Volume 3 solar thermal systems components and applications 3 11 – modeling and simulation of passive and active solar thermal systems Volume 3 solar thermal systems components and applications 3 11 – modeling and simulation of passive and active solar thermal systems Volume 3 solar thermal systems components and applications 3 11 – modeling and simulation of passive and active solar thermal systems 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 Comprehensive Renewable Energy, Volume 359 359 361 362 367 369 379 379 381 381 383 383 383 388 389 390 391 394 396 396 400 401 401 401 402 402 402 403 403 404 404 406 407 407 408 408 410 410 410 411 411 413 415 415 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 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 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: ‘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) 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 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 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 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 (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: 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 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 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 Generally, thermal mass has high thermal capacitance but low thermal resistance For example, a m2 concrete block that is 10 cm thick can store (for °C temperature rise) Q ¼ cp ρ Vol ΔT ¼ 840 J kg − ˚C  2200 kg m −  0:10 Â1 m2  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 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 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 T T ẳ x2 t ẵ1a 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 ẳ sT ẵ1b 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 ẳ c1 ex ỵ c2 e x ; where ẳ ẵ2a Note: the Laplace transform of eqn [1a] assumes the initial condition T{x, t = 0} = This is acceptable as the aim is to derive steady-periodic (or frequency domain) solutions Rewrite eqn [2a] as T fx; sg ẳ M coshxị ỵ N sinhxị ẵ2b Heat flux q΄ is obtained by differentiating eqn [2b]: q′ ¼ −k dT ⇒ q′ fx; sg ¼ −M k sinhxịN sinhxị dx ẵ3 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 ẳ L; sgẳM cos hLị ỵ N sin hLị q2 fx ẳ L; sgẳM k sin hLị ỵ N k cos hLị ẵ4 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): � � � � cos hðγLÞ sin hLị=k T1 T ẳ4 ẵ5 ksin hLị 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 ½6 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=u ½7 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 3 � � T2 T ỵ B D B 1=u D D=u T1 o o 5⋅ 5 ẳ4 ẳ ẵ8 q q q C D C C=uo ỵ 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 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 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 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 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 where D ẳ cos hxị sin hxị Bẳ k C ẳ ksin hxị ẵ9 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=u To ẳ ẵ10 C D qo qs After the multiplication, Ts is temporarily set to (short-circuit) to get the Norton equivalent source as Qeq ¼ −Yt To where the transfer admittance Yt is given by Yt ẳ A A cos hxị sin hxị ỵ k uo ½11 ½12a 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 ỵ k tan hxị A Ys ẳ ẵ12b uo tan hxị ỵ 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 ẳ A k tan hxị 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 ½12c ½12d 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 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 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 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 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 5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 Mass thickness (m) 0.2 0.3 0.4 0.5 0.6 Mass thickness (m) Figure 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 10 20 Harmonic number 30 Concrete Softwood 52.5 50 47.5 45 42.5 40 Figure Variation in self-admittance and its phase angle with harmonic number n 10 20 Harmonic number 30 Modeling and Simulation of Passive and Active Solar Thermal Systems 403 _ a is the actual collector flow rate per square meter of collector area (l s−1-m−2) and m _ s is the standard collector flow rate per where m square meter of collector area (10 l s−1−m−2) As can be understood from above, if in a solar system both airflow rate and storage size are different from the standard ones, two corrections must be done on the dimensionless parameter X, and the final X value to be used should be the uncorrected value multiplied by the two correction factors obtained from eqns [96] and [97] 3.11.6.3 Performance and Design of Solar Service Water Systems The solar contribution for liquid-based systems given by eqn [91] can also be used to estimate the performance of systems producing solar service water heating only The system configuration is shown in Figure 44 Although a liquid-based system is shown, air or water collectors can be used with the appropriate heat exchanger to transfer heat to the preheat storage tank Hot water from the preheat storage tank is then fed to a water heater, where its temperature can be increased to the required temperature if needed A tempering valve may also be used to maintain the supply temperature below a maximum temperature, but this mixing is usually done at the point of use by the user In this case, the performance of the solar water heating system is affected by the mains water temperature Tm and the minimum acceptable hot water temperature Tw Both affect the average system operating temperature and thus the collector energy losses Thus, the parameter X, which as was seen before accounts for the collector energy losses, needs to be corrected The additional correction factor for the parameter X is given by [49] Xc 11:6 1:18Tw ỵ 3:86Tm 2:32T a ẳ X 100 −T a ½98 where Tm is the mains water temperature (°C), Tw is minimum acceptable hot water temperature (°C), and T a is monthly average ambient temperature (°C) The correction factor given in eqn [98] is based on the assumption that the solar preheat tank is well insulated It should also be noted that tank losses from the auxiliary tank are not included in the f-chart correlations Therefore, the load should also include the losses from the auxiliary tank, and these can be estimated from the heat loss coefficient and tank area (UA) on the basis of the assumption that the entire tank is at the minimum acceptable hot water temperature, Tw The system performance is also based on the storage capacity of 75 