Problem 10.81 Challenge

กำหนดให้ \(ΔT = 10.0 - 0 = 10.0 °C, x_{1} = 4.00 cm, x_{2} = 8.00 cm,\)

\(L = 3.33\times 10^{5} J/kg, ρ = 917 kg/m^{3}\) และ \(k = 2.00 W/m·°C\)

จาก \(P = kA|\frac{dT}{dx}|\)

\(\frac{dQ}{dt}= kA|\frac{dT}{dx}|; P =\frac{dQ}{dt}\)

\(\frac{d(mL)}{dt}= kA\frac{\Delta T}{x}; Q = mL\)

\(L\frac{dm}{dt}= kA\frac{\Delta T}{x}\)

\(Lρ\frac{dV}{dt}= kA\frac{\Delta T}{x}; ρ =\frac{dm}{dV}\)

\(LρA\frac{dx}{dt}= kA\frac{\Delta T}{x}; dV = Adx\)

\(dt =\frac{ρL}{k\Delta T}xdx\)

\(_{t8.00 cm}\)

\(\int dt =\frac{ρL}{k\Delta T}\int xdx\)

\(_{t_{0}4.00 cm}\)

\(_{8.00 cm}\)

\(\Delta t =\frac{ρLx^{2}}{2k\Delta T}= 3.66\times 10^{4} s = 10.2 hr \;\blacksquare\)

\(_{4.00 cm}\)