Problems in Solar Engineering of Thermal Processes 1st Ed (highlighted) (2)

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APPENDIX A
Problems
The problems below are arranged by chapters. Most of them have quantitative
answers; a few have descriptive answers. Time is assumed to be solar time unless
otherwise stated.
A solution manual for the problems is available from the authors at the Solar
Energy Laboratory, University of Wisconsin, 1500 Johnson Drive, Mqdison, WI
53706.'
The most important equations for solution of these problems have been
programmed for hand-held calculators. For information on these programs
contact FCHART, P.O. Box, 5562, Madison, WI 53705.
1.1 From the diameter and effective surface temperature of the sun, estimate
the rate at which it -emits energy. What fraction of this emitted energy
is intercepted by the earth? Estimate the solar constant, given the mean
earth-sun distance.
1.2 What fraction of the extraterrestrial radiation is at wavelengths below
0.5 flm? 2 flm? What fraction is included in the wavelength range 0.5 flm
to 211m?
1.3 Divide the extraterrestrial solar spectrum into 10 equal increments of
energy. Specify a characteristic wavelength for each increment.
1.4 Calculate the angle of incidence of beam radiation at 1400 solar time
on January 20 at latitude 35°N on surfaces with the following orientation:
a Horizontal
b Tilted to south at slope of 40°
c At slope of 40°, but facing 25° west of south
d Vertical, facing south
e Vertical, facing west _
1.S Determine the sunset hour angle and day length for Madison and for
Miami, for the following dates: January I, March 22, July 1.
1.6 a When it is noon Pacific Standard Time in San Francisco (L = 122° W)
on March 1, what is the corresponding solar time?
b What Central Daylight.Time corresponds to solar noon in Madison
(L = 89° W) on September 30?
c When it is 10 a.m. Mountain Standard Time in Salt Lake City
(L = 112° W) on January 26, what is the Solar Time?
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654 Appendix A
I
1.7 Estimate Rb for a collector sloped 60° from horizontal, with y = 0°,
at Madison (rjJ = 43°) at 2:30 on March 5.
1.8 What is Rb for a collector at latitude 43° sloped 45°, with surface azimuth
angle of 15°, at 2: 30 solar time on March 5?
1.9 At Denver (latitude 40°), on December 16, what is the extraterrestrial
radiation on a horizontal surface, H Q ? What is the December monthly
mean extraterrestrial radiation on a horizontal surface, H Q ?
1.10 a What is the day's extraterrestrial radiation on a horizontal surface·
for Madison for June 16? How does this compare to the monthly
mean value?
b What is the H Q for Madison for January 26? What is fl Q for Madison
for January?
1.11 What is the angle of incidence of beam radiation on the aperture of a
concentrating collector located at rjJ = 35°, at 1500 solar time on February
15 if the collector is rotated about a single axis to minimize the angle of
incidence: .
a If the axis is horizontal and east-west?
b If the axis is horizontal and north-south?
c If the axis is parallel to the earth's axis?
1.12 Estimate the ratio of beam radiation on a collector tilted 45° toward the
south to that on a horizontal surface, if located at a latitude of 40° on
March 1, a at noon, b at 3:30 pm.
1.13 The plane of the collectors on the Arlington (WI) solar house is inclined
60° from the horizontal, is at a latitude of 43.3°N, and has a surface
azimuth angle of 0°. What is Rb for this surface on Feb. 7 at 11: 30 a.m.?
What will it be for the hour from 3 to 4 p.m. on July 4? If the collector
were tilted at 30°, what would be the value ofRb ? (Note: Do this problem
by equation and graphically.)
2.1 a For the clear day illustrated in Figure 2.5.1, determine the hourly and
daily solar radiation on a horizontal surface.
b For the cloudy day illustrated in Figure 2.5.1, determine the hourly
and daily solar radiation on a horizontal surface.
2.2 (. a For Poona, India estimate the monthly average radiation for January
and July from the average hours of sunshine data of Table 2.7.2.
b Estimate the monthly average radiation on a horizontal surface
in January and June in Madison starting with the average hours
of sunshine data from Table 2.7.1. Compare your results with
data from Appendix G.
2.3 Solar radiation on a horizontal surface integrated over the day of January
9th at Boulder (rjJ = 40°) is 4.48 MJjm 2 • What is the clearness index, K T ,
for that day? What is the estimated fraction of the day's energy which is
diffuse?
