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Universal graph for optimal dimensions of solar collector plate
SolarEnergy,Vol.29.No. 6, pp. 573-574,1982 Printedin GreatBritain.
0038.-092X/821120573-02503.00/0 © 19~2PergamonPress Ltd.
TECHNICAL NOTE Universal graph for optimal dimensions of solar collector plate M. KOVARIK Division of Mechanical Engineering, Commonwealth Scientific and Industrial Research Organization, Highett, Victoria, Australia
(Received 17 June 1981; accepted 3 March 1982) INTRODUCTION The cost of energy delivered by a solar collector depends on irradiance, cost of components and manufacture, thermal properties of materials employed, details of construction, mode of operation and external atmospheric conditions. It has been shown, however (Kovarik, [1]), that in a fiat plate collector, the dimensions, plate thickness and pipe spacing, which result in the lowest cost per unit of energy collected depend only on the material properties, unit costs of certain components and the coefficient of heat transfer between the collector plate and the ambient air. The conditions for the optimal dimensions are expressed in a pair of simultaneous transcendental equations which can be solved numerically or, more conveniently, by graphical methods. Due to the dependence on material properties, earlier methods require separate calculations or graphs for each material considered. In the following, a solution for optimal pipe spacing and plate thickness will be derived and presented in a form which permits the construction of a universal graph, valid for all materials and heat transfer coefficients.
and the fabrication costs resulting from the process of joining th pipe to the plate. The cost factor CA may include the unit cost of cover, envelope and insulation with the part of fabrication cost (if any) which is independent of plate thickness and pipe spacing. The cost factor Co may include the cost of plate material per unit volume plus the part of fabrication cost attributable to the plate thickness, expressed in dollars per unit area per unit thickness ($m-3). The output for unit cost is thus Q= q/C, or Q = (hrLp -t tanh p)/(Cp + CaL + C.LT).
(5)
The extreme value of Q is obtained at the dimensions L, T satisfying equations
COST OF COLLECTED ENERGY Let us consider a small part of a solar energy collector comprising a short length of pipe carrying the heat collecting fluid and the attached strip of absorber plate bounded by two parallel lines at right angles to the axis of the pipe in length and by two adjacent pipes in width. Let this strip be sufficiently narrow so that the temperature differences along any line parallel to the pipe axis are negligible. Let the total width of the strip not including the pipes be L. The total heat output of the collector will be the sum of heat outputs of such elements. Each element will deliver to the pipes on its ends, the heat flux
aQlOL =0
(6a)
oQ/oT = 0
(6b)
Performing the above differentiations leads to the following conditions for minimal cost:
(2L+3Y)I(2L+ Y) = S -I sinh S
(7a)
T = CJ(2C~L)
(7b)
Y = G/CA,
(g)
S = 21/2L 3/2X
(9)
X = (CoU/Gk) ~/2
(lO)
and
where
q=F'hr.L
(1)
per unit length in the direction of the pipe. Here, h, is the net heat influx retained by unit area of the plate at the temperature obtained at the plate-pipe junction and F is the fin efficiency defined [2] for the plate thickness T, conductivity k and plate-toambient-air heat transfer coefficient U as F = p-i tanh p
and
(2)
CONCLUSION The introduction of variables X and Y permits the representation of eqn (7a) as a plot with L as a single parameter, all the values of physical and economic factors being contained in the coordinates X and Y. This plot is given in Fig. 1. The optimal pipe spacing L can be obtained from the graph; the thickness T follows from eqn (7b). This equation can be rearranged to read
where p = (L 2 U[4kT)It2.
(2)
The cost of collector per unit length in the direction of the pipe can be represented by
C = G + CaL+ G L T
(4)
where the subscripted C's are the unit cost factors related to the unit length of pipe, unit area of plate and unit volume of plate, respectively. The cost factor Cp may include the cost per unit length of pipe
Cp = 2C~LT, i.e. the cost of the pipes is twice the cost of the plate at optimal configuration. This point is made in Kovarik[l] and applies to both
SE Vol. 29, No. 6--1 573
574
Technical Note
////////
0.1
O.OS 0.04, 0.03' 0.02,
Y(m) 0.01,
0.005
0.004" 0403° 0.002"
0.001
w ila|
0.5
2.
