For the set of manometer data below, calculate the pressure coefficient (Cp) for Tapping 4 on the outer wall. Give your answer to 3 decimal places.

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BlackTom AI · Accepted Answer

Here's how to approach the problem and what information you need to compute the pressure coefficient Cp for Tapping 4.

First, restating the task: you are asked to calculate the pressure coefficient Cp for Tapping 4 on the outer wall using the provided set of manometer data. The table shows for each tapping number the readings on three manometer limbs: Outer Wall I (mm), Inner Wall I (mm), and Radial Section I (mm). For tapping 4, these readings are: Outer Wall I = 165 mm, Inner Wall I = 73 mm, Radial Section I = 117 mm.

Key concept: Cp is defined as Cp = (p - p_ref) / (0.5 ρ U^2), where p is the local static pressure at the tapping, p_ref is a reference pressure (often the stagnation or ambient pressure, depending on the convention used in your course), ρ is the fluid density, and U is a characteristic flow velocity (often the free-stream velocity at the tap or a reference velocity corresponding to the dynamic pressure used in the calibration).

Next, identify what you must extract from the data:
- You need the local pressure p at tapping 4. In manometer data, p is typically inferred from the differential height readings (the limb extensions) combined with the known density of the manometer fluid and the pressure reference used in the setup. The table provides only the heights (I in mm) for various limbs; you must know how those heights convert to pressure differences (Δp) via the manometer equation Δp = ρ_m g Δh, where ρ_m is the manometer fluid density and Δh is the height difference between limbs.
- You also need p_ref, the chosen reference pressure for Cp. This is not explicitly given in the snippet. It could be the ambient/atmospheric pressure or the stagnation pressure, depending on the experiment protocol.
- You need the dynamic pressure q = 0.5 ρ U^2, which requires the fluid density ρ and the characteristic velocity U (or the corresponding measured/assumed velocity for the flow). These values are not present in the snippet and must be provided by the problem context (or derived from other measurements in the data set).

Because the necessary quantities ρ, U (or q), and the reference pressure p_ref aren’t included in the excerpt, you cannot compute a numeric Cp yet. Here is a structured procedure you would follow once those inputs are available:
1) Determine the pressure at tapping 4 (p4): use the manometer reading for tapping 4 and the appropriate differential relation. If the table gives a direct differential height relative to a reference limb, apply Δp = ρ_m g Δh with the corresponding height difference to obtain p4 relative to the reference pressure p_ref.
2) Compute the dynamic pressure q = 0.5 ρ U^2 using the given or provided ρ and U (or use the problem’s specified q or velocity profile).
3) Compute Cp for tapping 4: Cp4 = (p4 − p_ref) / q. If p4 was already expressed as a pressure difference relative to p_ref, then Cp4 = (p4_rel_to_ref) / q.
4) Round the result to three decimal places as required.

Important notes and common pitfalls:
- Do not mix up the sign of Δp: a higher pressure reading on the manometer limb corresponding to the local pressure will yield a positive Cp if the local pressure exceeds the reference; a lower local pressure yields a negative Cp.
- Ensure consistent units throughout: if heights are in millimeters, convert to meters for Δp if you are using SI units with ρ in kg/m^3 and g in m/s^2. Likewise, ensure ρ, U, and p_ref are in compatible units.
- If the problem provides a single Cp calibration or a map from each tapping’s limb reading to Cp directly, you should apply that mapping. In the absence of such a direct mapping, you must perform the differential-pressure-to-Cp conversion as described.

In summary, to proceed to a numeric Cp value for tapping 4, you must obtain (a) the pressure corresponding to tapping 4 from the manometer readings (and the reference pressure p_ref), (b) the fluid density ρ and a characteristic velocity U (or the provided dynamic pressure q), and (c) apply Cp4 = (p4 − p_ref) / (0.5 ρ U^2). If you supply the missing values (ρ, U or q, p_ref, and how to convert the 165 mm/73 mm/117 mm readings into a pressure difference), I can compute the Cp to three decimal places.

Thermofluids 2025 Fluids Pre-Lab Quiz Thermofluids 2025 Fluids Pre-Lab Quiz

For the set of manometer data below, calculate the pressure coefficient (Cp) for Tapping 4 on the outer wall. Give your answer to 3 decimal places.

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