2.4 The average daily solar radiation on a horizontal surface in Lander, WY
(rp = 42.8°) is 18.9 MJjm2 in March. How much of this is beam, and how
much is diffuse?
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Problems 655
2.5 At ¢ = 43 Oil December 22, K T was 0.63. Estimate the total radiation on a
horizontal surface in the hour from 10:00 to 11 :00 a.m.
2.6 Estimate the hourly beam and diffuse radiation on a horizontal surface
in Madison for May 18, 1960; H = 29.7 MJjm 2 • Assume symmetry about
solar noon.
2.7 From Madison, 43°N, 60° tilt, January 20, the total radiation on a hori-
zontal surface is 8.0 MJjm2 •
a Estimate the total horizontal radiation for 10 to 11.
b Estimate the beam and diffuse for 10 to 11.
c What is Rb for that hour?
d If all radiation is treated as beam, what is total radiation on the tilted
surface for that hour?
e If the diffuse radiation is considered to be independent of orientation,
what is I T for that hour?
f If the diffuse radiation is uniform over sky, and ground reflectance is
considered, what is IT for that hour? Let p be 0.2 (bare ground) and
0.7 (snow).
2.8 The day's radiation on a horizontal surface in Madison (¢ = 43°) on a
Dec. 22 is 8.80 MJjm2 . There is a fresh snow cover. Estimate the diffuse
radiation, ground-reflected radiation and the total radiation on a south-
facing vertical surface during the hour 11 to 12,
2.9 A collector is installed at Boulder (t/J = 40°) at a slope of 60°, facing
south. What will be the hourly beam and diffuse components of solar
radiation on this collector on January 13, if the total radiation on a
horizontal surface for that day is 10.9 MJjm2 , and gr0l!nd reflectance
is 0.7?
2.10 Estimate for May 11, for Madison, WI the hourly total radiation on
a a horizontal surface, and b on a surface tilted toward the equator with
a slope equal to the latitude. Weather bureau data show a total solar
radiation for the day of 622 caljcm2 •
Note: For purposes of comparison, the weather bureau hourly data
are as follows:
Time cal/cm2 Time cal/cm2
5-6 0 12-1 82
6-7 7 1-2 76
7-8 22 2-3 53
8-9 39 3-4 37
9-10 54 4-5 19
10-11 77 5-6 4
11-12 82 6-7 0
2.11 Estimate the radiation on the plane of a collector sloped at 60°, y = 0°,
¢ = 43.3°N, for the hour from It to 12 a.m. on Feb, 7, The total day's
radiation on a horizontal surface is 11.8 MJjm2 •
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656 Appendix A
2.12 What is the average daily radiation in March on a south-facing surface
tilted at a slope of 35° for Albuquerque?
2.13 For Albuquerque, plot the monthly average daily radiation as a function
of month for collectors with the following orientations:
a Horizontal surface.
South facing:
b Collector tilt equal to latitude (35°).
c Collector tilt of 50°.
d Collector tilt of 20°.
e Vertical collector.
Note: let ground reflectance be 0.2 for all months.
2.14 A window faces south at a location with latitude 36°. Estimate the
January average radiation on the window if K T for the month is 0.47.
2.15 F1 for a location at </J = 45° is 14.0 MJ/m 2 in March. For that month,
which has very little snow, estimate H T for a south facing surface with
f3 = 60°.
2.16 For Madison in October, for a south-facing surface sloped at 58°,
estimate Rand H T •
2.17 Estimate the monthly average radiation in February on a south facing
vertical collector-storage wall at </J = 43.3°N. The average radiation on a
horizontal surface in February is 9.2 MJ/m 2 •
2.18 Estimate the standard clear sky (i.e., 23 km visibility) beam and diffuse
radiation on a horizontal surfacefor December 23 for Minneapolis
(</J = 46.5", elevation 432 m) for each of the hours from sunrise to noon.
2.19 If the radiation on the surface of Problem 2.18 is 0.66 MJ/m2 for the
hour 10 to 11 a.m., estimate the diffuse on this horizontal surface.
