/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 109 A horizontal, \(25-\mathrm{mm}\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 109

A horizontal, \(25-\mathrm{mm}\) diameter cylinder is maintained at a uniform surface temperature of \(35^{\circ} \mathrm{C}\). A fluid with a velocity of \(0.05 \mathrm{~m} / \mathrm{s}\) and temperature of \(20^{\circ} \mathrm{C}\) is in cross flow over the cylinder. Determine whether heat transfer by free convection will be significant for (i) air, (ii) water, (iii) engine oil, and (iv) mercury.

Short Answer

Expert verified
In conclusion, heat transfer by free convection is significant only for engine oil. For air, water, and mercury, forced convection is the dominant mode of heat transfer.

Step by step solution

01

Calculate the Reynolds number for each fluid

The Reynolds number (Re) is calculated using the formula: \[Re = \frac{Vd}{谓}\] Where V is the fluid velocity, d is the diameter of the cylinder, and 谓 is the kinematic viscosity of the fluid. For each fluid, find the Reynolds number using the given properties: Air: \(Re_{air} = \frac{0.05 \mathrm{~m} / \mathrm{s} \times 0.025 \mathrm{~m}}{15.69 \times 10^{-6} \mathrm{~m}^2/\mathrm{s}} = 79.42\) Water: \(Re_{water} = \frac{0.05 \mathrm{~m} / \mathrm{s} \times 0.025 \mathrm{~m}}{1.11 \times 10^{-6} \mathrm{~m}^2/\mathrm{s}} = 1126.13\) Engine oil: \(Re_{oil} = \frac{0.05 \mathrm{~m} / \mathrm{s} \times 0.025 \mathrm{~m}}{0.077 \mathrm{~m}^2/\mathrm{s}} = 0.016\) Mercury: \(Re_{Hg} = \frac{0.05 \mathrm{~m} / \mathrm{s} \times 0.025 \mathrm{~m}}{1.21 \times 10^{-7} \mathrm{~m}^2/\mathrm{s}} = 103082.64\)
02

Calculate the Grashof number for each fluid

The Grashof number (Gr) is calculated using the formula: \[Gr = \frac{g \cdot 尾 \cdot (T_s - T_f) \cdot d^3}{谓^2}\] Where g is the acceleration due to gravity (\(9.81 m/s^2\)), 尾 is the thermal expansion coefficient, \(T_s\) is the surface temperature, \(T_f\) is the fluid temperature, d is the diameter of the cylinder, and 谓 is the kinematic viscosity of the fluid. For each fluid, find the Grashof number using the given properties: Air: \(Gr_{air} = \frac{9.81 \cdot \frac{1}{300} \cdot (35 - 20) \cdot (0.025)^3}{(15.69 \times 10^{-6})^2} = 0.027\) Water: \(Gr_{water} = \frac{9.81 \cdot \frac{1}{6300} \cdot (35 - 20) \cdot (0.025)^3}{(1.11 \times 10^{-6})^2} = 238.31\) Engine oil: \(Gr_{oil} = \frac{9.81 \cdot \frac{1}{3900} \cdot (35 - 20) \cdot (0.025)^3}{(0.077)^2} = 0.0002\) Mercury: \(Gr_{Hg} = \frac{9.81 \cdot \frac{1}{18500} \cdot (35 - 20) \cdot (0.025)^3}{(1.21 \times 10^{-7})^2} = 47963.87\)
03

Compare Reynolds and Grashof numbers

For each fluid, compare the Grashof number (Gr) to the Reynolds number (Re). Air: \(Gr_{air} \ll Re_{air}\) -> forced convection is dominant Water: \(Gr_{water} \ll Re_{water}\) -> forced convection is dominant Engine oil: \(Gr_{oil} \gg Re_{oil}\) -> free convection is significant Mercury: \(Gr_{Hg} \ll Re_{Hg}\) -> forced convection is dominant In conclusion, based on the comparison of the Grashof number and the Reynolds number for each fluid: (i) air - free convection is not significant (ii) water - free convection is not significant (iii) engine oil - free convection is significant (iv) mercury - free convection is not significant

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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 crucial dimensionless number in fluid dynamics that helps determine the flow regime of a fluid. It describes the ratio of inertial forces to viscous forces and is used to predict whether the flow will be laminar or turbulent.
In our exercise, the Reynolds Number (Re) is calculated using the formula:
  • \[ Re = \frac{Vd}{u} \]
where \( V \) is the fluid velocity, \( d \) is the diameter of the object the fluid flows over, and \( u \) is the kinematic viscosity of the fluid.
For example, in the case of air flowing over the cylinder, the Reynolds Number is calculated as \( 79.42 \), indicating a laminar flow. This helps us assess how momentum is transferred through the fluid and whether it is likely to have a streamlined or chaotic motion.
Grashof Number
The Grashof Number is another dimensionless number in fluid dynamics, used to assess the nature of buoyancy-driven or free convection. It represents the ratio of the buoyancy forces to viscous forces in a fluid flow.
The formula for Grashof Number (Gr) is:
  • \[ Gr = \frac{g \cdot \beta \cdot (T_s - T_f) \cdot d^3}{u^2} \]
where \( g \) is the acceleration due to gravity, \( \beta \) is the thermal expansion coefficient, \( T_s \) is the surface temperature, \( T_f \) is the fluid temperature, and \( u \) is the kinematic viscosity.
In this exercise, the Grashof Number for air is \( 0.027 \), which indicates that the effects of natural convection are negligible compared to forced convection. By comparing Grashof with Reynolds Numbers for different fluids, we can determine which type of convection is dominant.
Convection
Convection is the heat transfer process that occurs through fluid motion. It can be categorized into two types: free (or natural) convection and forced convection.
  • Free Convection: Occurs when fluid motion is caused by buoyancy forces that arise from temperature differences within the fluid itself.
  • Forced Convection: Occurs when an external source, such as a pump or fan, induces fluid motion over a surface.

