J. S. KIM and R. HAINES, Covestro, Baytown, Texas
Storage tanks filled with condensable vapors can be damaged due to sudden heavy rain if the venting devices are inadequate. If vapor condensation occurs in the tank, the thermal inbreathing rate can exceed the venting rates given by American Petroleum Institute (API) 2000, which recommends performing an engineering analysis of heated, uninsulated tanks with vapor space temperatures > 120°F. However, there is no specific guideline for the engineering analysis.
This article extends the simplified dynamic approach to vapor condensation inside storage tanks.1 This simplified dynamic approach is used for non-condensable vapors: on the contrary, this simplified approach considers the vapor condensation in the storage tanks and provides an estimate for the maximum thermal inbreathing rate without the numerous calculations required for the dynamic approach. Unlike existing models for vapor condensation, this simplified approach considers the effects of the vapor condensation using the tank inside condensation heat transfer coefficient, the vapor properties and the amount of rainwater. This simplified approach is also applicable to multiple components as well as single components. The calculation results are compared with published experimental results.2−5
Assumptions. The assumptions used in this simplified approach for storage tank thermal inbreathing rates are based on available API 2000 guidelines, as follows:6
Governing equations for the simplified approach. The thermal inbreathing rate for vapor condensation is highly dependent on the tank inside heat transfer coefficient, the amount of rainwater and the energy balance on the tank plate. A simplified approach is developed in this article to simplify the mathematics for the thermal inbreathing rate.
The amount of rain for 1 min is determined using Eq. 1.7 The rain angle is 60° from vertical based on a 10 m/sec wind velocity.
Mrain = (Rain Intensity / 60) × [(D2 × π / 4) + D × H × COS(∝ – 90°)] (1)
where:
Mrain = the amount of rain for 1 min, lb
Rain intensity = the rain flow density, lb/ft2/hr
D = the diameter of tank, ft
H = the height of tank, ft
∝ = the rain angle with respect to vertical, 60°.
The vapor condensation heat transfer coefficient seems to decrease after initial vapor condensation.2 The vapor condensation heat transfer coefficient is time-dependent and rapidly decreases with time. Due to the lack of suitable correlations, the proposed approach assumes that the maximum thermal inbreathing rate is reached around 1 min after the start of sudden heavy rain. This assumption is consistent with the literature.2
A two-step iterative calculation, described below, is required to find the tank thermal inbreathing rate at 1 min. Eq. 2 represents an energy balance on the tank plate. Step 1: Determine the final tank plate temperature using Eqs. 2–4 by rearranging those equations—for the first trial, assume that the final vapor temperature is equal to the initial vapor temperature. The mass of the tank plate should be based on the roof plate thickness, resulting in a conservative design. Step 2: Determine the new trial final vapor temperature using Eqs. 5–7. The Antoine equation is convenient to determine the actual vapor temperature at the calculated vapor pressure of condensable vapors using Eq. 6. The denominator of 10 in Eq. 7 is to avoid diverging for less volatile vapors. For volatile condensable vapors, the denominator can be 1. Repeat Steps 1 and 2 until there is no change in the new trial final vapor temperature.
Final Tplate = [(Q1 – Q2 / 60) + (Mplate × Cpplate × Start Tplate + Mrain × Cprain × Train )] / (Mplate × Cpplate + Mrain × Cprain ) (2)
Q1 = hi × f × Area × [((Start Tvapor + Final Tvapor ) / 2) – ((Start Tplate + Final Tplate ) / 2)] (3)
Q2 = h × Area × [((Start Tplate + Final Tplate ) / 2) – Tamb ] (4)
Mcond.vapor = [(Q1 / 60) – Mvapor × CPvapor × (Start Tvapor – Final Tvapor )] / ∆H (5)
P*at final T = P*at start T × [1 – ((Mcond.vapor × 10.73 × (460 + Start Tvapor ) × 100) / (Vtank × MW × 14.7 × x))] (6)
New Trial Final Tvapor = Final Tvapor – [(Final Tvapor – Tvapor at P*at final T ) / 10] (7)
Final Tplate = the final temperature of the tank plate, °F
Q1 = the heat gain of the tank plate from condensable vapors in the tank, Btu/hr
Q2 = the heat loss from tank plate to ambient air, Btu/hr
Area = the total tank surface area (shell and roof), ft2
Mplate = the mass of the tank plate (shell and roof), lb
Cpplate = the heat capacity of the tank plate, Btu/lb/°F
Start Tplate = the start temperature of the tank plate, °F
Cprain = the heat capacity of rain, Btu/hr/lb/°F
Train = the rain temperature, °F
hi = the tank inside heat transfer coefficient at initial conditions, Btu/hr/ft2/°F
f = the correction factor for the tank inside heat transfer coefficient at initial conditions
h = the ambient air heat transfer coefficient, 2.6415 Btu/lb/°F
Tamb = the ambient temperature, °F
Mvapor = the mass of total vapors in the vapor space, lb
CPvapor = the heat capacity of total vapors in the vapor space, Btu/lb/°F
Start Tvapor = the initial temperature of the vapor space at time 0, °F
Final Tvapor = the final temperature of the vapor space at 1 min, °F
Mcond.vapor = the amount of condensed vapors for 1 min, lb
∆H = the heat of condensation of condensable vapors at Start Tvapor , Btu/lb
P*at start T = the vapor pressure of condensable vapors at time 0, psia
P*at final T = the vapor pressure of condensable vapors at 1 min, psia
MW = the molecular weight of the condensable vapors, lb/mole
x = mole % of condensable vapors
New Trial Final Tvapor = the final vapor temperature for the next trial, °F
Tvapor at P*at final T = the actual vapor temperature at 1 min, °F.