2.20 Estimate the hourly beam radiation on the aperture of a collector which
rotates continuously about a horizontal north-south axis so as to track
the sun. The data base is measurements of normal incidence radiation
measured with a pyrheliometer and integrated over the hour, and is
indicated on the table below. The latitude is 38° and the date is January 7.
Hour 8-9 9-10 10-11
Ibll , MJ/m 2 0.35 0.70 2.66
11-12 12-1
3.05 3.30
1-2 2-3 3-4
3.19 1.80 1.42
3.1 Verify the values of the blackbody spectral emissive power as given in
Figure 3.4.1 for a T = 1000 K and A = 10 Jim; b T = 400 K and A = 5
Jim; and c T = 6000 K and A = 1 Jim.
3.2 What is the percentage of the blackbody radiation from a source at 300 K
in the wavelength region from 8 to 14 Jim? (This is the so-called" window"
in the earth's atmosphere.)
3.3 Write a computer subroutine to calculate the fraction of the energy from
a blackbody source at T in the wavelength interval a to b.
3.4 Calculate the energy transfer per unit area by radiation between two large
parallel flat plates. The temperature and emittance of one plate are 500 K
and 0.45 and for the other plate are 300 I$. and 0.2. What is the radiation
heat transfer coefficient?
3.
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Problellls 657
3.5 Calculate three different overall heat transfer coefficients for a plate at
50 C when exposed to an ambient temperature of 10 C. Base your three
results on three different approximations for the effective sky temper-
ature. Plate emittance is 0.88, the relative humidity is 70 percent, and the
wind heat transfer coefficient is 25 W/m 2 C.
3.6 Consider two large flat plates spaced L mm apart. One plate is at 100 C
and the other is at 50 C. Determine the convective heat transfer between
the plates for the following conditions:
a Horizontal, heat flow up, L = 20 mm.
b Horizontal, heat flow up, L = 50 mm.
e Inclined at 45°, heat flow up, L = 20 mm.
3.7 Compute the equilibrium temperature of a thin polished copper plate
1 m x 1 m x 1 mm, under the following conditions;
a In earth orbit, with solar radiation normal to a side of the plate.
Neglect the influence of the earth. See Table 4.5.1.
b Just above the earth's surface, with solar radiation normal to the plate
and the sun directly overhead. See Table 4.5.1.
Assume a the sky is clear the transmits 0.80 of the solar radiation;
b the" equivalent blackbody sky temperature" is 10 C less than the
ambient temperature; e the ambient air temperature is 25 C; d the
wind heat transfer coefficient is 23 W/m2 C; and e the earth's surface
is effectively a blackbody at 15 C.
3.8 Consider two thin circular disks, thermally isolated from each other,
and suspended horizontally, side by side, in the same plane, inside a
glass sphere on low conductance mounts. The sphere is filled with an
inert gas, such as dry nitrogen, to prevent deterioration of the surfaces.
Dimensions of the disks are identical. One disk is painted with black
paint (CXb = 0.95, 0b = 0.95) and the other with white paint (cxw = 0.35,
Ow = 0.95). The glass has a transmittance for solar radiation (Tg) of
0.90 and an emittance for long-wave radiation of 0.88. The convection
coefficient, il, between each of the disks and the glass cover is 16 W/m2 C.
(Note that the disks have two sides and that the edges can be neglected.)
When exposed to an unknown solar radiation on a horizontal surface,
G, the temperature of the white disks (Tw) is 5 C and the temperature of
the black disc (T,,) is 15 C. 'l,;mbient is 0 C.
a Write the energy balances for the black and white disk assuming the
glass cover is at a uniform temperature eTc).
b Derive an expression for the combined convection and radiation heat
transfer coefficien t.
e Using the result of b, derive an expression giving the incident solar
radiation as a function of the difference in temperature between the
black and white disks.
d What is the incident solar radiation, G, for the conditions stated
above?
3.9 Determine the convection heat transfer between two large flat plates
(covers in a collector) separated by a distance of 20 mm and inclined at
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658 Appendix A
an angle of 60°. The lower plate is at 150 C and the upper plate is at 60 C.
3.10 What is the convective heat transfer for the conditions of Problem 3.9
when a "slat" type honeycomb is inserted in the space between the
plates? The distance between the slats is 10 mm.
