Authors
| Ephraim M. Sparrow |
Edward G. Palmer |
| John A. Sipple |
SolarAttic |
| Department of Mechanical Engineering |
Minnesota |
| University of Minnesota |
|
| Minneapolis, MN 55455 |
|
First Published
Solar 95 Conference Proceedings
American Solar Energy Society
2400 Central Avenue, Suite G-1
Boulder, CO 80301
All Rights Reserved © 1995
ABSTRACT
A novel solar concept is the utilization of existing attic spaces as
solar collectors. A heat
exchanger situated in the attic facilitates the utilization of the solar-heated
attic air to create
useful energy products such as heated swimming-pool water and residential
hot water. To
enhance these products, a method is developed here to increase the energy
carried into the heat
exchanger by the solar-heated air. The basic idea is to utilize all parts
of the attic as a hot-air
reservoir rather than only the immediate neighborhood of the heat exchanger
inlet face. In the
practical realization of this idea, a flexible conduit attached to the heat
exchanger inlet is
deployed throughout the attic. The wall of the conduit is made permeable
to enable the ingestion
of air into the conduit from all neighborhoods along its length. The far
end of the conduit is
capped. An analytical model is developed which yields a specification of
the axial distribution
of the permeability needed to achieve axially uniform air ingestion. An apparatus
was built to
validate the model and its predictions. The measured axial pressure distributions
were in very
good agreement with that predicted from the analysis. This agreement validates
the model and
supports its further use as a design tool for enhancing the utilization of
attic-collected solar
energy. Key Words: solar energy, attic-collected solar, swimming-pool water
heating, residential
water heating.
There are numerous approaches for collecting and utilizing solar energy for
residential
purposes (1,2). These approaches can be classified as either active or passive.
Active solar is
characterized by the use of dedicated equipment for collecting the incident
flux of solar energy.
On the other hand, passive solar utilizes the physical structure of the residence
to perform the
collection function.
The concept to be dealt with here is based on using existing attic spaces
as solar
collectors. In that sense, it is definitely a passive approach. However,
some of the modes of
utilizing the attic-captured solar radiation are similar to those for active
systems. For instance,
the attic-collected energy can be used for heating swimming pool water and
even for heating
residential hot water.
The analysis of attic-based collection systems naturally subdivides itself
into two parts.
One part is concerned with the techniques and equipment (if any) by which
the actual energy
collection is accomplished. The other part has to do with the modalities
and equipment for
utilizing the collected solar energy. In this paper, attention will be focused
on the collection
function.
It is, of course, a common experience to encounter elevated temperatures
in an attic space
on whose roof or side walls solar energy is incident. Attic temperatures
of 120°F (49°C) are
commonplace and temperatures of 150°F (65°C) are possible. The
temperature level that is
attained depends on the intensity of the insolation, the presence or absence
of insulation and of a
radiation barrier at the interior surface of the roof, and the strength of
the wind-based, forced
convection heat transfer at the exterior surface of the roof. It is to be
expected that the
temperature will not be uniform throughout the air that fills the attic space.
The degree of non
uniformity will be influenced by the degree of venting, the geometry of the
attic, and the quality
of the insulation at the floor of the attic.
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At the heart of present systems for utilizing attic-collected solar energy
(3) is an air-to-
liquid (typically water) heat exchanger. In principle, the exchanger should
be positioned in that
part of the attic where the highest air temperatures occur. However, in practice,
the positioning
is often governed by geometric issues such as where the attic height is
sufficient to accommodate
the height of the exchanger. As a consequence, there is a possibility that
the exchanger may not
be situated optimally from the standpoint of energy utilization.
In connection with efficient energy utilization, there is an issue even more
basic than the
placement of the heat exchanger. That issue is the possible short circuiting
of the air discharged
from the heat exchanger directly back into the inlet face of the exchanger.
Such a flow pattern is
illustrated schematically in Fig. 1. As seen there, the discharge flow may
loop around to the inlet
without intervening contact with the hot surfaces of the attic, i.e., the
roof and, perhaps, the side
walls. Such contact is essential to efficient use of the attic-collected
solar energy. Indeed, in the
ideal manifestation of the utilization of such energy, the air discharged
from the heat exchanger
would experience numerous contacts with the hot walls before it is returned
to the inlet face of
the exchanger.
