# A Simple model

Our model assumes that the application is a solar-driven absorption cooling system displacing the electrical energy that would have been used to drive a conventional vapor-compression air conditioning system. It introduces a simple economic figure of merit that can be parameterized in terms of five di­mensionless quantities, all of order unity. These five parameters are used to characterize (1) the climate, (2) the local cost of electrical energy, (3) the collector performance, (4) the performance of the pro­posed absorption cooling system, and (5) the performance of the existing electrical cooling system.

We introduce a basic figure of merit that is the approximate 10-year cumulative value (V) of the electrical energy displaced by the solar system. This 10-year value is a useful benchmark number since it can also be thought of as the maximum collector cost consistent with a simple payback of 10 years. This parameter can be calculated as follows

V = V • EKhP • (C. O.P.)th (8.4)

where [annual average insolation available to the collector]

170 W/m2

[cost of electricity]

\$ 0.10/kW – hr

3.9

(C. O.P.)el

annual average thermal collection efficiency of the collector array.

(C. O.P.) and (C. O.P.)el are the coefficients of performance of the thermally driven absorption cooling chiller and the electrically driven vapor-compression system whose energy is being dis­placed, respectively.

(The coefficient of performance of a cooling system is the ratio of the quantity of heat re­moved from the cooled space to the energy that must be supplied to remove it.)

Note that E depends on the location (i. e., climate) and the collector type and deployment ge­ometry and K depends on the location (i. e., local economic conditions) and particular application.

The three quantities, E, K, and P, in Eq. (1) have been defined as dimensionless ratios relative to typical representative values for the corresponding quantities. In particular, note that E = 1.0 corresponds to a yearly average insolation of 170 W/m2, which in turn is what one would expect if one simply takes the local interplanetary solar constant of I = 1370 W/m2, and takes a global average by multiplying by 1/4 [the ratio of the area of the earth’s surface projected on a plane perpendicular to the earth-sun line (pre2) to the earth’s global surface area (4pre2)] and then again by a factor of 0.5 to account for weather effects. By multiplying by the number of seconds per year (about p x 107 sec), one finds that this is equivalent to an annual average of insolation on a horizontal plane of 5.34 GJ/m2-year. The value of E for a particular location and collector acceptance geometry then is a single parameter that adjusts up or down, usually by about a factor of 2, from this reference value. Similarly, the factors Kand P are defined so that they are equal to unity for “typical” values of the cost of electricity and the C. O.P. for electrical air conditioning. Ten cents per kW-hr represents the midrange of electricity costs for summer peaking situations, and a coefficient of performance of 3.9 is standard practice for commercial vapor-compression systems. Note that there is nothing magical about these normalization values except that they have been chosen so that V will be a representative “ballpark” quantity that characterizes the fundamental economics of solar cooling application. Note further that if other normalization quantities were chosen, it would modify Vo but would not change the value of V as determined by Eq. (8.4).

When the variables are normalized as defined, the quantity Vo turns out to be equal to about \$380; that is,

r 10 years 170(W/m2)(8.76 kW – hr/W – year)\$0.10/(kW – hr)

Vo= T9

= 381.85(\$ / m2 — 10 years)

= \$ 380/m2 – 10 years

where, in view of all the other uncertainties, we have rounded V to two significant figures. It is important to note that Vo is a numerical constant depending only on conversion factors and is independent of assumptions other than the normalizations discussed earlier. These normalizations have been taken so that the dimensionless quantities in Eq. (8.4) (E, K, h, and P) have typical val­ues ranging from ~0.5 to ~2.0 (of course, the efficiency h is always less than 1.0) so that there is a relatively small range of possible values for our basic economic figure of merit V, typically within a factor of 4 of F. As a very rough rule of thumb, for even marginal economic viability, the solar collectors used to drive the absorption system should have a cost per square meter at most equal to V, and the yearly maintenance costs associated with the collectors must be a very small fraction of

V (i. e., < 1%).

For example, although the thermal efficiency for a flat plate collector can be between 0.7 and 0.8 for heating domestic hot water, at temperatures around 100 °C, such as those required for driving a single-effect absorption chiller, its efficiency will generally be reduced to 0.1-0.3. The C. O.P. values for typical single-effect absorption chillers range from about 0.5 to 0.7. Thus, if for simplicity we take E, K, and h all to be unity, we find that for a single-effect chiller driven by flat plates, Vturns out to be about \$20-\$80 at a typical mid-latitude location. As a practical matter, installed flat plates collec­tors cost substantially more per square meter than this (perhaps \$300-\$500/m2). Thus, it is clear that the low C. O.P. values attainable by single-effect chillers combined with the low thermal efficiency of flat plate collectors at even moderately high temperatures together make them unsuitable for this ap­plication and have contributed to the perception that active solar cooling is not economical. Signifi­cantly higher collector array operating efficiencies and higher chiller C. O.P. values will be required.

Recently, the manufacturers of air conditioning equipment have developed and are begin­ning to market commercial-scale double-effect absorption chillers. These units have C. O.P. values of about 1.2 (a factor of nearly 2 better than the best old single-effect units) but require heat to be delivered at about 160-190 °C, well beyond the capabilities of flat plate or ordinary evacuated tubular collectors. The ICPC (see Chapter 3) has the potential to increase the value of Vfor solar absorption systems to a level closer to that required for economic viability. These collectors provide the only simple and most effective method for delivering solar thermal energy efficiently in the temperature range from 100 °C to about 300 °C without tracking. At 160 °C, they are expected to achieve efficiencies between 0.5 and 0.7, so that if, as above, we take E, K, and h all to be unity,

V would be between \$230 and \$320, which is a factor of 5-10 better than that for flat plate collec­tors with single-effect systems and is at least in the right ballpark to have an appreciable chance for economic viability.

Updated: August 19, 2015 — 1:15 am