Question

At a distance \(x\) down a vertical, isothermal flat plate on which a saturated vapor is condensing in a continuous film, the thickness of the liquid condensate layer is \(\delta\). The heat transfer coefficient at this location on the plate is given by (a) \(k_{l} / \delta\) (b) \(\delta h_{f}\) (c) \(\delta h_{f g}\) (d) \(\delta h_{\mathrm{s}}\) (e) none of them

Short answer

Answer: The correct expression for the heat transfer coefficient is given by the quotient of the thermal conductivity of the liquid (\(k_{l}\)) and the thickness of the liquid condensate layer (\(\delta\)): $$h = \frac{k_{l}}{\delta}$$.

Step-by-step solution

Identify the conceptual components

Since the problem concerns a vertical flat plate with condensing vapor, it is a film condensation problem. Additionally, we know that the plate is isothermal, which simplifies the problem. We need to find the heat transfer coefficient of the specified location on the plate, so let's recall the definition of the heat transfer coefficient and the variables needed: Heat transfer coefficient (h): It represents the heat transfer by convection between a surface and a fluid (in this case, the condensate layer), and is given by: $$h = \frac{q}{\Delta T}$$, where q is the heat transfer per unit area and \(\Delta T\) is the temperature difference between the surface and the fluid. Now let's review the given options: (a) \(k_{l} / \delta\): This term presents the quotient of the thermal conductivity of the liquid, represented by \(k_{l}\), and the thickness of the liquid layer, \(\delta\). Since it connects the ease of heat transfer (thermal conductivity) and the thickness of the condensate, it's a possible answer. (b) \(\delta h_{f}\): This term combines the thickness of the liquid layer (\(\delta\)) and the specific enthalpy of fusion (\(h_{f}\)). It doesn't relate to heat transfer or temperature, so we can rule this one out. (c) \(\delta h_{f g}\): This term consists of the thickness of the liquid layer (\(\delta\)) and the specific enthalpy of vaporization (\(h_{f g}\)). It also doesn't relate to heat transfer or temperature, so we can rule this one out as well. (d) \(\delta h_{\mathrm{s}}\): This term combines the thickness of the liquid layer (\(\delta\)) with the specific enthalpy of sublimation (\(h_{\mathrm{s}}\)). It also deals with enthalpy instead of heat transfer or temperature, so we rule this answer out. (e) None of them: This option is incorrect, as we identified in option (a) a potential relation between heat transfer and the quantities given.

Identify the correct expression

Based on the analysis in Step 1, we can now conclude that option (a) provides the correct expression for the heat transfer coefficient at the specified location on the isothermal flat plate. It is given by the quotient of the thermal conductivity of the liquid (\(k_{l}\)) and the thickness of the liquid condensate layer (\(\delta\)): $$h = \frac{k_{l}}{\delta}$$