![[Image]](imgs/Solar_pict0.gif)
Fig. 1 Representation of Flow Short-circuiting
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The foregoing discussion sets the stage for the subject matter of this paper.
The objective
of the paper is to propose and substantiate a method for creating a pattern
of fluid flow within the
attic space which maximizes the utilization of the attic-collected solar
energy.
A clear guideline for the attainment of extensive contact between the attic
airflow and its
bounding walls is to establish a large separation distance between the inflow
and outflow faces
of the heat exchanger. This can be accomplished in practice by ducting the
air via a conduit from
the outflow face of the heat exchanger to some location remote from the inflow
face, and
discharging it there. The thus-discharged air would then find its way to
the inlet on its own. An
alternative would be to attach one end of a conduit to the inflow face of
the exchanger and to
position the other end at a location remote from both the inflow face and
the outflow face. This
remote positioning would force the air that is discharged at the outflow
face to traverse a
considerable length of the attic before it reaches the open end of the conduit,
from where it would
be ducted directly to the inlet.
Both of these arrangements have a flaw which is conveniently discussed with
reference to
the latter of the two setups. In that case, the heat exchanger would be fed
by air that is drawn
only from the immediate neighborhood of the open end of the conduit. No air
would be taken
from the many neighborhoods situated adjacent to the cylindrical wall of
the conduit. The
proposed remedy is to make the conduit wall permeable by creating a set of
discrete holes in the
material from which the wall is made.
This is the concept that will be developed here. It will be developed in
accordance with
the intuitive perception that the greatest flux of energy carried into the
heat exchanger will be
achieved when the air is ingested uniformly into the permeable-walled conduit
at all locations
along the length of the conduit.
4 of 18
An analysis will now be made to determine the distribution of the holes in
the conduit
wall which provides the desired uniform ingestion. Subsequently, experiments
performed to
verify the results of the analysis will be described.
The fluid flow in the permeable-walled conduit will be analyzed by making
use of the
equations for mass and momentum conservation. To facilitate the analysis,
attention may be
directed to Fig. 2. Figure 2 is a schematic side view of the permeable-walled
conduit. As seen
there, the conduit has a diameter D and length L. The axial
coordinate in the flow direction is x,
where x = 0 corresponds to the end of the conduit that is farthest
from the heat exchanger and
x = L denotes the end of the conduit that interfaces with the inlet
of the heat exchanger. The
diagram also indicates that the far end of the conduit is capped. This practice
was adopted
because an open-ended conduit would necessarily ingest much more air at its
open end than
from other neighborhoods that are adjacent to the conduit wall. This situation
would not be
compatible with the aforementioned goal of uniform ingestion.
![[Image]](imgs/Solar_pict1.gif)
Fig. 2 Schematic Diagram of the Permeable-Walled Conduit
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At any cross section x, the mass flow rate of air is m(x).
The rate at which air passes into
the conduit through its permeable wall in a length dx is dm, subject to the
constraint
dm/dx = constant = M (1)
Since the x = 0 cross section is capped, m(0) = 0. It then
follows from equation (1) that
m(x) = [x/L]m(L) (2)
where m(L) is the rate of mass flow at the inlet of the heat exchanger
(i.e., at the exit of the
conduit). In the subsequent numerical evaluation of the analytical results,
m(L) will serve as a
prescribable parameter (in actuality, the volumetric counterpart of
m(x) will be prescribed).
The air that permeates through the wall of the conduit is driven by a pressure
difference.
Let p(amb) be the pressure in the attic space adjacent to the conduit
and p(x) be the pressure in
the conduit at a cross section x. Then, the local driving force for
permeation is p(amb) - p(x).
This pressure drop results from two processes: (a) the acceleration experienced
by the air as it
passes from the relatively quiescent attic ambient to the outer surface of
the permeable-walled
duct, and (b) the flow resistance of the permeable wall. These components
are, respectively, one
velocity head and one-half velocity head, where
velocity head = 0.5
p[V(wall,x)]2 (3)
in which V(wall,x) is the velocity of the air that is moving radially
inward through the permeable
wall at location x. Therefore,
p(amb) - p(x) = 1.5 velocity heads (4)
and from equations (3) and (4)
![[Image]](imgs/Solar_pict2.gif) (5)
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The next step is to write an expression for the ingestion rate dm.
