A hot-water heater is operated by solar power. If the solar collector has an area of 5.8 m2 and the power delivered by sunlight is 550 W/m2, how long will it take to increase the temperature of 1.0 m3 of water from 24°C to 59°C?
Recived solar power x time = C*M*deltaT
C = 4184 J/kg*C
M = 1000 kg
deltaT = 35 C
received solar power = 5.8 m^2 * 550 W/m^2 = 3190 Watts
Solve for time required.
It looks like about a half a day.
To calculate the time it takes to increase the temperature of water, we need to determine the amount of energy required and then calculate the time using the power delivered by the solar collector.
Step 1: Calculate the energy required
The energy required to increase the temperature of water can be calculated using the specific heat capacity of water and the change in temperature. The specific heat capacity of water is approximately 4.18 J/g·°C.
Given:
Volume of water = 1.0 m^3 = 1000 L = 1000 kg (since 1 L of water is equal to 1 kg)
Change in temperature = 59°C - 24°C = 35°C
Energy required = mass of water × specific heat capacity × change in temperature
Energy required = 1000 kg × 4.18 J/g·°C × 35°C
Step 2: Convert units
Since the specific heat capacity is given in grams, we need to convert the mass of water to grams.
1 kg = 1000 g
Energy required = 1000 g × 1000 × 4.18 J/g·°C × 35°C
Step 3: Calculate the total energy required
Energy required = 1000 × 1000 × 4.18 J × 35
Step 4: Calculate the time
Now that we know the energy required, we can calculate the time using the power delivered by the solar collector.
Given:
Solar collector area = 5.8 m^2
Power delivered by sunlight = 550 W/m^2
The power delivered by the solar collector can be calculated as:
Power delivered = Solar collector area × Power delivered by sunlight
Power delivered = 5.8 m^2 × 550 W/m^2
Finally, we can calculate the time using the formula:
Time = Energy required / Power delivered
Substituting the values:
Time = (1000 × 1000 × 4.18 J × 35) / (5.8 m^2 × 550 W/m^2)
Simplify the equation and cancel out the units:
Time = (1000 × 1000 × 4.18 × 35) / (5.8 × 550)
Solving this equation will give us the time required to increase the temperature of 1.0 m^3 of water from 24°C to 59°C.