal Systems

T2

Convection+
radiation

363

T1
Teo
Solar

Conducted
Room
side
Tr

Reflected
To
Stored
heat

Convection +
radiation

x
Figure 2 Heat exchanges in a wall layer with absorption of solar radiation (To, ambient temperature; Teo, sol-air temperature [3]).

For frequency domain analysis, none of these approximations are required; in periodic models however, the number of
harmonics employed must be kept within reasonable limits. Frequency domain analysis techniques with complex variables are
usually employed for steady-periodic analysis of multilayered walls. They provide a convenient means for periodic analysis, in
which the main parameters of interest are the magnitude and phase angle of room temperatures and heat flows.
First, the frequency domain solution of heat transfer in multilayered walls is determined (see Figure 2). Consider a slab and
assume one-dimensional transient conduction with uniform properties k, ρ, c, then
α

∂T
∂2 T
¼
∂x2
∂t

½1aŠ

where thermal diffusivity α = k/(ρc).
The application of a Laplace transform to eqn [1a] converts it to an ordinary differential equation as follows (s is the Laplace
domain variable):
d2 T
α 2 ¼ sT
½1bŠ
dx
This is an ordinary differential equation which may be solved for T(x) while keeping s as a constant:
rffiffiffiffiffi
s

T fx; sg ¼ c1 eγx þ c2 e − γx ; where γ ¼
α

½2aŠ

Note: the Laplace transform of eqn [1a] assumes the initial condition T{x, t = 0} = 0. This is acceptable as the aim is to derive
steady-periodic (or frequency domain) solutions.
Rewrite eqn [2a] as
T fx; sg ¼ M coshðγxÞ þ N sinhðγxÞ

½2bŠ

Heat flux q΄ is obtained by differentiating eqn [2b]:
q′ ¼ −k

dT
⇒ q′ fx; sg ¼ −M k γ sinhðγxÞ−N sinhðγxÞ
dx

½3Š

At each surface, the following temperatures and heat fluxes are obtained:
T1 f x ¼ 0; s g ¼ M
q′1 fx ¼ 0;sg¼−Nkγ
T2 fx ¼ L; sg¼M cos hðγLÞ þ N sin hðγLÞ
q′2 fx ¼ L; sg¼M k γ sin hðγLÞ þ N k γ cos hðγLÞ

½4Š

The above equations for the conditions at the two surfaces may be expressed in the so-called cascade equation matrix form [4] as

follows (assuming that heat flux q′ is positive into the wall on both sides):
2
3
�
�
�
�
cos
hðγLÞ
sin
hðγLÞ=kγ
T1
5⋅ T 2
¼4
½5Š
kγsin hðγLÞ cos hðγLÞ
−q′2
q1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
two port cascade matrix

The constant k is the thermal conductivity, x is the thickness, and γ is equal to (s/α)1/2, s being the Laplace transform variable and α
being the thermal diffusivity. For admittance calculations, s = jω, where j = √–1 and ω = 2π/P. For diurnal analysis, the period
P = 86 400 s. For a multilayered wall, the cascade matrices for each successive layer are multiplied to get an equivalent wall cascade
matrix that relates conditions at one surface of the wall to those at the other surface, thus eliminating all intermediate nodes with no
approximation required and no discretization:
�
� �
��
� �

��
�
A1 B1
A2 B2
AN BN
T
T1
¼
⋅
⋅⋅⋅⋅
⋅ N
½6Š
C1 D1
C2 D2
CN DN
−q′N
q′2


364

Components

The effective wall cascade matrix is expressed as follows:
�
� �
T1
A
¼
q′1

C

��
�
T
B
⋅ N′
−q N
D

�

The cascade matrix for a simple conductance (per unit area), u, can be shown to be given by

1
0

�
1=u
.
1

½7Š

Usually, the variables of primary interest are the surface temperatures of the room interior. Consider, for example, a wall made
up of an inner (room side) storage mass layer and insulation on the exterior. This can be represented by
2
3 2
3
2

3
�
�
�
�
�
�
T2
T
þ
B
D
B
1
1=u
D
D=u
T1
2
o
o
5⋅ 4
5⋅
4
5
¼4
¼
⋅
½8Š
−q ′2

−q ′2
q′ 1
C D
0 1
C C=uo þ D
|fflfflffl{zfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
mass
cascade
matrix

wall cascade matrix

insulation
and air cascade
matrix

3.11.2.1.1(i) Admittance transfer functions for walls
The above cascade equations for walls may be utilized to obtain frequency domain (admittance) transfer functions for walls that can
be used for steady-periodic analysis or controls and system dynamics studies.
Simple Fourier series models for outside temperature or sol-air temperature and solar radiation are used for steady-periodic
thermal analysis of wall heat flow. Frequency domain transfer functions such as the wall admittance are studied in terms of
magnitude and phase lag and are then used together with Fourier series models for weather variables to determine the
steady-periodic thermal response of walls. The technique is applied to passive solar analysis and design.
Significant insight into the dynamic thermal behavior of walls may be obtained by studying their admittance transfer functions
(magnitude and phase angle) as a function of frequency, thermal properties, and geometry.
Figure 3 shows conceptually how wall response to weather inputs (e.g., T sin(ωt)) may be obtained for one harmonic, and the
time lag between the input and output waves. For inputs with more than one harmonic, the total response may be obtained by
superposition of the response harmonics.

The thermal admittance of a wall is a transfer function parameter useful for analysis of the effects on room temperature of cyclic
variations in weather variables such as solar radiation, outside temperature, and dynamic heat flows under steady-periodic conditions.
There are two transfer functions of primary interest, namely, the self-admittance Ys relating the effect of a heat source at one
surface to the temperature of that surface and the transfer admittance Yt relating the effect of an outside temperature variation to the
resulting heat flow at the inside surface.
These two transfer functions are determined as demonstrated in the following model [5]. The wall in Figure 4 consists of insulation
and thermally nonmassive layers (low thermal capacity) with conductance u per unit area, and a thermally massive layer of thickness L.
The Norton equivalent network for a wall with a specified temperature on one side (such as basement temperature or
sol-air temperature) is obtained from the cascade form of the wall equations which relates temperature and heat flow at one
surface to those at the other surface. The cascade form of the equations is derived by first taking the Laplace transform of the
one-dimensional heat diffusion equation to obtain an ordinary differential equation (as previously described) which can then be
readily solved to relate heat flux and temperature at one surface of a one-dimensional medium to those at the other surface as
follows (based on eqn [5]):

Phase shift
(log)

Input wave
T Sin(wt)

time log = φ /ω
T
MEAN
Q
Q= Y ×T
Output wave
Q × sin (wt + φ )
Ultimate periodic response
TIME

Figure 3 Schematic of temperature and heat flow waves (Y = admittance transfer function with magnitude |Y| and phase angle φ).


Modeling and Simulation of Passive and Active Solar Thermal Systems

WALL SECTION

ELEMENTS
Qeq = Yt × To

L
u

Ti

Ti
YS

Qeq

ROOM
SIDE

To

365

resistance
twoport
heat source

temp source

Insulation

Equivalent
thermal network

Mass

Figure 4 Exterior wall with massive interior layer and equivalent thermal network (for a wall with incident solar radiation, replace To with the sol-air
temperature Teo).

