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Chapter 4: Problem 28

The highest heat flux that can be achieved in nucleate boiling (called \(q_{\max }-\) see the qualitative discussion in Section 9.1) depends upon \(\rho_{g}\), the saturated vapor density; \(h_{f g}\), the latent heat vaporization; \(\sigma\), the surface tension; a characteristic length, \(l\); and the gravity force per unit volume, \(g\left(\rho_{f}-\rho_{g}\right)\), where \(\rho_{f}\) is the saturated liquid density. Develop the dimensionless functional equation for \(q_{\max }\) in terms of dimensionless length.

Short Answer

Expert verified
The dimensionless functional equation is \[ \frac{q_{\max}}{h_{fg} \rho_g l} = f\left(\frac{g(\rho_f - \rho_g) l^2}{\sigma}\right) \].

Step by step solution

01

Identify the Relevant Variables

The variables influencing the heat flux under nucleate boiling conditions are given as follows.\- Saturated vapor density, \(\rho_g\)\- Latent heat of vaporization, \(h_{fg}\)\- Surface tension, \(\sigma\)\- Characteristic length, \(l\)\- Gravity force term, \(g(\rho_f - \rho_g)\)The dependent variable is \(q_{\max}\), the maximum heat flux during nucleate boiling.
02

Apply Buckingham 蟺 Theorem

According to the Buckingham \(\pi\) theorem, if we have \(n\) variables and these variables contain \(m\) fundamental dimensions, then we can form \((n - m)\) dimensionless groups.In this problem, the fundamental dimensions are mass (M), length (L), time (T), and temperature (\(\Theta\)). Since we have 6 variables \(q_{\max}, \rho_g, h_{fg}, \sigma, l, g(\rho_f - \rho_g)\) and 4 fundamental dimensions (M, L, T, \(\Theta\)), we expect to achieve \(6 - 4 = 2\) dimensionless groups.
03

Assign Dimensions to Each Variable

- \([q_{\max}] = \left[\frac{M}{LT^3}\right]\)- \([\rho_g] = [\rho_f] = \left[\frac{M}{L^3}\right]\)- \([h_{fg}] = \left[\frac{L^2}{T^2}\right]\)- \([\sigma] = \left[\frac{M}{T^2}\right]\)- \([l] = [L]\)- \([g] = \left[\frac{L}{T^2}\right]\)- \([g(\rho_f - \rho_g)] = \left[\frac{M}{L^2T^2}\right]\)
04

Construct Dimensionless Groups

A dimensionless group is formed by setting the product of the variables, each raised to a power, to have overall dimensionless expression. One common dimensionless group using surface tension, vapor density, and gravity is the Bond number:\[ Bo = \frac{g(\rho_f - \rho_g) l^2}{\sigma} \]Another dimensionless group can be formed using the latent heat, surface tension, and characteristic length:\[ \pi_1 = \left(\frac{q_{\max}}{h_{fg} \rho_g}\right) \]Both groups can represent the functional relationship for \(q_{\max}\) in terms of characteristic length \(l\).
05

Develop the Dimensionless Functional Equation

Combine the dimensionless groups to express \(q_{\max}\) in terms of them:\[ \frac{q_{\max}}{h_{fg} \rho_g l} = f\left(\frac{g(\rho_f - \rho_g) l^2}{\sigma}\right) \]This dimensionless equation provides a relationship where the left-hand side is a dimensionless form for heat flux, and the right-hand side depends on a combination of the Bond number representing the gravitational effects on the system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dimensionless Groups
When dealing with physical equations, dimensionless groups are incredibly useful. They help us simplify complex problems into terms that generally apply, no matter the scale of the system. In nucleate boiling problems, dimensionless groups allow us to fine-tune boil characteristics without getting bogged down by actual dimensions. For instance, using dimensionless numbers like the Bond number, we can compare the relative effects of forces in different scenarios.
  • This practice helps engineers and scientists design more efficient boiling systems.
  • Dimensionless groups often involve ratios of similar physical quantities, which cancel out units, providing pure numbers.
These pure numbers allow researchers to make predictions and optimizations in a broader context.
Latent Heat of Vaporization
The concept of latent heat of vaporization, denoted as \( h_{fg} \), is pivotal in the process of phase changes, particularly in nucleate boiling. It represents the energy required to transform a unit mass of a liquid into vapor without changing its temperature. This energy is absorbed from the surroundings, making it crucial for heat transfer processes.
  • The latent heat of vaporization is critical in determining maximum heat flux during boiling.
  • A higher \( h_{fg} \) generally means more energy is needed for phase change, potentially altering boiling dynamics.
Understanding this concept helps in designing systems where efficient heat management is essential, like in cooling systems and power generation.
Surface Tension
Surface tension is a fundamental concept in fluid mechanics and plays a significant role in nucleate boiling. It is the force per unit length that acts along a liquid's surface, pulling the molecules inward and minimizing the surface area. This property is crucial in forming bubbles during boiling.
  • Surface tension affects the shape and stability of bubbles forming in the liquid.
  • A key factor in forming dimensionless groups such as the Bond number, which describes gravitational effects versus surface tension.
A good grasp of how surface tension operates helps in predicting and controlling boiling behavior by understanding how forces are distributed around a bubble.
Buckingham 蟺 Theorem
The Buckingham 蟺 theorem is a cornerstone principle in the dimensional analysis field. This theorem provides a method to derive dimensionless numbers from the variables involved in physical problems. It states that if you have \( n \) variables and these are described in terms of \( m \) fundamental dimensions, you can form \( n - m \) dimensionless groups.
  • In nucleate boiling scenarios, the theorem leads to the formation of dimensionless groups like those involving \( q_{\text{max}} \) and the Bond number.
  • It simplifies the complexity of equations by transforming them into general dimensionless forms, which have broader application.
Utilizing the Buckingham 蟺 theorem allows engineers to scale up small-scale findings into real-world applications, making predictions across various conditions.

