Passive Heave Compensation Basics

How does passive heave compensation work for reduction of landing speed?

Heave means vertical motion, in our context vertical motion of the crane hook, caused by waves. Passive heave compensation can be thought of as a spring-mass-damper system with the objective of reducing wave induced motion below the compensator. The below simplified sketch illustrates our scenario:

The crane hook motion follows the sinusoidal given by $\zeta \cos(\omega t)$ , the PHC has stiffness $k$ , the water has mass density $\rho_w$ , while the payload has basic properties $\rho, m, A_\perp$ , respectively for payload mass density, mass and area perpendicular to the heave motion.

Since we in this example are assuming that the payload is subsea it is important to take into account buoyancy, drag and added mass that affects the payload. All these three effects can improve the performance of the PHC.

Added mass

m_A = \rho_w C_A V_R

Where $C_A$ is the added mass coefficient (can be found in DNV RP-N103) and $V_R$ is the reference volume.

Drag

F_D = \rho_w C_D A_\perp \dot z |\dot z|

Where $C_D$ is the drag coefficient and $\dot z$ is the payload vertical velocity.

Buoyancy

F_B = \rho_w V g

Where $V$ is the displaced volume of the payload and $g$ is the acceleration of gravity.

PHC gas spring

We can define an average stiffness of the PHC gas spring as the difference in force from equilibrium stroke to full stroke divided by the change in stroke:

k = \frac{p_1 A_0 – p_0 A_0}{\Delta S}

Where $p_0$ is the equilibrium pressure and $A_0$ is the piston area of the PHC.

Let us assume the following:

The full stroke length is $S$ , and we are at mid-stroke during equilibrium.
We use the ideal gas law with adiabatic compression to calculate the pressure change.
The equilibrium force should be equal to the force of gravity minus buoyancy.

From these assumptions, we then get:

k = \frac{(\rho - \rho_w) \, V \, g}{0.5 \, S} \left[ \left( \frac{V_{\mathrm{eq}}}{V_{\mathrm{eq}} - 0.5 \, A_0 \, S} \right)^\gamma - 1 \right]

$\gamma$ is the adiabatic exponent.

We could further assume that the equilibrium volume can be defined as:

V_{\mathrm{eq}} = (R – 0.5) \, A_0 \, S

Where $R$ is the gas-to-oil ratio, which typically is in the range 2–12 for a PHC. A larger value of $R$ corresponds to a softer spring.

We then get the following expression for the PHC stiffness $k$ :

k = \frac{2 (\rho - \rho_w) \, V \, g}{S} \left[ \left( \frac{R - 0.5}{R - 1} \right)^\gamma - 1 \right]

Which we can rewrite as:

k = \frac{2 \, m \, g}{S} \left( 1 - \frac{\rho_w}{\rho} \right) \left[ \left( \frac{R - 0.5}{R - 1} \right)^\gamma - 1 \right]

Hydraulic flow restriction of PHC

Fluid flow through a restriction is typically given as:

Q = A_f \, \alpha \, \sqrt{\frac{2 \, \Delta p}{\rho}}

Where:

$Q$ is the fluid volumetric flow,
$A_f$ is the smallest flow area,
$\alpha$ is the pressure loss coefficient, and
$\Delta p$ is the pressure loss.

Using this as a basis, we can find the force due to hydraulic restriction:

F_h = A_0 \, \Delta p = A_0 \, \frac{\rho}{2} \left( \frac{A_0 \, \dot{S}}{A_f \, \alpha} \right)^2

Also note that the sign of the hydraulic force will depend on extension or retraction of the rod.
Further, it may have a different magnitude if check valves are present.

The trick with this equation is to know $\alpha$ , which is not easy to calculate.
It should be found using CFD or measurements and may also have many variables.

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Seal friction of PHC

Seal friction is a very complicated topic. It depends on many factors such as:

Pressure of fluid
Pretension of elastic element
Seal material
Speed of piston or piston rod
Surface roughness
Fluid type
Width of seal
Seal configuration

It is too complicated to discuss details in this brief introduction.

Differential equation

Now let us use the above with the following assumptions:

Seal friction is ignored (in reality it can be significant).
Hydraulic restriction is ignored (usually this can be low if the PHC design is good).
Drag is ignored (for a high performing PHC this assumption is OK).
Ignore stiffness and damping of rigging/wire rope.
Ignore hydrodynamics of PHC.
Ignore self weight of PHC.

A more accurate numerical solution with everything included (and a more precise equation of state for the gas pressure) is available from Norwegian Dynamics.

We can apply Newton’s second law to the payload mass to find out how it moves relative to the crane hook.
Let us assume that downwards is the positive direction:

(m + m_A) \, \ddot z = m g – F_B – k \, [ z + z_0 + \zeta \cos(\omega t) ]

This has a steady-state solution given as:

z(t) = \frac{k \, \zeta}{(m + m_A)\, \omega^2 - k} \, \cos(\omega t)

What we want to know is the ratio between the payload motion and the hook motion.

\frac{z(t)}{\zeta \, \cos(\omega t)} = \frac{k}{(m + m_A)\, \omega^2 - k}

We can then change $\omega$ to $\frac{2 \pi}{T_P}$ , where $T_P$ is the wave period, and replace $k$ with our expression above:

\frac{z(t)}{\zeta \cos\!\left(\frac{2\pi t}{T_p}\right)} = \frac{1}{ \displaystyle \underbrace{\left(\frac{m+m_A}{m}\right)}_{\text{Added mass}} \underbrace{\frac{\rho}{\rho-\rho_w}}_{\text{Buoyancy}} \underbrace{\frac{2\pi^2}{g\,T_p^2}}_{\text{Wave period}} \underbrace{\frac{S}{\left[\left(\frac{R-0.5}{R-1}\right)^{\gamma}-1\right]}}_{\text{PHC}} -1 }

Based on this ratio we can define the passive heave compensation efficiency. If the ratio is 0 then the efficiency is 100%, if the absolute value is bigger than 1 then it means we will have resonance. The calculator below can be used as a rough indicator of passive heave compensation performance.

Passive Heave Compensation Efficiency Calculator

Payload mass m (tonnes):

Added mass mₐ (tonnes):

Payload density ρ (kg/m³):

Wave period Tₚ (s):

Stroke length S (m):

Gas-to-oil ratio R:

We can also define the natural period of the PHC as:

T_n = \pi \sqrt{ \frac{m + m_A}{m} \, \frac{\rho}{\rho - \rho_w} \, \frac{2S}{\,g\!\left(\left(\dfrac{R - 0.5}{R - 1}\right)^{\!\gamma} - 1\right)} }

Other subsea uses

Passive heave compensation units can also give other benefits for subsea installations:

Ability to maintain wire tension throughout the landing phase, which prevents sudden vessel heeling.
Mitigation of peak loads in the event of re-lifting of the payload.
Provide tensioning during subsea retrieval to prevent overloading when fixed to seabed.