Problems

Overall Heat Transfer Coefficient

$11\mathrm{.}1$ An industrial boiler consists of tubes inside of which flow hot combustion gases. Water boils on the exterior of the tubes. When installed, the clean boiler has an overall heat transfer coefficient of $300\frac{W}{m^2}*K.$ Based on experience, it is anticipated that the fouling factors on the inner and outer surfaces will increase linearly with time as $R_{f,i}^{\text{'}\text{'}}=a_{i}t$ and $R_{f,o}^{\text{'}\text{'}}=a_{o}t$ where $a_i=2\mathrm{.}5{10}^{-11}{m}^2*\frac{K}{W}*s$ and $a_o=1\mathrm{.}0{10}^{-11}{m}^2*\frac{K}{W}*s$ for the inner and outer tube surfaces, respectively. If the boiler

is to be cleaned when the overall heat transfer coefficient is reduced from its initial value by $25\%,$ how long after installation should the first cleaning be scheduled? The boiler operates continuously between cleanings.

$11\mathrm{.}2$ A type$-302$ stainless steel tube of inner and outer diameters $D_i=22mm$ and $D_o=27mm,$ respectively, is used in a cross$-$flow heat exchanger. The fouling factors, $R_f^{\text{'}\text{'}},$ for the inner and outer surfaces are estimated to be $0\mathrm{.}0004$ and $0\mathrm{.}0002m^2*\frac{K}{W},$ respectively.

Problems

$689$

$($a$)$ Determine the overall heat transfer coefficient based on the outside area of the tube, $U_o.$ Compare the thermal resistances due to convection, tube wall conduction, and fouling.

$($b$)$ Instead of air flowing over the tube, consider a situation for which the cross$-$flow fluid is water at $15C$ with a velocity of $V_o=1\frac{m}{s}.$ Determine the overall heat transfer coefficient based on the outside area of the tube, $U_o.$ Compare the thermal resistances due to convection, tube wall conduction, and fouling.

$11\mathrm{.}4$ A heat recovery device involves transferring energy from the hot flue gases passing through an annular region to pressurized water flowing through the inner tube of the annulus. The inner tube has inner and outer diameters of $24$ and $30$ mm and is connected by eight struts to an insulated outer tube of $60-mm$ diameter. Each strut is $3$ mm thick and is integrally fabricated with the inner tube from carbon steel $)=(50\frac{W}{m}*K.$