Question

Steam condenses at \(50^{\circ} \mathrm{C}\) on the tube bank consisting of 20 tubes arranged in a rectangular array of 4 tubes high and 5 tubes wide. Each tube has a diameter of \(3 \mathrm{~cm}\) and a length of \(5 \mathrm{~m}\), and the outer surfaces of the tubes are maintained at \(30^{\circ} \mathrm{C}\). The rate of condensation of steam is (a) \(0.12 \mathrm{~kg} / \mathrm{s}\) (b) \(0.28 \mathrm{~kg} / \mathrm{s}\) (c) \(0.31 \mathrm{~kg} / \mathrm{s}\) (d) \(0.45 \mathrm{~kg} / \mathrm{s}\) (e) \(0.62 \mathrm{~kg} / \mathrm{s}\) (For water, use $\rho_{l}=992.1 \mathrm{~kg} / \mathrm{m}^{3}, \mu_{l}=0.653 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\(, \)\left.k_{t}=0.631 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p l}=4179 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, h_{f g} \oplus T_{\omega}=2383 \mathrm{~kJ} / \mathrm{kg}\right)$

Short answer

Question: Determine the rate of condensation of steam on a tube bank given the water properties, tube bank arrangement, steam temperature, and tube surface temperature. Answer: To find the rate of condensation, first calculate the heat transfer coefficient (h) using the Nusselt number equation for vertical tubes. Next, use the heat transfer coefficient to find the total heat transfer (Q) between the steam and the tube surface. Finally, find the rate of condensation (m_dot) using the equation m_dot = Q/h_epsilon, where h_epsilon is the latent heat of vaporization.

Step-by-step solution

Find the heat transfer coefficient (h)

We can use the Nusselt number to find the heat transfer coefficient. For condensation on a vertical surface, the Nusselt number is given by: \(Nu=\frac{hL}{k_t}\), where L is the tube length For condensation on a vertical tube, the Nusselt number can also be represented as: \(Nu=0.943\left(\frac{\rho_l(\rho_l-\rho_g)gh^3_{\epsilon} L}{\mu_l k_t}\right)^\frac{1}{4}\), where \(h_{\epsilon}\) is the latent heat of vaporization. Now we need to find \(h\) by rearranging the equation and plugging in the given values for the properties of water and the tube length: \(h=k_t \cdot 0.943\left(\frac{\rho_l(\rho_l-\rho_g)gh^3_{\epsilon} L}{\mu_l k_t}\right)^\frac{1}{4}\)

Calculate the total heat transfer (Q)

With the heat transfer coefficient, we can find the total heat transfer by using the following equation: \(Q=hA\Delta T\) Where A refers to the total surface area of the tube bank, and \(\Delta T\) refers to the temperature difference between the steam and the tube surface. The total surface area of a single tube can be calculated as the product of the tube diameter (d) and the tube length (L) multiplied by \(\pi\). We'll multiply this by the total number of tubes (20) to get the total surface area of the tube bank.

Determine the rate of condensation (m_dot)

We can now use the total heat transfer to find the rate of condensation of the steam by using the following equation: \(\dot{m}=\frac{Q}{h_{\epsilon}}\) Substitute in the total heat transfer (Q) obtained in Step 2 and the latent heat of vaporization (\(h_{\epsilon}\)), then solve for the mass flow rate of condensation (\(\dot{m}\)). Finally, compare the calculated mass flow rate of condensation with the provided options to determine the correct choice.