Question

To cool a storehouse in the summer without using a conventional air- conditioning system, the owner decided to hire an engineer to design an alternative system that would make use of the water in the nearby lake. The engineer decided to flow air through a thin smooth 10 -cm-diameter copper tube that is submerged in the nearby lake. The water in the lake is typically maintained at a constant temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If air (1 atm) enters the copper tube at a mean temperature of \(30^{\circ} \mathrm{C}\) with an average velocity of \(2.5 \mathrm{~m} / \mathrm{s}\), determine the necessary copper tube length so that the outlet mean temperature of the air is \(20^{\circ} \mathrm{C}\).

Short answer

Answer: The necessary length of the copper tube to cool the air from 30°C to 20°C is approximately 4.52 meters.

Step-by-step solution

Calculate the heat transfer area

First, we need to calculate the heat transfer area (\(A\)) of the copper tube. To do this, we can use the following formula: \(A = \pi D L\) Where \(D\) is the diameter of the tube (0.10m) and \(L\) is the length of the tube, which we will find in step 4.

Calculate the heat transfer rate

The heat transfer rate (\(q\)) from the air to the lake water can be calculated using the formula: \(q = hA(T_{air,in} - T_{water})\) Where \(h\) is the convection heat transfer coefficient (1000 W/m²K), \(T_{air,in}\) is the inlet air temperature (30°C), and \(T_{water}\) is the lake water temperature (15°C). We will simplify the equation by substituting the area we derived in step 1: \(q = h(\pi DL)(T_{air,in} - T_{water})\)

Determine the amount of heat to be removed

In order to find the amount of heat (\(Q\)) that has to be removed to achieve the desired outlet air temperature (\(T_{outlet}\) = 20°C), we can use the following formula: \(Q = \dot{m} C_p (T_{air,in} - T_{outlet})\) Where \(\dot{m}\) represents the mass flow rate of the air, and \(C_p\) is the specific heat capacity of air at constant pressure, which is approximately 1006 J/kgK. To find the mass flow rate, we can use the following formula: \(\dot{m} = \rho AV\) Where \(\rho\) is the air density (approximately 1.2 kg/m³ at 1 atm and 20°C), \(A\) is the cross-sectional area of the tube, and \(V\) is the average velocity of the air: \(\dot{m} = \rho (\frac{\pi D^2}{4})V\)

Calculate the necessary tube length

To find the necessary tube length (\(L\)) to achieve the desired outlet air temperature, we can equate the heat transfer rate to the amount of heat to be removed: \(q = Q\) \(h(\pi DL)(T_{air,in} - T_{water}) = \dot{m} C_p (T_{air,in} - T_{outlet})\) Substitute the value of \(\dot{m}\) from step 3: \(h(\pi DL)(T_{air,in} - T_{water}) = \rho (\frac{\pi D^2}{4})V C_p (T_{air,in} - T_{outlet})\) Now, we can solve for \(L\) by dividing both sides by \(h(\pi D)(T_{air,in} - T_{water})\) \(L = \frac{\rho (\frac{\pi D^2}{4})V C_p (T_{air,in} - T_{outlet})}{h(\pi D)(T_{air,in} - T_{water})}\) After plugging in the given values, we have: \(L = \frac{(1.2)(\frac{\pi (0.1)^2}{4})(2.5)(1006)(30 - 20)}{(1000)(\pi (0.1))(30 - 15)}\) Finally, we can calculate \(L\): \(L \approx 4.52 \mathrm{m}\) So, the necessary copper tube length to achieve the desired outlet air temperature is approximately 4.52 meters.

Key concepts

Convection Heat Transfer

Convection heat transfer is a mechanism of heat transportation between a solid surface and a fluid, in which the heat is carried away by the movement of the fluid. It's a process that depends heavily on the flow characteristics and properties of the fluid involved. In our case, air is flowing through a copper tube submerged in lake water, creating a setting for convection heat transfer.
Here’s how convection works: when a fluid like air passes over a surface, it picks up heat from the surface if the surface is hotter. If the fluid is warmer, it transfers heat to the surface. The actual rate of heat transfer depends on the convection heat transfer coefficient ( $h$ ), which is a measure of how effectively heat is transferred between the two substances in contact.
The solution to convection heat transfer problems often revolves around understanding the interaction between the fluid and the surface, which involves calculating the heat transfer rate using the formula: $q = hA(T_{air,in} - T_{water})$ . Here, $A$ is the area through which heat transfer occurs, and the difference $T_{air,in} - T_{water}$ represents the temperature difference driving the heat transfer.

Copper Tube Heat Exchanger

A copper tube heat exchanger is frequently chosen for cooling systems because copper is an excellent thermal conductor, meaning heat moves through it easily. In the exercise problem, the copper tube acts as the barrier through which heat is transferred from the warm air to the cooler lake water outside.
The copper tube supplies a vast surface where the heat transfer occurs, provided by its outer length multiplied by its circumference. That's where the equation for the heat transfer area, \(A = \pi DL\), fits in. \(L\) is the length we want to find to ensure the required temperature drop of the air moving inside.
The key advantage of using copper is reduced energy loss during heat transfer, making it efficient for systems that rely on ambient natural cooling, such as using lake water. This makes the tube not only a conduit for air flow but a critical interface for heat exchange.
Understanding how efficiently this copper tube performs is crucial for designing an effective air cooling system, where the goal is to achieve a specific outlet air temperature with minimal energy input. Thus, the length and diameter of the tube are critical in determining overall system performance.

Air Cooling System Design

Designing an air cooling system leveraging natural resources like lake water can be an efficient solution. In this context, the system relies on principles of heat transfer, with a primary focus on ensuring adequate contact time and area between the air and the cooler environment.
The engineer's task is to calculate the needed dimensions, notably the copper tube's length, to achieve desired cooling results. The initial air temperature, velocity, and desired exit temperature are key parameters driving this design. By calculating the mass flow rate of air using \(\dot{m} = \rho AV\), engineers can understand how much air is passing through the system, accounting for density \(\rho\), and velocity \(V\) of air.
The overall design has to balance several factors: effective air temperature reduction while utilizing the minimal necessary tube length to optimize costs and materials. The length calculation \(L = \frac{\rho (\frac{\pi D^2}{4})V C_p (T_{air,in} - T_{outlet})}{h(\pi D)(T_{air,in} - T_{water})}\) reflects such considerations, highlighting the interaction of volumetric flow, heat capacity, temperature difference, and the heat conductivity of copper, leading to efficient cooling solutions by nature-focused engineering principles.