COMSOL 6.4 - Diaphragm Accumulator

Diaphragm accumulators are pressure vessels used in hydraulic systems. A rubber diaphragm separates the hydraulic fluid from an inert, compressible gas, typically nitrogen (Figure 1). Accumulators can serve various purposes. For example, they can provide a temporary energy storage for the hydraulic fluid and compensate for pressure and volume fluctuations, thereby dampening pulsations and shocks.

Figure 1: A diaphragm accumulator consisting of a steel pressure vessel with connections for nitrogen (top) and the hydraulic fluid (bottom). The fluids are separated by a rubber membrane.

The analysis of these accumulators involves solving a fluid–structure interaction (FSI) problem, where two types of fluids exert pressure on both sides of the diaphragm and the surrounding metallic walls.

If the pressure changes in the fluids are slow enough so that momentum and energy transfer remain minimal, it is not necessary to explicitly solve for the fluid domains. Instead, pressure changes can be computed through the equation of state.

This approach eliminates the need to mesh the fluid domain, significantly simplifying the modeling of diaphragm accumulators, especially when contact problems are involved.

Model Definition

Diaphragm accumulators can be manufactured in different ways (screwed, forged, or welded). In this example, the geometry is given for a welded design, where the diaphragm is pressed into the bottom part of the shell with a clamp ring before the accumulator is welded together.

The pressure vessel has a gas connection on the top side, and a hydraulic connection on the bottom side, where the hydraulic fluid (for example, mineral oil) enters the accumulator. The rubber diaphragm mounted in the clamp ring separates the fluids. A valve seat (also poppet or button) made from a thermoset is vulcanized into the membrane. When the accumulator is precharged, the valve seat covers the fluid connection.

The geometry of a small accumulator is illustrated in Figure 2. Some geometric details such as welds, threads, or valves are not modeled explicitly. The geometry is assumed to be axially symmetric.

Figure 2: Diaphragm accumulator geometry.

The accumulator walls and the valve seat are modeled as linear elastic materials with material properties listed in Table 1.

Table 1: Material Properties.
Property	Shell	Valve Seat
Young’s modulus	200 GPa	1.1 GPa
Poisson's ratio	0.3	0.35
Density	7850 kg/m3	1300 kg/m3

The diaphragm is made from nitrile butadiene rubber (NBR). It is modeled as an incompressible Neo-Hookean hyperelastic material with Lamé parameter μ = 0.7 MPa and a density ρ = 1100 kg/m3.

Diaphragm accumulators have different operating conditions. Initially, neither the gas nor hydraulic fluid are pressurized. Before operation, nitrogen is precharged to an operating pressure, p0, which is determined by the operating requirements of the hydraulic system. Typically, the precharge pressure is set between 70% and 90% of the hydraulic fluid’s minimum operating pressure.

During operation, the hydraulic fluid flows into the accumulator when the system pressure exceeds p0. The system’s minimum and maximum operating pressures are denoted p1 and p2, respectively. All pressures, pi (i = 0, 1, 2), are relative pressures with respect to the atmospheric pressure.

In this example, the accumulator is first precharged to a pressure equal to p0 in a stationary study. In a second time-dependent study, a temporary increase of the hydraulic pressure is considered, and the response of the diaphragm and gas cavities are analyzed. The hydraulic pressure is increased to a maximum pressure p2 and subsequently decreased to p1. The time-dependent pressure variation is shown in Figure 3.

Figure 3: Pressure variation in the hydraulic fluid.

Nitrogen is modeled as an ideal gas. As the compression and expansion of the gas take place relatively fast, adiabatic conditions are assumed; that is, heat transfer between the gas and the surrounding walls is assumed negligible. For an adiabatic process, the deformed volume of the cavity, V, is related to its initial volume, Vref, by the equation of state

Here, pref is the reference fluid pressure inside the undeformed cavity, p is the absolute pressure after deformation, and γ is the ratio of specific heats. For nitrogen γ = 1.4. Since p = pref + prel, the equation of state is equivalently written as

where prel is the relative pressure acting on all cavity walls. The volume in the reference and deformed configurations can be obtained using the divergence theorem (or Gauss’s theorem). More information on the volume computation can be found in the section Enclosed Fluids in the Structural Mechanics Theory chapter of the Structural Mechanics Module User’s Guide.

Results and Discussion

In the first study, the accumulator is precharged with nitrogen. There is no relative pressure acting on the hydraulic fluid side. Figure 4 shows the deformation of the diaphragm and the von Mises stress distribution after the precharge step.

Figure 4: von Mises stress distribution after precharging the accumulator with nitrogen gas.

In the second study, the effect of a time-dependent pressure change in the hydraulic fluid is analyzed. The pressure in the hydraulic fluid increases to the maximum pressure, p2, and the rubber membrane deforms significantly. The diaphragm bulges into the gas cavity, thereby compressing the nitrogen. The von Mises stress at p(t = 15 s) = p2 is shown in Figure 5.

Figure 5: von Mises stress at the maximum hydraulic pressure.

Once the hydraulic pressure exceeds the precharge pressure, there is a pressure equilibrium; that is, the gas pressure and fluid pressures balance each other as shown in Figure 6.

Figure 6: Pressure variation of the nitrogen gas and hydraulic fluid.

