Physics, 27.02.2020 03:51 rainbow7020
Circular copper rods of diameter D= 1.75 mm and length L= 25 mm are used to enhance heat transfer from a surface that is maintained at Ts,1= 100°C. One end of the rod is attached to this surface (at x= 0), while the other end (x= 25 mm) is joined to a second surface, which is maintained at Ts,2= 0°C. Air flowing between the surfaces (and over the rods) is also at a temperature of [infinity] T[infinity]= 0°C, and a convection coefficient of h= 125 W/m2·K is maintained. (1) What is the rate of heat transfer by convection from a single copper rod to the air, in W? (2) What is the total rate of heat transfer from a 1 m × 1 m section of the surface at 100°C, if a bundle of the rods is installed on 4-mm centers, in W?
Answers: 3
![\theta_{(x)} = C_1e^{mx}+C_2e^{-mx}\\\theta_{(x)} = \frac{100}{(1-e^{0.05m})} e^{mx}-\frac{100e^{0.05m}}{(1-e^{0.05m})} e^{-mx}\\=\frac{100}{(1-e^{0.05m})} [e^{mx}-e^{0.05m-mx}]](/tpl/images/0526/3298/81008.png)
![\frac{d}{dx}(\theta_{(x)}) = \frac{d}{dx}(\frac{100}{(1-e^{0.05m})}[e^{mx}-e^{0.05m-mx}] )\\\frac{d\theta_{(x)}}{dx} = \frac{100}{(1-e^{0.05m})}(me^{mx}-(-m)e^{0.05m-mx})\\\frac{d\theta_{(x)}}{dx} = \frac{100}{(1-e^{0.05m})}m(e^{mx}+e^{0.05m-mx})](/tpl/images/0526/3298/bc7cf.png)
![q_{cond}=-kA_c\frac{d \theta}{dx} =-k* \frac{\pi D^2}{4}*\frac{100}{(1-e^{0.05m})}m(e^{mx}+e^{0.05m-mx})\\=\frac{-100k(\frac{\pi D^2}{4} )m}{(1-e^{0.05m})}[e^{mx}+e^{0.05m-mx}]\\=\frac{-(25 \pi)kmD^2}{(1-e^{0.05m})}[e^{mx}+e^{0.05m-mx}]](/tpl/images/0526/3298/6fde4.png)
![q_{cond,i}=\frac{-(25 \pi)kmD^2}{(1-e^{0.05m})}[e^{mx}+e^{0.05m-mx}]\\=\frac{-(25 \pi)*(400)*(26.73)*(0.00175)^2}{(1-e^{0.05*26.73})}[e^{26.73*0}+e^{(0.05*26.73)-(26.73*0)}]\\=\frac{-2.572}{-2.806}[1+3.806}]=4.405W](/tpl/images/0526/3298/97010.png)
![q_{cond,o}=\frac{-(25 \pi)kmD^2}{(1-e^{0.05m})}[e^{mx}+e^{0.05m-mx}]\\=\frac{-(25 \pi)*(400)*(26.73)*(0.00175)^2}{(1-e^{0.05*26.73})}[e^{(26.73*0.025)}+e^{(0.05*26.73)-(26.73*0.025)}]\\=\frac{-2.252}{-2.806} (3.903)=3.13W](/tpl/images/0526/3298/38453.png)
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