l m−2 of the collector aperture area and on a typical hot water load profile, with other consumption profiles producing only small effects on the system performance For different storage capacities, the correction given by eqn [92] can be used 3.11.6.4 General Remarks The f-chart design method is used to estimate quickly the long-term performance of solar systems of standard configurations The input data needed are: • The monthly average radiation and temperature • The monthly load required to heat a building and provide hot water • The collector performance parameters obtained from standard collector tests There are a number of general assumptions made for the development of the f-chart method such as: • The systems are well-built • System configuration and control are close to the ones considered in the development of the method • Flow rate in the collectors is uniform Relief valve Tempering valve Hot water supply Collector array Preheat storage tank Collector-storage heat exchanger Collector pump Water heater Auxiliary Storage pump Water supply Figure 44 Schematic diagram of the standard of water heating system configuration 404 Components If a system differs considerably from these conditions, then the f-chart method will not give accurate results The f-chart can be used as a design tool for residential space and domestic water heating systems of standard configuration in which the minimum temperature at the load is near 20 °C, and energy above this temperature value is useful The f-chart method should not be used for the design of systems that require minimum temperatures that are substantially different from this minimum value For example, it cannot be used for solar air-conditioning systems using absorption chillers, for which the minimum load temperature is 80–90 °C It should also be noted that because of the nature of the input data used, there are a number of uncertainties in the results obtained by the f-chart method The first is related to the nature of the meteorological data used, especially when horizontal radiation data are converted into radiation falling on the collector surface, which is usually inclined This is due to the fact that average weather data are used, which may differ considerably from real values, and that all days were considered symmetrical about solar noon The second uncertainty is related to the fact that solar systems are assumed to be well-built and to have well-insulated storage tanks and no leaks in the systems, which is not valid because most air systems leak to some extent, leading to a degraded performance Additionally, all liquid storage tanks are assumed to be fully mixed, which is not very correct, as all tanks show some degree of stratification This however leads to conservative long-term performance predictions as the collector inlet temperature is overestimated The final uncertainty is related to the building and hot water loads, which strongly depend on the variable weather conditions and on the habits of the occupants Despite these limitations, the f-chart method is a handy method that can be used for the easy and quick design of residential-type solar heating systems When the system under study satisfies the main assumptions made in the development of the method, quite accurate results are obtained Additionally, the method can be used in a programmable calculator or a computer to give results very quickly 3.11.7 Utilizability Method Due to its limitations, the f-chart method cannot be used for systems in which the minimum temperature supplied to a load is not near 20 °C Most of the systems that cannot be simulated with f-chart however can be modeled with the utilizability method or its enhancements, presented in this section The utilizability method is a design technique used for the calculation of long-term thermal collector performance for certain types of systems Initially originated by Whillier [54] and called the Φ-curve method, it is based on solar radiation statistics and the necessary calculations that have to be done at hourly intervals about solar noon each month Subsequently, the method was generalized to be used for any time of year and geographic location by Liu and Jordan [55] The generalized Φ curves of Liu and Jordan, generated from daily data, enable the calculation of utilizability curves for any location and tilt, by knowing only the clearness index, KT Subsequently, Klein [56] and Collares-Pereira and Rabl [57] eliminated the necessity of hourly calculations and introduced the monthly average daily utilizability, Φ, which reduced the complexity of the original method and improved the utility of the method Here again, only equations are given and not the actual graphs, which can be obtained from the original sources or by using a spreadsheet program to plot the equations as in f-chart 3.11.7.1 Hourly Utilizability The utilizability method is based on the notion that only radiation that is above a critical or threshold intensity is useful Therefore, the utilizability Φ is defined as the fraction of insolation incident on a collector’s surface that is above a given threshold or critical value A solar collector can give useful heat only if solar radiation is above a critical level, given by Gtc ẳ FR UL Ti Ta ị FR ị ẵ99 This is the radiation level at which the absorbed solar radiation and loss terms are equal It is obtained from the standard collector equation given in eqn [73], for Qu = and solving for Gt, now called Gtc, the critical incident total radiation Similarly, hourly values of total radiation can be used with symbols It and Itc When radiation is incident on the tilted surface of a collector, the utilizable energy for any hour is (It – Itc)+ The + sign indicates that the utilizable energy can only be positive or zero The fraction of the total energy for the hour that is above the critical level is called the hourly utilizability and is given by Φh ¼ ðIt −Itc ịỵ It ẵ100 It should be noted that utilizability can also be defined in terms of rates, using the irradiance on tilted surface Gt and its critical value Gtc, but as radiation data are usually available on an hourly basis, the hourly values are preferred This is also in agreement with the basis of the concept used for the development of the method Modeling and Simulation of Passive and Active Solar Thermal Systems 405 The utilizability for a single hour is not a very useful quantity, whereas utilizability for a particular hour of a month having N days in which the average radiation for the hour is It is very useful This is given by N 1X ðIt −Itc ịỵ N It ẳ ẵ101 The average utilizable