Flow of Hydraulic Fluid Through Orifices and Gaps

I. Flow of Hyd Fluid Through Orifices

Orifices can be classified into three types: when the length-to-diameter ratio l/d ≤ 0.5, it is a thin-walled orifice; when l/d > 4, it is a long orifice; and when 0.5 < l/d ≤ 4, it is a short orifice. Understanding these classifications helps engineers predict how hyd fluid will behave in different system configurations.

Flow Calculation for Thin-Walled Orifices

Figure 2-16 shows a typical thin-walled orifice with a sharp edge on the inlet side. Due to inertial effects, the hyd fluid stream contracts when passing through the orifice, creating a maximum contraction of the flow cross-section just downstream of the orifice. For thin-walled circular orifices, when the ratio of the upstream channel diameter to the orifice diameter d₁/d > 7, the contraction of the stream is not affected by the inner wall of the upstream channel, and this is called complete contraction. Conversely, when d₁/d < 7, the upstream channel guides the hyd fluid into the orifice, resulting in incomplete contraction.

Applying the Bernoulli equation to the fluid between the upstream cross-section 1-1 and the downstream cross-section 2-2 of the orifice, we have:

Where h_wg represents the local energy head loss, which includes two components: the local pressure head loss h₁ when the cross-section suddenly contracts and the local pressure head loss h₂ when the cross-section suddenly expands. For hyd fluid systems, these loss coefficients are critical factors in accurate flow prediction.

Since A₂ ≤ A₁, we can substitute the appropriate relationships into the Bernoulli equation, considering that A₁ ≈ A₂, so u₁ ≈ v₂, α₁ = α₂; and h₁ = h₂. This results in:

Where Δp is the pressure difference across the orifice, Δp = P₁ - P₂; and C_v is the orifice velocity coefficient, C_v = 1/√(1 + ζ).

The flow rate equation for a thin-walled orifice is thus:

Where C_d is the flow coefficient, C_d = C_vC_c; C_c is the contraction coefficient, C_c = A₂/A₀ = (d₂/d₀)²; A₂ is the area of the contracted section; and A₀ is the area of the orifice cross-section, A₀ = πd₀²/4. The values of C_c, C_v, and C_d for hyd fluid can be determined experimentally.

For fully contracted flow (when the ratio of pipe diameter to orifice diameter d₁/d₀ > 7), C_c = 0.61~0.63, C_v = 0.97~0.98, giving C_d = 0.6~0.62. For incompletely contracted flow (when d₁/d₀ < 7), C_d = 0.7~0.8. Because the flow path is very short, the flow rate of hyd fluid through thin-walled orifices is not sensitive to changes in oil temperature, resulting in stable flow, making them suitable for use as throttle orifices.

Flow Through Short and Long Orifices

The flow rate through short orifices can be calculated using the same formula as for thin-walled orifices, but with a different flow coefficient C_d, typically taken as 0.82. Short orifices are easier to manufacture than thin-walled orifices and are suitable for use as fixed restrictors in hyd fluid systems.

Flow through long orifices is mostly laminar due to increased viscous effects. The flow calculation can use the previously derived laminar flow formula for circular pipes (Equation 2-28):

The flow rate through long orifices depends on the viscosity of the hyd fluid. When the oil temperature changes, the viscosity of the hyd fluid changes, and thus the flow rate also changes. This characteristic is significantly different from that of thin-walled orifices.

A general formula can be derived to summarize the flow equations for various orifice types:

Where A₀ and Δp are the orifice cross-sectional area and pressure difference across the orifice; K is a coefficient determined by the orifice shape, size, and fluid properties - for long orifices, K = d²/(32μl), and for thin-walled and short orifices, K = C_d√(2/ρ); and m is an exponent determined by the orifice length-to-diameter ratio - m = 0.5 for thin-walled orifices and m = 1 for long orifices. This general formula is often used to analyze the flow-pressure characteristics of orifices in hyd fluid systems.

II. Flow of Hyd Fluid Through Gaps

Gaps (or clearances) generally exist between various parts of hydraulic devices, especially between parts with relative motion. The flow of hyd fluid through these gaps results in leakage, known as gap flow. Due to the narrow nature of these passages, the flow of hyd fluid is significantly influenced by the wall surfaces, and thus the flow regime in gaps is always laminar.

There are two types of gap flow: one caused by the pressure difference across the gap, known as pressure flow; the other caused by the relative motion of the two walls forming the gap, known as shear flow. These two types of flow often coexist in hyd fluid systems.

(A) Flow Through Parallel Plate Gaps

Parallel plate gaps can be formed by two fixed parallel plates or by two parallel plates with relative motion. Understanding how hyd fluid behaves in these configurations is crucial for minimizing leakage in hydraulic systems.

