Design and Simulation of a Net Zero Energy Healthy Home

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    DESIGN AND SIMULATION OF A

    NET ZERO ENERGY HEALTHY HOME IN MONTREAL

    Type of Paper: Refereed

    Jos A. Candanedo1, Sevag Pogharian2, Andreas K. Athienitis1, Andr Fry3

    1Dept. of Building, Civil and Environmental Eng., Concordia University, Montral, Canada

    Tel.: (514)-848-2424, ext. 7080, e-mail:[email protected] Pogharian Design, Montral, Canada

    Tel.: (514)-935-5210, Fax: (514)-935-9672, e-mail: [email protected] Inc., Sorel-Tracy, Qubec, Canada

    Tel.: (450-780-0608), Fax: (450-780-0062), e-mail: [email protected]

    ABSTRACT

    In 2006, Canada Mortgage and Housing Corporation

    (CMHC) launched the Net Zero Energy Healthy

    Housing (NZEHH) competition (recently renamed

    EQuilibrium Housing), an initiative to encourage the

    design of energy efficient and environmentally

    friendly homes. This paper presents the energy

    design process ofAlstonvale Net Zero House, one of

    the winning designs of this competition. Key features

    of the house include a unique passive solar design

    with adjustable shutter area, a BIPV/T roof with heat

    recovery linked to an air-water heat pump and a large

    thermal storage tank connected to a floor heating

    system. This paper presents the models and

    simulations that were performed to optimise the

    proposed design.

    INTRODUCTION

    The objectives of theEQuilibrium Housing initiative

    can be briefly summarised as follows:

    1. To develop grid-tied homes with net zero, or nearnet zero, energy consumption over a twelve

    month period.

    2. To achieve high resource efficiency during itsconstruction and operation.

    3. To obtain a high quality indoor environment.4. To reach specified levels of affordability and

    marketability.

    CMHC formulated a point system attributing

    weighting factors to the variables that were

    considered for the evaluation of the projects.

    Alstonvale Net Zero House, the design solution

    discussed in this article, is one of the twelve selected

    teams from across Canada to continue to the next

    phase: the construction of the house.

    Team Montral ZERO, the team that designed

    Alstonvale Net Zero House, began work circa August

    2006. The team included architects, engineers in

    diverse disciplines (civil, mechanical, and electrical),

    consultants, water management expert and suppliers.

    Most of the team members had previous experience

    in the design of energy efficient homes.

    The location chosen for the design was the town of

    Hudson, Qubec (N4527, W749), a suburb of

    Montral. The objective market for this house is a

    mid-income Canadian family.

    DESIGN PROCESS: INITIAL

    APPROACH

    Although general guidelines exist for designing an

    energy efficient and environmentally friendly house

    in a cold climate (good level of insulation, windows

    facing south, etc.), an optimal solution requires a

    methodical approach, taking numerous design

    constraints into consideration.

    In a traditional approach to building design all the

    professionals involved work sequentially: thearchitect makes a conceptual design, the structural

    engineer then performs the structural calculations,

    and finally the electrical and mechanical engineer

    each make their contributions. However, the

    complexity of the design of a net-zero energy home

    requires that all the professionals involved

    collaborate very closely from the beginning.

    One of the requirements of the competition was the

    organisation of a design charrette. This word has

    been used traditionally by architects to refer to an

    intensive brainstorming session in which several

    professionals contribute ideas.

    Figure 1. Rendering of Alstonvale Net Zero House

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    The charrette for the design ofAlstonvale Net Zero

    House took place on October 10th

    and 11th

    2006. The

    first architectural sketches and design proposals were

    presented during this charrette, and subjected to the

    scrutiny of the team members. General consensus

    was found on the following recommendations:

    1. A high-quality building envelope.2. Implementation of passive heating and cooling

    design strategies.

    3. Use of technologies taking advantage ofrenewable energies (photovoltaic panels, solar

    thermal energy).

    4. Energy efficient lighting and appliances.5. Adequate construction materials to guarantee a

    healthy indoor environment.

    6. Restriction of the dimensions of the house to aminimum size guaranteeing health and comfort of

    the occupants.

    7. Allowing for future expansions of the house byleaving a section of the interior unfinished.

    8. Employing standard construction methods asmuch as possible.

    9. Water management strategies to reduce waterwaste.

    Other design decisions were not so straightforward.