energy for the month is NIt Φ These calculations can be done for all hours of the month with the results added up to get the utilizable energy of the month Another parameter required in these calculations is the dimensionless critical radiation level defined as Xc ẳ Itc It ẵ102 For each hour, the monthly average hourly radiation incident on the collector is given by [50] � � � � � ỵ cosị cosị ỵ HrG It ẳ Hr H D rd RB ỵ HD rd 2 ½103 By dividing by H and considering the definition of the monthly average clearness index, K T , defined as the ratio of the monthly average total insolation on a terrestrial horizontal surface to the monthly average daily total insolation on an extraterrestrial horizontal surface, H=Ho : �� � � H ỵ cosị cosị H I t ẳ K T Ho ỵ rG ẵ104 r D rd ịRB ỵ D rd 2 H H The ratios r and rd can be estimated by the Collares-Pereira and Rabl [58] and Liu and Jordan [52] correlations given in the following equations: Liu and Jordan [52] correlation for rd (ratio of hourly diffuse radiation to daily diffuse radiation) is given by rd ¼ � π � 24 cosðhÞ −cosðhss Þ � � 2πhss sinðhss Þ − cosðhss Þ 360 ½105 Collares-Pereira and Rabl [58] correlation for r (ratio of hourly total radiation to daily total radiation) is given by rẳ ỵ coshịị 24 coshị coshss ị ẳ ỵ coshịịr 2hss 24 coshss ị sinhss ị 360 where ẳ 0:409 ỵ 0:5016 sinhss 60ị ẳ 0:6609 0:4767 sinhss 60ị ½106a ½106b ½106c where hss is the sunset hour angle (degrees), obtained from cos(hss) = –tan(L) tan(δ), where L is the local latitude and δ is the declination, and h is the hour angle (degrees) at the midpoint of each hour (degrees) Liu and Jordan [55] constructed a set of Φ curves for various values of K T With these curves, it is possible to predict the utilizable energy at a constant critical level by knowing only the long-term average radiation Finally, Clark et al [59] developed a simple procedure to estimate the generalized Φ functions given by if Xc ≥� Xm > > > � X > > < 1− c if Xm ¼ Xm ẳ ẵ107a " #1 = � > > > �� X � c > > otherwise : j g j g ỵ ỵ 2gị Xm where gẳ Xm ẳ 1:85 ỵ 0:169 Rh kT Xm Xm ỵ 0:0696 ẵ107b cosị kT 0:981 kT cos ị ẵ107c The monthly average hourly clearness index, kT , is given by kT ẳ I Io ẵ108 406 Components and can be estimated using eqns [105] and [106] as kT ¼ I r r H ¼ KT ¼ ẳ ẵ ỵ coshị K T rd rd Ho Io ½109 The parameters α and β can be estimated from eqns [106b] and [106c], respectively The monthly average daily total insolation on an extraterrestrial horizontal surface H o can be estimated from � � ��� � � � 360N hss 24 3600Gsc Ho ẳ sinLị sinị ẵ110 þ 0:033 cos cosðLÞ cosðδÞ sinðhss Þ þ π 365 180 where hss is the sunset hour in degrees The unit of eqn [110] is J m−2 The ratio of monthly average hourly radiation on a tilted surface to that on a horizontal surface,Rh , is given by Rh ¼ It I ẳ t I rH ẵ111 The hourly utilizability is used to obtain values every hour, which means that three to six hourly calculations are required per month if hour pairs are used (morning hours similar to afternoon hours with respect to solar noon) For surfaces having an azimuth angle equal to zero, that is, those facing the equator, the monthly average daily utilizability, Φ, presented in the following section, can be used, which is a simpler way of calculating the useful energy as hour pairs can be used However, for surfaces that have a certain azimuth angle or for processes that have critical radiation levels that vary consistently during the days of a month, the hourly Φ curves need to be used 3.11.7.2 Daily Utilizability It is clear from the hourly utilizability method that a large amount of calculations are required in order to use the Φ curves For this reason, Klein [56] developed the monthly average daily utilizability Φ concept This is defined as the sum over all hours and days of a month of the radiation falling on a titled surface that is above a given threshold or critical value, which is similar to the one used in the Φ method, divided by the monthly radiation In equation form, it is given by Φ¼ X X days hours It Itc ịỵ NHt ẵ112 The monthly utilizable energy is then equal to the product NHt Φ The value of Φ for a month depends on the distribution of hourly values of radiation in that month Klein [56] assumed that all days are symmetrical about solar noon, which means that Φ depends on the distribution of daily total radiation, that is, the relative frequency of occurrence of below average, average, and above average daily radiation values Because of this assumption, any departure from this symmetry within days leads to increased values of Φ, which means that the Φ calculated will give conservative results Klein developed the correlations of Φ as a function of K T , a dimensionless critical radiation level X c and a geometric factor R=Rn The R parameter is the monthly ratio of radiation on a tilted surface to that on a horizontal surface Ht =H, given by � � � � � � H H H ỵ cosị cosị ẵ113 ỵ G R ẳ t ẳ D RB ỵ D 2 H H H where Ht is the monthly average daily total radiation on a tilted surface and RB is the monthly mean beam radiation tilt factor given by eqn [86a] For the hour centered about noon, the ratio of radiation on a tilted surface to that on a horizontal surface for an average day of the month, Rn, is given by � � �� � � � ỵ cosị cosị rd ; n HD rd ; n HD It RB ; n ỵ ỵ G ẵ114 Rn ẳ ẳ I n rn H rn H 2 where rd,n and rn are obtained from eqns [105] and [106], respectively, at solar noon (h = 0°) It should be noted that Rn is calculated for a day that has a total radiation equal to the monthly average daily total radiation, that is, a day for which H ¼ H The term HD/H is given from Erbs et al [60] as For hss 81.4 HD 1:0 0:2727KT ỵ 2:4495KT2 11:9514KT3 ỵ 9:3879KT4 for KT < 0:715 ẳ ẵ115a 0:143 for KT ≥ 0:715 H For hss > 81.4° HD ¼ H 1:0 ỵ 0:2832KT 2:5557KT2 ỵ 0:8448KT3 0:175 for KT ≥ 0:722 for KT < 0:722 ½115b 407 Modeling and Simulation of Passive and Active Solar Thermal Systems Another parameter required is the monthly average critical radiation level X c , defined as the ratio of the critical radiation level to the noon radiation level on a day of the month in which the radiation is the same as the monthly average In equation form, it is given by Xc ¼ Itc rn Rn H ½116 The procedure followed by Klein [56] is for a given K T and a set of days is established that have the correct long-term average distribution of KT values The radiation in each of the days in a sequence is divided into hours, and these hourly values of radiation are used to find the total hourly radiation on a tilted surface, It Subsequently, critical radiation levels are