�

T1
q ′1

�

�
¼

D B
C D

��

T2
−q ′2

�

where
D ¼ cos hðγxÞ
sin hðγxÞ
B¼
kγ
C ¼ kγsin hðγxÞ

½9Š

and q′ is assumed to be positive into the slab (on both sides). As explained above, the cascade matrix for a multilayered wall is
obtained by multiplying the cascade matrices for consecutive layers. Usually, the temperatures of interest are either the inside or the
outside temperatures. In this way, wall intermediate layer nodes and their temperatures are eliminated. A linear subnetwork
connected to a network at only two terminals (a port) can be represented by its Norton equivalent, consisting of a heat source
and an admittance connected in parallel between the terminals [5].
The admittance is the subnetwork equivalent admittance as seen from the connection port (the two terminals), and the heat source is
the short-circuited heat flow at the port. Consider, for example, the wall in Figure 4, assumed to be made up of an inner layer of storage
mass of uniform thermal properties and an insulation layer with negligible thermal capacity, also of uniform thermal properties. The
region behind the thermal mass may be represented by equivalent conductance U in series with the outside temperature To (for exterior
walls the sol-air temperature Teo). The conductance U combines the insulation resistance and a film coefficient. The determination of Ys
(called the wall self-admittance) and the equivalent source Qsc produced by the transformation will now be explained. The first step is
obtaining the total cascade matrix by multiplying the cascade matrix for the storage mass layer by that for u (note: u = U/A):
�
� �
��
��
�
Ts
D B

1 1=u
To
¼
½10Š
C D
0 1
−qo
qs
After the multiplication, Ts is temporarily set to 0 (short-circuit) to get the Norton equivalent source as
Qeq ¼ −Yt To
where the transfer admittance Yt is given by
Yt ¼

−A
A cos hðγxÞ sin hðγxÞ
þ
kγ
uo

½11Š

½12aŠ

The transfer admittance has been multiplied by the area A to obtain its total value. To obtain Ys, To is temporarily set to 0, and the
admittance as seen from the interior surface is obtained, yielding (after multiplying by A)
�u
�
o
A
þ kγ tan hðγxÞ

A
Ys ¼
½12bŠ
uo
tan hðγxÞ þ 1
kγA
If there is no thermal mass, the simple equality Ys = –Yt = u0 is obtained. A similar result is derived for windows in eliminating all
nodes exterior to the inner glazing. An important result is obtained for an infinitely thick wall or a wall with no heat loss at the back
(adiabatic surface, or high amount of insulation uo ≈ 0); in this case, Ys is given by
Ys ¼ A k γ tan hðγxÞ
Thick walls have admittance that is close to that given by the above equation. When the penetration depth, given by
sffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi
2k
2α
d¼
¼
ω
cp ρω
is significantly less than the wall thickness, then the wall behaves like a semi-infinite solid.

½12cŠ

½12dŠ


366

Components

The magnitude and phase angle (and time lag/lead) of a transfer function such as Ys and Yt are computed by means of complex

variables.

6

20
Concrete
Softwood

Concrete
Softwood

15

Time lead (hr)

Self-admittance mag. |Ys| W/K

3.11.2.1.1(i)(a) Analysis Substantial insight into wall and building thermal behavior may be gained by studying the magnitude
and phase angle of important transfer functions such as Ys and Yt [6]. The time lead ds of Ys is the time difference between the peak of
a sinusoidal input function, such as solar radiation in the case of the room interior surface, and the resulting peak of the interior
surface temperature Ti. Now, consider the variation of wall thermal admittance with thermal mass thickness L for the fundamental
frequency (one cycle per day, n = 1) for unit wall area. Note that the diurnal (n = 1) frequency is important in the analysis of variables
with a dominant diurnal harmonic such as solar radiation. High frequencies are important in analyzing the effect of varying heat
inputs such as those due to on/off cycling of a furnace.
Compare two walls, one with a concrete interior and the other with a softwood interior. The exterior insulating layer of both
walls has insignificant thermal capacity, and its thermal resistance is 2.5 RSI. The concrete is assumed to have a specific heat capacity
of 800 J kg−1 °C−1, a density of 2200 kg m−3, and a thermal conductivity of 1.7 W m−1 K−1. The softwood has a specific heat capacity
of 1360 J kg−1 °C−1, a density of 630 kg m−3, and a thermal conductivity of 0.13 W m−1 K−1.
The results presented below are specific to this concrete, but they generally indicate the expected trends for concrete, brick, and
masonry-type materials. Note that the thermal conductivity of these materials increases with moisture content and density. Figure 5

shows an extremely important result in steady-periodic analysis of building thermal response – the fact that there exists a certain
wall thermal mass thickness that will reduce room temperature fluctuations most – in this case for L = 0.2 m for concrete,
corresponding to the maximum admittance. Therefore, this is the optimum thermal mass thickness for passive solar design because
the dominant harmonic component of solar radiation is that corresponding to one cycle per day.
As indicated in Figure 6, the magnitude of wall admittance (for mass thickness of 20 cm) increases with frequency (decreases with
period). The magnitude of wall admittance is also higher for concrete than for softwoods. Thus, the inside room temperature
fluctuations are smaller for high-frequency fluctuations in internal heat gains in the case of the concrete wall. For harmonic numbers
higher than about eight – that is, periods less than 3 h – the wall behaves like an infinitely thick solid; in this case, the phase angle is 45°.
The variation of transfer admittance Yt with mass thickness is depicted in Figure 7. In this case, the magnitude of Yt decreases with
thickness, and therefore, fluctuations in the sol-air temperature are significantly modulated as they are transmitted to the room interior.
This is a well-known phenomenon, efficiently employed in traditional architecture in adobe buildings. The time lag of the heat gains
transmitted (q = Yt  Teo) into the interior is the time lag of Yt. This time lag increases to about 7.5 h for a mass thickness of 30 cm.

10

4

3

5

0

5

0

0.1

0.2

0.3

0.4

0.5

2

0.6

0

0.1

Mass thickness (m)

0.2

0.3

0.4

0.5

0.6

Mass thickness (m)

Figure 5 Variation in self-admittance and its time lead with mass thickness and material type for the one cycle per day harmonic.

55
Concrete
Softwood

Phase angle (degrees)

Self-admittance mag. |Ys| W/K

140

105

70

35

0

0

10
20
Harmonic number

30

Concrete
Softwood

52.5
50
47.5
45
42.5
40

Figure 6 Variation in self-admittance and its phase angle with harmonic number n.

0

10
20
Harmonic number

30


367

20
Concrete
Softwood

0.3

0.2

0.1

0.1

Transfer admittance time lag (h)

Transfer admittance mag. |Yt| W/K

Modeling and Simulation of Passive and Active Solar Thermal Systems

Concrete
Softwood

10

0

−10

−20

0.2

0.1
0.2
Mass thickness (m)

Mass thickness (m)
Figure 7 Variation in transfer admittance and its time lag with mass thickness.

400
Concrete

Softwood

Concrete
Softwood

Phase angle (degrees)

Transfer admittance mag. |Yt| W/K

0.4

0.3

0.2

0.1

0

0

8
16
Harmonic number n

24

300

200

100

0

0

10
Harmonic number n

20

Figure 8 Variation in transfer admittance magnitude with harmonic number n.

The time delay in the transmission of a heat wave through a wall is another positive effect of thermal mass in addition to the
attenuation of temperature swings. Thus, the peak heat gains through the structure coincide with cooler outside conditions when
natural ventilation may be employed to reduce total instantaneous cooling loads.
Figure 8 shows that the magnitude of the transfer admittance decreases relatively fast with increasing harmonic number n and
decreasing period. Thus, the heat gain fluctuations transmitted into the room as a result of sol-air temperature fluctuations are
significantly reduced at high frequencies. For example, a temperature fluctuation (amplitude of wave) of 10 °C in sol-air tempera­
ture will result in a heat gain fluctuation of about: 0.1W °C−1 Â 10 °C = 1W per square meter of wall.

