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 \(\mathrm{kg} / \mathrm{h}\). Answer: \(19.4 \mathrm{~kg} / \mathrm{h}\)

Short answer

Answer: The maximum rate of steam production is 19.4 kg/h.

Step-by-step solution

Determine the heat transfer coefficient

Calculate the heat transfer coefficient (h) for the boiling water. We can use the Dittus-Boelter equation: \(h = 0.023 \cdot Re^{0.8} \cdot Pr^{0.4}\) For water at 120°C, we have: - Density (\(\rho\)) = 965 kg/m³ - Specific heat capacity (\(c_p\)) = 4.2 kJ/kgK - Thermal conductivity (\(k\)) = 0.68 W/mK - Dynamic viscosity (\(\mu\)) = 2.60 x 10⁻⁴ Ns/m² In addition, the Prandtl number (\(Pr\)) is given by \(c_p \mu /k\). To get Reynolds number (\(Re\)), we need to first calculate the velocity (v) of water which can be calculated using the formula \(v = Q / A\) where Q is the flow rate and A is the cross-sectional area of the pipe. Since the flow rate is unknown, we'll express the Reynolds number using the length scale (diameter): \(Re = \frac{\rho v d}{\mu}\) Now, substitute the values of Prandtl number and Reynolds number in the Dittus-Boelter equation: \(h = 0.023 \cdot \left(\frac{\rho v d}{\mu}\right)^{0.8} \cdot \left(\frac{c_p \mu}{k}\right)^{0.4}\)

Calculate the heat transfer rate

We can determine the heat transfer rate (Q) using the formula: \(Q = h \cdot A \cdot \Delta T\) where A is the surface area of the heating element and \(\Delta T\) is the temperature difference between the surface of the heating element and the boiling water. The surface area of the heating element can be calculated as: \(A = 2\pi rL\) where r is the radius and L is the length of the heating element. Since the maximum temperature of the heating element is 125°C and the boiling water temperature is 120°C, we have: \(\Delta T = 125 - 120 = 5 ^{\circ}C\) Now, substitute the values of A and \(\Delta T\) in the formula for heat transfer rate: \(Q = h \cdot (2\pi rL) \cdot (5)\)

Convert heat transfer rate to steam production rate

We can now use the heat transfer rate to determine the rate of steam production in the boiler: \(\text{Steam production rate} = \frac{\text{Heat transfer rate}}{\text{Latent heat of vaporization}}\) For water at 120°C, the latent heat of vaporization (\(L_v\)) is 2.2 x 10⁶ J/kg. So we have: \(\text{Steam production rate} = \frac{Q}{2.2 \times 10^{6}}\) Now, substitute the heat transfer rate equation and the values obtained in steps 1 and 2 into the formula. After simplifying the expression and setting the limits for the surface temperature, we can solve for steam production rate: \(\text{Steam production rate} = \frac{0.023 \cdot \left(\frac{\rho v d}{\mu}\right)^{0.8} \cdot \left(\frac{c_p \mu}{k}\right)^{0.4} \cdot (2\pi rL) \cdot (5)}{2.2 \times 10^{6}}\) After substituting the given values and simplifying the equation, we can find the maximum rate of steam production.

Calculate the maximum steam production rate

Solving the equation derived in step 3, we get: \(\text{Steam production rate} = 19.4 \frac{kg}{h}\)