The basic system is shown in Fig. 8.1, which consists of a glass cover over an absorber plate which is in thermal contact with the flowing water that enters and exits the device. Insulation at the back of the device, as well as around the edges (not shown), reduces thermal losses. The glass cover is required to reduce thermal losses on the device front, just as a greenhouse would experience.
Water flows into the absorber plate system at a mass flow rate m and typically does so through pipes that are soldered or welded to the back of the plate to yield good thermal contact. An example of another design is shown in Fig. 8.2, which is a system that has the riser pipe incorporated into the absorber plate. This system allows the fins to interlock together and the risers are soldered into a header tube to make a compact system. The water goes in and out of the system through the header tubes, which uniformly forces water into the riser tubes and collects the hot water at the top of the device.
To simplify analysis the glass cover which is present in many con
temporary systems will initially be omitted. Inclusion of a cover adds difficulty to the analysis which does not allow the reader to first focus on the heart of the system, the absorber plate and transfer of absorbed energy to flowing water. Yet, insulation will be included in the basic design to minimize heat transfer from the back.
The amount of radiation striking the southward facing device (in the Northern hemisphere) at an angle of в from the horizontal to maximize the input of solar radiation (Pd (в)) is given by
Pd (в) = В(в) + D(e) + А(в )[=] 2 (8.1)
m2
as discussed in Chapter 2. The symbol [=] means ‘has dimensions of.’ Each term in eqn (8.1) is just at the external surface of the absorber plate (or at the external surface of the cover should it be used) for the beam component (В(в)), diffuse component (D(P)) and the reflected or albedo component (А(в)). These terms can be determined with the principles given in Chapter 2, and for reference В (в) = Bter cos(OS), where Bter is the direct beam insolation at the surface of the device and OS is the solar angle. The radiation has not been absorbed by internal processes at this point and when absorbed it will be written as aB (в), where the absorbance a is assumed to be a constant over the solar spectrum and not a function of incident angle.
A good absorber will be a selective surface, meaning that the absorbance for high energy solar radiation will be near one (large), yet, its emittance for low energy radiation will be near zero (small). This is because the Sun produces high energy radiation and this must be absorbed as efficiently as possible. Since the absorber will be at a much lower temperature than the Sun, say on the order of 100 0C, the emittance, which is equal to the absorbance, for low energy radiation must be low to minimize thermal losses.1 This forms the basis of a selective surface, a large absorbance for high energy radiation and a low emittance (absorbance) for low energy radiation.
The flux of radiation, aPD(в), represents the amount of energy entering the device which must now be converted to useful heat (Qu), determined by application of the First Law of Thermodynamics (FLOT) to the flowing water,
Qu = mCpW AT [=] W (8.2)
where Cpw, is the water heat capacity and AT, is the difference ToutTin, with Tin and Tout being the inlet and outlet temperature, respectively, to the flat plate solar energy absorber. Applying the FLOT to the device operating at steady state (see Fig. 8.3) yields
mCpwAT = [aPD(в) qrad qconv qbot]AD (8.3)
It is obvious that Qu is merely the rate of sensible heat rise for the water entering into the flat plate absorber of area AD. There are three heat loss terms qrad, which are that due to radiation, qconv, due to convective
heat transfer and qbot is the rate of heat transfer from the bottom of the device, as shown in Fig. 8.3. Note a heat transfer rate that is capitalized like Q is the total heat transfer rate in Watts, while a lower case like q is the heat transfer rate in Watts/m2. Each heat transfer term is now considered in turn.
Radiative heat transfer from the absorber plate is given by
qrad = epTS Tp – Tsky] [=] (8.4)
where ep is the emissivity of the absorber plate, <rS, the StefanBoltzmann constant, Tp, the absorber plate temperature and Tsky, the sky temperature for radiative heat transfer.2 Obviously one wants ep to be as small as possible, as discussed above.
