Now that we know how a chimney works we can use the above analysis to determine a way to cool a house with a chimney! This is similar to the cooling effect mentioned in the above example where cooler, outside air is directly brought into the house. Air circulation is promoted through the house or building by heating air in a chimney with the Sun to draw air into the building. These types of air conditioners have been used for centuries, if not millennia, to draw air through a house and promote cooling by convection and/or a temperature decrease. The solar chimney has the added advantage that the mass flow rate of air though the building increases with power input from the Sun, which is what is wanted; an increased cooling effect when it is required.

Moreover, if the outside air is too hot during the day a damper is used to stop air flow. When the outside air cools down the damper is opened and air is then drawn through the building to provide cooling. This is accomplished by having a large thermal mass in the solar chimney that can store energy to provide it to the air, even when the Sun has set.

A schematic of a solar chimney is given in Fig. 7.2 demonstrating the design variables including the azimuthal angle, 9Z, and the angle

perpendicular to the device area, 6S, relative to the radiation received by the device, PD (в), where в is the angle the device makes with the horizontal. The chimney is assumed to be perpendicular to the ground and so в is 90°, which is not always the case.

A key component of the solar chimney is the transparent window of area Ad that allows the radiation to heat the air (or thermal mass) within the chimney. We will assume that any radiation transmitted through the window will be absorbed by the air and that it has the value Pd (в), which will be discussed in greater detail below. The analysis is begun by applying the First Law of Thermodynamics (FLOT) between stations 1 and 3 to arrive at what will be called the operating line

m [CP [T3 – Ti] + gH] = PD (в)Ad operating line (7.16)

where kinetic energy effects have been neglected, a good assumption, which is why applying the FLOT between stations 1 and 2 and 2 and 3 was not considered. The heat capacity of air is given the symbol CP and the temperatures of the two stations are Ti and T3 as discussed above. The heat capacity was assumed constant over that temperature range and CP [T3 – T{] approximated the change in enthalpy. In many solar chimneys there is a black absorber plate on the wall opposite to the transparent material, to absorb the solar radiation, only to re-emit it to the surrounding air.

Unfortunately, there is only one equation and two unknowns: m and AT = T3 – T1. In this case the FLOT is called the operating line, now another equation is required.

A design line is needed which could be developed to include detailed calculations of heat transfer to the air within the solar chimney and how that affects m. Here we will assume that all the radiation reaching the air within the chimney will be absorbed by the air to give a maximum possible heating effect. This is not a poor assumption, merely a necessary one. In general the solar chimney will not be thermally isolated from the environment, making detailed heat transfer calculations difficult and full of many assumptions.

So, how can one determine the solar chimney performance? Fortunately we have eqns (7.8) or (7.15) to provide the design line that is based on the Bernoulli equation and fluid mechanics, the result of which is shown below, where wind effects have not been included,

AT

m = p3 2 gH A design line (7.8)

Ti

The solar chimney performance is now determined by the intersection of the operating line and the design line to find the operating point. In other words, the two simultaneous equations are solved to find m and AT, which define the operating point. The following example demonstrates this principle.

Find the operating point of a solar chimney which receives 500 W/m2 of radiation and has a device area (window) of area 10 m2. The exit area from the 10 m high chimney is 1.5 m2 while the ambient air temperature is 35 0 C. Neglect the effect of wind and the empirical constant Cc is assumed to be equal to one.

First the operating line needs to be calculated. The heat capacity of air can be found in Appendix B and will be assumed to be constant with a value of 1007 J/kg-K. Various values of AT (i. e. T3-Ti) will be assumed and the mass flow rate determined with eqn (7.16). The result of this calculation is given in Fig. 7.3.

Now the design line can be calculated from eqn (7.8) by assuming various values of AT and calculating m. This is also shown in the figure and the intersection of the two lines defines the operating point. For the conditions here, the mass flow rate is about 2 kg/s and the temperature rise is 2 0C.

The density of air at 350C is 1.17 kg/m3 using the data in Appendix B, so, the volumetric flow rate of air into the house is 1.7 m3/s. Assuming there is a central hallway in the house, through which the air is drawn, with dimensions of 2 m by 3 m, this corresponds to a velocity of about 300 mm/s, which is substantial. Of course, if the air is forced to be drawn from an underground chamber the cooling effect can be even greater (see Exercise 7.3). This will certainly reduce the mass flow rate though, as the frictional drag of the air with the walls in the chamber will produce an additional pressure drop that will have to be considered.

The operating point in the above example was found by the intersection of the operating and design lines, however, it is possible to combine eqns (7.8) and (7.16) to find

which can be written

m3 + 3pm + 2q = 0 (7.18)

where

_ 2[gHp3A]2 p 3CpTi

_ Pd (в) AdgH[p3 A]2

9 _ Cp Ti

This third-order equation can be solved and the only real solution is

For the conditions encountered in most solar chimneys one can approximate sinh(^) with 2 exp(^) according to its definition (as well as sinh(^/3) with the appropriate relation) to find

This equation could have been determined by noting the relative order of magnitude for each term in eqn (7.18) and eliminating the 3pm term.

Both eqns (7.19) and (7.20) have p3 as a variable that will be a function of the temperature rise within the chimney and is a function of the mass flow rate. Thus, a circular solution of these equations with eqn (7.8) or

(7.15) (assuming vw is zero) will have to be done. Yet, the temperature rise in solar chimneys is small and one can use p1 to find the mass flow rate with little error, especially for a solar chimney. The mass flow rate at the operating point in Example 7.2 is found to be 2.09 kg/s using eqn (7.19) and 2.12 kg/s from eqn (7.20). The difference between the two is a mere 1-2%, which is good enough for design calculations!

The equation for the approximate operating point has further use since one can determine how the solar chimney operating point scales with design variables as given here,

m – Pd (в)1/3AD/3H1/3 A2/3 (7.21)

This has a different scaling to the design variables in eqn (7.9) for the chimney. It is amazing how little the mass flow rate, and hence the cooling effect, is affected by the power input Pd (в)Ad as well as the chimney height H. Again, as with the normal chimney, the variable to which the mass flow rate is most sensitive is the chimney area A.

Previous experimental studies have shown that

m – Pd(в)°’46-0’57Н0’60A0’71-0’76 ( 7.22)

While the scaling with the chimney area A is close to that predicted by the simple model discussed here, the scaling with Pd (в) and H is not close. The reason is certainly due to the simplicity of the model presented where heat transfer effects have not been considered. This can be seen in how the efficiency of an experimental device n scales with power input

affected by the power with a fairly small power law exponent. Yet, it was assumed here that the efficiency was 100% and that all the power from the Sun was absorbed by the air, making the efficiency independent of PD(в) or to scale as PD (в)0.