Question

To warm up some milk for a baby, a mother pours milk into a thin-walled glass whose diameter is 6 cm. the height of the milk in the glass is 7 cm. she then places the glass into a large pan filled with hot water at 60°C. the milk is stirred constantly, so that its temperature is uniform at all times. if the heat transfer coefficient between the water and the glass is 120 W/(m²K), determine how long it will take for the milk to warm up from 3°C to 38°C. take the properties of the milk to be the same as those of water. can the milk in this case be treated as a lumped system? why? the following is given for milk: k = 0.607 W/m/K, density is 997 kg/m³ and specific heat capacity is 4.18 KJ/kg/K.

Answer 1

The calculation involves using Newton's law of cooling and an energy balance to find the heating time required for the milk and determining if lumped system analysis is applicable by analyzing the Biot number.

Step-by-step explanation:

The student's question involves calculating the time required for heating milk from 3°C to 38°C in a glass within a pan of hot water and determining if the lumped system analysis can be used. To solve this, we need to calculate the heat transfer rate using Newton's law of cooling and then the time using the energy balance equation on the milk. Since the milk is constantly stirred, it maintains a uniform temperature. Assuming the properties of milk are the same as water, the lactating mother can use the heat transfer coefficient, thermal conductivity, density, and specific heat capacity of water to calculate the warming time.

Now, a lumped system analysis assumes that the temperature gradient within an object is negligible compared to the gradient between the object and its surroundings. To justify its use, the Biot number (Bi) should be much less than 1. The Biot number is the ratio of internal resistance to heat conduction within the body to the external resistance to heat transfer by convection. It is given by Bi = hLc/k, where h is the heat transfer coefficient, Lc is the characteristic length, and k is the thermal conductivity.

In this case, the characteristic length can be taken as the diameter of the glass, divided by 2 to get the radius as a representative length scale since heat transfer is radially symmetric. If we calculate the Biot number and it is much less than 1, the lumped system approach is valid, which simplifies the analyses considerably.