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The effect of tube spacing on the cost-effectiveness of a flat-plate solar collector
Renewabh: Energy Vol. 2, No. 6, pp, 603 61)6. 1992 Printed in Great Britain.
0960 1481/92 $5.00 + . 0 0 Pergamon Press Ltd
THE E F F E C T O F T U B E S P A C I N G O N THE C O S T - E F F E C T I V E N E S S OF A F L A T - P L A T E S O L A R COLLECTOR D. M . GHAMARI a n d R. A. WORTH Technology Department, University of the South Pacific, P.O. Box 1168, Suva, Fiji
(Received 3 February 1992 ; accepted 27 FebruaO' 1992) Abstract---A graph of collector efficiency versus tube spacing was obtained experimentally for a flat-plate copper collector. The graph was found to be approximately linear. From its slope, the optimum tube spacing for maximum cost-effectiveness was estimated, using actual material costs. For Fiji prices it was found to be in the region of 160 mm.
1. INTRODUCTION
The fin efficiency is given by :
The design of solar energy systems is concerned with obtaining useful energy as cheaply as possible. Duffle and Beckman [1] observed that "it may be desirable to design a collector with an efficiency lower than is technologically possible, if the cost is signifcantly reduced". The energy input to the system is free, and it is immediately obvious that a solar collector design with an efficiency of, say, 4 0 % , costing x dollars per unit area of collector, is preferable to a design with an efficiency of 80% costing 3x dollars per unit area, unless space is an overriding factor. Considerable research has been aimed at predicting the efficiency of solar collectors, defined as the ratio of the useful energy gained over a specified time period to the incident solar energy over the same time period. However, the concatenation of that research with cost-effective design has to some extent been neglected. Duffle and Beckman defined a collector efficiency factor as "the ratio of the actual useful energy gain to the useful energy gain that would result if the collector absorbing surface had been at the local fluid temperature". They derived the following expression for the efficiency factor, F ' :
tanh [( UL/k• ) t/2( W-- D)/2]
F --
F' =
(2)
where k = thermal conductivity of the sheet material ; c5 = sheet thickness. It is apparent from this expression that if the tube spacing is reduced, the efficiency factor will increase. F r o m the point of view of efficiency, therefore, it would seem logical that the tubes should be spaced as close together as physically possible. In the extreme case, the tubes would be touching and the plate would become redundant. Obviously as the number of tubes increases, so does the cost of the collector. There will be an optimum tube spacing, at which the cost-effectiveness of the collector will be a maximum. The costeffectiveness of the collector may be defined as the rate of useful energy output divided by the collector cost. 2. ANALYSIS The graph shown in Fig. 1 was obtained by putting the following values into eq. 1 : UL = 8 W / m 2 C ,
I/UL
(UL/kfi) '"2(W-D)/2
(1)
D = 0.016m,
E 1 , , ]i W UL{D+(W_D)IF+Cb+~hf,
k6=O.3W/C, Di = 0.0145m,
cb=30W/mC,
hf.i = 1 0 0 W / m C .
These values would be reasonable for a single-glazed copper collector. It can be seen that the graph is very close to a straight line. Duffle and Beckman evaluated eq. 1 for a range of values of k6, UL and ff.~, and their results show that, except for small values of tube spacing, the graphs of efficiency factor against spacing could
where UL = collector overall loss coefficient; W = distance between tubes; D = tube outer diameter; D~ = tube inner diameter; F-= standard fin efficiency; Cb = bond conductance; hr.~ = heat transfer coefficient between the fluid and the tube wall. 603
604
D. M. GHAMARIand R. A. WORTH 1.0
conditions, the useful energy output, in watts, is
lid CZ) I--
where L = collector length ; S = solar intensity. The cost-effectiveness of the collector is
~0,9 >U Z I..d U
(6)
q = #wLS
e = q/ c -
0.8
labW
(#o - a W ) L S
(c~ + ct/w)
(7)
(W/S)
This expression may be differentiated with respect to W, and the result equated to zero, in order to find the optimum spacing
0.7 0.05 TUBE
0.10
0.15
0.20 de
SPACINGIM
Fig. 1. Typical theoretical graph of collector efficiency factor versus tube spacing, according to eq. 1.
