91Ó°ÊÓ

A two-tube-pass, single-shell-pass heat exchanger transfers heat from the collector loop to the storage tank in a solar heating system for a building in Denver. The maximum heat transfer rate to be accommodated is \(24 \mathrm{~kW}\) (for a \(50 \mathrm{~m}^{2}\) collector at noon on a sunny day in fall or spring). Water at \(1 \mathrm{~kg} / \mathrm{s}\) from the storage tank enters the tubes at \(50^{\circ} \mathrm{C}\). The collector loop contains \(50 \%\) ethylene glycol aqueous solution flowing at \(0.66 \mathrm{~kg} / \mathrm{s}\) and enters the shell at \(67.5^{\circ} \mathrm{C}\). The tube bundle consists of 50 tubes of \(24 \mathrm{~mm}\) O.D. and \(2 \mathrm{~mm}\) wall thickness, for which an overall heat transfer coefficient based on outside area of \(835 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) is estimated. Determine the length of tubes required. For the glycol solution use \(c_{p}=3690 \mathrm{~J} / \mathrm{kg}\)

A 2 kg/s stream of water is to be heated from \(30^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\) by 5 kg/s of exhaust gases available at \(200^{\circ} \mathrm{C}\). A finned-tube cross-flow exchanger is available for which a \(1 \mathrm{~m}^{2}\) cross-sectional area is provided for the gas flow. For the resulting gas velocity, an overall heat transfer coefficient of \(160 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) is estimated. If both streams can be considered to be unmixed, calculate the heat transfer area required (i) using the LMTD method. (ii) using the \(\varepsilon-N_{\text {tu }}\) method. The specific heat of the exhaust gases can be taken as \(1100 \mathrm{~J} / \mathrm{kg} \mathrm{K}\).

In a laundry, \(67^{\circ} \mathrm{C}\) dirty wash water is dumped into the drain, and \(70^{\circ} \mathrm{C}\) clean water is required. Presently \(15^{\circ} \mathrm{C}\) water is heated in an electric hot water heater, and the electricity costs 9 cents/ \(\mathrm{kW}\). The water is required at a rate of \(5000 \mathrm{~kg} / \mathrm{h}\), 12 hours per day, 312 days/yr. To conserve energy it is proposed to install a counterflow heat exchanger to preheat the feed to the electric water heater. The installation will cost $$\$ 20,000$$ plus $$\$ 900$$ per square meter of heat exchanger surface. The interest rate to amortize the investment over 12 years is \(10 \%\) per annum. Taxes and insurance are expected to have a fixed cost of \(\$ 500\) per annum plus \(\$ 50 / y r\) per square meter of heat exchanger surface. If the overall heat transfer coefficient is estimated to be \(1000 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\), determine the optimal heat transfer area of the exchanger and the corresponding net annual savings.