Task 1: Solar heating systems
A solar heating system is used to produce hot water for showering, see (the
fancy) Figure 1. The systems consists of several components, of which the most
important ones are the solar collector, the storage tank, the electric heater and
the mixing valve. The solar collector absorbs ̇q(t) = (300 + d) exp{−0.2(t −
12)2}[W/m2] heat per square meter, where t is the time in hours [0..24] and d is
4th digit of your student number. The hot water storage tank has a volume of
0.5m3. Water enters the tank with a temperature of 15oC and should leave the
tank with a temperature above 60oC (to avoid Legionella), for safety reasons
the water temperature should never exceed 80oC. The electric heater has a
power ̇We of 2kW and can be turned on/off at every instant. The water leaving
the storage tank is mixed with cold water of 15oC to a temperature of 41oC.
Everyday between 7 and 7.30am and between 11 and 11:30pm 100 liter of tap
water is used for showering. Assumptions:
Figure 1: A sketch of the solar heating system.
• Solar collector surface is 4.?m2 where the question marks denotes the last digit
of your student number.
• Heat capacity of water is 4.18kJ/kgK, density of water is 998kg/m3 and pressure
in the system is 4.5 bar.
• Initially the temperature inside the vessel is 15oC, the vessel is always fully
mixed.
• All the solar heat is transferred to the storage tank with a temperature above
the temperature in the storage tank.
• There are no heat losses to the surroundings. Volume of flow and pressure losses
in the piping system can be ignored.
1
Calculate the running costs for a full week if the electricity price is 0.20Euro/KWh,
between 7am and 7pm and 0.14 Euro/KWh between 7pm and 7am. (Initially
at 0:00am of the first day the temperature in the vessel is 15oC). Show graphs
of the temperature inside- and the water flow through the vessel as a function
of time.
Hints
Important equations
̇m1h1 + ̇m2h2 = ̇m3h3, ̇m2 = ̇m′
2 (1)
dUvessel
dt = ̇Qs − ̇We + ̇m2 ∗ (h2 − h′
2) (2)
If the system is fully mixed we can assume that u′
2 = u. For constant density
and pressure dH/dt = dU/dt
