Question

A 7 -cm-external-diameter, 18 -m-long hot-water pipe at \(80^{\circ} \mathrm{C}\) is losing heat to the surrounding air at \(5^{\circ} \mathrm{C}\) by natural convection with a heat transfer coefficient of \(25 \mathrm{W} / \mathrm{m}^{2} \cdot^{\circ} \mathrm{C}\) Determine the rate of heat loss from the pipe by natural convection, in kW.

Short answer

Answer: The rate of heat loss from the pipe by natural convection is approximately 7.356 kW.

Step-by-step solution

Determine the surface area of the pipe

We can find the surface area (A) of the pipe using the following formula: \(A = 2\pi rL\) where r is the radius and L is the length of the pipe. Since we are given the external diameter (7 cm), we can find the radius as follows: \(r = \dfrac{D}{2} = \dfrac{7}{2} = 3.5\,\text{cm} = 0.035\,\text{m}\) Now, we can calculate the surface area: \(A = 2\pi (0.035\,\text{m})\cdot (18\,\text{m}) = 3.927\,\text{m}^2\)

Determine the temperature difference between the pipe and the surrounding air

Given, the temperature of the pipe (\(T_{pipe}\)) is \(80^{\circ} \mathrm{C}\), and the temperature of the surrounding air (\(T_{air}\)) is \(5^{\circ} \mathrm{C}\). The temperature difference is: \(\Delta T = T_{pipe} - T_{air} = 80^{\circ} \mathrm{C} - 5^{\circ} \mathrm{C} = 75^{\circ} \mathrm{C}\)

Determine the rate of heat loss by natural convection

We can now calculate the rate of heat loss by convection using the formula: Rate of heat loss = \(hA(T_{pipe} - T_{air})\) Substituting the given values into the formula, we get: Rate of heat loss = \((25 \mathrm{W} / \mathrm{m}^{2} \cdot^{\circ} \mathrm{C} )(3.927\,\text{m}^2)(75^{\circ} \mathrm{C}) = 7355.625\,\text{W}\) To convert this to kilowatts, divide by 1000: Rate of heat loss = \(\dfrac{7355.625\,\text{W}}{1000} = 7.356\,\text{kW}\) The rate of heat loss from the pipe by natural convection is approximately 7.356 kW.

Key concepts

Heat Transfer Coefficient

The heat transfer coefficient (\( h \)) is a measure of a material's ability to transfer heat through convection. It represents how effectively heat is conveyed from a hot object to a cooler fluid (which could be liquid or gas) that is in contact with the object's surface. In the context of natural convection, this coefficient is influenced by properties of the fluid, such as viscosity and thermal conductivity, as well as the geometry and orientation of the surface.

When calculating the rate of heat loss from objects like the hot-water pipe in our exercise, we use the heat transfer coefficient to quantify the convective heat transfer per unit area per degree of temperature difference between the object's surface and the surrounding fluid. The units of the heat transfer coefficient are typically \( W/m^2\cdot^\circ C \).

This coefficient is crucial because it allows us to use a simplified formula to predict heat loss rates, assuming that the coefficient remains constant over the surface of the object and throughout the temperature range involved.

Surface Area Calculation

The surface area calculation is a fundamental step in determining the potential for heat exchange between a solid object and its environment. The total area of the object that is exposed to the surrounding medium directly influences the amount of heat that can be transferred. In our pipe scenario, the surface area calculation requires knowledge of the object's geometry.

For a cylindrical object like a pipe, the surface area can be found using the formula \( A = 2\pi rL \), where \( r \) is the radius and \( L \) is the length. The radius is half of the diameter, and for our pipe with a 7 cm diameter, the radius would be 3.5 cm or 0.035 m when converted to meters, since \( 1 \text{cm} = 0.01 \text{m} \).

Once the radius and length are known, the surface area is determined by plugging these values into the formula. Manipulating the surface area directly affects the heat transfer since a larger area allows for more heat to be exchanged.

Temperature Difference

The temperature difference (\( \Delta T \)) between two environments is the driving force behind heat transfer by natural convection. It's the difference in temperature between a hot surface and the cooler surrounding fluid. In our pipe example, the temperature difference is the gap between the pipe's temperature and the air temperature. The larger this difference, the greater the potential for heat transfer.

In thermal calculations, we express the temperature difference in degrees Celsius (℃) or Kelvin (K), as both scales have the same size degrees. The formula to find \( \Delta T \) is \( \Delta T = T_{pipe} - T_{air} \), where \( T_{pipe} \) is the temperature of the pipe's surface, and \( T_{air} \) is the air temperature around the pipe. A larger \( \Delta T \) typically indicates a more vigorous convection process and hence, an increased rate of heat loss.

Understanding the importance of temperature difference is key because it affects both the direction and magnitude of heat transfer—heat flows naturally from a warmer to a cooler area, and the efficiency of this flow depends on how much warmer the heat source is compared to its surroundings.