High-Pressure Accumulator Design Theory

3.3.1   The Role of the High-Pressure Accumulator

In theory, every hydraulic rock breaker needs a variable-pressure accumulator — especially a large high-pressure accumulator.

The high-pressure accumulator, installed at the system inlet of a hydraulic rock breaker, serves three purposes:

(1) To balance the surplus and deficit of system supply and oil consumption. When the pump discharge is greater than the system oil consumption, the high-pressure accumulator absorbs the surplus discharge and acts as an oil storage device. When the pump discharge is less than the system oil consumption, it discharges oil to supplement the deficit, acting as an oil discharge device. The high-pressure accumulator plays the role of balancing flow surplus and deficit in the system, and is an important component for stable system operation.

(2) To absorb system pressure fluctuations and reduce small pressure spikes, protecting pipelines and hydraulic components and increasing their service life.

(3) In the design of hydraulic impact mechanisms using abstract variable theory, it aids in realising the equivalent force. As long as the accumulator is designed correctly, the accurate equivalent force can be obtained, ensuring that the system achieves the required kinematics and dynamics.

Given the important role of the high-pressure accumulator in the hydraulic rock breaker system — and especially its special function of ensuring the system achieves the required kinematics and dynamics — establishing a correct high-pressure accumulator design theory and method is very important.

3.3.2   Effective Discharge Volume of the Accumulator

Effective discharge volume is an important performance parameter of the accumulator and also the basis for accumulator design calculations. When a hydraulic rock breaker runs at steady state, the maximum oil volume that the accumulator stores and discharges in one cycle is called the effective discharge volume, denoted ΔV.

The effective discharge volume ΔV is related to the kinematics characteristics. When the pump flow is fixed and the hydraulic rock breaker's structure and kinematics are fixed, impact energy W_H, frequency f_H, and effective discharge volume ΔV are all necessarily fixed. So when designing the accumulator, the effective discharge volume is already known. How to calculate ΔV will be introduced in later chapters.

3.3.3   Calculating the Effective Volume (Charge Volume) Vₐ of the Accumulator

The basis for calculating accumulator effective volume V_a is its real effective discharge volume ΔV. When ΔV is working inside the accumulator it necessarily causes the system oil pressure to change, and the equivalent force F_g must be maintained. Therefore, the accumulator design calculation method that satisfies the above requirements must be studied. The pressure(force)–volume diagram of the accumulator during operation is shown in Fig. 3-2.

Although the working frequency of a hydraulic rock breaker is not very high, the nitrogen compression and expansion process inside it is also quite rapid, with insufficient time to exchange heat with the surroundings; it can therefore be treated as an adiabatic process. From the gas state equation:

p₁V^k₁ = p₂V^k₂ = p_aV^k_a                                                             (3.12)

where: p_a — charge pressure, i.e. the pressure of the sealed gas;

       V_a — charge volume, i.e. the accumulator volume when the piston is at the impact point (generally the maximum working volume V_amax);

       p₂ — maximum working pressure;

       V₂ — volume corresponding to p₂ (generally the minimum working volume V_2min);

       p₁ — minimum working pressure;

       V₁ — volume corresponding to p₁, V₁ < V_a.

In Eq. (3.12), k = 1.4 is the adiabatic exponent. Clearly:

ΔV = V₁ − V₂                                                                      (3.13)

From Eq. (3.12):

V₁ = V_a (p_a / p₁)^1/k                                                                (3.14)

V₂ = V₁ (p₁ / p₂)^1/k                                                                (3.15)

Substituting into Eq. (3.13) gives:

ΔV = V_a (p_a / p₁)^1/k [1 − 1 / (p₂ / p₁)^1/k]                                     (3.16)

In Eq. (3.16), let p_a / p₁ = a = 0.8 to 1; and the gas working pressure ratio γ = p₂ / p₁, typically γ = 1.2 to 1.45, chosen based on the working characteristics of the hydraulic rock breaker. When a = 1, the minimum working pressure of the piston equals the charge pressure (p_a = p₁); in this state V₁ = V_a. To prevent the accumulator membrane from touching the base at the minimum working pressure of the hydraulic rock breaker — which would shorten service life — a should be set to less than 1.