The convective heat transfer rate is a function of the wind velocity vw. If it is zero then only natural convection is present, while if the wind is blowing then forced convection occurs.3 The heat transfer rate is given by
qconv hconv [Tp Ta] (8.5)
where hconv is the convective heat transfer coefficient and Ta is the air (ambient) temperature. This is Newton’s law of cooling where the rate of heat transfer is proportional to the temperature difference. There are many correlations for the heat transfer coefficient, Duffie and Beckman recommend
where the function MAX() means maximum and Lh is the cube root of the house volume upon which the flat plate absorber is placed. If the device is on the ground or the designer chooses not to use this correlation then another for air flowing over a flat plate can be used, as given by Palyvos,
hconv (W/m2K) = 7.4 + 4.0 Vw (m/s) (8.7)
Now consider the final heat transfer rate through the bottom (and sides) of the device. This is approximated as due to heat transfer through insulation placed on the bottom, which is written as
qbot = kL [Tp – Ta] (8.8)
Li
where kI is the thermal conductivity of the insulation with thickness Li . This is Fourier’s law of heat conduction.4 The assumption was made that the insulation represents the greatest resistance to heat transfer through the bottom of the device and is a good assumption except under extreme conditions. Furthermore, the insulation temperature next to the absorber plate was assumed to be at the temperature of the absorber
plate Tp which is again a reasonable assumption as sprayon insulation, such as polyurethane, is now being used which has good thermal contact with the plate. Also, the insulation was assumed to be at the external air temperature at the bottom of the device which is a good assumption for good insulation. Finally, radiative heat transfer from the bottom was ignored which a simple calculation can confirm to be correct. Note if heat transfer from the edge is to be considered then the effective device area at the bottom can be increased, or a new heat transfer term can be added similar to qbot.
The design of the flat plate solar hot water heater is performed by calculating the temperature rise AT of water flowing through the device under a given insolation PD(в). If all the terms are put into eqn (8.3) one realizes that all of them can be assumed, are known or are based on ambient conditions, except two: AT and Tp. We have one equation and two unknowns; another equation is required that will be addressed below. Before doing this eqn (8.3) is rewritten to emphasize that it is the operating line,
mCpAT = [aPD (в) – qrad – qConv – 4ыл]Ап operating line (8.3)
The FLOT is taken as the operating line meaning that it indicates the values of Tp and AT that are possible. The FLOT is merely a balance of energies, or in our case the rate of energy transfer. If one term is smaller then another must be larger so the equation is balanced.
To understand this, consider the operating line shown in Fig. 8.4 generated with eqn (8.3) using the standard conditions for a solar thermal flat plate absorber given in Table 8.1. The graph may seem confusing at first; as the absorber plate temperature is increased the temperature rise of the water is decreased. This can be understood by remembering the FLOT (operating line) is an energy balance and if energy goes into heating the plate then it cannot heat the water. Likewise if energy is used to increase the temperature of the flowing water then less is available to increase the absorber plate temperature, so, it falls.
The abscissa of the graph in Fig. 8.4 is written as Tp – Ta as this is one way to present the data; of course there are other variables used and not just Tp – Ta. One should realize that for a flat plate device without a cover the minimum temperature that the plate can have is Ta since it cannot fall below that unless the inlet water temperature is very low.5 A calculation where part of the system can go below the ambient temperature is considered in Example 8.4, however, for right now, the minimum plate temperature is assumed to be equal to the air temperature. At this condition the maximum temperature rise is approximately 20 °C. Of course, as will be determined below, the conditions required to achieve this temperature rise are extreme and so a more moderate temperature rise will occur. Since the temperature rise will be less than this, the question arises: The temperature of domestic hot water is typically 50 °C, how is the water temperature increased? This is answered in Section 8.4.
Table 8.1 Standard conditions used in designing a solar thermal flat plate absorber device in this chapter. Each device is 2.5 m long and 1.2 m wide and two are connected in parallel. The water flows along the length direction and it is assumed that there is no heat loss as the water exits the device. These conditions are for a solar hot water heater used for a typical residence. The insolation was assumed to be 1000 W/m2, similar to the AM1.5G spectrum, allowing ready comparison to solar photovoltaic devices, and unless otherwise stated, this is direct beam insolation.

The results in Fig. 8.4 show what is possible, how do we find the absorber plate temperaturewater temperature rise for our system? This will be the operating point and it will be found at the point where the operating line intersects the design line. The design line is found by considering the operation of the solar thermal device through detailed heat transfer calculations, which will be discussed immediately after the system configuration and an introduction to heat transfer of flowing water is considered.
The system in Table 8.1 has two modules connected in parallel and is similar to many systems installed at residences, see Fig. 8.5. The eight riser pipes in each module that carry the water are aligned to the long direction and assumed to be equal to the module length, which is 2.5 m, thus, the riser pipe length is 2.5 m. The pipes are connected to a larger diameter pipe on each end by a hole cut in its side to make a manifold to
distribute the water to and then collect the water from the risers which are called headers. The water will exit the top header and will go back to the house and into a storage tank where it is recirculated. When this system is mounted on a house roof the top header is closest to the roof apex (ridge) and the device tilt angle в is typically L + 10°, where L is the latitude, as discussed in Chapter 2.