Once the final vapor temperature is determined, the tank thermal inbreathing rate is obtained using Eq. 8:
VIT = [(Mcond.vapor / ρcond.vapor ) + Vtk × ((Start Tvapor – Final Tvapor ) / (460 + Start Tvapor ))] × 60 (8)
VIT = the maximum thermal inbreathing rate, sft3/hr of air
Vtk = the tank volume, ft3
ρcond.vapor = the density of condensed vapors at Final Tvapor , lb/ft3.
Interestingly, the influence of the vapor properties (heat of condensation and vapor density) on thermal inbreathing rate is small because the value changes of ∆H × ρ in the various vapor space contents are relatively small, as shown in TABLE 1. From TABLE 1, it is obvious that the vapor pressure of the condensable vapors significantly contributes to the thermal inbreathing rate, while the heat of condensation and condensable vapor density are insignificant. The mole percentage of condensable vapors in the tank is a major influence on the thermal inbreathing rate. Generally, a higher mole percentage of condensable vapors results in a higher thermal inbreathing rate. If the thermal inbreathing rate calculated is less than the thermal inbreathing rate for dry air,1 the condensable vapor is non-volatile. The thermal inbreathing rate for dry air should be the minimum design value for non-volatile vapors.
Tank inside condensation heat transfer coefficient. One of the key parameters in Eq. 3 is the tank inside heat transfer coefficient for vapor condensation. The tank inside heat transfer coefficient of 0.8805 Btu/hr/ft2/°F (5 W/m2/K) is considered reasonable for dry air; however, there is no available tank inside heat transfer coefficient for condensable vapors in API 2000.
A few existing models use a tank inside vapor heat transfer coefficient and neglect the heat conduction, though the condensation mass flux is included in the mass balance.8 For vapor condensation, the heat transfer coefficient is significantly increased due to film condensation at the tank wall or homogeneous condensation in the vapor space.
FIG. 1 was developed based on the information for condensation heat transfer coefficients for condensable vapors with non-condensable gasses by Kern,9 who proposed the condensation heat transfer coefficients for oil vapors and steam in the presence of non-condensable gasses. Kern indicates that many condensers designed using FIG. 1 have operated successfully. This article uses the heat transfer coefficient of condensable vapors containing non-condensable vapors at the starting vapor space temperature. The heat transfer coefficient is applicable to multiple components as well as single components.
Eqs. 9 and 10 best represent FIG. 1 as a function of the mole percentage of condensables. The mole percentage of condensables is determined by the vapor pressure of the saturated condensable vapors. The vapor space of storage tanks is generally filled with condensable vapors and non-condensable air or nitrogen.
For water vapor (steam) (Eq. 9):
hi = 0.000346 × x3 – 0.022947 × x2 + 2.574234 × x + 0.8805 (9)
For organic compounds (hydrocarbon) (Eq. 10):
hi = 0.000248 × x3 – 0.014980 × x2 + 2.060429 × x + 0.8805 (10)
The tank inside condensation heat transfer coefficients from Eqs. 9 and 10 are effective for a short time at the beginning of condensation. The initial vapor condensation spontaneously forms a temperature gradient in the vapor space, resulting in a significant reduction of the heat transfer coefficients. Estimating the condensation heat transfer coefficient is quite complex and published information for the tank thermal inbreathing is rare. Therefore, it is assumed that the condensation heat transfer coefficient during the initial 1-min period is approximately hi × 0.65(f, correction factor) for saturated vapors. For unsaturated vapors, the condensation heat transfer coefficient is approximately hi × 0.4(f, correction factor).
Unheated storage tanks can be considered filled with unsaturated vapors. If the amount of condensed vapors at the initial vapor temperature is not less than the mass of all condensable vapors or the final temperature of vapor space at 1 min is lower than the initial temperature of vapor space by 7°F, the condensation heat transfer coefficient is approximately hi × 0.15(f, correction factor). A low heat transfer coefficient is expected for small tanks due to the small amount of condensable vapors. When correcting the tank inside condensation heat transfer coefficient, it is important to consider nonuniform vapor temperatures in the tank since the simplified approach is based on a uniform vapor temperature. The correction factors are based on the analysis of existing literature models and experiment data.3,4,10 More experimental data are needed to further optimize the correction factors.
