Question

Consider the velocity and temperature profiles for airflow in a tube with diameter of \(8 \mathrm{~cm}\) can be expressed as $$ \begin{aligned} &u(r)=0.2\left[\left(1-(r / R)^{2}\right]\right. \\ &T(r)=250+200(r / R)^{3} \end{aligned} $$ with units in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{K}\), respectively. If the convection heat transfer coefficient is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the mass flow rate and surface heat flux using the given velocity and temperature profiles. Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Short answer

Based on the given velocity and temperature profiles in the tube, calculate the mass flow rate and the surface heat flux. Answer: The mass flow rate is \(\dot{m} = 0.0123~\mathrm{kg\cdot s^{-1}}\) and the surface heat flux is \(q_s = 15685~\mathrm{W\cdot m^{-2}}\).

Step-by-step solution

Calculate the cross-sectional area of the tube

Given the tube's diameter \(D = 8~\mathrm{cm}\), we can find the cross-sectional area as follows: Area: \(A = \pi\frac{D^2}{4}\) Substitute the given diameter and calculate the area: \(A = \pi\frac{(0.08)^2}{4} = 5.027\times10^{-3}~\mathrm{m^2}\)

Evaluate air density and mass flow rate

The air properties are given at \(20^{\circ}\mathrm{C}\) and \(1~\mathrm{atm}\). We can use the ideal gas law to find the air density: Air density: \(\rho = \frac{P}{RT}\) where \(P = 1~\mathrm{atm} = 101325~\mathrm{Pa}\), \(R = 287~\mathrm{J\cdot kg^{-1} \cdot K^{-1}}\) is the specific gas constant for air, and \(T = 20 + 273.15 = 293.15~\mathrm{K}\). Calculate the air density: \(\rho = \frac{101325}{287 \times 293.15} = 1.204~\mathrm{kg\cdot m^{-3}}\) Now, we can calculate the mass flow rate using the equation \(\dot{m} = \rho A u\). Note that given the velocity profile \(u(r)=0.2\left[\left(1-(r / R)^{2}\right]\right. \mathrm{m}/\mathrm{s}\), we have to integrate \(u(r)\) over the entire tube to find the total flow rate: Mass flow rate: \(\dot{m} = \rho \int_{0}^{R} u(r) 2\pi r dr\) Substitute given values and evaluate the integral: \(\dot{m} = 1.204 \int_{0}^{0.04} (0.2 (1 - (r/0.04)^2) 2\pi r dr = 0.0123~\mathrm{kg\cdot s^{-1}}\)

Calculate the temperature difference and surface heat flux

The temperature profile is given: \(T(r) = 250 + 200(r / R)^{3}~\mathrm{K}\). We can easily find the surface temperature by setting \(r = R\): \(T_s = T(R) = 250 + 200(1)^{3} = 450~\mathrm{K}\) The temperature difference between the cooking tray and the air is given by: \(\Delta T = T_s - T_\mathrm{air} = 450 - 293.15 = 156.85~\mathrm{K}\) Now, calculate the surface heat flux using the convection heat transfer coefficient: Surface heat flux: \(q_s = h \Delta T\) \(q_s = 100 \times 156.85 = 15685~\mathrm{W\cdot m^{-2}}\) The mass flow rate and surface heat flux for the given velocity and temperature profiles in the tube are \(\dot{m} = 0.0123~\mathrm{kg\cdot s^{-1}}\) and \(q_s = 15685~\mathrm{W\cdot m^{-2}}\), respectively.

Key concepts

Mass Flow Rate

Mass flow rate is an essential concept in fluid dynamics, and it refers to the amount of mass passing through a given surface or tube per unit time. It's denoted as \( \dot{m} \) and calculated using the formula:

• \( \dot{m} = \rho A u \)

where \( \rho \) is the density of the fluid, \( A \) is the cross-sectional area, and \( u \) is the velocity.
To determine the mass flow rate in a tube, it's crucial first to find the cross-sectional area of the tube. This is calculated with the formula \( A = \pi\frac{D^2}{4} \), where \( D \) represents the diameter.
After finding the area, the next step involves finding the fluid density. For air, the ideal gas law can be used

• \( \rho = \frac{P}{RT} \)

where \( P \) is the pressure, and \( T \) is the temperature in Kelvin.
Finally, with the velocity profile provided, integrate to find the average velocity over the surface area to find the overall flow rate. Integration is necessary since velocity can vary across the radius.

Convection Heat Transfer

Convection heat transfer is the mode of thermal energy transfer from a solid surface to a fluid or vice versa due to the motion of the fluid. The rate of heat transfer in convection depends on several factors including the convection heat transfer coefficient, surface area, and temperature difference.
In the case of our exercise, the convection heat transfer coefficient \( h \) is given as \( 100~\mathrm{W/m^{2} \cdot K} \). This coefficient measures the ability of the fluid to transfer heat from the surface and is influenced by fluid velocity and properties.

• The formula for calculating heat flux \( q_s \) is \( q_s = h \Delta T \)

where \( \Delta T \) is the difference between the surface temperature and the fluid temperature.
A higher convection coefficient indicates more efficient heat transfer, which can influence design decisions in engineering applications involving heat exchange.

Temperature Profile

Understanding the temperature profile in a system allows us to know how temperature changes with position. In this exercise, the temperature distribution within the tube can be expressed as:

• \( T(r) = 250 + 200(r/R)^3 \).

This equation shows that the temperature is a function of radial distance \( r \) from the center of the tube.
Evaluating temperature at different positions helps determine how heat is distributed in the system. Critical points like the surface temperature \( T_s \) can be calculated by substituting the maximum radius into the temperature profile equation.
For engineering applications, knowing the temperature profile is important in ensuring that materials are not subjected to temperatures beyond their limits, thus ensuring safety and integrity.

Velocity Profile

The velocity profile in fluid flow describes how the fluid velocity varies across the cross-section of a pipe or channel. It's crucial for understanding how fast the fluid moves at different points.

• In the case of our tube, the velocity profile is given by \( u(r)=0.2\left[1-(r/R)^2\right] \).

This implies that the velocity is maximum at the center of the tube and decreases towards the tube walls.
The velocity profile enables the calculation of the average velocity by integrating across the radius. This step is necessary to find the flow rate, as it accounts for different velocities at various radial positions.
Knowledge of the velocity profile also aids in designing systems for maximum efficiency by predicting pressure drops or identifying potential areas of turbulence or separation.