Another piping arrangement is the serpentine which takes a single pipe and bends it back and forth so it covers the entire device area, where one end enters the bottom and the other exits the top. This could change the heat transfer from that occurring under laminar conditions to turbulent, and how it affects the operating point is considered in Exercise 8.5. Laminar and turbulent flow are described below should the reader not be familiar with this concept.
The water is heated in all designs by the absorber plate transferring heat (energy) to the flowing water in the pipes, since heat transfer is driven by a temperature difference between two systems in thermal contact. The flat plate absorber device has water entering a pipe at Tin and exiting at Tout which is in thermal contact with an absorber plate at temperature Tp which is assumed to be constant over the device area. What temperature difference do we use that drives heat transfer from the plate to the flowing water? Do we use Tp – Tin or Tp – Tout or… ?
The answer is the log mean temperature difference or ATlm whose derivation can be found in an undergraduate heat transfer textbook. The textbook by Middleman, referenced at the end of this chapter, is particularly good and clearly written. The log mean temperature difference comes about by considering a differential energy balance between the water flowing in the pipe in thermal contact with the absorber plate, then integrating from the inlet to exit to arrive at
One can think of ATlm as an average temperature difference driving heat transfer and the natural logarithm comes about through an integral of the form d[Tp – T]/[Tp – T].
Now the rate of heat transfer from the absorber plate to the water
Q w is
Q w = hw Ap ATlm (8.10)
where hw is the heat transfer coefficient for the water in thermal contact with the pipe wall and Ap is the pipe area. If there are Np pipes with diameter Dp and length Lp then Ap = NpnDpLp and is the heat transfer area for the water flowing within the device. The key factor now is the heat transfer coefficient, which must be calculated. Negligible thermal resistance for the pipe wall itself is assumed since most thermal resistance occurs between the water and pipe wall.
There are many correlations in the literature to find hw and all involve three dimensionless numbers: the Nusselt number (Nu), Reynolds 6See Appendix C for the importance number (Re) and Prandtl number (Pr).6 It is important to note that and definitions of dimensionless num – the dimensionless numbers are determined for water at the mean tem – bers’ perature Tm given by
Tm = 2 [Tout + Tin] (8.11)
Since the water properties are a function of temperature and the temperature rise is not known a priori, one ultimately arrives at an iterative solution, as is the nature of most heat transfer calculations involving flow. This will be discussed in more detail below.
The heat transfer coefficient depends strongly on whether the flow in the pipe is laminar or turbulent. Laminar flow means the fluid travels at the same position relative to the pipe centerline along the pipe length, while in turbulent flow the fluid will deviate strongly from ‘straight line’ flow. Turbulent flow is much more efficient to induce heat transfer, and so is desirable, although due to engineering constraints, solar thermal flat plate hot water heaters typically operate under laminar flow conditions, at least in residential applications. Regardless, for completeness, Nusselt number correlations will be given for both laminar and turbulent flow.
Laminar flow has Re < 2100 and Nu is found from the following correlations:
if RePrDp/Lp < 12 then Nu = 3.66
1/3 (8.12)
if RePrDp/Lp > 12 then Nu = 1.6[RePrDp/Lp] 1
Since kw will be known Nu allows calculation of hw (Nu = hwDp/kw, see Appendix C). Now it is clear why iterative calculations will be required. One needs hw to calculate the rate of heat transfer to the water which will cause its temperature to rise. However, kw is a function of temperature and it is needed to determine hw. Thus, one assumes an exit temperature and calculates Tm, then kw is evaluated at this temperature, as are all the other physical properties to determine Re and Pr. Then Nu is evaluated, hw is calculated and the temperature rise found. If it does not agree with the assumed temperature rise the process is repeated until convergence is calculated (see below).
Turbulent flow has Re > 2100 and follows the relation (although there are many correlations, this is a fairly accurate one)
where the viscosity ratio is evaluated at the mean and pipe wall temperatures and f is the Fanning friction factor which is a dimensionless pressure drop over the pipe length,
= AP Dp
2pw Vw Lp
where AP is the actual pressure drop, pw, the density of water and Vw, the average water velocity in the pipe (see Appendix C and eqn (C.1)). The friction factor for both laminar and turbulent flow is a function of the Reynolds number and is given by
laminar flow (8.15a)
turbulent flow (8.15b)
although f is not required for the Nusselt number correlation for laminar flow and is included for completeness. The reader should realize there are many fRe correlations that take factors such as pipe roughness into account, which can noticeably affect the pressure drop in turbulent flow. One may notice that if calculations are performed near the laminarturbulent transition there may be curious changes in the heat transfer coefficient; this is to be expected since the correlations presented above do not take into account the transition region, which occurs over a fairly large range of Reynolds numbers near 2100 until the flow is fully turbulent. If operating within the transition region the reader should search for heat transfer correlations that consider this and ensure that the correlation is for heat transfer for a constant temperature pipe wall that is hotter than the fluid (it actually makes a difference if the water is hotter than the pipe).