3.11.2.1.2

Building transient response analysis

Transient thermal analysis of walls or zones may be performed with the following objectives:
1. Peak heating/cooling load calculations.
2. Calculation of dynamic temperature variation within walls, including solar effects, room temperature swings, and condensation on
wall interior surfaces; two-dimensional steady-state temperature profiles in walls (e.g., for investigation of thermal bridge effects).

For a multilayered wall, an energy balance is applied at each node at regular time intervals to obtain the temperature of the nodes as
a function of time. These equations may be solved with the implicit method as a set of simultaneous equations or with the explicit
method which involves a forward progression in time from a set of initial conditions. Mixed differencing schemes are also often
used in building simulation.
Here we consider the finite-difference thermal network approach. In this approach, each wall layer is discretized (divided) into a
number of sublayers (regions). Each region is represented by a node and is assumed to be isothermal. Each node (i) has a thermal
capacitance (Ci) associated with it and resistances connecting it to adjacent nodes.
Wall transient thermal response analysis with finite-difference techniques may generally provide a more accurate estimation of
temperatures and heat flows owing to the capability to model nonlinear effects such as convection and radiation. One disadvantage
is that the initial conditions are usually unknown. Thus, the simulation is repeated until a steady-periodic response is obtained.
In the transient one-dimensional finite-difference method, each wall layer is represented by one or more sublayers (regions) –
also known as control volumes. Each region is represented by a central node with a thermal capacitance C connected to two thermal


368

Components

L
ho

hi
43

TR

qsolar
Concrete
modeled
by two layers

To

2 1

Transparent insulation
Gap
Concrete
(thickness L)

τα qsolar
Ri

4

Rc2

Rc1

3

1 Rgap

Rins

1/ho

C2

C3

TR

Rb

2

To

Figure 9 Wall with transparent insulation and its thermal network.

resistances, each equal to half the R-value of the layer (Figure 9). Thus, the finite-difference thermal network model for the wall
consists of two capacitances for the concrete thermal capacity and interconnecting thermal resistances. The energy balance for the
thermal network is as follows:
�
X Tj ; p −Ti ; p �
Δt
Ti ; p þ 1 ¼
qi þ
þ Ti ; p
½13Š
Rt ; j
Ci
Subscript i indicates the node for which the energy balance is written, and j indicates all nodes connected to node i, while p is the
time interval; qi represents a heat source at node i, such as solar radiation.
The time step is selected on the basis of the following condition for numerical stability:
0
1
B Ci C
C

Δt ≤ minB
@P 1 A
j
Ri ; j

½14Š

for all nodes i.
The explicit finite-difference method is particularly suitable for the modeling of nonlinear heat diffusion problems such as heat
transfer through the building envelope. It can easily accommodate nonlinear heat transfer coefficients and control actions.
τα qsolar
1

Rins

Rgap

1/ho

1

Ra

To

Ri

TR

Figure 10 Simplified thermal network for Figure 9.

4

Rc2

3

Rc1

C3

Teq

Rb

2

C2

1

Ra

Teq


Modeling and Simulation of Passive and Active Solar Thermal Systems

L = 0.20 m

A = 1⋅m2

Concrete
properties:

c = 800⋅

hi = 10

τα = 0.7

degC

k = 1.7⋅

watt
m⋅degC

ρ = 2200⋅

watt

ho = 20

m2⋅degC

kg
m3

density

conductivity

Ra =

R gap = 0.3 m2

watt

R gap + R ins +

R c1 =

joule

watt
m2⋅degC

effective transmittance−absorptance of transparent insulation
system and concrete face)

R ins= 0.5⋅m2

R c=

concrete thickness and face area (perpendicular to heat flow)

kg⋅degC
specific heat

Film coefficients:

369

1

h
o

R a = 0.85

A
L

degC

transparent insulation and gap resistances

watt

degC

resistance from outside to concrete face

watt

R b=

total concrete resistance

k⋅ A
Rc

R c1 = 0.059

2

L
C2 = ρ⋅c⋅ ⋅A
2

degC
R c2 =

watt

C2 = 1.76× 105

Rc

Rc

R b = 0.029

4
Ri =

4

joule

C3 = C2

degC

degC
watt

1
A ⋅ hi

R i = 0.1

degC
watt

thermal capacitance
for each layer

Stability test
TS = ⎛⎜

C2
1

+
⎜
⎝ R a+ R b R c1

(

⎞
⎟
+
⎟
R c1 R c2+ R i ⎠
C3

1

1

1

The time step Dt should be lower than
the minimum of the two values in the
vector TS

)

TS = 9.704 × 103 7.118 × 103 s
Δtcritical = min(TS)
i = 0., 1.. 96

Δtcritical = 7.118× 103 s
number of time steps

Δt = 1800s
t = 0 s, 1800 s.. 172 800 s

Figure 11 Computations of finite-difference model in MathCad [6].

Figure 9 shows the thermal network for a wall with transparent insulation. The wall consists of an exterior layer of transparent
insulation, an air cavity, and a thermal storage layer of concrete. The thermal circuit represented in Figure 9 can be simplified by
using the equivalent sol-air temperature and equivalent resistance. The simplification and the resulting thermal network are shown
in Figure 10.
The specific calculations for the above model developed in MathCad [6] are shown in Figures 11 and 12, and the resulting
temperature variation of a south-facing wall for two winter days with high solar gains is shown in Figure 13. For summer
simulations, a shutter is assumed to be closed in the air cavity, reflecting outwards 90% of transmitted solar gains.

3.11.2.1.3

Simplified analytical direct-gain room model and solution (passive)

An analytical model can provide insight into passive solar design and quick comparison of design options. A simple analytical
model based on the admittance method is presented below. The room is shown schematically in Figure 14. The walls (all of which
are external) are assumed to be made up of an inner lining of storage mass material of uniform thermal transport properties and
outer insulating layers with negligible thermal storage capacity. Walls with storage mass are assumed to be at the same surface
temperature, and they are thus treated as one exterior wall, which is modeled as a two-port distributed element as previously


370

Components

π
rad
w=2
⋅
86 400 s

Assume

To(t) = ⎛⎜5⋅cos ⎜⎛ w ⋅ t + 3⋅

⎝

⎝

π

frequency based on period of one day

⎞
⎞
⎟ − 10⎟ ⋅degC
⎠

outside temperature

4⎠

incident solar radiation
modeled as half-sinusoid

f(t) = 500⋅cos [ w ⋅( t − 43 200 s)]⋅watt

qsolar(t) = if(f(t)>0⋅watt, f(t), 0⋅watt)

Teq(t) = To(t) +qsolar(t)⋅τα⋅Ra

equivalent 'sol-air' temperature at node 1 (concrete surface)

TR = 22⋅degC

room temperature

Initial estimates of temperatures

⎛ T20 ⎞ ⎛ 10⎞
⎜
⎟=
⋅degC
⎜ T30 ⎟ ⎝⎜ 10⎠⎟
⎝
⎠

Simulation for nodes with capacitances:

⎛ T2i+1
⎜
⎜ T3i+1
⎝

⎡⎢ Δt ⎛ Teq (i⋅Δt ) − T2i T3i − T2i ⎞
⎤
⋅⎜
+
⎟ + T2i
⎥

⎞ ⎢C2 ⎝
⎥
Rc1 ⎠
Ra + Rb
⎟=
⎢
⎥
⎟
T R − T3i ⎞
T2 − T3i
⎥
⎠ ⎢ Δt ⋅ ⎛⎜ i
+
⎟ + T3i ⎥
⎢ C3
R
+
R
R
c1
i
c2
⎠
⎦
⎣
⎝

Finite-difference simulation

(i is present time, and i + 1 the next time step)

Calculaiton of intermediate temperatures

T4i = TR+Ri⋅

T3i − T R
Rc2 + Ri

T1i = T2i + Rb ⋅

Teq (i⋅Δt ) − T2i
Ra + R b

Figure 12 Computations of finite-difference model in MathCad.

described in section 3.11.2.1.1, and its internal surface is assumed to be uniformly irradiated by solar radiation. The important
temperatures are also shown in Figure 14: the storage mass internal surface temperature Ts, the (double-glazed) window internal
glazing temperature Twi, the room air temperature Tai, and the outdoor air temperature To.
The thermal network, shown in Figure 15, contains several conductances; the equivalent conductance Uinf due to infiltration
heat loss links Tai and To. Convective conductances Usa and Uaw link Tai with Ts and Twi, respectively, while the radiative
conductance Usw links Ts and Twi. Combined radiative and convective conductances Uwi and Uwo thermally link Twi with Two
and To, respectively. The components outside the thermal mass have been replaced in Figure 16 by an equivalent conductance Uo in
series with the sol-air temperature Teo associated with the external wall surface. The conductance Uo combines the insulation
resistance Rins and the external surface radiative–convective conductance (equal to Aoho). While the model as just described assumes
a uniformly distributed storage mass, by a small extension it can model a situation where some of the walls (normally the ceiling)
are nonmassive, by modeling them as conductances in parallel with Uinf. Therefore, Uinf is replaced by Uinf + Unonmassive.
The network also contains several heat sources. The major source Ss is the solar radiation transmitted through the window and
absorbed by the storage mass. Sources Swi and Swo are the rates at which solar radiation is absorbed (uniformly) by the inner and
outer glazings, respectively.
A delta to star transformation permits the network in Figure 15 to be reduced to that in Figure 16. This transformation has permitted

a ‘natural’ representative room temperature Tei to be obtained at an important node; Tei is a weighted average of Tai, Twi, and Ts:
Tei ¼
where

Uew Twi þ Uea Tai þ Ues Ts
Uew þ Uea þ Ues

�
�
Uaw
þ Uaw
Uew ¼ Usw 1 þ
Usa
�
�
Usa
þ Usa
Uea ¼ Uaw 1 þ
Usw
�
�
Usa
Ues ¼ Usw 1 þ
þ Usa
Usw

½15aŠ

½15bŠ
½15cŠ

½15dŠ


Modeling and Simulation of Passive and Active Solar Thermal Systems

−6
−8
−10
−12
−14
−16

800

Solar radiation incident (W)

Outside temperature (°C)

−4

0

10

20 30 40
Time (hour)

600
400
200

0

50

10

20 30 40
Time (hour)

50

150

50

Heat flow into room (W)

40

Temperature (°C)

0

30

20

10

30

T1
T2
T3
T4

40
Time (hour)

100

50

0

50

30
40
Time (hour)

50

Figure 13 Weather conditions and temperature variation under clear winter conditions for a wall with transparent insulation.

To

Twi

Ts

Two
Tai

INSULATION

STORAGE
MASS
Figure 14 Direct-gain room with uniformly distributed thermal storage mass.

Uinf

Tai
Uaw

Usa
Uo
Teo

Ts
STORAGE
MASS

Tw
Ss

Usw

Swi

Figure 15 Thermal network for room with wall model by distributed RC element.

Uwi
Swo

Two

Uwo
To

371


372

Components

Uinf

Tai

Uo
Teo

Uea

Ues

Ts

Uew

Twi

Uwi

Two

Uwo

Tei

STORAGE
MASS

Swi

Ss

Swo

To

Figure 16 Thermal network after delta-to-star transformation.

Ts
Q T,s

Tei

Ywall

Ss

Ues

Sei

Ueo

To

Figure 17 Simplified thermal network.

The temperature Tei can serve as a representative ‘sensed’ temperature because it should closely follow the effective temperature that
would be felt by an occupant who senses, through radiant exchange, the temperature of the room surfaces, and through convective
exchange, the room air temperature.
More simplifications are applied to obtain the simplified network in Figure 17. The first transforms the minor sources Swi and
Swo to the equivalent source Sei given by
Sei ¼

Uew ½Swi ðhi þ ho Þ þ Swo hi Š
hi ho Aw þ ðhi þ ho ÞUew

½16aŠ

Sei ¼

Uew ½ Swi ðhi þ ho Þ þ Swo hi Š
½ hi ho Aw þ ðhi þ ho ÞUew Š

½16bŠ

Source Sei represents the portion of solar radiation, which is absorbed in the glazings and then transferred to the room interior by
thermal radiation or convection. After Sei has been obtained, the equivalent conductance Ueo coupling Tei and To can be determined as
Ueo ¼ �

1
1
�þ�
�
1
1
1
1
1
þ
þ
þ
Uea Uinf
Uew Uwi Uwo

½17Š

Ueo ¼ �

1
1
�þ�
�
1

1
1
1
1
þ
þ
þ
Uew Uwi Uwo
Uea Uinf

½18Š

(If there is no mass on the ceiling, the ceiling is modeled as a pure conductance in parallel with Uinf, and its conductance is added to
Uinf in eqn [18]). Both Sei and Ss are proportional to the solar irradiance on the window, so that Sei is expected to be proportional to
Ss. It was found that Sei is typically about 4% of the magnitude of Ss (for standard double glazing and midlatitudes in winter). Sei can
be transformed to node s and added to Ss. Thus, Ss is multiplied by the appropriate transformation factor so as to include the effect
of Sei; this factor is equal to [1 + ((Sei / Ss)Ues /(Ues + Ueo))].
The final simplification is replacement of the wall by its Norton equivalent, which is determined as explained above. The variable
of interest here is the interior surface temperature Ts, unimportant nodes are eliminated by this transformation, and the resulting
network consists of only two-terminal components. These two simplifications yield the network in Figure 17. The Norton
equivalent consists of Ywall (called the wall self-admittance in the previous section) and the equivalent source QT,s given by:
QT ; s ¼ −Yseo Teo

½19Š

As
�
As cos hðγxÞ sin hðγxÞ
þ
kγ

Uo

½20Š

where the transfer admittance Yseo is given by
Yseo ¼ − �

The wall self-admittance is given by

Ywall

where E = tan h(y).