There are two considerations for choosing γ: when γ is large, because the accumulator works in an adiabatic state, the temperature rises sharply, which can cause premature deterioration of the accumulator membrane or even burn it out; but increasing γ can effectively reduce the effective volume V_a of the accumulator, which is very beneficial for reducing the structural size of the accumulator. The designer must weigh the pros and cons and decide based on the application conditions; therefore:

ΔV = V_aa^1/k (1 − 1 / γ^1/k)                                                       (3.17)

From Eq. (3.17), the effective volume of the accumulator can be found:

V_a = ΔVγ^1/k / [a^1/k (γ^1/k − 1)]                                                   (3.18)

Eq. (3.18) shows that, from the effective discharge volume ΔV, the corresponding charge volume can be found, to ensure that the designed kinematics and ΔV are achieved. In practice, the effective discharge volume ΔV is the oil that the accumulator supplements to the piston during the power stroke, to make up for the pump's insufficient supply.

For the design calculation of effective discharge volume ΔV, please refer to Section 7.5. To satisfy the requirements of optimal design, for different design objectives, the calculation of effective discharge volume ΔV changes with the selected α_u (see Sections 7.2.5 and 7.27a).

3.3.4   Calculating Minimum Working Pressure p₁ and Charge Pressure pₐ

At this point, although V_a has been found and can be used to design the structural parameters of the accumulator, the design calculation task for the accumulator is not yet complete. The most critical issue is how to control the oil pressure to ensure that the equivalent force is achieved; and only by achieving the equivalent force can the designed kinematics be guaranteed, which in turn guarantees ΔV. In other words, there is a corresponding relationship between ΔV and F_g.

It must be pointed out that when V_a is a fixed value, p₁, p₂, and p_a can have many combinations, realising multiple equivalent forces, multiple dynamics, and multiple kinematics — i.e. multiple ΔV values. The following task is, given a fixed V_a, to find the combination of p₁, p₂, and p_a that can achieve the required equivalent force F_g and ΔV. Because when p_a changes, W_H, f_H, ΔV, p₁, and p₂ all change accordingly. In other words, there must be a charge pressure p_a that can guarantee achieving the equivalent pressure p_g. Of course, the basis for finding p_a is p₁ and p₂, i.e. the equivalent pressure p_g. Once the relationships among these parameters are understood, the method for finding p₁, p₂, and p_a from the equivalent pressure p_g can be studied.

Fig. 3-2 describes the p–V diagram of the high-pressure accumulator during operation. Based on this diagram, and combining with the equivalent force principle — the work done by the varying force equals the work done by the equivalent force — we have:

p_g ΔV = ∫_V₂^V₁p dV                                                                 (3.19)

In Eq. (3.19):

p = C / V^k

Substituting into Eq. (3.19) and integrating:

p_g ΔV = C ∫_V₂^V₁ dV / V^k = 1 / (1 − k)  (p₁V^k₁V^1−k₁ − p₂V^k₂V^1−k₂)    (3.20)

Therefore:

p_g ΔV = 1 / (1 − k)  (p₁V₁ − p₂V₂)                                         (3.21)

Eliminating V₁ and V₂ by substitution and substituting Eq. (3.17) gives:

p_g = p₁ / (k − 1)  ·  (γ − γ^1/k) / (γ^1/k − 1)                                (3.22)

After rearranging:

p₁ = p_g (k − 1)  (γ^1/k − 1) / (γ − γ^1/k)                                       (3.23)

In Eq. (3.23), p_g is the equivalent pressure applied to the piston pressure-bearing face. Considering system pressure losses, it should be expressed as the system rated pressure p_g = p_H / K. The p₁ and p₂ obtained in this way will be closer to the actual values. Therefore:

p₁ = (p_H / K)(k − 1)(γ^1/k − 1) / (γ − γ^1/k)                                   (3.24)

p₂ = γp₁                                                                             (3.25)

p_a = ap₁                                                                             (3.26)

In Eq. (3.24), the resistance coefficient accounting for system pressure losses is K = 1.1 to 1.2.

When the high-pressure accumulator of a hydraulic rock breaker operates at these parameters, it guarantees that the equivalent force motion effect is achieved, that the designed kinematics are realised, and that the required impact energy and impact frequency are delivered. In this way a complex calculation problem is simplified and a nonlinear problem is linearised.

Based on the above, the hydraulic impact device (hydraulic rock drill and hydraulic rock breaker) — a nonlinear system — is converted into a linear system. From a theoretical perspective, the piston can move over stroke S according to any pattern whatsoever, as long as it can be controlled and, at the impact point, reaches the required maximum velocity v_m — all of this is feasible. For every piston motion pattern, there must be a corresponding force variation pattern; the two are related as cause and effect. In other words, whatever motion pattern the piston has, a corresponding force variation pattern must be applied to it — force is the cause, motion is the effect.

Of course, after designing the optimal motion pattern, the corresponding force variation pattern can also be found, thus raising two theoretical topics for hydraulic rock breaker research: the kinematics and dynamics of the hydraulic rock breaker.