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Chapter 7: Problem 139

Air at \(25^{\circ} \mathrm{C}\) flows over a 5 -cm-diameter, 1.7-m-long smooth pipe with a velocity of \(4 \mathrm{~m} / \mathrm{s}\). A refrigerant at \(-15^{\circ} \mathrm{C}\) flows inside the pipe and the surface temperature of the pipe is essentially the same as the refrigerant temperature inside. The drag force exerted on the pipe by the air is (a) \(0.4 \mathrm{~N}\) (b) \(1.1 \mathrm{~N}\) (c) \(8.5 \mathrm{~N}\) (d) \(13 \mathrm{~N}\) (e) \(18 \mathrm{~N}\) (For air, use \(\nu=1.382 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \rho=1.269 \mathrm{~kg} / \mathrm{m}^{3}\) )

Short Answer

Expert verified
Answer: To determine the drag force acting on the pipe, follow these steps: 1. Calculate the Reynolds number using the given values: \(Re = \dfrac{\rho v d}{\nu}\). 2. Determine the flow regime (laminar, transitional, or turbulent) based on the calculated Reynolds number. 3. Identify the drag coefficient (\(C_D\)) based on the flow regime. 4. Calculate the drag force using the formula: \(F_D = \dfrac{1}{2} \rho v^2 A C_D\). 5. Compare the calculated drag force with the given options to determine the correct answer.

Step by step solution

01

Calculate the Reynolds Number

First, we will compute the Reynolds number using the formula: \(Re = \dfrac{\rho v d}{\nu}\) Where \(Re\) is the Reynolds number, \(\rho\) is the density of air, \(v\) is the flow velocity, \(d\) is the diameter of the pipe, and \(\nu\) is the kinematic viscosity of air. Given values are \(\rho= 1.269 \mathrm{~kg/m}^{3}\), \(v = 4 \mathrm{~m/s}\), \(d = 0.05 \mathrm{~m}\), and \(\nu = 1.382 \times 10^{-5} \mathrm{~m}^{2}/\mathrm{s}\).
02

Determine the Flow Regime

We will now use the Reynolds number calculated in Step 1 to determine the flow regime. As a rule of thumb: - If \(Re < 2000,\) the flow is laminar. - If \(2000 < Re < 4000,\) the flow is transitional. - If \(Re > 4000,\) the flow is turbulent.
03

Determine the Drag Coefficient

Once we know the flow regime, we have to determine the drag coefficient (\(C_D\)) using the appropriate correlation: - For laminar flow (\(Re < 2000\)), \(C_D = \dfrac{24}{Re}\) - For turbulent flow (\(Re > 4000\)), Blasius correlation is \(C_D = 0.079 \cdot Re^{-0.25}\)
04

Calculate the Drag Force

Now that we have the drag coefficient, we can calculate the drag force using the formula: \(F_D = \dfrac{1}{2} \rho v^2 A C_D\) Where \(F_D\) is the drag force, \(A\) is the area of the pipe (cross-sectional), and \(C_D\) is the drag coefficient found in step 3.
05

Choose the Correct Answer

Using the calculated drag force from step 4, we can now compare the value to the given options to choose the correct answer.

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Key Concepts

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

Flow Regime
Understanding the flow regime is crucial in fluid dynamics as it impacts how fluid behaves when moving across surfaces. The flow regime is determined by the Reynolds number (Re), which is calculated using the formula:\[Re = \dfrac{\rho v d}{u}\]where:
  • \(\rho\) is the fluid density,
  • \(v\) is the fluid velocity,
  • \(d\) is the characteristic length (diameter of the pipe in our case),
  • \(u\) is the kinematic viscosity.

