91Ó°ÊÓ

Experiments have been conducted to determine local heat transfer coefficients for flow perpendicular to a long, isothermal bar of rectangular cross section. The bar is of width \(c\) parallel to the flow, and height \(d\) normal to the flow. For Reynolds numbers in the range \(10^{4} \leq R_{d} \leq 5 \times 10^{4}\), the face-averaged Nusselt numbers are well correlated by an expression of the form The values of \(C\) and \(m\) for the front face, side faces, and back face of the rectangular rod are found to be the following: \begin{tabular}{llll} \hline Face & cld & \(\boldsymbol{C}\) & \(\boldsymbol{m}\) \\ \hline Front & \(0.33 \leq\) cld \(51.33\) & \(0.674\) & \(1 / 2\) \\ Side & \(0.33\) & \(0.153\) & \(2 / 3\) \\ Side & \(1.33\) & \(0.107\) & \(2 / 3\) \\ Back & \(0.33\) & \(0.174\) & \(2 / 3\) \\ Back & \(1.33\) & \(0.153\) & \(2 / 3\) \\ \hline \end{tabular} Determine the value of the average heat transfer coefficient for the entire exposed surface (that is, averaged over all four faces) of a \(c=40\)-mm-wide, \(d=30\)-mm-tall rectangular rod. The rod is exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{x}=300 \mathrm{~K}\). Provide a plausible explanation of the relative values of the face-averaged heat transfer coefficients on the front, side, and back faces.

Step by step solution

01

Calculate the Reynolds number

To determine the Reynolds number (\(R_d\)), we use the following formula: \[R_d = \frac{V d}{\nu}\] Where \(V\) is the flow velocity, \(d\) is the height of the rod normal to the flow, and \(\nu\) is the kinematic viscosity of air. The kinematic viscosity of air at room temperature (\(T_x = 300K\)) is approximately \(15 \times 10^{-6} m^2/s\). We are given \(V=10 m/s\) and \(d=30mm=0.03m\), so we can calculate the Reynolds number: \[R_d = \frac{10 \times 0.03}{15 \times 10^{-6}}\]

02

Determine the Nusselt number

Since the Reynolds number is within the given range, we can use the provided expressions for the Nusselt numbers. We will determine the Nusselt number for each face (\(Nu_{Front}\), \(Nu_{Side_1}\), \(Nu_{Side_2}\), \(Nu_{Back_1}\), and \(Nu_{Back_2}\)) using the values of \(C\), \(m\), and \(R_d\) from the table: \[Nu_{Face} = C \cdot R_d^m\]

03

Calculate the heat transfer coefficients

Now that we have found the Nusselt numbers, we can calculate the heat transfer coefficients (\(h_{Front}\), \(h_{Side_1}\), \(h_{Side_2}\), \(h_{Back_1}\), and \(h_{Back_2}\)) for each face using the definition of Nusselt number: \[Nu_{Face} = \frac{h_{Face} \cdot d}{k}\] Here, \(k\) is the thermal conductivity of the fluid (air). For air at room temperature, \(k \approx 0.0262 W/(m \cdot K)\).

04

Find the average heat transfer coefficient

To find the average heat transfer coefficient for the entire exposed surface, we need to calculate the weighted average of the heat transfer coefficients: \[h_{avg} = \frac{h_{Front}A_{Front} + h_{Side_1}A_{Side_1} + h_{Side_2}A_{Side_2} + h_{Back_1}A_{Back_1} + h_{Back_2}A_{Back_2}}{A_{total}}\] Where \(A_{total}\) is the total exposed surface area of the rod.

05

Provide explanation to face-averaged heat transfer coefficients

The difference in the face-averaged heat transfer coefficients can be explained based on the direction of the flow and the nature of the airflow over the rod surfaces. The front face is directly impacted by the flow, generating a turbulent boundary layer, causing higher heat transfer on that face. On the side faces, the flow is less turbulent and the air travels across a larger area, resulting in lower heat transfer coefficients. For the back face, the flow has already interacted with the front and side faces, and this results in lower face-averaged heat transfer coefficient as compared to the front face.

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Key Concepts

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

Understanding the Reynolds Number

The Reynolds number (\textbf{Re}) is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces and is used to determine whether a fluid flow will be laminar or turbulent.

\textbf{Re} is given by the equation \(Re = \frac{\rho V L}{\mu}\) or in terms of kinematic viscosity \(u\), \(Re = \frac{VL}{u}\), where \(\rho\) is the fluid density, \(V\) is the velocity of the fluid with respect to a surface, \(L\) is a characteristic length, and \(\mu\) and \(u\) are the dynamic and kinematic viscosities, respectively.

In the context of the problem, the Reynolds number is calculated using the height of the rectangular rod as the characteristic length (\textbf{d}) and the velocity of the air flow (\textbf{V}). This number helps us classify the airflow around the rod as laminar or turbulent. In turn, this influences the convective heat transfer process, as turbulent flow generally enhances heat transfer due to the increased mixing and disruption of the thermal boundary layer.

Deciphering the Nusselt Number

The Nusselt number (\textbf{Nu}) is another dimensionless parameter in heat transfer that measures the enhancement of heat transfer through a fluid as a result of convection, compared to heat transfer through conduction alone.

The Nusselt number is defined as \(Nu = \frac{hd}{k}\), where \(h\) represents the convective heat transfer coefficient, \(d\) is the characteristic length (in this case, the height of the rod), and \(k\) is the thermal conductivity of the fluid. Higher \textbf{Nu} values indicate more effective convective heat transfer, as they imply a larger contribution from convection relative to conduction.

By using this concept, with given constants \textbf{C} and \textbf{m}, and the previously calculated Reynolds number, we can determine the heat transfer characteristics for each face of the rod. The correlation provided in the exercise essentially empowers us to establish a relationship between the Nusselt number and the flow conditions (embodied in the Reynolds number) for the specific geometry of the rod.

Convective Heat Transfer Coefficients

Convective heat transfer measures the movement of heat between a solid surface and a fluid (liquid or gas) when they are at different temperatures. When the fluid moves, it carries heat with it, which results in convective heat transfer. The convective heat transfer coefficient (\textbf{h}) quantifies how effective this process is and is dependent on a number of factors including fluid velocity, viscosity, and temperature.

To calculate this coefficient, one needs the Nusselt number we learned about previously. The convective heat transfer coefficient is inversely proportional to the thermal boundary layer thickness: thinner layers result in higher \(h\) values because the resistance to heat transfer is lower.

In our problem, after determining the Nusselt number for each face of the rod, we use it to find the associated heat transfer coefficients. By finding a weighted average of these coefficients, considering the area each face takes up, we arrive at the average heat transfer coefficient for the entire rod. This value represents the overall efficiency of the rod in transferring heat to the air flow in cross-section, considering the distinct heat transfer properties of each face.