In cold climates, including the northern United States, a house can be built with very large windows facing south to take advantage of solar heating. Sunlight shining in during the daytime is absorbed by the floor, interior walls, and objects in the room, raising their temperature to 37.0°C. If the house is well insulated, you may model it as losing energy by heat steadily at the rate 5 575 W on a day in April when the average exterior temperature is 4°C and when the conventional heating system is not used at all. During the period between 5:00 p.m. and 7:00 a.m., the temperature of the house drops and a sufficiently large "thermal mass" is required to keep it from dropping too far. The thermal mass can be a large quantity of stone (with specific heat 850 J/kg · °C) in the floor and the interior walls exposed to sunlight. What mass of stone is required if the temperature is not to drop below 19.0°C overnight?

Question

In cold climates, including the northern United States, a house can be built with very large windows facing south to take advantage of solar heating. Sunlight shining in during the daytime is absorbed by the floor, interior walls, and objects in the room, raising their temperature to 37.0°C. If the house is well insulated, you may model it as losing energy by heat steadily at the rate 5 575 W on a day in April when the average exterior temperature is 4°C and when the conventional heating system is not used at all. During the period between 5:00 p.m. and 7:00 a.m., the temperature of the house drops and a sufficiently large "thermal mass" is required to keep it from dropping too far. The thermal mass can be a large quantity of stone (with specific heat 850 J/kg · °C) in the floor and the interior walls exposed to sunlight. What mass of stone is required if the temperature is not to drop below 19.0°C overnight?

Answer 1

To determine the mass of stone required to keep the temperature from dropping below 19.0°C overnight, we need to calculate the energy stored in the thermal mass during the day and then use it to calculate how much energy it can release during the night.

First, let's calculate the energy absorbed by the thermal mass during the day. We can use the formula:

Energy absorbed = Mass * Specific heat * Change in temperature

The change in temperature is the difference between the initial temperature (37.0°C) and the final temperature (19.0°C). The specific heat of stone is given as 850 J/kg · °C.

Let's plug in the values:

Energy absorbed = Mass * 850 J/kg · °C * (37.0°C - 19.0°C)

Now, to calculate the energy released by the thermal mass overnight, we multiply the energy absorbed by the heat loss rate:

Energy released = Energy absorbed - Heat loss rate * time

Since we already know the heat loss rate (5,575 W) and the duration overnight (14 hours), we can plug in these values:

Energy released = Energy absorbed - 5,575 W * 14 hours

Now, we can set the energy released equal to zero since we want the temperature to remain above 19.0°C:

Energy released = 0

By equating the energy released to zero and solving the equation, we can determine the mass of the stone required:

Mass * 850 J/kg · °C * (37.0°C - 19.0°C) = 5,575 W * 14 hours

Simplifying the equation, we have:

Mass = (5,575 W * 14 hours) / (850 J/kg · °C * (37.0°C - 19.0°C))

Now you can calculate the mass of stone required by plugging in the values and performing the calculation.