3.11 Determining the heat transfer coefficient for air flowing by forced con-
vection in aIm wide by 2 m long by 15 mm deep channel. The flow rate
is 0.012 kg/so What is the heat transfer coefficient if the plate spacing is
halved? What is the heat transfer coefficient if the mass flow rate is
doubled? 7;. = 25 C.
3.12 What is the convective heat transfer coefficient due to a 7 m/s wind over
a free standing 2 m by 3 m collector array? Assume the air temperature
is 20 C.
3.13 Estimate the heat transfer coefficient due to a 10 m/s wind flowing over a
collector array. The collectors are mounted on a building with dimensions
10 m x 12 m x 4 m.
3.14 Estimate the pressure drop in a pebble bed with a 3 m by 4 m flow area and
with a 2 mjlow length. The flow rate through the bed is 1.1 kg/so The
pebbles are 0.02 m diameter river washed gravel with a void fraction of
0.45. Use an average air temperature of 40 C.
3.15 Estimate the rock diameter required for a pressure drop of 55 Pa for the
conditions of Problem 3.14. Assume the pebble void fraction remains at
0.45 for all pebble sizes.
4.1 Consider a surface that has been prepared for use in outer space and has
the following spectral characteristics:
p = 0.10
p = 0.90
for
for
For Ac = 1, 2, and 3 Jim, calculate the equilibrium temperature of the
plate. Assume the sun can be approximated by a blackbody at 6000 K
and that the solar flux on the plate is 1353 W/m 2 • Also assume the back
side of the plate is perfectly insulated.
4.2 A selective surface for solar collector absorber plates has the characteristic
a = e = 0.95
a = e = 0.05
for
for
o < A < 1.8 11m
1.8 < A < CD 11m
Assume the sun to be a blackbody emitter at a temperature of 6000 K.
Calculate the absorptance of this surface. If the surface is at 150 C,
calculate its emittance.
4.3 Determine the solar absorptance of the black chrome (after the humidity
test) surface given in Figure 4.6.3. Use the extraterrestrial solar spectrum
for this calculation.
4.4 What is the absorptance of the surface of Problem 4.3 for the terrestrial
solar radiation distribution of Table 2.6.1 ?
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Problems 659
4.5 What is the emittance of the surface of Problem 4.3 at a temperature of
350 C?
4.6 For curve C of Figure 4.6.2, calculate the emittance at temperatures
of a 300 K; b 500 K; and c 1000 K.
5.1 Calculate the reflectance of one glass surface for angles of incidence of
a 10°, b 30°, c 50°, and d 70°. (Index of refraction = 1.526.)
5.2 Calculate the transmission of three nonabsorbing glass covers at angles
of 10 and 70° and compare your results to Figure 5.1.3.
5.3 For glass with K = 20 m -1 and 2.0 mm thick, calculate the transmission
of two covers at a normal incidence and b at 50°.
5,4 Calculate the (ta) product for a two-glass-cover collector (KL = 0.0370
per plate) with a flat black collector plate surface for radiation incident
on the collector at an angle of a 25° b 60°.
5.5 Estimate the transmittance of a single cover with KL = 0.0370 for
diffuse radiation from the sky and for diffuse radiation from the ground.
The slope of the cover is a 45° and b 90°.
5.6 What is the transmittance for solar radiation of a collector cover with
index of refraction 1.60 at an angle of incidence of 58°? Thecover is
2 mm thick and the extinction coefficient is 10 m -1. If the refractive index
is 1.40 what will be the transmittance?
5.7 Estimate the radiation absorbed by a collector under the following
conditions: I b. T = 4.1 MJ/m2, I d = 0.4 MJ/m2, OT =: 25°, f3 = 45°, t
is given on Figure 5.3.1 with KL =: 0.037 and one cover, a = 0.93 and in-
" dependent of angle, ground reflectance equal to 0.2.
5.8 Estimate the absorbed solar radiation and the monthly average trans-
mittance-absorptance product for December for a vertical south-facing
collector-storage wall at Albuquerque if the angular absorptance char-
acteristic of the black surface is that shown in Figure 4.7.1 with an = 0.95.
The two covers have KL =: 0.0125 per plate. The ground reflectance
is 0.4.
