In Shah, L. J. & Furbo, S. (2003) and Shah, L. J. & Furbo, S. (2004), a theoretical model for calculating the thermal performance of evacuated collectors with tubular absorbers was developed. The principle in the model was that flat plate collector performance equations were integrated over the whole absorber circumference. In this way, the transverse incident angle modifier was eliminated. The model was valid only for vertically tilted pipes.

In this section, the principle of the model will shortly be summarized. Further, the newest development that improves the model to be able to also take tilted pipes into calculation will be described.

Generally, for a solar collector without reflectors and without parts of the collector reflecting solar radiation to other parts of the collector, the performance equation can be written as:

TOC o "1-5" h z P. = P + Pd + P8I — Ploss (1) or more detailed described:

P« = Ab■-F’-(та).-K9’Rb’Gb + A. — F ’-(ха). — Kw — FcVG + A.-F’-(та).- — Fc_8^ — A. U — (Tta — T.) (2) where Ke is the incident angle modifier defined as:

Ke=1 — tan‘ (jj (3)

The incident angle modifiers for diffuse radiation, Ke, d, and ground reflected radiation, Ke, gr, are evaluated by equation 3 using 0=n/3.

To calculate the thermal performance of the evacuated tubes, the general performance equations (1) and (2) have been integrated over the whole absorber circumference. This means that the tube is divided into small "slices”, and each slice is treated as if it was a flat plate collector. In this way, the transverse incident angle modifier is eliminated. For describing the solar radiation on a tubular geometry, this method has previously been used by Pyrko J. (1984). .

Integrating over the absorber area, the performance equation can be described as: P. = j(Pb + Pa + P„ — Pioss К (4) where, Pioss = JA.-UL-(Ttn — T.)-^ = JLVUL-(Tta — T.)a = 2-*-LVUL-(Tta — T.) (5) Pa = JA.-F’-(xa)e-Ke, a-Fc_s-Ga-a5 = 2-*VL-F4xa)e-KM-Ga — JF^ (6) Pg, = JA.’F’-(xa)e-K„ gr-F^-Ggr-^ = JF^ (7) Ggr =Pgr-(Gb + Ga) (8) Fc_s = 0.5 — F1_2 (9) II © 1 (10)

SHAPE * MERGEFORMAT

Power from beam radiation on collector/tube, Pb:

The power contribution from the beam radiation can be written as:

0 <Vl-y0 <k : Pb = } F’-(xa)e-Gb-Ab-K9= F’-(xa)e-Gb-LV} K9^-d?

To To

0 <Yo-у, <k : Pb = ] F’-(xa)e-Gb-Ab-K9 ^-d? = F’-(xa)e-Gb-L-rp-} K9^-d?

Fig. 3 shows an example where a part of one tube vector N tube is shaded and a part is exposed to beam ’ ‘

radiation. In order to determine the size of the area exposed to beam radiation, the points P0 and P1 must be determined.

Since P0 is located where the solar vector and the tube vector are at right angles to each other, P0, described by the angle y0, can be determined by the scalar product of the two vectors:

S-N = |s|-|N|-cos (J = o ^ sin 6,-cos уs’cos ( “-Ps j cos уo + sin 6,-sin уs-sin у o + cos 6,-sin ( “-Ps ]’sin Yo — o =

Since the equation for Yo involves the tangens function, the equation will return two solutions. Based on information on the position of the sun, the correct solution is found.

Fig. 4: Illustration of the

Equations(15) (16) and (17) together give four equations shaded area and the area to the four unknowns: T, Y1, xn and zn. Solving for Y1 gives: exp0sed t0 beam rndiati0n.

(19)

1

v 2 s) tan (6z )-sin (ys — yf)

К

4 2 2 2 2

4 — ^^3 ^^"1 ^^3 K2 ^^3

From equation (18) it appears that there are two solutions for y1. Based on information on the position of Y0, the correct solution is found.

The incident angle, в, and the geometric factor, Rb:

The incident angle, 0, can be described as:

C0S(6) = sin(62)-COs(ys — У f)-COS I -2 — p, J’COS(y actual) + — y f)-sin(y„cl„1) + COs(62)-sin I jC°S(y aCMJ

Solving the performance equation:

In order to evaluate the performance of the tubular collector on a yearly basis, the above theory is implemented into a Trnsys type. All the integrals can be solved analytically, except the integral in equation (11), which is solved by using the trapezoidal formula for solving integrals numerically. 360 integration steps are used in the numerical integration. Taking the collector capacity into account, the collector outlet temperature is evaluated by: