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Chapter 7: Problem 75

Consider a sphere with a diameter of \(20 \mathrm{~mm}\) and a surface temperature of \(60^{\circ} \mathrm{C}\) that is immersed in a fluid at a temperature of \(30^{\circ} \mathrm{C}\) and a velocity of \(2.5 \mathrm{~m} / \mathrm{s}\). Calculate the drag force and the heat rate when the fluid is (a) water and (b) air at atmospheric pressure. Explain why the results for the two fluids are so different.

Short Answer

Expert verified
The drag force and heat rate for a sphere submerged in water and air were calculated using fluid properties, Reynolds number, drag coefficient, and Nusselt number. The values are significantly different due to the differences in density, viscosity, thermal conductivity, and heat capacity between the two fluids. Water has higher density and viscosity, leading to higher Reynolds number, drag coefficient and ultimately higher drag force. Similarly, the higher thermal conductivity of water results in a higher Nusselt number and convective heat transfer coefficient, leading to a higher heat rate.

Step by step solution

01

Find fluid properties

Look up the following fluid properties for both water and air at atmospheric pressure and given temperature: 1. Density (\( \rho \)) 2. Viscosity (\( \mu \)) 3. Thermal conductivity (\( k \)) 4. Specific heat capacity (\( c_p \)) Using a reference table or online resources, find the properties for both fluids: (a) Water at \(30^{\circ} \mathrm{C}\) (b) Air at \(30^{\circ} \mathrm{C}\) Remember that these values are required for further calculations.
02

Calculate the Reynolds number

Determine the Reynolds number (Re) using the following equation: \[ Re = \frac{\rho vD}{\mu} \] where: - \(D\) is the diameter of the sphere (\(20 \, mm = 0.02 \, m\)) - \(v\) is the velocity of the fluid (\(2.5 \, m/s\)) - \(\rho\) and \(\mu\) are the density and viscosity of the fluid Calculate the Reynolds number for both fluids: (a) Water (b) Air
03

Calculate the drag coefficient

Using the Reynolds number, find the drag coefficient (\(C_D\)) for the sphere. The following correlation can be used for a sphere: \[C_D = \frac{24}{Re} \left(1 + \frac{1}{6}Re^{2/3}\right)\] Calculate the drag coefficient for both fluids: (a) Water (b) Air
04

Calculate the drag force

Determine the drag force (\(F_D\)) using the following equation: \[F_D = \frac{1}{2} \rho v^2 C_D A\] where: - \(A\) is the projected area of the sphere given by \(A = \frac{\pi D^2}{4}\) Calculate the drag force for both fluids: (a) Water (b) Air
05

Calculate the Nusselt number

Determine the Nusselt number (\(Nu\)) using the following correlation for a sphere: \[Nu = 2 + 0.4 Re^{1/2}Pr^{1/3}\] where \(Pr\) is the Prandtl number given by: \[Pr=\frac{\mu c_p}{k}\] Calculate the Nusselt number for both fluids: (a) Water (b) Air
06

Calculate the heat rate

Determine the heat rate (\(q\)) using the following equation: \[q = h A \Delta T\] where: - \(h\) is the convective heat transfer coefficient given by \(h = \frac{k}{D} Nu\) - \(\Delta T\) is the temperature difference between the sphere and the fluid (\(60^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C} = 30 \, K\)) Calculate the heat rate for both fluids: (a) Water (b) Air
07

Compare the results for water and air

Compare the drag force and heat rate calculated for water and air. Explain the differences by considering the differences in fluid properties like density, viscosity, thermal conductivity, and heat capacity. Discuss the roles played by Reynolds number, drag coefficient, Nusselt number, and convective heat transfer coefficient in determining the drag force and the heat rate.

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Key Concepts

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

Drag Force
Drag force is a resistance force caused by the motion of a body through a fluid, such as air or water. It acts in the opposite direction to the object's motion and is influenced by several factors:
  • Velocity: The force increases with the square of the velocity of the object.
  • Surface Area: A larger area leads to higher drag.
  • Fluid Properties: Density and viscosity of the fluid affect the drag.
The drag force, \(F_D\), can be calculated using:\[F_D = \frac{1}{2} \rho v^2 C_D A\]Here, \(C_D\) is the drag coefficient, \(\rho\) is the density of the fluid, \(v\) is the velocity, and \(A\) is the projected area. A sphere's area is given by \(A = \frac{\pi D^2}{4}\).
By calculating the drag force for water and air, we find differences due to variations in their densities and viscosities. Water, being denser, often exerts a larger drag force compared to air.
Reynolds Number
The Reynolds number (Re) helps predict flow patterns in different fluid flow situations. It is a dimensionless quantity that provides insight into whether the flow will be laminar or turbulent. It is calculated as follows:
\[Re = \frac{\rho v D}{\mu}\]Where:
  • \(\rho\) is the fluid density.
  • \(v\) is the velocity of the fluid.
  • \(D\) is the characteristic length, such as the diameter of the object.
  • \(\mu\) is the fluid's viscosity.
In general, a low Reynolds number indicates laminar flow, and a high Reynolds number suggests turbulent flow.
For spheres, the drag coefficient changes with the Reynolds number, affecting the calculation of drag force. Water usually has a higher Reynolds number due to its higher density and lower viscosity compared to air, influencing the flow characteristics around the sphere.
Nusselt Number
The Nusselt number (Nu) is a dimensionless number that describes the rate of heat transfer through convection compared to conduction. For a sphere, it's calculated using:
\[Nu = 2 + 0.4 Re^{1/2} Pr^{1/3}\]Where:
  • \(Re\) is the Reynolds number.
  • \(Pr\) is the Prandtl number, given by \(Pr = \frac{\mu c_p}{k}\), with \(c_p\) as specific heat and \(k\) as thermal conductivity.
A higher Nusselt number indicates more efficient convective heat transfer. In our exercise, calculate this for both water and air to find differences in heat transfer rates.
Typically, water with its higher thermal conductivity and specific heat capacity results in a larger Nusselt number compared to air, enabling better heat transfer under similar conditions.
Convective Heat Transfer Coefficient
The convective heat transfer coefficient, \(h\), quantifies the heat transferred between a surface and a fluid per unit area per unit temperature difference. It's essential in calculating the heat rate and is found using:\[h = \frac{k}{D} Nu\]Where:
  • \(k\) is the thermal conductivity of the fluid.
  • \(D\) is the diameter of the sphere.
  • \(Nu\) is the Nusselt number.
By knowing \(h\), we can calculate the heat rate \(q\) using:\[q = h A \Delta T\]Here, \(A\) is the surface area and \(\Delta T\) is the temperature difference between the sphere and fluid.
Water, with its higher heat transfer coefficient due to its thermal properties, typically provides more efficient heat transfer than air. This results in different heat rates for the sphere when submersed in these fluids.

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Most popular questions from this chapter

Consider the packed bed of aluminum spheres described in Problem \(5.12\) under conditions for which the bed is charged by hot air with an inlet velocity of \(V=1 \mathrm{~m} / \mathrm{s}\) and temperature of \(T_{g, i}=300^{\circ} \mathrm{C}\), but for which the convection coefficient is not prescribed. If the porosity of the bed is \(\varepsilon=0.40\) and the initial temperature of the spheres is \(T_{i}=25^{\circ} \mathrm{C}\), how long does it take a sphere near the inlet of the bed to accumulate \(90 \%\) of its maximum possible energy?

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