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A streamlined strut supporting a bearing housing is exposed to a hot airflow from an engine exhaust. It is necessary to run experiments to determine the average convection heat transfer coefficient \(\bar{h}\) from the air to the strut in order to be able to cool the strut to the desired surface temperature \(T_{x}\). It is decided to run mass transfer experiments on an object of the same shape and to obtain the desired heat transfer results by using the heat and mass transfer analogy. The mass transfer experiments were conducted using a half-size model strut constructed from naphthalene exposed to an airstream at \(27^{\circ} \mathrm{C}\). Mass transfer measurements yielded these results: \begin{tabular}{rr} \hline \multicolumn{1}{c}{\(\boldsymbol{\boldsymbol { e } _ { \boldsymbol { L } }}\)} & \(\overline{\boldsymbol{S h}}_{\boldsymbol{L}}\) \\ \hline 60,000 & 282 \\ 120,000 & 491 \\ 144,000 & 568 \\ 288,000 & 989 \\ \hline \end{tabular} (a) Using the mass transfer experimental results, determine the coefficients \(C\) and \(m\) for a correlation of the form \(\overline{S h}_{L}=C R e_{L}^{m} S c^{1 / 3}\). (b) Determine the average convection heat transfer coefficient \(\bar{h}\) for the full-sized strut, \(L_{H}=60 \mathrm{~mm}\), when exposed to a free stream airflow with \(V=60 \mathrm{~m} / \mathrm{s}\), \(T_{\infty}=184^{\circ} \mathrm{C}\), and \(p_{\infty}=1 \mathrm{~atm}\) when \(T_{s}=70^{\circ} \mathrm{C}\). (c) The surface area of the strut can be expressed as \(A_{s}=2.2 L_{H} \cdot l\), where \(l\) is the length normal to the page. For the conditions of part (b), what is the change in the rate of heat transfer to the strut if the characteristic length \(L_{H}\) is doubled?

Step by step solution

01

Write down the given correlation

We are given a correlation of the form \(\overline{S h}_{L}=C R e_{L}^{m} S c^{1 / 3}\).

02

Express variables as provided in the experimental results

We are provided with the experimental results having \(\overline{S h}_L\) for various \(Re_L\). The Schmidt number \(Sc\) remains constant for the given fluid. So we can rewrite the given correlation as: \[\overline{S h}_{L}=C R e_{L}^{m}\]

03

Use the least-squares method to find coefficients C and m

Let's express the given correlation as: \[\log{\overline{Sh}_L} = \log{C} + m\log{Re_L}\] Now, treating \(\log{\overline{Sh}_L}\) as Y and \(\log{Re_L}\) as X, we can use the least-squares method to find the coefficients \(C\) and \(m\) by solving two equations with two unknowns. The least-squares method gives us the following equations: \[\sum Y = n\log{C} + m\sum X\] \[\sum XY = \log{C}\sum X + m\sum X^2\] where n is the number of experimental results. Plug in provided experimental results, to find the values of the coefficients \(C\) and \(m\). (b) Determining the average convection heat transfer coefficient \(\bar{h}\)

04

Use heat and mass transfer analogy

The given mass transfer correlation can be translated into a heat transfer correlation using the heat and mass transfer analogy. The analogy is given as follows: \[\overline{S h}_{L} \equiv \overline{N u}_{L}\] \[\overline{N u}_{L}=C R e_{L}^{m} P r^{1 / 3}\] where \(\overline{N u}_{L}\) is the Nusselt number and \(P r\) is the Prandtl number.

05

Calculate \(\bar{h}\) using Nusselt number and provided conditions

We can express the Nusselt number as: \[\bar{h} = \frac{\overline{N u}_{L} k}{L_H}\] Calculate the Nusselt number using the provided conditions for the full-sized strut, i.e., \(V = 60~m/s\), \(T_\infty = 184^\circ C\), and \(p_\infty = 1~atm\). After plugging all values into the equation, we can find the average convection heat transfer coefficient, \(\bar{h}\). (c) Change in the rate of heat transfer when the characteristic length is doubled

06

Express the heat transfer rate

The heat transfer rate can be expressed as: \[\dot{Q} = \bar{h} A_s (T_s - T_\infty)\]

07

Calculate the rate of heat transfer for both cases

Calculate the rate of heat transfer for \(L_H=60~mm\) and for double the length (\(2L_H\)). Use the given formula for the surface area \(A_s=2.2 L_H \cdot l\).