Figure 7 shows the volume of the hydraulic fluid that enters the accumulator. At a pressure p = p2, a volume of 94.4 ml has entered the accumulator. During the adiabatic compression of the nitrogen chamber, the gas temperature increases significantly as shown in Figure 8.

Figure 7:

Volume of hydraulic fluid entering the accumulator.

Figure 8: Temperature change of the nitrogen gas.

Notes About the COMSOL Implementation

To account for the initial strain state due to press fitting the clamp ring, an node is added. Here, the circumferential strain resulting from the press fit is entered.

The cavities for the nitrogen gas and hydraulic fluid are modeled with the built-in node in the Solid Mechanics interface.

The feature automatically computes the volume of the undeformed and deformed configurations. The advantage of this approach is such that these domains do not need to be meshed.

The pressure load is known for the hydraulic fluid. It can be applied with an ordinary node. However, it is also possible to apply this load with the subnode under . The advantage with this approach is that the volume change is automatically tracked.

Nonlinear_Structural_Materials_Module/Hyperelasticity/diaphragm_accumulator

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Geometry 1

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file diaphragm_accumulator_geom_sequence.mph.

Global Definitions

Import the geometry sequence for the diaphragm accumulator.

Geometry Parameters

In the window, under click .

In the window for , type Geometry Parameters in the text field.

Operating Parameters

Next, add auxiliary parameters for the operating pressures.

In the toolbar, click

and choose > .

In the window for , type Operating Parameters in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
para	1	1	Stepping parameter
p0	0.9*p1	2.7E6 Pa	Precharge pressure
p1	30[bar]	3E6 Pa	Minimum operating pressure
p2	3*p1	9E6 Pa	Maximum operating pressure
t_end	30[s]	30 s	Time

Interpolation 1 (int1)

Add an interpolation function for the prescribed hydraulic fluid pressure.

In the toolbar, click

and choose > .

In the window for , locate the section.

In the text field, type pressure.

In the table, enter the following settings:


t	f(t)
0	0
t_end*0.5	p2
t_end*0.9	p1
t_end	p1

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Function	Unit
pressure	Pa

In the table, enter the following settings:


Argument	Unit
t	s

Click

Definitions

Contact Pair 1 (p1)

In the toolbar, click

and choose .

In the window for , locate the section.

Click

In the dialog, type 13, 60, 65, 73 in the text field.

Click .

In the window for , locate the section.

Click

In the dialog, type 2, 25, 40, 61, 63 in the text field.

Click .

Solid Mechanics (solid)

Linear Elastic Material 1

As the clamp ring has been pressfitted, add an approximate initial strain.

In the window, under > click .

Initial Stress and Strain 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Domain 8 only.

Locate the section. Specify the ε0 matrix as


0	0	0
0	0.01[mm]/R	0
0	0	0

Contact 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

In the fp text field, type 5.

Hyperelastic Material 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundary 31 only.

Enclosed Cavity, Nitrogen Gas

Add an node to model the compressible nitrogen gas. Select the boundaries, which are enclosing the gas in order to compute the volume change, and thus compute the pressure change.

In the toolbar, click

and choose .

In the window for , type Enclosed Cavity, Nitrogen Gas in the text field.

Locate the section. Click

In the dialog, type 5, 9, 10, 16, 34-36, 40-42, 47, 52, 55, 59, 66, 67, 69, 70, 73, 75 in the text field.

Click .

As the selected boundaries do not form a watertight volume, mark the selection as open and select a reference point. The reference point defines a virtual cut plane parallel to the R-axis. In axisymmetry, only the Z-coordinate of the chosen point is taken into account.

In the window for , locate the section.

From the list, choose .

Locate the section. Click to select the

toggle button.

Select Point 13 only.

Prescribed Pressure 1

Precharging the accumulator involves two study steps. First, the known precharge pressure is applied with a subnode. The precharge pressure and the computed volume then serve as the new reference configuration for the equation of state when the hydraulic pressure is applied.

In the toolbar, click

and choose .

In the window for , locate the section.

In the p text field, type p0*para.

Fluid 1

The compression and expansion of the nitrogen gas is assumed to be adiabatic. Change the reference pressure and reference volume in order to set the new reference state for the analysis involving the hydraulic fluid.

In the window, click .

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

Enclosed Cavity, Hydraulic Fluid

In the toolbar, click

and choose .

In the window for , type Enclosed Cavity, Hydraulic Fluid in the text field.

Locate the section. From the list, choose .

Locate the section. Click to select the

toggle button.

Select Point 15 only.

Locate the section. Click

In the dialog, type 2, 11, 13, 25, 60, 61, 63, 65 in the text field.

Click .

Fluid 1

In the window, right-click and choose .

Prescribed Pressure 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the p text field, type pressure(t).

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Right-click and choose .

In the toolbar, click

to close the window.

Materials

Structural steel (mat1)

In the window for , locate the section.

From the list, choose .

Nitrile butadiene rubber (NBR)

In the window, right-click and choose .

In the window for , type Nitrile butadiene rubber (NBR) in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Lamé parameter μ	muLame	0.7[MPa]	N/m²	Lamé parameters
Density	rho	1100	kg/m³	Basic

Thermoset

In the window, expand the > > node.

Right-click and choose .

In the window for , type Thermoset in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	1.1[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.35	1	Young’s modulus and Poisson’s ratio
Density	rho	1300	kg/m³	Basic