An outline of all sections included in this SRS is recorded here for easy reference.
This section records information for easy reference.
The unit system used throughout is SI (Système International d'Unités). In addition to the basic units, several derived units are also used. For each unit, the Table of Units lists the symbol, a description, and the SI name.
Symbol | Description | SI Name |
---|---|---|
°C | temperature | centigrade |
J | energy | joule |
kg | mass | kilogram |
m | length | metre |
s | time | second |
W | power | watt |
The symbols used in this document are summarized in the Table of Symbols along with their units. The choice of symbols was made to be consistent with the heat transfer literature and with existing documentation for solar water heating systems. The symbols are listed in alphabetical order. For vector quantities, the units shown are for each component of the vector.
Symbol | Description | Units |
---|---|---|
AC | Heating coil surface area | m2 |
ACmax | Maximum surface area of coil | m2 |
Ain | Surface area over which heat is transferred in | m2 |
Aout | Surface area over which heat is transferred out | m2 |
AP | Phase change material surface area | m2 |
AR | Aspect ratio | -- |
ARmax | Maximum aspect ratio | -- |
ARmin | Minimum aspect ratio | -- |
C | Specific heat capacity | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CL | Specific heat capacity of a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CS | Specific heat capacity of a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CV | Specific heat capacity of a vapour | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPL | Specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPS | Specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
Ctol | Relative tolerance for conservation of energy | -- |
CW | Specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CWmax | Maximum specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CWmin | Minimum specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPLmax | Maximum specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPLmin | Minimum specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPSmax | Maximum specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPSmin | Minimum specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
D | Diameter of tank | m |
E | Sensible heat | J |
EP | Change in heat energy in the PCM | J |
EW | Change in heat energy in the water | J |
EPmeltinit | Change in heat energy in the PCM at the instant when melting begins | J |
g | Volumetric heat generation per unit volume | \(\frac{\text{W}}{\text{m}^{3}}\) |
Hf | Specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}}\) |
Hfmax | Maximum specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
Hfmin | Minimum specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
h | Convective heat transfer coefficient | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hC | Convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hCmax | Maximum convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hCmin | Minimum convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hmin | Minimum thickness of a sheet of PCM | m |
hP | Convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hPmax | Maximum convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hPmin | Minimum convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
L | Length of tank | m |
Lmax | Maximum length of tank | m |
Lmin | Minimum length of tank | m |
m | Mass | kg |
mP | Mass of phase change material | kg |
mW | Mass of water | kg |
MINFRACT | Minimum fraction of the tank volume taken up by the PCM | -- |
n̂ | Unit outward normal vector for a surface | -- |
Q | Latent heat | J |
QP | Latent heat energy added to PCM | J |
q | Heat flux | \(\frac{\text{W}}{\text{m}^{2}}\) |
qC | Heat flux into the water from the coil | \(\frac{\text{W}}{\text{m}^{2}}\) |
qin | Heat flux input | \(\frac{\text{W}}{\text{m}^{2}}\) |
qout | Heat flux output | \(\frac{\text{W}}{\text{m}^{2}}\) |
qP | Heat flux into the PCM from water | \(\frac{\text{W}}{\text{m}^{2}}\) |
q | Thermal flux vector | \(\frac{\text{W}}{\text{m}^{2}}\) |
S | Surface | m2 |
T | Temperature | °C |
ΔT | Change in temperature | °C |
Tboil | Boiling point temperature | °C |
TC | Temperature of the heating coil | °C |
Tenv | Temperature of the environment | °C |
Tinit | Initial temperature | °C |
Tmelt | Melting point temperature | °C |
TmeltP | Melting point temperature for PCM | °C |
TP | Temperature of the phase change material | °C |
TW | Temperature of the water | °C |
t | Time | s |
tfinal | Final time | s |
tfinalmax | Maximum final time | s |
tmeltfinal | Time at which melting of PCM ends | s |
tmeltinit | Time at which melting of PCM begins | s |
tstep | Time step for simulation | s |
V | Volume | m3 |
VP | Volume of PCM | m3 |
Vtank | Volume of the cylindrical tank | m3 |
VW | Volume of water | m3 |
η | ODE parameter related to decay rate | -- |
π | Ratio of circumference to diameter for any circle | -- |
ρ | Density | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρP | Density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρPmax | Maximum density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρPmin | Minimum density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρW | Density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρWmax | Maximum density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρWmin | Minimum density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
