Question

When a saturated vapor condenses on a vertical, isothermal flat plate in a continuous film, the rate of heat transfer is proportional to (a) \(\left(T_{s}-T_{\text {sat }}\right)^{1 / 4}\) (b) \(\left(T_{s}-T_{s a t}\right)^{1 / 2}\) (c) \(\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\) (d) \(\left(T_{s}-T_{\text {sat }}\right)\) (e) \(\left(T_{s}-T_{\text {sat }}\right)^{2 / 3}\)

Short answer

Answer: (e) \(\left(T_{s}-T_{\text {sat }}\right)^{2 / 3}\)

Step-by-step solution

Understand Nusselt's film condensation theory

Nusselt's film condensation theory is a well-established theory for calculating heat transfer in film condensation processes. According to this theory, the heat transfer is related to the difference between the plate temperature \(T_s\) and the saturation temperature \(T_{sat}\) of the fluid.

Identify the expression for the heat transfer coefficient

Nusselt's film condensation theory suggests that the heat transfer coefficient, \(h\), can be represented in terms of the difference between the surface temperature and the saturation temperature of the working fluid as: \( h = C \cdot \left( \rho_l - \rho_v \right) \cdot g \cdot \left(T_s - T_{sat} \right)^{(1/3)} \) where \(C\) is a constant, \(\rho_l\) and \(\rho_v\) are the densities of the liquid and vapor phases respectively, and \(g\) is the acceleration due to gravity.

Relate heat transfer rate to the heat transfer coefficient

The rate of heat transfer, \(q\), can be expressed in terms of the heat transfer coefficient as: \(q = h \cdot A \cdot \left(T_s - T_{sat} \right)\) Substituting the expression for \(h\) from Step 2, we get: \(q = C \cdot A \cdot \left( \rho_l - \rho_v \right) \cdot g \cdot \left(T_s - T_{sat} \right)^{(1/3)} \cdot \left(T_s - T_{sat} \right) \) This simplifies to: \(q = C \cdot A \cdot \left( \rho_l - \rho_v \right) \cdot g \cdot \left(T_s - T_{sat} \right)^{(4/3)} \)

Compare the given options to the correct expression

Now, examining the options given in the problem statement, we can compare them to the correct expression derived above: (a) \(\left(T_{s}-T_{\text {sat }}\right)^{1 / 4}\): incorrect, as the exponent is \((4/3)\). (b) \(\left(T_{s}-T_{\text {sat }}\right)^{1 / 2}\): incorrect, as the exponent is \((4/3)\). (c) \(\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\): incorrect, as the exponent is \((4/3)\). (d) \(\left(T_{s}-T_{\text {sat }}\right)\): incorrect, as the exponent is \((4/3)\). (e) \(\left(T_{s}-T_{\text {sat }}\right)^{2 / 3}\): correct, as the exponent is \((4/3)\), and the rate of heat transfer is proportional to \((T_s - T_{sat})^{(4/3)}\). Therefore, the correct answer is (e) \(\left(T_{s}-T_{\text {sat }}\right)^{2 / 3}\).