HW11.2 Experiments show that the pressure drop for flow through an orifice plate of diameter d mounted in a length of pipe of diameter D may be expressed as Δp = p1 - p2 = f(ρ, μ,... HW11.2 Experiments show that the pressure drop for flow through an orifice plate of diameter d mounted in a length of pipe of diameter D may be expressed as Δp = p1 - p2 = f(ρ, μ, V, d, D). You are asked to organize some experimental data. Obtain the resulting dimensionless parameters.
Understand the Problem
The question is asking to organize experimental data related to the pressure drop for flow through an orifice plate. Specifically, it wants to derive dimensionless parameters from the given formula, which involves the pressures and characteristics of the flow.
Answer
Dimensionless parameters: $Re = \frac{\rho V D}{\mu}$, $P_d = \frac{\Delta p d^2}{\rho V^2}$, $A = \frac{d^2}{D^2}$.
Answer for screen readers
The resulting dimensionless parameters are:
- Reynolds number: $Re = \frac{\rho V D}{\mu}$
- Pressure drop ratio: $P_d = \frac{\Delta p d^2}{\rho V^2}$
- Orifice area ratio: $A = \frac{d^2}{D^2}$
Steps to Solve
- Identify the Variables We need to identify the dimensions and units of each variable in the equation. The variables are:
- $\Delta p$ (pressure drop) with units of pressure (e.g., Pa)
- $\rho$ (density) with units of mass/volume (e.g., kg/m³)
- $\mu$ (dynamic viscosity) with units of mass/(length·time) (e.g., Pa·s or kg/(m·s))
- $V$ (velocity) with units of length/time (e.g., m/s)
- $d$ (diameter of the orifice) with units of length (e.g., m)
- $D$ (diameter of the pipe) with units of length (e.g., m)
- List the Dimensions Next, we list the dimensions of each variable in terms of fundamental dimensions: mass (M), length (L), and time (T).
- $\Delta p$: $[M L^{-1} T^{-2}]$
- $\rho$: $[M L^{-3}]$
- $\mu$: $[M L^{-1} T^{-1}]$
- $V$: $[L T^{-1}]$
- $d$: $[L]$
- $D$: $[L]$
- Form Dimensionless Parameters To form dimensionless parameters, we typically combine different variables in such a way that their units cancel out. The common dimensionless groups in fluid mechanics include:
- Reynolds number: $Re = \frac{\rho V D}{\mu}$
- Pressure drop ratio: $P_d = \frac{\Delta p d^2}{\rho V^2}$
- Orifice area ratio: $A = \frac{d^2}{D^2}$
- Combine for Dimensionless Forms Now we construct the dimensionless parameters:
-
Calculate $Re$: $$ Re = \frac{\rho V D}{\mu} $$
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Calculate $P_d$: $$ P_d = \frac{\Delta p d^2}{\rho V^2} $$
-
Calculate $A$: $$ A = \frac{d^2}{D^2} $$
- Verify Dimensional Consistency Check that each of these parameters is indeed dimensionless (i.e., has no units). Each result needs to simplify to 1 when substituted with the original dimensions.
The resulting dimensionless parameters are:
- Reynolds number: $Re = \frac{\rho V D}{\mu}$
- Pressure drop ratio: $P_d = \frac{\Delta p d^2}{\rho V^2}$
- Orifice area ratio: $A = \frac{d^2}{D^2}$
More Information
These dimensionless parameters help characterize the behavior of fluid flow through an orifice plate. The Reynolds number, for example, indicates whether the flow is laminar or turbulent. Understanding these parameters is crucial in fluid dynamics and applications in engineering.
Tips
- Failing to include all relevant variables in the dimensionless parameter calculations.
- Miscalculating dimensions and not simplifying to ensure they are dimensionless.
- Mixing units incorrectly, which can lead to incorrect dimension assignments.