τ | Dummy variable for integration over time | s |
τPL | ODE parameter for liquid PCM | s |
τPS | ODE parameter for solid PCM | s |
τW | ODE parameter for water related to decay time | s |
ϕ | Melt fraction | -- |
∇ | Gradient | -- |
Abbreviation | Full Form |
---|---|
A | Assumption |
DD | Data Definition |
GD | General Definition |
GS | Goal Statement |
IM | Instance Model |
LC | Likely Change |
ODE | Ordinary Differential Equation |
PCM | Phase Change Material |
PS | Physical System Description |
R | Requirement |
RHS | Right-Hand Side |
RefBy | Referenced by |
Refname | Reference Name |
SRS | Software Requirements Specification |
SWHS | Solar Water Heating System |
TM | Theoretical Model |
UC | Unlikely Change |
Uncert. | Typical Uncertainty |
Due to increasing costs, diminishing availability, and negative environmental impact of fossil fuels, the demand is high for renewable energy sources and energy storage technology. Solar water heating systems incorporating phase change material (PCM) use a renewable energy source and provide a novel way of storing energy. Solar water heating systems incorporating PCM improve over the traditional solar water heating systems because of their smaller size. The smaller size is possible because of the ability of PCM to store thermal energy as latent heat, which allows higher thermal energy storage capacity per unit weight.
The following section provides an overview of the Software Requirements Specification (SRS) for solar water heating systems incorporating PCM. The developed program will be referred to as Solar Water Heating System (SWHS). This section explains the purpose of this document, the scope of the requirements, the characteristics of the intended reader, and the organization of the document.
The primary purpose of this document is to record the requirements of SWHS. Goals, assumptions, theoretical models, definitions, and other model derivation information are specified, allowing the reader to fully understand and verify the purpose and scientific basis of SWHS. With the exception of system constraints, this SRS will remain abstract, describing what problem is being solved, but not how to solve it.
This document will be used as a starting point for subsequent development phases, including writing the design specification and the software verification and validation plan. The design document will show how the requirements are to be realized, including decisions on the numerical algorithms and programming environment. The verification and validation plan will show the steps that will be used to increase confidence in the software documentation and the implementation. Although the SRS fits in a series of documents that follow the so-called waterfall model, the actual development process is not constrained in any way. Even when the waterfall model is not followed, as Parnas and Clements point out parnasClements1986, the most logical way to present the documentation is still to "fake" a rational design process.
The scope of the requirements includes thermal analysis of a single solar water heating tank incorporating PCM. This entire document is written assuming that the substances inside the solar water heating tank are water and PCM.
Reviewers of this documentation should have an understanding of heat transfer theory from level 3 or 4 mechanical engineering and differential equations from level 1 and 2 calculus. The users of SWHS can have a lower level of expertise, as explained in Sec:User Characteristics.
The organization of this document follows the template for an SRS for scientific computing software proposed by koothoor2013, smithLai2005, smithEtAl2007, and smithKoothoor2016. The presentation follows the standard pattern of presenting goals, theories, definitions, and assumptions. For readers that would like a more bottom up approach, they can start reading the instance models and trace back to find any additional information they require.
The goal statements are refined to the theoretical models and the theoretical models to the instance models. The instance models to be solved are referred to as IM:eBalanceOnWtr, IM:eBalanceOnPCM, IM:heatEInWtr, and IM:heatEInPCM. The instance models provide the ordinary differential equations (ODEs) and algebraic equations that model the solar water heating systems incorporating PCM. SWHS solves these ODEs.
This section provides general information about the system. It identifies the interfaces between the system and its environment, describes the user characteristics, and lists the system constraints.
Fig:SysCon shows the system context. A circle represents an external entity outside the software, the user in this case. A rectangle represents the software system itself (SWHS). Arrows are used to show the data flow between the system and its environment.
SWHS is mostly self-contained. The only external interaction is through the user interface. The responsibilities of the user and the system are as follows:
The end user of SWHS should have an understanding of undergraduate Level 1 Calculus and Physics.
There are no system constraints.