The value of the Reynolds number helps categorize the flow:
  • Laminar flow: \(Re < 2000\)
  • Transitional flow: \(2000 < Re < 4000\)
  • Turbulent flow: \(Re > 4000\)
In this exercise, after calculating, you might find the flow to be either laminar, transitional, or turbulent. This classification is essential as it dictates which formulas and correlations to use in subsequent calculations, like determining drag coefficient and drag force.
Drag Coefficient
The drag coefficient (C_D) is a dimensionless number that characterizes the drag or resistance of an object as it moves through a fluid. The drag coefficient depends heavily on the flow regime.For laminar flow, where \(Re < 2000\), the drag coefficient is calculated using a simplified expression: \(C_D = \dfrac{24}{Re}\).This reflects the steady and orderly behavior of fluid molecules that resist motion minimally.

Once the flow becomes turbulent (\(Re > 4000\)), the drag coefficient might be calculated using correlations like the Blasius equation: \(C_D = 0.079 \cdot Re^{-0.25}\).This accounts for the chaotic and irregular motion in turbulent regimes, leading to greater resistance.Understanding which formula to use for the drag coefficient helps in determining how much force will be exerted by the fluid on the object, which is crucial for engineering designs.
Drag Force
Drag force is the resistive force exerted by a fluid on a moving object in the direction opposite to the object's motion. Calculating drag force involves using the drag coefficient obtained from the previous section. The equation to find drag force (F_D) is:\[F_D = \dfrac{1}{2} \rho v^2 A C_D\]where:
  • \(\rho\) is the fluid density,
  • \(v\) is the fluid velocity,
  • \(A\) is the cross-sectional area of the object exposed to the fluid,
  • \(C_D\) is the drag coefficient.

The drag force tells us about energy loss due to friction between the fluid and the object. Engineers need this information to design objects that minimize drag when necessary, such as in automotive and aerospace industries, or to maximize it, as in the case of parachutes.
Kinematic Viscosity
Kinematic viscosity (u) is a measure of a fluid's internal resistance to flow under gravitational forces. It's calculated by dividing the dynamic viscosity of the fluid by its density. In engineering practices, knowing the kinematic viscosity is crucial for determining the Reynolds number and analyzing the flow regime.Kinematic viscosity is expressed in terms of \(m^2/s\), reflecting how easily a fluid flows when subject to stress. In simpler terms, fluids with high kinematic viscosity flow more slowly compared to those with low kinematic viscosity. This property is especially vital in situations where fluid flows over surfaces, as it influences the formation of boundary layers and affects drag calculations.When calculating the Reynolds number, the kinematic viscosity is instrumental in helping determine if the flow is laminar or turbulent, thereby impacting the choice of appropriate models and correlations for further analysis.

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Most popular questions from this chapter

During a plant visit, it was noticed that a 12-m-long section of a \(10-\mathrm{cm}\)-diameter steam pipe is completely exposed to the ambient air. The temperature measurements indicate that the average temperature of the outer surface of the steam pipe is \(75^{\circ} \mathrm{C}\) when the ambient temperature is \(5^{\circ} \mathrm{C}\). There are also light winds in the area at \(10 \mathrm{~km} / \mathrm{h}\). The emissivity of the outer surface of the pipe is \(0.8\), and the average temperature of the surfaces surrounding the pipe, including the sky, is estimated to be \(0^{\circ} \mathrm{C}\). Determine the amount of heat lost from the steam during a 10 -h-long work day. Steam is supplied by a gas-fired steam generator that has an efficiency of 80 percent, and the plant pays \(\$ 1.05 /\) therm of natural gas. If the pipe is insulated and 90 percent of the heat loss is saved, determine the amount of money this facility will save a year as a result of insulating the steam pipes. Assume the plant operates every day of the year for \(10 \mathrm{~h}\). State your assumptions.

