IEC 61511 is the standard that set the requirements for Safety Instrumented Systems (SIF). As part of this standard, a SIF is required to act within adequate response time to bring the process or a system to a safe state. This is called response time of a SIF and the time available to act before anything dangerous happens is called process safety time (PST). (See part 1, part 2, part 3, part 4  part 5, part 7 and part 8 for more information).

PST calculation and definition is a mandatory requirement for all the instrumented functions to which IEC 61511 is applicable.

This part contains a sample calculation to show a simple calculation with assumptions, available data and calculation procedure.

Definition of SIF loop
In the previous parts (See part 1, 2, 3, 4 and 5), I tried to define PST and calculation methodology and at the end how to assess and explain the results.
In this part, I will try to define and explain a sample calculation.

Consider a temperature switch after a gas heater) and after a knock-out drum, as shown in figure below that is providing the gas to the final consumer. This temperature switch protects the downstream heater #2 piping and stop flow to the final user (e.g. a compressor seal system).

###### Figure 1 LTSD downstream heater #2 protects downstream users from condensation

One of the causes of low temperature is failure of heater #2. In this case the gas may condense in the downstream heater’s piping and may damage the end user (e.g. a compressor seals) leading to loss of containment and gas flow to the environment with risk of fire and explosion.

In other words, the intention of LTSD after heater #2 is to prevent the temperature reaching the gas dew point after knockout vessel/heater #2.

## Assumptions and known data

Followings are assumptions and data are valid for the system described/shown in the Figure 1:

1. Minimum gas temperature before the heater #2 is assumed to be 40 °C. Minimum backup gas temperature before PDCV is 3°C (trip setting of LTSD-1).
2. Maximum flow through PDCV is 417.8 kg/h equal to 7.286 m3/h, based on Heat and Mass Balance.
3. The outlet pressure of PDCV is 75 barg.
4. Using a simulation software, one can calculate gas physical properties at 75 barg and 50°C, Molecular weight of gas us 18.25 kg/kmole.
5. Gas molecular weight is 18.25 kg/kmole, Gas Cp/Cv=1.49 and Cp= 2441 J/kg K.
6. Gas kinematic viscosity is 0.000014 kg/m.s (N.s/m2). Gas density is 57.4 kg/m3. Gas thermal conductivity is 0.042 W/m K.
7. The temperature sensing element is a thermowell that measures the fluid (gas temperature).
8. The pipe wall temperature and thermowell temperature are at equilibrium.
9. The pipe wall and fluid form an adiabatic system with no heat transfer to surroundings.
10. Complete instantaneous failure of the upstream heater #2 is allowed for.
11. The gas will lose a negligible amount of energy and will not change temperature in traversing the pipe segment.
12. Fouling has no effect on heat transfer.
13. The conduction within the pipe wall is instantaneous therefore the pipe will be of uniform temperature.
14. Calculation of a simplistic process safety time well above the expected system response time is sufficient.
15. Gas dew point is assumed as design temperature to prevent condensation in the downstream system. The dew point of the gas after heater #2 is assumed as 40 °C (worst case).
16. There is also a backup gas to the system. The dew point of the backup gas is assumed (-10 °C) worst case.
17. LTSD-2 has a trip set point of +50°C.
18. Outlet of Heater #2 is traced, however no credits have been taken for the electrical heat tracing of the line downstream of the Heater #2.
19. Downstream pipe has initial the trip temperature.
20. Downstream pipe (designated by D, in formulas) is 1″-Stainless steel, Schedule 40S and piping length between the gas heater and the user is assumed to be 3m.
21. Pipe thermal conductivity (Kn) is taken from engineeringtoolbox.com and is 14.4 W/m K.
22. Initial pipe wall temperature is 50°C.
23. Using the gas volumetric flow and pipe diameter, gas velocity v is 3.63 m/s. This velocity will be used in Re number calculation.
24. Cold gas scenario PST is more severe than cold back-up gas PST, as the trip set point has the lowest margin for the cold gas scenario. Only the cold gas scenario after Heater #2 is calculated.
25. Heat transfer coefficient for forced convection of gases is typically in the region of 10 – 1000 W/m².K. Later on we need this range to see if our calculated h is within the range.
26. Stopping end user takes 10 seconds (e.g. closing a valve or stopping a motor).
27. A thermowell/Temperature transmitter will detect a temperature within 1 seconds.
28. Logic solver (ESD system) is assumed to react within 0.3 seconds).