--(G +ct/W)aLS+(IJo--aW)LSc,/W
dW -
F" = C ' - a ' W
(3)
where a' is the slope of the approximate straight line and C ' is the intercept on the F ' axis. It is reasonable to expect that the corresponding graph of collector efficiency versus tube spacing would show a similar form, since for a given collector design under specific operating conditions, the collector efficiency will be equal to the efficiency factor multiplied by some constant. Therefore the graph of collector efficiency versus tube spacing could be represented by It = Y0 - a W
(4)
where /~ = collector efficiency; Y0 = collector efficiency for zero tube spacing ; a = slope. If a can be found by experiment, then its value can be used to determine the optimum tube spacing, in terms of cost-effectiveness, as follows. Let the collector width be w. If the tube spacing is W, then the number of tubes is n = w / W . The collector cost per unit length is c = wc~ + n c ,
(5)
where c~ = cost of sheet/unit area ($/m 2) ; ct = cost of tube/unit length (S/m). The value of ct should include the cost per unit length of soldering the tube to the sheet. For a given collector design operating under specific
(8)
For a maximum -(c~+ct/W)a+(#o-aW)ct/W
reasonably be represented by straight lines (though not necessarily passing through the point F ' = 0 at W = 0). The deviation from straight lines is greater for large values of hf,~. The graph of efficiency factor versus tube spacing, for a given collector design and under specific operating conditions, could be represented by the equation
2
(Cs + c , / W ) 2
2= 0
(9)
which gives acs W 2 + 2ac~ W - p oc~ = O.
(1 O)
Solving this quadratic equation for W will yield the optimum value of tube spacing.
3. EXPERIMENTAL
An experimental graph of collector efficiency, y, versus tube spacing, W, was required. It would have been expensive and time-consuming to manufacture a number of different collectors having a variety of tube spacings. Instead, one collector was manufactured, having a single tube, and with a width of 460 m m and a length of 970 ram, as shown in Fig. 2. The tube and sheet were copper. The thickness of the sheet was 0.52 ram. The outer diameter of the tube was 15 mm, and the inner diameter was 13.6 mm. The collector was painted matt black, and placed inside a wooden box with a glass cover. The mounting arrangement was such that the panel assembly could be adjusted so that the collector was perpendicular to the incident solar radiation. A rod was attached to the panel in a perpendicular position, and its shadow was used to gauge whether the panel was perpendicular to the sun's radiation. Water was allowed to flow through the copper tube at a rate which was slow enough to ensure that an easily measurable temperature difference was set up between the inlet and the outlet. Temperatures at inlet (Ti) and outlet (To) were measured using thermocouples after steady state conditions had been established. The flow rate of water was determined by collecting the flow in a measuring cylinder over a given time period. A solarimeter was used to measure the
The effect of tube spacing on the cost-effectiveness of a flat-plate solar collector
605
DETfllLS nF TUBE AND SHEET
Fig. 2. Details of the collector. solar intensity, a n d a vane a n e m o m e t e r to measure the wind speed. The useful heat o u t p u t was calculated as the product o f mass flow rate, specific heat a n d t e m p e r a t u r e rise. The energy input was calculated as the p r o d u c t o f collector area and solar intensity. Hence the collector efficiency was determined. The collector was removed from the assembly, a n d its width was reduced by shearing a n a r r o w strip off each side of the sheet. Subsequently the collector was remounted, a n d its efficiency determined. This was repealed for 12 collector widths. The results are presented in T a b l c 1. The experiments were all carried o u t w h e n there was no visible cloud between the collector a n d the sun, and at similar times o f day so t h a t solar intensity was a p p r o x i m a t e l y constant, The experiments were conducted u n d e r conditions when wind speeds were
Table I. Experimental results
as similar as possible. F o r each experiment strips o f a l u m i n i u m foil were placed o n the glass cover o n either side of the collector, so t h a t radiation not incident on the collector would be reflected. F r o m the results of Table l a g r a p h o f collector efficiency versus tube spacing was plotted, which is s h o w n in Fig. 3. 4. RESULTS Figure 3 shows the experimental points, together with the best straight line t h r o u g h the points (determined using the least m e a n squares method.) The intercept on the efficiency axis at zero spacing, t~0, is [).96 and the (negative) slope, a, of the straight line is 1.12. The cost of the copper sheet (in Fiji dollars) was F$107.80 per square metre, a n d the cost o f the c o p p e r
the effect of panel width on absorber efficiency
Panel width (m)
Solar intensity (W/m 2)
Mass flow rate (g/s)
Inlet temp. (C)
Outlet temp. (C)
Efficiency
0.46 0.43 0.40 0.37 0.34 0.31 0.28 0.25 0.22 0.19 0.16 0.13
879 991 964 906 987 955 877 964 1044 1044 1003 1026
2.28 2.37 2.40 2.40 2.40 2.40 2.33 2.33 2.10 2.03 1.83 2.03
33.6 31.6 33.5 31.7 32.7 33.9 32.6 36.3 30.5 33.7 36.2 33.6
52.5 51.I 52.0 49.7 51.5 51.0 48.0 52.7 49.0 50.7 52.2 46.0
0.46 0.47 0.50 0.56 0.58 0.60 0.63 0.68 0.73 0.75 0.79 0.82
D. M. GHAMARIand R. A. WORTH
606
Solving the quadratic, and taking the positive value of W, it is found that the optimum tube spacing is 0.162 m.
