The performances of the proposed novel developed hybrid system are evaluated theoretically. This section includes the mathematical models of all component of the novel developed hybrid system proposed in this study; photovoltaic/thermal panels, solar dish concentrator collector, hot water tank, air–water heat exchanger, solar dryer with energy storage materials, developed solar distiller with energy storage materials and air injection system, and the external condenser.
Photovoltaic/thermal modeling
This section presents detailed equations for mathematical models used to predict the performance of photovoltaic/thermal panels51,52:
Thermal efficiency of photovoltaic/thermal collector (\({\eta }_{PVT,th}\)) is calculated as:
$${\eta }_{PVT,th}=\frac{{\dot{Q}}_{u,PVT}}{I(t). {A}_{PVT}},$$
(1)
where \({A}_{PVT}\) is the surface area of photovoltaic/thermal panel (m2), I(t) is the intensity solar rays (W/m2), \({\dot{Q}}_{u,PV}\) is the useful heat energy of photovoltaic/thermal collector (W) which evaluated as:
$${Q}_{u,PVT}={\dot{m}}_{a}. {C}_{p,a}. \left({T}_{a, out}-{T}_{a,in}\right),$$
(2)
$${Q}_{u, PVT}={A}_{PVT}. {F}_{R}.\left[I-{U}_{L, PVT}.\left({T}_{PV,cell}-{T}_{amb}\right)\right],$$
(3)
where \({\dot{m}}_{a}\) and \({C}_{p,a}\) are the mass flow rate of inlet air to PVT collector (kg/s), air specific heat (J/kg.oC), \({T}_{a,in}\) and \({T}_{a, out}\) are the air temperatures at entering and outlet of PVT collector (oC), \({T}_{amb}\) is the ambient air temperature (oC), \({U}_{L,PVT}\) is overall heat losses of PVT collector (W/m2.oC), \({T}_{PV,cell}\) is the cell temperature of PV panel (oC), and \({F}_{R}\) is the heat removal factor which calculated as:
$${F}_{R}=\frac{{\dot{m}}_{a} .{C}_{p,a}}{{A}_{PVT}. {U}_{L,PVT}}\left[1-{e}^{\left(\frac{{A}_{PVT}. {U}_{L,PVT}. {F}{\prime}}{{\dot{m}}_{a}. {C}_{p,a}}\right)}\right].$$
(4)
Electrical efficiency of photovoltaic/thermal collector (\({\eta }_{PVT,ele}\)) is calculated as:
$${\eta }_{PVT,ele}=\frac{{P}_{MPP}}{I(t). {A}_{PVT}},$$
(5)
$${\eta }_{PVT,ele}={\eta }_{ref}.\left[1-{\beta }_{ref}\left({T}_{PV,cell}-{T}_{ref}\right)\right].$$
(6)
where \({P}_{MPP}\) is the electrical power at MPP, \({\eta }_{ref}\) is the reference efficiency, and \({\beta }_{ref}\) is the PV temperature coefficient.
Solar dish concentrator modeling
This section includes comprehensive mathematical model equations that are used to forecast solar dish concentrator performance when utilized with a solar thermal receiver, assuming a constant heat flow over the absorber surface53:
Available solar irradiation at collector aperture (\({Q}_{s}\)) is calculated as:
$${Q}_{s}={A}_{ap}. I\left(t\right).$$
(7)
Absorbed solar thermal energy using receiver (\({Q}_{abs, rec.}\)) is calculated using solar collector optical efficiency (ηopt)53:
$${Q}_{abs, rec.}={\eta }_{opt} . {Q}_{s}.$$
(8)
The useful thermal energy from the receiver of solar dish concentrator (\({Q}_{u, rec.}\)) calculated as follows53:
$${Q}_{u, rec.}={\dot{m}}_{oil} {. C}_{p,oil} .\left({T}_{oil,out}-{T}_{oil, in}\right)= {Q}_{abs,rec.}-{Q}_{loss, rec}.$$
(9)
Heat losses from receiver to ambient (\({Q}_{loss, rec.}\)) is calculated as53:
$${Q}_{loss, rec.}={Q}_{conv.}+{Q}_{rad.}={A}_{rec, out}. {h}_{air} . \left({T}_{m,rec.}-{T}_{amb}\right)+{A}_{rec, out}. {\varepsilon }_{r} . \sigma . \left({{T}_{m,rec.}}^{4}-{{T}_{amb}}^{4}\right),$$
(10)
where \({A}_{ap}\) is the effective aperture area of dish concentrator (m2), \({\dot{m}}_{oil}\) is the hot oil mass flow rate (kg/s), \({. C}_{p,oil}\) is the oil specific heat capacity (J/kg oC), \({T}_{oil, in}\) and \({T}_{oil,out}\) is the entering and outlet hot oil temperature of spiral coil (˚C), \({A}_{rec, out}\) is the outer surface area of receiver (m2), \({T}_{m,rec.}\) is the receiver mean temperature (oC), \({h}_{air}\) is coefficient of air convection heat transfer (W/m2 K) which calculated as53:
$${h}_{air}=2.8+\left(3.{V}_{wind}\right).$$
(11)
The convective heat coefficient of oil inside the spiral coil (\({h}_{oil}\)) is calculated as53:
$${h}_{oil}=\frac{{Q}_{u, rec.}}{{A}_{coil}. \left({T}_{m,rec.}-{T}_{m,oil}\right)}=\frac{Nu. k}{{D}_{coil,inner}}.$$
(12)
Mean oil temperature (\({T}_{m,oil}\)) is calculated as:
$${T}_{m,oil}=\frac{{T}_{oil,out}+{T}_{oil, in}}{2}.$$
(13)
Nusselt number Nu is calculated as follows53:
$$Nu=0.023. {Re}^{0.8}{. Pr}^{0.4}.$$
(14)
Reynolds number (\(Re\)) and Prandtl number (\(Pr\)) are calculated as follows:
$$Re=\frac{4 . {\dot{m}}_{oil}}{\pi {. D}_{coil,inner} . {\mu }_{oil}},$$
(15)
$$Pr=\frac{{\mu }_{oil}. {C}_{p,oil}}{k}.$$
(16)
Hot water storage tank modeling
This section contains comprehensive mathematical model equations that are used to forecast hot water storage tank performance. Providing that the hot water in the storage tank is spread evenly54:
The energy balance equations of hot water storage tank written as follows:
$${\dot{m}}_{oil}{. C}_{p,oil}. \left({T}_{oil, in}-{T}_{oil,out}\right)={\dot{m}}_{hw}{. C}_{p,hw}. \frac{{dT}_{hw}}{dt}+{\dot{m}}_{hw}{. C}_{p,hw}. \left({T}_{hw, out}-{T}_{hw,in}\right)+{U}_{loss}{ .A}_{s}. \left({T}_{hw,avg}-{T}_{amb}\right).$$
(17)
Heat exchanger modeling
This section provides comprehensive numerical equations for forecasting water–air heat exchanger performance. Assuming neglected the thermal heat losses to its surrounding54:
The energy balance equations of heat exchanger written as:
$${\dot{m}}_{hw}{. C}_{p,hw}. \left({T}_{hw,in}-{T}_{hw,out}\right)={\dot{m}}_{a}{. C}_{p,a} .\left({T}_{a,out}-{T}_{a,in}\right),$$
(18)
where \({\dot{m}}_{hw}\) and \({C}_{p,hw}\) are the mass flow rate (kg/s) and specific enthalpy (J/kg) of the hot water, \({T}_{hw,in}\) and \({T}_{hw,out}\) are the inlet and outlet hot water temperatures (oC), respectively.
Modeling of indirect solar dryer with energy storage materials
This section presents detailed equations for mathematical models applied in predicting the performance of indirect solar dryer with energy storage materials. Neglecting the thermal energy losses from the dryer to its surrounding55,56:
The energy balance equations of an indirect solar dryer using materials for energy storage written as:
$${\dot{m}}_{a}{.C}_{p,a}{.(T}_{a,in}-{T}_{a,out})= {\dot{Q}}_{d,m }+{\dot{Q}}_{PCM },$$
(19)
where; \({\dot{Q}}_{d,m}\) is the thermal energy required to drying the product material which calculated as55,56:
$${\dot{Q}}_{d,m}={m}_{i,pm}{.C}_{p,pm}{.(T}_{pm.f}-{T}_{pm.i})=MR .{h}_{fg}.$$
(20)
The thermal energy added to energy storage materials (Phase change materials) (\({\dot{Q}}_{PCM}\)) is calculated as55,56:
$${\dot{Q}}_{PCM }={m}_{PCM}.\left[{C}_{p,PCM,s}{(T}_{PCM,m}-{T}_{PCM.i})+ {h}_{fg, PCM}+{C}_{p,PCM,l}{(T}_{PCM,f}-{T}_{PCM,m})\right],$$
(21)
where mi,pm and \({C}_{p,pm}\) are the initial mass (kg) and specific heat of product material (J/kg K), \({T}_{pm.i}\) and \({T}_{pm.f}\) are the initial and final temperature of product materials (K), \(MR\) the amount of water removed in kilogram’s from the product ingredients additionally and \({h}_{fg}\) is latent heat at average temperature [\({(T}_{a,in}+{T}_{a,out})/2\)] (J/kg), \({m}_{PCM}\) is the mass of energy storage materials (kg), \({C}_{p,PCM,s}\) and \({C}_{p,PCM,l}\) are the specific heat of phase change materials in solid and liquid state (J/kg K), \({T}_{PCM.i}\), \({T}_{PCM,m}\), and \({T}_{PCM,f}\) are the initial, melting, and final temperature of phase change materials (K), and \({h}_{fg, PCM}\) is the latent heat of phase change materials (J/kg).
