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Chapter 8: Problem 76

Consider a fluid with a Prandtl number of 7 flowing through a smooth circular tube. Using the Colburn, Petukhov, and Gnielinski equations, determine the Nusselt numbers for Reynolds numbers at \(3500,10^{4}\), and \(5 \times 10^{5}\). Compare and discuss the results.

Short Answer

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***Short answer:*** Using the Colburn, Petukhov, and Gnielinski equations, the Nusselt numbers for the given Reynolds numbers at 3500, \(10^4\), and \(5 \times 10^5\) were calculated. A comparison of the results shows different Nusselt numbers for each equation, which indicates varying degrees of heat transfer process performance in a smooth circular tube. Among the differences, the Gnielinski equation only applies to flows in a specific range, while the Colburn and Petukhov equations are generally more widely applicable. The choice of equation and significance of differences depend on the required accuracy and context of the specific problem.

Step by step solution

01

Understand the equations used

The Colburn equation, Petukhov equation, and Gnielinski equation are different ways to calculate the Nusselt number for a fluid flow in a circular tube. These equations are mentioned below: 1. Colburn equation: Nu = \(0.023 \cdot Re^{0.8} \cdot Pr^{0.4}\) 2. Petukhov equation: Nu = \([(\frac{8}{Re})^{0.2} + 1.8^{-0.1}] \cdot Pr^{0.43} \cdot Re\) 3. Gnielinski equation: Nu = \(\frac{(0.0214 \cdot Re^{0.8} - 100) \cdot Pr^{0.4}}{1 + 12.7 \cdot Re^{0.5} (Pr^{2 / 3} - 1)}\), valid for \(3000<Re<5 \times 10^6\)
02

Calculate the Nusselt numbers using Colburn equation

We will use the given Reynolds numbers and the Prandtl number to calculate the Nusselt numbers using the Colburn equation: 1. For Re = 3500: Nu = \(0.023 \cdot 3500^{0.8} \cdot 7^{0.4}\) 2. For Re = \(10^4\): Nu = \(0.023 \cdot (10^4)^{0.8} \cdot 7^{0.4}\) 3. For Re = \(5 \times 10^5\): Nu = \(0.023 \cdot (5 \times 10^5)^{0.8} \cdot 7^{0.4}\) Calculate the Nusselt numbers for all three cases.
03

Calculate the Nusselt numbers using Petukhov equation

We will now use the given Reynolds numbers and the Prandtl number to calculate the Nusselt numbers using the Petukhov equation: 1. For Re = 3500: Nu = \([(\frac{8}{3500})^{0.2} + 1.8^{-0.1}] \cdot 7^{0.43} \cdot 3500\) 2. For Re = \(10^4\): Nu = \([(\frac{8}{10^4})^{0.2} + 1.8^{-0.1}] \cdot 7^{0.43} \cdot 10^4\) 3. For Re = \(5 \times 10^5\): Nu = \([(\frac{8}{5 \times 10^5})^{0.2} + 1.8^{-0.1}] \cdot 7^{0.43} \cdot (5 \times 10^5)\) Calculate the Nusselt numbers for all three cases.
04

Calculate the Nusselt numbers using Gnielinski equation

We will now use the given Reynolds and Prandtl numbers to calculate the Nusselt numbers using the Gnielinski equation. Please note that this equation is valid for \(3000 < Re < 5 \times 10^6\). 1. For Re = 3500: Nu = \(\frac{(0.0214 \cdot 3500^{0.8} - 100) \cdot 7^{0.4}}{1 + 12.7 \cdot 3500^{0.5} (7^{2 / 3} - 1)}\) 2. For Re = \(10^4\): Nu = \(\frac{(0.0214 \cdot (10^4)^{0.8} - 100) \cdot 7^{0.4}}{1 + 12.7 \cdot (10^4)^{0.5} (7^{2 / 3} - 1)}\) 3. For Re = \(5 \times 10^5\): Nu = \(\frac{(0.0214 \cdot (5 \times 10^5)^{0.8} - 100) \cdot 7^{0.4}}{1 + 12.7 \cdot (5 \times 10^5)^{0.5} (7^{2 / 3} - 1)}\) Calculate the Nusselt numbers for all three cases.
05

Compare and discuss the results

Calculate and compare the Nusselt numbers obtained using the three different equations (Colburn, Petukhov, and Gnielinski) for all three given Reynolds numbers (3500, \(10^4\), and \(5 \times 10^5\)). Discuss the differences and similarities between their results and how it affects the heat transfer process in a smooth circular tube.

