In previous post, I shared how to calculate reciprocating compressor power if number of compression stage is two and three, respectively. In this post, I will use the same problem/example to estimate reciprocating compressor power if number of compression stage is 4. In the end of this posting series, I will show you the difference of each stage and to see how important to determine number of compression stages.

The method used to calculate reciprocating compressor is the same, whether it is two-stage, three-stage, or four-stage. There are several steps which is repetitive.

__Example__

__Example__

*Compress 2 MMscfd of gas measured at 14.65 psia and 60 ^{o}F. Intake pressure is 100 psia, and intake temperature is 100^{o}F. Discharge pressure is 900 psia. The gas has a specific gravity of 0.80 (23 MW). What is the required horse power?*

__Assuming number of stages is 4__

__Solution__

__Solution__

**Step 1 – Calculate Overall Compression Ratio**

Overall compression ratio is ratio of discharge pressure in absolute pressure and suction pressure in absolute pressure. So,

Pressure ratio = 900 psia/100 psia = 9

**Step 2 – Decide Number of Stage **

Four compression stages.

**Step 3 – Determine Compression Ratio Per Stage**

For four-stage compression, ratio per stage is = (overall compression ratio) ^ (1/4)

Ratio per stage = (9)^(1/4) = 1.73

**Step 4 – Determine 1**^{st} Stage Discharge Pressure

^{st}Stage Discharge Pressure

1^{st} stage discharge pressure = compression ratio per stage x suction pressure

So, 1^{st} stage discharge pressure = 1.73 x 100 psia = 173 psia

**Step 5 – Determine Suction Pressure for 2**^{nd} Stage

^{nd}Stage

Assuming pressure drop = 5 psi,

Then 2^{nd} stage suction pressure = 1^{st} stage discharge pressure – pressure drop = 173 – 5 = 168 psi

**Step 6 – Determine Compression Ratio for 2**^{nd} Stage

^{nd}Stage

As we considered pressure drop between 1^{st} stage discharge pressure and 2^{nd} stage suction pressure, compression ratio of 2^{nd} stage is no longer the same as compression ratio of 1^{st} stage.

For 4-stage compression, compression ratio for 2^{nd} stage = (discharge pressure/2^{nd} stage suction pressure) ^ (1/3)

Compression ratio for 2^{nd} stage = (900/168) ^(1/3) = 1.75

**Step 7 – Determine 2**^{nd} Stage Discharge Pressure

^{nd}Stage Discharge Pressure

2^{nd} stage discharge pressure = compression ratio per at 2^{nd} stage compression x suction pressure for 2^{nd} stage

2^{nd} stage discharge pressure = 1.75 x 168= 294 psia

**Step 8 – Determine k value (heat-capacity ratio)**

K value or heat-capacity ratio is function of molecular weight and temperature. We will determine K value by curve. For most compression application, 150^{o}F curve will be adequate. This should be checked after determining the average cylinder temperature.

So, for molecular weight 23 and temperature of 150^{o}F, we get K value of 1.21.

**Step 9 – Determine discharge temperature of 1**^{st} stage compression

^{st}stage compression

Discharge temperature of compression can also be estimate by using the following equation.

By using equation above:

t_{d} at 1^{st} stage compression = (100 + 460) x (1.73) ^ [(1.21-1)/1.21] – 460

(1.73 is compression ratio at first-stage compression)

t_{d} = 156^{o}F

**Step 10 – Determine cylinder temperature of 1**^{st} stage compression

^{st}stage compression

Cylinder temperature of 1^{st} stage compression is average of suction temperature (100^{o}F) and discharge temperature of 1^{st} stage compression (156^{o}C). Therefore, cylinder temperature of 1^{st} stage compression is 128^{o}F.

**Step 11 – Determine discharge temperature of 2**^{nd} stage compression

^{nd}stage compression

Step 11 is the same as Step 9. Assuming interstage cooling to 120^{o}F.

