Question

In a house achieving a heat loss rate of \(200 \mathrm{~W} /{ }^{\circ} \mathrm{C}\) equipped only with two \(1,500 \mathrm{~W}\) space heaters, what is the coldest it can get outside if the house is to maintain an internal temperature of \(20^{\circ} \mathrm{C} ?\)

Short answer

Answer: The coldest possible outside temperature is 5°C.

Step-by-step solution

Calculate total heat provided by space heaters

First, let's determine the total heat being provided by the two space heaters. Each space heater has a power of 1500 W, so we can find the sum of their powers: $$Total\_heat = 1500 \mathrm{~W}+ 1500 \mathrm{~W}$$

Calculate the maximum heat loss

Now we need to find the maximum heat loss the house can have while still maintaining an internal temperature of 20°C. We know that the heat loss rate is 200 W/°C, therefore maximum heat loss is equal to the total heat provided by the space heaters: $$Maximum\_heat\_loss = Total\_heat$$

Calculate the maximum temperature difference between inside and outside

Next, we'll need to find the maximum temperature difference between the inside and outside of the house. We can do this by dividing the maximum heat loss by the heat loss rate: $$Maximum\_temperature\_difference = \frac{Maximum\_heat\_loss}{Heat\_loss\_rate}$$

Determine the coldest possible outside temperature

Finally, we can determine the coldest possible outside temperature by subtracting the maximum temperature difference from the internal temperature: $$Coldest\_outside\_temperature = Internal\_temperature - Maximum\_temperature\_difference$$ Now let's use these steps to solve the problem.

Calculate total heat provided by space heaters

$$Total\_heat = 1500 \mathrm{~W}+ 1500 \mathrm{~W} = 3000 \mathrm{~W}$$

Calculate the maximum heat loss

$$Maximum\_heat\_loss = Total\_heat = 3000 \mathrm{~W}$$

Calculate the maximum temperature difference between inside and outside

$$Maximum\_temperature\_difference = \frac{Maximum\_heat\_loss}{Heat\_loss\_rate} = \frac{3000 \mathrm{~W}}{200 \mathrm{~W}/^{\circ}\mathrm{C}} = 15^{\circ}\mathrm{C}$$

Determine the coldest possible outside temperature

$$Coldest\_outside\_temperature = Internal\_temperature - Maximum\_temperature\_difference = 20^{\circ}\mathrm{C} - 15^{\circ}\mathrm{C} = 5^{\circ}\mathrm{C}$$ Therefore, the coldest it can get outside while still maintaining an internal temperature of 20°C is 5°C.

Key concepts

Space Heaters

Space heaters are portable devices used to heat small areas. In homes, they serve as a convenient way to provide warmth during cold seasons, offering an immediate and direct heat source. Power is a key factor in their efficiency, often measured in watts (W). For example, our problem involves two 1,500 W space heaters, making a total heating capacity of 3,000 W combined.
These devices work by converting electrical energy into heat, assisting in maintaining a comfortable indoor temperature.

• Safety: They usually include features like automatic shut-off and tip-over protection.
• Usage: Best used in enclosed spaces to maximize heat retention.
• Types: Options include ceramic, oil-filled, and infrared heaters.

Choosing the right heater depends on factors such as room size, insulation quality, and personal preference.

Heat Loss Rate

The heat loss rate represents how fast heat escapes a building, measured in watts per degree Celsius (W/°C). In this exercise, the house has a heat loss rate of 200 W/°C. This means for every degree the outside temperature drops below the inside, the house loses 200 W of heat.
Understanding this rate helps calculate how much heat a home needs to maintain a desired temperature. Key aspects include:

• Construction Materials: Quality of insulation and materials affect heat loss.
• Windows and Doors: Poorly sealed windows and doors can increase heat loss significantly.
• Ventilation: While necessary, it can contribute to heat escape if not managed properly.

Managing these factors effectively reduces the heat loss rate, improving energy efficiency and comfort.

Temperature Difference Calculation

The temperature difference calculation helps in determining the outer weather conditions one can withstand while keeping a room at a certain temperature. To find this, you divide the total provided heat by the heat loss rate.
In our problem, a total of 3,000 W is provided by the heaters, with a house heat loss rate of 200 W/°C.
Using the formula: \[\text{Maximum Temperature Difference} = \frac{\text{Total Heat}}{\text{Heat Loss Rate}}\]we derive:\[\text{Maximum Temperature Difference} = \frac{3,000 \text{ W}}{200 \text{ W/°C}} = 15 °C\]
This calculation indicates how many degrees of outdoor temperature drop is sustainable while keeping the internal setting at 20°C. It helps in energy planning and ensuring comfort without excessive energy costs.