This section first presents the problem description, which gives a high-level view of the problem to be solved. This is followed by the solution characteristics specification, which presents the assumptions, theories, and definitions that are used.
A system is needed to investigate the effect of employing PCM within a solar water heating tank.
This subsection provides a list of terms that are used in the subsequent sections and their meaning, with the purpose of reducing ambiguity and making it easier to correctly understand the requirements.
The physical system of SWHS, as shown in Fig:Tank, includes the following elements:
PS1: Tank containing water.
PS2: Heating coil at bottom of tank. (qC represents the heat flux into the water from the coil.)
PS3: PCM suspended in tank. (qP represents the heat flux into the PCM from water.)
Given the temperature of the heating coil, the initial conditions for the temperature of the water and the temperature of the phase change material, and the material properties, the goal statements are:
The instance models that govern SWHS are presented in the Instance Model Section. The information to understand the meaning of the instance models and their derivation is also presented, so that the instance models can be verified.
This section simplifies the original problem and helps in developing the theoretical models by filling in the missing information for the physical system. The assumptions refine the scope by providing more detail.
This section focuses on the general equations and laws that SWHS is based on.
Refname | TM:consThermE |
---|---|
Label | Conservation of thermal energy |
Equation | \[-∇\cdot{}\symbf{q}+g=ρ C \frac{\,\partial{}T}{\,\partial{}t}\] |
Description | |
Notes |
The above equation gives the law of conservation of energy for transient heat transfer in a given material. For this equation to apply, other forms of energy, such as mechanical energy, are assumed to be negligible in the system (A:Thermal-Energy-Only). |
Source | |
RefBy |
Refname | TM:sensHtE |
---|---|
Label | Sensible heat energy |
Equation | \[E=\begin{cases} {C^{\text{S}}} m ΔT, & T\lt{}{T_{\text{melt}}}\\ {C^{\text{L}}} m ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\ {C^{\text{V}}} m ΔT, & {T_{\text{boil}}}\lt{}T \end{cases}\] |
Description | |
Notes |
Sensible heating occurs as long as the material does not reach a temperature where a phase change occurs. A phase change occurs if T = Tboil or T = Tmelt. If this is the case, refer to TM:latentHtE. |
Source | |
RefBy |
Refname | TM:latentHtE |
---|---|
Label | Latent heat energy |
Equation | \[Q\left(t\right)=\int_{0}^{t}{\frac{\,dQ\left(τ\right)}{\,dτ}}\,dτ\] |
Description | |
Notes |
Q is the change in thermal energy (latent heat energy). Q(t) = ∫0t\(\frac{\,dQ\left(τ\right)}{\,dτ}\) dτ is the rate of change of Q with respect to time τ. t is the time elapsed, as long as the phase change is not complete. The status of the phase change depends on the melt fraction (from DD:meltFrac). Latent heating stops when all material has changed to the new phase. |
Source | |
RefBy |
Refname | TM:nwtnCooling |
---|---|
Label | Newton's law of cooling |
Equation | \[q\left(t\right)=h ΔT\left(t\right)\] |
Description | |
Notes |
Newton's law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings. h is assumed to be independent of T (from A:Heat-Transfer-Coeffs-Constant). ΔT(t) = T(t)−Tenv(t) is the time-dependant thermal gradient between the environment and the object. |
Source |
incroperaEtAl2007 (pg. 8) |
RefBy |
This section collects the laws and equations that will be used to build the instance models.