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Most popular questions from this chapter

Steam condenses on the inside of a small pipe, keeping it at a specified temperature, \(T_{i}\). The pipe is heated by electrical resistance at a rate \(\dot{q} \mathrm{~W} / \mathrm{m}^{3}\). The outside temperature is \(T_{\infty}\) and there is a natural convection heat transfer coefficient, \(\bar{h}\) around the outside. (a) Derive an expression for the dimensionless expression temperature distribution, \(\Theta=\left(T-T_{\infty}\right) /\left(T_{i}-T_{\infty}\right)\), as a function of the radius ratios, \(\rho=r / r_{o}\) and \(\rho_{i}=r_{i} / r_{o}\); a heat generation number, \(\Gamma=\dot{q} r_{o}^{2} / k\left(T_{i}-T_{\infty}\right)\); and the Biot number. (b) Plot this result for the case \(\rho_{i}=2 / 3, \mathrm{Bi}=1\), and for several values of \(\Gamma\). (c) Discuss any interesting aspects of your result.

A large freezer's door has a \(2.5 \mathrm{~cm}\) thick layer of insulation \(\left(k_{\text {in }}=0.04 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\right)\) covered on the inside, outside, and edges with a continuous aluminum skin \(3.2 \mathrm{~mm}\) thick \(\left(k_{\mathrm{Al}}=165\right.\) \(\left.\mathrm{W} / \mathrm{m}^{2} \mathrm{~K}\right)\). The door closes against a nonconducting seal \(1 \mathrm{~cm}\) wide. Heat gain through the door can result from conduction straight through the insulation and skins (normal to the plane of the door) and from conduction in the aluminum skin only, going from the skin outside, around the edge skin, and to the inside skin. The heat transfer coefficients to the inside, \(\bar{h}_{i}\), and outside, \(\bar{h}_{o}\), are each \(12 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\), accounting for both convection and radiation. The temperature outside the freezer is \(25^{\circ} \mathrm{C}\), and the temperature inside is \(-15^{\circ} \mathrm{C}\). a. If the door is \(1 \mathrm{~m}\) wide, estimate the one-dimensional heat gain through the door, neglecting any conduction around the edges of the skin. Your answer will be in watts per meter of door height. b. Now estimate the heat gain by conduction around the edges of the door, assuming that the insulation is perfectly adiabatic so that all heat flows through the skin. This answer will also be per meter of door height.

One end of a copper rod \(30 \mathrm{~cm}\) long is held at \(200^{\circ} \mathrm{C}\), and the other end is held at \(93^{\circ} \mathrm{C}\). The heat transfer coefficient in between is \(17 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) (including both convection and radiation). If \(T_{\infty}=38^{\circ} \mathrm{C}\) and the diameter of the rod is \(1.25 \mathrm{~cm}\), what is the net heat removed by the air around the rod? [19.13 W.]

Steam condenses in a \(2 \mathrm{~cm}\) I.D. thin-walled tube of \(99 \%\) aluminum at 10 atm pressure. There are circular fins of constant thickness, \(3.5 \mathrm{~cm}\) in diameter, every \(0.5 \mathrm{~cm}\) on the outside. The fins are \(0.8 \mathrm{~mm}\) thick and the heat transfer coefficient from them \(\bar{h}=6 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) (including both convection and radiation). What is the mass rate of condensation if the pipe is \(1.5 \mathrm{~m}\) in length, the ambient temperature is \(18^{\circ} \mathrm{C}\), and \(\bar{h}\) for condensation is very large? \(\left[\dot{m}_{\text {cond }}=0.802 \mathrm{~kg} / \mathrm{hr}\right.\). \(]\)

Develop the dimensionless temperature distribution in a spherical shell with the inside wall kept at one temperature and the outside wall at a second temperature. Reduce your solution to the limiting cases in which \(r_{\text {outside }} \gg r_{\text {inside }}\) and in which \(r_{\text {outside }}\) is very close to \(r_{\text {inside. Discuss these limits. }}\)
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