.
.
3.
.
4 . .5
.
.
10
1o;o2o50
X (m "1"s) Fig. 1. Optimal pipe spacing L [mm].
the plate of uniform thickness and an optimally tapered collecting fin.
plate dimension is low in the vicinity of optimal design point and, therefore, the readability of the graph is sufficient for its purpose. NOMENCLATURE
EXAMPLE
CA Cp C~ k U X Y L T
60 $m -2 3 $m -I 36000 $m -3 385Wm-I K -I 4.2W m-2 K -I 11.44m -3/2 0.05 m 175 mm 0.238 mm (from eqn 7(b).
The optimal plate has an output per unit cost, from eqn (5) in W per cent:
C cost of collector per unit length of pipe, $ m -]. CA Cp C~ hr k L
Q T U X Y
cost factor related to unit area of plate, $ m -~. cost factor of collector related to unit length of pipe, $ m -l. cost factor related to unit volume of plate, $ m -3. heat influx density retained by the absorber plate at the temperature of the plate-pipe junction, W m -2. thermal conductivity of plate material, W m -] K -I. pipe spacing, m. output per unit cost, W$ -~. collector plate thickness, m. heat transfer coefficient, plate to ambient, W m -2 K -~. graph coordinate defined by eqn (10), m -3/2. graph coordinate defined by eqn (8), m.
Q = 1.047h~ REFERENCES
For a near-optimal collector of the same design using plate tl-,ickness of 0.2(0.3)ram, the corresponding value would be 1.044. hr (1.042. hr), whereas a far-from-optimal thickness of ;mm yields a value of Q = 0.859. hr W per cent. The example suggests that the sensitivity of performance to
1. M. Kovarik, Optimal distribution of heat conducting material in the finned pipe solar energy collector. Solar Energy 21, 477-484 (1978). 2. H. C. Hottel and B. B. Woertz, Performance of plat-plate solar heat collectors. Trans. ASMF, 64, 91-104 (1942).
0038.-092X/821120573-02503.00/0 © 19~2PergamonPress Ltd.
TECHNICAL NOTE Universal graph for optimal dimensions of solar collector plate M. KOVARIK Division of Mechanical Engineering, Commonwealth Scientific and Industrial Research Organization, Highett, Victoria, Australia
(Received 17 June 1981; accepted 3 March 1982) INTRODUCTION The cost of energy delivered by a solar collector depends on irradiance, cost of components and manufacture, thermal properties of materials employed, details of construction, mode of operation and external atmospheric conditions. It has been shown, however (Kovarik, [1]), that in a fiat plate collector, the dimensions, plate thickness and pipe spacing, which result in the lowest cost per unit of energy collected depend only on the material properties, unit costs of certain components and the coefficient of heat transfer between the collector plate and the ambient air. The conditions for the optimal dimensions are expressed in a pair of simultaneous transcendental equations which can be solved numerically or, more conveniently, by graphical methods. Due to the dependence on material properties, earlier methods require separate calculations or graphs for each material considered. In the following, a solution for optimal pipe spacing and plate thickness will be derived and presented in a form which permits the construction of a universal graph, valid for all materials and heat transfer coefficients.
and the fabrication costs resulting from the process of joining th pipe to the plate. The cost factor CA may include the unit cost of cover, envelope and insulation with the part of fabrication cost (if any) which is independent of plate thickness and pipe spacing. The cost factor Co may include the cost of plate material per unit volume plus the part of fabrication cost attributable to the plate thickness, expressed in dollars per unit area per unit thickness ($m-3). The output for unit cost is thus Q= q/C, or Q = (hrLp -t tanh p)/(Cp + CaL + C.LT).