Refname | GD:rocTempSimp |
---|---|
Label |
Simplified rate of change of temperature |
Equation | \[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] |
Description | |
Source | -- |
RefBy |
Integrating TM:consThermE over a volume (V), we have:
\[-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\]Applying Gauss's Divergence Theorem to the first term over the surface S of the volume, with q as the thermal flux vector for the surface and n̂ as a unit outward normal vector for a surface:
\[-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\]We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:
\[{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\]Where qin, qout, Ain, and Aout are explained in GD:rocTempSimp. The integral over the surface could be simplified because the thermal flux is assumed constant over Ain and Aout and 0 on all other surfaces. Outward flux is considered positive. Assuming ρ, C, and T are constant over the volume, which is true in our case by A:Constant-Water-Temp-Across-Tank, A:Temp-PCM-Constant-Across-Volume, A:Density-Water-PCM-Constant-over-Volume, and A:Specific-Heat-Energy-Constant-over-Volume, we have:
\[ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\]Using the fact that ρ=m/V, Equation (2) can be written as:
\[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\]Refname | GD:htFluxWaterFromCoil |
---|---|
Label |
Heat flux into the water from the coil |
Units |
\(\frac{\text{W}}{\text{m}^{2}}\) |
Equation | \[{q_{\text{C}}}={h_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\] |
Description | |
Notes |
qC is found by assuming that Newton's law of cooling applies (A:Newton-Law-Convective-Cooling-Coil-Water). This law (defined in TM:nwtnCooling) is used on the surface of the heating coil. |
Source | |
RefBy |
Refname | GD:htFluxPCMFromWater |
---|---|
Label | Heat flux into the PCM from water |
Units |
\(\frac{\text{W}}{\text{m}^{2}}\) |
Equation | \[{q_{\text{P}}}={h_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)\] |
Description | |
Notes |
qP is found by assuming that Newton's law of cooling applies (A:Law-Convective-Cooling-Water-PCM). This law (defined in TM:nwtnCooling) is used on the surface of the phase change material. |
Source | |
RefBy |
This section collects and defines all the data needed to build the instance models.
Refname | DD:waterMass |
---|---|
Label | Mass of water |
Symbol | mW |
Units | kg |
Equation | \[{m_{\text{W}}}={V_{\text{W}}} {ρ_{\text{W}}}\] |
Description | |
Source | -- |
RefBy |
Refname | DD:waterVolume.pcm |
---|---|
Label | Volume of water |
Symbol | VW |
Units | m3 |
Equation | \[{V_{\text{W}}}={V_{\text{tank}}}-{V_{\text{P}}}\] |
Description | |
Notes |
Based on A:Volume-Coil-Negligible. Vtank is defined in DD:tankVolume. |
Source | -- |
RefBy |
Refname | DD:tankVolume |
---|---|
Label | Volume of the cylindrical tank |
Symbol | Vtank |
Units | m3 |
Equation | \[{V_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L\] |
Description | |
Source | -- |
RefBy |
Refname | DD:balanceDecayRate |
---|---|
Label |
ODE parameter for water related to decay time |
Symbol | τW |
Units | s |
Equation | \[{τ_{\text{W}}}=\frac{{m_{\text{W}}} {C_{\text{W}}}}{{h_{\text{C}}} {A_{\text{C}}}}\] |
Description | |
Source | |
RefBy |
Refname | DD:balanceDecayTime |
---|---|
Label |
ODE parameter related to decay rate |
Symbol | η |
Units | Unitless |
Equation | \[η=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}}\] |
Description | |
Source | |
RefBy |
Refname | DD:balanceSolidPCM |
---|---|
Label | ODE parameter for solid PCM |
Symbol |
τPS |
Units | s |
Equation | \[{{τ_{\text{P}}}^{\text{S}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}}{{h_{\text{P}}} {A_{\text{P}}}}\] |
Description | |
Source | |
RefBy |
Refname | DD:balanceLiquidPCM |
---|---|
Label | ODE parameter for liquid PCM |
Symbol |
τPL |
Units | s |
Equation | \[{{τ_{\text{P}}}^{\text{L}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{L}}}}{{h_{\text{P}}} {A_{\text{P}}}}\] |
Description | |
Source | |
RefBy |
Refname | DD:htFusion |
---|---|
Label | Specific latent heat of fusion |
Symbol | Hf |
Units |
\(\frac{\text{J}}{\text{kg}}\) |
Equation | \[{H_{\text{f}}}=\frac{Q}{m}\] |
Description | |
Notes |
The specific latent heat of fusion (also known as the enthalpy of fusion) of a substance is the heat energy required (Q) to change the state of a unit of the mass (m) of the substance from solid to liquid, at constant pressure. |
Source |
bueche1986 (pg. 282) |
RefBy |
Refname | DD:meltFrac |
---|---|
Label | Melt fraction |
Symbol | ϕ |
Units | Unitless |
Equation | \[ϕ=\frac{{Q_{\text{P}}}}{{H_{\text{f}}} {m_{\text{P}}}}\] |
Description | |
Notes |
The value of ϕ is constrained to 0 ≤ ϕ ≤ 1. |
Source | |
RefBy |
Refname | DD:aspectRatio |
---|---|
Label | Aspect ratio |
Symbol | AR |
Units | Unitless |
Equation | \[\mathit{AR}=\frac{D}{L}\] |
Description | |
Source | -- |
RefBy |
This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.
