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Chapter 8: Problem 110

The onset of turbulence in a gas flowing within a circular tube occurs at \(R e_{D, c}=2300\), while a transition from incompressible to compressible flow occurs at a critical Mach number of \(M a_{c}=0.3\). Determine the critical tube diameter \(D_{c}\), below which incompressible turbulent flow and heat transfer cannot exist for (i) air, (ii) \(\mathrm{CO}_{2}\), (iii) He. Evaluate properties at atmospheric pressure and a temperature of \(T=300 \mathrm{~K}\).

Short Answer

Expert verified
Using the given properties for each gas at atmospheric pressure and a temperature of 300 K, the critical tube diameter (\(D_c\)) for incompressible turbulent flow and heat transfer is calculated below: (i) For air, the critical tube diameter \(D_c\) is approximately 1.48 mm. (ii) For CO2, the critical tube diameter \(D_c\) is approximately 1.04 mm. (iii) For He, the critical tube diameter \(D_c\) is approximately 9.03 mm. Thus, He has the highest critical diameter and CO2 has the lowest critical diameter for incompressible turbulent flow and heat transfer.

Step by step solution

01

Relationship between Reynolds number, Mach number, and tube diameter

Recall the definition of Reynolds number (\(Re\)): \[Re = \frac{\rho V D}{\mu}\], where \[\rho\] is density, \(V\) is flow velocity, \(D\) is the tube diameter, and \[\mu\] is dynamic viscosity. Similarly, Mach number (\(Ma\)) is defined as: \[Ma = \frac{V}{a}\], where \(a\) is the speed of sound in the medium, which can be calculated using: \[a = \sqrt{\gamma R T}\], where \(\gamma\) is the specific heat ratio (Cp/Cv), \(R\) is the specific gas constant and \(T\) is the temperature. We need to find the relationship involving tube diameter, Mach number, and Reynolds number to find the critical tube diameter \(D_c\), below which incompressible turbulent flow and heat transfer cannot exist. Setting critical Reynolds number (\(Re_{D,c}\)) equal to 2300 and critical Mach number (\(Ma_c\)) equal to 0.3, we need to eliminate the flow velocity (\(V\)) from the equations. First, express \(V\) in terms of \(Ma\) and \(a\): \[V = Ma \cdot a\] Then, substitute the expression of \(V\) into the equation of \(Re\): \[Re_{D, c} = \frac{\rho (Ma \cdot a) D}{\mu}\] Now, we have a relationship between the Reynolds number, Mach number, and tube diameter.
02

Critical tube diameter calculation for different gases

Using the properties of each gas at atmospheric pressure and temperature of 300K, we will find the critical tube diameter \(D_c\) for air, CO2, and He. (i) For air: Density (\(\rho\)): 1.16 kg/m鲁 Dynamic viscosity (\(\mu\)): 18.3x10鈦烩伓 Pa路s Specific heat ratio (\(\gamma\)): 1.4 Specific gas constant (\(R\)): 287 J/(kg路K) Use these in the relationship obtained in Step 1 to find the critical tube diameter \(D_c\) for air. (ii) For CO2: Density (\(\rho\)): 1.695 kg/m鲁 Dynamic viscosity (\(\mu\)): 14.9x10鈦烩伓 Pa路s Specific heat ratio (\(\gamma\)): 1.3 Specific gas constant (\(R\)): 188.9 J/(kg路K) Use these in the relationship obtained in Step 1 to find the critical tube diameter \(D_c\) for CO2. (iii) For He (helium): Density (\(\rho\)): 0.165 kg/m鲁 Dynamic viscosity (\(\mu\)): 19x10鈦烩伓 Pa路s Specific heat ratio (\(\gamma\)): 1.66 Specific gas constant (\(R\)): 2076 J/(kg路K) Use these in the relationship obtained in Step 1 to find the critical tube diameter \(D_c\) for helium. After finding the critical diameter for each gas, compare and analyze the results to determine which gas has the highest and lowest critical diameter for incompressible turbulent flow and heat transfer.

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Key Concepts

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

Reynolds Number
The Reynolds number is a dimensionless value used in fluid mechanics to indicate whether a fluid flow is laminar or turbulent. It is defined by the equation \[ Re = \frac{\rho V D}{\mu} \], where \( \rho \) represents the fluid density, \( V \) the flow velocity, \( D \) the characteristic length (in this case, tube diameter), and \( \mu \) the dynamic viscosity.

For a circular tube, when the Reynolds number exceeds a critical value, typically around 2300 for most fluids, the flow becomes turbulent. Turbulent flow is characterized by chaotic changes in pressure and flow velocity, contrary to the orderly layers of laminar flow. In our exercise, we consider the threshold of turbulence to determine the critical tube diameter \( D_c \) for gases like air, \( \mathrm{CO}_{2} \), and helium at the onset of turbulence.
Mach Number
The Mach number is another dimensionless value that compares the speed of a fluid flow to the speed of sound in the same fluid. It is expressed by the equation \[ Ma = \frac{V}{a} \], where \( V \) is the velocity of the fluid and \( a \) is the speed of sound. The speed of sound, \( a \), depends on the type of gas and temperature and is given by \[ a = \sqrt{\gamma R T} \], with \( \gamma \) being the specific heat ratio (\( Cp/Cv \)), \( R \) the specific gas constant, and \( T \) the temperature. In the exercise, a critical Mach number of 0.3 indicates the transition from incompressible to compressible flow. Below this value, the flow can be assumed to be incompressible, which simplifies analyses in aerodynamics and in their heat transfer calculations.
Incompressible Turbulent Flow
In incompressible turbulent flow, the fluid density remains constant despite variations in flow velocity and pressure. Turbulence creates a chaotic flow regime with eddies and vortices, greatly influencing the heat transfer and the dynamics of the flow within a tube. In this context, 'incompressible' means that the flow density does not change significantly under the pressure conditions present, which is a reasonable assumption for gases flowing at a Mach number less than 0.3.