08

Determine the change in heat transfer rates

Subtract the initial rate of heat transfer from the rate of heat transfer when the characteristic length is doubled to find the change in the rate of heat transfer.

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Key Concepts

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

Heat and Mass Transfer Analogy

The concept of heat and mass transfer analogy is based on the similarities between the transfer of heat and the transfer of mass. This analogy is often used to simplify complex problems by establishing correlations that can predict behavior in one domain by observing the other. This is useful in various engineering applications, where running experiments for both heat and mass transfer individually can be cumbersome and expensive. The critical point of this analogy is that under certain circumstances, the equations governing heat and mass transfer are mathematically similar. For instance, the Nusselt number (heat transfer) and the Sherwood number (mass transfer) can be analogous when phenomena such as molecular diffusion are similar in both processes. This means:

• Heat transfer by convection can be inferred from mass transfer data and vice versa.
• You can use Sherwood number correlations to obtain Nusselt numbers, reducing the need for separate experiments.

This analogy becomes incredibly valuable in experimental settings, allowing engineers to design more efficient systems by understanding one domain and inferring characteristics in another.

Nusselt Number

The Nusselt number (\( ar{N u}_{L} \)) is a dimensionless parameter used in heat transfer to quantify the enhancement of heat transfer through a fluid as compared to conduction. It essentially measures the effectiveness of convective heat transfer occurring within the fluid or because of the fluid flow. The definition is given by: \[\bar{N u}_{L} = rac{hL}{k}\]Where:

• \( h \) is the convective heat transfer coefficient
• \( L \) is the characteristic length
• \( k \) is the thermal conductivity of the fluid

Using the Nusselt number, you can predict how efficiently heat will be transferred from a solid surface to a fluid. Higher Nusselt numbers indicate higher convective heat transfer. It plays a vital role in designing cooling and heating systems, for instance, in electronic devices, where maximizing convective heat transfer is crucial.

Reynolds Number

Reynolds number (\( Re_L \)) is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is critical in determining whether the flow is laminar or turbulent, both of which have direct influences on heat and mass transfer. It is calculated as:\[Re_L = rac{ ho VL}{u}\]Where:

• \( ho \) is the fluid density
• \( V \) is the velocity of the fluid
• \( L \) is the characteristic length
• \( u \) is the kinematic viscosity

Reynolds number helps identify the regime of the fluid flow:

• Low \( Re_L \) signifies laminar flow, characterized by smooth and orderly flow, which is less efficient for mixing and heat transfer.
• High \( Re_L \) indicates turbulent flow, which enhances mixing and improves heat transfer efficiency due to increased convection.

Understanding the flow regime is crucial for optimizing heat exchange in applications such as radiator designs, where turbulence is often desired to improve heat dissipation.

Experimental Correlation

Experimental correlation refers to the relationship between different dimensionless numbers obtained through empirical methods. These correlations allow engineers to predict heat and mass transfer processes in real-world scenarios.When conducting experiments, like the one described involving the half-size model strut, the goal is to obtain reliable data that can be generalized in a usable form for larger or different systems. This involves establishing a correlation of the form: \[\overline{S h}_{L}=C R e_{L}^{m} S c^{1 / 3}\]Here, \(C \) and \(m \) are constants derived from experimental data. The process includes:

• Gathering measurements under controlled conditions
• Using techniques like the least-squares method to fit data and derive constant values (\( C \) and \( m \) )
• Utilizing the correlation to forecast results for various conditions without running exhaustive tests.

Experimental correlations are especially useful as they bridge the gap between theoretical analysis and practical application, making them indispensable in design and analysis of engineering systems. They help predict outcomes, streamline design processes, and improve system efficiency.