1.0
>-
u
0.8
Z W
\\
5. DISCUSSION
0.7 U
,\
b_ 0.6 h LJ
0.5
\
0.4
O.O 0.I
0.2
0.3
0.4
0.5
PANEL WIIITH/II
Fig. 3. Experimental results of collector efficiencyversus tube spacing, showing the best straight line through the points. tubing was F$5.30 per metre. (F$1 = US$0.71.) The latter included the cost of soldering the tube to the sheet. Substituting the above values into eq. 10 we have 120.7W2+ 11.88W--5.093 = 0.
(11)
A simple method of determining the optimum spacing of tubes in a fiat-plate solar collector has been described. The actual value of optimum spacing will depend on many factors, including the local cost of materials, the prevailing solar intensity, the thickness of the sheet and the dimensions of the tubing, the heat transfer coefficient between the water and the tube wall, and the overall loss coefficient. However for a given design, which is to be operated in a particular region of the world, it is possible to optimize the tube spacing, given the local cost of materials. For the design described, operating in Fiji, the optimum tube spacing is in the region of 160 ram. REFERENCE
1. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. John Wiley, New York (1980).
0960 1481/92 $5.00 + . 0 0 Pergamon Press Ltd
THE E F F E C T O F T U B E S P A C I N G O N THE C O S T - E F F E C T I V E N E S S OF A F L A T - P L A T E S O L A R COLLECTOR D. M . GHAMARI a n d R. A. WORTH Technology Department, University of the South Pacific, P.O. Box 1168, Suva, Fiji
(Received 3 February 1992 ; accepted 27 FebruaO' 1992) Abstract---A graph of collector efficiency versus tube spacing was obtained experimentally for a flat-plate copper collector. The graph was found to be approximately linear. From its slope, the optimum tube spacing for maximum cost-effectiveness was estimated, using actual material costs. For Fiji prices it was found to be in the region of 160 mm.
1. INTRODUCTION
The fin efficiency is given by :
The design of solar energy systems is concerned with obtaining useful energy as cheaply as possible. Duffle and Beckman [1] observed that "it may be desirable to design a collector with an efficiency lower than is technologically possible, if the cost is signifcantly reduced". The energy input to the system is free, and it is immediately obvious that a solar collector design with an efficiency of, say, 4 0 % , costing x dollars per unit area of collector, is preferable to a design with an efficiency of 80% costing 3x dollars per unit area, unless space is an overriding factor. Considerable research has been aimed at predicting the efficiency of solar collectors, defined as the ratio of the useful energy gained over a specified time period to the incident solar energy over the same time period. However, the concatenation of that research with cost-effective design has to some extent been neglected. Duffle and Beckman defined a collector efficiency factor as "the ratio of the actual useful energy gain to the useful energy gain that would result if the collector absorbing surface had been at the local fluid temperature". They derived the following expression for the efficiency factor, F ' :
tanh [( UL/k• ) t/2( W-- D)/2]
F --
F' =
(2)
where k = thermal conductivity of the sheet material ; c5 = sheet thickness. It is apparent from this expression that if the tube spacing is reduced, the efficiency factor will increase. F r o m the point of view of efficiency, therefore, it would seem logical that the tubes should be spaced as close together as physically possible. In the extreme case, the tubes would be touching and the plate would become redundant. Obviously as the number of tubes increases, so does the cost of the collector. There will be an optimum tube spacing, at which the cost-effectiveness of the collector will be a maximum. The costeffectiveness of the collector may be defined as the rate of useful energy output divided by the collector cost. 2. ANALYSIS The graph shown in Fig. 1 was obtained by putting the following values into eq. 1 : UL = 8 W / m 2 C ,
I/UL
(UL/kfi) '"2(W-D)/2
(1)
D = 0.016m,
E 1 , , ]i W UL{D+(W_D)IF+Cb+~hf,
k6=O.3W/C, Di = 0.0145m,
cb=30W/mC,
hf.i = 1 0 0 W / m C .
