To cool a summer home without using a vapor- compression refrigeration cycle, ai

Question

To cool a summer home without using a vapor- compression refrigeration cycle, air is routed through a copper pipe (k = 400 W/m-K. D_i = 0.15 m. D_o = 0.16 m) that is submerged in an adjoining body of water. The water temperature is nominally at T = 17 degree C, and a convection coefficient of h_o = 1500W/m^2-K is maintained at the outer surface of the pipe. Due to the high conductivity of the pipe and the high heat transfer coefficient ho you can assume a constant surface temperature in the pipe equal to the water temperature. If air from the home enters the pipe at a temperature of T_m/s = 28 degree C and a volumetric lion rate of = 0.028 m^3/s, what pipe length L. is needed to provide a discharge temperature of T_m, theta = 21 degree C? Properties: Air rho = 1.155 kg/m^3, C_p = 1007 J/Kg/m^3 mu = 183.6 times 10^-7 N s/m^3, k = 0.0261 W/m-K, Pr = 0.707.

Explanation / Answer

Case of constant surface temperature: Solution of ncropera Dewit used with a mean temperature.

dTm/dx = P/(m'Cp) h( Ts-Tm)

where P is the perimeter length, h the convection coeff ( constant ovr the length as temp of water is constant)

solvinf ( Ts-Tmo)/(Ts-Tm1) = exp(-PLh/m'Cp)  

Using h as th eheat coeff, but a composite coeff including the wall conductivity couuld be used

using Tm0 =21 , Tm1 =28, Ts=17

4/11 =exp (( -pi*.16*L *1500/(1.155*.028*1007))

taking logs and solving for L gives L = .0437 m

Using this as an approximate length , may continue with the composite heat transfer coeff, hk/( k+lh)

geting 1.647 m

---etc.

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