Question

A \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) horizontal flat heater is used for vaporizing refrigerant-134a at \(350 \mathrm{kPa}\). The heater is supplied with \(0.35 \mathrm{MW} / \mathrm{m}^{2}\) of heat flux, and the surface temperature of the heater is \(25^{\circ} \mathrm{C}\). If the experimental constant in the Rohsenow correlation is \(n=1.7\), determine the value of the coefficient \(C_{s f}\). Discuss whether or not the Rohsenow correlation is applicable in this analysis.

Short answer

Based on the Rohsenow correlation, the coefficient \(C_{sf}\) for the given scenario with a horizontal flat heater and refrigerant-134a is approximately \(0.0042\). The correlation appears to be reasonably applicable in this case due to the surface temperature of the heater being not far from the saturation temperature and vaporization of the refrigerant-134a. However, it is important to verify the applicability for such heating conditions using more specific correlations for refrigerants or conducting experiments.

Step-by-step solution

Calculate the temperature difference

First, let's find the temperature difference across the interface. The surface temperature of the heater is \(25^{\circ} \mathrm{C}\). To find the interface temperature, we need to find the saturation temperature at the given pressure of \(350 \mathrm{kPa}\). Using refrigerant-134a property tables or steam tables, we can find the saturation temperature to be approximately \(38^{\circ} \mathrm{C}\). Thus, the temperature difference is: $$\Delta T = T_{sat} - T_s = 38 - 25 = 13^{\circ} \mathrm{C}$$

Find the latent heat of vaporization for refrigerant-134a

Now we need to find the latent heat of vaporization (\(h_{fg}\)) for refrigerant-134a at the given pressure. Using refrigerant-134a property tables or steam tables, we can find that: $$h_{fg} \approx 240 \mathrm{~kJ/kg}$$

Convert the heat flux to the appropriate unit

We need to convert the heat flux from \(\mathrm{MW/m^2}\) to \(\mathrm{kJ/(m^2s)}\). Given the heat flux of \(0.35 \mathrm{MW}/\mathrm{m^2}\): $$q = 0.35 \times 10^3 \mathrm{~kJ/(m^2s)} = 350 \mathrm{~kJ/(m^2s)}$$

Solve for the coefficient \(C_{sf}\)

Now, we can use the Rohsenow correlation to solve for the coefficient \(C_{sf}\): $$q = C_{sf}\Delta T^n h_{fg}$$ Rearranging the equation to solve for \(C_{sf}\): $$C_{sf} = \frac{q}{\Delta T^n h_{fg}}$$ Plugging in the values from the previous steps: $$C_{sf} = \frac{350}{(13)^{1.7} \times 240} \approx 0.0042$$ So, the value of the coefficient \(C_{sf}\) is approximately \(0.0042\).

Discuss the applicability of Rohsenow correlation

The Rohsenow correlation is an empirical correlation that is widely used to predict the rate of heat transfer during boiling, particularly for pool boiling. However, it may not be applicable in all scenarios, especially when the system deviates significantly from the original experimental conditions. In this case, it seems reasonable to apply Rohsenow correlation since the surface temperature of the heater is not far from the saturation temperature and the refrigerant-134a experiences vaporization. However, it's important to check with more specific correlations for refrigerants or conduct experiments to verify the applicability for such heating conditions.

Key concepts

Understanding the Rohsenow Correlation

The Rohsenow correlation is a key empirical formula used to estimate the heat transfer coefficient for boiling processes. It helps in predicting the heat flux required to sustain a specified temperature difference between a heated surface and the boiling fluid.
This correlation is especially popular in analyzing pool boiling scenarios. The general form of the Rohsenow equation is:

• \( q = C_{sf} \, \Delta T^n \, h_{fg} \)

Here, \( q \) represents the heat flux, \( C_{sf} \) is the surface-fluid coefficient, \( \Delta T \) is the temperature difference, \( n \) is the experimentally determined constant, and \( h_{fg} \) is the latent heat of vaporization.
The correlation assumes that the boiling heat transfer is controlled by the natural convection up to a nucleate boiling point, making it quite valuable for engineering applications across different fluids and surface conditions.

Latent Heat of Vaporization Explained

The latent heat of vaporization is a vital concept in thermodynamics and heat transfer. It refers to the amount of heat required to transform a liquid into a vapor without changing its temperature. This happens at a constant pressure.
For refrigerant-134a, as in the problem above, the latent heat of vaporization is approximately 240 kJ/kg at 350 kPa. This energy is consumed in breaking the intermolecular forces that hold the liquid molecules together.

• Occurs at a constant pressure
• Necessary for phase change from liquid to vapor
• Involved in natural convection during boiling

Understanding this property is essential because it directly impacts the efficiency and capacity of heat exchangers and refrigeration systems.

Delving into Saturation Temperature

Saturation temperature is the boiling point temperature at a specific pressure. For instance, under atmospheric pressure, water boils at 100 °C. However, the saturation temperature can vary with changes in external pressure.
In the provided exercise, the saturation temperature of refrigerant-134a at 350 kPa is found to be approximately 38 °C. This temperature marks the onset of boiling or vaporization, where a higher energy input (latent heat) is required to transition the liquid into vapor.

• Directly related to boiling point
• Varies with ambient pressure
• Critical in determining boiling processes for various substances

This concept is crucial for designing systems where controlled vaporization is necessary under specific conditions.

What is Heat Flux?

Heat flux is a measure of the heat transfer rate per unit area across a surface. It's expressed in watts per square meter (W/m²) or kilojoules per square meter per second (kJ/m²·s).
In the context of this heater application, a heat flux of 0.35 MW/m² (or 350 kJ/m²·s) is supplied to sustain the vaporization of refrigerant-134a. The calculation and understanding of heat flux are essential for designing efficient heating systems.

• Measures rate of heat transfer
• Expressed in W/m² or kJ/m²·s
• Important for thermal system design and analysis

Accurately calculating heat flux helps in determining how effectively a surface can transfer energy to a fluid, especially during boiling processes.

The Basics of Pool Boiling

Pool boiling describes the process where a liquid is resting on a heated surface and begins to boil. Boiling occurs when the liquid reaches its saturation temperature, and heat input causes vapor bubbles to form and rise.
In this scenario, the heat is supplied from the bottom surface, usually a horizontal heater as in the exercise, and heat is transferred by convection and phase change processes.
Factors to consider in pool boiling include:

• The surface roughness of the heater
• The pressure and temperature conditions
• The fluid properties, such as wettability and heat capacity

Understanding pool boiling is crucial in fields like refrigeration, where the refrigerant is boiled over a heated surface, making it an important area of study in heat transfer engineering.