In the provided exercise, we need to determine if free convection will have a significant role. By evaluating the Grashof and Reynolds Numbers, it is evident that for fluids like engine oil, free convection plays a major part due to high buoyancy forces relative to viscous forces. A focus on convection ensures efficient thermal management in applications such as cooling systems and heating devices.
Fluid Dynamics
Fluid Dynamics is a branch of physics that focuses on the study of the movement of fluids (liquids and gases) and the forces acting upon them. Understanding fluid dynamics is vital for designing systems in energy generation, aerospace, automotive industries, among others.
It involves analyzing fluid flow problems using various dimensionless numbers, such as the Reynolds and Grashof Numbers, to characterize and predict flow behavior.
  • Applications: Include optimizing airflow in buildings, design of aircraft wings, and improving vehicle aerodynamics.
  • Tools: Computational fluid dynamics (CFD) is often used to simulate fluid flow and improve design processes.
In educational exercises like ours, understanding these principles helps predict how different fluids will behave over surfaces, indicating efficient methods for heat transfer. Proper analysis aids in optimizing engineering and environmental systems.

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Most popular questions from this chapter

Common practice in chemical processing plants is to clad pipe insulation with a durable, thick aluminum foil. The functions of the foil are to confine the batt insulation and to reduce heat transfer by radiation to the surroundings. Because of the presence of chlorine (at chlorine or seaside plants), the aluminum foil surface, which is initially bright, becomes etched with in- service time. Typically, the emissivity might change from \(0.12\) at installation to \(0.36\) with extended service. For a \(300-\mathrm{mm}\)-diameter foil-covered pipe whose surface temperature is \(90^{\circ} \mathrm{C}\), will this increase in emissivity due to degradation of the foil finish have a significant effect on heat loss from the pipe? Consider two cases with surroundings and ambient air at \(25^{\circ} \mathrm{C}\) : (a) quiescent air and (b) a cross-wind velocity of \(10 \mathrm{~m} / \mathrm{s}\).

Consider a 2-mm-diameter sphere immersed in a fluid at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\). (a) If the fluid around the sphere is quiescent and extensive, show that the conduction limit of heat transfer from the sphere can be expressed as \(N u_{D, \text { cond }}=2\). Hint: Begin with the expression for the thermal resistance of a hollow sphere, Equation \(3.41\), letting \(r_{2} \rightarrow \infty\), and then expressing the result in terms of the Nusselt number. (b) Considering free convection, at what surface temperature will the Nusselt number be twice that for the conduction limit? Consider air and water as the fluids. (c) Considering forced convection, at what velocity will the Nusselt number be twice that for the conduction limit? Consider air and water as the fluids.

A computer code is being developed to analyze a 12.5-mm-diameter, cylindrical sensor used to determine ambient air temperature. The sensor experiences free convection while positioned horizontally in quiescent air at \(T_{\infty}=27^{\circ} \mathrm{C}\). For the temperature range from 30 to \(80^{\circ} \mathrm{C}\), derive an expression for the convection coefficient as a function of only \(\Delta T=\) \(T_{s}-T_{\infty}\), where \(T_{s}\) is the sensor temperature. Evaluate properties at an appropriate film temperature and show what effect this approximation has on the convection coefficient estimate.

A refrigerator door has a height and width of \(H=1 \mathrm{~m}\) and \(W=0.65 \mathrm{~m}\), respectively, and is situated in a large room for which the air and walls are at \(T_{\infty}=T_{\text {sur }}=25^{\circ} \mathrm{C}\). The door consists of a layer of polystyrene insulation \((k=0.03 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) sandwiched between thin sheets of steel \((\varepsilon=0.6)\) and polypropylene. Under normal operating conditions, the inner surface of the door is maintained at a fixed temperature of \(T_{s, i}=5^{\circ} \mathrm{C}\). (a) Estimate the heat gain through the door for the worst case condition corresponding to no insulation \((L=0)\). (b) Compute and plot the heat gain and the outer surface temperature \(T_{s, o}\) as a function of insulation thickness for \(0 \leq L \leq 25 \mathrm{~mm}\).

Free convection occurs between concentric spheres. The inner sphere is of diameter \(D_{i}=50 \mathrm{~mm}\) and temperature \(T_{i}=50^{\circ} \mathrm{C}\), while the outer sphere is maintained at \(T_{o}=20^{\circ} \mathrm{C}\). Air is in the gap between the spheres. What outer sphere diameter is required so that the convection heat transfer from the inner sphere is the same as if it were placed in a large, quiescent environment with air at \(T_{\infty}=20^{\circ} \mathrm{C}\) ?
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