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) \(\delta h_{g}\) (e) none of them

Short answer

#Answer# The correct formula for the heat transfer coefficient in this case is (a) \(k_{l} / \delta\).

Step-by-step solution

Recall the definition of the heat transfer coefficient

The heat transfer coefficient (\(h\)) can be defined as the proportionality constant between the heat transfer rate (\(q\)) and the temperature difference (\(\Delta T\)) between the surfaces or across the fluid, as follows: $$q = h \cdot A \cdot \Delta T$$, where \(A\) is the area through which the heat is being transferred.

Recall the heat transfer rate equation for condensation

The heat transfer rate in condensation can be defined through this standard equation: $$q = k_{l} \frac{\Delta T}{\delta} \cdot A$$, where \(k_l\) is the thermal conductivity of the liquid and \(\delta\) is the thickness of the liquid condensate layer.

Compare the given options to the standard definition

We'll now compare the given options to the standard heat transfer coefficient definition and heat transfer rate equation to determine the correct formula for the heat transfer coefficient. (a) \(k_{l} / \delta\): Comparing the equation for the heat transfer rate (\(q = k_{l} \frac{\Delta T}{\delta} \cdot A\)) with the definition of the heat transfer coefficient (\(q = h \cdot A \cdot \Delta T\)), we can see that the correct heat transfer coefficient should be \(h = k_{l} / \delta\). So, the correct answer is (a) \(k_{l} / \delta\). (b) \(\delta h_{f}\), (c) \(\delta h_{f g}\), (d) \(\delta h_{g}\), and (e) none of them do not represent the correct heat transfer coefficient based on the standard definition and heat transfer rate equation, so they are not the correct answer.

Key concepts

Condensation Heat Transfer

Condensation heat transfer is a process where vapor changes into a liquid on a surface, releasing heat in the process. This form of heat transfer is crucial in many industrial applications such as in power plants, refrigeration systems, and the process industry. When vapor comes into contact with a cooler surface, it cools down and turns into a liquid, forming a condensate film.

For condensation to occur efficiently, the surface must be below the saturation temperature of the vapor. The heat transferred from the vapor to the surface during condensation directly depends on the temperature difference between the vapor and the surface. It's quantified by a heat transfer coefficient, thoroughly explained in the exercise solutions.

An essential aspect of enhancing condensation heat transfer is to minimize resistance to heat flow. This can involve promoting dropwise condensation, where droplets form and quickly run off the surface, instead of filmwise condensation, where a continuous film of condensate develops. The latter tends to have a higher thermal resistance, leading to a less effective heat transfer process.

Thermal Conductivity

Thermal conductivity, denoted by the symbol \(k\), is a material property that measures a substance's ability to conduct heat. It's quantified in watts per meter-kelvin (W/m·K). High thermal conductivity means that the material is an excellent heat conductor, like metals, while low thermal conductivity indicates that the material is a poor heat conductor, like insulators.

Understanding thermal conductivity is vital because it aids in selecting suitable materials for heat exchangers and insulation. In the context of the condensate layer formed during vapor condensation, the thermal conductivity of the liquid layer is a key factor that influences the heat transfer rate. The better the liquid's ability to conduct heat, the more efficient the heat transfer will be, assuming the thickness of the condensate layer remains constant.

As illustrated in the solution steps for the exercise, the thermal conductivity of the liquid, \(k_l\), directly participates in the expression for the heat transfer rate during condensation. It's vital for students to grasp the role of \(k_l\) to understand the core mechanisms behind condensation heat transfer effectively.

Liquid Condensate Layer Thickness

The thickness of the liquid condensate layer, denoted by \(\delta\), is a critical parameter in the analysis of heat transfer during condensation. The condensate layer is the liquid film that forms on the surface where condensation is occurring, and its thickness can significantly affect the heat transfer rate.

When the condensate layer is thin, it provides less thermal resistance, allowing for more efficient heat transfer. As the film grows thicker, the resistance to heat flow increases, and the overall heat transfer rate decreases. That’s why in the problem from the textbook, the heat transfer coefficient - which is inversely proportional to resistance - is inversely proportional to the condensate layer thickness as given by the expression \(h = k_{l} / \delta\).

It's also worth noting that the layer thickness can change over the area of the condensing surface. For instance, gravity affects the thickness of the liquid film by pulling the condensate down, thus increasing the film thickness as you move further down the surface. This variation is an integral part of the exercise, prompting students to consider how changes in layer thickness can influence heat transfer calculations and, in turn, the efficiency of systems that rely on condensation.