The available energy calculated so far describes the heat supply from the absorber sheet metal to the fluid at a point y on the collector. This heat supply leads to a local rise in temperature of the fluid, which depends on the mass flow through the fluid pipe.
The calculation of the rise in temperature over the complete collector length then enables the calculation of the available useful power from the collector. For an absorber pipe the following energy balance results for a total mass flow through the collector m, i. e. a mass flow per pipe of fnjN:
with the boundary condition
The differential equation is solved by substitution
W = Ота-Ut (Tf – To)
and separation of the variables:
The boundary condition results in
W|y=0 = C, = Ота-Ut (Tf in – T0) (3.44)
and thus for the fluid temperature at any point y in the direction of flow:
To determine the fluid output temperature, the collector length L is used for y. The product of the number of tubes N, absorber strip width W and collector length L thereby corresponds to the collector surface A.
The available power of the collector Qu can thus be represented analytically as a function of the fluid inlet temperature, the ambient temperature and the irradiance.
Qu = m cp (Tf, out – Tf, i„ )
The first term of the equation, normalised to the collector surface A, is defined after Duffie-Beckmann as the heat dissipation factor FR. It indicates the ratio of the actual available power to the attainable available power, if the complete absorber were at the cold fluid inlet temperature.
The heat dissipation factor, which depends on the mass flow and geometrically on the collector efficiency factor, leads to a simple available-power equation:
Q. = AFr (Gra – Ut (Tf Jn – To))
The thermal efficiency is determined by the ratio of available power per square metre collector surface and irradiance.
The mean fluid temperature of the collector is obtained by integrating the fluid temperature using Equation (3.45) over the collector length.
The mean absorber temperature is obtained by equating the available power equation as a function of the input temperature and as a function of the mean absorber temperature.
Calculation of the available power and temperatures of a flat plate collector at G = 800 W/m2 irradiance, ambient temperature To = 10°C and an inlet temperature into the collector Tf in from the lower part of the storage tank at 30°C. To represent the influence of the mass flow on the available power and the temperature conditions, the calculation for a low-flow system with m = 10 kg/m2h is to be carried out, and also for a standard system with m = 50 kg/m2h.
The optical efficiency n0 =ш = 0.9×0.9 = 0.81 and the entire calorific loss Ut with 4 W/m2K are given. The data of the collector are as follows:
Width of the absorber strip W 15 cm
Length of the absorber strip L 2.5 m
External pipe diametre D (DN8, 1mm wall strength) 8 mm
Heat conductivity of the absorber sheet metal 1copper 385 W/mK
Sheet thickness d 0.5 mm
Contact resistance 1/1con? eff 0
Heat transfer coefficient 1000 W/пЖ
FrL=10_g = 2.9 (1 – exp (0.94/2.9)) = 0.8
Heat dissipation factor FR: m
FrI=50A = 14.5(1 – exp (0.94/14.5)) = 0.91
The heat dissipation factor FR is mass-flow dependent.
The available power and efficiency improve with rising mass flow. At typical flow conditions of collectors between 10 and 50 kg/m2h, the efficiency varies by 12%.
Mm=10 A – = 0’57
Mm=50 A = 0.64
Tfo„, = 69°C (39°C)
The advantage of the smaller mass flows shows up particularly in the outlet temperature. During a single flow-through with the low-flow system, a rise in temperature of 39 K is achieved, with the higher through flows only 9 K.
Mean fluid temperature: Tf = 51.1°C (31.5°C)
Mean absorber temperature: Ta = 58.4°C (42.8°C)
Between the mean absorber temperature and mean fluid temperature there is a difference of 7.3 or 8.3 K, respectively.