These values would be reasonable for a single-glazed copper collector. It can be seen that the graph is very close to a straight line. Duffle and Beckman evaluated eq. 1 for a range of values of k6, UL and ff.~, and their results show that, except for small values of tube spacing, the graphs of efficiency factor against spacing could
where UL = collector overall loss coefficient; W = distance between tubes; D = tube outer diameter; D~ = tube inner diameter; F-= standard fin efficiency; Cb = bond conductance; hr.~ = heat transfer coefficient between the fluid and the tube wall. 603
604
D. M. GHAMARIand R. A. WORTH 1.0
conditions, the useful energy output, in watts, is
lid CZ) I--
where L = collector length ; S = solar intensity. The cost-effectiveness of the collector is
~0,9 >U Z I..d U
(6)
q = #wLS
e = q/ c -
0.8
labW
(#o - a W ) L S
(c~ + ct/w)
(7)
(W/S)
This expression may be differentiated with respect to W, and the result equated to zero, in order to find the optimum spacing
0.7 0.05 TUBE
0.10
0.15
0.20 de
SPACINGIM
Fig. 1. Typical theoretical graph of collector efficiency factor versus tube spacing, according to eq. 1.
--(G +ct/W)aLS+(IJo--aW)LSc,/W
dW -
F" = C ' - a ' W
(3)
where a' is the slope of the approximate straight line and C ' is the intercept on the F ' axis. It is reasonable to expect that the corresponding graph of collector efficiency versus tube spacing would show a similar form, since for a given collector design under specific operating conditions, the collector efficiency will be equal to the efficiency factor multiplied by some constant. Therefore the graph of collector efficiency versus tube spacing could be represented by It = Y0 - a W
(4)
where /~ = collector efficiency; Y0 = collector efficiency for zero tube spacing ; a = slope. If a can be found by experiment, then its value can be used to determine the optimum tube spacing, in terms of cost-effectiveness, as follows. Let the collector width be w. If the tube spacing is W, then the number of tubes is n = w / W . The collector cost per unit length is c = wc~ + n c ,
(5)
where c~ = cost of sheet/unit area ($/m 2) ; ct = cost of tube/unit length (S/m). The value of ct should include the cost per unit length of soldering the tube to the sheet. For a given collector design operating under specific
(8)
For a maximum -(c~+ct/W)a+(#o-aW)ct/W
reasonably be represented by straight lines (though not necessarily passing through the point F ' = 0 at W = 0). The deviation from straight lines is greater for large values of hf,~. The graph of efficiency factor versus tube spacing, for a given collector design and under specific operating conditions, could be represented by the equation
2
(Cs + c , / W ) 2
2= 0
(9)
which gives acs W 2 + 2ac~ W - p oc~ = O.
(1 O)
Solving this quadratic equation for W will yield the optimum value of tube spacing.