1. Flow Through Fixed Parallel Plate Gaps

Figure 2-18 shows the flow through a fixed parallel plate gap. Let the gap thickness be δ, width be b, length be l, and the pressures at the two ends be P₁ and P₂. Considering a small parallelepiped (with volume bdxdy) in the gap, the pressure forces on its left and right ends are p and p+dp, and the frictional forces on its upper and lower surfaces are τ and τ+dτ. The force equilibrium equation is:

After rearrangement:

From the relationship between shear stress and velocity gradient for hyd fluid, τ = μdu/dy, the equation can be transformed to:

Integrating this equation twice with respect to y gives:

Where C₁ and C₂ are integration constants. Applying the boundary conditions y = 0, u = 0 and y = δ, u = 0 to Equation (2-37) gives:

Additionally, in gap flow, the pressure gradient dp/dx along the x-direction is constant:

Substituting these relationships into Equation (2-37) gives:

The flow rate of hyd fluid in pressure-driven flow through a fixed parallel plate gap is thus:

From Equation (2-39), it can be seen that the flow rate of hyd fluid through a fixed parallel plate gap under pressure difference is proportional to the cube of the gap thickness δ. This indicates that the size of gaps in hydraulic components has a significant impact on their leakage rate, a critical consideration in hyd fluid system design.

2. Flow Through Relatively Moving Parallel Plate Gaps

As shown in Figure 2-19, when one plate is fixed and the other moves with velocity u₀, due to the viscosity of the hyd fluid, the fluid adhering to the moving plate moves with velocity u₀, while the fluid adhering to the fixed plate remains stationary. The velocity of the fluid layers in between varies linearly, creating shear flow. Since the average velocity of the fluid is u = u₀/2, the flow rate of hyd fluid through the gap due to the relative motion of the plates is:

Equation (2-40) gives the flow rate of hyd fluid in shear flow between parallel plates. In general, flow through relatively moving parallel plate gaps involves both pressure flow and shear flow. Therefore, the total flow rate through relatively moving parallel plate gaps is the algebraic sum of the pressure flow and shear flow:

Where u₀ is the relative velocity between the parallel plates. The "+" sign is used when the direction of movement of the long plate relative to the short plate is the same as the direction of the pressure difference, and the "-" sign when the directions are opposite. This combined flow behavior is essential to understand when analyzing hyd fluid systems with moving components.

(B) Flow Through Annular Gaps

In hydraulic components, annular gaps exist between the piston and cylinder bore in hydraulic cylinders, and between the spool and valve bore in hydraulic valves. There are two cases for annular gaps: concentric and eccentric, each with different flow formulas that are critical for predicting hyd fluid leakage.

1. Flow Through Concentric Annular Gaps

Figure 2-20 shows flow through a concentric annular gap. The diameter of the cylinder is d, the gap thickness is δ, and the gap length is l. If the annular gap is unrolled circumferentially, it is equivalent to a parallel plate gap. Therefore, substituting πd for b in Equation (2-41) gives the flow rate formula for a concentric annular gap with relative motion between the inner and outer surfaces:

When the relative velocity u₀ = 0, this becomes the flow rate formula for a concentric annular gap with no relative motion between the inner and outer surfaces:

2. Flow Through Eccentric Annular Gaps

If the inner and outer circles of the annulus are not concentric, with an eccentricity e (Figure 2-21), an eccentric annular gap is formed. The flow rate formula is:

Where δ is the gap thickness when the inner and outer circles are concentric; and ε is the relative eccentricity, which is the ratio of the eccentricity e to the concentric annular gap thickness δ, ε = e/δ.

From Equation (2-44), when ε = 0, it becomes the flow rate formula for a concentric annular gap. When ε = 1, under maximum eccentricity, the pressure flow is 2.5 times that of a concentric annular gap. This demonstrates that in hydraulic components, to reduce leakage through annular gaps, the mating parts should be as concentric as possible, which is a key design principle for hyd fluid systems.

Understanding both orifice and gap flows is fundamental to hydraulic system design. The behavior of hyd fluid in these restrictive passages directly impacts system efficiency, control accuracy, and leakage characteristics. Engineers must carefully consider these factors when designing valves, cylinders, and other components that rely on controlled hyd fluid flow.

The equations presented provide a foundation for calculating and predicting hyd fluid behavior, but real-world applications often require empirical adjustments and consideration of additional factors such as fluid temperature, contamination, and surface roughness. Nevertheless, these fundamental principles remain essential for any engineer working with hyd fluid systems.