With the convective heat transfer coefficients determined above, the heat transfer coefficient for radiation hr and the total heat transfer coefficient Ut between the absorber and environment, the efficiency factor F can be calculated using Equation (3.113). The heat transfer coefficient for radiation is calculated on the simplified assumption that the absorber and gap rear wall can be described as infinitely expanded flat-parallel surfaces, and the ribs are not considered.
Due to the small emissivities of the duct confinement surfaces (ea, eb = 0.04-0.1) and the small temperature difference between the surfaces, typically 5K, these assumptions produce sufficiently exact results (Ta and Tb in Kelvin).
hr = —– ( + Tb2 ) + Tb) (3.131)
— + — -1
The heat transfer coefficient Ut is calculated, as with the flat plate collector, in simplified fashion as the total of the front, side and rear wall losses, i. e. of the temperature node of the absorber to the environment; the side and rear wall losses Us and Ub are temperature-independent and can be set as constant.
Ut = Uf + Ub + Us (3.132)
The heat transfer coefficient through the transparent cover takes into account wind influences (hcw) and radiation losses to the sky (hg. sky) and is, as with the flat plate collectors, calculated iteratively as a function of the glas cover temperature Tg.
h + h h + h,
c, a-g r, a-g c, w r, g-sky
Since all the heat transfer coefficients are temperature-dependent, first temperatures for all surfaces and the mean fluid temperature must be given. With the initial temperature field all coefficients are then calculated, the collector efficiency factor F’ is determined and the available power is calculated as a function of the fluid input temperature:
Q„ = AFr (g(та)-Ut (TfJn – T0)) (3.134)
From the available power, the mean temperatures for the absorber Ta, flow channel rear wall Tb, fluid Tf and glass cover Tg are then calculated similarly to water-through flowed collectors.
The temperature of the flow channel rear wall is calculated by resolving the energy balance Equation (3.111) with the mean fluid temperature used for Tf.
With these temperatures, in the next iteration new heat transfer coefficients are calculated. The iteration is continued until the change in the temperature field becomes negligibly small.
Calculation of the available power, the outlet temperatures and the thermal efficiency for a facade- integrated air collector at 800 W/m2 irradiance and an ambient temperature of 10°C. The ambient temperature equals the inlet temperature Tf;n in the collector. The air duct geometry corresponds to the examples already calculated with a collector length of 2.5 m, and the heat transfer coefficient of the rear Ub is a constant 0.65 W/пЖ (side losses ignored).
Table 3.11: Result table for air collector efficiencies.
From the results, the efficiency rise with the change of the flow from laminar (No. 1, 2) into turbulent conditions (No. 3-5) is clearly evident. The first two simulations with laminar flow differ by the emission coefficient of the absorber ea, which in the case of the selective coating is 0.1 and with the black absorber 0.9. Due to the selective coating, the heat transfer coefficient falls from 6.6 to 4.2 W/m2K and the efficiency rises by 17%. At higher flow rates of 2.5 m/s, the influence of the outside wind velocity was examined. A reduction of 3 m/s to 1 m/s leads to a reduction of the Ut-value of 0.8 W/m2K, and an efficiency improvement of 3%. The selective coating brings a further 6% efficiency improvement.
Figure 3.44: Rise in temperature (continuous lines) and efficiency (broken lines) of an air collector.
The boundary conditions for these temperature-rise or efficiency calculations as a function of the specific flow rate (in m3/h per m2 of collector surface) are: irradiance of 800 W/m.2, 10°C ambient temperature and 3 m/s wind velocity.
3.2.3 Design of the air circuit
The volumetric air flow of the collector depends on the desired application. While during pure fresh air pre-heating, high flow rates of > 60 m3/m2h with good thermal efficiency are favourable, direct heating or hypocaust applications require high rises in temperature and thus low specific flow rates (20-40 m3/m2h).
With direct heating applications, the outlet temperature determined by the surface- specific flow rate must be limited for reasons of comfort, in houses to 45°C, in industrial applications to a maximum of 60°C. The temperature limitation can take place either by flow rate regulation or by the addition of cold air.