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) \(\left(T_{s}-T_{\mathrm{sat}}\right)^{1 / 2}\) (c) \(\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\) (d) \(\left(T_{s}-T_{\text {sat }}\right)\) (e) \(\left(T_{s}-T_{\mathrm{sat}}\right)^{2 / 3}\)

Short answer

Answer: (c) \(\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\)

Step-by-step solution

Nusselt's Condensation Theory

In the case of condensation on a vertical plate, Nusselt's condensation theory provides a relationship between the heat transfer coefficient, h, and the difference between the surface temperature and the saturation temperature. The Nusselt's theory states that: \(hL=0.943\left[\frac{g \rho_{L}\left(\rho_{L}-\rho_{V}\right)c_{p}\left(T_{s}-T_{\text {sat}}\right)\Lambda L^{3}}{\mu_{L} \Lambda}\right]^{1 / 4}\) where, \(h\) - Heat transfer coefficient (W/m²K) \(L\) - Length of the vertical flat plate (m) \(g\) - Gravitational acceleration (9.81 m/s²) \(\rho_{L}\) - Density of the liquid (kg/m³) \(\rho_{V}\) - Density of the vapor (kg/m³) \(c_{p}\) - Specific heat at constant pressure (J/kgK) \(T_{s}\) - Surface temperature (K) \(T_{\text{sat}}\) - Saturation temperature (K) \(\Lambda\) - Latent heat of condensation (J/kg) \(\mu_L\) - Dynamic viscosity of the liquid (Pa·s)

Determining the Proportionality Between the Rate of Heat Transfer and the Difference in Temperatures

The rate of heat transfer, \(q\), is given by the product of the heat transfer coefficient, h, and the difference in temperatures: \(q = hA\left(T_{s} - T_{\text{sat}}\right)\) Substituting the expression for h from Nusselt's condensation theory: \(q=0.943A\left(\frac{g \rho_{L}\left(\rho_{L}-\rho_{V}\right)c_{p}\left(T_{s}-T_{\text {sat}}\right)L^{3}\Lambda}{\mu_{L} \Lambda}\right)^{1 / 4}\left(T_{s}-T_{\text {sat}}\right)\) Canceling out the common terms and rearranging, we get: \(q\propto\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\) Comparing this expression with the given options, the correct answer is (c) \(\left(T_{s}-T_{\text {sat }}\right)^{3 / 4}\).

Key concepts

Heat Transfer Coefficient

When discussing heat transfer processes, the heat transfer coefficient (often symbolized as h) is a crucial concept. It quantifies the capacity of a material to conduct heat. Specifically, it's a measure of the rate at which heat is transferred per unit area per unit temperature difference. Imagine you're holding a hot cup of coffee; the sensation of warmth you feel on your hand is partly due to the heat transfer coefficient of the cup material. In mathematical terms, the heat transfer coefficient is expressed in watts per square meter kelvin (W/m²K).

In the context of Nusselt's condensation theory, the heat transfer coefficient becomes particularly important when a vapor, such as steam, condenses into liquid form on a surface. If that surface is a vertical plate, the heat transfer coefficient describes how efficiently the latent heat of condensation is dispersed away from the plate. A higher coefficient suggests that the plate is very effective at spreading out heat, cooling the condensate more rapidly.

Condensation on Vertical Plate

Condensation on a vertical plate occurs when vapor in contact with the surface cools down and transforms into liquid. This change from gas to liquid, or condensation, releases heat—referred to as the latent heat of condensation. Nusselt's condensation theory specifically looks at this process happening on an isothermal vertical plate.

The surface temperature of the plate (T_s) and the saturation temperature of the vapor (T_sat) play key roles in determining how the condensate film forms and flows down the plate. The theory predicts that the film thickness, the speed at which it moves, and, crucially, the rate at which heat is transferred, are all related to a balance of forces and heat transfer properties, which are encapsulated by the Nusselt number—a dimensionless parameter.

Latent Heat of Condensation

Latent heat is energy released or absorbed by a body during a constant-temperature process. The latent heat of condensation is the amount of energy released when a substance transitions from its gaseous phase to a liquid phase at its saturation temperature.

This released energy is significant in the context of condensation on a vertical plate as it dictates the rate of heat transfer away from the plate. When a vapor condenses, it releases latent heat, which must be removed to sustain the condensation process. The latent heat has a direct impact on calculating the heat transfer coefficient in Nusselt's equation, influencing how heat is conducted away from the condensate film forming on the plate.

Temperature Difference in Heat Transfer

The temperature difference in heat transfer refers to the difference in temperature between two spaces or substances that are exchanging heat. In the case of condensation on a vertical plate, we look at the temperature difference between the plate's surface temperature (T_s) and the saturation temperature of the vapor (T_sat).

This temperature differential is crucial in driving the condensation process. Heat naturally flows from the warmer surface to the cooler liquid condensate. As Nusselt's theory suggests, the temperature difference affects the rate at which heat transfers. The relationship presented in Nusselt's theory exhibits that the rate of heat transfer due to condensation is not linear but proportional to the temperature difference raised to a power of 3/4.