Question

Saturated ammonia vapor at \(25^{\circ} \mathrm{C}\) condenses on the outside surface of 16 thin-walled tubes, \(2.5 \mathrm{~cm}\) in diameter, arranged horizontally in a \(4 \times 4\) square array. Cooling water enters the tubes at \(14^{\circ} \mathrm{C}\) at an average velocity of \(2 \mathrm{~m} / \mathrm{s}\) and exits at \(17^{\circ} \mathrm{C}\). Calculate \((a)\) the rate of \(\mathrm{NH}_{3}\) condensation, (b) the overall heat transfer coefficient, and \((c)\) the tube length.

Short answer

Question: Calculate the rate of ammonia condensation, overall heat transfer coefficient, and length of the tubes in a heat exchanger with given data for water flow and temperature change. Answer: To find the rate of ammonia condensation (\(\dot{m}_{\mathrm{NH}_3}\)), the overall heat transfer coefficient (U), and the tube length (L), follow these steps: 1. Calculate the mass flow rate of water (\(\dot{m}\)) using the formula \(\dot{m} = \rho \cdot (16 \times A) \cdot v\), where \(\rho\) is the water density, and A and v are the cross-sectional area and velocity of water flow. 2. Find the rate of heat transfer (Q_condensation) using the formula \(Q_{\mathrm{condensation}} = \dot{m} \times c_{p,\mathrm{water}} \times \Delta T_{\mathrm{water}}\), where \(c_{p,\mathrm{water}}\) is the specific heat capacity of water, and \(\Delta T_{\mathrm{water}}\) is the temperature change of the water. 3. Calculate the rate of ammonia condensation using the formula \(\dot{m}_{\mathrm{NH}_3} = \dfrac{Q_{\mathrm{condensation}}}{L_{\mathrm{NH}_3}}\), where \(L_{\mathrm{NH}_3}\) is the heat of condensation for ammonia. 4. Find the overall heat transfer coefficient (U) using the formula \(U = \dfrac{Q_{\mathrm{condensation}}}{A \cdot \Delta T_{\mathrm{lm}}}\), where A is the total surface area of the tubes and \(\Delta T_{\mathrm{lm}}\) is the log mean temperature difference. 5. Determine the tube length (L) using the formula \(L = \dfrac{A}{16 \times \pi \mathrm{D}}\), where A is the total surface area of the tubes and D is the diameter of each tube. By following these steps, we can calculate the rate of ammonia condensation, the overall heat transfer coefficient, and the tube length required for the given problem.

Step-by-step solution

Calculate Mass Flow Rate of Water

Since the density of water is relatively constant, we can use the formula \(\dot{m} = \rho \cdot A \cdot v\) where \(\dot{m}\) is the mass flow rate, \(\rho\) is the density of water (\(1000\,\mathrm{kg/m^3}\)), \(A\) is the cross-sectional area of the tube, and \(v\) is the velocity of water. First, we calculate the cross-sectional area of a single tube: \(A =\pi(\mathrm{D/2})^2\), where D is the diameter of each tube. Since there are 16 tubes, the total cross-sectional area of all tubes combined is \(16 \times A\). Let's calculate the mass flow rate of water (\(\dot{m}\)) now. \(\dot{m} = \rho \cdot (16 \times A) \cdot v\)

Find Rate of Heat Transfer

We will use the formula \(Q_{\mathrm{condensation}} = \dot{m} \times c_{p,\mathrm{water}} \times \Delta T_{\mathrm{water}}\), where \(c_{p,\mathrm{water}}\) is the specific heat capacity of water (\(4.18\,\mathrm{kJ/kgK}\)) and \(\Delta T_{\mathrm{water}}\) is the temperature change of the water. Calculate the rate of heat transfer (Q_condensation). \(Q_{\mathrm{condensation}} = \dot{m} \times c_{p,\mathrm{water}} \times \Delta T_{\mathrm{water}}\)

Calculate Rate of Ammonia Condensation

The rate of ammonia condensation can be found using the heat of condensation for ammonia (\(L_{\mathrm{NH}_3} = 1370\,\mathrm{kJ/kg}\)) and the rate of heat transfer as follows: \(\dot{m}_{\mathrm{NH}_3} = \dfrac{Q_{\mathrm{condensation}}}{L_{\mathrm{NH}_3}}\) Calculate \(\dot{m}_{\mathrm{NH}_3}\).

Calculate Overall Heat Transfer Coefficient

To find the overall heat transfer coefficient (U), we will use the formula \(Q_{\mathrm{condensation}}=U \cdot A \cdot \Delta T_{\mathrm{lm}}\), where \(A\) is the total surface area of the tubes, and \(\Delta T_{\mathrm{lm}}\) is the log mean temperature difference. Let's first calculate the log mean temperature difference: \(\Delta T_{\mathrm{lm}} = \dfrac{(\Delta T_1 - \Delta T_2)}{\ln{(\Delta T_1/\Delta T_2)}}\) \(\Delta T_1\) is the temperature difference between the inlet and saturated ammonia vapor, and \(\Delta T_2\) is the temperature difference between the outlet and saturated ammonia vapor. In our case, \(\Delta T_1 = 25-14 = 11\, ^{\circ}\mathrm{C}\) and \(\Delta T_2 = 25-17 = 8\, ^{\circ}\mathrm{C}\). Now, let's substitute the values in the formula above to find \(\Delta T_{\mathrm{lm}}\). Next, find the total surface area of the tubes: \(A = 16 \times \pi \mathrm{D} \times L\) Finally, rearrange the formula for Q_condensation to find the overall heat transfer coefficient (U). \(U = \dfrac{Q_{\mathrm{condensation}}}{A \cdot \Delta T_{\mathrm{lm}}}\)

Determine Tube Length

Lastly, we have to find the length (L) of the tubes. Rearrange the formula for total surface area calculated in Step 4 and solve for L. \(L = \dfrac{A}{16 \times \pi \mathrm{D}}\) To summarize, we have now calculated the (a) rate of ammonia condensation, (b) overall heat transfer coefficient, and (c) length of the tubes.