For an axial length dx,
the surface area of the conduit is pDdx. A fraction of this area
b(x) is open to permit ambient air
to pass into the conduit, where b is recognized to be a function of
x. Therefore, in the length dx,
d(free flow area) = b(x)pDdx
(6)
It follows that the rate of ingestion dm in an axial length dx
is
dm = p[V(wall,x)]b(x)pDdx (7)
Then, with equation (5),
![[Image]](imgs/Solar_pict3.gif) (8)
The desired design condition is that dm/dx = constant = M (equation
(1). Equation (8) can then
be solved for b (x)
![[Image]](imgs/Solar_pict4.gif) (9)
Equation (9) is a means for obtaining numerical values for the free flow
area factor b, provided
that the axial pressure distribution p(x) is known. The next part
of the analysis deals with p(x).
The momentum conservation principle is the basis for the derivation of
p(x). The
derivation is facilitated by reference to Fig. 3, which shows a control volume
( CV) and
nomenclature. Conservation of momentum may be written in any coordinate
direction. For
present purposes, the appropriate direction is the x-direction. The momentum
conservation
principle states that the difference between the rate of x-momentum that
is carried out of a
control volume and the rate of x-momentum that is carried into the control
volume has to equal
the net x-direction force. Figure 3 identifies the outflowing and inflowing
x-moments and the x-
direction stresses.
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![[Image]](imgs/Solar_pict5.gif)
Fig. 3 Control Volume
The velocity V denotes the cross-sectional average velocity. The pressures
p and (p + dp) act
on the cross-sectional area of the control volume, while the shear stress
T acts on the cylindrical
area that forms the interface between the control volume and the conduit
wall. The areas
are, respectively,
cross-sectional area = ÆD2 Æ 4
(10)
area of CV-wall interface = ÆDdx
(11)
With these, the mathematical representation of the x-momentum balance can
be written as
(m + dm) (V +dV) - mV = {[p - (p + dp)]ÆD2 Æ 4} - TÆDdx
(12)
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At any cross section,
m(x) = rV(x)ÆD2Æ4
(13)
and
dV = dmÆ[rÆD2Æ4]
(14)
When equations (13) and (14) are introduced into the momentum balance (12),
there follows,
-dp/dx = 2m[dm/dx]Ær[ÆD2Æ4]2 +
4TÆD
(15)
Equation (15) is much simplified when it is recognized that dm/dx = constant
= M. To
proceed, the shear stress appearing in the last term of equation (15) may
be eliminated by
introducing the friction factor f via the definition
4T = [rV2Æ2]f
(16)
or, upon elimination of V by means of equation (13),
4T = {m2Æ2r[ÆD2Æ4]2}f
(17)
The friction factor f is a function of the Reynolds number Re
of the air flowing in the conduit.
There are many available algebraic relationships between the friction factor
and the Reynolds
number. The well-established Blasius formula will be used here because it
enables a closed-
form, analytical, algebraic solution to be obtained for p(x). The
Blasius formula can be written
as
(18)
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and the Reynolds number is
Re = 4m Æ µÆD
(19)
After substitution of equations (17), (18) and (19) and the introduction
of dimensionless
variables, the governing equation (15) for pressure distribution emerges
in a remarkably simple
form:
(20)
where
X = x/L, Re(L) = 4m(L) Æ µÆD
(21)
and
P(X) = [p(amb) - p(x)] Æ 0.5r[V(L)]2
(22)
V(L) = m(L) Æ r[ÆD2Æ4]
(23)
Equation (20) is easily integrated to give P(x). Since it is a
first-order differential
equation, one boundary condition must be provided. For the contemplated design
methodology,
the state of the air flowing into the heat exchanger would be specified.
This includes the desired
volumetric flow rate in cubic feet per minute (which is equivalent to
V(L)) and the allowable
pressure drop [p(amb) - p(L)]. With these, the value of P(1)
can be specified, and this serves
as the boundary condition for the differential equation (20).
