# Accumulators sizing for energy storage apps.

This week I worked on an accumulator sizing calculator for BOPs, so I have recovered my thermodynamic knowledge and just would like to note somewhere the materials summary and my thoughts about generic accumulator sizing for energy storage applications.

## Let’s get generic formula

First of all, all accumulator calculations based on a generic formula of the polytropic process (Ideal Gas Law):

##pV^{n}=const \hspace{35pt} (1)##

Where the specific value of #n# corresponds to particular cases:
#n=0# for isobaric process
#n=+\infty# for isochoric process
#n=1# for isothermal process
#n=\gamma# for adiabatic process

Now, take a look at the image and apply this formula for the accumulator’s 3 major work stages:

Sizing Accumulators

 An accumulator is filled with Nitrogen. No work pressure is applied. #p_{0}# – pre-charge Nitrogen pressure: #p_{0}=0.9 \cdot p_1# (for energy storage applications). #V_{0}# – Accumulator’s full volume – this number we have to get by calculation. Min work pressure is applied. #p_{1}# – pre-charge Nitrogen pressure at MIN work pressure: #p_{1}=p_{min}# #V_{1}# – Nitrogen volume at MIN work pressure Max work pressure is applied. #p_{2}# – pre-charge Nitrogen pressure at MAX work pressure: #p_{2}=p_{max}# #V_{2}# – Nitrogen volume at MAX work pressure

So, now the formula (1) for the accumulator will look:

##p_0V_0^{n}=p_1V_1^{n}=p_2V_2^{n} \hspace{35pt} (2)##

The volume we need for our consumers is a difference between the volumes at min work pressure and max work pressure:

##\Delta V=V_1-V_2##

Using the relationship of volumes from the formula (2):

##V_1^n=V_0^n\cdot \frac{p_0}{p_1} \hspace{35pt} V_2^n=V_0^n\cdot \frac{p_0}{p_2}##
##V_1=\sqrt[n]{V_0^n\cdot \frac{p_0}{p_1}} \hspace{35pt} V_2=\sqrt[n]{V_0^n\cdot \frac{p_0}{p_2}}##
##V_1=V_0\cdot\sqrt[n]{\frac{p_0}{p_1}} \hspace{35pt} V_2=V_0\cdot\sqrt[n]{\frac{p_0}{p_2}}##
##V_1=V_0\cdot \left(\frac{p_0}{p_1}\right)^{\frac{1}{n}} \hspace{35pt} V_2=V_0\cdot \left(\frac{p_0}{p_2}\right)^{\frac{1}{n}}##
##\Delta V=V_1-V_2=V_0\cdot \left(\frac{p_0}{p_1}\right)^{\frac{1}{n}}-V_0\cdot \left(\frac{p_0}{p_2}\right)^{\frac{1}{n}}##
##\Delta V=V_0\cdot \left[ \left(\frac{p_0}{p_1}\right)^{\frac{1}{n}}-\left(\frac{p_0}{p_2}\right)^{\frac{1}{n}} \right]##

Remember, #V_{0}# is an accumulator’s full volume, so now we can get the minimum required accumulator size:

##V_0=\frac{\Delta V}{\left(\frac{p_0}{p_1}\right)^{\frac{1}{n}}-\left(\frac{p_0}{p_2}\right)^{\frac{1}{n}}} \hspace{35pt} (3)##

But before we do that let’s understand what kind of processes we can expect during the accumulator’s work cycles.

## Slow gas expansion and compression – Isothermal process

In the simplest case, an accumulator works at the same temperature as it was filled with Nitrogen, with slow gas expansion and compression.
In this case, we can assume the process is isothermal #n=1# and #T=const#, so formula (3) is simplified to:

##V_0=\frac{\Delta V}{\frac{p_0}{p_1}-\frac{p_0}{p_2}} \hspace{35pt} (4)##

But in the reality, accumulators never work at the same temperature they were filled. Usually, they are filled at the specific temperature of ambient (mostly in the shop @70*F), so when the accumulator starts to work for example at night, or in winter time, at the start of an operation the Nitrogen pre-charge pressure is different from when accumulator was filled.

So, before using the formula (4) we have to correct Nitrogen pre-charge pressure for at min ambient temperature using formula as for the isochore process (when the gas area of the accumulator is pre-charged at high temperature and then subjected to a pressure change at constant volume due to the environment temp decreasing):

##\frac{p}{T}=const##
where #T# – temperature in Kelvins. So, for temperature in Fahrenheit:

##p_{0@T_{min}}=p_{0@T_{fill}} \cdot \frac{T_{min}+459.67}{T_{fill}+459.67}##

For temperature in Celsius:

##p_{0@T_{min}}=p_{0@T_{fill}} \cdot \frac{T_{min}+273}{T_{fill}+273}##

When the corrected Nitrogen pre-charge pressure is determined, we can use this value in the formula (4):

##V_0=\frac{\Delta V}{\frac{p_{0@T_{min}}}{p_1}-\frac{p_{0@T_{min}}}{p_2}} \hspace{35pt} (5)##

There is no sense to check #V_0# at max ambient temp because the result value will be always bigger at min ambient temp.