Air \((\operatorname{Pr}=0.7, k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) at \(200^{\circ} \mathrm{C}\) flows across 2-cm-diameter tubes whose surface temperature is \(50^{\circ} \mathrm{C}\) with a Reynolds number of 8000 . The Churchill and Bernstein convective heat transfer correlation for the average Nusselt number in this situation is $$ \mathrm{Nu}=0.3+\frac{0.62 \mathrm{Re}^{0.5} \mathrm{Pr}^{0.33}}{\left[1+(0.4 / \mathrm{Pr})^{0.67}\right]^{0.25}} $$ (a) \(8.5 \mathrm{~kW} / \mathrm{m}^{2}\) (b) \(9.7 \mathrm{~kW} / \mathrm{m}^{2}\) (c) \(10.5 \mathrm{~kW} / \mathrm{m}^{2}\) (d) \(12.2 \mathrm{~kW} / \mathrm{m}^{2}\) (e) \(13.9 \mathrm{~kW} / \mathrm{m}^{2}\)

Exposure to high concentration of gaseous ammonia can cause lung damage. To prevent gaseous ammonia from leaking out, ammonia is transported in its liquid state through a pipe \(\left(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i, \text { pipe }}=2.5 \mathrm{~cm}\right.\), \(D_{o, \text { pipe }}=4 \mathrm{~cm}\), and \(\left.L=10 \mathrm{~m}\right)\). Since liquid ammonia has a normal boiling point of \(-33.3^{\circ} \mathrm{C}\), the pipe needs to be properly insulated to prevent the surrounding heat from causing the ammonia to boil. The pipe is situated in a laboratory, where air at \(20^{\circ} \mathrm{C}\) is blowing across it with a velocity of \(7 \mathrm{~m} / \mathrm{s}\). The convection heat transfer coefficient of the liquid ammonia is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Calculate the minimum insulation thickness for the pipe using a material with \(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\), while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\).

A glass \((k=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) spherical tank is filled with chemicals undergoing exothermic reaction. The reaction keeps the inner surface temperature of the tank at \(80^{\circ} \mathrm{C}\). The tank has an inner radius of \(0.5 \mathrm{~m}\) and its wall thickness is \(10 \mathrm{~mm}\). Situated in surroundings with an ambient temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the tank's outer surface is being cooled by air flowing across it at \(5 \mathrm{~m} / \mathrm{s}\). In order to prevent thermal burn on individuals working around the container, it is necessary to keep the tank's outer surface temperature below \(50^{\circ} \mathrm{C}\). Determine whether or not the tank's outer surface temperature is safe from thermal burn hazards.

Hydrogen gas at \(1 \mathrm{~atm}\) is flowing in parallel over the upper and lower surfaces of a 3-m-long flat plate at a velocity of \(2.5 \mathrm{~m} / \mathrm{s}\). The gas temperature is \(120^{\circ} \mathrm{C}\) and the surface temperature of the plate is maintained at \(30^{\circ} \mathrm{C}\). Using the EES (or other) software, investigate the local convection heat transfer coefficient and the local total convection heat flux along the plate. By varying the location along the plate for \(0.2 \leq x \leq 3 \mathrm{~m}\), plot the local convection heat transfer coefficient and the local total convection heat flux as functions of \(x\). Assume flow is laminar but make sure to verify this assumption. 7-31 Carbon dioxide and hydrogen as ideal gases at \(1 \mathrm{~atm}\) and \(-20^{\circ} \mathrm{C}\) flow in parallel over a flat plate. The flow velocity of each gas is \(1 \mathrm{~m} / \mathrm{s}\) and the surface temperature of the 3 -m-long plate is maintained at \(20^{\circ} \mathrm{C}\). Using the EES (or other) software, evaluate the local Reynolds number, the local Nusselt number, and the local convection heat transfer coefficient along the plate for each gas. By varying the location along the plate for \(0.2 \leq x \leq 3 \mathrm{~m}\), plot the local Reynolds number, the local Nusselt number, and the local convection heat transfer coefficient for each gas as functions of \(x\). Discuss which gas has higher local Nusselt number and which gas has higher convection heat transfer coefficient along the plate. Assume flow is laminar but make sure to verify this assumption.
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