Question

Water is boiled at atmospheric pressure by a horizontal polished copper heating element of diameter \(D=0.5\) in and emissivity \(\varepsilon=0.05\) immersed in water. If the surface temperature of the heating element is \(788^{\circ} \mathrm{F}\), determine the rate of heat transfer to the water per unit length of the heating element.

Short answer

Question: Determine the rate of heat transfer to the water per unit length of the heating element with the following given values: diameter = 0.5 in, emissivity = 0.05, and surface temperature = 788°F. Answer: The rate of heat transfer to the water per unit length of the heating element is approximately 0.3787 W/m.

Step-by-step solution

Convert the given temperature from Fahrenheit to Kelvin

First, we need to convert the given surface temperature of the heating element from Fahrenheit to Kelvin using the following formula: \(T_K = \frac{5}{9} (T_F-32) + 273.15\) where \(T_F\) is the temperature in Fahrenheit and \(T_K\) is the temperature in Kelvin. \(T_K = \frac{5}{9} (788 - 32) + 273.15\) \(T_K \approx 694.15 \, K\) The surface temperature of the heating element in Kelvin is approximately 694.15 K.

Apply Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the heat transfer per unit area from a black body is proportional to the fourth power of the temperature, and it can be described by the equation: \(q = \varepsilon \sigma (T^4 - T_{\infty}^4)\) where \(q\) is the heat transfer per unit area, \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant (\(\sigma = 5.67 \times 10^{-8} W/(m^2K^4)\)), and \(T\) and \(T_{\infty}\) are the temperatures of the heating element and the surrounding fluid (water) in Kelvin. However, since water is in contact with the heating element, we assume that the water temperature is at the boiling point. At atmospheric pressure, water boils at 100°C, or 373.15 K. So \(T_{\infty} = 373.15 \, K\). Now we can plug in the given values: \(q = 0.05 \times 5.67 \times 10^{-8} \times (694.15^4 - 373.15^4)\) \(q \approx 9.55 \, W/m^2\)

Determine the heat transfer rate per unit length

Next, we need to determine the heat transfer rate per unit length of the heating element. We can do this by multiplying the heat transfer rate per unit area (\(q\)) by the circumference of the heating element (\(2 \pi r\)): \(Q' = q (2 \pi r)\) where \(Q'\) is the heat transfer rate per unit length, \(q\) is the heat transfer rate per unit area, and \(r\) is the radius of the heating element. First, we will convert the diameter from inches to meters: \(D = 0.5\) in \(= 0.5 \times 0.0254\) m \(\approx 0.0127\) m Then, we can calculate the radius \(r = D/2 \approx 0.00635\) m. Now, we can plug in the values: \(Q' = 9.55 \times (2 \pi \times 0.00635)\) \(Q' \approx 0.3787 \, W/m\) The rate of heat transfer to the water per unit length of the heating element is approximately 0.3787 W/m.

Key concepts

Stefan-Boltzmann Law

To understand how heat transfer works in this context, we first dive into the Stefan-Boltzmann law. This law relates to black body radiation, stating that the thermal radiation emitted by a black body is proportional to the fourth power of its absolute temperature. Although the copper heating element isn't a perfect black body, this concept gives us a core understanding of radiation heat transfer.
The formula expressed by the Stefan-Boltzmann law is:

• \(q = \varepsilon \sigma (T^4 - T_{\infty}^4)\)

Here, \(\varepsilon\) represents emissivity, \(\sigma\) the Stefan-Boltzmann constant \((5.67 \times 10^{-8} W/(m^2K^4))\), and \(T\) the surface temperature in Kelvin. This relationship highlights that as the temperature differences between the object and its surroundings increase, the heat transfer rate rises notably due to the fourth power dependence. This powerful concept helps calculate the exact heat transferred from the heating element to the surrounding water.

Emissivity

Emissivity is pivotal in understanding the efficiency of a material to emit thermal radiation. It is a dimensionless quantity ranging from 0 to 1. A perfect black body has an emissivity of 1, emitting the maximum radiation possible at its temperature.
Copper, however, being a real-world material, does not have an emissivity of 1. Plants, metals or polished surfaces tend to have lower emissivity values. Our exercise uses an emissivity value of 0.05, representing that polished copper does not emit much heat radiation compared to a black body.
Emissivity becomes crucial when applying the Stefan-Boltzmann law because it adjusts the idealized scenario (a black body) to practical situations where materials emit different levels of radiation. This adjustment makes the heat transfer calculations more accurate and reflective of reality.

Boiling Point

The boiling point is a fundamental concept establishing the temperature at which a substance changes from a liquid to a gas at a given pressure. For water at atmospheric pressure, this boiling point is 100°C or 373.15 K. This temperature is essential in the exercise as it defines the temperature of the surrounding fluid \(T_{\infty}\).
When heat is applied to water, like in the context of our heating element, the heat causes the water molecules to gain kinetic energy. Once water reaches its boiling point, this kinetic energy is enough to break the intermolecular bonds, causing water to evaporate.
In the exercise, when calculating the heat transfer using the Stefan-Boltzmann law, the boiling point is used to establish \(T_{\infty}\), the temperature parameter for the surrounding substance, which is imperative for accurate computations.

Unit Conversion

Unit conversion is crucial in physics problems, allowing us to maintain consistency across different measurement systems. Measurements like temperature, length, and energy often require conversions to use consistent units throughout calculations. In this specific exercise, several conversions are necessary:

• Temperature conversion from Fahrenheit to Kelvin using the formula: \(T_K = \frac{5}{9} (T_F-32) + 273.15\).
• Length conversion from inches to meters: using the conversion factor \(1 \text{ inch} = 0.0254 \text{ meters}\).

These conversions harmonize the units, making it easier to employ physical constants like the Stefan-Boltzmann constant in calculations where SI units are typically used. Consistent units are especially crucial when calculating physical quantities, avoiding errors that can occur from mismatched unit calculations. By converting units before computation, we ensure accuracy and compatibility within the mathematical equations.