8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimizedto achieve minimum first cost. During the daHide image transcript8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimized to achieve minimum first cost. During the day the temperature of water in the storage vessel is elevated from 25°C (the minimum useful temperature) to tmax, as shown in Fig. 8-15b. The collector receives 260 W/m² of solar ener- gy, but there is heat loss from the collector to ambient air by convection. The convection coefficient is 2 W/(m² · K), and the average temperature differ- ence during the 10-hour day is (25 + tmax)/2 minus the ambient temperature of 10°C. The energy above the minimum useful temperature of 25°C that is to be stored in the vessel during the day is 200,000 kJ. The density of water is 1000 kg/m³, and its specific heat is 4.19 kJ/(kg · K). The cost of the solar collector in dollars is 20A, where A is the area in square meters, and the cost of the storage vessel in dollars is 101.5V, where V is the volume in cubic meters. (a) Using A and V as the variables, set up the objective function and constraint to optimize the first cost. (b) Develop the Lagrange multiplier equations and verify that they are satisfied by V = 1.2 m³ and A = 29.2 m2. Ans.: (a) y = 20A + 101.5V , subject to A(230 – 47.7/V ) = 5555. solve for part b 8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimizedto achieve minimum first cost. During the day the temperature of water inthe storage vessel is elevated from 25°C (the minimum useful temperature) tomax, as shown in Fig. 8-15b. The collector receives 260 W/m² of solar ener-gy, but there is heat loss from the collector to ambient air by convection. Theconvection coefficient is 2 W/(m² K), and the average temperature differ-ence during the 10-hour day is (25+ max)/2 minus the ambient temperatureof 10°C.The energy above the minimum useful temperature of 25°C that is tobe stored in the vessel during the day is 200,000 kJ. The density of water is1000 kg/m³, and its specific heat is 4.19 kJ/(kg K). The cost of the solarcollector in dollars is 20A, where A is the area in square meters, and the costof the storage vessel in dollars is 101.5V, where V is the volume in cubicmeters.(a) Using A and V as the variables, set up the objective function andconstraint to optimize the first cost.(b) Develop the Lagrange multiplier equations and verify that they aresatisfied by V 1.2 m³ and A = 29.2 m².Ans.: (a) y== 20A + 101.5V, subject to A(230 - 47.7/V) = 5555.CollectorStoragetankӨTemperature, °CImax25°C, lowest useful temperature10°C ambientTime10 h = 36,000 s(a)FIGURE 8-15(b)(a) Collector and storage tank (b) temperature variation in Prob. 8.8.

The objective function is the total cost of the solar collector and the storage vessel. This is…

8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimized to achieve minimum first cost. During the day the temperature of water in the storage vessel is elevated from 25°C (the minimum useful temperature) to max, as shown in Fig. 8-15b. The collector receives 260 W/m² of solar ener- gy, but there is heat loss from the collector to ambient air by convection. The convection coefficient is 2 W/(m² K), and the average temperature differ- ence during the 10-hour day is (25+ max)/2 minus the ambient temperature of 10°C. The energy above the minimum useful temperature of 25°C that is to be stored in the vessel during the day is 200,000 kJ. The density of water is 1000 kg/m³, and its specific heat is 4.19 kJ/(kg K). The cost of the solar collector in dollars is 20A, where A is the area in square meters, and the cost of the storage vessel in dollars is 101.5V, where V is the volume in cubic meters. (a) Using A and V as the variables, set up the objective function and constraint to optimize the first cost. (b) Develop the Lagrange multiplier equations and verify that they are satisfied by V 1.2 m³ and A = 29.2 m². Ans.: (a) y = = 20A + 101.5V, subject to A(230 - 47.7/V) = 5555. Collector Storage tank Ө Temperature, °C Imax 25°C, lowest useful temperature 10°C ambient Time 10 h = 36,000 s (a) FIGURE 8-15 (b) (a) Collector and storage tank (b) temperature variation in Prob. 8.8.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimized to achieve minimum first cost. During the da Hide image transcript 8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimized to achieve minimum first cost. During the day the temperature of water in the storage vessel is elevated from 25°C (the minimum useful temperature) to tmax, as shown in Fig. 8-15b. The collector receives 260 W/m² of solar ener- gy, but there is heat loss from the collector to ambient air by convection. The convection coefficient is 2 W/(m² · K), and the average temperature differ- ence during the 10-hour day is (25 + tmax)/2 minus the ambient temperature of 10°C. The energy above the minimum useful temperature of 25°C that is to be stored in the vessel during the day is 200,000 kJ. The density of water is 1000 kg/m³, and its specific heat is 4.19 kJ/(kg · K). The cost of the solar collector in dollars is 20A, where A is the area in square meters, and the cost of the storage vessel in dollars is 101.5V, where V is the volume in cubic meters. (a) Using A and V as the variables, set up the objective function and constraint to optimize the first cost. (b) Develop the Lagrange multiplier equations and verify that they are satisfied by V = 1.2 m³ and A = 29.2 m2. Ans.: (a) y = 20A + 101.5V , subject to A(230 – 47.7/V ) = 5555. solve for part b

8.8. A solar collector and storage tank, shown in Fig. 8-15a, is to be optimized
to achieve minimum first cost. During the day the temperature of water in
the storage vessel is elevated from 25°C (the minimum useful temperature) to
max, as shown in Fig. 8-15b. The collector receives 260 W/m² of solar ener-
gy, but there is heat loss from the collector to ambient air by convection. The
convection coefficient is 2 W/(m² K), and the average temperature differ-
ence during the 10-hour day is (25+ max)/2 minus the ambient temperature
of 10°C.
The energy above the minimum useful temperature of 25°C that is to
be stored in the vessel during the day is 200,000 kJ. The density of water is
1000 kg/m³, and its specific heat is 4.19 kJ/(kg K). The cost of the solar
collector in dollars is 20A, where A is the area in square meters, and the cost
of the storage vessel in dollars is 101.5V, where V is the volume in cubic
meters.
(a) Using A and V as the variables, set up the objective function and
constraint to optimize the first cost.
(b) Develop the Lagrange multiplier equations and verify that they are
satisfied by V 1.2 m³ and A = 29.2 m².
Ans.: (a) y
=
= 20A + 101.5V, subject to A(230 - 47.7/V) = 5555.
Collector
Storage
tank
Ө
Temperature, °C
Imax
25°C, lowest useful temperature
10°C ambient
Time
10 h = 36,000 s
(a)
FIGURE 8-15
(b)
(a) Collector and storage tank (b) temperature variation in Prob. 8.8.

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