6.1 Compare the value of UL calculated with Equation 6.4.7 to the graphs
of Fig. 6.4.4 at
a illY = 5 W/m2 C, Gp = 0.95, Tp = 60 C, 1;, = 10 C
b illY = 20 W/m2 C, Gp = 0.1, Tp = 100 C, 1;, =: 40 C
6.2 Calculate the overall loss coefficient for a flat-plate solar collector located
in Madison, Wisconsin, and tilted toward the equator with a slope equal
to the latitude. Assume a single-glass cover 25 mm above the absorber
plate (glass hemispherical emittance =: 0.88), a wind speed of 6.5 mis,
an absorber plate long-wavelength emittance of 0.11, and 70 mm of
rockwool insulation at the rear having a conductivity of 0.034 W/m c.
The mean absorber plate temperature is 100 C and the ambient temper-
ature is 25 C. Neglect edge effects and absorption of solar radiation by
glass. The collectors are mounted on a house with a volume of 300 m3 .
6.3 For a wind of 5 mis, an ambient air temperature of 10 C, and an average
plate temperature of 80 C, calculate the top loss coefficient for a collector
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660 Appendix A
having a single plastic cover with a transmittance for infrared radiation
of 0.30 and an emittance of 0.63. The slope is 45°, the plate-cover spacing
is 25 mm and the plate emittance is a 0.95, and b 0.10. The collector is
I m by 2 m and is mounted on a rack on a flat roof.
6.4 A flat plate solar collector has 2 glass covers, a black absorber with
Gp = 0.95, mean plate temperature of 110 C at an ambient temperature
of 10 C and a wind loss coefficient of 10 W1m 2 C. Estimate its top loss
coefficient. If the back of the collector is insulated with 50 mm of mineral
wool insulation of k of 0.035 W1m C, what is its overall loss coefficient?
(Neglect edge effects.) The slope is 45°.
6.5 Calculate UL for the collector in Figure 6.4.3a if it is insulated on the back
with 50 mm of rockwool (k = 0.045 W/m C). Neglect edge losses.
6.6 A collector has one glass cover and the plate emittance is 0.10. Wind
speed is 5 mis, average plate temperature is 75 C and ambient temperature
is 10 C. The back loss coefficient is 0.5 W/m2 C. The collectors are mount-
ed on a house 12 m long, 8 m wide, and 4 m high. Slope = 45°. Plate-
cover spacing = 25 mm. What is UL for this collector?
6.7 A tube and sheet collector is made entirely of copper. The i in. diameter
tubes are on 9 in. centers and soldered to the collector plate. The thickness
of the copper plate is 0.027 in. The collector overall energy loss coefficient,
UL' is 1.4 BTUfhr ft2 F. Calculate the fin efficiency factor, F.
6.8 Consider a flat-plate collector with a fin and tube type absorber plate.
Assume UL = 8.0 W/m
2 C, the plate is 0.5 mm thick, the tube center-to-
center distance is 100 mm, and the heat transfer coefficient inside the
20 mm diameter tubes is 300 W/m 2 C. Assume bond conductance is high.
Calculate the collector efficiency factor, F', for a copper fins, b aluminum
fi ns, and e steel fins.
6.9 An aluminum collector absorber plate has a thickness of 1 mm and a
conductivity of2l1 W1m c. For a tube spacing of200mm, a tube diameter
of 10 mm, hli = 300 W/m
2 Cand a UL of4 W/m 2 C, F' is 0.88. If the plate
were made of steel rather than aluminum (i.e., k is 48 W1m C), calculate
F'.
6.10 A flat-plate water heating collector absorber plate is copper and is 1.0 mm
thick. Tubes 10 mm in diameter are spaced 160 mm apart. The overall
collector loss coefficient is 3.0 W/m2 C and the inside heat transfer co-
efficient is 300 W1m2 c. The solder bond between the plate and tubes is
5 mm wide and averages 2 mm thick. The solder has a conductivity of
20 Wlm C. What is F' for the collector?
6.11 Estimate the useful output ofa solar collector when FR UL = 6.3 W1m2 C,
FR(m) = 0.83, 7; = 56 C, T" = 14 C, IT = 3.4 MJ/m2.
6.12 The solar energy absorbed by a solar collector, S, and the ambient
temperatures are given in the table below. The collector has UL = 5.2
W1m2 C and F R = 0.92. Determine the useful output of the collector for
the day in question with a constant inlet temperature of 35 C.
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