The solution for the pressure distribution which incorporates the known boundary
condition is
P(X) - P(1) = 2[X2-1] + 0.3636[LÆD]
[Re(L)]-0.25[X2.75-1]
(24)
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To extract a more physical form of equation (24), use may be used of the
identity
[p(x) - p(L)]/[p(amb) - p(L)] = [P(X) - P(1)]/P(1)
(25)
Equation (25) enables the pressure distribution along the conduit to be presented
in
dimensionless form as a function of x/L. For the determination
of the free flow area factor b(x),
which is needed for the specification of the holes in the surface of the
conduit, the quantity
[p(amb) - p(x)] is required as input to equation (9). To extract this
pressure difference from
equation (24), it may be noted that
[p(amb) - p(x)] = [P(X)/P(1)] [p(amb) - p(L)]
(26)
where [p(amb) - p(L)] is a specified design parameter.
The simplicity of the algebraic equations which convey the results precludes
the need for
an exhaustive graphical presentation. Such a presentation would be mandatory
had a numerical
solution (e.g., finite-difference or finite-element) been required. In view
of this, the graphical
presentation of results will be confined to the operating conditions of a
complementary
experimental study which will now be described.
A 25-foot-long (7.6 m), 10-inch diameter (0.25 m), commercially available
conduit was
used for the experiments. It consisted of a tightly stretched, impermeable
plastic skin firmly
supported by an embedded helical wire. In its as-arrived state, the conduit
was open at both
ends. The blower procured for the experiments was situated in a rectangular
housing, the inlet
face of which was a 12.5-inch-diameter (0.32 m) circle, and the exit face
of which was a 14-inch
(0.36 m) square. The blower was driven by a 1/12th hp electric motor. A conical
transition
piece was fabricated to facilitate the mating of one end of the conduit with
the inlet face of the
blower.
11 of 18
Two quantities were measured in order to test the analytical predictions:
(a) volumetric
air flow rate and (b) the pressure distribution p(x)along the length
of the conduit. The
volumetric flow rate was determined by a velocity traverse across the fully
exposed exit face of
the blower housing. For this purpose, the exit face was subdivided into a
7 by 7 grid to define 49
equally deployed measurement sites. The local velocities on this grid were
measured with a hot-
film anemometer whose electronics provided a digital readout which was manually
recorded.
The 49 velocity values were integrated numerically to yield the volumetric
flow rate in cfm
(cubic feet per minute).
The static pressure at any axial station x was determined by introducing
an L-shaped Pitot
tube into the conduit at the desired station. The holes through which the
probe was inserted will
be described shortly. During the period of its insertion into the conduit,
the probe was hand-held.
It was oriented so that the sensing portion of the L was aligned with the
axis of the conduit, with
the impact opening facing upstream. Only the pressure at the static holes
of the probe was read
and recorded. All pressures were measured with respect to the ambient pressure
p(amb). The
pressure meter was electronic and of the capacitance type. It provided a
voltage output that was
displayed by a digital voltmeter. The pressure meter/voltmeter combination
was capable of
resolving pressure differentials as small as 0.00005 inches of water. To
put this resolution into
perspective, it may be noted that the pressure difference [p(amb) -
P(L)] for the experiments was
about 0.2 inches of water.
A possible operating point was selected from the pressure vs flowrate
characteristic curve
for the blower. That operating point was approximately 600 cfm and approximately
0.2 inches
of water. This information, together with the diameter and length of the
conduit and the
expected density and kinematic viscosity of the air, was sufficient to enable
the determination of
the free flow area ratio b from equations (9) and (25) (recall that
the distribution of b is
determined to achieve a uniform ingestion of air into the conduit all along
its length).
12 of 18
The resulting distribution of b as a function of position along the
duct is presented in Fig.
4, where x = 0 is at the capped end of the conduit (the end farthest from
the blower inlet), while x
= L is at the blower inlet. As seen there, only a very small fraction of
the surface area of the
conduit wall need be open to achieve the desired uniform ingestion. The largest
bvalue in
evidence in Fig. 4 is about 0.03. The second noteworthy feature of Fig. 4
is that the required
open-area fraction diminishes in the flow direction, i.e., b decreases
by a factor of four. The
physical reason for this behavior is that the local pressure difference
[p(amb) - p(x)] , which
drives the ingestion, increases in the flow direction, so that less open
surface area is needed to
pass the ingested flow.
The predicted dimensionless pressure distribution along the length of the
conduit which
corresponds to the b(x) of Fig. 4 is shown in Fig. 5. The ordinate
of the figure is the ratio of the
pressure drop [p(x) - p(L)] between x = x and x = L to the overall
pressure drop [p(amb) - p(L)].
