Fire Heat Flux to Un-Wetted Vessel

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Fire Heat Flux to Un-Wetted Vessel

Saeid Rahimi

05-Nov-2010

Introduction

For unwetted vessels, API-521 section 5.15.1.2.2 states that “recent calculation indicates that heat flux of the fire is in the range ofapproximately 80kW/m2 to 100kW/m2. In absence of any guideline in depressuring standards and design practices, many people are

using this value as net heat input to the gas inside the vessel for depressuring calculation of unwetted (gas filled) vessels. This

technical note reviews the adequacy of this range and recommends alternative way.

Fire heat flux

As discussed in technical note “Effect of Different Parameters on Depressuring Calculation Results” below equation is usually used

to define heat input to gas filled vessels:

q = C1 + C2 time + C3 (C4 – Vessel Temp) + C5 LiqVol(time = t) / LiqVol(time = 0) (1)

where

C1 = Q A

Q = 80-100 kW/m2 

C2 = C3 = C4 = C5 = 0

Using this equation with above coefficients resolves the problem of setting fire heat flux but introduces new questions and  problems. For example:

•  This value seems to be high at first glance compared to wetted vessel heat flux. According to above equation (q = 100 A), anunwetted vessel with surface area of 1 m2 can absorb 100kW while wetted vessel’s heat absorption formula (q = 43.2 F A0.82)

gives 43.2kW (assuming F=1.0). Higher heat flux through unwetted surface than wetted one totally contradicts with heat

transfer fundamentals.

•  Considering constant heat flux of 100kW/m2 will result in highly unreasonable final temperature. As vessel gas content is

depleted by time during depressuring, vessel temperature increases drastically. Using constant heat flux without taking gas andmetal temperature changes into account is not right

Following section introduces a method for calculating heat transfer rate from fire to gas inside the vessel as a function of wall and gas temperature and gas physical properties.

Heat transfer rate

Since the heat transfer coefficient from fire to vessel metal and metal conductivity is much higher than internal heat transfer

coefficient, the overall heat transfer rate from fire to vessel gas is mainly controlled by vessel internal heat transfer coefficient.Internal heat transfer takes place between vessel metal and gas according to below equation:

TAhq  ∆= (2) 

When fire starts, vessel gas and metal are in operating temperatureand heat transfer rate is zero. The temperature of plate in contact with

fire increases rapidly and reaches an equilibrium temperature where

heat gain from fire is equal to heat loss to surrounding (vessel insideand outside). During this period, internal heat transfer is high because

of high differential temperature between vessel metal and gas. This

equilibrium temperature is assumed to be 1100°F (593°C) as advised 

 by API-521. When vessel wall is at 593°C, internal heat transfer 

reduces by time and ultimately stops when gas temperature reaches plate temperature. Figure 1 depicts the trend of main heat transfer 

 parameters as a function of time.

Figure 1 – The profile of heat transfer parameters 

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To simplify the heat transfer calculation, the first minutes of fire during which plate temperature is increasing to 593°C has beenignored and it is assumed that plate temperature and remains constant during depressuring. For such condition (isothermal surface),

free convection heat transfer coefficient can be calculated using below typical equation:

( )mPr Gr C Nu= (3)

Where,

2

32 xTgGr µ∆βρ= ,

k CPr 

 pµ= ,k xh Nu= , gw TTT −=∆  

Vessel type Gr Pr C m x

Vertical104-109 

109-1013 

0.59

0.10

1/4

1/3 L

Horizontal104-109 

109-1013 

0.53

0.13

1/4

1/3 D

Therefore internal heat transfer coefficient can be calculated by equation (4).

m

 p

m

2

32

k 

CxTg

x

k Ch ⎟⎟ ⎠

 ⎞⎜⎜⎝ 

⎛ µ⎟⎟ ⎠

 ⎞⎜⎜⎝ 

⎛ 

µ

∆βρ= (4)

ReplacingavT

1=β in equation (4) where

2

TTT

gwav

+= generates equation (5).

(5)

Equation (5) shows that systems with higher gas density (higher pressure, higher molecular weight, lower temperature) have

higher heat transfer rate. For most process vessels, flow on the inner surface of vessel will be turbulent (109 < Gr Pr < 1013).

Laminar flow (104 < Gr Pr < 109) is observed in very low pressure (basically atmospheric) systems.

Case study

Below table shows the fire heat flux for a horizontal vessel with total surface area of 20m2, containing methane initially at

operating temperature and pressure of 0°C and of 100bar when wall temperature is at 593°C and gas temperature increases.