(5)
The extreme value of Q is obtained at the dimensions L, T satisfying equations
COST OF COLLECTED ENERGY Let us consider a small part of a solar energy collector comprising a short length of pipe carrying the heat collecting fluid and the attached strip of absorber plate bounded by two parallel lines at right angles to the axis of the pipe in length and by two adjacent pipes in width. Let this strip be sufficiently narrow so that the temperature differences along any line parallel to the pipe axis are negligible. Let the total width of the strip not including the pipes be L. The total heat output of the collector will be the sum of heat outputs of such elements. Each element will deliver to the pipes on its ends, the heat flux
aQlOL =0
(6a)
oQ/oT = 0
(6b)
Performing the above differentiations leads to the following conditions for minimal cost:
(2L+3Y)I(2L+ Y) = S -I sinh S
(7a)
T = CJ(2C~L)
(7b)
Y = G/CA,
(g)
S = 21/2L 3/2X
(9)
X = (CoU/Gk) ~/2
(lO)
and
where
q=F'hr.L
(1)
per unit length in the direction of the pipe. Here, h, is the net heat influx retained by unit area of the plate at the temperature obtained at the plate-pipe junction and F is the fin efficiency defined [2] for the plate thickness T, conductivity k and plate-toambient-air heat transfer coefficient U as F = p-i tanh p
and
(2)
CONCLUSION The introduction of variables X and Y permits the representation of eqn (7a) as a plot with L as a single parameter, all the values of physical and economic factors being contained in the coordinates X and Y. This plot is given in Fig. 1. The optimal pipe spacing L can be obtained from the graph; the thickness T follows from eqn (7b). This equation can be rearranged to read
where p = (L 2 U[4kT)It2.
(2)
The cost of collector per unit length in the direction of the pipe can be represented by
C = G + CaL+ G L T
(4)
where the subscripted C's are the unit cost factors related to the unit length of pipe, unit area of plate and unit volume of plate, respectively. The cost factor Cp may include the cost per unit length of pipe
Cp = 2C~LT, i.e. the cost of the pipes is twice the cost of the plate at optimal configuration. This point is made in Kovarik[l] and applies to both
SE Vol. 29, No. 6--1 573
574
Technical Note
////////
0.1
O.OS 0.04, 0.03' 0.02,
Y(m) 0.01,
0.005
0.004" 0403° 0.002"
0.001
w ila|
0.5
2.
.
.
3.
.
4 . .5
.
.
10
1o;o2o50
X (m "1"s) Fig. 1. Optimal pipe spacing L [mm].
the plate of uniform thickness and an optimally tapered collecting fin.
plate dimension is low in the vicinity of optimal design point and, therefore, the readability of the graph is sufficient for its purpose. NOMENCLATURE
EXAMPLE
CA Cp C~ k U X Y L T
60 $m -2 3 $m -I 36000 $m -3 385Wm-I K -I 4.2W m-2 K -I 11.44m -3/2 0.05 m 175 mm 0.238 mm (from eqn 7(b).
The optimal plate has an output per unit cost, from eqn (5) in W per cent:
C cost of collector per unit length of pipe, $ m -]. CA Cp C~ hr k L
Q T U X Y
cost factor related to unit area of plate, $ m -~. cost factor of collector related to unit length of pipe, $ m -l. cost factor related to unit volume of plate, $ m -3. heat influx density retained by the absorber plate at the temperature of the plate-pipe junction, W m -2. thermal conductivity of plate material, W m -] K -I. pipe spacing, m. output per unit cost, W$ -~. collector plate thickness, m. heat transfer coefficient, plate to ambient, W m -2 K -~. graph coordinate defined by eqn (10), m -3/2. graph coordinate defined by eqn (8), m.
Q = 1.047h~ REFERENCES
For a near-optimal collector of the same design using plate tl-,ickness of 0.2(0.3)ram, the corresponding value would be 1.044. hr (1.042. hr), whereas a far-from-optimal thickness of ;mm yields a value of Q = 0.859. hr W per cent. The example suggests that the sensitivity of performance to
1. M. Kovarik, Optimal distribution of heat conducting material in the finned pipe solar energy collector. Solar Energy 21, 477-484 (1978). 2. H. C. Hottel and B. B. Woertz, Performance of plat-plate solar heat collectors. Trans. ASMF, 64, 91-104 (1942).