A comparison of the simplified approach with experiments. The following three examples from literature are chosen to validate this simplified approach on experiments.2,3,4,5 TABLE 2 shows the validation details for the three examples.
1) Details of Example 1 (Holtkoetter’s small-scale experiments):
2) Details of Example 2 (Schmidt’s small scale experiments):
3) Details of Example 3 (Schmidt’s large scale experiments):
For Example 1, three experiments for water, isopropanol and methanol were performed indoors with a very small spray water rate of 10 kg/hr. This leads to similar tank inside heat transfer coefficients for water, isopropanol and methanol because the small spray water rate limits the tank inside heat transfer coefficient (cooling rate) according to Eq. 11. Therefore, the corresponding thermal inbreathing rates calculated deviate significantly from the experimental values. Reproducing the experimental values with the spray water rate of 10 kg/hr is unlikely because the spray water rate does not provide enough cooling rate for the experimental results. Abou-Chakra used a spray water rate of 454 kg/m2/hr in his validation.5 The measurements of Example 1 are questionable.
For Example 2, three experiments for dry air, water and methanol were performed outdoors with a very high spray water rate of 2 m3/hr. The water spray rate exceeds the rain intensity of 225 kg/m2/hr typically assumed by the API 2000. The prediction of the previous simplified dynamic approach comes close to the experiment value with dry air. It is evident that solar radiation does not noticeably affect the thermal inbreathing rate. Results of the simplified approach for condensable vapors are consistent with the measurements and conservative. In addition, Example 2 indicates that the measurements of Example 1 are inaccurate.
For Example 3, three experiments for water were performed outdoors in a horizontal tank. The spray water rate (7 m3/hr and 9 m3/hr) does not limit the tank inside heat transfer coefficient. It is assumed that the tank inside heat transfer coefficient for condensable vapors in a horizontal tank can be reduced by 35% to reflect the horizontal installation and water sprayed vertically on the tank in the absence of wind. The simplified approach provides a conservative estimation of the thermal inbreathing rate.
The lowest thermal inbreathing rate for Water-V001 is affected by the lowest vapor temperature, the highest spray water temperature and the lowest spray water rate. The measurement value for Water-V005 appears unexpected. The experiment initially with the highest vapor temperature should be the highest thermal inbreathing rate since the vapor pressure of condensable vapors significantly impacts the thermal inbreathing rate. However, the lower measurement value may be due to unsaturated vapors in the tank based on the evaluation of experiment data.4 The calculated value with the simplified approach for unsaturated vapors is close to the measurement value. The inbreathing rates for all three cases appear to be less than the inbreathing rate for dry air; the inbreathing rate calculated using the simplified dynamic approach is 1,273 ft3/hr of air for Water-V005. The tank’s thermal inbreathing rate for dry air should be a lower bound for relief device sizing.
The tank inside heat transfer coefficient for small spray water rates can be calculated by following the energy balance on the rapid cooling tank system. The value of hi used should not be greater than the calculated value from Eq. 11:
hiused = [ (Mrain × Cprain × (Final Tplate – Train ) × 60) / (Area × ((Start Tvapor + Final Tvapor) / 2) – ((Start Tplate + Final Tplate ) / 2))] (11)
Takeaway. This simplified approach shows that the thermal inbreathing rates for vapor condensation can exceed the venting rates based on API 2000. The vapor condensation would result in undersized breathing valves, especially for the tanks uninsulated with high vapor pressure vapors. Unlike the existing models for vapor condensation, this simplified approach directly considers the effects of the vapor condensation using the tank inside condensation heat transfer coefficient, the vapor properties and the amount of rainwater. This simplified approach is applicable to multiple components as well as single components.
As validated by recent experiments, this simplified approach is a suitable engineering tool to size the inbreathing devices for uninsulated tanks with condensable vapors. In the simplified approach, the use of the tank inside condensation heat transfer coefficient and the amount of rainwater is satisfactory for vapor condensation because the results compared with published recent experiments are consistent and conservative. More volatile condensable vapors result in higher thermal inbreathing rates. Generally, the thermal inbreathing rate increases with an increase in the vapor pressure of condensable vapors. For non-volatile saturated vapors, the vapor condensation does not increase the thermal inbreathing rate. However, the tank thermal inbreathing rate for dry air should be a lower bound for the tank inbreathing device to be sized. HP
LITERATURE CITED
Jung Seob Kim is a Principal Pressure Safety Engineer at Covestro and has more thn 40 yr of experience in different roles within the petrochemical industry, including with ioMosaic, SK E&C USA, Bayer Technology Services, Samsung BP Chemicals and Samsung Engineering. He earned a BS degree in chemical engineering from the University of Seoul. Kim is a member of AIChE and is a registered Professional Engineer in the State of Texas.
Ryan J. Haines is a Process Safety Engineer at Covestro. He has 6 yr of experience in different roles, primary focusing on reliability and process engineering. He earned a BS degree in chemical engineering from Texas A&M University.