The goals GS:Predict-Water-Temperature, GS:Predict-PCM-Temperature, GS:Predict-Water-Energy, and GS:Predict-PCM-Energy are solved by IM:eBalanceOnWtr, IM:eBalanceOnPCM, IM:heatEInWtr, and IM:heatEInPCM. The solutions for IM:eBalanceOnWtr and IM:eBalanceOnPCM are coupled since the solutions for TW and TP depend on one another. IM:heatEInWtr can be solved once IM:eBalanceOnWtr has been solved. The solutions of IM:eBalanceOnPCM and IM:heatEInPCM are also coupled, since the temperature of the phase change material and the change in heat energy in the PCM depend on the phase change.
Refname | IM:eBalanceOnWtr |
---|---|
Label |
Energy balance on water to find the temperature of the water |
Input |
mW, CW, hC, AP, hP, AC, TP, tfinal, TC, Tinit |
Output | TW |
Input Constraints | \[{T_{\text{C}}}\gt{}{T_{\text{init}}}\] |
Output Constraints | |
Equation | \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η \left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right)\] |
Description | |
Notes |
TP is defined by IM:eBalanceOnPCM. The input constraint Tinit ≤ TC comes from A:Charging-Tank-No-Temp-Discharge. τW is calculated from DD:balanceDecayRate. η is calculated from DD:balanceDecayTime. The initial conditions for the ODE are TW(0) = TP(0) = Tinit following A:Same-Initial-Temp-Water-PCM. The ODE applies as long as the water is in liquid form, 0 < TW < 100 (°C) where 0 (°C) and 100 (°C) are the melting and boiling point temperatures of water, respectively (from A:Water-Always-Liquid and A:Atmospheric-Pressure-Tank). |
Source | |
RefBy |
IM:eBalanceOnWtr, IM:eBalanceOnPCM, UC:No-Internal-Heat-Generation, FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values |
To find the rate of change of TW, we look at the energy balance on water. The volume being considered is the volume of water in the tank VW, which has mass mW and specific heat capacity, CW. Heat transfer occurs in the water from the heating coil as qC (GD:htFluxWaterFromCoil) and from the water into the PCM as qP (GD:htFluxPCMFromWater), over areas AC and AP, respectively. The thermal flux is constant over AC, since the temperature of the heating coil is assumed to not vary along its length (A:Temp-Heating-Coil-Constant-over-Length), and the thermal flux is constant over AP, since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (A:Perfect-Insulation-Tank). Since the assumption is made that no internal heat is generated (A:No-Internal-Heat-Generation-By-Water-PCM), g = 0. Therefore, the equation for GD:rocTempSimp can be written as:
\[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}} {A_{\text{C}}}-{q_{\text{P}}} {A_{\text{P}}}\]Using GD:htFluxWaterFromCoil for qC and GD:htFluxPCMFromWater for qP, this can be written as:
\[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\]Dividing Equation (2) by mW CW, we obtain:
\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\]Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by hC AC / hC AC yields:
\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}} {A_{\text{C}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\]Rearranging this equation gives us:
\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\]By substituting τW (from DD:balanceDecayRate) and η (from DD:balanceDecayTime), this can be written as:
\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\]Finally, factoring out \(\frac{1}{{τ_{\text{W}}}}\), we are left with the governing ODE for IM:eBalanceOnWtr:
\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}+η \left({T_{\text{P}}}-{T_{\text{W}}}\right)\right)\]Refname | IM:eBalanceOnPCM |
---|---|
Label |
Energy Balance on PCM to find temperature of PCM |
Input |
TmeltP, tfinal, Tinit, AP, hP, mP, CPS, CPL |
Output | TP |
Input Constraints | \[{{T_{\text{melt}}}^{\text{P}}}\gt{}{T_{\text{init}}}\] |
Output Constraints | |
Equation | \[\frac{\,d{T_{\text{P}}}}{\,dt}=\begin{cases} \frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ \frac{1}{{{τ_{\text{P}}}^{\text{L}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ 0, & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases}\] |
Description | |
Notes |
TW is defined by IM:eBalanceOnWtr. The input constraint Tinit ≤ TmeltP comes from A:PCM-Initially-Solid. The temperature remains constant at TmeltP, even with the heating (or cooling), until the phase change has occurred for all of the material; that is as long as 0 < ϕ < 1. ϕ (from DD:meltFrac) is determined as part of the heat energy in the PCM, as given in (IM:heatEInPCM). τPS is calculated in DD:balanceSolidPCM. τPL is calculated in DD:balanceLiquidPCM. The initial conditions for the ODE are TW(0) = TP(0) = Tinit following A:Same-Initial-Temp-Water-PCM. |
Source | |
RefBy |
IM:eBalanceOnWtr, UC:No-Internal-Heat-Generation, UC:No-Gaseous-State, FR:Output-Values, FR:Find-Mass, FR:Calculate-Values, FR:Calculate-PCM-Melt-End-Time, and FR:Calculate-PCM-Melt-Begin-Time |