subtracted from the It values and summed as shown in eqn [112] to get the Φ values The Φ curves calculated in this manner can be obtained from graphs or from the flowing relation: �� � � h i Rn ẳ exp AỵB ẵ117a X c ỵ CX c R where A ẳ 2:943 9:271K T ỵ 4:031K T ẵ117b B ẳ 4:345 ỵ 8:853K T 3:602K T ẵ117c C ẳ 0:170 0:306K T ỵ 2:936K T ẵ117d Both the and Φ concepts can be applied in a variety of designs such as heating systems and passive heated buildings, where the unutilizable energy (excess energy) that cannot be stored in the building mass can be estimated 3.11.7.3 Design of Active Systems with Utilizability Methods The method can be developed for an hourly or daily basis as shown in the following sections 3.11.7.3.1 Hourly utilizability Another way of defining utilizability is that it is the fraction of incident solar radiation that can be converted into useful heat It is the fraction utilized by an ideal collector that has no optical losses and a heat removal factor of unity, that is, FR(τα) = 1, operating at a fixed temperature difference between the inlet and the ambient temperatures As can be understood, the utilizability of this collector is lower than since thermal losses always exist The equation that relates the rate of useful energy collection by a flat-plate solar collector Qu to the design parameters of the collector and meteorological conditions has been reported by Hottel and Whillier [61] This equation is given by eqn [73] and can be expressed in terms of the hourly radiation incident on the collector plane, It, as Qu ẳ Ac FR ẵIt ị UL Ti Ta ị ỵ ẵ118 where Ac is collector area (m2), FR is collector heat removal factor, It is total radiation incident on the collector surface per unit area (kJ m−2), (τα) is effective transmittance–absorptance product, UL is energy loss coefficient (kJ m−2-K−1), Ti is inlet collector fluid temperature (°C), and Ta is ambient temperature (°C) As was seen before, the radiation level must exceed a critical value before useful output is produced This critical level is found by setting Qu in eqn [118] to zero This is given in eqn [99], but in terms of the hourly radiation incident on the collector plane is given by Itc ¼ FR UL ðTi −Ta Þ FR ðταÞ ½119 Therefore, the useful energy gain can be written in terms of the critical radiation level as Qu ẳ Ac FR ịIt Itc ịỵ ẵ120 The + sign in eqns [118] and [120] and in the following equations indicates that only positive values of Itc should be considered By considering that the critical radiation level is constant for a particular hour of the month having N days, the monthly average hourly output for this hour is Ac FR ị X It Itc ịỵ ẵ121 Qu ẳ N N By considering that the monthly average radiation for this particular hour is It , the average useful output can be expressed by Qu ẳ Ac FR ịIt where Φ is given by eqn [101] ½122 408 Components This can be estimated from eqn [107] given earlier for the dimensionless critical radiation level Xc, given by eqn [102], which can be written in terms of the collector parameters, using eqn [119], as Xc ¼ Itc FR UL ðTi −Ta Þ ¼ ðταÞ It I FR ðταÞ n ðταÞ n t ½123 where (τα)/(τα)n can be determined for the mean day of the month (Table 4) and the appropriate hour angle, and can be estimated with the incidence angle modifier equation obtained from the performance testing of the collector [50] If Φ is known, the utilizable energy is It Φ The hourly utilizability is used to estimate the output of processes that have a critical radiation level, Xc, that changes considerably during the day because of collector inlet temperature variation Although the utilizability method is a very powerful tool, caution is required for possible wrong use Due to finite storage capacity, the critical level of collector inlet temperature for liquid-based domestic solar heating systems varies considerably during the month; therefore, the Φ curves method cannot be applied directly Exceptions to this are air heating systems during winter, where the inlet air temperature to the collector is the return air from the house, and systems with seasonal storage, where due to its size, the storage tank temperature shows small variations during the month [50] 3.11.7.3.2 Daily utilizability As mentioned in Section 3.11.7.2, the use of Φ curves involves a large number of calculations Klein [56] and Collares-Pereira and Rabl [58, 62] simplified the calculations for systems for which it is possible to use a critical radiation level for all hours of the month Daily utilizability is defined as the ratio of the sum, over all hours and all days of a month, of the radiation on a tilted surface that is above a critical level to the monthly radiation, given in eqn [112] The critical level Itc is similar to eqn [119], but in this case, the monthly average (τα) product must be used and the inlet and ambient temperatures are representative temperatures for the month: � � FR UL Ti T a Itc ẳ ẵ124 ị FR ị n ðτα Þ n In eqn [124], the term ðταÞ= ðταÞ n can be estimated with eqn [85] The monthly average critical radiation ratio is the ratio of the critical radiation level, Itc, to the noon radiation level for a day on which the total radiation for the day is the same as the monthly average [50] In equation form, this is given as � � FR UL Ti −T a Itc FR ị Xc ẳ ẳ ẵ125 rn Rn H rn Rn K T Ho The monthly average daily useful energy gain is then given by Qu ¼ Ac FR ịHt ẵ126 Finally, the daily utilizability can be obtained from eqn [117] Even though monthly average daily utilizability reduces greatly the complexity of the method, quite a lot of calculations can still be required, especially when monthly average hourly calculations need to be estimated It should be noted that the majority of the methods described so far for computing solar energy utilizability have been derived for North-American data based on clearness index, which is the parameter used to indicate dependence on the climate Carvalho and Bourges [63] applied some of these methods to European and African locations By comparing the results obtained with values from long-term measurements, they found that these methods can give acceptable results when the actual monthly average daily irradiation on the considered surface is known 3.11.8 The Φ ¯, f-Chart Method The utilizability design concept is useful when the collector operates at a known critical radiation level during a specific month In actual systems however, the collectors are connected to a storage tank, and thus, the monthly sequence of weather and load-time distributions cause a changeable storage tank temperature, which leads to a variable critical