��
�
�
Uo
þ kγE
As
As
�
�
¼ ��
Uo
Eþ1
kγAs

½21Š

Modeling and Simulation of Passive and Active Solar Thermal Systems

373

3.11.2.1.3(i) Important design approximation
For rooms with well-insulated walls and high mass (more than 10 cm of concrete or equivalent), the approximate wall
self-admittance can be calculated with the following equation:
Ywall ≈ A kγ tan hðγxÞ

½22aŠ

For very thick mass (more than 25 cm of concrete or equivalent), the wall may be approximated to a semi-infinite solid with
admittance equal to
Ywall; semi-inf ≈ A kγ

½22bŠ

In both of the above cases, the transfer admittance (for all frequencies apart from the mean) is negligible.
The solutions for Ts and Tei are readily obtained from a heat balance at the two nodes (note that Sei, a small source, is not
explicitly included in the heat balance as it has been transformed to node s and added to Ss). The storage mass surface temperature is
found to be given by
Ts ðωÞ ¼

Ss þ QT ; s þ UL To
Ys

½23aŠ

where Ys is the total room admittance as seen from the port formed by the reference and the Ts node, given by
Ys ¼ Ywall þ UL

½23bŠ

1
1
1
þ
Ues Ueo

½23cŠ

where
UL ¼

The conductance UL represents the total loss conductance between the storage mass interior surface and the ambient temperature To.
The representative room temperature (expected to be close to the operative room temperature [3]) can be shown to be as follows:
Tei ðωÞ ¼ ðTF Þ eis ðSs þ QT ; s Þ þ

ðUeo To Þ
Yei

½23dŠ

where
ðTF Þ eis ¼ �

1
�
Ywall
þ 1 Ueo

UL

½23eŠ

Ywall Ues
Ywall þ Ues

½23f Š

and
Yei ¼ Ueo þ

The transfer function (TF)eis determines the effect of the primary source Ss, while Yei is the total room admittance as seen from the
port formed by the Tei node and reference node.
3.11.2.1.3(ii) Source models
Steady-state periodic conditions are assumed in computing the room temperature swing (difference between maximum and
minimum) in Tei or Ts over a particular day. The period is assumed to be 1 day, implying that the day in question has been
preceded by identical days. Although deviations from the 1-day periodic assumption (e.g., previous day overcast and current day
clear) would cause errors in the ‘mean’ value of Tei for the day considered, the variation of the waveform of Tei about its mean should
not be affected, and therefore, the difference between its maximum and minimum points should be close to the actual swing.
Another option would have been to include the effect of the previous day, that is, to assume a 2-day periodicity; the present method
can be readily generalized to include such a model. Note that because of the 1-day periodic assumption, weather input is needed
only for the particular day in question.
Fourier series models are required for the sources Ss, Teo, and To, in order to obtain the time domain solution for Ts and Tei under
periodic conditions. For equatorial-facing windows (e.g., south-facing in the northern hemisphere), the primary source Ss can be
closely modeled over the daytime as a half sinusoid. The constraints on the half sinusoid are chosen to satisfy two conditions: first,
that it have an integrated energy equal to the actual solar radiation Ha absorbed in the room over the day, and second, that it must
start at sunrise and terminate at sunset. Thus, Ss for a single day is defined by
Ss ¼ 0;

forj t j ≤ ts

and
�
Ss ¼

� �
�
πt
π Ha
cos
;
2ts
ts

for jt j > ts

½24aŠ


374

Components

where t is the time measured from the solar noon (t is negative in the morning, positive in the afternoon), and ts is the value of t at
sunset. This expression for Ss can be approximated by the following truncated Fourier series:
X
Ss ¼ qo þ
qn cosðωn tÞ
½24bŠ

n

where
ωn ¼

2πn
Ha
; qo ¼
and qn ¼ Ha f n
td
td

½24cŠ

and fn being given by
�
�
�
�
td −4ts n
td þ 4ts n
sin π
sin π
2td
2td
fn ¼
þ
ðtd þ 4ts nÞ
ðtd −4ts nÞ

½24dŠ

As can be seen, the relative magnitude of the harmonics is determined completely by the day length 2ts. The number of harmonics
necessary to model Ss was found to increase with a decrease in the day length because of the more abrupt increase in the absorbed
radiation at sunrise and its more rapid decrease at sunset. For windows that are not south-facing, the Fourier coefficients can be
determined by direct numerical integration of the instantaneous absorbed irradiation. Various methods can be used for determining
the total daily irradiation absorbed Ha. Either real weather may be employed, which is cumbersome, or instead the daily clearness
index may be used to generate values of the hourly clearness index.
The variation of the ambient temperature To was modeled by a single sinusoid, with maximum at 3.00 p.m. and minimum at
3.00 a.m. (solar times). Thus, if Tom is the daily average of To, and ΔTo the daily range of To then
�
� �
π�
ΔTo
To ¼ Tom þ
cos ω1 t −
½25Š
2
4
The model does not distinguish between the differing orientations of the exterior wall surfaces, and hence, it is not possible
to accurately model the effect of fluctuations in the solar radiation absorbed by the wall exterior surfaces. A sensitivity
analysis showed that the effect of this absorbed solar radiation on the temperature swing is small, provided there is at least a
medium amount of insulation (about 5 RSI) behind the mass. Thus, the sol-air temperature Teo can be modeled as
equivalent to To.
3.11.2.1.3(iii) Periodic solution
The time domain solution for the representative temperature Tei is obtained after substituting the source models in eqn [26] and
evaluating magnitudes and phase angles for the complex transfer functions. Since the variable of interest here is fluctuations, the
mean source terms can be ignored. The following solution is obtained for the variation of Tei about its mean Teim:
�
Tei ðtÞ−Teim ¼

�
N
X
�
ΔTo Yseo1 ðTF Þ eis1
ΔTo Ueo
π�
cos ω1 t−φei1 4 þ
þ
ðTFÞ eisn qn cosðωn t þ φeisn Þ
π
2Yei1
2 cosðω1 t þ φeis1 þ φseo1 4 Þ n ¼ 1

½26Š

where (TF)eisn and φeisn are the magnitude and phase, respectively, of (TF)eis for ω = ωn. Similar definitions apply to Yei1 and φei1, and
Yseo1 and φseo1. The variation in Ts is given by
�
Ts ðtÞ−Tsm ¼

� �
N
�
X
ΔTeo Yseo1
qn
ΔTo UL
cos ω1 −φs1 − ðπ = 4 Þ þ

þ
−
ðπ
=
4
Þ
2Ys1
ð2Ys1 Þcosðω1 þ φs1 þ φseo1
Þ n ¼ 1 Ysn cosðωn t þ φsn Þ

½27Š

The swings in Ts and Tei can be determined by differentiating the appropriate equation, finding the two zeroes corresponding to the
times at which the maximum and minimum temperatures occur, and then substituting these times back into the original equation.
3.11.2.1.3(iv) Approximate design method for temperature swings
For a well-insulated room with massive walls, the maximum temperature swings can be estimated with the following approxima­
tion that adjusts the response to the first (fundamental) harmonic to include higher harmonic effects:
Temperature swing ¼ 2Fh

jq1 j
jYwall j

½28Š

where the magnitude of the first harmonic of solar gains |q1| is estimated from eqn [24a].
Fh = 1.12 is a factor to approximate for the effect of higher harmonics, and |Ywall| may be estimated by
�
j Ywall j ¼ j Akγ j j tanhðγxÞ j ¼ A k

