Question

Water is boiled at \(90^{\circ} \mathrm{C}\) by a horizontal brass heating element of diameter \(7 \mathrm{~mm}\). Determine the maximum heat flux that can be attained in the nucleate boiling regime.

Short answer

The maximum heat flux that can be attained in the nucleate boiling regime is approximately 14,497.6 W/m².

Step-by-step solution

Write down the Rohsenow nucleate boiling correlation

The Rohsenow correlation for nucleate boiling is given by the following equation: \(q'' = C_{sf} \cdot h_{fg} \cdot (\frac{\sigma}{g(\rho_l-\rho_g)})^{1/2} \cdot m^n \cdot \Delta T\) where \(q''\) = heat flux \((W/m^2)\), \(C_{sf}\) = dimensionless constant, \(h_{fg}\) = specific enthalpy of vaporization \((J/kg)\), \(\sigma\) = surface tension \((N/m)\), \(g\) = acceleration due to gravity \((m/s^2)\), \(\rho_l\) = liquid density \((kg/m^3)\), \(\rho_g\) = vapor density \((kg/m^3)\), \(m\) = liquid-vapor mass qualities, \(n\) = a dimensionless constant, and \(\Delta T\) = excess temperature \((K)\).

Find the excess temperature

Excess temperature (\(\Delta T\)) is the difference between the saturation temperature of water at a given pressure and the surface temperature of the heating element. In this case, water is boiled at \(90^{\circ} \mathrm{C}\), which is \(363.15 \mathrm{K}\) in Kelvin. Assuming atmospheric pressure, the saturation temperature of water is \(100^{\circ} \mathrm{C}\) or \(373.15 \mathrm{K}\). So, \(\Delta T = T_{sat} - T_{surface} = 373.15 - 363.15 = 10 \mathrm{K}\).

Find the properties of water at the given temperature

We need the following properties of water at \(90^{\circ} \mathrm{C}\) (calculated at the saturation temperature): \(h_{fg} = 2.257 \times 10^6 \mathrm{\ J/kg}\) (enthalpy of vaporization), \(\sigma = 0.059 \mathrm{\ N/m}\) (surface tension), \(\rho_l = 971.8 \mathrm{\ kg/m^3}\) (liquid density), \(\rho_g = 0.597 \mathrm{\ kg/m^3}\) (vapor density).

Select appropriate values of constants and dimensionless groups

For nucleate boiling of water on a smooth horizontal brass surface, the following values are recommended: \(C_{sf} = 0.0064\), \(m = 2\), and \(n = 1\).

Calculate the maximum heat flux

Now, we can plug all the values into the Rohsenow correlation to find the maximum heat flux: \(q'' = C_{sf} \cdot h_{fg} \cdot (\frac{\sigma}{g(\rho_l-\rho_g)})^{1/2} \cdot m^n \cdot \Delta T\) \(q'' = 0.0064 \cdot 2.257 \times 10^6 \cdot (\frac{0.059}{9.81(971.8-0.597)})^{1/2} \cdot 2^1 \cdot 10\) \(q'' = 14,497.6 \mathrm{\ W/m^2}\) Thus, the maximum heat flux that can be attained in the nucleate boiling regime is approximately \(14,497.6 \mathrm{\ W/m^2}\).

Key concepts

Rohsenow Correlation

The Rohsenow Correlation is a key concept in understanding nucleate boiling. It provides a way to calculate the heat flux during this process using various properties of the system. In nucleate boiling, bubbles form at the microscale and detach from a heated surface. The Rohsenow Correlation considers both fluid and surface characteristics, making it highly versatile for different liquids and surfaces.

The general form of the Rohsenow Correlation for heat flux is: \[ q'' = C_{sf} \cdot h_{fg} \cdot \left(\frac{\sigma}{g(\rho_l-\rho_g)}\right)^{1/2} \cdot m^n \cdot \Delta T \] Key variables include:

• \(C_{sf}\): The dimensionless surface fluid constant, which varies depending on the fluid and surface type.
• \(h_{fg}\): The specific enthalpy of vaporization, crucial as it denotes the energy required for phase change.
• \(\Delta T\): The temperature difference between the heated surface and the saturation temperature.

The correlation helps to estimate the efficiency and limits of boiling systems in industries such as power generation and chemical processing.

Heat Flux Calculation

Calculating heat flux in nucleate boiling provides insight into the energy transfer efficiency of a boiling process. Heat flux \(q''\), expressed in \(W/m^2\), is the rate of heat energy transfer per unit area. This is a fundamental aspect of thermal engineering when designing heating elements and systems.

In the given problem, heat flux calculation makes use of the Rohsenow Correlation formula. Essential parameters like the surface tension \(\sigma\), and the densities \(\rho_l\) and \(\rho_g\) need to be known. These affect the buoyancy and surface behavior of the liquid during boiling.

The accuracy of heat flux calculations hinges on precise measurements and the correct use of constants and material properties. For brass heating elements with water, typical values might include the constant \(C_{sf} = 0.0064\), excess temperature of 10 (K), and generally accepted thermodynamic properties of water. Calculating \(q''\) is vital as it helps prevent scenarios like burnout, where excessive heat flux damages components.

Boiling Heat Transfer

Boiling heat transfer is the mechanism through which heat is transferred from a surface into a liquid, causing a phase change from liquid to vapor. This process is highly efficient due to the large enthalpy of vaporization associated with water and other liquids. During nucleate boiling, the focus is on how bubbles form and transfer heat from a heated surface to the surrounding liquid.

The efficiency of boiling heat transfer is determined by factors including the surface material, liquid properties, and temperature differences involved. Nucleate boiling, specifically, occurs at relatively low temperature excess compared to film boiling, making it desirable for applications requiring high heat flux.

Key concepts in boiling heat transfer include:

• Bubble Dynamics: Formation and detachment of bubbles help disperse heat throughout the liquid.
• Enthalpy of Vaporization: Indicates energy absorbed in converting a liquid to vapor.
• Thermal Conductivity: Influences how quickly heat moves through a substance or a surface.

These aspects affect how heat effectively moves and changes the state of water, providing enhanced energy transfer solutions.

Thermodynamic Properties of Water

The thermodynamic properties of water are crucial for accurately predicting and analyzing nucleate boiling scenarios. Key properties include the enthalpy of vaporization, surface tension, and specific densities of liquid and vapor at a given temperature.

Water is often preferred in heat transfer applications due to its high specific heat and latent heat of vaporization. These characteristics enable efficient thermal management and energy usage. In the context of the exercise, knowing the enthalpy of vaporization \(h_{fg} = 2.257 \times 10^6 \ J/kg\) and surface tension \(\sigma = 0.059 \ N/m\) is imperative for applying the Rohsenow Correlation correctly.

With water being isotropic and having well-documented thermodynamic data, it simplifies the modeling of boiling processes, ensuring designs are predictable and scalable. Knowing density values like \(\rho_l = 971.8 \ kg/m^3\) and \(\rho_g = 0.597 \ kg/m^3\) helps model the fluid dynamics accurately, enabling tailored and optimal engineering solutions.