To find the rate of change of TP, we look at the energy balance on the PCM. The volume being considered is the volume of PCM (VP). The derivation that follows is initially for the solid PCM. The mass of phase change material is mP and the specific heat capacity of PCM as a solid is CPS. The heat flux into the PCM from water is qP (GD:htFluxPCMFromWater) over phase change material surface area AP. The thermal flux is constant over AP, since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume (A:No-Internal-Heat-Generation-By-Water-PCM), g = 0, the equation for GD:rocTempSimp can be written as:
\[{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}} {A_{\text{P}}}\]Using GD:htFluxPCMFromWater for qP, this equation can be written as:
\[{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\]Dividing by mP CPS we obtain:
\[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\]By substituting τPS (from DD:balanceSolidPCM), this can be written as:
\[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\]Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that CPS is replaced by CPL, and thus τPS is replaced by τPL. Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible (A:Volume-Change-Melting-PCM-Negligible).
In the case where TP = TmeltP and not all of the PCM is melted, the temperature of the phase change material does not change. Therefore, d TP / d t = 0.
This derivation does not consider the boiling of the PCM, as the PCM is assumed to either be in a solid state or a liquid state (A:No-Gaseous-State-PCM).
Refname | IM:heatEInWtr |
---|---|
Label | Heat energy in the water |
Input |
Tinit, mW, CW, mW |
Output | EW |
Input Constraints | |
Output Constraints | |
Equation | \[{E_{\text{W}}}\left(t\right)={C_{\text{W}}} {m_{\text{W}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)\] |
Description | |
Notes |
The above equation is derived using TM:sensHtE. The change in temperature is the difference between the temperature at time t (s), TW and the initial temperature, Tinit (°C). This equation applies as long as 0 < TW < 100°C (A:Water-Always-Liquid, A:Atmospheric-Pressure-Tank). |
Source | |
RefBy |
Refname | IM:heatEInPCM |
---|---|
Label | Heat energy in the PCM |
Input |
TmeltP, tfinal, Tinit, AP, hP, mP, CPS, CPL, TP, Hf, tmeltinit |
Output | EP |
Input Constraints | \[{{T_{\text{melt}}}^{\text{P}}}\gt{}{T_{\text{init}}}\] |
Output Constraints | |
Equation | \[{E_{\text{P}}}=\begin{cases} {{C_{\text{P}}}^{\text{S}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}} {m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{\text{P}}}\left(t\right), & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases}\] |
Description | |
Notes |
The above equation is derived using TM:sensHtE and TM:latentHtE. EP for the solid PCM is found using TM:sensHtE for sensible heating, with the specific heat capacity of the solid PCM, CPS (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\)) and the change in the PCM temperature from the initial temperature (°C). EP for the melted PCM (TP > EPmeltinit) is found using TM:sensHtE for sensible heat of the liquid PCM plus the energy when melting starts, plus the energy required to melt all of the PCM. The energy required to melt all of the PCM is Hf mP (J) (from DD:htFusion). The change in temperature is TP−TmeltP (°C). EP during melting of the PCM is found using the energy required at the instant melting of the PCM begins, EPmeltinit plus the latent heat energy added to the PCM, QP (J) since the time when melting began tmeltinit (s). The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state (A:No-Gaseous-State-PCM) (A:PCM-Initially-Solid). |
Source | |
RefBy |
IM:eBalanceOnPCM, UC:No-Gaseous-State, FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values |
The Data Constraints Table shows the data constraints on the input variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable. The uncertainty column provides an estimate of the confidence with which the physical quantities can be measured. This information would be part of the input if one were performing an uncertainty quantification exercise. The constraints are conservative to give the user of the model the flexibility to experiment with unusual situations. The column of typical values is intended to provide a feel for a common scenario. The column for software constraints restricts the range of inputs to reasonable values. (*) These quantities cannot be equal to zero, or there will be a divide by zero in the model. (+) These quantities cannot be zero, or there would be freezing (A:PCM-Initially-Solid). (++) The constraints on the surface area are calculated by considering the surface area to volume ratio. The assumption is that the lowest ratio is 1 and the highest possible is \(\frac{2}{{h_{\text{min}}}}\), where hmin is the thickness of a "sheet" of PCM. A thin sheet has the greatest surface area to volume ratio. (**) The constraint on the maximum time at the end of the simulation is the total number of seconds in one day.