The computation of mass of water removed (MR) from the product materials is given by:
$$MR=\frac{{m}_{i,pm}({MC}_{i,pm}-{MC}_{f,pm})}{(100-{MC}_{f,pm})},$$
(22)
where \({MC}_{i,pm}and {MC}_{f,pm}\) are initial and final moisture content in the product materials.
Modeling of solar distiller with PCM and air injection system
This section presents detailed mathematical model equations used to predict the performance of developed solar distiller with air injection system energy storage materials.
Analytically we solved the glass cover’s energy balance equations, basin water, absorber plate, and energy storage reservoir to determine the glass cover’s temperature, basin water, and absorber plate. The energy balance equations of various component of developed solar distiller with air injection system energy storage materials are expressed as follows57:
Glass cover
$$I\left(t\right). {A}_{\text{g}}. {\alpha }_{\text{g}}+\left({h}_{rw}+{h}_{cw}+{h}_{ew}\right).{A}_{w}.\left({T}_{w}-{T}_{g}\right)={h}_{c,g-a}.{A}_{g}.\left({T}_{g}-{T}_{a}\right)+{h}_{r,g-s}.{A}_{g}.\left({T}_{g}-{T}_{sky}\right)+{m}_{\text{g}} {C}_{p\text{g}}\frac{d{T}_{\text{g}}}{dt}.$$
(23)
The empirical relation of convective heat transfer coefficient (\({h}_{cw}\)), evaporative heat transfer coefficient (\({h}_{ew}\)), and radiative heat transfer coefficient (\({h}_{rw}\)) between the free surface of basin water and inner surface of glass cover are expressed as follows:
$${h}_{cw}=0.884{\left[\left({T}_{w}-{T}_{\text{g}}\right)+\frac{({P}_{w}-{P}_{\text{g}})({T}_{w}+273)}{\left(268.9\times {10}^{3}-{P}_{w}\right)}\right]}^{1/3},$$
(24)
$${h}_{ew}={16.28\times {10}^{-3}.h}_{cw}. ({P}_{w}-{P}_{\text{g}})/\left({T}_{w}-{T}_{\text{g}}\right),$$
(25)
$${h}_{rw}=\upsigma . {\upvarepsilon }_{\text{eff}}.\frac{\left[{\left({T}_{\text{w}}+273\right)}^{4}-{\left({T}_{\text{g}}+273\right)}^{4}\right]}{\left({T}_{\text{w}}-{T}_{\text{g}}\right)}.$$
(26)
Basin water
$$I\left(t\right) {A}_{w} {\alpha }_{w} {\uptau }_{\text{g}}+{h}_{c,ab-w}.{A}_{ab}.\left({T}_{ab}-{T}_{w}\right)+{\dot{m}}_{a} {h}_{a,in}=\left({h}_{rw}+{h}_{cw}+{h}_{ew}\right).{A}_{w}.\left({T}_{w}-{T}_{g}\right)+{m}_{w} {C}_{pw}\frac{d{T}_{w}}{dt}+{\dot{m}}_{a} {h}_{a,out},$$
(27)
where \({h}_{c,ab-w}\) is the coefficient of convection heat transfer between absorber and water which is calculated as:
$${h}_{c,ab-w}=0.54\left(\frac{{k}_{w}}{x}\right).{\left(Gr. {Pr}_{w}\right)}^{0.25}.$$
(28)
Grashof number defined as,
$$Gr=\frac{\beta g{d}^{3}{{\rho }_{v}}^{2}\Delta T}{{{\mu }_{v}}^{2}}.$$
(29)
Effective temperature difference (\(\Delta T\)) is calculated as:
$$\Delta T=\frac{\left({T}_{w}-{T}_{g}\right)+\left({P}_{w}-{P}_{g}\right).\left({T}_{w}+273\right)}{\left(268.9\times {10}^{3}-{P}_{w}\right)}.$$
(30)
Prandtl number defined as:
$$Pr=\frac{{\mu }_{v}{C}_{v}}{{k}_{v}},$$
(31)
Mean temperature of vapor film on basin water surface given by:
$${T}_{i}=\frac{{T}_{w}+{T}_{g}}{2},$$
(32)
