ASHRAE 2001 HVAC Fundamentals Handbook

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Unsteady Flow Conduit flows are not always steady. In a compressible fluid, the acoustic velocity is usually high and the conduit length is rather short, so the time of signal travel is negligibly small. Even in the incompressible approximation, system response is not instantaneous. If a pressure difference ∆p is applied between the conduit ends, the fluid mass must be accelerated and wall friction overcome, so a finite time passes before the steady flow rate corresponding to the pressure drop is achieved.

Near the start of the straight conduit, the layer is very thin (and laminar in all probability), so the uniform velocity core outside has a velocity only slightly greater than the average velocity. As the layer grows in thickness, the slower velocity near the wall requires a velocity increase in the uniform core to satisfy continuity. As the flow proceeds, the wall layers grow (and the centerline velocity increases) until they join, after an entrance length Le . Application of the Bernoulli relation of Equation (5) to the core flow indicates a decrease in pressure along the layer.

The elbow meter employs the pressure difference between inside and outside the bend as the metering signal (Murdock et al. 1964). A momentum analysis gives the flow rate as 2 R 2 ∆p πD Q theor = ---------- -------  ---------- 4 2D  ρ  (40) where R is the radius of curvature of the bend. Again, a discharge coefficient Cd is needed; as in Figure 18, this drops off for the lower Reynolds numbers (below 105). 13 2 ∆p D ρL – 32νθ V = ------  --------- 1 – ------ exp  --------------2   L  32µ ∆p D (45a) For long times (θ → ∞), this indicates steady velocity as 2 2 ∆p D ∆p R V ∞ = ------  --------- = ------  ------ L  32µ L  8µ (45b) as by Equation (8).

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