Var | Physical Constraints | Software Constraints | Typical Value | Uncert. |
---|---|---|---|---|
AC | AC > 0 | AC ≤ ACmax | 0.12 m2 | 10% |
AP | AP > 0 | VP ≤ AP ≤ \(\frac{2}{{h_{\text{min}}}}\) Vtank | 1.2 m2 | 10% |
CPL | CPL > 0 | CPLmin < CPL < CPLmax | 2270 \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) | 10% |
CPS | CPS > 0 | CPSmin < CPS < CPSmax | 1760 \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) | 10% |
CW | CW > 0 | CWmin < CW < CWmax | 4186 \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) | 10% |
D | D > 0 | ARmin ≤ D ≤ ARmax | 0.412 m | 10% |
Hf | Hf > 0 | Hfmin < Hf < Hfmax | 211600 \(\frac{\text{J}}{\text{kg}}\) | 10% |
hC | hC > 0 | hCmin ≤ hC ≤ hCmax | 1000 \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) | 10% |
hP | hP > 0 | hPmin ≤ hP ≤ hPmax | 1000 \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) | 10% |
L | L > 0 | Lmin ≤ L ≤ Lmax | 1.5 m | 10% |
TC | 0 < TC < 100 | -- | 50 °C | 10% |
Tinit | 0 < Tinit < Tmelt | -- | 40 °C | 10% |
TmeltP | 0 < TmeltP < TC | -- | 44.2 °C | 10% |
tfinal | tfinal > 0 | tfinal < tfinalmax | 50000 s | 10% |
tstep | 0 < tstep < tfinal | -- | 0.01 s | 10% |
VP | 0 < VP < Vtank | VP ≥ MINFRACT Vtank | 0.05 m3 | 10% |
ρP | ρP > 0 | ρPmin < ρP < ρPmax | 1007 \(\frac{\text{kg}}{\text{m}^{3}}\) | 10% |
ρW | ρW > 0 | ρWmin < ρW ≤ ρWmax | 1000 \(\frac{\text{kg}}{\text{m}^{3}}\) | 10% |
The Data Constraints Table shows the data constraints on the output variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable.
Var | Physical Constraints |
---|---|
TW | Tinit ≤ TW ≤ TC |
TP | Tinit ≤ TP ≤ TC |
EW | EW ≥ 0 |
EP | EP ≥ 0 |
A correct solution must exhibit the law of conservation of energy. This means that the change in heat energy in the water should equal the difference between the total energy input from the heating coil and the energy output to the PCM. This can be shown as an equation by taking GD:htFluxWaterFromCoil and GD:htFluxPCMFromWater, multiplying each by their respective surface area of heat transfer, and integrating each over the simulation time, as follows:
\[{E_{\text{W}}}=\int_{0}^{t}{{h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)}\,dt-\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\]In addition, the change in heat energy in the PCM should equal the energy input to the PCM from the water. This can be expressed as
\[{E_{\text{P}}}=\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\]Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as "sanity" checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than Ctol FR:Verify-Energy-Output-Follow-Conservation-of-Energy.
This section provides the functional requirements, the tasks and behaviours that the software is expected to complete, and the non-functional requirements, the qualities that the software is expected to exhibit.
This section provides the functional requirements, the tasks and behaviours that the software is expected to complete.