3. EXPERIMENTAL
An experimental graph of collector efficiency, y, versus tube spacing, W, was required. It would have been expensive and time-consuming to manufacture a number of different collectors having a variety of tube spacings. Instead, one collector was manufactured, having a single tube, and with a width of 460 m m and a length of 970 ram, as shown in Fig. 2. The tube and sheet were copper. The thickness of the sheet was 0.52 ram. The outer diameter of the tube was 15 mm, and the inner diameter was 13.6 mm. The collector was painted matt black, and placed inside a wooden box with a glass cover. The mounting arrangement was such that the panel assembly could be adjusted so that the collector was perpendicular to the incident solar radiation. A rod was attached to the panel in a perpendicular position, and its shadow was used to gauge whether the panel was perpendicular to the sun's radiation. Water was allowed to flow through the copper tube at a rate which was slow enough to ensure that an easily measurable temperature difference was set up between the inlet and the outlet. Temperatures at inlet (Ti) and outlet (To) were measured using thermocouples after steady state conditions had been established. The flow rate of water was determined by collecting the flow in a measuring cylinder over a given time period. A solarimeter was used to measure the
The effect of tube spacing on the cost-effectiveness of a flat-plate solar collector
605
DETfllLS nF TUBE AND SHEET
Fig. 2. Details of the collector. solar intensity, a n d a vane a n e m o m e t e r to measure the wind speed. The useful heat o u t p u t was calculated as the product o f mass flow rate, specific heat a n d t e m p e r a t u r e rise. The energy input was calculated as the p r o d u c t o f collector area and solar intensity. Hence the collector efficiency was determined. The collector was removed from the assembly, a n d its width was reduced by shearing a n a r r o w strip off each side of the sheet. Subsequently the collector was remounted, a n d its efficiency determined. This was repealed for 12 collector widths. The results are presented in T a b l c 1. The experiments were all carried o u t w h e n there was no visible cloud between the collector a n d the sun, and at similar times o f day so t h a t solar intensity was a p p r o x i m a t e l y constant, The experiments were conducted u n d e r conditions when wind speeds were
Table I. Experimental results
as similar as possible. F o r each experiment strips o f a l u m i n i u m foil were placed o n the glass cover o n either side of the collector, so t h a t radiation not incident on the collector would be reflected. F r o m the results of Table l a g r a p h o f collector efficiency versus tube spacing was plotted, which is s h o w n in Fig. 3. 4. RESULTS Figure 3 shows the experimental points, together with the best straight line t h r o u g h the points (determined using the least m e a n squares method.) The intercept on the efficiency axis at zero spacing, t~0, is [).96 and the (negative) slope, a, of the straight line is 1.12. The cost of the copper sheet (in Fiji dollars) was F$107.80 per square metre, a n d the cost o f the c o p p e r
the effect of panel width on absorber efficiency
Panel width (m)
Solar intensity (W/m 2)
Mass flow rate (g/s)
Inlet temp. (C)
Outlet temp. (C)
Efficiency
0.46 0.43 0.40 0.37 0.34 0.31 0.28 0.25 0.22 0.19 0.16 0.13
879 991 964 906 987 955 877 964 1044 1044 1003 1026
2.28 2.37 2.40 2.40 2.40 2.40 2.33 2.33 2.10 2.03 1.83 2.03
33.6 31.6 33.5 31.7 32.7 33.9 32.6 36.3 30.5 33.7 36.2 33.6
52.5 51.I 52.0 49.7 51.5 51.0 48.0 52.7 49.0 50.7 52.2 46.0
0.46 0.47 0.50 0.56 0.58 0.60 0.63 0.68 0.73 0.75 0.79 0.82
D. M. GHAMARIand R. A. WORTH
606
Solving the quadratic, and taking the positive value of W, it is found that the optimum tube spacing is 0.162 m.
1.0
>-
u
0.8
Z W
\\
5. DISCUSSION
0.7 U
,\
b_ 0.6 h LJ
0.5
\
0.4
O.O 0.I
0.2
0.3
0.4
0.5
PANEL WIIITH/II
Fig. 3. Experimental results of collector efficiencyversus tube spacing, showing the best straight line through the points. tubing was F$5.30 per metre. (F$1 = US$0.71.) The latter included the cost of soldering the tube to the sheet. Substituting the above values into eq. 10 we have 120.7W2+ 11.88W--5.093 = 0.
(11)
A simple method of determining the optimum spacing of tubes in a fiat-plate solar collector has been described. The actual value of optimum spacing will depend on many factors, including the local cost of materials, the prevailing solar intensity, the thickness of the sheet and the dimensions of the tubing, the heat transfer coefficient between the water and the tube wall, and the overall loss coefficient. However for a given design, which is to be operated in a particular region of the world, it is possible to optimize the tube spacing, given the local cost of materials. For the design described, operating in Fiji, the optimum tube spacing is in the region of 160 ram. REFERENCE
1. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. John Wiley, New York (1980).