Question #163371

1. Consider a square plate of 0.5 m x 0.5 m in a room at 30 C. One side of the plate is kept at a uniform temperature of 74 C while the other side is insulated. Determine the rate of heat transfer from the plate by free convection under the following three situations: (i) The plate is kept vertical (ii) The plate is kept horizontal with hot surface facing up (iii) The plate is kept horizontal with hot surface facing down.

Expert's answer

The properties of air at the film temperature of

"T_f = \\large\\frac{(T_s + T_{\\infin})}{2}" = "\\large\\frac{30+74}{2}" = "52^oC \\space" and 1 atm here

"k= 0.02808 W\/m ^o C"

"P_r= 0.7202"

"\\nu = 1.896*10^{-5} \\frac{m^2}{s}"

"\\beta = \\large\\frac{1}{T_f}" "=\\large\\frac{1}{333K}"

(a) Vertical. The characteristic length in this case is the height of the plate, which is L = 0.5m. The Rayleigh number is

"Ra_D = \\large\\frac{g\\beta (T_s - T_{\\infin})L^3}{V^2}""Pr = \\large\\frac{9.81*\\frac{1}{333}(74-30)*0.5^3}{(1.896*10^{-5})^2}""(0.722) = 3.254*10^{8}"

The natural convection Nusselt number

"Nu = \\lbrace 0.825 + \\large\\frac{0.387Ra_D^{\\frac{1}{6}}}{\\lbrack1+(\\frac{0.492}{Pr})^{\\frac9{16}}\\rbrack^{\\frac{8}{27}}} \\rbrace^2" "=\\lbrace 0.825 + \\large\\frac{0.387(3.254*10^{8})^{\\frac{1}{6}}}{\\lbrack1+(\\frac{0.492}{0.7202})^{\\frac9{16}}\\rbrack^{\\frac{8}{27}}} \\rbrace ^2" "=" 102.6

"h = \\frac{k}{D}Nu = \\frac{0.02808 W\/m ^oC}{0.5m}(102.6) = 5.76 \\frac{W}{m} C"

"A_s = L^2 = (0.5m)^2 = 0.25m^2"

and

"Q = hA_s(T_s - T_{\\infin}) = (5.76 \\frac{W}{m} C)(0.25m^2)(74-30)^oC = 63.36W"

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