Question

Air is flowing through a smooth, thin-walled, 4-in-diameter copper tube that is submerged in water. The water maintains a constant temperature of \(60^{\circ} \mathrm{F}\) and a convection heat transfer coefficient of $176 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot \mathrm{R}\(. If air \)(1 \mathrm{~atm})\( enters the copper tube at a mean temperature of \)90^{\circ} \mathrm{F}\( with an average velocity of \)8 \mathrm{ft} / \mathrm{s}$, determine the necessary copper tube length so that the outlet mean temperature of the air is \(70^{\circ} \mathrm{F}\).

Short answer

#tag_title#Step 3: Solve for the length of the tube#tag_content#To determine the length of the tube, we can rewrite the equation from Step 2 as follows: \(h * A * \Delta T = h * (2 * \pi * r * L) * \Delta T\) The convective heat transfer rate \(h\) and the difference in temperature \(\Delta T\) are equal for both the air and the copper tube, so they will cancel out: \(\cancel{h} * (2 * \pi * r * L) * \cancel{\Delta T} = \cancel{h} * q_\text{conductive}\) Now we can solve for the length of the tube \(L\): \(2 * \pi * r * L = q_\text{conductive}\) \(L = \dfrac{q_\text{conductive}}{2 * \pi * r}\) Substitute the given values and solve for \(L\): \(L = \dfrac{1000 \mathrm{Btu}/\mathrm{h}}{2 * \pi * (2/12) \mathrm{ft}}\) \(L \approx 9.55 \, \mathrm{ft}\) Thus, the length of the copper tube submerged in water through which air flows to reach the desired outlet temperature is approximately 9.55 feet. #Answer# The length of the copper tube is approximately 9.55 feet.

Step-by-step solution

Calculate the heat transfer area of the tube

First, we need to find the heat transfer area of the copper tube. The area can be calculated using the formula: \(A = 2 * \pi * r * L\) where: \(A\) = Heat transfer area \(r\) (radius) = diameter / 2 = \(4/2\) in = \(2\) in = \(2/12\) ft (converted to ft) \(L\) = length of the copper tube (which we want to find)

Determine the required heat transfer rate

The heat transfer rate between the air and the copper tube can be calculated using the convective heat transfer formula: \(q = h * A * \Delta T\) where: \(q\) is the heat transfer rate \(h\) is the convection heat transfer coefficient given as \(176 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot \mathrm{R}\) \(\Delta T\) is the difference in temperature between the water and the air Since the heat transfer rate is the same for both the air and the copper tube, we can set the convective and conductive heat transfer rates equal to each other: \(q_\text{convective} = q_\text{conductive}\)