    For instance, several alternatives were discussed

    concerning the heating and cooling systems, as well

    as the method for distributing the heat in the

    building. The general consensus was that building

    simulations, also an important requirement in the

    competition, were needed to decide on these andother key issues such as the values of several

    important parameters (size of the PV system, window

    sizes, R-values of the building envelope components,

    thermal mass of the building, etc.).

    SIMULATIONS

    HOT2000 Simulations

    CMHC rules required all the contestants to submit

    numerical simulations of the performance of the final

    design of the house. HOT2000, a program developed

    by NRCan as an assessment tool for professionals in

    the building industry, was the mandatory tool for two

    simulations (http://oee.nrcan.gc.ca). The first one

    required that parameters such as the HVAC system,

    temperature set-points and hot water consumption be

    fixed in order to evaluate the performance of the

    building envelope alone. HOT2000 simulations

    generate an index called EGH (which stands for

    Energuide for Houses, the name of a previous

    government incentive program) which is taken as an

    indicator of the performance of the building

    envelope. The minimum acceptable value of EGH

    was 82.

    The second simulation was intended to account for

    the effect of using renewable energies and alternative

    HVAC systems, while keeping adequate levels of

    comfort and health for the building occupants, and

    guaranteeing that reasonable assumptions of

    energy usage were employed. A new index, labelled

    EGH*, was calculated using the results of the second

    simulation, as follows:

    Annual Estimated Energy Consumption* 100 20

    Reference Energy ConsumptionEGH

    =

    (1)

    The reference energy consumption, calculated as a

    property of a given house, depends on the volume of

    the building, the temperature of the tap water, the

    number of heating degree-days, and the intended

    heating system (i.e., furnace or electric).

    The minimum acceptable value of EGH* was 90;

    however, the competitions marking system was non-

    linear. For instance, EGH* ratings of 90, 95 and 98

    correspond to 20%, 45% and 72% of the marks

    assigned to the energy performance section. Thiscircumstance made it extremely desirable to reach or

    approach EGH* = 100, corresponding to net zero

    energy performance.

    CMHC also required that the parameters employed

    for the second simulation be used to determine the

    necessity of a cooling system. No cooling system

    was required for cooling loads below 1500 MJ per

    year.

    The HOT2000 simulations were very useful as a

    decision making tool for some critical aspects of the

    building envelope. For instance, the size of the

    overhangs on the south facing faade had to be

    increased in order to obviate the need for an air

    conditioning system. It was also found that above a

    critical R-value (near R-30) the benefit of increasing

    the insulation in the walls was very marginal, and not

    economically justifiable because of the cost of

    additional insulation in walls.

    Custom Simulations

    Although the submission of the simulations in

    HOT2000 was a competition requirement,

    supplementary simulations performed in other

    software packages could also be submitted.

    HOT2000 cannot represent the performance of new

    renewable energy systems (such as BIPV/T) and

    passive solar behaviour of the custom is not

    accurately represented in this bin method-based

    software. Also, HOT2000 can not accurately model

    thermal mass and overheating. It was found that

    other tools, more suitable for this purpose, were

    therefore necessary.

    Mathcad 2001, a general purpose mathematical

    programming tool, was employed for simulating the

    most innovative elements ofAlstonvale Net Zero

    House. Mathcad has been used as a tool for building

    simulation (Athienitis, 1994; Athienitis, 1999;Tzempelikos, 2005). An advantage of using Mathcad

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    is the flexibility that it allows for incorporating new

    models.

    The forcing functions (i.e., heat source and

    temperatures) are based on hourly data of solar

    radiation and temperature from a typical

    meteorological year file (TMY2) for Montral

    (TRNSYS 16). This kind of file includes relatively

    extreme conditions, such as very cold or very hot

    days, but maintains the yearly average values

    corresponding to a given location. Since the time step

    used in the calculations was 150 s (2.5 min), linear

    interpolation functions were used to estimate

    intermediate values between the hourly values

    available in the TMY2 file. This procedure generated

    a set of approximately 17,300 data points per

    variable per month.

    The Perez model (Perez et al., 1990) is a useful tool

    for calculating solar radiation incident on surfaces

    having any given orientation, based on two inputvalues: beam radiation and diffuse horizontal

    radiation. The Perez model was used to calculate the

    value of solar radiation for the north, south, east and

    west walls, and north and south roofs. Sol-air

    temperatures were calculated for every wall, based

    on the incident solar radiation, the exterior

    temperature and the external heat transfer

    coefficients.

