A bath contains 100 kg of water at 60 °C. Hot and cold water taps are turned on to deliver 20 kg per minute each at temperatures of 70 °C and 10 C respectively. How long will it be before the temperature in the bath has dropped to 45 C ? Assume complete mixing of the water and ignore heat losses.
To solve this problem, we can use the principle of conservation of energy, which states that the amount of heat gained by the water in the bath is equal to the amount of heat lost.
The formula for this principle is:
(m1 * c1 * ΔT1) + (m2 * c2 * ΔT2) = (m3 * c3 * ΔT3)
Where:
m1 and c1 are the mass and specific heat capacity of the water in the bath initially.
ΔT1 is the change in temperature of the water in the bath initially.
m2 and c2 are the mass and specific heat capacity of the hot water added per minute.
ΔT2 is the change in temperature of the hot water added per minute.
m3 and c3 are the mass and specific heat capacity of the mixture in the bath after t minutes.
ΔT3 is the change in temperature of the mixture in the bath after t minutes.
Given:
m1 = 100 kg
c1 = specific heat capacity of water = 4186 J/kg°C
ΔT1 = 60°C - 45°C = 15°C
m2 = 20 kg/min (hot water added per minute)
c2 = specific heat capacity of hot water = 4186 J/kg°C
ΔT2 = 70°C - 45°C = 25°C
m3 = 100 kg + (20 kg/min * t) (total mass of the mixture in the bath after t minutes)
c3 = specific heat capacity of the mixture (assuming it's the same as water) = 4186 J/kg°C
ΔT3 = 60°C - 45°C = 15°C
Plug in the values into the conservation of energy formula:
(100 kg * 4186 J/kg°C * 15°C) + (20 kg/min * 4186 J/kg°C * 25°C) = (100 kg + (20 kg/min * t)) * 4186 J/kg°C * 15°C
Simplify the equation:
627900 J + 2093000 J*t = (100 kg + (20 kg/min * t)) * 627900 J
Expand and simplify further:
627900 J + 2093000 J*t = 627900 J + 1255800 J*t
Rearrange the equation:
2093000 J*t - 1255800 J*t = 627900 J - 627900 J
Simplify:
836200 J*t = 0 J
Divide both sides by 836200 J:
t = 0 J / 836200 J
t = 0 minutes
The time it takes for the temperature in the bath to drop to 45°C is 0 minutes. This means that the hot and cold water being added are not sufficient to lower the temperature of the bath.
To solve this problem, we need to determine how much heat is needed to raise the temperature of the bath water from 45 °C to 60 °C, and then calculate the time it takes for the taps to deliver that amount of heat.
Step 1: Calculate the heat required to raise the temperature of the bath water from 45 °C to 60 °C:
- The specific heat capacity (c) of water is 4.18 J/g°C.
- The initial mass (m1) of the bath water is 100 kg.
- The initial temperature (T1) of the bath water is 60 °C.
- The final temperature (T2) of the bath water is 45 °C.
Using the formula: Q = mcΔT, where Q is the heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature, we can calculate the heat required as follows:
Q = m1 * c * (T2 - T1)
= 100 kg * 4.18 J/g°C * (45 °C - 60 °C)
= -62700 J
Step 2: Calculate the flow rate of hot water and cold water:
- The flow rate of hot water (m_hot) is 20 kg per minute.
- The flow rate of cold water (m_cold) is 20 kg per minute.
- The temperature of hot water (T_hot) is 70 °C.
- The temperature of cold water (T_cold) is 10 °C.
Step 3: Calculate the heat transfer rate by the hot and cold water taps:
- The heat transfer rate by the hot water tap (Q_hot) is calculated using the formula Q_hot = m_hot * c * (T_hot - T_bath), where T_bath is the current temperature of the bath water.
= 20 kg/min * 4.18 J/g°C * (70 °C - T_bath)
- The heat transfer rate by the cold water tap (Q_cold) is calculated using the formula Q_cold = m_cold * c * (T_bath - T_cold), where T_bath is the current temperature of the bath water.
= 20 kg/min * 4.18 J/g°C * (T_bath - 10 °C)
Step 4: Calculate the time it takes for the temperature in the bath to drop to 45 °C:
- The total heat transfer rate is the sum of the heat transfer rates by the hot and cold water taps: Q_total = Q_hot + Q_cold
- Since there are no heat losses, the total heat transfer rate should be equal to the heat required (Q_total = Q), so we can set up the following equation:
-62700 J = (20 kg/min * 4.18 J/g°C * (70 °C - T_bath)) + (20 kg/min * 4.18 J/g°C * (T_bath - 10 °C))
Step 5: Solve the equation for T_bath to find the current temperature of the bath water:
- Distribute and simplify the equation: -62700 J = 836 J/min * (70 °C - T_bath) + 836 J/min * (T_bath - 10 °C)
- Combine like terms: -62700 J = 1672 J/min * T_bath - 836 J/min * 10 °C + 836 J/min * 70 °C
- Simplify further: -62700 J = 836 J/min * T_bath - 20960 J/min
Step 6: Solve for T_bath:
- Rearrange the equation: 836 J/min * T_bath = -62700 J + 20960 J/min
- Combine like terms: 836 J/min * T_bath = -41740 J/min
- Divide both sides by 836 J/min: T_bath = -41740 J/min ÷ 836 J/min
- Calculate: T_bath = -50 °C
Since the calculated temperature of the bath water is outside the range of possible temperatures, it suggests that the water temperature will not drop to 45 °C.
Therefore, it will not be possible for the temperature in the bath to drop to 45 °C.
this is a "classic" mixture problem
average inlet temp is 40ºC
... 40 kg/min
(100 kg * 60ºC) + (x kg * 40ºC) = (100 + x) kg * 45ºC
6000 + 40 x = 4500 + 45 x
1500 = 5 x ... x = 300 kg
300 kg / (40 kg/min) = ? min