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A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3 and 3.8 kJ kg–1 K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________
Correct answer is between '4,4.2'. Can you explain this answer?
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A high pressure dairy homogenizer operates under upstream and downstre...
$^{-3}$ and 3.93 kJ kg$^{-1}$ K$^{-1}$ respectively. The homogenizer is adiabatic and the milk experiences an isentropic compression during the homogenization process.
(a) What is the temperature rise of the milk during homogenization?
(b) What is the power required by the homogenizer to operate at this condition?
(c) If the homogenizer is 80% efficient, what is the power input required to operate the homogenizer?
(d) What is the overall heat transfer coefficient of the homogenizer if the milk outlet temperature is 50°C and the ambient temperature is 20°C?
**Solution:**
(a) The process is adiabatic and isentropic, therefore, we can use the isentropic relation for an ideal gas to determine the temperature rise:
$$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma - 1}{\gamma}}$$
where $T_1$ and $T_2$ are the initial and final temperatures of the milk, $P_1$ and $P_2$ are the initial and final pressures, and $\gamma$ is the ratio of specific heats. For whole milk, we can assume $\gamma = 1.4$.
Substituting the given values, we get:
$$\frac{T_2}{T_1} = \left(\frac{40}{200}\right)^{\frac{0.4}{1.4}} = 0.702$$
Therefore, the temperature rise of the milk is:
$$\Delta T = T_2 - T_1 = T_1 (0.702 - 1) = -0.298 T_1$$
We can rearrange the equation to find $T_1$:
$$T_1 = \frac{\Delta T}{-0.298} = \frac{(50 - 20)}{-0.298} = 101.34 \text{°C}$$
Therefore, the initial temperature of the milk is 101.34°C.
(b) The power required by the homogenizer can be calculated using the following equation:
$$P = \rho Q C_p \Delta T$$
where $\rho$ is the density of the milk, $Q$ is the volumetric flow rate, $C_p$ is the specific heat capacity of the milk, and $\Delta T$ is the temperature rise of the milk.
Substituting the given values, we get:
$$P = 1030 \times 0.03 \times 3.93 \times (50 - 101.34) = -59.5 \text{ kW}$$
Note that the negative sign indicates that the homogenizer is doing work on the milk.
(c) If the homogenizer is 80% efficient, the power input required to operate the homogenizer can be calculated as:
$$P_{in} = \frac{P}{\eta} = \frac{-59.5}{0.8} = -74.4 \text{ kW}$$
Therefore, the power input required to operate the homogenizer is 74.4 kW.
(d) The overall heat transfer coefficient of the homogenizer can be
(a) What is the temperature rise of the milk during homogenization?
(b) What is the power required by the homogenizer to operate at this condition?
(c) If the homogenizer is 80% efficient, what is the power input required to operate the homogenizer?
(d) What is the overall heat transfer coefficient of the homogenizer if the milk outlet temperature is 50°C and the ambient temperature is 20°C?
**Solution:**
(a) The process is adiabatic and isentropic, therefore, we can use the isentropic relation for an ideal gas to determine the temperature rise:
$$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma - 1}{\gamma}}$$
where $T_1$ and $T_2$ are the initial and final temperatures of the milk, $P_1$ and $P_2$ are the initial and final pressures, and $\gamma$ is the ratio of specific heats. For whole milk, we can assume $\gamma = 1.4$.
Substituting the given values, we get:
$$\frac{T_2}{T_1} = \left(\frac{40}{200}\right)^{\frac{0.4}{1.4}} = 0.702$$
Therefore, the temperature rise of the milk is:
$$\Delta T = T_2 - T_1 = T_1 (0.702 - 1) = -0.298 T_1$$
We can rearrange the equation to find $T_1$:
$$T_1 = \frac{\Delta T}{-0.298} = \frac{(50 - 20)}{-0.298} = 101.34 \text{°C}$$
Therefore, the initial temperature of the milk is 101.34°C.
(b) The power required by the homogenizer can be calculated using the following equation:
$$P = \rho Q C_p \Delta T$$
where $\rho$ is the density of the milk, $Q$ is the volumetric flow rate, $C_p$ is the specific heat capacity of the milk, and $\Delta T$ is the temperature rise of the milk.
Substituting the given values, we get:
$$P = 1030 \times 0.03 \times 3.93 \times (50 - 101.34) = -59.5 \text{ kW}$$
Note that the negative sign indicates that the homogenizer is doing work on the milk.
(c) If the homogenizer is 80% efficient, the power input required to operate the homogenizer can be calculated as:
$$P_{in} = \frac{P}{\eta} = \frac{-59.5}{0.8} = -74.4 \text{ kW}$$
Therefore, the power input required to operate the homogenizer is 74.4 kW.
(d) The overall heat transfer coefficient of the homogenizer can be
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A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer?
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A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer?.
A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A high pressure dairy homogenizer operates under upstream and downstream pressures of 200 and 40 bar respectively homogenizing 30 L of whole milk per hour. Density and specific heat capacity of whole milk are 1030 kg m–3and 3.8 kJ kg–1K–1, respectively. Assuming complete energy conservation, the temperature rise of whole milk in degree Celsius is ____________Correct answer is between '4,4.2'. Can you explain this answer?.
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