    The area of the windows for every orientation was

    introduced as an input to calculate the incoming solar

    heat gains.

    Figure 2 shows the result of the Mathcad simulation

    for the interior temperature, for the first 15 days of

    February. Passive solar heat gains account for most

    of the reduction of the heating load. Auxiliary

    heating is needed on days 35 and 38.

    BIPV/T System

    Several alternatives were considered as the main

    source of heat for the space heating system: a ground

    source heat pump, a heat pump linked to a Canadian

    tunnel system, and a BIPV/T system, also with a heat

    pump.

    In the BIPV/T system considered (Figure 3), a cavity

    is located underneath the PV panels. At the bottom of

    the cavity, there is an absorber plate made of

    corrugated metal. When solar radiation allows,

    exterior air is drawn upwards, recovering some heat

    from the PV panels. A glazed section following the

    PV panels is intended to further increase the

    temperature of the air in the cavity during the winter.

    The hot air exiting the cavity can be used as the heat

    source of an air-to-water heat pump. When its

    temperature is high enough (typically higher than

    40C), it can be passed through an air-to-water heat

    exchanger, without resorting to the refrigerationcycle. A large water tank is used as a heat reservoir.

    Some benefits of the BIPV/T with glazed air

    collector at the top coupled to an air-water heat pump

    as compared to a separate geothermal heat pump

    were apparent: the cost of digging or drilling is

    avoided; advantage is taken of the existing framing

    system to support the PV panels; and the cooling

    effect due to the circulating air improves theelectrical conversion efficiency of the PV panels.

    Obviously, the BIPV/T system can only recover heat

    when solar radiation is available. The interaction of

    the BIPV/T system with the storage tank, and with

    the rest of the house, is critical for assessing the

    performance of the system. It was necessary to

    determine whether the heat supplied during the

    heating season would justify relying on the BIPV/T

    system as the primary heating source (the installation

    of an auxiliary heating system, employing renewable

    fuels, had already been decided).

    As previously mentioned, HOT2000 is not the ideal

    tool for simulating the capabilities of a BIPV/T

    system coupled to a heat pump. The closest module

    available in HOT2000 was an air-source heat pump.

    Glazed

    area

    PVpan

    els

    Airflo

    w

    Absorbing Plate

    Insulation

    Figure 3. Conceptual representation of the PV-

    thermal system intended for EQuilibrium House #1.

    Figure 2. Exterior temperature from TMY2 file

    (Toutit) and simulated interior temperature (Tinit),

    February 1st to February 15th.

    32 34 36 38 40 42 44 4630

    20

    10

    0

    10

    20

    3030

    30

    Tinit

    Toutit

    4631 LSTit

    day

    Absorber Plate

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    aspect ratio of the cavity is much larger than unity

    (as in this case). The presence of the PV-panel

    framing together with the unavoidable surface

    imperfections, and the corrugated surface of the

    absorbing plate, strongly suggests the use of a

    correlation that takes into consideration the surface

    friction factor (or equivalent friction factor, sincethe cross section is not uniform) for the

    determination of the Nusselt number. For instance, it

    is not appropriate to use the common Dittus-Boelter

    correlation, which is intended for fully developed

    turbulent flow in smooth pipes (Incropera and

    DeWitt, 2002; Lienhard and Lienhard, 2006). The

    problem is further complicated as the Reynolds

    number is in the vicinity of the critical region

    between laminar and turbulent flow for most of the

    flow rates considered. In the laminar flow regime,

    the thermal boundary layer is not fully developed for

    most of the cavitys length, and it is reasonable to

    expect relatively high heat transfer coefficients.

    Depending on the flow regime, two correlations were

    used for calculating the heat transfer coefficient in

    the cavity (Lienhard and Lienhard, 2006): the

    Gnielinski correlation for turbulent regime:

    ( )

    ( ) ( )0.5 2/ 3

    / 8 ( 1000) Pr

    1 12.7 /8 Pr 1

    D

    D

    f ReNu

    f

    =

    + (2)

    and a correlation for thermally developing, laminar

    flow

    1/ 3

    2/ 30.06683.6570.04

    D

    GzNu

    Gz= + +(3)

    In both cases, the hydraulic diameter was used as the

    length scale.

    The exterior convection heat transfer coefficient on

    the surface of the PV panels and glazing is a strong

    function of the wind speed. The McAdams

    dimensional equation (Duffie and Beckman, 2006),

    based on wind speed, was used for these calculations.

