Question

A series of experiments were conducted by passing ${40}^{\circ }\mathrm{C}$ air over a long $25\mathrm{mm}$ diameter cylinder with an embedded electrical heater. The objective of these experiments was to determine the power per unit length required $(\dot{W}/L)$ 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: $$\overline{)\begin{array}{lccccc}V(\mathrm{m}/\mathrm{s})& 1& 2& 4& 8& 12\\ \dot{W}/L(\mathrm{W}/\mathrm{m})& 450& 658& 983& 1507& 1963\end{array}}$$ (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 $h(\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K})$ 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 of $h=C{V}^m$. 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

Answer: To determine the convection heat transfer coefficient (h) and constants C and n, follow these steps: 1. Calculate the convection heat transfer coefficient (h) for each velocity (V) using the formula h = (power per unit length) / (surface area × temperature difference). 2. Tabulate the calculated results for each velocity (V) and heat transfer coefficient (h). 3. Plot a graph with x-axis as V(m/s) and y-axis as h(W/m²·K) using the computed data points. 4. To find the constants C and n, plot h vs. V on log-log coordinates, choose a C value that matches the data for V = 1 m/s, and vary the value of n to obtain the best fit line for the given data.

Step-by-step solution

Calculating the convection heat transfer coefficient (h)

For each velocity (V), we will calculate the convection heat transfer coefficient (h) by using the formula: $$h=\frac{\dot{W}/L}{A(\mathrm{\Delta }T)}$$ Here, $\dot{W}/L$ = power per unit length (given in the table), A = surface area of the cylinder, ΔT = temperature difference between the surface and air. First, we should calculate the surface area of the cylinder: A = πD × L (For per unit length, L = 1) A = π × 0.025 m Now, we should calculate the temperature difference: ΔT = (300°C - 40°C) = 260°C = 260 K For each velocity, we can calculate 'h' using the above formula.

Tabulating the results

Now, we will tabulate the results of the calculated convection heat transfer coefficients (h) for each velocity (V): | V (m/s) | Power per unit length (W/m) | h (W/m²·K) | |---------|---------------------------------|--------------| | 1 | 450 | h1 | | 2 | 658 | h2 | | 4 | 983 | h3 | | 8 | 1507 | h4 | | 12 | 1963 | h5 |

Plotting the graph

Plot a graph with the x-axis as V (m/s) and the y-axis as h(W/m²·K). Plot the computed data points (V, h) on the graph. Make sure to use a computer-generated graph for the display.

Determining Constants C and n

We have the relation, h = CV^n. To find the constants C and n, we must plot h vs. V on log-log coordinates. Choose a C value that matches the data for V = 1 m/s. Now, vary the value of n to obtain the best fit line for the given data. Once we have the values of constants C and n, the relation h = CV^n can be used to predict the convection heat transfer coefficient for different air velocities.

Key concepts

Convective Heat Transfer

Convective heat transfer refers to the mode of energy transfer between a solid surface and the adjacent fluid when there is relative motion between them. This process is governed by the convection heat transfer coefficient, denoted as $h$, which quantifies the rate at which heat is transferred per unit area and per unit temperature difference between the surface and the fluid.

In the context of the given exercise, the objective was to ascertain the required power per unit length $(\dot{W}/L)$ that maintains a cylinder's surface temperature at a constant level when air at different velocities is passed over it. Using the formula $h=\frac{\dot{W}/L}{A(\mathrm{\Delta }T)}$, where $A$ is the surface area and $\mathrm{\Delta }T$ is the temperature difference, students can compute the value of $h$ for varying velocities of air flow.

To ensure comprehension, it's helpful to visualize the cylinder as an object being evenly heated while air flows over it. The faster the air flows, the more heat is removed from the cylinder, and consequently, this affects the value of the convection heat transfer coefficient. Understanding these principles is essential for various engineering applications, such as designing heating and cooling systems.

Heat Transfer Experiments

Heat transfer experiments, like the one described in the problem, are crucial for determining heat transfer coefficients and for validating theoretical models in real-world scenarios. These experiments involve precise measurements of temperature, heat input (power), and fluid flow rates.

In the described experiment, a cylinder with a heater embedded within it is exposed to air at different velocities. The power required to maintain the cylinder's surface temperature is measured at each velocity. This data is then used to calculate the heat transfer coefficient, which in turn reflects the efficiency of the convective heat transfer process at each condition.

To aid students' understanding, it is important to highlight that experiments like these often assume steady-state conditions (no change in temperature with time) and that other modes of heat transfer, such as radiation, are considered negligible to focus solely on the convective effects. Real experiments would also entail detailed setup and consistent measurements to ensure credible results.

Log-Log Coordinate Plotting

Log-log coordinate plotting is a powerful mathematical tool used to analyze the relationship between variables where both axes are in a logarithmic scale. It's particularly useful when dealing with data that follows a power law relationship, which is common in physics and engineering.

In the given exercise, students are asked to display the relationship between the heat transfer coefficient $h$ and the air flow velocity $V$ using a log-log plot. By plotting $\log (h)$ versus $\log (V)$, students can identify a linear relationship if the data follows a power law, as described by the equation $h=C{V}^n$. A straight line on a log-log plot indicates a power-law behavior which simplifies the analysis and helps determine the constants $C$ and $n$.

It is essential for students to grasp that the log-log plot simplifies the process of determining the slope of the line, which corresponds to the exponent of the velocity $n$, and the y-intercept, which correlates with the constant $C$. Mastery of this plotting technique allows for easier prediction and scaling of heat transfer behaviors in different contexts.