radiation level The f-chart has been developed to overcome the restriction of a constant critical level but is only applicable to systems delivering a load near 20 °C The utilizability concept described in the previous section has been combined with the f-chart by Klein and Beckman [64] to produce the Φ, f-chart design method for closed-loop solar energy systems (Figure 45) In this system, the storage tank is assumed to be pressurized or filled with a liquid of high boiling point so that no energy dumping occurs through the relief valve, and the auxiliary heater is in parallel with the solar system as shown The new method is not restricted to loads at 20 °C In these systems, the energy supplied to the load must be above a specified minimum useful temperature, Tmin (not necessarily 20 °C), and it must be used at a constant thermal efficiency or coefficient of performance so that the load on the solar system can be estimated The return temperature from the load is always at or above Tmin This method cannot be applied to a heat pump or heat engine because their Modeling and Simulation of Passive and Active Solar Thermal Systems Relief valve 409 Auxiliary Energy supplied at T > Tmin Collector array Energy system Storage tank Collector-storage heat exchanger Load Load heat exchanger Storage pump Collector pump Figure 45 Schematic diagram of a closed-loop solar energy system performance varies with the temperature level of supplied energy It is very useful, however, for absorption refrigerators, industrial process heating, and space heating systems [50] The maximum monthly average daily energy that can be delivered from the system shown in Figure 45 is given by X Qu ẳ Ac FR ịHt max ẵ127 This is the same as eqn [126] except that Φ is replaced with Φ max , which is defined as the maximum daily utilizability estimated from the minimum monthly average critical radiation ratio, given by X c; ¼ FR UL ðTmin − T a Þ FR ðταÞ rn Rn K T Ho ½128 Klein and Beckman [64] correlated the results of many detailed simulations of the system shown in Figure 45, for various storage size–collector area ratios, with two dimensionless variables These variables are similar to the ones used in the f-chart but are not the same In this method, the f-chart dimensionless parameter Y is replaced by Φ max Y, given by max Y ẳ max Ac FR ịNHt L ½129 and the f-chart dimensionless parameter X is replaced by a modified dimensionless variable X′, given by X′ ¼ Ac FR UL 100ịt L ẵ130 In fact, the modification in the dimensionless variable X given in eqn [77] is that the parameter (Tref – T a ) is replaced with an empirical constant 100 and FR is used instead of FR′ The Φ, f-charts can be obtained from actual charts or from the following analytical eqn [64]: � � f ẳ max Y 0:015ẵ exp3:85 f ị exp 0:15X ị R0:76 ẵ131 s where Rs is the ratio of standard storage heat capacity per unit of collector area of 350 kJ m−2− °C−1 to the actual storage capacity given by [64] Rs ¼ 350 Mcp Ac ½132 where M is the actual mass of storage capacity (kg) Since f is included in both sides of eqn [131], it must be estimated by trial and error Because actual Φ, f-charts are given for various storage capacities and the user has to interpolate, the use of eqn [131] is preferred The Φ, f-chart method is used in the same way as the f-chart method The values of Φ max , Y, and X΄ need to be calculated from the long-term radiation data for the particular location and load patterns As in the case of the f-chart method, fL is the average monthly contribution of the solar system, and the monthly values can be summed and divided by the total annual load to obtain the annual fraction F It should be noted that the Φ, f-chart method overestimates the monthly solar fraction, f This is due to assumptions made that no losses occur from the storage tank and that the heat exchanger is 100% efficient These assumptions require certain corrections, which are presented in the following subsections 410 3.11.8.1 Components Storage Tank Losses Correction The rate of energy lost from the storage tank to the environment, which is considered to be at temperature Tenv, is given by _ st ẳ UA ị s Ts Tenv ị Q ẵ133 By integrating eqn [133], the storage tank losses for the month can be obtained This is done by considering that (UA)s and Tenv are constant for the month � � Qst ẳ UAị s T s Tenv t ẵ134 where T s is the monthly average storage tank temperature (°C) The total load on the solar system is the actual load plus the storage tank losses, although the storage tanks are usually well insulated, and therefore, the storage tank losses are small Generally, the tank temperature rarely drops below the minimum temperature Finally, the fraction of the total load supplied by solar energy, including storage tank losses, is given by f TL ẳ Ls ỵ Qst Lu ỵ Qst ẵ135 where Ls is the solar energy supplied to the load (GJ) and Lu is the useful load (GJ) Therefore, first Qst is estimated, and then fTL is obtained from the Φ, f-charts as usual The solar fraction f can also be represented by Ls/Lu, that is, the ratio of solar energy supplied to the load to the useful load, by using eqn [135]: � � Qst Qst f ¼ f TL ỵ ẵ136 Lu Lu Storage tank losses are estimated by assuming that the tank remains at Tmin during the month, or that the average tank temperature is equal to the monthly average collector inlet temperature, T i , which is estimated by the Φ chart method Finally, once fTL is known, the average daily utilizability is given by [64] ẳ f TL Y ẵ137 For the estimation of tank losses by using eqn [134], Klein and Beckman [64] recommend the use of the mean of Tmin and T i The process is iterative, that is, T s is assumed and is used to estimate Qst From this, the fTL is estimated with the Φ, f-charts; subsequently, Φ is estimated from eqn [137], and X c is obtained from the Φ charts The temperature T i can then be estimated by correcting the originally assumed value with the ratio of the new and original value of Xc Subsequently, the new average tank temperature is estimated as the mean of T s and T i This new value of T s is compared with the initially assumed value, and a new iteration is carried out if necessary Finally, eqn [136] is used to estimate the solar fraction f 3.11.8.2 Heat Exchanger Correction As the heat exchanger adds a thermal resistance between the tank and the load, it increases the storage tank temperature The presence of a heat exchanger leads to a reduction in the useful energy collection, as higher collector inlet temperatures are present, and the storage tank losses are increased The average increase in tank