Symbol | Description | Units |
---|---|---|
AC | Heating coil surface area | m2 |
AP | Phase change material surface area | m2 |
Atol | Absolute tolerance | -- |
CPL | Specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPS | Specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CW | Specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
D | Diameter of tank | m |
Hf | Specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}}\) |
hC | Convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hP | Convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
L | Length of tank | m |
Rtol | Relative tolerance | -- |
TC | Temperature of the heating coil | °C |
Tinit | Initial temperature | °C |
TmeltP | Melting point temperature for PCM | °C |
tfinal | Final time | s |
tstep | Time step for simulation | s |
VP | Volume of PCM | m3 |
ρP | Density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρW | Density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
This section provides the non-functional requirements, the qualities that the software is expected to exhibit.
This section lists the likely changes to be made to the software.
This section lists the unlikely changes to be made to the software.
The purpose of the traceability matrices is to provide easy references on what has to be additionally modified if a certain component is changed. Every time a component is changed, the items in the column of that component that are marked with an "X" should be modified as well. Tab:TraceMatAvsA shows the dependencies of the assumptions on each other. Tab:TraceMatAvsAll shows the dependencies of the data definitions, theoretical models, general definitions, instance models, requirements, likely changes, and unlikely changes on the assumptions. Tab:TraceMatRefvsRef shows the dependencies of the data definitions, theoretical models, general definitions, and instance models on each other. Tab:TraceMatAllvsR shows the dependencies of the requirements and goal statements on the data definitions, theoretical models, general definitions, and instance models.
The purpose of the traceability graphs is also to provide easy references on what has to be additionally modified if a certain component is changed. The arrows in the graphs represent dependencies. The component at the tail of an arrow is depended on by the component at the head of that arrow. Therefore, if a component is changed, the components that it points to should also be changed. Fig:TraceGraphAvsA shows the dependencies of assumptions on each other. Fig:TraceGraphAvsAll shows the dependencies of data definitions, theoretical models, general definitions, instance models, requirements, likely changes, and unlikely changes on the assumptions. Fig:TraceGraphRefvsRef shows the dependencies of data definitions, theoretical models, general definitions, and instance models on each other. Fig:TraceGraphAllvsR shows the dependencies of requirements and goal statements on the data definitions, theoretical models, general definitions, and instance models. Fig:TraceGraphAllvsAll shows the dependencies of dependencies of assumptions, models, definitions, requirements, goals, and changes with each other.
For convenience, the following graphs can be found at the links below:
This section contains the standard values that are used for calculations in SWHS.
Symbol | Description | Value | Unit |
---|---|---|---|
ACmax | maximum surface area of coil | 100000 | m2 |
ARmax | maximum aspect ratio | 100 | -- |
ARmin | minimum aspect ratio | 0.01 | -- |
Ctol | relative tolerance for conservation of energy | 0.001% | -- |
CWmax | maximum specific heat capacity of water | 4210 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CWmin | minimum specific heat capacity of water | 4170 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPLmax | maximum specific heat capacity of PCM as a liquid | 5000 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPLmin | minimum specific heat capacity of PCM as a liquid | 100 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPSmax | maximum specific heat capacity of PCM as a solid | 4000 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
CPSmin | minimum specific heat capacity of PCM as a solid | 100 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
Hfmax | maximum specific latent heat of fusion | 1000000 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
Hfmin | minimum specific latent heat of fusion | 0 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
hCmax | maximum convective heat transfer coefficient between coil and water | 10000 | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hCmin | minimum convective heat transfer coefficient between coil and water | 10 | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hPmax | maximum convective heat transfer coefficient between PCM and water | 10000 | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
hPmin | minimum convective heat transfer coefficient between PCM and water | 10 | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
Lmax | maximum length of tank | 50 | m |
Lmin | minimum length of tank | 0.1 | m |
MINFRACT | minimum fraction of the tank volume taken up by the PCM | 1.0⋅10-6 | -- |
tfinalmax | maximum final time | 86400 | s |
ρPmax | maximum density of PCM | 20000 | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρPmin | minimum density of PCM | 500 | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρWmax | maximum density of water | 1000 | \(\frac{\text{kg}}{\text{m}^{3}}\) |
ρWmin | minimum density of water | 950 | \(\frac{\text{kg}}{\text{m}^{3}}\) |