Understanding Head Loss in Laminar Flow Between Parallel Plates

The question asks how head loss varies with the distance 'h' for laminar flow between two parallel plates separated by a total distance of 2h. To answer this, we need to examine the governing equations for laminar flow in such a geometry.

Laminar Flow Between Parallel Plates

For steady, incompressible, laminar flow between two parallel plates, separated by a distance 2h, the flow is driven by a pressure gradient. Assuming the flow is in the x-direction and the plates are parallel to the x-z plane, located at y = +h and y = -h, the velocity profile $u(y)$ is parabolic. The pressure drop ($\Delta P$) over a length $L$ for this flow is given by the formula:

$$\Delta P = \frac{12 \mu L V_{avg}}{(2h)^2}$$

where:

• $\Delta P$ is the pressure drop
• $\mu$ is the dynamic viscosity of the fluid
• $L$ is the length of the flow section
• $V_{avg}$ is the average velocity of the fluid
• $2h$ is the distance between the plates

Simplifying the formula based on the separation $2h$:

$$\Delta P = \frac{12 \mu L V_{avg}}{4h^2} = \frac{3 \mu L V_{avg}}{h^2}$$

This equation shows the pressure drop ($\Delta P$) varies inversely with the square of 'h' (half the separation distance).

Relating Pressure Drop to Head Loss

Head loss ($h_f$) is related to the pressure drop ($\Delta P$) by the equation:

$$h_f = \frac{\Delta P}{\rho g}$$

where:

• $\rho$ is the density of the fluid
• $g$ is the acceleration due to gravity

Substituting the expression for $\Delta P$ from the laminar flow between parallel plates:

$$h_f = \frac{1}{\rho g} \left( \frac{3 \mu L V_{avg}}{h^2} \right) = \frac{3 \mu L V_{avg}}{\rho g h^2}$$

Analyzing the Variation of Head Loss

The formula for head loss is $h_f = \frac{3 \mu L V_{avg}}{\rho g h^2}$. Assuming the fluid properties ($\mu$, $\rho$), the flow length ($L$), and the average velocity ($V_{avg}$) (which relates to flow rate) are constant, the head loss ($h_f$) is directly proportional to $1/h^2$.

$$h_f \propto \frac{1}{h^2}$$

This means that head loss varies inversely as the square of 'h', where 'h' is half the distance between the parallel plates. Since the total separation is $2h$, 'h' represents a measure directly related to the separation. Therefore, as the separation distance increases (meaning 'h' increases), the head loss decreases, and specifically, it decreases rapidly with the square of 'h'.

Conclusion

Based on the derivation from the fundamental principles of laminar flow between parallel plates, the head loss is found to be inversely proportional to the square of 'h' (half the separation distance). Thus, head loss varies inversely as h².

Comparing this to the given options:

• Option 1: directly as h (Incorrect)
• Option 2: inversely as h (Incorrect)
• Option 3: directly as h² (Incorrect)
• Option 4: inversely as h² (Correct)

Revision Table: Laminar Flow Head Loss

Additional Information on Laminar Flow and Head Loss

Laminar flow is characterized by smooth, orderly fluid motion, occurring at low Reynolds numbers. For internal flows like between parallel plates or inside pipes, the Reynolds number depends on velocity, characteristic length, and kinematic viscosity.

The characteristic length for flow between parallel plates separated by $2h$ is often taken as the hydraulic diameter, which is $D_h = \frac{4A}{P}$, where $A$ is the flow area and $P$ is the wetted perimeter. For a rectangular channel of width $W$ and height $2h$, $A = W \times 2h$ and $P = 2(W + 2h)$. If $W \gg 2h$ (wide parallel plates), $P \approx 2W$, and $D_h \approx \frac{4(W \times 2h)}{2W} = 4h$. The Reynolds number $Re = \frac{\rho V_{avg} D_h}{\mu} = \frac{V_{avg} D_h}{\nu}$. Laminar flow typically occurs for $Re$ below a certain critical value (e.g., often cited around 1400 for parallel plates using $D_h=4h$, or $Re^* = \rho V_{max} h / \mu < 1000$ using $h$ and maximum velocity).

Head loss represents the energy lost per unit weight of fluid due to viscous effects as the fluid flows. In laminar flow, this loss is directly proportional to the viscosity and the flow velocity (or flow rate) and inversely proportional to the geometry's size squared (for plates) or to the power of four (for pipes).

The formula $\Delta P = \frac{3 \mu L V_{avg}}{h^2}$ applies specifically to fully developed laminar flow between infinite parallel plates. 'Fully developed' means the velocity profile does not change along the flow direction. 'Infinite' refers to the width being much larger than the separation, so edge effects are negligible.