Question

A 65 -cm-long, 2-cm-diameter brass heating element is to be used to boil water at \(120^{\circ} \mathrm{C}\). If the surface temperature of the heating element is not to exceed \(125^{\circ} \mathrm{C}\), determine the highest rate of steam production in the boiler, in \(\mathrm{kg} / \mathrm{h}\).

Short answer

Answer: The highest rate of steam production in the boiler is approximately 3.64 kg/h.

Step-by-step solution

Calculate the surface area of the heating element

The heating element is a cylindrical shape with length L = 65 cm and diameter D = 2 cm. First, we need to find the surface area (A) of the heating element. A cylinder's surface area can be calculated using the following formula: $$A = 2\pi r L$$ where r is the radius of the cylinder. Since the diameter is given, we can find the radius by dividing the diameter by 2: $$r = \frac{D}{2} = \frac{2\,\text{cm}}{2} = 1\,\text{cm}$$ Now, we can calculate the surface area of the heating element: $$A = 2\pi (1\,\text{cm})(65\,\text{cm}) = 130\pi\,\text{cm}^{2}$$

Determine the temperature difference

We have the surface temperature (\(T_{s}\)) of the heating element limited to \(125^{\circ}C\) and the given water boiling temperature (\(T_{w}\)) is \(120^{\circ}C\). The temperature difference (\(\Delta T\)) between the heating element and the water can be calculated as follows: $$\Delta T = T_{s} - T_{w} = 125^{\circ}C - 120^{\circ}C = 5^{\circ}C$$

Calculate the heat transfer rate

We can use the heat transfer coefficient formula to find the heat transfer rate (Q) from the heating element to the water. The formula is: $$Q = hA\Delta T$$ In this exercise, we'll assume the heat transfer coefficient (h) for brass is 111 W/(m^2K). To convert it to SI units, we need to multiply the surface area by 0.0001 to convert from cm^2 to m^2. $$A = 130\pi\,\text{cm}^{2} \times 0.0001 = 0.0130\pi\,\text{m}^{2}$$ Now we can calculate the heat transfer rate (Q): $$Q = (111\,\text{W/m}^{2}\text{K})(0.0130\pi\,\text{m}^{2})(5\,\text{K}) \approx 2282.58\,\text{W}$$

Find the heat capacity of water

We need to find the amount of heat required to convert 1 kg of water at \(120^{\circ}C\) to steam. We'll use the latent heat of vaporization (L) for water, which is approximately 2257 kJ/kg at \(100^{\circ}C\). Due to the given boiling temperature, we'll assume the heat capacity (C) required is approximately the same.

Calculate the maximum rate of steam production

Finally, we can calculate the maximum rate of steam production (m) in kg/h by dividing the heat transfer rate (Q) by the heat capacity (C) and converting the rate to kg/h: $$m = \frac{Q}{C\times 10^3} \times 3600 = \frac{2282.58\,\text{W}}{2257\,\text{kJ/kg}\times 10^3} \times 3600\,\text{h} = 3.64\,\text{kg/h}$$ So, the highest rate of steam production in the boiler is approximately 3.64 kg/h.

Key concepts

Latent Heat of Vaporization

The latent heat of vaporization is a crucial concept when studying the conversion of a substance from a liquid phase to a gas phase without a temperature change. It represents the amount of energy needed to change one kilogram of a substance from liquid to vapor at constant temperature and pressure. For our day-to-day experience, water is the most commonly encountered example.

Essentially, this latent heat is the energy required to break intermolecular bonds that hold water molecules together in a liquid state. When these bonds break, the substance can transition to a gaseous state, forming vapor or steam. In the context of our exercise, the latent heat of vaporization for water at the boiling point, which is typically around 2257 kJ/kg at 100°C, is utilized to ascertain how much energy is needed to produce steam from water at 120°C.

To calculate steam production in a boiler, as seen in the exercise, this latent heat value is crucial. It is the denominator in our final step, factoring into the calculation to give us the rate at which steam is produced. Without understanding the latent heat of vaporization, it would be impossible to reliably calculate the steam production rate, or to gauge the efficiency of heating elements in boilers and other similar devices.

Heat Transfer Coefficient

Understanding the heat transfer coefficient is vital for engineers and physicists when they are dealing with heat exchange between surfaces and fluids. Its role in calculations is to quantify how effectively heat is transferred from a solid surface to a surrounding fluid or vice versa. In SI units, it is expressed as watts per square meter-kelvin (W/m²K).

The heat transfer coefficient depends on a variety of factors, including the nature of the heat transfer process (e.g., conduction, convection, or radiation), properties of the fluid (such as viscosity and thermal conductivity), the velocity of the fluid, and the surface characteristics (like roughness).

In our exercise, the heat transfer coefficient of brass is essential to determine the heat transferring from the heating element to water within the boiler. Higher coefficients indicate more efficient transfer of heat, which facilitates a faster rate of steam production. This coefficient multiplies with the surface area of the element and the temperature difference to calculate the heat transfer rate, as demonstrated in Step 3 of the solution.

Cylindrical Surface Area Calculation

The surface area of a cylinder is one of the foundational topics in geometry and is essential for multiple practical applications, including the calculation of heat transfer in engineering problems like the one presented in the exercise.

To calculate the surface area of a cylinder, one needs to understand that a cylinder has two congruent circular bases and a curved surface that connects them. The formula to calculate the surface area of the cylindrical part (excluding the bases) is given by the product of the circumference of the base (which is the circle's perimeter, or \(2\text{π}r\), where \(r\) is the radius) and the cylinder's height (or length, in the case of a horizontal cylinder).

In our exercise, we focused on the side surface area since that's the part of the brass element in contact with the water. The formula \(A = 2\text{π}rL\) directly relates to the rate at which heat can be transferred from the element to the water, because the larger the surface area, the more opportunity for heat transfer. As such, calculations involving this formula become essential for estimating the potential of a heating element's performance when designing boilers, radiators, and similar systems.