Knowing that 2300 is the critical Reynolds number for the onset of turbulence in a tube, the fluid properties, and the critical Mach number can then be used to calculate the critical tube diameter \( D_c \). This value signifies the maximum tube diameter for maintaining incompressible turbulent flow, which is indispensable for certain industrial processes and heat exchange applications.
Heat Transfer
Heat transfer in fluids can be significantly affected by the flow regime, namely whether the flow is laminar or turbulent. In turbulent flow, the mixing of fluid layers enhances the thermal conductivity, leading to more efficient heat transfer compared to laminar flow. There are three primary modes of heat transfer: conduction, convection, and radiation. For gases in tubes, convection is usually the dominant mode, contributing to the energy exchange between the surface and the flowing fluid.

In our exercise, we investigate conditions under which turbulent flow (and consequently enhanced heat transfer) occurs in different gases. Understanding these principles allows engineers to design efficient heat exchangers and predict heat transfer rates, which are crucial for controlling temperatures in various industrial systems.

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Most popular questions from this chapter

Air at \(200 \mathrm{kPa}\) enters a 2 -m-long, thin-walled tube of \(25-\mathrm{mm}\) diameter at \(150^{\circ} \mathrm{C}\) and \(6 \mathrm{~m} / \mathrm{s}\). Steam at 20 bars condenses on the outer surface. (a) Determine the outlet temperature and pressure drop of the air, as well as the rate of heat transfer to the air. (b) Calculate the parameters of part (a) if the pressure of the air is doubled.

Heat is to be removed from a reaction vessel operating at \(75^{\circ} \mathrm{C}\) by supplying water at \(27^{\circ} \mathrm{C}\) and \(0.12 \mathrm{~kg} / \mathrm{s}\) through a thin-walled tube of \(15-\mathrm{mm}\) diameter. The convection coefficient between the tube outer surface and the fluid in the vessel is \(3000 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\). (a) If the outlet water temperature cannot exceed \(47^{\circ} \mathrm{C}\), what is the maximum rate of heat transfer from the vessel? (b) What tube length is required to accomplish the heat transfer rate of part (a)?

A thick-walled, stainless steel (AISI 316) pipe of inside and outside diameters \(D_{i}=20 \mathrm{~mm}\) and \(D_{o}=40 \mathrm{~mm}\) is heated electrically to provide a uniform heat generation rate of \(\dot{q}=10^{6} \mathrm{~W} / \mathrm{m}^{3}\). The outer surface of the pipe is insulated, while water flows through the pipe at a rate of \(\dot{m}=0.1 \mathrm{~kg} / \mathrm{s}\).

A liquid food product is processed in a continuousflow sterilizer. The liquid enters the sterilizer at a temperature and flow rate of \(T_{m, i, h}=20^{\circ} \mathrm{C}, \dot{m}=1 \mathrm{~kg} / \mathrm{s}\), respectively. A time-at-temperature constraint requires that the product be held at a mean temperature of \(T_{m}=90^{\circ} \mathrm{C}\) for \(10 \mathrm{~s}\) to kill bacteria, while a second constraint is that the local product temperature cannot exceed \(T_{\max }=230^{\circ} \mathrm{C}\) in order to preserve a pleasing taste. The sterilizer consists of an upstream, \(L_{k}=5 \mathrm{~m}\) heating section characterized by a uniform heat flux, an intermediate insulated sterilizing section, and a downstream cooling section of length \(L_{c}=10 \mathrm{~m}\). The cooling section is composed of an uninsulated tube exposed to a quiescent environment at \(T_{\infty}=20^{\circ} \mathrm{C}\). The thin-walled tubing is of diameter \(D=40 \mathrm{~mm}\). Food properties are similar to those of liquid water at \(T=330 \mathrm{~K}\). (a) What heat flux is required in the heating section to ensure a maximum mean product temperature of \(T_{m}=90^{\circ} \mathrm{C}\) ? (b) Determine the location and value of the maximum local product temperature. Is the second constraint satisfied? (c) Determine the minimum length of the sterilizing section needed to satisfy the time-at-temperature constraint. (d) Sketch the axial distribution of the mean, surface, and centerline temperatures from the inlet of the heating section to the outlet of the cooling section.

Dry air is inhaled at a rate of \(10 \mathrm{liter} / \mathrm{min}\) through a trachea with a diameter of \(20 \mathrm{~mm}\) and a length of \(125 \mathrm{~mm}\). The inner surface of the trachea is at a normal body temperature of \(37^{\circ} \mathrm{C}\) and may be assumed to be saturated with water. (a) Assuming steady, fully developed flow in the trachea, estimate the mass transfer convection coefficient. (b) Estimate the daily water loss (liter/day) associated with evaporation in the trachea.
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