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Chapter 6: Problem 74

Saturated liquid water at \(5^{\circ} \mathrm{C}\) is flowing over a flat plate at a velocity of \(1 \mathrm{~m} / \mathrm{s}\). Using EES (or other) software, determine the effect of the location along the plate \((x)\) on the velocity and thermal boundary layer thicknesses. By varying \(x\) for \(0

Short Answer

Expert verified
Answer: As the distance along the flat plate (\(x\)) increases, both the velocity and thermal boundary layer thicknesses grow. However, the velocity boundary layer thickness is typically larger than the thermal boundary layer, indicating that the momentum and thermal effects diffuse at different rates, with the velocity effect spreading more rapidly than the thermal effect.

Step by step solution

01

Calculate the properties of water at the given conditions

First, we need to obtain the properties of saturated liquid water at \(5^{\circ} \mathrm{C}\). Using software or a steam table, we can find the density (\(\rho\)), viscosity (\(\mu\)), specific heat (\(c_p\)), and thermal conductivity (\(k\)) of the water.
02

Determine the Reynolds number (\(\mathrm{Re}\))

Using the properties obtained in Step 1, we can calculate the Reynolds number at each location along the plate (\(x\)) using the formula: \(\mathrm{Re}=\frac{\rho V x}{\mu}\) where \(V\) is the velocity of the water, and \(x\) is the distance from the leading edge of the plate.
03

Calculate the velocity and thermal boundary layer thicknesses

The velocity and thermal boundary layers can be calculated using the following formulas: Velocity boundary layer thickness (\(\delta\)): \(\delta=\frac{5x}{\sqrt{\mathrm{Re}}}\) Thermal boundary layer thickness (\(\delta_t\)): \(\delta_t=\frac{x}{\sqrt{\mathrm{Re} \cdot \mathrm{Pr}}}\) where \(\mathrm{Pr}\) is the Prandtl number calculated using: \(\mathrm{Pr}=\frac{\mu c_p}{k}\)
04

Vary \(x\) and determine the boundary layers for each value

Now we can vary the distance \(x\) along the plate from \(0\) to \(0.5 \mathrm{~m}\) and calculate the Reynolds number, velocity boundary layer thickness (\(\delta\)), and thermal boundary layer thickness (\(\delta_t\)) for each location.
05

Plot the results

With the boundary layer thicknesses calculated as a function of \(x\), we can now plot the results. The x-axis will represent the distance (\(x\)) along the plate, and the y-axis will show the boundary layer thicknesses. Two curves will be displayed on the plot, one for the velocity boundary layer thickness (\(\delta\)) and another for the thermal boundary layer thickness (\(\delta_t\)).
06

Discuss the results

Analyzing the plot, we can see how the velocity and thermal boundary layer thicknesses change as we move along the flat plate (\(x\)). As \(x\) increases, both the velocity and thermal boundary layers grow. It can be observed that the velocity boundary layer thickness is typically larger than the thermal boundary layer. This is because the momentum and thermal effects diffuse at different rates in the fluid, with the velocity effect spreading more rapidly than the thermal effect. This provides insight into the mixing characteristics and heat transfer behavior of the fluid flow over the flat plate.

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Most popular questions from this chapter

What is external forced convection? How does it differ from internal forced convection? Can a heat transfer system involve both internal and external convection at the same time? Give an example.

A 5 -mm-thick stainless steel strip \((k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=\) \(8000 \mathrm{~kg} / \mathrm{m}^{3}\), and \(\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) is being heat treated as it moves through a furnace at a speed of \(1 \mathrm{~cm} / \mathrm{s}\). The air temperature in the furnace is maintained at \(900^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the furnace length is \(3 \mathrm{~m}\) and the stainless steel strip enters it at \(20^{\circ} \mathrm{C}\), determine the surface temperature gradient of the strip at mid-length of the furnace. Hint: Use the lumped system analysis to calculate the plate surface temperature. Make sure to verify the application of this method to this problem.

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