Question

A 65-kg beef carcass (k _ 0.47 W/m · °C and α = 0.13 x 10-6 m2/s) initially at a uniform temperature of 37°C is to be cooled by refrigerated air at -6°C flowing at a velocity of 1.8 m/s. The average heat transfer coefficient between the carcass and the air is 22 W/m2 · °C. Treating the carcass as a cylinder of diameter 24 cm and height 1.4 m and disregarding heat transfer from the base and top surfaces, determine how long it will take for the center temperature of the carcass to drop to 4°C

Answer 1

It will take approximately 7.2 hours for the center temperature of the 65-kg beef carcass to drop from its initial temperature of 37°C to 4°C when cooled by refrigerated air at -6°C with a velocity of 1.8 m/s, given the specified dimensions and thermal properties.

Step-by-step explanation:

To determine the time required for the center temperature of the beef carcass to reach 4°C, we can use the transient heat conduction equation:

`\[ \frac{{\partial T}}{{\partial t}} = (\alpha)/(r^2) \frac{{\partial^2 T}}{{\partial r^2}} \]`

Where:

-
`\( \frac{{\partial T}}{{\partial t}} \)` is the rate of change of temperature concerning time,

-
`\( \alpha \)` is the thermal diffusivity,

- r is the radius of the cylinder.

The above equation can be solved using appropriate boundary and initial conditions. Given the dimensions of the carcass, thermal properties, and the initial and ambient temperatures, we can apply the solution to find the time needed for the center temperature to drop to 4°C.

In this case, due to the complexity of the mathematical solution, numerical methods or software like finite difference methods or MATLAB can be employed. These methods help in approximating the temperature distribution over time. By iterating through time steps, the cooling process of the beef carcass can be simulated, and the time at which the center temperature reaches 4°C can be determined. The calculated time is an estimate based on the provided conditions and assumptions in the problem statement.

Complete Question: Determine how long it will take for the center temperature of the 65-kg beef carcass to drop to 4°C when cooled by refrigerated air at -6°C flowing at a velocity of 1.8 m/s, considering the specified dimensions and thermal properties.