Question

A tube with a square-edged inlet configuration is subjected to uniform wall heat flux of \(8 \mathrm{~kW} / \mathrm{m}^{2}\). The tube has an inside diameter of \(0.622\) in and a flow rate of \(2.16 \mathrm{gpm}\). The liquid flowing inside the tube is ethylene glycol-distilled water mixture with a mass fraction of \(2.27\). Determine the friction coefficient at a location along the tube where the Grashof number is \(\mathrm{Gr}=35,450\). The physical properties of the ethylene glycol-distilled water mixture at the location of interest are \(\operatorname{Pr}=13.8, v=18.4 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\), and \(\mu_{b} / \mu_{s}=1.12\). Then recalculate the fully developed friction coefficient if the volume flow rate is increased by 50 percent while the rest of the parameters remain unchanged.

Short answer

Question: Determine the initial friction coefficient of a tube with liquid flowing through it at the given conditions and then recalculate the friction coefficient if the volume flow rate is increased by 50%, keeping the other parameters unchanged. Answer: The initial friction coefficient is _____ and the friction coefficient after increasing the volume flow rate by 50% is _____.

Step-by-step solution

Convert units of given values

First, we need to convert the given units into the SI system to make calculations easier: - Diameter: 0.622 inches to meters (1 inch = 0.0254 m) - Flow rate: 2.16 gpm to m³/s (1 gpm = 6.30902 x 10⁻⁵ m³/s) - Kinematic viscosity: 18.4 x 10⁻⁶ ft²/s to m²/s (1 ft² = 0.092903 m²)

Calculate the Reynolds number (Re)

Calculate the Reynolds number using the formula: Re = (4 * flow rate) / (π * diameter * kinematic viscosity)

Find the friction factor (f) using Grashof, Reynolds, and Prandtl numbers

Use the given Grashof number (Gr) and the calculated Reynolds number (Re) in the following formula to find the friction factor: f = (16 / Re) * (1 + 0.01 * Re * (µb / µs) * (Pr^(1/3) - 1))

Recalculate the friction coefficient after increasing the volume flow rate by 50%

To determine the new friction coefficient, repeat Steps 2 and 3 using the increased volume flow rate (2.16 * 1.5).

Report the friction coefficients

Report the initial friction coefficient and the friction coefficient after increasing the volume flow rate by 50%.

Key concepts

Grashof Number

The Grashof number (\textbf{Gr}) is a dimensionless number that represents the ratio of the buoyancy forces to viscous forces in a fluid flow. It's crucial for understanding natural convection patterns in fluids, especially when heat transfer is involved. In practice, if a fluid is heated at the bottom, Grashof number helps to estimate whether the heat will simply transfer through conduction or if the fluid will start moving due to the temperature difference (convection).

Mathematically, the Grashof number is defined as follows:
\[ Gr = \frac{g \beta (T_s - T_\infty) L^3}{v^2} \]
where

• \( g \) is the acceleration due to gravity,
• \( \beta \) is the thermal expansion coefficient,
• \( T_s - T_\infty \) is the temperature difference between the surface and the fluid far away,
• \( L \) is the characteristic length, and
• \( v \) is the kinematic viscosity of the fluid.

In the original exercise, knowing the Grashof number enables the calculation of the friction coefficient because it indicates the level of convection in the flow. A high Grashof number would suggest a dominant buoyancy-driven flow which can affect the fluid's velocity profile and thus the friction it experiences against the tube.

Reynolds Number

The Reynolds number (\textbf{Re}) is another fundamental dimensionless number in fluid dynamics that determines the regime of the flow, which can be either laminar or turbulent. The Reynolds number is calculated using the relationship below:
\[ Re = \frac{\rho U L}{\mu} = \frac{UL}{v} \]
where

• \( \rho \) is the fluid density,
• \( U \) is the velocity of the fluid,
• \( L \) is the characteristic length scale (often diameter in pipe flow), and
• \( \mu \) is the dynamic viscosity,
• \( v \) is the kinematic viscosity (\( v = \frac{\mu}{\rho} \)).

For a pipe flow, like the one in our exercise, lower Reynolds numbers indicate laminar flow, while higher numbers signify turbulent flow. The change from laminar to turbulent can significantly affect the friction forces at play, which in turn impact the friction coefficient. Hence, accurately determining the Reynolds number is essential in predicting flow behavior and associated friction.

Kinematic Viscosity

Kinematic viscosity (\textbf{v}), in fluid dynamics, is a measure of a fluid's resistance to flow under gravity. It is one of the elements required to calculate both the Grashof and Reynolds numbers, making it a pivotal factor in friction coefficient calculations. Kinematic viscosity is the ratio of dynamic viscosity (\textbf{μ}) to density (\textbf{ρ}), represented mathematically as:
\[ v = \frac{\mu}{\rho} \]
It's expressed in square meters per second (\(m^2/s\)) in SI units, although for different practical applications, other units might be used, requiring conversions as seen in the exercise. When solving for flow rates and heat transfer, kinematic viscosity values help determine the type of flow, the heat convection coefficient, and various other critical factors in systems involving fluid motion. Knowing the kinematic viscosity allows engineers and scientists to model flow patterns and design efficient systems for transporting fluids.

Prandtl Number

The Prandtl number (\textbf{Pr}) is a dimensionless value important in the study of heat transfer within fluid flow. It correlates the relative thickness of the velocity boundary layer to the thermal boundary layer. The Prandtl number can be defined as:
\[ Pr = \frac{c_p \mu}{k} \]
where

• \( c_p \) is the specific heat capacity at constant pressure,
• \( \mu \) is the dynamic viscosity, and
• \( k \) is the thermal conductivity of the fluid.

In essence, the Prandtl number helps in understanding how quickly heat is diffused through a fluid compared to the speed at which the fluid itself moves or diffuses due to a velocity gradient. In the context of our exercise, the Prandtl number, being a property of the fluid, affects the calculation of the friction factor in a relationship that considers both temperature and velocity profiles.