    Incorporating an additional vertical glazing section

    has been suggested as a method for recovering heat

    during the winter (Pantic, 2007), when the solaraltitudes are low, while avoiding overheating during

    the summer months.

    Figure 5 compares the outlet temperature of the

    cavity and the exterior temperature, for a particular

    configuration with 1.4 m of vertical section for the

    first 15 days of February. According to this

    simulation, the temperature rise within the cavity can

    exceed 50 C.

    Figure 6, showing the temperature rise for 3 days in

    winter and one day in late spring, illustrates the

    effect of the vertical section. During the winter

    months, the vertical section keeps warming the air(although the change in slope is hardly noticeable).

    However, in June, the presence of the vertical section

    reduces the exit temperature.

    In general, high temperatures are desirable in order to

    have higher COP values when operating the heat

    pump, but higher temperatures are usually associated

    with low flow rates, and consequently, lower heat

    removal from the roof.

    For the basic simulations it was assumed that the fan

    of the heat pump would drive the flow through the

    system, overcoming the pressure drop of its own heatexchanger. The pressure drop across the BIPV/T

    system is small due to the low velocities, large cross

    sectional area of the cavity and short hydraulic length

    of the roof. The design also considered bypassing the

    first heat exchanger when the heat pump is running.

    A separate variable speed fan is being considered for

    controlling the flow rate, with the possibility of also

    bypassing the heat pump heat exchanger. The proper

    size of the fan is currently under study.

    Figure 6. Simulated temperature rise for Julian days

    356 (December 22nd

    ), 4 (January 4th

    ), 40 (February

    10th), and 161 (June 10th).

    0 0.93 1.85 2.78 3.7 4.63 5.55 6.48 7.40

    10

    20

    30

    40

    50

    6060

    0

    T356dn

    T4dn

    T40dn

    T161dn

    7.40

    coordn

    Figure 5. Exterior temperature (TMY2) and

    simulated exit temperature of the BIPV/T cavity

    from February 1st to February 15th. Roof slope =

    45, air speed = 0.5 m/s, cavity height = 2 in, length

    of PV section = 5 m, length of glazing section = 1m, length of vertical section = 1.4 m.

    32 34 36 38 40 42 44 4630

    20

    10

    0

    10

    20

    30

    40

    50

    60

    60

    30

    Texitit

    Toutit

    4631 tit

    day

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    FEATURES OF THE FINAL DESIGN

    The final configuration of the mechanical system is

    shown in Figure 7. After numerous simulations the

    final design ofAlstonvale Net Zero House included:

    1. A 5.5 kW photovoltaic system will beinstalled on the roof (45 tilt angle, due

    south azimuth). According to a mandatory

    RETScreen simulation performed for this

    competition, this system should generate

    approximately 6745 kWhr annually.

    2. A two stage, 3.5 ton heat pump (14 kW)will be used to recover the heat from the

    roof and use it to heat the water of a large

    storage tank.

    3. The temperature of the storage tank for theradiant floor heating system will often be

    below the required temperature for domestic

    hot water (55 C). For this reason, and toserve as a significant supplementary energy

    source, the inclusion of a hydronic solar

    collector was deemed appropriate. A 40-gal

    tank will be used to store the domestic hot

    water. The main storage tank and the small

    DHW will be linked through a coil (Figure

    7) which will permit, if necessary, heat

    transfer both tanks. The use of efficient

    faucets and nozzles, and a heat exchanger,

    will allow the reduction of the daily use of

    domestic hot water to 120 L. The annual

    water heating load will be about 9,000 MJ.

    4. The annual consumption of the main

    electrical appliances has been estimated at

    1,435 kWhr (5166 MJ), or 3.93 kWhr per

    day. Adding lighting and other loads, the

    use of electricity for uses other than

    ventilation or heating are about 4,358 kWhr

    per year (12130 MJ) or 11.94 kWhr/day.

    Moderate use of non-essential loads (stereo,TV, computer, coffee maker, etc.), from a

    few minutes to a couple of hours daily, has

    been assumed. The house has been designed

    for two adults and two children occupying it

    50% of the time.

    5. In order to improve thermal comfort aradiant floor heating system will be the main

    heating method. An additional hydronic

    heating coil will provide supplementary

    heating to the ventilation air.

    6. No air conditioning system will be installed.A solar chimney on the roof of the house,

    with an opening to the east, is intended totake advantage of low pressures due to the

    winds coming from the west thereby

    enhancing the effect of natural convection

    to remove hot air from the house.

