Question

Consider a sphere and a cylinder of equal volume made of copper. Both the sphere and the cylinder are initially at the same temperature and are exposed to convection in the same environment. Which do you think will cool faster, the cylinder or the sphere? Why?

Short answer

Answer: The cylinder cools faster because it has a larger surface area exposed to the convection medium, which results in a higher heat transfer rate through convection as compared to the sphere with a smaller surface area.

Step-by-step solution

Understanding heat transfer through convection

In convection, heat is transferred between a solid object and a fluid medium (such as air). The heat transfer rate depends on the surface area of the object and the temperature difference between the object and the surrounding environment. So, to determine which of the two shapes cools faster, we need to compare their surface areas.

Derive expressions for the surface areas of a sphere and a cylinder

Let's denote the radius of the sphere as r_s and the radius and height of the cylinder as r_c and h_c, respectively. We can express the surface areas of a sphere (A_s) and a cylinder (A_c, excluding the top and bottom surfaces) as follows: A_s = 4 * pi * r_s^2 A_c = 2 * pi * r_c * h_c

Derive expressions for the volumes of a sphere and a cylinder

Let's denote the volumes of the sphere (V_s) and the cylinder (V_c) as follows: V_s = (4/3) * pi * r_s^3 V_c = pi * r_c^2 * h_c Since the volumes of the sphere and the cylinder are equal, we have: V_s = V_c => (4/3) * pi * r_s^3 = pi * r_c^2 * h_c

Relate the height of the cylinder to its radius

Rearrange the equation in Step 3 to solve for the height of the cylinder in terms of its radius: h_c = ((4 * r_s^3) / (3 * r_c^2))

Calculate the surface area of the cylinder in terms of the radius of the sphere

Replace the height of the cylinder in the expression for the cylinder's surface area (excluding the top and bottom surfaces) from Step 2 with the expression derived in Step 4: A_c = 2 * pi * r_c * (((4 * r_s^3) / (3 * r_c^2)))

Comparing surface areas

Comparing the surface areas of the sphere and the cylinder from Steps 2 and 5, we see that they are not directly dependent on each other. However, since they are made of the same material and have the same initial temperature, the rate of heat loss is determined by the ratio of their surface areas. A_s/A_c = (4 * pi * r_s^2) / (2 * pi * r_c * (((4 * r_s^3) / (3 * r_c^2)))) Simplifying the above expression, we find that: A_s/A_c = (2 * r_s) / ((4 * r_s) / (3 * r_c))

Conclusion

Since the volumes of the sphere and the cylinder are equal, and the surface area of the sphere is always smaller than that of the cylinder, the sphere will cool more slowly because it has less surface area exposed to the convection medium. Therefore, the cylinder will cool faster than the sphere.