temperature that is necessary to supply the required energy load is given by [64] T ẳ f L=tL L Cmin ẵ138 where ΔtL is the number of seconds during a month for which the load is required (s), εL is the effectiveness of the load heat exchanger, and Cmin is the minimum capacitance of the two fluid streams in the heat exchanger (W °C−1) Finally, the temperature difference found by eqn [138] is added to Tmin to find the monthly average critical radiation from eqn [128] 3.11.9 Modeling and Simulation of Solar Energy Systems Although the simple methods described so far have been proved to be quite accurate and can be carried out with hand calculations, the most accurate way to estimate the performance of solar processes is with detailed simulation The initial step in modeling a system is the derivation of a structure to be used to represent the system However, the structure that represents the system should not be confused with the real system, as this will always be an imperfect copy of reality; nevertheless, the system structure will foster an understanding of the real system In this process, system boundaries consistent with the problem being analyzed need to be established first This is done by specifying the items, processes, and effects that are internal and external to the system Two basic types of methods – simplified and detailed – are generally considered Simplified analysis methods have the advantages of fast computational speed, low cost, rapid turnaround (which is especially important during iterative design phases), Modeling and Simulation of Passive and Active Solar Thermal Systems 411 and easy use by persons with little technical experience Their disadvantages include limited flexibility for design optimization, lack of control over assumptions, and a limited selection of systems that can be analyzed [50] If the system application under consideration, configuration, or load characteristics is significantly nonstandard, a detailed computer simulation may be required to achieve accurate results Computer modeling of thermal systems presents many advantages, the most important of which are the following [50]: Elimination of the expense of building prototypes Organization of complex systems in an understandable format Provision of thorough understanding of system operation and component interactions Possible optimization of system components Estimation of the amount of energy delivered from the system Provision of temperature variations of the system Estimation of the effects of design variable changes on system performance by using the same weather conditions Simulations can provide valuable information on the long-term performance of solar systems and can provide information on transient system performance These simulations include variations in temperature, which may reach values above the degradability limit (e.g., of selective coating) and boiling of water with consequent heat dumping through the relief valve Usually, detailed models and fine time steps specified by the user require intensive calculations, which increase the time required to compute the results There are a number of programs that were developed over the years for the modeling and simulation of solar systems Some of the most popular ones – F-chart, TRNSYS, and WATSUN – are described briefly in this section 3.11.9.1 The F-Chart Program Although the f-chart method presented in Section 3.11.6 is simple in concept, the required calculations are tedious, particularly with respect to the manipulation of radiation data The use of computers greatly reduces the effort required The program F-chart [65], which is provided by the developers of TRNSYS, is very easy to use and gives predictions very quickly As in the basic method, the model is accurate only for solar heating systems of a type comparable to the model assumed in the development of the f-chart The F-chart program is written in BASIC It can be used to estimate the long-term performance of solar systems that use flat-plate, evacuated-tube collectors, compound parabolic concentrators, and one- or two-axis tracking concentrating collectors Additionally, the program includes the design of other solar systems like a swimming pool heating system that provides estimates of the energy losses from the swimming pool The solar energy systems that can be handled by the program are as follows: • • • • • • • • Pebble-bed storage space and domestic water heating systems Water storage space and domestic water heating systems Active collection with building storage space heating systems Direct-gain passive systems Collector-storage wall passive systems Swimming pool heating systems General heating system such as process heating systems Integral collector-storage domestic water heating systems The F-chart program can also perform economic analysis of the systems The program, however, does not provide the flexibility of detailed simulations and performance investigations as TRNSYS does 3.11.9.2 The TRNSYS Simulation Program TRNSYS, the transient systems simulation program that has been commercially available since 1975, was developed by the interna tional cooperation between the United States, France, and Germany This program, currently in version 17, was originally developed at the University of Wisconsin by the members of the Solar Energy Laboratory and written in FORTRAN TRNSYS remains one of the most flexible energy simulation software packages, which facilitates the addition of mathematical models and add-on components, and has multizone building model capability and the ability to interface with other simulation programs The program was originally developed for use in solar energy applications but has now been extended to include a large variety of thermal and other processes such as hydrogen production, PVs, and many others The program consists of many subroutines that model subsystem components The mathematical models for the subsystem components are given in terms of their ordinary differential or algebraic equations With a program such as TRNSYS, which has the capability of interconnecting system components in any desired manner, solving differential equations and facilitating information output, the entire problem of system simulation reduces to a problem of identifying all the components that form the particular system and formulating a general mathematical description of each [66] Users can also create their own programs, which are no longer required to be recompiled together with all other program subroutines but can just be created as a dynamic link library (DLL) file with any FORTRAN