Calculation
The intention of calculation is to calculate the heat loss required to reach gas dew point (40 °C)
For this calculation we can use the simple heat transfer formula: q=m X Cp X dT.

Heat loss required to reach fuel gas dew point (40 °C)
Pipe is 1 inch, schedule 40S and 3m length. It has a nominal diameter of 0.03 m, with a weight of 2.54 kg/m (Pipe data from arvindpipe.com).
Pipe mass is then 3 m x 2.54 kg/m = 7.6 kg
Cp of stainless steel (data from: www.engineeringtoolbox.com) is 0.49 kJ/kg.K
Starting Temperature T1= 50 °C and Trip set point is 40°C.
Heat loss required to reach 40°C will then be -37 kJ (=7.6 kg x . 0.49 kJ/kg.K x (50-40) K)

Heat transfer coefficient hci
For this part of calculation, we must calculate heat transfer coefficient inside the pipe because of forced convection.
Heat transfer between the gas and the steel body is determined by the Dittus-Boelter equation:

The Dittus-Boelter correlation is valid for turbulent flow where Re > 10,000 and 0.6 < Pr < 160.
The Reynolds number is calculated as follows:

The Prandtl number is calculated as follows:

Where:
ν is momentum diffusivity
α is thermal diffusivity
μ is Viscosity
kg is Thermal conductivity
ρ is Gas density @ P and T
v is Gas velocity through pipe
D is pipe diameter
Cp is gas specific heat capacity

Using the information in assumption section above:
Re=385,242 and Pr=0.84. Then the Nu number will be 631.9 and hence hci=989.1 W/m².K.
This value is within the range of expected convective heat transfer coefficient (assumption 24).

Heat transfer from pipe to gas
Now that the forced convection heat transfer coefficient is calculated, we need to calculate how much heat is taken away by gas flow.
Initial pipe wall temperature is 50°C. Gas flows at a temperature of 40°C.

Overall heat transfer coefficient comprises of two parts: forced convection heat transfer (hci) inside pipe and pipe conduction heat transfer (Sn/Kn).

Using the formula:

Then U is 803 W/m2 K.

The heat taken away by flowing gas can be calculated by general heat transfer formula:

Pipe diameter is 0.03 m with a length of 3m then:
A (heat transfer area) is A= 3.14 x D x L = (3.14 x 3 x 0.03) = 0.3 m2
From assumption section: t1 = 40 °C and t2=50 °C
Then heat carried away by gas flow equals to:
803 W/m2 K x 0.3 m2 x (- 10) °C= – 2409 W = – 2409 J/s = – 2.4 kJ/s

Time to cool down the piping

We have calculated the heat required to be taken away from pipe to reach to 40 °C, as well as heat that gas is taking away while flowing in the 1” pipe.
If we divide these two values then the time that it takes to cool down the piping after heater #2 from 50°C to 40°C is calsulated.

Process Safety Time (PST) = (- 37 kJ) / (-2.4 kJ/s) = 18.5 seconds = 0.3 minutes

SIF response time
Using assumption for SIF components, SIF response is:
Thermowell response time + logic solver response time + final element response time equals to:
1+0.3+10 = 11.4 seconds

Comparison of SIF response time with PST
SIF response time is 11.4 seconds and PST calculated is 18.5 seconds. This means the SIF will react fast enough to bring the process to a safe state.
This is (11.4/18.5=0.6) or 60% of the PST available which is another check to see if calculated PST is acceptable (Refer to part 4).
Conclusion
Response time is less than PST and hence the SIF can act within time to prevent reaching the lower temperature (dangerous temperature) downstream heater #2.