    7. The south facing windows have been set toapproximately 29% of the habitable area

    (172 m2), or 43% of the south faade, to

    take advantage of solar heat gains. Several

    measures are taken to avoid overheating due

    to this rather large glazing area: a) a large

    thermal mass behind the windows, b)

    controllable roller blinds, working under an

    Figure 7. Mechanical System of Alstonvale Net Zero House.

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    anticipatory control algorithm, and c)

    adequately sized overhangs (especially for

    summer). The fact that most of the south

    facing windows are located in front of a

    central hall, also helps to distribute the heat

    more evenly in the house.

    8. A fireplace or furnace fed by a renewablefuel will provide auxiliary heating.

    9. The fresh air intake will pass through a heatrecovery ventilator (HRV).

    KEY FINAL RESULTS

    The HOT2000 simulations corresponding to the final

    design produced the results shown in Table 1.

    The space heating consumption corresponds to the

    electricity consumption of the heat pump (6,872 MJ)

    plus the energy consumed by the fuel of the auxiliary

    heating system (2,057 MJ). If ethanol is used as the

    fuel, this would correspond to about 90 L per year.

    The space heating load is much larger than the

    effective energy used to supply it. According to the

    HOT2000 simulation, the gross annual heating loadis nearly 89,299 MJ. Passive solar gains through the

    south-facing windows reduce this load by providing

    54,174 MJ for the space heating load. Internal loads

    account for 12,157 MJ. The rest of the heating load

    (20,910 MJ) is provided by the heat pump. The

    effective COP of the heat pump would then be

    20,910/6,872 = 3.04. A sizeable amount of energy,

    2,841 MJ per year, is used to drive the ventilation

    system.

    According to this simulation, domestic water heating

    would use approximately 496 MJ of electric energy

    per year. The rest of the load is provided by the solarcollector.

    The custom simulations provided similar results: the

    annual energy consumption is about 25,546 MJ, of

    which 9,856 MJ correspond to space heating. The

    simulations estimate less electric energy consumption

    for space heating, but the use of the auxiliary heating

    source (renewable fuel) is larger (more than 5,000

    MJ).

    Perhaps the most relevant fact from both the custom

    and HOT2000 simulations is the realization that they

    mutually validate their results of heating loads.

    EGH and EGH* ratings

    The EGH rating ofAlstonvale Net Zero House was

    estimated to be about 85. The EGH* rating obtained

    from the HOT2000 simulation was 99.6. Based on

    the numbers from the Mathcad simulation, the EGH*

    rating would be about 101 (more energy is obtained

    from renewable sources than it is consumed annually

    by the house).

    CONCLUSIONS

    This paper presents salient features of the design

    process ofAlstonvale Net Zero House. Details of a

    supporting simulation in Mathcad are presented and

    discussed.

    BIPV/T roofs, coupled with a heat pump and a

    storage system, are a realistic alternative as a heat

    source for Canadian homes.

    The construction ofAlstonvale Net Zero House has

    been scheduled to begin this year. We are lookingforward to the monitoring of the performance of the

    house.

    BIPV/T systems have the potential to become an

    important technology for residences in the Canadian

    climate. Further research is recommended to evaluate

    several design alternatives such as the use of multiple

    inlets for the air in order to increase the efficiency of

    heat recovery and the incorporation of fins of

    different shapes. A discussion on the design of solar

    air collectors (akin to BIPV/T systems) is presented

    by Duffie and Beckman (2006).

    The house control system should take into accountflow rate variation (and associated changes in the

    exit temperature of the cavity and heat recovered),

    heat pump COP, storage tank and indoor air

    temperatures. In particular, weather forecast data,

    essential for predictive control, would improve the

    performance of the system.

    ACKNOWLEDGEMENTS

    The collaboration of all the members of team

    Montreal ZERO is gratefully acknowledged. The

    authors would like to express their recognition to

    CMHC for its commitment towards energy efficient,green buildings. Financial support of this work was

    provided in part by NSERC through the Solar

    Buildings Research Network.

