A water bath contains 15kg of water at 50°C. If hot and cold taps are turned on such that they deliver water at the rate of 30kg per minute.
Determine the temperature of the mixture after 20minutes, if the temperature of the hot and cold water is 100°C and 15°C respectively?
{(15 * 50) + [20 * 30 * [(100 + 15) / 2]]} / (total mass)
To determine the temperature of the mixture after 20 minutes, we need to consider the heat gained by the water bath and the heat lost by the taps water during this time.
First, let's calculate the heat gained by the water bath:
Heat gained = mass * specific heat capacity * temperature rise
The mass of water in the water bath is 15 kg, and the specific heat capacity of water is 4.18 J/g°C. The temperature rise is calculated by subtracting the initial temperature (50°C) from the final temperature of the mixture.
Now, let's calculate the heat lost by the taps water:
Heat lost = mass * specific heat capacity * temperature difference
The mass of the taps water is 30 kg per minute. In 20 minutes, the total mass of the taps water would be 30 kg/minute * 20 minutes = 600 kg. The specific heat capacity of water is 4.18 J/g°C. The temperature difference is calculated by subtracting the initial temperature of the taps water (15°C) from the final temperature of the mixture.
Next, we need to equate the heat gained by the water bath to the heat lost by the taps water:
Heat gained = Heat lost
mass * specific heat capacity * temperature rise = mass * specific heat capacity * temperature difference
Since the mass and the specific heat capacity are the same on both sides of the equation, we can cancel them out:
temperature rise = temperature difference
Finally, we can calculate the final temperature of the mixture:
Final temperature = initial temperature of the water bath + temperature rise
Let's plug in the values:
temperature rise = 100°C - 15°C = 85°C
Final temperature = 50°C + 85°C = 135°C
Therefore, the temperature of the mixture after 20 minutes is 135°C.