A 3-phase 400-W Grundfos (UPS32-120F) centrifugal pump is used for water circulation in the solar field. Control of collector outlet temperature is achieved by modulating the pump flow rate using a variable speed drive.
Pump similarity laws can be used to extend the information given by the pump manufacturer of discharge (Q), head (H) and power (P) as function of the rotating speed (ω). According to [17], when the fluid density and the pump impeller diameter are constant, the pump similarity laws can be reduced to:
$$ Q_{2} \, = \, Q_{1} \, \left( {\omega_{2} \, / \, \omega_{1} } \right) $$
(3)
$$ H_{2} \, = \, H_{1} \, \left( {\omega \, _{2} \, / \, \omega \, _{1} } \right)^{2} $$
(4)
$$ P_{2} \, = \, P_{1} \, \left( {\omega \, _{2} \, / \, \omega \, _{1} } \right)^{3} $$
(5)
where subscripts 1 and 2 refer to pump parameters at two different speeds.
The three-phase induction motor model can be formulated, following [18]. The torque is proportional to the square of speed:
$$ T \, = \, K \, * \, \omega^{2} $$
(6)
where T is the electromagnetic torque produced by the motor, ω is the speed of the rotor and K is a constant of proportionality. Since the supply frequency is 50 Hz, Eq. (7) is used to determine the synchronous speed for the 2-pole machine [19]:
$$ N_{\text{s}} \, = \frac{120f}{P} $$
(7)
where Ns is the synchronous speed of the motor, f is the frequency of the supply voltage and p is the number of poles. Thus, the synchronous speed is 3000 rpm, or 314.16 rad/s. The nominal torque of the motor can be calculated as shown in Eq. (8). This yields a value of 1.27 N m for the 400-W motor.
$$ T_{\text{n}} = \frac{{P_{\text{n}} }}{{\omega_{\text{n}} }} $$
(8)
where Tn is the nominal torque, Pn is the horsepower rating and ωn is the synchronous speed of the motor. Hence, the value of k in (6) is found to be 1.29 × 10−5 N m s2. This value of k is multiplied with the square of the rotor speed and fed back to the torque input of the motor, to create the model for a centrifugal pump type load [19].
HDH unit
The model of the system consists of different heat exchangers and other components like a flow mixer and a flow diverter. The present model uses the methodology and governing equations suggested in [14]. Figure 2 shows the model components of HDH unit. The saltwater enters the WDU in the condensation chamber in which the saltwater is preheated by the hot air and then passes a flow mixer where it is mixed up with a part of the hot water coming from the evaporating chamber. After passing the flow mixer, the water passes again another heat exchanger to raise its temperature even more before entering the external heat exchanger of the WDU in which the water is heated up with an external heat source to the maximum temperature in the system. It leaves the external heat exchanger and sprays down in the evaporating chamber in which the energy is transferred from the hot water to the cold air. Then, it passes a flow diverter, which splits the saltwater flow into two different volume flows.
The heat exchangers are modelled using the following differential equation:
$$ \frac{{{\text{d}}T_{{w , {\text{out}}}} }}{{{\text{d}}t}} = \frac{{\dot{m}_{w} }}{{m_{w, m} }}\left( {T_{{w,{\text{in}}}} - T_{{w,{\text{out}}}} } \right) \, - \frac{kA}{{C_{p,w} m_{w,m } }}\vartheta_{\text{cond}} \left( {T_{{w,{\text{in}}}} ,T_{{w,{\text{out}}}} ,T_{{a,{\text{in}}}} ,T_{{a,{\text{out}}}} } \right) $$
(9)
$$ \frac{{{\text{d}}T_{{a,{\text{out}}}} }}{{{\text{d}}t}} = \frac{{\dot{m}_{a} }}{{m_{a, m} }}\left( {T_{{a,{\text{in}}}} - T_{{a,{\text{out}}}} } \right) \, + \frac{kA}{{C_{p,a} m_{a,m } }}\vartheta_{\text{cond}} \left( {T_{{w,{\text{in}}}} ,T_{{w,{\text{out}}}} ,T_{{a,{\text{in}}}} ,T_{{a,{\text{out}}}} } \right) $$
(10)
$$ \vartheta_{\text{cond}} = \frac{{T_{{w,{\text{in}}}} - T_{{a,{\text{out}}}} - T_{{w,{\text{out}}}} - T_{{a,{\text{in}}}} }}{{Ln \frac{{(T_{{w,{\text{in}}}} - T_{{a,{\text{out}}}} )}}{{(T_{{w,{\text{out}}}} - T_{{a,{\text{in}}}} )}}}} $$
(11)
HDH model parameters are listed in Table 3. The flow diverter and the flow mixer are implemented using CARNOT toolbox.