    NOMENCLATURE

    COP Coefficient of Performance of the heat pump

    f Darcy-Weisbach friction factor, defined by

    this expression:

    ( )2 / 2

    P D

    L V

    Gz Graetz number ( /RePrD x )

    Annual Energy Consumption MJ kWhrSpace Heating 8929 2480.3

    Water Heating 496.3 137.9

    Ventilation 2841 789.2Lighting/appliances 15688 4357.8

    Total 27954.3 7765.1

    Table 1. Annual Energy Consumption

    (HOT2000 Simulation)

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    DNu Local Nusselt number (hD/k)

    DNu Average Nusselt number (hD/k)

    Pr Prandtl number ( / )

    DRe Local Reynolds number ( /VD )

    TMY2 Typical meteorological year file (2nd

    generation)

    REFERENCES

    Athienits, A.K.. 1994. Buiding Thermal Analysis,

    Mathcad Electronic Book.

    Athienitis, A.K. 1999. Thermal Analysis of

    Buildings in a Mathematical Programming

    Environment and Applications. Building and

    Environment, Vol. 34, pp. 401-415.

    Duffie, J.A.; Beckman, W.A. 2006. Solar

    Engineering of Thermal Processes. Third

    Edition. John Wiley & Sons, New Jersey, USA.

    Incropera, F.P., DeWitt, D.P. 2002. Fundamentals of

    Heat and Mass Transfer. Fifth Edition. John

    Wiley & Sons.

    Jones, O.C. 1976. An Improvement in the

    Calculation of Turbulent Friction in Rectangular

    Ducts. ASME Journal of Fluids Engineering,

    Vol. 98, pp. 173-181.

    Lienhard, J. IV; Lienhard, J. V. 2006. A Heat

    Transfer Textbook. Third Edition. Internet

    Book. Phlogiston Press.

    Pantic, S. 2007. Energy Analysis of Photovoltaic

    Thermal System Integrated with Roof and

    HVAC System. Masters Thesis. Concordia

    University, Montral, Canada.

    Perez, R., Ineichen, P., Seals, R. 1990. Modeling

    Daylight Availability and Irradiance

    Components from Direct and Global Irradiance.

    Solar Energy Vol.44, No.5, pp. 271-289.

    TRNSYS 16. 2004. Transient Simulation Studio.

    Tzempelikos, A. 2005. A Methodology for

    Integrated Daylighting and Thermal Analysis ofBuildings. PhD Thesis. Concordia University,

    Montral, Canada.

    www.builditsolar.com/Projects/SpaceHeating/SolarS

    hed/solarshed.htm

    http://oee.nrcan.gc.ca

    www.solarhouse.com

    www.stsscoinc.com

    APPENDIX: THERMAL NETWORKS

    The mathematical model developed in Mathcad 2001

    makes use of the thermal network concept, so named

    because of its reliance on analogies between heat

    transfer phenomena and electric circuits. Electric

    potential and current are analogous to temperaturedifference and heat flux. Similarly to electric

    resistors, thermal resistances provide a good model

    to calculate the heat flux between points at different

    temperatures. Thermal capacitances adequately

    describe the heat storage capacity of materials and

    the resulting time lag between heat flux and their

    temperature change. Despite the simplicity of their

    operating principle, thermal networks are a powerful

    tool for treating complex heat transfer problems.

    Electric circuit analysis techniques can be transposed

    and applied for simplifying and solving thermal

    networks.

    Finite volume numerical analysis methods can also

    be used to study the transient response of the thermal

    network to forcing functions, such as exterior or

    interior heat sources, or temperature variations. The

    heat balance principle is used to write the equations.

    For the Alstonvale Net Zero House simulations, a

    fully explicit scheme was used: the temperatures

    corresponding to a future time step (p+1) are

    functions only of the temperatures of the current time

    step (p). For instance, for the nodes having a thermal

    capacitance the energy balance equation in thermal

    network form is given by (Athienitis, 1994):

    ( )11

    np p p p p

    i i ij j i i

    ji

    tT Q U T T T

    C

    +

    =

    = + +

    (4)

    Where Ti refers to the temperature of the node, Cito

    its capacitance, Qi represents all the incoming heat

    sources, Tj refers to the temperature of adjacent

    nodes, andUij refers to the conductances between Ti

    andTj.

    Figure 8 shows a typical thermal sub-network

    employed for the construction of the larger model for

    the entire building. Node 15 and 16 represent the

    state of two planes in the wall. C15 represents thethermal inertia of the wall. Thermal resistances

    connect nodes 15 and 16, and node 15 with the

    exterior. A current source corresponds to the solar

    the solar heat gains incident on the walls interior

    surface, while a voltage source stands for the sol-

    air temperature on the exterior.

    R15_16 1516 R15_o

    C15 Teo_northS16

    Figure 8. Typical network model for a wall.