First Law of Thermodynamics in its Eulerian Description, applied to an open system:

dE_system/dt = Q_dot_in - Q_dot_out + Q_dot_gen - Q_dot_accum + (other forms of energy, e.g. gravitation potential energy, nuclear source, etc..)

In this case, the Energy Flux of the work's fluid stream need to be added, i.e. the Enthalpy difference between the entering and exiting fluid.

Therefore, completing the above equation, we should write :

dE_system/dt = Q_dot_in - Q_dot_out + Q_dot_gen - Q_dot_accum + H_dot_in - H_dot_out

Just in mention of a few remarks on the above equation:

- All terms in the Eq. are in [W]

- The subscript "dot" means: each quantity is an Energy Rate (Energy per unit of time)

- The expression is the general form which can be used for the, either general scenario of Transient State or the particular case of Steady State.

Regards !

Dear Amos Ali

The second part of your question has to be considered in the Q_dot_rad, which is inside the term "Q_dot_in" of the above equation.

Which is calculated from the Stefan-Boltzmann's Law of Radiation.

The solar radiation not absorbed by the absorber's surface due to its optical properties: (emissivity, reflectivity, transmissivity and absorptivity) is considered in the epsilon greek letter in the Stefan-Boltzmann's law :

Q_dot_rad = eps*A*F*sigma*(T_sup^4 - T_sky^4)

which is the emissivity of the surface

remember that for any surfaces: Reflectivity + Transmissivity + Absorptivity = 1. (rho + tau + alpha = 1)

and for opaque surfaces and in thermal equilibrium:

tau = 0

alpha + rho = 1 ,and

epsilon=alpha.

The other consideration involving the collected/absorbed solar radiation is the one that has to be made due to the orientation and shape of the surface.

This factor is considered in the letter "F" in the Stefan-Boltzmann's equation, and it is called the "View Factor", or "Shape Factor". It's a geometric factor which depends on the shape of the surface, its orientation with respect to the radiation soruce and (sometimes) the surface's emissivity. You can easily find corresponding expressions of the View Factor to your case in a Heat Transfer HANDBOOK.

Best Regards !

ok. sir

Dear Franklin

Thank you for your response

I can sand you my study: Equations and yelds of parabolic concentrators, that is developed for my thermionic converter which however does not work due to the space charge effect. It has a glass vacuum tube and internal reflectig screen. You should adapt the coefficients and the configuration to your case. Look at add files. This study it has been developed for extremely high temperatures, in the case of applications at lower temperatures the reflective pattern is probably not necessary but the covering with a glass is sufficient.

Dear Massimo,

Thank you for the materials

In the document I sent to you, energy balancing was developed with taking into account only irradiation, because, in my case, there is no convection or conduction, because the collector is under vacuum. There is not treatment of the containment glass because, in my case, it is transparent for both incoming and outgoing radiation. In the case of a collector without cover, you must remove the radiation patterns as the amount of energy lost by radiation will be small. In the air you need to add the subtracted energy share by convection. You can improve the uptake coefficient by blackening the collector but it can still be improved with a selective treatment (big alpha for the solar spectrum, and epsilon small at the design temperature). The same effect is obtained with the blackened collector protected by a cover glass that is transparent to solar radiation and opaque to infrared, with the further advantage of limiting convection or eliminating it if a manifold locked in vacuum glass tubes is also made partial. Good work and I wish you great success.

Thank you

A complete study on solar energy systems can be found in:

"Solar energy thermal processes" Duffie, Beckman 1974.

Another concentration system is very interesting. It is called:

nonimaging optics or anidolic optics

https://en.wikipedia.org/wiki/Nonimaging_optics

This allows us to tolerate much wider entry angles

compared to the classic parabolic concentrator that must

be aimed precisely, more precisely, the higher the concentration coefficient.

The second advantage is that a flat panel can be made, covered by a glass,

much less cumbersome and easy to clean than parabolic cylindrical mirrors

with focus far from the mirror and without coverage.

There is a complete discussion in:

"Optical and thermal properties of compound parabolic concentrators" 1976

that you find in the attached file.

It is made up of two half linear parabolic mirrors with the two focus shifted,

which can be easily made with silver-plated molded plastic

with the same economical technology of car headlights.

From geomertical and also thermodynamic considerations

it is shown that the maximum theoretical optical concentration coefficient

for linear concentration systems is 200.

But practically it is not easy to exceed 100 suns.

While for optical disk concentrators, the theoretical concentration limit is 40000

but even here it is not easy to overcome the 20,000 suns and practically can be reach 10000.