compiler and put in a specified directory Each component has a unique TYPE number, which relates the component to a subroutine, that is, the model of that component The UNIT number is used to identify each component (which can be used more than once) Although two or more 412 Components system components can have the same TYPE number, each must have a unique UNIT number Once all the components of the system have been identified and as the mathematical description of each component is available, it is necessary to construct an information flow diagram for the system The purpose of the information flow diagram is to facilitate identification of the components and the flow of information between them Each component requires a number of constant PARAMETERS and time-dependent INPUTS, and produces time-dependent OUTPUTS An information flow diagram shows the manner in which all system components are interconnected A given OUTPUT may be used as an INPUT to any number of other components Generally, simulations require some components that are not ordinarily considered as part of the system Such components are utility subroutines and output-producing devices TRNSYS subsystem components include solar collectors, auxiliary heaters, heating and cooling loads, differential controllers, thermostats, pumps, pebble-bed storage, hot water cylinders, relief valves, heat pumps, and many more Some of the main ones are shown in Table [67] There are also components for processing radiation data, performing integrations, and handling input and output Time steps down to 1/1000 h (3.6 s) can be used Additionally, the program can read actual weather and other data, which makes it very flexible In addition to the main TRNSYS components, an engineering consulting company specializing in the modeling and analysis of innovative energy systems and buildings, called Thermal Energy System Specialists (TESS), developed libraries of components for Table Main components in standard library of TRNSYS 17 Building loads and structures Energy/degree-hour house Roof and attic Detailed zone Overhang and wingwall shading Thermal storage wall Attached sunspace Detailed multizone building Controller components Differential controllers Three-stage room thermostat Proportional-integral-derivative (PID) controller Microprocessor controller Collectors Flat-plate collector Performance map solar collector Evacuated-tube solar collector Compound parabolic collector Electrical components Regulators and inverters Photovoltaic array Photovoltaic–thermal collector Wind energy conversion system Diesel engine generator set Power conditioning Lead–acid battery Heat exchangers Constant effectiveness heat exchanger Counter flow heat exchanger Cross-flow heat exchanger Parallel-flow heat exchanger Shell and tube heat exchanger Waste heat recovery HVAC equipment Auxiliary heater Dual-source heat pump Cooling towers Single-effect absorption chiller Hydronics Pump Fan Pipe (Continued) Modeling and Simulation of Passive and Active Solar Thermal Systems Table 413 (Continued) Duct Pressure relief valve Output devices Printer Online plotter Histogram plotter Simulation summary Physical phenomena Solar radiation processor Collector array shading Psychrometrics Weather data generator Refrigerant properties Undisturbed ground temperature profile Thermal storage Stratified fluid storage tank Rock-bed thermal storage Variable volume tank Utility components Data file reader Quantity integrator Calling Excel Calling Energy Equation Solver (EES) Calling MATLAB Weather data reading Standard format files User format files use in TRNSYS The TESS libraries, currently in version 3, provide a large variety of components on loads and structures, thermal storage, ground coupling, geothermal, optimization, collectors, HVAC, utility, controls, and many more The German research institute (DLR) also developed a TRNSYS model library for solar thermal electric components (STEC) The library includes models of all the components required to model a solar electricity generation system TRNSYS used to be a nonuser-friendly program; however, the two latest versions of the program (since version 16) operate in a graphic interface environment called the ‘simulation studio’ developed by Center Scientifique at Technique du Batiment (CSTB) from France In this environment, icons of components are dragged and dropped in the working project area from a list and connected together according to the real system configuration This is done in a way similar to the way piping and control wires are connecting the real system components Each icon represents the detailed program of each component and requires a set of inputs (from other components or data files) and a set of constant parameters, which are specified by the user Each component gives a set of output parameters, which can be plotted, saved in a file, or used as input in other components Therefore, the procedure is to identify the components required, drag and drop them in the working project area, and connect them together to form the model of the system to be simulated Subsequently, parameters and inputs need to be introduced by the user Additionally, by double-clicking with the mouse on the lines connecting the various components, the user can specify which outputs of one component are inputs to the other The project area must also contain a weather processing component, printers, and plotters through which the output data are viewed and/or saved to data files, if this is required by the user The model diagram of a solar water heating system, which includes two plotters to visually see the running output of the system and two printers to record the produced output, is shown in Figure 46 More details about the TRNSYS program can be found in the program manual and in Reference 68 There are numerous applications of the program in the literature A characteristic change made to version 17, compared with version 16, is the new simulation studio which can create colored connection loops to enable easy interpretation of the system in the simulation Additionally, for complex projects with numerous connections and components, it allows the user to focus exclusively on one component and its connections, making the identification of any errors easy Equations in the new version can be written as lines of code in a single editor window as text, which saves time for users who implement many equations in a simulation 3.11.9.3 WATSUN Simulation Program The WATSUN program simulates only active solar systems It was developed originally by the WATSUN Simulation Laboratory of the