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Using A and V as the variables, set up the objective function and constraint to optimize the first cost. Set it up so that there is one constraint equation.

Using A and V as the variables, set up the objective function and constraint to optimize the first cost. Set it up so that there is one constraint equation.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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A solar collector and storage tank, shown in Figure (a) below, 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 Figure (b). The collector receives 260 W/m² of solar energy, 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 difference 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 101.5C, where V is the volume in cubic meters. Storage tank Collector (a) Temperature, C tmax 25°C, lowest useful temperature i 10°C ambient Time (b) Using A and V as the variables, set up the objective functio cost. Set it up so that there is one constraint equation. 10 h = 36,000 s and constraint to optimize the first
Transcribed Image Text:A solar collector and storage tank, shown in Figure (a) below, 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 Figure (b). The collector receives 260 W/m² of solar energy, 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 difference 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 101.5C, where V is the volume in cubic meters. Storage tank Collector (a) Temperature, C tmax 25°C, lowest useful temperature i 10°C ambient Time (b) Using A and V as the variables, set up the objective functio cost. Set it up so that there is one constraint equation. 10 h = 36,000 s and constraint to optimize the first
Expert Solution
Step 1: introduction

Given:

water temperature = T = 250C

Solar Energy: 260 W/m2

convection coefficient = h = 2 W/(m2.k)

ambient temperature = Tamb = 100C

density of water = 1000 kg/m3

specific heat = 4.19 KJ/ kg.K


To be find:

Using A and V as variables, setup the objective function and constraint to optimize the first cost

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