Well, formula (5) works well for the isothermal process with slow gas expansion and compression (operation > 1 min) with the stable temperature of Nitrogen that equals ambient temperature. But what about rapid processes?

## Rapid processes – adiabatic process

For rapid processes (operation < 1 min), the temperature of nitrogen changes but there is no transfer of heat to or from the system. This is an adiabatic process with polytropic exponent #n=1.4# (for Nitrogen as a diatomic gas) and generic formula (3) will look:

##V_0=\frac{\Delta V}{\left(\frac{p_{0@T_{min}}}{p_1}\right)^{\frac{1}{1.4}}-\left(\frac{p_{0@T_{min}}}{p_2}\right)^{\frac{1}{1.4}}} \hspace{35pt} (6)##

This is already much better, we got formulas (5) and (6) for slow and rapid processes. But the problem is, both these formulas are based on generic formula (1) for the ideal gas. The real gas requires correction of result accumulator volume because real gas has different behavior, especially at high pressures or too low temperatures.

There are two ways we can proceed to make the correction.

## Real gas – correction factor

The most popular way is to use the special diagram to get the correction factor:

Correction factor for isothermal and adiabatic processes

1. Calculate the ratio of Max/Min pressure, #p_2/p_1#
2. On the graph find the intersection of #p_2/p_1# and the maximum working system pressure #p_2#, which is shown as a curve on the graphs for either an isothermal or adiabatic change of condition.
3. Project the intersection point to the Y-axis to determine the appropriate correction factor, #C_i# or #C_a#
4. Get volume of accumulator for the real Nitrogen gas:

##V_{0 real}=V_0 \cdot C_i \hspace{10pt} or \hspace{10pt} V_{0 real}=V_0 \cdot C_a##

Note: Do not forget, #V_{0 real}# is the minimum required full volume of the accumulator, so accumulator size to be #>V_{0 real}#.

## Calculation via Nitrogen density for isothermal process

Dealing with API 16ST (Recommendations for Coiled Tubing Well Control Equipment Systems) I learned the interesting way of accumulator sizing calculations for the isothermal process: via Nitrogen density.

Let’s get formula for that.

As we know, the equation of state for gases is the Clapeyron-Mendeleev:

##p \cdot V=\frac{m}{M}\cdot RT##
where:
#M# – Gas molar mass = 28.02 (for Nitrogen as a diatomic gas)
#V# – Gas volume, liters
#p# – Gas pressure, in atm
#R# – Molar gas constant = 8.314 J/(mol*K)
#T# – Gas temperature in Kelvins

On another hand, #\rho=m/V#, so we can write the equation of Clapeyron-Mendeleev via density:

##p = \rho \cdot \frac {RT}{M}##

Let’s put pressures from this formula into formula (4) for the isothermal process:

##V_0=\frac{\Delta V}{\frac{\rho_0 \cdot \frac {RT}{M}}{\rho_1 \cdot \frac {RT}{M}}-\frac{\rho_0 \cdot \frac {RT}{M}}{\rho_2 \cdot \frac {RT}{M}}}##

Values #M# and #R# are constant, and value of the temperature #T# is also constant for the isothermal process so we can cut all of them:

##V_0=V_{0 real}=\frac{\Delta V}{\frac{\rho_0}{\rho_1}-\frac{\rho_0}{\rho_2}} \hspace{10pt} or \hspace{10pt} V_0=\frac{\Delta V}{\rho_0 \left( \frac{1}{\rho_1}-\frac{1}{\rho_2} \right)} \hspace{10pt} (7)##
where:
#\rho_0# – density of pre-charged Nitrogen at #p_0# (Nitrogen pre-charged pressure)
#\rho_1# – density of pre-charged Nitrogen at #p_1=p_{min}# (at MIN work pressure)
#\rho_2# – density of pre-charged Nitrogen at #p_2=p_{max}# (at MAX work pressure)

The idea is to take the density of the real (not the ideal!) Nitrogen gas and therefore no correction factors are required.
You can ask – what are the benefits of this method? Where to take real Nitrogen gas density?

The density of the real Nitrogen gas for the various pressures at the temperature of interest can be found in the NIST Chemistry WebBook: https://webbook.nist.gov/chemistry/fluid/. This is an open-source database for different gases and it is free for use… for now, but maybe this statement will change in the future.

With this method, you can get the most realistic value of the minimum required accumulator size #V_{0 real}#.

