Question

A series of experiments were conducted by passing \(40^{\circ} \mathrm{C}\) air over a long 25 -mm-diameter cylinder with an embedded electrical heater. The objective of these experiments was to determine the power per unit length required \((W / L)\) to maintain the surface temperature of the cylinder at \(300^{\circ} \mathrm{C}\) for different air velocities \((V)\). The results of these experiments are given in the following table: $$ \begin{array}{lccccc} \hline V(\mathrm{~m} / \mathrm{s}) & 1 & 2 & 4 & 8 & 12 \\ W / L(\mathrm{~W} / \mathrm{m}) & 450 & 658 & 983 & 1507 & 1963 \\ \hline \end{array} $$ (a) Assuming a uniform temperature over the cylinder, negligible radiation between the cylinder surface and surroundings, and steady-state conditions, determine the convection heat transfer coefficient \((h)\) for each velocity \((V)\). Plot the results in terms of $h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\( vs. \)V(\mathrm{~m} / \mathrm{s})$. Provide a computer- generated graph for the display of your results, and tabulate the data used for the graph. (b) Assume that the heat transfer coefficient and velocity can be expressed in the form \(h=C V^{n}\). Determine the values of the constants \(C\) and \(n\) from the results of part (a) by plotting \(h\) vs. \(V\) on log-log coordinates and choosing a \(C\) value that assures a match at \(V=1 \mathrm{~m} / \mathrm{s}\) and then varying \(n\) to get the best fit.

Short answer

Based on the provided solution for determining the convection heat transfer coefficient (h) for each air velocity (V) and finding the constants C and n for the equation \(h = CV^n\), briefly explain the importance of these constants in understanding heat transfer. The constants C and n in the equation \(h = CV^n\) play an essential role in understanding the relationship between the convective heat transfer coefficient (h) and the air velocity (V). In this context, C represents the proportionality constant, which influences the overall heat transfer for a specific system (e.g., heated cylinder). On the other hand, n reflects the power-law exponent that describes how sensitive the heat transfer coefficient (h) is to changes in the velocity (V). By evaluating these constants, we gain insights into how the convective heat transfer is influenced by the air velocity, allowing us to optimize the heat transfer process for various applications in engineering and energy management.

Step-by-step solution

Recall the convection heat transfer equation

We can describe the convection heat transfer between the cylinder and the air as: \(Q = hA(T_s - T_\infty)\), where \(Q\) is the heat transfer rate, \(h\) is the convective heat transfer coefficient, \(A\) is the surface area of the cylinder, \(T_s\) is the temperature of the cylinder's surface, and \(T_\infty\) is the temperature of the surrounding air.

Calculate the heat transfer rate per unit length

We are given the power per unit length \((W/L)\), which can be represented as \(q = Q/L\). Therefore, we can calculate the heat transfer rate \((Q)\) for each air velocity as follows: \(Q = (W/L) * L\). We do not need to calculate \(Q\) explicitly because we will use \(W/L\) in the next steps.

Calculate the surface area of the cylinder per unit length

The surface area of the cylinder per unit length \((A/L)\) can be calculated as: \(\frac{A}{L} = \pi D\), where \(D\) is the diameter of the cylinder.

Calculate the convection heat transfer coefficient \((h)\) for each velocity \((V)\)

For each velocity, we can rearrange the convective heat transfer equation and solve for \(h\): \(h = \frac{Q}{A(T_s - T_\infty)}\). Substitute the values for each \(V\), \(W/L\), and \(A/L\) and plug in the given temperatures \((T_s = 300 ^\circ\mathrm{C}\) and \(T_\infty = 40 ^\circ\mathrm{C})\) to find \(h\) values.

Plot the graph of \(h\) vs. \(V\)

Use a computer software to create a scatter plot of \(h\) vs. \(V\) with the calculated \(h\) values and the given \(V\) values from the table.

Find the constants \(C\) and \(n\) by using a log-log plot

Create a log-log plot of the calculated \(h\) values as a function of \(V\). Find the best-fit line, with the equation \(h = CV^n\), by choosing a \(C\) value that matches at \(V = 1 \mathrm{~m/s}\) and adjusting \(n\) for the best fit. Following these steps, you will determine the convection heat transfer coefficient for each air velocity and the fitting constants \(C\) and \(n\) for the convection heat transfer expression.