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Two different poppet valve designs implement the same flow path. The two valve designs are shown in the figures below. The Case A uses a large diameter to guide the sliding element when compared to Case B.

Determine an expression for the flow force acting on the valve poppet respectively for Case A and Case B. Assume that the valve poppet angle θ perfectly guides the flow at the valve exist section Ω.

Given:

Poppet valve geometry for two cases (figures above): flow rate Q; exit area Ω; jet force angle θ; and valve pressure drop Δp = p₁ − p₂

Find:

The expression and the direction of the flow force for Case A and Case B.