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Chapter 6: Problem 70

Consider a laminar boundary layer flow over a flat plate. Determine the \(\delta / \delta_{t}\) ratios for air (at 1 atm), liquid water, isobutane, and engine oil, and mercury. Evaluate all properties at \(50^{\circ} \mathrm{F}\).

Short Answer

Expert verified
Answer: The ratio of boundary layer thickness (\(\delta\)) to thermal boundary layer thickness (\(\delta_t\)) indicates the relative thicknesses of the momentum and thermal boundary layers during laminar flow for each fluid at 50°F. Higher ratios suggest a larger difference between the two layers, while lower ratios indicate more comparable thicknesses for the momentum and thermal boundary layers. This information can help in understanding the fluid flow and heat transfer characteristics in various applications involving these fluids at 50°F.

Step by step solution

01

Introduce the boundary layer and thermal boundary layer thicknesses

The boundary layer thickness, \(\delta\), is a measure of how far fluid properties are affected by the presence and physical characteristics of the solid boundary (e.g., a flat plate). The thermal boundary layer thickness, \(\delta_t\), is a measure of how far fluid properties are affected by the temperature variations of the solid boundary.
02

Find the ratio of boundary layer thickness to thermal boundary layer thickness

To find the ratio \(\delta / \delta_t\), we need to use the Prandtl Number, \(\mathrm{Pr}\), which is the ratio of momentum diffusivity to thermal diffusivity. The Prandtl Number is given by: $$ \mathrm{Pr} = \frac{\text{momentum diffusivity}}{\text{thermal diffusivity}} = \frac{\nu}{\alpha} $$ Where \(\nu\) is the kinematic viscosity and \(\alpha\) is the thermal diffusivity. The relationship between boundary layer thickness and thermal boundary layer thickness can be written as: $$ \frac{\delta}{\delta_t} = \sqrt{\mathrm{Pr}} $$ So, to find the ratio \(\delta / \delta_t\), we calculate the Prandtl number using given fluid properties and find their square root.
03

Gather fluid properties at the given temperature

In order to calculate the Prandtl number, we need to find properties like dynamic viscosity (\(\mu\)), specific heat (\(c_p\)), density (\(\rho\)), and thermal conductivity (\(k\)) for air, water, isobutane, engine oil, and mercury at \(50^{\circ} \mathrm{F}\) (or \(10^{\circ} \mathrm{C}\)). These properties can be found in fluid properties tables or handbooks found online or in a library.
04

Calculate the Prandtl number

Once the fluid properties are obtained, calculate the kinematic viscosity (\(\nu\)) and thermal diffusivity (\(\alpha\)) using the following formulas: $$ \nu = \frac{\mu}{\rho} \quad \text{and} \quad \alpha = \frac{k}{\rho c_p} $$ Then, calculate the Prandtl number for each fluid: $$ \mathrm{Pr} = \frac{\nu}{\alpha} $$
05

Determine the ratio of boundary layer thickness to thermal boundary layer thickness

Now that we have the Prandtl number for each fluid, calculate the ratio \(\delta / \delta_t\) using: $$ \frac{\delta}{\delta_t} = \sqrt{\mathrm{Pr}} $$ Perform this calculation for all the fluids (air, water, isobutane, engine oil, and mercury) and compare the ratios.
06

Interpret the results

The \(\delta / \delta_t\) ratios provide information about the relative thicknesses of the boundary layer and the thermal boundary layer during laminar flow for each fluid. Higher ratios indicate a larger difference between the two layers, while lower ratios indicate a more comparable thickness for the momentum and thermal boundary layers. The ratios can help in understanding the fluid flow and heat transfer characteristics in various applications involving these fluids at \(50^{\circ} \mathrm{F}\).

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Most popular questions from this chapter

For the same initial conditions, one can expect the laminar thermal and momentum boundary layers on a flat plate to have the same thickness when the Prandtl number of the flowing fluid is (a) Close to zero (b) Small (c) Approximately one (d) Large (e) Very large

Consider two identical small glass balls dropped into two identical containers, one filled with water and the other with oil. Which ball will reach the bottom of the container first? Why?

The coefficient of friction \(C_{f}\) for a fluid flowing across a surface in terms of the surface shear stress, \(\tau_{s}\), is given by (a) \(2 \rho V^{2} / \tau_{w}\) (b) \(2 \tau_{w} / \rho V^{2}\) (c) \(2 \tau_{w} / \rho V^{2} \Delta T\) (d) \(4 \tau_{w} / \rho V^{2}\) (e) None of them

Consider a 5-cm-diameter shaft rotating at \(4000 \mathrm{rpm}\) in a \(25-\mathrm{cm}\)-long bearing with a clearance of \(0.5 \mathrm{~mm}\). Determine the power required to rotate the shaft if the fluid in the gap is (a) air, (b) water, and (c) oil at \(40^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

In which mode of heat transfer is the convection heat transfer coefficient usually higher, natural convection or forced convection? Why?
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