Note that the pressure p(0) at the capped end x = 0 is slightly below
p(amb). The pressure p(x)
drops off relatively slowly in the neighborhood of the capped end but drops
off relatively rapidly
near the exit of the conduit.
The b(x) information given in Fig. 4 was used as the guideline for
cutting circular holes
in the surface of the conduit. The holes were cut with a sharp-bladed Exacto
knife in conjunction
with appropriately sized circular templates. The diameters of the individual
holes ranged from
1.375 inches (3.5 cm) near the capped end to 0.25 inches (0.64 cm) near the
exit end. The far
end of the conduit was capped with a disk of rigid cardboard held in place
with duct tape.
The experiments were actually run at a volumetric flow rate of about 630
cfm. Two
replicate data runs were performed on successive days. The measured axial
pressure
distributions are plotted in Fig. 6, where they are compared with the predictions
of the analysis.
The ordinate variable is the difference between the local pressure in the
conduit and the ambient
pressure, i.e., [p(x) - p(amb)] . This pressure difference is negative,
a pressure drop. As
expected, the pressure drop increases in the flow direction.
13 of 18
Inspection of the figure reveals that the replicate data runs show excellent
mutual
agreement. Perhaps of greater significance is the observation that the predicted
pressure
distribution (the solid curve) is in very good agreement with the experimental
data. This level of
agreement lends strong support for the analytical model. The model then qualifies
as a design
tool for attaining enhanced utilization of attic collected solar energy.
![[Image]](imgs/Solar_pict6.gif)
Fig. 4 Axial Distribution of the Free-Flow Surface Area
Fraction
14 of 18
![[Image]](imgs/Solar_pict7.gif)
Fig. 5 Predicted Axial Pressure Distribution
15 of 18
![[Image]](imgs/Solar_pict8.gif)
Fig. 6 Comparison of Measured and Predicted Pressure
Distributions
16 of 18
This work was undertaken to develop a method for increasing the utilization
of attic-
collected solar energy. The basic idea is to draw air from all parts of the
attic space into the inlet
face of a heat exchanger which facilitates the utilization of the solar-heated
attic air. In the
practical realization of this idea, a flexible conduit attached to the inlet
face of the exchanger is
deployed throughout the attic space. The wall of the conduit is made permeable
to enable air to
be drawn into it from all attic neighborhoods along its length. The permeability
is adjusted so
that air is ingested uniformly along the entire length of the conduit, and
the far end of the conduit
is capped.
An analytical model was developed to determine the axial variation of the
conduit-wall
permeability needed to achieve the uniform ingestion condition. The model
yielded closed-form,
algebraic solutions for both the axial distributions of the permeability
and the static pressure.
An experimental facility was fabricated to test the validity of the model
and its results. A
conduit was made permeable in accordance with the prescription provided by
the analysis, and
axial pressure distributions were measured. The measured pressure distributions
were found to
be in very good agreement with that predicted by the analysis. This level
of agreement validated
the model and supports its further use as a design tool to enhance the
utilization of attic-collected
solar energy.
17 of 18
| D |
Diameter of conduit |
| f |
Friction factor |
| L |
Length of conduit |
| M |
Rate of ingestion per unit length |
| m(x) |
Air flow rate at any x |
| m(L) |
Air flow rate at conduit exit |
| P(X) |
Dimensionless pressure, equation (22) |
| p(amb) |
Ambient pressure in attic |
| p(x) |
Static pressure at x |
| p(L) |
Static pressure at conduit exit |
| Re |
Reynolds number |
| T |
Wall shear stress |
| X |
Dimensionless axial coordinate, x/L |
| x |
Axial coordinate |
| V |
Air velocity |
| V(wall,x) |
Air velocity passing through conduit wall; |
| V(x) |
Axial velocity at x |
| b |
Fraction of wall surface area that is open to airflow |
| µ |
Air viscosity |
| r |
Air density |
REFERENCES
(1) |
Duffie, J. A. and Beckman, W. A., Solar Engineering
of Thermal
Processes, John Wiley & Sons, New York, 1991. |
(2) |
Kreith, F and Kreider, J. F., Principles of Solar
Engineering, McGraw-Hill, New York, 1978. |
(3) |
PCS1 SolarAttic Pool Heater Manual, SolarAttic, Elk
River,
Minnesota, 1991. |
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