2π

Pα

�1 = 2 �

cosh X−cos X
cosh X þ cos X

�1 = 2

½29aŠ


Modeling and Simulation of Passive and Active Solar Thermal Systems

with

�
X ¼ 2L

π
ðPαÞ

�1 = 2

The phase angle of Ywall is given by
Phase ðYwall Þ ¼ 45˚ þ tan − 1

�

375

½29bŠ

sin X
sinh X

�
½29cŠ

Note that the first term (45°) in eqn [29c] corresponds to the phase angle for kγ, and it is equivalent to a time lead of 3 h (15° are
equivalent to 1 h for a period of 24 h).
3.11.2.1.3(v) Detailed frequency domain zone model and building transfer functions
Building heat exchanges may be represented by a thermal network, and transfer functions are obtained by performing an energy
balance at all nodes in the Laplace domain. Both lumped and distributed elements can be considered with this approach. Simple
models, as in the previous section which do not represent in detail infrared radiation heat exchanges between room interior surfaces,
can usually be solved analytically.
The thermal network model for a typical room over a basement (Figure 18), with one window and convective auxiliary heating,
is depicted in Figure 19. The thermally massive walls are modeled by a two-port distributed element, while the room air and
lightweight room contents are modeled by a lumped thermal capacitance. Although this capacitance has no effect on load
calculations because of the relatively low frequencies involved, it is important to include it for short-term (high frequency) control
studies. Each two-port element represents the equivalent two-port for each wall, obtained after multiplying the cascade matrices for
each massive and nonmassive layer. The resistances connecting node 1 (room air) to the interior surfaces represent convective
conductances given by
U1j ¼ Að j Þ Â hcð j Þ

½30aŠ

The radiation conductances interconnecting room interior surface nodes 2–8 are given by
Uij ¼ Að i Þσ4Tm 3 F Ã ij

½30bŠ

is a linearization factor that is based on an estimated mean temperature, Tm. The
where σ is the Stefan–Boltzman constant and
radiation exchange factors F*(i,j) between the pair of surfaces under consideration (i and j) are determined from the radiation view
factors, Fij (also denoted by F(i,j)) and the radiative properties of the room surfaces as follows:
4Tm3

F Ã ði; jÞ ¼

mði; jÞεi εj
ρi

½30cŠ

where [m] = [M]−1; the elements of matrix [M] are given by M(I, j) – ρiF(i,j), with I(i, j) = 1 if i = j; otherwise, I(i,j) = 0 (identity matrix).
The energy balances at the room interior nodes for both models are readily obtained after replacing each wall with its Norton
equivalent subnetwork, which consists of an equivalent heat source Qsc and a self-admittance Yeq, thereby eliminating all exterior

T Sensor
Tai
Room
under
consideration
Floor mass

PI
Control

S

C
R

Coil
Fan

Tb
Basement

Figure 18 Schematic of one zone (heated by convective heating) over a basement.


376

Components

8

ELEMENTS
resistance
twoport

2

3

7

6

4

heat source

floor

temp source

5

1 = room air

T0

qaux
3

1

2

Te03

To
4

Te04

8
Teo8

5
Te05

7
6
TBASEMENT

Te06
REF TEMP.
Figure 19 Detailed thermal network model of zone in Figure 18 (node 1, room air; nodes 2–8, room interior surfaces).

nodes without discretizing the massive elements. The equivalent source Qsc is equal to the wall transfer admittance times an external
specified temperature. For the floor with self-admittance Yfs and transfer admittance Yft, Qsc = –TbYft (the negative sign is because of
the sign convention used).
The energy balance for the model (with summations ΣUiJ over J = 1–8) is as follows:
3 2
3
2
32
Q1
T1
sCa þ ΣU1J þ Uinf −U12
−U13
−U14
−U15
−U16
−U17
−U18
6

7
6
6 −U12
76 T2 7 6 Q2 7
Y2s þ ΣU2J −U23
−U24
−U25
−U26
−U27
−U28
7
6
76
6 −U13
7 T3 7 6 Q3 7
−U23
Y
þ
ΣU3J
−U34
−U35
−U36
−U37
−U38
3s
7 6
7
6
76
6

7
6
7
6 −U14
7
−U46
−U47
−U48
−U24
−U34
Y4s þ ΣU4J −U45
6
76 T4 7 ¼ 6 Q4 7
6 −U15
76 T5 7 6 Q5 7
−U25
−U35
−U45
Y5s þ ΣU5J −U56
−U57
−U58
6
76
7 6
7
6 −U16
76 T 7 6 Q 7
−U26
−U36
−U46

−U56
Y6s þ ΣU6J −U67
−U68
6
76 6 7 6 6 7
4 −U17
54 T 5 4 Q 5
−U27
−U37
−U47
−U57
−U67
Y7s þ ΣU7J −U78
7
7
−U18
−U28
−U38
−U48
−U58
−U68
−U78
Y8s þ ΣU8J
T8
Q8
½31aŠ
or
½ Y ŠN  N f T gN ¼ f Q gN
where [Y] is the admittance matrix, {T} is the temperatures vector, and {Q} is the source vector. The solution for {T} in the
frequency domain is obtained by

½ZŠ fQg ¼ fTg
½31bŠ
where [Z] = [Y]−1.
The elements of the admittance matrix may be obtained by inspection from the topology of the network. The diagonal entry
Y(i,i) is equal to the sum of the component admittances connected to node i. Off-diagonal entry Y(i,j) is the sum of component
admittances/conductances connected between nodes i and j, multiplied by –1. The heat source vector element Q(i) is the sum of the
heat sources connected to node j (positive if directed to the node). As can be seen, for thermal networks the admittance matrix has
certain important characteristics: (1) it is symmetric, with all off-diagonal elements Y(i,j) being real and equal to – (conductance
Uij), and (2) all capacitances and all self-admittances appear in the diagonal entries, which are consequently complex. The transfer


Modeling and Simulation of Passive and Active Solar Thermal Systems

377

functions of interest are the elements of the inverse of [Y], that is, the impedance transfer functions Z(i,j). The temperature of node I
for each frequency is given by
N
X
TðIÞ ¼
Z ðI; JÞ QðJÞ
½31cŠ
J¼1

For room air temperature, I is set equal to 1 in the above equation, which is determined for each frequency (harmonic) of interest.
The functions Z(I,J) are determined at specific frequencies (s = jωn) by inverting the admittance matrix [Y]. The operative tempera­
ture, Te, is a scalar function of nodal temperatures {T(1) … T(8)}.
Building transfer functions generally provide the response of interest – heat flow or temperature for unit heat input or unit
temperature change at a node in the thermal network. The most important transfer function required in the present method is the
impedance transfer function:

Z ðI; JÞ ¼

TðIÞ
QðJÞ

½32Š

which represents the temperature change for node I due to unit heat input at node J for a given frequency. Thus, for heat input Q(J)
the room temperature change ΔT(1) (1 = room air node) is equal to
ΔTð1Þ ¼ Zð1; JÞ Â QðJÞ

½33Š

It is often useful to determine a transfer function not only for individual room temperatures but also for an effective room
temperature such as the operative temperature [3]. The operative temperature is defined as the uniform temperature of an enclosure
in which an occupant would exchange the same amount of heat by radiation plus convection as in the actual nonuniform
environment. The operative temperature is given by
Te ¼

hr Tmr þ hc Tai
hr þ hc

½34aŠ

where Tai is the air temperature, Tmr is the mean radiant temperature, and hr and hc are radiative and convective coefficients,
respectively, for a person or object (sensor). The operative temperature transfer functions X(I) are given by
XðIÞ ¼

Te
QðIÞ

½34bŠ

and represent the effect of a source Q(I) acting at node I on the operative temperature.
3.11.2.1.3(vi) Analysis of building transfer functions
Substantial insight into building thermal behavior may be obtained by studying the magnitude and phase angle of the important
transfer functions. Consider, for example, the transfer functions Z11 and Z17 in the detailed model (Z11 and Z12, respectively, in
the simple model); these represent the effects of heat sources at node 1 (room air) and 7 (floor), respectively, on the temperature of
node 1 (in both cases all other sources set to zero):
Z11ðsÞ ¼

T1ðsÞ
Q1ðsÞ

½35aŠ

Z17ðsÞ ¼

T1ðsÞ
Q1ðsÞ

½35bŠ

The magnitude of Z17(jω) may be used to determine the approximate room temperature swings due to solar radiation absorbed at
the floor surface as follows.
If S7 represents solar radiation absorbed at the floor interior surface and 3S7(jω1)3 represents the magnitude of its
fundamental harmonic, the approximate temperature swing amplitude is given by 3Z17(jω1)3 Â 3S7(jω1)3. Perhaps a more
significant result is the time delay between the peak of S(t) (noon for south-facing windows) and the resulting peak of the
room temperature; this is approximately equal to φ17/ω1 (seconds), where φ17 is the phase angle of Z17 (φ17 = tan−1
Im(Z17)/Re(Z17)).