The specific heat (\({C}_{v}\)), thermal conductivity (\({k}_{v}\)), density (\({\rho }_{v}\)), and viscosity (\({\mu }_{v}\)) of water vapor are estimated as follows:
$${C}_{v}=999.2+0.143{T}_{i}+1.0101\times {10}^{-4}{{T}_{i}}^{2}-6.7581\times {10}^{-8}{{T}_{i}}^{3},$$
(33)
$${k}_{v}=0.0244+0.7673\times {10}^{-4} {T}_{i},$$
(34)
$${\rho }_{v}=\frac{353.44}{{T}_{i}+273},$$
(35)
$${\mu }_{v}=1.718\times {10}^{-5}+4.62\times {10}^{-8} {T}_{i}.$$
(36)
The cubical expansion coefficient of vapor (\(\beta\)) is calculated as:
$$\beta =\frac{1}{{T}_{i}+273}.$$
(37)
The vapor pressure at mean temperature of vapor film is calculated as:
$${P}_{i}=exp\left(25.317-\frac{5144}{{T}_{i}+273}\right).$$
(38)
Absorber plate
$$I\left(t\right) {A}_{ab} {\alpha }_{ab} {\uptau }_{\text{g}} {\uptau }_{w}=\left({h}_{rw}+{h}_{cw}+{h}_{ew}\right).{A}_{ab}.\left({T}_{ab}-{T}_{w}\right)+{A}_{ab}.\left(\frac{{k}_{PCM}}{{x}_{PCM}}\right).\left({T}_{ab}-{T}_{PCM}\right)+{m}_{ab} {C}_{p,ab}\frac{d{T}_{ab}}{dt}.$$
(39)
Phase change material (PCM)
$$\left(\frac{{k}_{PCM}}{{x}_{PCM}}\right).\left({T}_{ab}-{T}_{PCM}\right)=\left(\frac{{k}_{ins}}{{x}_{ins}}\right).\left({T}_{PCM}-{T}_{a}\right)+\left(\frac{{M}_{equ}}{{A}_{ab}}\right)\left(\frac{d{T}_{PCM}}{dt}\right).$$
(40)
where \({M}_{equ}\) is the equivalent heat capacity of PCM given by:
$${M}_{equ}={m}_{PCM}\times {C}_{s,PCM} for {T}_{PCM}<{T}_{m},$$
$${M}_{equ}={m}_{PCM}\times {L}_{PCM} for {T}_{m}\le {T}_{PCM}\le {T}_{m}+\delta ,$$
$${M}_{equ}={m}_{PCM}\times {C}_{L,PCM} for {T}_{PCM}>{T}_{m}+\delta .$$
(41)
The distillate water productivity \({\dot{m}}_{ew}\) is calculated as,
$${\dot{m}}_{ew}=\frac{{h}_{\text{ew}}.{A}_{w}.({T}_{\text{w}}-{T}_{\text{g}})\times 3600}{{h}_{\text{fg}}}.$$
(42)
The latent heat of water vapor (\({h}_{\text{fg}}\)) is defined as:
$${h}_{\text{fg}}=3.1615\left({10}^{6}-761.6 {T}_{i}\right) for {T}_{i}>70^\circ C$$
$${h}_{\text{fg}}=2.4935\left({10}^{6}-947.79 {T}_{i}+0.13132 {{T}_{i}}^{2}-0.004774{{ T}_{i}}^{3}\right) for {T}_{i}<70^\circ .$$
(43)
Modeling of condenser
This section presents detailed mathematical model equations used to forecast the condenser’s performance57:
The energy balance of condenser is calculated as:
$${\dot{m}}_{cw} {C}_{p,cw}\left({T}_{cw.out}-{T}_{cw,in}\right)={\dot{m}}_{a}. \left({H}_{a,in}-{H}_{a,out}\right).$$
(44)
The specific enthalpy of air \({H}_{a}\) (J/kg) is calculated by:
$${H}_{a}=\left[1.006 {T}_{a}+{\omega }_{a}.\left(2501+1.86 {T}_{a}\right)\right]\times {10}^{3}.$$
(45)
Air humidity ratio \({\omega }_{a}\) is estimated from:
$${\omega }_{a}=\frac{0.622 {P}_{s}}{{P}_{a}-{P}_{s}}$$
(46)
Partial vapor pressure of air \({P}_{s}\) is estimated as:
$${P}_{s}= 0.611379\text{exp}\left(A{T}_{a}-B{{T}_{a}}^{2}+C{{T}_{a}}^{3}\right),$$
(47)
where; \(A=0.0723669; B=2.78793\times {10}^{-4}; C=6.76138\times {10}^{-7}\)
The rate of freshwater productivity of condenser (\({\dot{m}}_{fw}\)) is estimated by:
$${\dot{m}}_{fw}={\dot{m}}_{a}.\left({\omega }_{a,in}-{\omega }_{a,out}\right).$$
(48)
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