University of Waterloo, Canada, in the early 1970s and subsequently in the 1980s [69] The program complexity fills the gap 414 Components Plotter Weather Pump Collectors Tee piece Type2b Tank Load profile Daily load Daily Integration Plotter Diverter Daily results Efficiencies Simulation Integration Totals Figure 46 Model diagram of a solar water heating system in simulation studio between the simple tools used for quick assessments and the more complete, full simulation programs that provide more flexibility but are harder to use The complete list of systems that can be simulated by the original program is as follows: • • • • • • • DHW systems with stratified storage tanks or without storage and heat exchangers Phase-change system for water boiling Sun Switch system, stratified tank with heater Swimming pool heating system (indoor/outdoor) Industrial process heat system-reclaim before or after the collector, with and without storage Variable volume tank-base system Space heating system for one-room building Recently, Natural Resources Canada (NRCan) developed a new version of the program called WATSUN 2009 [70] This new version is also used for the design and simulation of active solar systems and is freely available from the NRCan website [70] Both versions of the program focus on the hourly simulation of solar energy systems and use similar equations for the modeling of basic components; however, the new program was rewritten from scratch, in C++, using object-oriented techniques The program currently models two different kinds of systems: water heating systems with and without storage The latter actually covers a variety of system configurations in which the heat exchanger can be omitted, the auxiliary tank/heater can be replaced with an online heater, and the preheat tank can be either fully mixed or stratified Simple entry tables are used where the main parameters of the system (collector size, collector performance equation, tank size, etc.) can be entered easily The program simulates the interactions between the system and its environment by using an hourly time step However, the program can sometimes break down when sub-hourly intervals are required by the numerical solver, usually when ON/OFF controllers change state WATSUN is a ready-made program that the user can learn and operate easily It combines collection, storage, and load information provided by the user with hourly weather data for a specific location and calculates the system state every hour The output of the program provides information necessary for long-term performance calculations The program models each compo nent of the system, such as collector, pipes, tanks, and so on,individually and provides globally convergent methods to calculate their state As was seen above, WATSUN uses hourly weather data on solar radiation on the horizontal plane, dry-bulb ambient tempera ture, and in the case of unglazed collectors, wind speed At the moment, WATSUN TMY files and comma or blank-delimited ASCII files are recognized by the program Modeling and Simulation of Passive and Active Solar Thermal Systems 415 The system is an assembly of solar collectors, storage tanks, and load devices that the user wants to assess The WATSUN simulation interacts with the user through a series of files Such a file includes a collection of information, labeled and placed in a specific location Files are also used by the program to input and output information There is however one input file defined by the user, called the simulation data file, which includes all the system input data supplied by the user The file is made up of data blocks that contain groups of related parameters The simulation data file controls also the simulation The parameters in this file specify the simulation period, weather data, and output options The simulation data file also contains information about the physical characteristics of the collector, storage device(s), heat exchangers, and load When the simulation is finished, the program produces three output files: a listing file, an hourly data file, and a monthly data file The outputs of the program include a summary of the simulation and a file containing simulation results summed by month The energy balance of the system on a monthly basis includes solar gains, energy delivered, auxiliary energy, and parasitic gains from pumps This file can be readily imported into a spreadsheet program for further analysis and plotting of graphs WATSUN gives also the option to output data on a sub-hourly basis, which enables the user to analyze the result of the simulation in greater detail and facilitates comparison with monitored data when these are available Another use of WATSUN is the simulation of active solar systems for which monitored data are available This can be done either for validation purposes or to identify areas of improvement for the system For this purpose, WATSUN allows the user to enter monitored data from a separate file called the ‘alternate input file’ Monitored climatic data, energy collected, and other data can be read from the alternate input file and override the values normally used in the program The program can also print out strategic variables (such as collector temperature and temperature of water delivered to the load) on an hourly basis for comparison with monitored values The program was validated against the TRNSYS program by Thevenard [71] by using several test cases These programto-program comparisons were very satisfactory and gave prediction differences in yearly energy delivered of less than 1.2% for all configurations tested 3.11.10 Limitations of Simulations Simulations are powerful tools for solar systems design and offer, as was seen above, a number of advantages; there are, however, limits to their use For example, it is easy to make mistakes, such as assume wrong constants or parameters and neglect important performance factors As in all other engineering calculations, a high level of skill and scientific judgment is required in order to produce accurate and useful results [66] Generally, in simulations, it is very difficult to represent in detail some of the phenomena taking place in real systems Additionally, practical problems cannot be easily modeled or accounted for, for example, bad system installation, plugged pipes, leaks, problematic operation of controllers, scale on heat exchanger surfaces, and poor insulation of collectors and other equipment Simulation programs deal only with the thermal behavior of the processes; however, mechanical and 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