3.11.2.1.3(vi)(a) Results The room considered in the example has dimensions 7.3 m wide by 2.4 m high and the north–south
depth is 6.7 m. The south-facing double-glazed window area is 11.1 m2. The thermal mass on the floor is 4 cm-thick concrete with
thermal conductivity equal to 1.8 W m−2 K−1, density 2242 kg m−3, and specific heat capacity 840 J kg−1 K−1. The interior lining on
the vertical walls and on the ceiling is a 13 mm-thick gypsum board. The insulation is 3.5 RSI on the vertical walls, 7.4 RSI on the
ceiling, and 1 RSI on the floor, which connects to a basement.
Figures 20 and 21 depict the magnitude and phase variation (Bode plots) for Z11 and Z17, respectively, in the case of a carpeted
floor and concrete floor, respectively, showing actual discrete frequency responses and fitted third-order transfer functions. Figure 5
shows the Nyquist plot, that is, imaginary versus real components, for Z11 of the room with a concrete floor and for Z17 of the
room with the carpeted floor. Figure 20 shows that the room response can be approximately separated into a short-term dynamics


378

Components
BODE PLOT

Z11 CARPET FLOOR

M = 2, N = 3

MAGNITUDE RATIO (MR)
10

PHASE LAG (PL) (DEGREES)

0
−20

1

−40
−60

0.1

−80
0.01
0.1

1

10

−100
1000

100

FREQUENCY (CYCLES PER DAY)
MR-FITTED FUNCTION

MR-EXACT

PL-FITTED FUNCTION

PL-EXACT

Figure 20 Transfer function plots (magnitude and phase) for Z11 (fitted third-order transfer functions also shown).

Z17 CONCRETE FLOOR

MAGNITUDE RATIO (MR)
10

PHASE LAG (PL) (DEGREES)
0

1

−20

0.1

−40
−60

0.01

−80

1.000 × 10−3

−100

1.000 × 10−4

−120

1.000 × 10−5
1.000 × 10−6

0.1

−140
1

10

100

1000

−160
10000

FREQUENCY (CYCLES PER DAY)

MR FOR FITTED FUNC.

MR Exact

PL FOR FITTED FUNC.

PL

Figure 21 Transfer function plots (magnitude and phase) for Z17 (fitted third-order transfer functions also shown).

high-frequency range and a low-frequency range; in the high-frequency short-term dynamics region, the room air thermal
capacitance is significant, and the difference between Z11 for the concrete floor and the carpeted one is small in both phase and
magnitude, that is, the effect of thermal mass is minimal in this region. The separation between short-term and long-term building
thermal dynamics begins at frequencies of approximately 35 cycles per day or periods of 41 min. The separation is also indicated in

Figure 5(a) by the Nyquist plot of Z11; the small cusp is associated with the short-term dynamics, which is the frequency range
within which the room air (and furniture, etc.) thermal capacity is important. Simulations by the authors with different construc­
tions showed similar results for Z11, which represents the effects of convective heat gains or losses. Short-term dynamics are
particularly important for feedback control studies. For lower frequencies such as one cycle per day, the magnitude and phase of Z11
and Z17 is a strong function of room construction, and there is a significant difference between the response of the massive
(concrete) construction and the nonmassive (carpet) one.
An example of a third-order fitted function for Z17 obtained with the above technique is given in Figure 21. As can be seen, this
third-order fit is good in both magnitude and phase, the error being typically less than 2%. The fitted Laplace transfer function for Z17 is
Z17 ðsÞ ¼

0:008 04 þ 10:5 s þ 55:2 s2 −10:5 s3
1 þ 4:01 Â 104 s þ 6:64 Â 107 s2 þ 2:63 Â 109 s3

3.11.2.1.3(vii) Heating/cooling load and room temperature calculation
Heating/cooling load and associated room temperature calculations are performed with the same building transfer functions
employed in frequency response and thermal control studies. These computations are performed by means of discrete Fourier series.


Modeling and Simulation of Passive and Active Solar Thermal Systems

379

The building transfer functions are calculated at discrete frequencies (s = jωn), and a discrete Fourier transform (DFT) of the weather
data is performed. For example, convective auxiliary heating is given by
Qð1Þ ¼ q aux þ q int þ q eq

½36Š

where qint represents the convective portion of internal gains and qeq is an equivalent source representing heat flow
due to infiltration and is given by UinfTo. Therefore, by substituting eqn [36] in eqn [31c] and assuming that the

room air temperature T(1) is specified, the auxiliary heating/cooling power qaux may be determined at each frequency of interest as
(
)
N
X
Tð1Þ − Zð1; 1Þ½ qint þ Uinf To Š−
Zð1; JÞQðJÞ
J¼ 2

qaux ðjωÞ ¼

½37Š

Zð1; 1Þ

where all quantities are evaluated as complex numbers for s = jω (N = number of nodes). Each source or specified temperature is
represented by a discrete Fourier series (DFT), and the time domain variation of qaux is obtained through an inverse discrete Fourier
transform (IDFT). For design day analysis, five to nine harmonics are usually adequate. These are the harmonics necessary for
adequate representation of the inputs, that is, heat sources such as absorbed solar radiation, internal gains, and ambient
temperature To. One advantage of this approach over more commonly used methods is that the superposition principle is applied
directly. Therefore, effects of various inputs may be studied separately, or a passive analysis (qaux = 0) can be easily performed – in
this case eqn [31c] is directly applied. Note that the air temperature T(1) in eqn [37] can be a profile, that is, it may vary with time.
Thus, optimum set-point profile variations may be determined to optimize solar gain utilization. The discrete Fourier series
approach is described in more detail by Athienitis and Sanatmouris [1], including a model for a proportional control source in
the thermal network and a technique for modeling time-varying parameters, such as a conductance representing infiltration based
on the substitution network theorem.
3.11.2.1.3(viii) Discrete Fourier series method for simulation
Steady-periodic conditions are usually assumed; for example, if the simulation is to be performed for a week, it is assumed that all
previous weeks have been identical to the week considered. The steps needed for a periodic steady-state solution are as follows:
1. Select the number N of harmonics to perform the analysis. If n represents a harmonic number and P is the time length of the

simulation or analysis (e.g., a day or a week), then a harmonic’s frequency ωn is equal to 2πn/P.
2. Obtain the appropriate discrete Fourier series representations for the sources. An arbitrary source M(t) is represented by a
complex Fourier series (IDFT) of the form
MðtÞ ¼

N
X

mn ðj ωn Þ exp ðj ωn tÞ

½38aŠ

n¼ −N

the complex coefficients mn(jωn) being determined numerically by a DFT:
K
½ ΣMð tkÞexpð− j ωn tk Þ Š
mn ¼
P
k¼1