Use equation as stated in Step 9, we can get discharge temperature:

t_{d} at 2^{nd} stage compression = (120 + 460) x (1.75) ^ [(1.21-1)/1.21] – 460

(1.75 is compression ratio at second-stage compression)

t_{d} = 179^{o}F

**Step 12 – Determine cylinder temperature of 2**^{nd} stage compression

^{nd}stage compression

Cylinder temperature of 2^{nd} stage compression is average of suction temperature to 2^{nd} stage compression (120^{o}F) and discharge temperature of 2^{nd} stage compression (179^{o}C). Therefore, cylinder temperature of 2^{nd} stage compression is 179^{o}F.

**Step 13 – Determine Compression Ratio for 3**^{rd} Stage

^{rd}Stage

We considered there is pressure drop between 2^{nd} stage discharge pressure and 3^{rd} stage suction pressure. Therefore, compression ratio of 3^{rd} stage may no longer the same as compression ratio of 1^{st} stage or 2^{nd} stage

For 4-stage compression, compression ratio for 3^{rd} stage = (discharge pressure/3^{nd} stage suction pressure)

3^{rd} stage suction pressure = 2^{nd} stage discharge pressure – pressure drop = 294 (see Step 7) – 5 = 289 psig

Compression ratio for 3^{rd} stage = (900/289) ^ (1/2) = 1.76

**Step 14 – Determine Discharge Temperature of 3**^{rd} stage compression

^{rd}stage compression

Use equation as stated in Step 9, we can get discharge temperature:

t_{d} at 3^{rd} stage compression = (120 + 460) x (1.76) ^ [(1.21-1)/1.21] – 460

(1.76 is compression ratio at third stage compression, and 120^{o}F is suction temperature to 3^{rd} stage compression which is equal to outlet cooler temperature)

t_{d} = 180^{o}F

**Step 15 – Determine Discharge Pressure at 3**^{rd} Stage Compression

^{rd}Stage Compression

Discharge pressure at 3^{rd} stage compression = suction pressure at 3^{rd} stage x compression ratio for 3^{rd} stage = 289 x 1.76 = 510 psia (see Step 13)

**Step 16 – Determine Compression Ratio for 4**^{th} Stage

^{th}Stage

We considered there is pressure drop between 3^{rd} stage discharge pressure and 4^{th} stage suction pressure. Therefore, compression ratio of 4^{th} stage may no longer the same as compression ratio of 1^{st} stage, 2^{nd} stage, or 3^{rd} stage.

For 4-stage compression, compression ratio for 4^{th} stage = (discharge pressure/4^{th} stage suction pressure)

4^{th} stage suction pressure = 3^{rd} stage discharge pressure – pressure drop = 510 (see Step 15) – 5 = 505 psig

Compression ratio for 4^{th} stage = (900/505) = 1.78

**Step 17 – Determine Discharge Temperature of 4**^{th} stage compression

^{th}stage compression

Use equation as stated in Step 9, we can get discharge temperature:

t_{d} at 3^{rd} stage compression = (120 + 460) x (1.78) ^ [(1.21-1)/1.21] – 460

(1.78 is compression ratio at fourth stage compression, and 120^{o}F is suction temperature to 4^{th} stage compression which is equal to outlet cooler temperature)

t_{d} = 181^{o}F

**Step 18 – Estimate compressibility factor (Z factor) at suction and discharge condition at each compression stage**

From calculation above, we get:

1^{st} stage (P suction = 100 psia, T suction = 100^{o}F)

1^{st} stage (P discharge = 173 psia, T discharge = 156 ^{o}F) ____ see Step 4 and Step 9

2^{nd} stage (P suction = 168 psia, T suction = 120^{o}F) ____ see Step 5. Assuming interstage cooling to 120^{o}F

2^{nd} stage (P discharge = 294 psia, T discharge = 179^{o}F ____ (see step 7 and step 11)

