Problem 34 - A Heat Transfer Textbook

Question

A \(17^{\circ} \mathrm{C}\) sheet of water, \(\Delta_{1} \mathrm{~m}\) thick and moving at a constant speed \(u_{\infty} \mathrm{m} / \mathrm{s}\), impacts a horizontal plate at \(45^{\circ}\), turns, and flows along it. Develop a dimensionless equation for the thickness \(\Delta_{2}\) at a distance \(L\) from the point of impact. Assume that \(\delta \ll \Delta_{2}\). Evaluate the result for \(u_{\infty}=1 \mathrm{~m} / \mathrm{s}, \Delta_{1}=0.01 \mathrm{~m}\), and \(L=0.1 \mathrm{~m}\), in water at \(27^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

At \(L = 0.1\), approximate the thickness \(\Delta_2 = 0.01 \text{ m}\).

Step 1: Establish Initial Conditions

Given that the water initially approaches the plate at an angle of 45 degrees, we need to decompose the initial velocity of the water sheet into components parallel and perpendicular to the plate. The initial velocity components can be calculated as:- Perpendicular component: \(u_{\perp} = u_{\infty}\sin(45^\circ) = \frac{u_{\infty}}{\sqrt{2}}\)- Parallel component: \(u_{\parallel} = u_{\infty}\cos(45^\circ) = \frac{u_{\infty}}{\sqrt{2}}\)

Step 2: Conservation of Mass

Apply the principle of conservation of mass to relate the initial and final thickness of the water film. Under steady conditions and assuming the flow is incompressible, the mass flow rate must remain constant along the flow:\[ \Delta_{1} u'_{\parallel} = \Delta_{2} u_{\parallel} \]Where \(u'_{\parallel}\) is the velocity just upon impacting the plate and \(u_{\parallel}\) is the velocity at distance \(L\). Since the inflow is redirected along the plate, \(u'_{\parallel} = u_{\infty}\). Thus:\[ \Delta_{1} \frac{u_{\infty}}{\sqrt{2}} = \Delta_{2} u_{L} \]

Step 3: Dimensionless Formulation

Aiming for a dimensionless equation, express all variables in terms of dimensionless groups. Introduce non-dimensional terms, such as the Reynolds number based on the initial film thickness \( \text{Re}_{\Delta_1} = \frac{u_{\infty} \Delta_1}{u} \), and similar for \( \Delta_2 \). However, focus on expressing thickness in terms of measurable parameters:\[ \frac{\Delta_{2}}{\Delta_{1}} = \frac{1}{\sqrt{2}} \frac{u_{\infty}}{u_{L}} \] Assuming negligible viscous effects at intersection of the components, the dimensionless thickness expression shows dependency on velocity and angle, neglecting immediate Reynolds dependency.

Step 4: Calculate using Given Values

Insert the given values into your dimensionless equation. With \(u_{\infty} = 1 \text{ m/s}\), \(\Delta_1 = 0.01 \text{ m}\), and \(L = 0.1 \text{ m}\). Assuming the velocity largely remains similar along the streamline (as total dissipation isn't given), approximate:\[ \Delta_2 \approx \Delta_1 = 0.01 \text{ m} \] Given the simplifications (e.g., neglecting narrowing effects due to angle or velocity change at \(L\)), the assumed linear drop in velocity is not modeled. The thin film approximation heavily depends on external data for exact precision.

Key Concepts

Conservation of Mass

The principle of conservation of mass is a fundamental concept that states mass cannot be created or destroyed in an isolated system. In the context of fluid dynamics, particularly when dealing with flows like the water sheet impacting the plate in this problem, the conservation of mass is crucial.

For incompressible flow, such as water, the volume flow rate must remain constant as the water sheet transitions over the plate. This is because the density of water does not change significantly with pressure or temperature.

We apply this principle by ensuring the mass flow rate coming into a system equals the mass flow rate coming out. Here, it is expressed through the thickness transition from \(\Delta_1\) to \(\Delta_2\), connected through velocity components parallel to the plate.

• The initial condition before impact uses \(\Delta_1 u'_{\parallel}\).
• After the impact, this is represented as \(\Delta_2 u_{\parallel}\).

This relationship allows us to solve for changes in thickness resulting from the flow’s persistence on a new trajectory, while maintaining mass. Ensuring mass conservation at each stage preserves the characteristic volume flow and supports calculating the flow behavior post-impact.

Velocity Components

Understanding velocity components is essential when analyzing fluid flow, particularly when the flow direction changes, as with the water sheet hitting the plate at a 45-degree angle.

Velocity components can be broken down into their perpendicular and parallel components based on the direction of the surface and the initial flow direction. Here, the impact angle necessitates calculating both components for a complete picture:

• The perpendicular component, \(u_{\perp} \) \, will influence how the flow interacts with the surface upon impact. It's calculated using: \( u_{\perp} = u_{\infty}\sin(45^\circ) = \frac{u_{\infty}}{\sqrt{2}} \).
• The parallel component, \(u_{\parallel} \) \, describes the flow along the plate, given by: \( u_{\parallel} = u_{\infty}\cos(45^\circ) = \frac{u_{\infty}}{\sqrt{2}} \).

This approach aids in breaking down the flow into manageable parts when calculating subsequent behavior, such as changes in film thickness or flow perturbations. For complex systems, these components give insight into how fluids interact with surfaces, affecting pressure distributions and shear forces, key for designing systems involving fluid impact.

Dimensionless Equation

Dimensionless equations are used to simplify complex physical relationships into more universal terms without units, benefiting engineers and scientists by enabling direct comparison across different systems.

In this exercise, transitioning the equation into a dimensionless form provides insights that are less dependent on specific system scales, focusing purely on behavior and interactions. Here, converting variables to dimensionless form such as the ratio \(\frac{\Delta_2}{\Delta_1}\) helps relate changes in thickness directly to velocity components and impact conditions.

The expression reveals patterns that result from interaction nuances, like angle of attack or velocity ratios, without dimensionally causing bias. This is beneficial as it's:

• Independent from physical units, simplifying comparability.
• Useful in scaling analysis, impacting design and experimentation.

Such equations reduce complexity, promoting clearer understanding by highlighting underlying governing dynamics rather than system-specific details alone.

Incompressible Flow

Incompressible flow is a simplifying assumption where the fluid density is considered constant. This is applicable in liquids like water under typical conditions, where density changes are negligible despite pressure or temperature fluctuations.

In this problem, assuming water as incompressible means the volume of liquid remains mostly unchanged as it flows and interacts with surfaces like the plate. Such assumptions underpin the conservation laws applied in calculations here:

• It simplifies mathematical modeling by eliminating terms related to density variations.
• As density remains constant, changes in velocity are directly translating into geometric changes like thickness adjustments in films or sheets.

Understanding and applying the concept of incompressibility allows focusing on variables that influence thickness changes and direction change without dealing with complex compressibility factors, particularly useful in many engineering problems involving liquids.