Question

Consider cooling of butter in cartons that are 280 mm long, 370 mm wide, and 185 mm high. The butter is initially at 30°C, and air at 1°C is blown over the cartons to cool them. The flow of air over the cartons is turbulent with velocities of 2.5 m/s over the top and bottom of the cartons, 1 m/s along the sides of the cartons, and 0.4 m/s at the front and back of the cartons with film temperature of air at 10°C. Determine the length of time taken to chill the butter to 4°C?

Answer 1

The length of time taken to chill the butter to 4°C is 0.00317 seconds.

Step-by-step explanation:

To find the length of time taken to chill the butter to 4°C, we need to calculate the rate of heat transfer by convection between the air and the butter. The formula for heat transfer by convection is Q = hAΔT, where Q is the heat transfer rate, h is the convective heat transfer coefficient, A is the surface area, and ΔT is the temperature difference.

We can use the average velocity of the air over the cartons to determine the convective heat transfer coefficient. The average velocity is the sum of the velocities at each side, divided by the number of sides. In this case, the average velocity is (2.5 + 2.5 + 1 + 1 + 0.4 + 0.4) / 6 = 1.18 m/s.

Now, we can calculate the convective heat transfer coefficient using the formula h = 0.664 * v^(1/2), where v is the velocity. Plugging in the average velocity, we get h = 0.664 * (1.18)^(1/2) = 0.875 W/(m²·K).

Next, we calculate the surface area of the cartons. The surface area is equal to the sum of the areas of all six sides of the cartons. Plugging in the dimensions, we get A = 2*(280*370 + 280*185 + 370*185) = 0.6792 m².

Now we can calculate the rate of heat transfer by convection using the equation Q = hAΔT. The temperature difference ΔT is 30°C - 4°C = 26°C. Plugging in the values, we get Q = 0.875 * 0.6792 * 26 = 14.9892 W.

Finally, we can calculate the time taken to chill the butter by dividing the heat transfer rate by the total heat required to chill the butter. The total heat required can be calculated using the specific heat capacity of butter, which is 2.03 J/(g·°C), and the mass of the butter. Let's assume the mass of the butter is 100 g. The total heat required is then 100 * 2.03 * (30 - 4) = 4721.4 J.

Dividing the heat transfer rate by the total heat required, we get the time taken to chill the butter to 4°C: 14.9892 / 4721.4 = 0.00317 s.