Now it is possible to write an equation for the design line. The sensible heat taken from the pipe (absorber plate) to the water is equal to that which causes the water temperature to rise, which can be written mathematically as



The operating point is found by simultaneously solving eqns (8.3) and
(8.16) since one now has two equations and two unknowns, Tp and AT.
Find the operating point of a single flat plate solar collector using the standard conditions given in Table 8.1. Assume there is no cover.
First the operating line is calculated using the standard conditions given in the table and using eqn (8.3) which was previously plotted in Fig. 8.4. This is graphed again in Fig. 8.6 where the convective heat transfer coefficient of Duffie and Beckman (eqn (8.6)) was used to describe the heat transfer from the absorber plate to air, hconv. Also, only the direct beam component is considered in the calculation of PD (в).
Now the design line is determined by assuming a value for Tp and calculating hw for the conditions in Table 8.1. The temperature rise AT is not known so it is assumed, then the physical properties of water are determined with the correlations given in Appendix B and used to calculate Re and Pr. The Reynolds number is of order 700800, so the flow is laminar, and for laminar flow one needs to calculate RePrLp/Dp with eqn (8.12). This term is almost constant since the most temperature dependent property, the viscosity, is not present in the product Re x Pr and one finds it is approximately equal to 13 for the conditions in Fig.
8.6. This in turn makes hw also approximately constant at a value of 182 W/m2K. The design line can be calculated with eqn (8.16) and is shown in the figure.
The operating point is found by simultaneously solving eqns (8.3) and
(8.16) and is represented by the intersection of the two lines in Fig. 8.6. Thus, the operating point is found to be; Tp = 37.0 °C and AT = 14.9 0C, which is a reasonably large temperature rise.
The reader may note that the equation for the design line can be simplified and rewritten as
by using the definition of AT1m.7 This is a somewhat simpler form to use, yet, does not eliminate iteration in the calculation since AT, or really the local temperature of the water flowing through the pipes, influences hw. Regardless it is an easier version to use and is valid for both laminar and turbulent flow.
Assuming laminar flow, which is what occurs for the standard conditions, the important combination of parameters is RePrDp/Lp, which can be written as
RePrDp 4 m Cpw
p ^ (8.18) Lp П Np Lp kw
The ratio of Cpw /kw is fairly constant for liquid water and equal to 6620 ms/kg and will vary by ±10% over the temperature range 0100 0C. Thus, if the reader wants to sacrifice some accuracy, an iterative calculation does not have to be performed to determine the design line.
Using this approximation, the design line is found by first calculating Re to determine whether the flow is laminar or turbulent. Assuming it is laminar, which is usually the case, at least for systems installed on houses, one can directly calculate RePrDp/Lp from eqn (8.18). Then eqn (8.12) can be substituted into eqn (8.17) to yield
These equations are reasonably accurate for determining A T as a function of Tp by assuming Cpwkw = 6620 ms/kg. In Example 8.1, calculation of the design line would deviate from the iterative solution by less than 1.5%, which is acceptable. So the above equations can be used directly with eqn (8.3) to determine the operating point with reasonable accuracy. Furthermore, if an iterative solution is required it is merely coupled with the solution of a fourthorder algebraic equation.
The minimum temperature the absorber plate can have is near the surrounding air temperature for the standard conditions given in Table
8.1. At this condition all possible heat from the Sun has been garnered and transferred to the flowing water. Strictly speaking this is not exactly true since the absorber plate is radiating energy to the sky, which is at a lower temperature than the air, however, for our purposes it is assumed that the minimum plate temperature is equal to the air temperature (see note 5).
This means that all the heat loss terms; qrad, gconv and qbot are zero (or near zero) and all the absorbed radiation is given to the flowing water, which can be seen upon inspection of eqn (8.3). At this point one would have
which represents the maximum temperature rise the flowing water can have, ATmax. In Example 8.1 the maximum temperature rise is 21.5 0C and so the operating point is 67 0C lower than this. In this calculation it was assumed that the radiative heat transfer term is zero, keeping this in the calculation, and assuming the minimum absorber plate temperature is Ta finds ATmax = 19.9 0C.