### Sample Problem Statement

Design a pressure relief valve for the following service.

Protected equipment : Separator vessel

Relief service : Fuel Gas

Reason for relief : Blocked gas discharge

Relieving Rate : 10,000 kg/hr

Gas Density : 4.1 kg/m^{3}

Ratio of specific heats for the gas (C_{P}/C_{V}) : 1.55

Compressibility factor of gas = 0.95

Molecular weight of gas = 20 gm/mole

Relieving temperature = 20^{0}C

Set pressure : 4.5 barg

Accumulation : 10%

Back pressure at relief valve discharge : 2.1 barg

Type of relief valve : Balanced Bellows

### Solution

This sample problem for relief valve sizing calculation can be solved in following steps, which are based on pressure relief valve sizing procedures described in API RP 520 Part I. The first step is to determine whether the gas flow type is critical or sub-critical.

### Step1

The first step is to determine if the flow conditions are critical or sub-critical. For this purpose the critical flow pressure (P_{cf}) downstream to the relief valve has to be calculated using the following equation from API RP 520 Part I,

P_{1} : Relieving pressure upstream to relief valve in bara

P_{1} = 5.96 bara (including 10% accumulation)

k : Ratio of specific heats of the gas .. (C_{P}/C_{V})

k = 1.55

P_{cf} : Critical flow pressure for the given flow conditions in bara

P_{cf} = 3.0 bara

If the pressure downstream or back pressure (3.113 bara) of the relief valve is greater than the critical flow pressure (3.0 bara), then the flow is said to be of sub-critical type. In the present case, back pressure (3.113 bara) > P_{cf} (3.0 bara). Hence the flow is of sub-critical type.

### Step2

Next step is to use the relief valve sizing procedure given in the API 520 part I for sub-critical flow type gas relief valve. The following equation is used,

A : orifice are requirement in mm^{2}

W : flowrate in kg/hr

W = 10,000 kg/hr

T : Temperature in Kelvin

T = 273.16 + 20 = 293.26 K

Z : compressibility factor of gas

Z = 0.95

M : Molecular weight of the gas

M = 20 gm/mole

P_{1} : Relieving pressure upstream to relief valve in bara

P_{1} = 5.96 bara (including 10% accumulation)

P_{2} : back pressure downstream to relief valve in bara

P_{2} = 3.11 bara

K_{d} : Coefficient of discharge

K_{d} = 0.975 for gas service

K_{C} : combination correction factor for use of rupture discs

K_{C} = 1 (for absence of rupture disc upstream to the valve)

F_{2} : Coefficient of sub-critical flow

k : Ratio of specific heats of the gas .. (C_{P}/C_{V})

k = 1.55

r : ratio of back pressure to relieving pressure

r = P_{2}/P_{1} = 0.58

hence, F_{2} = 0.5985

Plugging this data into equation (1), the minimum required orifice area for the relief valve is calculated to be,

A = 34.07 cm^{2}

### Step3

The next higher available orifice area should be selected for the pressure relief valve to be actually installed. Referring to the table of standard orifice sizes given in EnggCyclopedia's relief valve sizing calculator, next high available orifice area is selected to be A = 41.20 cm^{2} and the corresponding designating letter is 'P'. Available valve size is '4P6', which corresponds to different nozzle sizes for suction and discharge nozzles of the pressure relief valve. For a '4P6' valve suction nozzle is of 4" size and discharge nozzle of 6" size.