½38bŠ

where M(tk) is the value of M at time tk corresponding to point k (for a total of K values over the time length P). The number of
harmonics N cannot exceed K/2.
3. Determine the discrete frequency response Z(jωn) of the output of interest to unit input at each node. The periodic response to
each source is obtained by superposition of the output harmonics using complex (phasor) multiplication. The total response to
more than one input is determined by a double summation for all inputs Q(J) and all frequencies of interest ωn. For example, for
the room air temperature T1(t) is obtained by

T1ðtÞ ¼

N
X
n¼ −N

(

8
X

)
Zð1; J; j ωn ÞQn ðJ; j ωn Þ exp ðj ωn tÞ

½38cŠ

J¼1

3.11.3 PV/T Systems and Building-Integrated Photovoltaic/Thermal (BIPV/T) Systems
3.11.3.1

Integration of Solar Technologies into the Building Envelope and BIPV/T

Integration improves the cost-effectiveness by having the PV panels provide additional functions, which involve active solar heating
and daylighting (see Figures 22 and 23 for two Canadian examples of a near net-zero energy house and a University building in
Montreal with a BIPV/T façade). The following are some recognized methods of beneficial integration:


380

Components

(a)

(b)

(c)

08/15/2007

Figure 22 EcoTerra home and its BIPV/T modular roof. (a) EcoTerraTM home with fully BIPV/T system (top roof). (b) Construction of BIPV/T roof
module. (c) BIPV/T roof module delivered on-site for assembly of house.

Figure 23 JMSB BIPV/T façade and schematic (Concordia University, Montreal). John Molson School of Business (JMSB) building BIPV/T system (top
right section of façade) and schematic illustrating the system concept: 70% of the transpired collector cladding area (288 m2) is covered by specially
designed PV modules; the system generates up to about 25 kW electricity and 75 kW of thermal energy used to directly heat ventilation air [7].

1. Integrating the PV panels into the building envelope (BIPV). This strategy could involve, for example, replacing roof shingles or wall
cladding with PV panels. It has significant advantages over the more usual ‘add-on’ strategy. Not only does it eliminate an extra
component (e.g., roof asphalt shingles), but it also eliminates penetrations of a preexisting envelope that are required in order to
attach the panel to the building. (It is understood that the components replaced are not windows, as this is covered by Method 3,
below.) Architectural and aesthetic integration is a major requirement in this type of BIPV system. Not only can this strategy lead to
much higher levels of overall performance, but it can also provide enhanced durability: one International Energy Agency (IEA) study
[8] found that PV systems can last 50 years, whereas curtain wall components such as sealed glazing units may only last 25 years.
2. Integrating heat collection functions into the PV panel (BIPV/T). PV panels typically convert from about 6% to 18% of the incident
solar energy to electrical energy, and the remaining solar energy (normally lost as heat to the outdoor environment) is available
to be captured as useful heat. In this strategy, a coolant fluid, such as water or air, is circulated next to the panel, extracting useful
heat. The coolant also serves to lower the temperature of the panel; this is beneficial, because panel efficiency increases at lower
panel temperatures. This strategy can be adopted in either in an open-loop or closed-loop configuration. In one open-loop
configuration, outdoor air is passed under PV panels, and the recovered heat can be used for space heating, preheating of

ventilation air, or heating DHW – either directly or through a heat pump.
3. Integrating light transmission functions into the PV panel (BIPV/L). This strategy uses special PV panels (semitransparent PV
windows) that transmit sunlight. As was the case for the previous strategy, this strategy draws on the fact that only a fraction
of the incident solar energy is converted to electricity, and the remainder can be used for other purposes – in this case for useful
light, thereby saving on the energy that electrical lights would otherwise draw. Thin-film PV cells that let some sunlight through
are commercially available for this purpose. A major challenge is limiting the temperature rise of the windows and controlling
the impact of the associated heat gains during times when building cooling is required. Compared with normal windows, these
windows have a reduced light transmission and can therefore function as shading devices.
Appropriate modeling of building-integrated solar energy systems (thermal, electric, hybrid, daylighting) is essential for the
designing of high-performance solar buildings. These systems will play a major role in achieving the net-zero energy goal and


Modeling and Simulation of Passive and Active Solar Thermal Systems

381

need to be carefully selected, modeled, and sized for an accurate design. Different simulation tools include different technologies
and simulate building-fabric energy transfer with different levels of detail. They also utilize different techniques to model the
transient response of buildings and their systems to changes in internal and external thermal loads.
At the early stage of the design, a simplified software tool may provide enough accuracy to size a BIPV or a solar thermal system
as it provides monthly estimates of energy generated. However, a BIPV/T system that generates both electricity and heat requires
estimation of the heat recovered and its potential uses – heating ventilation air, heating water, or space heating (directly or through a
heat pump). To properly simulate these systems, tools characterized by a high-integrity representation of the dynamic and
connected processes are required.

3.11.3.2

A Simplified Open-Loop PV/T Model

A simplified model for a PV/T façade or roof with the exterior layer being PV panels is described below to calculate the PV

temperature and the heat recovered.
Consider a façade with PV panels as the exterior layer; this façade may be represented by the thermal network model shown in
Figure 24.
The mean air temperature Tma is determined from a differential analysis that finds the air temperature as a function of vertical
distance x. It is assumed that the air speed is constant, that is, air is drawn into the window by a fan in the HVAC system fresh air
intake. The actual air temperature T(x) is then used to determine the Tma. This is then employed to find the correct values of Tpv and
Tb, which are used to fine-tune the calculations. Considering an element dx in the vertical direction, gives the following:
M  c  ρ  dT ¼ ðW  dx  hðTpv −TÞÞ þ ðW  dx  hðTb −TÞÞ

½39aŠ

where M = flow rate = V Â A (V is average velocity and A is cross-sectional area) and W is the width of façade. Note that the convective
coefficient h is an average for both cavity surfaces (in reality it will generally be higher on the hotter surface).
Note that this simple model assumes an equal convective heat transfer coefficient h for both cavity surfaces. The following
ordinary differential equation is obtained:
d
a T þ 2T ¼ Tb þ Tpv
½39bŠ
dx
with a = (M  c  ρ)/(W  h)
An exponential variation is obtained for the air temperature as follows:
�
�
Tb þ Tpv −X⋅2
Tpv þ Tb
TðxÞ ¼
þ To −
e a
2
2

½39cŠ

The PV and back panel temperatures are obtained as
Tma  Ub þ TR  U3 þ Tpv  Ur
U3 þ Ub þ Ur

½39dŠ

Uo  To þ Ua  Tma þ Ur  Tb þ Spv
Uo þ Ua þ Ur

½39eŠ

Tb ¼
Tpv ¼

where U represents conductance between the various nodes (Uo = Aho, Ur = Ahr, Ua = Ub = Ah, and U3 is negligible).

3.11.3.3

Transient and Steady-State Models for Open-Loop Air-Based BIPV/T Systems

BIPV/T systems produce thermal and electrical energy, and have lower effective system costs than do stand-alone PV systems. The
BIPV/T system absorbs solar energy on the top surface, which includes the PV panels and generates electricity while also heating air

PV
qu

Te

To

Uo

Tb

Ua

Ub

U3

TR

Tma
Spv
Ur

RSI 1

Tin

Figure 24 Thermal network model of façade with exterior PV (assuming isothermal surfaces); node b indicates the back panel interior surface (assumed
to be an insulating layer with a thermal resistance of RSI 1).