Question

10-59 The Reynolds number for condensate flow is defined as \(\operatorname{Re}=4 \dot{m} / p \mu_{l}\), where \(p\) is the wetted perimeter. Obtain simplified relations for the Reynolds number by expressing \(p\) and \(\dot{m}\) by their equivalence for the following geometries: \((a)\) a vertical plate of height \(L\) and width \(w,(b)\) a tilted plate of height \(L\) and width \(W\) inclined at an angle \(u\) from the vertical, \((c)\) a vertical cylinder of length \(L\) and diameter \(D,(d)\) a horizontal cylinder of length \(L\) and diameter \(D\), and (e) a sphere of diameter \(D\).

Short answer

Question: Write the simplified expressions for the Reynolds number for condensate flow in the following geometries: (a) vertical plate, (b) tilted plate, (c) vertical cylinder, (d) horizontal cylinder, and (e) sphere. Answer: (a) Vertical plate: $\operatorname{Re} = \frac{2 \rho L w v}{(L+w) \mu_l}$ (b) Tilted plate: $\operatorname{Re} = \frac{2 \rho L_p W v}{(L_p+W) \mu_l}$ (c) Vertical cylinder: $\operatorname{Re} = \frac{2 \rho D v}{\mu_l}$ (d) Horizontal cylinder: $\operatorname{Re} = \frac{2 \rho D v}{\mu_l (1 + 2L/\pi D)}$ (e) Sphere: $\operatorname{Re} = \frac{8 \rho D^2 v}{3 \mu_l}$

Step-by-step solution

(a) Vertical plate

For a vertical plate of height \(L\) and width \(w\), the wetted perimeter \(p\) is given as \(p=2(L+w)\). The mass flow rate \(\dot{m}\) is given by \(\rho L w v\), where \(\rho\) is the density of the fluid, and \(v\) is the velocity. Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho L w v}{2(L+w) \mu_l} = \frac{2 \rho L w v}{(L+w) \mu_l}$$

(b) Tilted plate

For a tilted plate of height \(L\) and width \(W\) inclined at an angle \(u\) from the vertical, we first find the projected length \(L_p = L \cos u\). The wetted perimeter \(p\) is given as \(p=2(L_p+W)\). The mass flow rate \(\dot{m}\) is given by \(\rho L_p W v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho L_p W v}{2(L_p+W) \mu_l} = \frac{2 \rho L_p W v}{(L_p+W) \mu_l}$$

(c) Vertical cylinder

For a vertical cylinder of length \(L\) and diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D\). The mass flow rate \(\dot{m}\) is given by \(\rho \pi (D/2)^2 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho \pi (D/2)^2 v}{\pi D \mu_l} = \frac{2 \rho D v}{\mu_l}$$

(d) Horizontal cylinder

For a horizontal cylinder of length \(L\) and diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D + 2L\). The mass flow rate \(\dot{m}\) is given by \(\rho \pi (D/2)^2 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho \pi (D/2)^2 v}{(\pi D + 2L) \mu_l} = \frac{2 \rho D v}{\mu_l (1 + 2L/\pi D)}$$

(e) Sphere

For a sphere of diameter \(D\), the wetted perimeter \(p\) is given as \(p=\pi D\). The mass flow rate \(\dot{m}\) is given by \(\rho (4/3)\pi (D/2)^3 v\). Thus, the Reynolds number is: $$\operatorname{Re} = \frac{4 \rho (4/3)\pi (D/2)^3 v}{\pi D \mu_l} = \frac{8 \rho D^2 v}{3 \mu_l}$$

Key concepts

Condensate Flow

Condensate flow refers to the movement of a liquid, typically water, that has condensed from a vapor state. This is common in heat exchangers and cooling systems. The flow behavior is vital in designing efficient thermal systems. When studying condensate flow, understanding various factors such as geometry and fluid properties is essential.
The geometry of the surface affects how the condensate is collected and flows. For instance:

• Vertical plates allow condensate to flow down due to gravity.
• Tilted surfaces alter flow paths depending on inclination.
• Cylinders and spheres involve more complex flow patterns due to curvature.

By examining the condensate flow, engineers can optimize heat transfer efficiency, reducing energy consumption and enhancing system operation.

Wetted Perimeter

The wetted perimeter is the part of a surface in contact with a fluid. In flow analysis, it helps determine the area that offers frictional resistance to flow.
For different geometries, the wetted perimeter varies:

• For a vertical plate, it is calculated as the total perimeter in contact with the flow, often expressed as \(p = 2(L + w)\).
• In cylinders, it can become more complex, considering circular shapes where \(p = \pi D\).
• Spheres experience a similar circular approach, but the flow interaction changes due to 3D curvature.

Understanding the wetted perimeter is crucial for calculating Reynolds number, which helps in evaluating the flow regime, whether laminar or turbulent.

Mass Flow Rate

Mass flow rate represents the amount of mass passing through a section per unit time. It's a key factor in determining how much fluid is moving and affects pressure and velocity within a system.
In condensate flow, the mass flow rate \(\dot{m}\) varies based on the geometry of the object:

• For a plate, it might be expressed as \(\rho L w v\), involving height \(L\), width \(w\), fluid density \(\rho\), and velocity \(v\).
• For a cylindrical shape, the expression might become \(\rho \pi (D/2)^2 v\), where \(D\) represents the diameter.

The accurate calculation of mass flow rate ensures that thermal systems are designed with efficiency and reliability, minimizing potential losses or issues.

Fluid Dynamics

Fluid dynamics is the study of fluids in motion and is essential in understanding real-world problems involving condensate flow. Key aspects such as velocity, pressure, and flow patterns are analyzed to predict how fluids behave in different situations.
In fluid dynamics, the Reynolds number is crucial as it determines the flow regime:

• Laminar flow indicates smooth and orderly motion, occurring at low Reynolds numbers.
• Turbulent flow is chaotic and irregular, typically happening at high Reynolds numbers.

By measuring parameters like the wetted perimeter and mass flow rate, engineers use fluid dynamics to design systems that operate effectively under various conditions. This ensures proper management of flow, reducing energy wastage and enhancing performance.