3^{rd} stage (P suction = 289 psia, T suction = 120^{o}F) ____ see Step 13. Assuming interstage cooling to 120^{o}F

3^{rd} stage (P discharge = 510 psia, T discharge = 180^{o}F ____ (see step 14 and Step 15)

4^{th} stage (P suction = 505 psia, T suction = 120^{o}F) ____ see Step 16. Assuming interstage cooling to 120^{o}F

4^{th} stage (P discharge = 900 psia, T discharge = 181^{o}F ____ (see step 17)

Based on operating condition above, we will check compressibility factor based on charts. __Please note that each chart is function of gas molecular weight. So, ensure you use proper chart. __

In this example, molecular weight of gas is 23. So, I will use compressibility factor for gas with molecular weight 23.20.

1^{st} stage, Z factor at suction condition = 0.98

1^{st} stage, Z factor at discharge condition = 0.95

Average Z factor = (0.98 + 0.96)/2 = 0.97

Calculate the same thing for 2^{nd} stage,

2^{nd} stage, Z factor at suction condition = 0.97

2^{nd} stage, Z factor at discharge condition = 0.95

Average Z factor = (0.95 + 0.935)/2 = 0.96

Calculate the same thing for 3^{rd} stage,

3^{rd} stage, Z factor at suction condition = 0.93

3^{rd} stage, Z factor at discharge condition = 0.91

Average Z factor = (0.95 + 0.935)/2 = 0.92

Calculate the same thing for 4^{th} stage,

4^{th} stage, Z factor at suction condition = 0.87

4^{th} stage, Z factor at discharge condition = 0.80

Average Z factor = (0.95 + 0.935)/2 = 0.84

**Step 19 – Estimate BHP per MMcfd from Chart **

BHP per MMcfd is a function of compression ratio and K value. BHP per MMcfd read from figure below use a pressure base of 14.4 psia.

For compression ratio 1.74, and k value 1.21, we get BHP per MMcfd is 36 (1^{st} stage)

For compression ratio 1.75, and k value 1.21, we get BHP per MMcfd is 36.5 (2^{nd} stage)

For compression ratio 1.77, and k value 1.21, we get BHP per MMcfd is 36 (3^{rd} stage)

For compression ratio 1.79, and k value 1.21, we get BHP per MMcfd is 37 (4^{th} stage)

**Step 20 – Check correction factor for low intake pressure and for specific gravity**

Based on figure below, for ratio of compression 36; 36.5; and 37, correction factor for low intake pressure is 1.

Based on figure below, for ratio of compression 36; 36.5; and 37, correction factor for specific gravity is 1.

**Step 21 (Last Step) – Calculate BHP for each stage and total BHP**

BHP = (BHP/MMcfd) x (P_{L}/14.4) x (T_{S}/T_{L}) x (Z _{average}) x (MMcfd)

P_{L} is pressure base, which is 14.65 psia

T_{s} is suction temperature, which is 100^{o}F (560^{o}R)

T_{L} is temperature base, which is 60^{o}F (520^{o}R)

For 1^{st} stage, BHP is 36 x (14.65/14.4) x (560/520) x 0.97 x 2 = 76.12

For 2^{nd} stage, T suction is 120^{o}F (580^{o}R)

Using the same equation, BHP is 36.5 x (14.65/14.4) x (580/520) x 0.96 x 2 = 79.11

For 3^{rd} stage, T suction is 120^{o}F (580^{o}R)

Using the same equation, BHP is 36 x (14.65/14.4) x (580/520) x 0.92 x 2 = 75.17

For 4^{th} stage, T suction is 120^{o}F (580^{o}R)

Using the same equation, BHP is 37 x (14.65/14.4) x (580/520) x 0.84 x 2 = 70.12

Total BHP is 76.12 + 79.11 + 75.17 + 70.12 = 300.51

Based on calculation above, we can illustrate the results below.

I also tried using Ariel Commercial to check if the result is similar. I found that the BHP is 312.01. The difference is about 4%.