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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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Most popular questions from this chapter

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