Steam condenses at \(50^{\circ} \mathrm{C}\) on the outer surface of a horizontal tube with an outer diameter of \(6 \mathrm{~cm}\). The outer surface of the tube is maintained at \(30^{\circ} \mathrm{C}\). The condensation heat transfer coefficient is (a) \(5493 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (b) \(5921 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (c) \(6796 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (d) \(7040 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (e) \(7350 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (For water, use \(\rho_{l}=992.1 \mathrm{~kg} / \mathrm{m}^{3}, \mu_{l}=0.653 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), \(\left.k_{l}=0.631 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p l}=4179 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, h_{f g} \oplus T_{\text {satl }}=2383 \mathrm{~kJ} / \mathrm{kg}\right)\) 10-130 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 \(6 \mathrm{~cm}\) and a length of \(3 \mathrm{~m}\), and the outer surfaces of the tubes are maintained at \(30^{\circ} \mathrm{C}\). The rate of condensation of steam is (a) \(0.054 \mathrm{~kg} / \mathrm{s}\) (b) \(0.076 \mathrm{~kg} / \mathrm{s}\) (c) \(0.315 \mathrm{~kg} / \mathrm{s}\) (d) \(0.284 \mathrm{~kg} / \mathrm{s}\) (e) \(0.446 \mathrm{~kg} / \mathrm{s}\) (For water, use \(\rho_{l}=992.1 \mathrm{~kg} / \mathrm{m}^{3}, \mu_{l}=0.653 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), \(\left.k_{l}=0.631 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p l}=4179 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, h_{f g \otimes T_{\text {sat }}}=2383 \mathrm{~kJ} / \mathrm{kg}\right)\)

Step by step solution

01

Temperature Difference

The temperature difference, \(\Delta T\), between the steam and outer surface of the tube is given by: \(\Delta T = T_{steam} - T_{outer}\) where \(T_{steam}\) is the steam temperature and \(T_{outer}\) is the temperature of the outer surface. \(\Delta T = 50^{\circ}\mathrm{C} - 30^{\circ}\mathrm{C} = 20^{\circ}\mathrm{C}\) #Step 2: Calculate the Nusselt number for condensation on the outer surface of the tube#

02

Nusselt Number

The Nusselt number for condensation on the outer surface of a horizontal tube is given by the following equation: \(Nu = 0.54 \left(\frac{k_{l}}{6\cdot10^{-2}\mathrm{m}}\left[\frac{g \rho_{l}^{2}(h_{fg} + 0.68c_{p l}\Delta T)\Delta T}{\mu_{l}k_{l}}\right]^{1/4}\right) \cdot \mathrm{m}\) Using the given properties of water and the temperature difference, we can calculate the Nusselt number. #Step 3: Calculate the heat transfer coefficient using the Nusselt number#

03

Heat Transfer Coefficient

The heat transfer coefficient \(h\) is given by the formula: \(h = \frac{Nu \times k_{l}}{6\cdot10^{-2}\mathrm{m}}\) By plugging in the Nusselt number from Step 2, we obtain the heat transfer coefficient, which will be one of the given options (a)-(e). #Step 4: Calculate the area of the tube's outer surface#

04

Surface Area

The surface area of a tube can be calculated as: \(A = n \times \pi D L\) where \(n\) is the number of tubes, \(D\) is the diameter of each tube, and \(L\) is the length of each tube. In this case, \(A = 20 \times \pi (6\cdot10^{-2}\mathrm{m})(3\mathrm{m})\) #Step 5: Calculate the rate of heat transfer#

05

Rate of Heat Transfer

The rate of heat transfer \(q\) for condensation on the outer surface of the tube is given by the formula: \(q = h \times A \times \Delta T\) Using the area calculated in Step 4 and heat transfer coefficient from Step 3, we can calculate the rate of heat transfer. #Step 6: Calculate the rate of condensation of steam#

06

Rate of Condensation

The rate of condensation, \(m'\), can be found using the formula: \(m' = \frac{q}{h_{f g}}\) where \(h_{f g}\) is the enthalpy of vaporization at the saturation temperature. By plugging in the values for \(q\) and \(h_{fg}\), we can obtain the rate of condensation, which is one of the given options (a)-(e).

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