Problem 1 - Basic Heat And Mass Transfer

Question

Two \(1 \mathrm{~m}\)-long concentric cylinders form an annular gap \(1 \mathrm{~mm}\) wide. The outer cylinder is stationary. The inner cylinder has a radius of \(10 \mathrm{~cm}\) and rotates at\(1000 \mathrm{rpm}\). Determine the power dissipated by viscous dissipation in the gap if the fluid is (i) air, (ii) water, (iii) SAE 50 oil, all at \(300 \mathrm{~K}\) and 1 atm pressure.

Step-by-Step Solution

Answer

The power dissipation is minimal for air, moderate for water, and significantly higher for SAE 50 oil due to its higher viscosity.

Step 1: Understand the Problem

We have two concentric cylinders with an annular gap where the inner cylinder rotates, causing viscous dissipation in the fluid between them. We need to calculate the power dissipated for different fluids.

Step 2: Gather Necessary Information and Formulas

To solve the problem, we need to find the shear stress and power dissipated. The formula for power dissipated due to viscous dissipation is \( P = T \cdot \omega \), where \( T \) is the torque and \( \omega \) is the angular velocity. Torque, \( T \), is given by \( T = \tau \cdot A \cdot r \), where \( \tau \) is the shear stress, \( A \) is the contact area, and \( r \) is the radius of the inner cylinder. Shear stress, \( \tau \), for a Newtonian fluid is \( \tau = \mu \cdot \frac{du}{dy} \), where \( \mu \) is the dynamic viscosity, \( du \) is the change in velocity, and \( dy \) is the gap width.

Step 3: Calculate Angular Velocity

The inner cylinder rotates at \(1000 \text{ rpm}\). Convert this to radians per second (\( \text{rad/s} \)) by multiplying \(1000\) by \(\frac{2\pi}{60}\) to obtain \( \omega = \frac{1000 \times 2 \pi}{60} \approx 104.72 \text{ rad/s} \).

Step 4: Calculate Power for Each Fluid

Using the known viscosities for air, water, and SAE 50 oil at 300 K, calculate the shear stress, torque, and finally the power dissipated using the formulas from Step 2. The viscosities are: air = 1.84e-5 Pa·s, water = 8.9e-4 Pa·s, and SAE 50 oil = 0.455 Pa·s. The power dissipated is calculated for each fluid individually.

Step 5: Compute the Shear Rate

From the geometry, the velocity gradient \( \frac{du}{dy} = \frac{102.72}{0.001} = 104720 \text{ s}^{-1} \). Apply this gradient with each fluid's viscosity to find the shear stress, \( \tau \).

Step 6: Calculate Torque and Power for Each Fluid

For each fluid, calculate the torque: \( T = \tau \cdot A \cdot r \). With \( \tau = \mu \cdot 104720 \), \( A = 2\pi\times0.1^2 \), and \( r = 0.1 \text{ m} \), get the power as \( P = T \cdot \omega \). For air, water, and SAE 50 oil, use their specific viscosities found in Step 4.

Step 7: List Dissipated Power for Each Fluid

Calculate and compare the power dissipated for the three fluids. Air, water, and SAE 50 oil each show increasing power dissipation due to increasing viscosity.

Key Concepts

Concentric Cylinders

In the given exercise, you are dealing with concentric cylinders. This means you have two cylinders, one inside the other, sharing the same central axis. The outer cylinder remains stationary, while the inner cylinder is set into rotation. The space between these cylinders creates an annular gap, which allows fluid to fill and flow through it. In this scenario, the inner cylinder has a radius of 10 cm, and the gap between the cylinders is only 1 mm wide. The movement of the inner cylinder through this fluid-filled gap causes shear, which leads to viscous dissipation—a form of energy loss due to the fluid's resistance to flow. Understanding this setup is key to visualizing the forces and energy at play in the problem.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. In this exercise, it is a crucial aspect because different fluids have different viscosities, which affect how much energy is dissipated as the inner cylinder rotates. Viscosity can be tough to grasp, but simply put, it's how 'thick' or 'thin' a fluid feels.

• Air: A very low viscosity of 1.84e-5 Pa·s means it offers minimal resistance.
• Water: Has a higher viscosity of 8.9e-4 Pa·s compared to air, hence more resistance.
• SAE 50 Oil: With a viscosity of 0.455 Pa·s, it has the highest resistance among the three fluids.

Viscosity directly influences shear stress: the higher the viscosity, the greater the shear stress for the same rate of deformation. This is why different fluids will dissipate different amounts of energy in the gap.

Shear Stress

Shear stress is the force per unit area exerted by the fluid as layers of it move relative to one another. For Newtonian fluids like those involved here (air, water, and SAE 50 oil), shear stress is directly proportional to the viscosity and the velocity gradient, denoted by the formula \( \tau = \mu \cdot \frac{du}{dy} \). The velocity gradient \( \frac{du}{dy} \) represents how quickly the velocity changes across the gap. It is a constant value of 104720 s\(^{-1}\) due to the geometry and rotation rate.
As viscosity (bC) increases, shear stress increases. Thus, oil, having a higher viscosity, will experience greater shear stress compared to air or water. This directly affects the torque applied to the inner cylinder, influencing how much power is dissipated.

Angular Velocity

Angular velocity is a measure of how fast an object rotates. In this problem, it describes how quickly the inner cylinder spins around its axis. Given that the inner cylinder rotates at 1000 revolutions per minute (rpm), you need to convert this to radians per second (rad/s), as it's the standard unit of angular velocity in physics.
Convert rpm to rad/s using the conversion factor: \( \omega = \frac{1000 \times 2 \pi}{60} \approx 104.72 \text{ rad/s} \) where \( \omega \) is the angular velocity. This conversion provides the rotational speed necessary to compute shear stress and power dissipation.
A faster angular velocity means a higher rate of shear in the fluid, which in turn affects how much power is dissipated through viscous dissipation due to friction within the fluid's layers.