Problem 14 - Basic Heat And Mass Transfer

Question

Show that for fully developed laminar flow in an annular duct the velocity profile is given by $$ u=\frac{R_{o}^{2}}{4 \mu}\left(-\frac{d P}{d x}\right)\left[1-\left(\frac{r}{R_{o}}\right)^{2}+\frac{1-\gamma^{2}}{\ln (1 / \gamma)} \ln \frac{r}{R_{o}}\right] $$ where \(\gamma=R_{i} / R_{o}\) and \(R_{i}, R_{o}\) are the inner and outer radii, respectively. Hence show that the bulk velocity is $$ u_{b}=\frac{R_{o}^{2}}{8 \mu}\left(-\frac{d P}{d x}\right)\left[1+\gamma^{2}-\frac{1-\gamma^{2}}{\ln (1 / \gamma)}\right] $$ and the location where the maximum velocity occurs is given by $$ \frac{r}{R_{o}}=\left(\frac{1-\gamma^{2}}{2 \ln (1 / \gamma)}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The velocity profile and bulk velocity expressions are derived using integration, boundary conditions, and calculus. Location of maximum velocity is determined by setting the velocity gradient to zero.

Step 1: Understand Fully Developed Laminar Flow in an Annulus

In this problem, we need to consider the flow between two concentric cylinders where the flow is fully developed, meaning the velocity profile doesn't change along the flow direction. The inner radius of the duct is \( R_i \), the outer radius is \( R_o \), and the flow is laminar and steady.

Step 2: Establish the Governing Equation

For laminar flow in a cylindrical coordinate system, the velocity profile \( u(r) \) is determined by the balance of pressure forces and viscous resistance. The governing equation is derived from the Navier-Stokes equations simplified for axisymmetric, steady, and fully developed flow:\[\frac{1}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) = \frac{1}{u} \frac{dP}{dx}\]where \( u \) is the kinematic viscosity, and \( \frac{dP}{dx} \) is the pressure gradient.

Step 3: Integrate the Equation Twice

First, integrate the differential equation with respect to \( r \):\[\frac{du}{dr} = \frac{1}{r} \left( C_1 + \frac{r^2}{2 \mu} \frac{dP}{dx} \right)\]Integrate again to find \( u(r) \):\[u(r) = \frac{1}{4 \mu} \frac{dP}{dx} r^2 + C_1 \ln r + C_2 \]

Step 4: Apply Boundary Conditions

Apply boundary conditions: At \( r = R_o \), \( u = 0 \) and similarly, at \( r = R_i \), \( u = 0 \). Use these to solve for constants \( C_1 \) and \( C_2 \). Substituting back will lead to the given velocity profile of the annular flow:\[u(r) = \frac{R_o^2}{4 \mu} \left(-\frac{dP}{dx}\right) \left[1-\left(\frac{r}{R_o}\right)^2+\frac{1-\gamma^2}{\ln (1 / \gamma)} \ln \frac{r}{R_o}\right].\]

Step 5: Derive the Expression for Bulk Velocity

The bulk velocity \( u_b \) is the average velocity across the annular cross-section. It can be found by integrating the velocity profile over the area and dividing by the area:\[ u_b = \frac{1}{A} \int_{R_i}^{R_o} 2 \pi r u(r) dr \]Calculating this integral using the derived \( u(r) \) and simplifying yields:\[ u_{b} = \frac{R_{o}^{2}}{8 \mu} \left(-\frac{dP}{dx}\right) \left[1+\gamma^2 - \frac{1-\gamma^2}{\ln (1 / \gamma)}\right].\]

Step 6: Find the Location of Maximum Velocity

The maximum velocity occurs where the derivative of \( u(r) \) with respect to \( r \) is zero. Set the derivative \( \frac{du}{dr} = 0 \) and solve for \( r/R_o \):\[ \frac{du}{dr} \Rightarrow 2 \frac{r}{R_o^2} - \frac{1-\gamma^2}{r \ln(1/\gamma)} = 0 \]Solving this results in the expression for the location of maximum velocity:\[ \frac{r}{R_o} = \left(\frac{1-\gamma^2}{2 \ln (1/\gamma)}\right)^{1/2}. \]

Key Concepts

Velocity Profile Derivation

In the study of fluid dynamics, particularly for laminar flow in an annular duct, deriving the velocity profile is crucial. We start with the Navier-Stokes equations which describe how pressure, viscous forces, and fluid inertia interact. For fully developed laminar flow between two concentric cylinders, these equations simplify significantly.
The simplified form is:

• Involves balance of forces: pressure forces equal the viscous resistance.
• Governing equation becomes axisymmetric, ideal for cylindrical coordinates.
• A steady state implies no change along the flow direction.

The resulting equation is often written as:\[\frac{1}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) = \frac{-1}{\mu} \frac{dP}{dx}\]Integrating twice with respect to radial position, \( r \), yields the velocity profile. Here, boundary conditions (velocity equals zero at the walls) are crucial to solve for integration constants. The solution gives:\[u(r) = \frac{R_o^2}{4 \mu}\left(-\frac{dP}{dx}\right) \left[1-\left(\frac{r}{R_o}\right)^2 + \frac{1-\gamma^2}{\ln(1/\gamma)} \ln\frac{r}{R_o}\right]\]where \( \gamma = R_i/R_o \), representing the ratio of inner to outer radii.

Bulk Velocity Calculation

The bulk velocity in a flow system, also known as the average velocity, is important for understanding the flow rate of fluids in conduits. To calculate this for annular ducts, we use the average value of the velocity profile derived earlier.
The bulk velocity, \( u_b \), is determined by integrating the velocity profile, \( u(r) \), over the cross-sectional area and then dividing by that area. Mathematically, this is expressed as:

• Area of annular section needs consideration: \( \pi (R_o^2 - R_i^2) \).
• Integral accounts for the radial distribution of velocities.

The formula derived for bulk velocity is:\[u_b = \frac{R_o^2}{8 \mu}\left(-\frac{dP}{dx}\right)\left[1+\gamma^2 - \frac{1-\gamma^2}{\ln(1/\gamma)}\right]\]This result reflects how the distribution of velocities across the duct affects the overall flow.

Maximum Velocity Location

In annular duct flow, identifying where the maximum velocity occurs helps in understanding the flow's behavior. This is not trivially at the center as in many simple duct flows due to the presence of the inner cylinder. Instead, the location depends on the geometric configuration.
To find where the maximum velocity happens:

• Take the derivative of the velocity profile \( \frac{du}{dr} \).
• Set this derivative to zero to locate the maxima.

The equation becomes:\[2\frac{r}{R_o^2} - \frac{1-\gamma^2}{r \ln(1/\gamma)} = 0\]Solving yields the position:\[\frac{r}{R_o} = \left(\frac{1-\gamma^2}{2 \ln (1/\gamma)}\right)^{1/2}\]This formula lets engineers and scientists determine exactly where in the annulus the highest speed is reached.

Navier-Stokes Equations

The Navier-Stokes equations form the cornerstone of fluid dynamics. They describe the motion of viscous fluid substances. In the context of an annular duct, these equations simplify under certain conditions.

• The flow is steady: no time-dependence.
• It is fully developed: profile doesn’t change in the flow direction.
• The geometry allows axisymmetric simplifications.

In laminar conditions, these equations are reduced to:\[\frac{1}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) = \frac{-1}{\mu} \frac{dP}{dx}\]This particular formulation emerges by balancing inertial forces with pressure gradients and viscosity. Understanding these helps to grasp how pressure differences drive the flow and how viscosity resists this motion, setting the stage for deriving detailed velocity profiles.