Tg °C 0 100 200 300 400 500

Tav °C 297 347 397 447 497 547

ρ at Tav  kg/m3  33.555 30.675 28.279 26.250 24.507 22.990

Cp at Tav J/kg °C 3254.5 3421.2 3589.5 3754.6 3913.4 4063.4

µ at Tav  kg/m s 1.97E-05 2.07E-05 2.18E-05 2.28E-05 2.39E-05 2.49E-05

k at Tav J/m s °C 8.38E-02 9.29E-02 1.02E-01 1.12E-01 1.22E-01 1.32E-01

Gr - 3.70E+12 1.71E+13 9.71E+12 5.29E+12 2.60E+12 9.48E+11

Pr - 0.765 0.764 0.764 0.765 0.766 0.767

Gr Pr - 2.83E+12 1.30E+13 7.42E+12 4.05E+12 1.99E+12 7.27E+11

h W/m2 °

C 308.1 284.0 259.1 231.7 199.0 154.1

Q kW/m2  182.74 140.03 101.85 67.90 38.42 14.34

NOMENCLATURE 

A Vessel surface area exposed to fire, m2 

C Constant

Cp Gas heat capacity, J/kg K 

D Vessel diameter, mL Vessel length, m

g Gravity acceleration, 9.81m/s2 

Gr Grashof number, dimensionless

h Heat transfer coefficient, J/s m2 K 

k Gas thermal conductivity, J/s m K 

m Constant

 Nu Nusselt number, dimensionless

Pr  Prandtl number, dimensionless

q Heat transfer rate, J/sec (W)

Q Fire heat flux (q/A), kW/m2

Tg Vessel gas temperature, K 

Tw Vessel wall temperature, K 

Tav Gas average temperature, K x Characteristic dimension, m

β Cubical expansion coefficient, 1/K 

µ Gas dynamic Viscosity, kg/m s

ρ Gas density, kg/m3 

m

av3m

1

1m

1

 p2

T

T

x

k CgCh

⎟⎟⎟⎟⎟⎟

 ⎠

 ⎞

⎜⎜⎜⎜⎜⎜

⎝ 

⎛ 

⎟⎟ ⎠

 ⎞⎜⎜⎝ 

⎛  ∆

µ

ρ=

⎟ ⎠

 ⎞⎜⎝ 

⎛ −

⎟ ⎠

 ⎞⎜⎝ 

⎛ −

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According to the results, fire heat flux can be initially double of API range and it approaches zero at higher gas temperature. Heatflux calculated in above example is almost the highest value possible because other hydrocarbon (ethane and heavier) cannot be

maintained in gas phase unless pressure is reduced or temperature is increased which will result in lower heat transfer rate. Air

remains in gas phase at higher pressure than 100 bar but air heat capacity is mush lesser than methane at the same pressure causing

lower heat transfer rate. Below table shows the effects of these parameters on fire heat flux.

Conclusion

Fire heat flux of 80 to 100kW/m2 is just an approximate range which may result in under-designed or over-designed depressuring

system depending on system condition. This value is certainly conservative for relatively low pressure systems but for high pressure

system it may cause undersized depressuring facilities. Therefore it should be verified on case by case basis.

Fire heat flux for unwetted vessels can be defined in better way by using “fire” model (equation (1)) in hysys with below setting:

C3 = h A kW/°C

C4 = 593°C

C1 = C2 = C5 = 0

This is nothing but reproducing equation (2) by using available tools in Hysys.This is the most accurate way of setting fire heat input for unwetted vessels as

gas temperature is limited to plate temperature and heat input is a function of 

time as the gas temperature (“Vessel Temp” in equation shown in Figure 2) ischanging by time.

Using maximum heat transfer coefficient from above table to set C3 is

conservative approach for depressuring calculation because of two reasons:

•  Heat transfer coefficient decreases as system pressure reduces during

depressuring.

•  Heat transfer coefficient decreases as gas temperature increases duringdepressuring.

So for above example, C3 = 0.308 kW/m2°C x 20 = 6.0 kW/°C. Figure 2 – Hysys fire heat flux equation

Contact

Please feel free to contact [email protected] or  [email protected] should you have any comment, question or

feedback.

Composition Pressure (bar) Temperature(°C) Max heat flux (kW/m2)

Methane 100 0 182.7

Methane 10 -50 43.2Methane 1 0 6.6

Ethane 20 0 76.8

Air 100 0 102.9

Air 1 0 3.6