2. (2 pts) Cooling of a Spherical Object in Lumped Capacitance Model

For Bi=h(V/A)/k ≤ 0.1, the transient temperature inside a solid object that is cooling under conditions

of constant properties and no internal heat generation may be written as:

T(t)-To=(To-Tz)e-t/t,

where 7 is a characteristic time constant for the exponential decay of the temperature difference between

the solid and the fluid.

A. Using the solution from lecture or the text, find an expression for T in terms of the solid density

p, the specific heat Cp, the volume-to-area ratio V/A, and the film heat-transfer coefficient h.

B. For a spherical object made of mild steel (1% Carbon) falling through water, what is the value

of t if the diameter is 0.1 cm? Evaluate all physical properties at 293 K. Hint: To determine the

heat-transfer coefficient h, you must first find the terminal velocity of the sphere in water and

then calculate the heat-transfer coefficient from an appropriate correlation (assume that forced

convection dominates over free convection and radiation). For the terminal velocity, remember

that drag and buoyancy are in balance, then start by assuming a drag coefficient Cp = 0.44 for

spherical objects and perform an iteration using Figure 12.4 of Welty et al. if necessary. Use

the Ranz and Marshall correlation (Eq. 20-38 in Welty et al.) for the heat loss.

C. Repeat part (B) but for a silver object. Which material has the larger value of T, and why?

D. For silver, calculate Nu, again, but this time using the Whitaker correlation (20-37 of Welty et

al), with T = 293 K and T, = 313 K. By what % does this value differ from that from the

Ranz and Marshall correlation?