Basic Examination of Gasoline Losses in Tank Headspace

Ted Tower, Ph.D.

These calculations were done to shed some light on the speculation that there is a significant cost associated with the headspace above gasoline storage tanks, whether in a vehicle or a storage unit.

Any liquid will come to equilibrium with the airspace above it, but recall that this is not necessarily "lost" in any sense, it is merely in equilibrium with its surrounding environment. Minimizing vapor loss is advantageous, but as you will see below, even the loss of this mass has near negligible cost.

Courtesy: http://www.simageconsulting.com

Tank
Ideal Gas Law
Octane Mass in Headspace
Octane Volume in Headspace
Cost Range
Summary
References

Tank

Whether we are interested in a vehicle gas tank or an underground storage tank, there are only two variables that need to be specified, the storage temperature and the headspace volume.

For an example problem, we shall assume a temperature of 20 degrees Celsius (68F) and a headspace volume of 15 gallons (i.e. a nearly empty moderate sized passenger vehicle).

T=20;          % temperature of environment of interest [degC]
headspace=15;  % headspace in gallons

Ideal Gas Law

We shall assume the gases and their mixtures are ideal, which tends to be the case for hydrocarbons and air at standard conditions. The Ideal Gas Law can be stated as such: PV=nRT or n=PV/RT, where P=pressure, V=volume, R=gas constant, T=temperature, and n=moles of gas.

V=headspace*3.785;   % volume of interest (converted to [L])
P=1;           % atmospheric pressure [atm]
R=8.2057e-2;   % gas constant [L.atm/K/mol]
MW=114.23;     % molecular weight of octane [g/mol]

Octane Mass in Headspace

Using the Ideal Gas Law we can easily calculate the fraction of octane in the headspace or the mass of octane itself.

1 atm=101.325kPa

x=(octane_vp(273+T)/101.325)/P;  % molar fraction (i.e. # molecules of octane
                                 % compared to what 1 atm of air can hold)
n=x*P*V/(R*(T+273));             % moles of octane (i.e. # of molecules)
g_octane=n*MW;                   % mass of octane [g]

Octane Volume in Headspace

We can use published tabular data of temperature/density data to calculate the volume of the would-be liquid at this temperature.

Additionally, we shall assume a nominal $3.00/gallon cost of gasoline.

For T=20C and a completely empty 15-gallon tank, but a headspace that is fully saturated with octane, this amounts to less than half a penny ($0.0042). Filling up a half-empty tank would "save" even less.

odens=[0 0.719;10 0.711;20 0.702;30 0.694;40 0.685];
densliq=interp1(odens(:,1),odens(:,2),T); % g/cc
v_octane=g_octane/densliq/1000/3.785;     % converted to gallons

ppg=3;   % price per gallon ($);

% The "cost" of this amount of octane that occupies the headspace is thus:
vapor_cost=v_octane*ppg;
disp(['There is $' num2str(vapor_cost,2) ' worth of octane in headspace.']);

There is $0.0042 worth of octane in headspace.

Cost Range

Lastly, for completeness, calculate for an entire range of temperatures and volumes. The figure below can be read by looking up the tank volume on the vertical axis, the storage temperature along the horizontal axis and reading the vapor cost at the intersection.

Please note that an underground storage tank will almost never be in the high temperature ranges in most areas of the temperate climate.

Trng=[1 10 20 30 39];Hrng=logspace(1,5,5);
[T,H]=meshgrid(Trng,Hrng);
cost=octane_calc(T,H);

figure(1);clf;set(gcf,'Color',[1 1 1]);
imagesc(cost);
set(gca,'Xtick',1:length(Trng),'XTickLabel',num2str(Trng'));
set(gca,'YTick',1:length(Hrng),'YTickLabel',num2str(Hrng'));
xlabel('Temperature [degC]');ylabel('headspace [gal]');
title('Cost of octane in storage headspace [$]');

[tx,ty]=meshgrid(1:length(Trng),1:length(Hrng));
txt=num2cell(cost);k=find(cost);
for i=1:length(k),txt{k(i)}=num2str(txt{k(i)},2);end;
txt=strcat('$',txt);
h=text(tx(:),ty(:),txt(:));
set(h,'HorizontalAlignment','center','Color','white');

Summary

In conclusion, it is apparent that the costs due to the vapor content, even from a nearly empty tank is negligible for passenger vehicles.

References

All the tables and constants can easily be found on the intranet and wikipedia, but some material properties used can be found here.

source: http://cheric.org/kdb/kdb/hcprop/showprop.php?cmpid=8

Octane has an approximate density of 0.70 g/cc, but we will use a temperature dependent density of octane.

source: http://www.engineersedge.com/fluid_flow/fluid_data.htm