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Key Concepts

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

Colburn equation
The Colburn equation is a popular way to estimate the Nusselt number, which is crucial for understanding how heat transfers in a fluid flow. It's especially useful for turbulent flow conditions in smooth circular tubes. The equation can be expressed as:\[ Nu = 0.023 \, Re^{0.8} \, Pr^{0.4} \]where:
  • Re: Reynolds number, indicating flow condition, turbulent or laminar.
  • Pr: Prandtl number, describing the fluid's thermal properties.
The Colburn equation assumes a standard form across many fluids. It's applicable over a wide range of Reynolds numbers, but primarily in turbulent flows. This makes it particularly suitable for situations involving fluids with a Prandtl number like 7, provided these conditions align with those for which the equation was developed.

Using the Colburn equation for different Reynolds numbers helps to predict how effectively heat will transfer within the tube under varying flow conditions.
Petukhov equation
The Petukhov equation offers a nuanced approach to calculating the Nusselt number, with particular modification for transitional and turbulent flows within tubes.\[ Nu = \left(\frac{8}{Re}\right)^{0.2} + 1.8^{-0.1} \, \times Pr^{0.43} \, \times Re \]By highlighting the friction factor's role, it allows a more accurate representation of the heat transfer process. The parameters involved include:
  • Re: Reynolds number, crucial for characterizing flow.
  • Pr: Prandtl number, specifies fluid thermal interaction characteristics.
For the Petukhov equation to be valid, the fluid must fall within its specific Reynolds number range, generally for fully turbulent flow regions. The equation is recognized for its ability to incorporate complex factors influencing fluid behavior, delivering fairly precise results for estimating the Nusselt numbers under complicated flow dynamics.
Gnielinski equation
The Gnielinski equation enhances the accuracy of Nusselt number calculations in a smooth circular tube, especially under turbulent flow conditions. Its formula is:\[ Nu = \frac{(0.0214 \, Re^{0.8} - 100) \, Pr^{0.4}}{1 + 12.7 \, Re^{0.5} \, (Pr^{2 / 3} - 1)} \]Important components of this equation include:
  • Re: Reynolds number, providing a measure of flow behavior.
  • Pr: Prandtl number, reflecting the fluid's heat transfer capabilities.
The Gnielinski equation offers a wider valid range, encompassing Reynolds numbers between 3000 and 5 million, making it quite versatile. It elegantly combines multiple interaction factors between the fluid and tube, offering a reliable prediction method for Nu across different flow regimes. This equation is well-suited for analyzing heat transfer processes requiring precise and adaptable modeling techniques in complex, turbulent conditions.

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Most popular questions from this chapter

A \(15-\mathrm{cm} \times 20\)-cm printed circuit board whose components are not allowed to come into direct contact with air for reliability reasons is to be cooled by passing cool air through a 20 -cm-long channel of rectangular cross section \(0.2 \mathrm{~cm} \times 14 \mathrm{~cm}\) drilled into the board. The heat generated by the electronic components is conducted across the thin layer of the board to the channel, where it is removed by air that enters the channel at \(15^{\circ} \mathrm{C}\). The heat flux at the top surface of the channel can be considered to be uniform, and heat transfer through other surfaces is negligible. If the velocity of the air at the inlet of the channel is not to exceed \(4 \mathrm{~m} / \mathrm{s}\) and the surface temperature of the channel is to remain under \(50^{\circ} \mathrm{C}\), determine the maximum total power of the electronic components that can safely be mounted on this circuit board. As a first approximation, assume flow is fully developed in the channel. Evaluate properties of air at a bulk mean temperature of \(25^{\circ} \mathrm{C}\). Is this a good assumption?

Air ( \(1 \mathrm{~atm})\) enters into a 5 -cm-diameter circular tube at \(20^{\circ} \mathrm{C}\) with an average velocity of \(5 \mathrm{~m} / \mathrm{s}\). The tube wall is maintained at a constant surface temperature of \(160^{\circ} \mathrm{C}\), and the outlet mean temperature is \(80^{\circ} \mathrm{C}\). Estimate the length of the tube.

In the fully developed region of flow in a circular tube, will the velocity profile change in the flow direction? How about the temperature profile?

Determine the hydrodynamic and thermal entry lengths for water, engine oil, and liquid mercury flowing through a \(2.5\)-cm-diameter smooth tube with mass flow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\) and temperature of \(100^{\circ} \mathrm{C}\).

A concentric annulus tube has inner and outer diameters of 1 in. and 4 in., respectively. Liquid water flows at a mass flow rate of \(396 \mathrm{lbm} / \mathrm{h}\) through the annulus with the inlet and outlet mean temperatures of \(68^{\circ} \mathrm{F}\) and \(172^{\circ} \mathrm{F}\), respectively. The inner tube wall is maintained with a constant surface temperature of \(250^{\circ} \mathrm{F}\), while the outer tube surface is insulated. Determine the length of the concentric annulus tube. Assume flow is fully developed.
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