Do not forget, this method works only for isothermal process (slow gas expansion and compression and at #T=const#)!

## Calculation example

Given:
BOP emergency work requires 16.8 US gal of oil. The BOP operates between 3000 psi and 1600 psi. The operation ambient temperature range is 15°F..120°F. Accumulators are charged with nitrogen in the shop at 70°F. We need to find:
– Nitrogen pre-charge pressure;
– get min volume of accumulators;
– select accumulators size and quantity.

Solution:
#p_1 = 1600 psi#
#p_2 = 3000 psi#
#\Delta V = 16.8 \hspace{5pt}USG#
#T_{min} = 15^\circ F#
#T_{max} = 120^\circ F#
#T_{fill} = 70^\circ F#

From the API, the max allowable time accumulators should be charged to max pressure should not exceed 10 mins. Moreover, BOP rams do not work rapidly, therefore we can assume an isothermal process for filling accumulators and make calculations by both methods and compare results.

Method 1. Using formula (5)

The pressure we need to fill accumulators at 70°F, for energy storage applications:
##p_{0@T_{fill}} = 0.9 \cdot p_1 = 0.9 \cdot 1600 = 1440 psi##

Nitrogen pre-charge pressure at worst case scenario (at min ambient temperature):

##p_{0@T_{min}} = p_{0@T_{fill}} \cdot \frac{T_{min}+459.67}{T_{fill}+459.67} = 1440 \cdot \frac{15+459.67}{70+459.67} = 1290 psi##

The minimum required accumulator volume, based on ideal gas formula (5):

##V_0 = \frac{\Delta V}{\frac{p_{0@T_{min}}}{p_1}-\frac{p_{0@T_{min}}}{p_2}} = \frac{16.8}{\frac{1290}{1600}-\frac{1290}{3000}} = 47.13 \hspace{5pt}USG##

To get this volume for for the real gas, lets determine the pressures ratio first:

## \frac{p_2}{p_1} = \frac{3000}{1600} = 1.875 ##

Next, using graph for the isothermal process, we can find correction factor #C_i = 1.05#.

Therefore real gas volume:

##V_{0 real} = V_0 \cdot C_i = 47.13 \cdot 1.05 = 49.48 \hspace{5pt}USG ##

With this value of accumulators volume we can assume the min quantities of 10 USG and 15 USG accumulators are:

## N_{10} = \frac{49.48}{10} = 4.9 pcs. \rightarrow 5 pcs. ##
## N_{15} = \frac{49.48}{15} = 3.3 pcs. \rightarrow 4 pcs. ##

Method 2. Using formula (7)

The pressure we need to fill accumulators at 70°F, for energy storage applications:
##p_{0@T_{fill}} = 0.9 \cdot p_1 = 0.9 \cdot 1600 = 1440 psi##

Nitrogen pre-charge pressure at worst case scenario (at min ambient temperature):

##p_{0@T_{min}} = p_{0@T_{fill}} \cdot \frac{T_{min}+459.67}{T_{fill}+459.67} = 1440 \cdot \frac{15+459.67}{70+459.67} = 1290 psi##

The minimum required accumulator volume, based on ideal gas formula (7):

##V_0=\frac{\Delta V}{\rho_0 \left( \frac{1}{\rho_1}-\frac{1}{\rho_2} \right)}##

where all density values wi take from the NIST Chemistry WebBook: https://webbook.nist.gov/chemistry/fluid/:

#\rho_0 = 7.2801# – density of Nitrogen at pre-charged pressure #p_{0@T_{min}} = 1290 psi# and at #T_{min} = 15^\circ F#
#\rho_1 = 9.0062# – density of Nitrogen at min work pressure #p_1 = 1600 psi# and at #T_{min} = 15^\circ F#
#\rho_2 = 16.0340# – density of Nitrogen at max work pressure #p_2 = 3000 psi# and at #T_{min} = 15^\circ F#

##V_{0 real}=\frac{16.8}{7.2801 \left( \frac{1}{9.0062}-\frac{1}{16.0340} \right)} = 47.42 \hspace{5pt} USG ##

With this value of accumulators volume we can assume the min quantities of 10 USG and 15 USG accumulators are:

## N_{10} = \frac{47.42}{10} = 4.74 pcs. \rightarrow 5 pcs. ##
## N_{15} = \frac{47.42}{15} = 3.16 pcs. \rightarrow 4 pcs. ##

Which method you trust and will work with – this is your personal decision. I believe in data from NIST database because they are more close to reality rather than using a correction factor. But like you see, the difference in the real volume #V_{0 real}# using both methods is approx 4% and the final qty of accumulators is the same (actually, in the reality, especially for BOP energy storage I prefer to proceed with # N_{15} + 1 = 5 # pcs. accumulators, just for safety).

Please let me know if you find any mistakes in this article.

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