Question

In a production facility, 1.2 -in-thick, \(2-\mathrm{ft} \times\) 2-ft square brass plates \(\left(\rho=532.5 \mathrm{lbm} / \mathrm{ft}^{3} \text { and } c_{p}=\right.\) \(0.091 \mathrm{Btu} / \mathrm{lbm} \cdot^{\circ} \mathrm{F}\) ) that are initially at a uniform temperature of \(75^{\circ} \mathrm{F}\) are heated by passing them through an oven at \(1300^{\circ} \mathrm{F}\) at a rate of 450 per minute. If the plates remain in the oven until their average temperature rises to \(1000^{\circ} \mathrm{F}\), determine ( \(a\) ) the rate of heat transfer to the plates in the furnace and ( \(b\) ) the rate of entropy generation associated with this heat transfer process.

Short answer

Question: Calculate the rate of heat transfer and entropy generation in the given problem. Answer: The rate of heat transfer to the brass plates is 7,894,600 Btu/min, and the rate of entropy generation is 5.214 Btu/R·min.

Step-by-step solution

Calculate the volume and mass of a single brass plate

We are given the dimensions and density of the brass plates. First, let's calculate the volume of a single plate: Volume \(= L \times W \times H = 2 \ \mathrm{ft} \times 2 \ \mathrm{ft} \times 0.1 \ \mathrm{ft} = 0.4 \ \mathrm{ft}^3\) Now, we can calculate the mass of a single plate using the given density: Mass \(= \rho \times \text {Volume} = 532.5 \ \mathrm{lbm/ft^3} \times 0.4 \ \mathrm{ft^3} = 213 \ \mathrm{lbm}\)

Calculate the total heat needed to heat a single brass plate

We can use the specific heat capacity of the brass to calculate the heat needed to heat the brass plate to the final temperature: \(Q = m \times c_p \times (\text {Final Temperature} - \text {Initial Temperature}) = 213 \ \mathrm{lbm} \times 0.091 \ \mathrm{Btu/lbm \cdot ^\circ F} \times (1000^\circ \mathrm{F} - 75^\circ \mathrm{F}) = 17543.55 \ \mathrm{Btu}\)

Calculate the rate of heat transfer

There are 450 plates being heated per minute, and thus we can find the rate of heat transfer to the plates \(\dot{Q} = \frac{\text {Heat per plate} \times \text {Number of plates per minute}}{\text {Time}} = \frac{17543.55 \ \mathrm{Btu} \times 450}{1 \ \mathrm{min}} = 7894600 \ \mathrm{Btu/min}\)

Calculate the rate of entropy generation

The rate of entropy generation can be found using the heat transfer rate and temperatures of the oven and the brass plates: \(\dot{S}_\text{generation} = \frac{\dot {Q}_\text{brass}}{T_\text{brass}} - \frac{\dot {Q}_\text{oven}}{T_\text{oven}} = \frac{7894600 \ \mathrm{Btu/min}}{(1000 + 459.67) \ \mathrm{R}} - \frac{7894600 \ \mathrm{Btu/min}}{(1300 + 459.67) \ \mathrm{R}} = 5.214 \ \mathrm{Btu/R \cdot min}\)

Results:

So, the rate of heat transfer to the plates is \(\dot {Q} = 7894600 \ \mathrm{Btu/min}\) and the rate of entropy generation is \(\dot {S}_\text{generation} = 5.214 \ \mathrm{Btu/R \cdot min}\).