A water film flows down a 0.500 m wide heated strip of length 2.00 m

A water film flows down a 0.500 m wide heated strip of length 2.00 m. At x = 0, its mass flow rate is 0.0200 kg/s and its temperature is 25.0 degrees C. The strip delivers a uniform heat flux q_w'' = 2400 W/m^2. Air at 35.0 degrees C, 101.3 kPa, and 15.0% relative humidity flows over the film; use h = 18.0 W/(m^2 K) and k_c = 0.0150 m/s. Assume the liquid is perfectly mixed across its thickness at each x, axial conduction is negligible, and the local free surface is at the bulk-liquid temperature T(x). Per unit interfacial area, N_W = k_c c ln[(1-y_W,infinity)/(1-y_W,s)], where y_W,infinity = 0.150 p_sat(308.15 K)/P, y_W,s = p_sat(T)/P, c = P/[R(T + T_air)/2] with temperatures in kelvin, and ln[p_sat(Pa)] = 23.1964 - 3816.44/(T[K] - 46.13). The governing balances are d m_dot/dx = -b M_W N_W and m_dot c_p(dT/dx) = b[q_w'' + h(T_air - T) - M_W N_W Delta H_vap]. Use b = 0.500 m, c_p = 4180 J/(kg K), M_W = 0.018015 kg/mol, Delta H_vap = 2.40 x 10^6 J/kg, and R = 8.314 J/(mol K). Numerically determine the outlet liquid temperature, the outlet mass flow rate, the fraction of the inlet water evaporated, and the position at which the liquid first reaches 40.0 degrees C, if it does so within the strip.

Treat \(\dot m(x)\) and \(T(x)\) as the two state variables of a coupled initial-value problem.