The temperature rise of water from a flat plate solar energy collector is small and of order 20 °C which is not hot enough for residential hot water which is delivered at approximately 50°C. This shortcoming is overcome by recirculating the water within a tank, as shown in Fig. 8.15. The useful heat is constantly delivered to the water in the system and it gradually rises in temperature with time. How is the temperature rise found?
The strict answer to this question is that coupled differential equations would have to be solved; one for the flat plate solar collector and the other for the tank. The differential equation is the Unsteady State First Law of Thermodynamics (US-FLOT) which is written
where m0 is the total mass of water in the system, H, the enthalpy, v, the velocity of fluid entering the system, g, the gravitational acceleration, h, the height, Q, the sum of all heat flows to and from the system and Wm, all mechanical work terms. The subscripts ‘1’ and ‘2’ are for conditions entering and exiting the system, respectively. The form of this equation would make the operating line a function of time, which is difficult to analyze.
This challenge is simplified by making the pseudo-steady state approximation or PSSA. The PSSA is frequently used for reaction sequences where one reaction is much faster than another. Consider this reaction sequence
A ^ B (slow reaction) 
B + C ^ D (fast reaction) 
If reaction  is much faster than , the reactants and product concentration in  can be assumed to be constant and it is at equilibrium. The kinetics for the much slower reaction  can then be solved.
For the system at hand, the flat plate solar energy collector and the water storage tank, the collector has much faster kinetics than the tank since the mass of water in the pipes within the collector is much less with a much faster average velocity than that in the tank. So, the operating point for the collector will not change, as a first approximation, for a given temperature change of the water stored within the tank. Note though that the water input temperature to the collector will slowly vary which could affect the operating point since the physical properties of water will change. This will be ignored here and the operating point for the collector having a water input temperature at the standard condition, 15°C as a stand-alone system will be used.
The energy input jar|PD(в) and output qout, which is the sum of qtop and qbot, are shown in the figure and it is their difference multiplied by the device area AD which gives the useful heat, written as
Qu = mCpwAT = 0.06 kg/s x 4195 J/kg-K x 15.8 °C = 3977 W (8.2)
for the standard conditions with a single cover. This is System I in the figure to give the rate of energy added to the tank.
Now one can apply the US-FLOT, eqn (8.40), to the System II, noting there is no mass flow into or out of the system nor mechanical work, to arrive at
d(m0lT) = – dt =Qu
If a pump were used to recirculate the water then there would be a work term included in the above equation. The enthalpy H is written as Cpw [T – T0], where T is the temperature at any time and To is the initial temperature taken as the reference temperature for the enthalpy
(i. e. the enthalpy is assumed to be zero at To). Substituting this in the equation it can then be easily integrated to yield
T = To + Q^ t (8.41)
This equation can be used to estimate the temperature of water in the tank or the amount of time it takes to reach a certain temperature. Inherent assumptions in arriving at this final equation are that the heat capacity of water is constant with temperature, a good assumption; the insolation is constant, probably not a good assumption over a day; and the water in the tank is well mixed so the temperature throughout the tank is homogeneous.
Find the amount of time it takes to heat 250 kg of water in a tank that is initially at 15 ° C to 50 ° C for a flat plate solar energy collector using the standard conditions given in Table 8.1 Assume there is a cover.
One can take eqn (8.41) and rearrange it to
where Tf is the final temperature. The time to achieve the given temperature is easily calculated by
This is an incredibly short time! In reality the insolation will be below 1000 W/m2 and not constant, which will increase the time. Typically it takes a day or so for the tank to achieve operating temperature.