MEMORANDUM

Subject:	Fuel Tank Temperature generation for evaporative emissions
modeling in MOVES

From:		Prashanth Gururaja

		Assessment and Standards Division

To:		EPA Air Docket OAR-2005-0161

For MOVES to estimate fuel-related evaporative emissions for a vehicle
in a given day, determining the fuel tank temperature throughout the day
is vital.  Therefore, we have developed a mathematical algorithm that
will calculate fuel tank temperature based on various inputs, such as
daily ambient temperature profile and the vehicle’s trip times
(driving periods).  

Data sources

CRC E-9 Diurnal tests

EPA MSOD historical compliance testing

CRC E-41 Hot Soak, Running Loss, and Diurnal tests

Certification and sales data from manufacturers

Input parameters

Hourly ambient temperature profile

Key on and key off times 

HourDayID (Hour x of day y) of first KeyON

Vehicle Type (LDT/LDV)

Pre-enhanced or enhanced evaporative emissions control system (model
year)

General steps

Define input parameters 

Fuel tank temperature is computed up to the start of the first trip,
assuming that the vehicle has been parked for a long time (overnight). 
This is done through the block diagram in Figure 1 on the next page,
which represents the differential equation in (1).  All soaks (hot and
cold) are calculated using this portion of the algorithm.

Next, for each trip, the fuel tank temperature is computed for the
operation period and the corresponding soak period after the key off for
that trip.  It computes fuel tank temperature until the start of the
next trip (next key on), at which point this step is repeated, or until
the end of the modeling period.  The fuel tank temperature during
operation is calculated using SAE equations (3) and (4).  The fuel tank
temperature during soak is calculated as in step 1, but with the initial
temperature (Ti) changed to the temperature at the end of each trip, and
the time interval modified to accommodate the key on/off times.

The results of steps 2 and 3 are joined together to get a calculated
fuel tank temperature profile for the given day.

Calculating soak temperatures (as a function of ambient temperature)

We averaged all the vehicles’ hourly tank temperature test results
from CRC E-9, E-41, and MSOD to get an average 24-hour fuel tank
temperature profile for each of the temperature cycles tested (60-84,
72-96, 82-96 °F).  We then used the following equation to model tank
temperature as a function of ambient temperature.  

 ,                      (1)     

where Ttank is the fuel tank temperature, Tair is the ambient
temperature, and k is a constant proportionality factor.  This equation
represents Newton’s law of cooling, which, in this case, states that
the rate of fuel tank temperature change is proportional to the
difference between the ambient temperature and the fuel tank
temperature.  This relationship is a common occurrence in nature, and we
used this as our starting point.  However, in our scenario, the input
ambient temperature is not constant, linear or easily definable by a
mathematical equation.  We were able to validate our theoretical
approach using the data-averaged tank temperature profiles.  Matlab and
Simulink were used to simulate this equation for each of the three
temperature test cycles (as the ambient temperature inputs), and
subsequently to determine the constant k that would match the
simulations with the test data.  A value of k=1.4 matched the simulated
curve with the average test data within 2% for each hour.  As a result
of this validation and its theoretical basis, we decided to implement
this equation to model the fuel tank temperature profile for any given
ambient temperature profile.

There was very little distinction made between hot soak and cold soak
calculations.  We assumed that during either soak, the only factor
affecting fuel tank temperature was the ambient temperature profile and
the fuel tank temperature at the start of the soak.  The block diagram
below simplifies the equation into several mathematical steps, which are
explained below.  

Figure 1

The time interval for which this part of the algorithm is used depends
on the key on and key off times.  Since this equation can be used only
for cold soaks and hot soaks (all parked conditions), it applies for the
following time intervals only:

from the start of the day to the first trip,

from all key off to key on times, and

from the last key off to the end of the desired modeling period.

Mathematical steps

At time t0 = 0 or KeyOFF (start of soak), Ttank = Ti.  This value will
either be the ambient temperature (at the very start of the model) or
the fuel tank temperature at the end of a trip.

Then, for all t > 0 or KeyOFF, the next tank temperature is calculated
in this manner:

      	(2a)	or

 		(2b)

(Tair – Ttank) is a function of time.  Since analytical integration is
too complicated (the input ambient temperature data is highly non-linear
and in tabular form), numerical integration should be used to perform
this step.  The method of numerical integration varies based on the
accuracy desired.  The above method in (2b) represents the Euler method,
one of the simplest methods of integration, and one of the few possible
methods MOVES can apply.  MOVES uses this method, with Δt = 15 minutes
for cold soaks, and Δt = 1 minute for hot soaks.

Calculating fuel tank temperatures during operation

We used data from CRC E-65 and certification data from the top ten
selling LDVs and top ten selling LDTs to determine the temperatures
encountered during operation.  We initially separated LDVs and LDTs
since most LDTs have larger fuel tanks than the LDVs, which we suspected
could possibly affect operation fuel tank temperature.  

Operation periods (trips) are relatively short compared to the length of
the day or modeling period.  Therefore, even though the fuel tank
temperature profile during operation is not exactly linear, assuming a
linear increase in temperature makes calculations easier without
compromising accuracy.  However, the increase in temperature during
operation ΔTtank depends on the temperature at the start of operation. 
It also depends on vehicle type.  The convention used in this algorithm
is that ΔTtank applies over a 4300 second period, which is the length
of the running loss test done by manufacturers for certification.  To
find ΔTtank, we must first find ΔTtank95, the average increase in tank
temperature at a standard 4300-second @ 95°F ambient temperature
running loss test.

If the vehicle is evap-enhanced, then ΔTtank95 = 24°F.  Sales data of
the vehicles in consideration were used to weight the vehicles’
temperature rises to model the fleet.

If the vehicle is pre-enhanced, the vehicle type affects ΔTtank95. 
Estimated from CRC E-65:

If LDV, then ΔTtank95 = 35°F.

If LDT, then ΔTtank95 = 29°F.

We can use these values for ΔTtank95 for 95F to calculate the ΔTtank
for other starting fuel tank temperatures (other trips) using the
following equation:

      (3)  

Since this gives us the increase in tank temperature, we can create a
simple linear function that models fuel tank temperature for each trip.

         (4)2

The 4300/3600 appears since the running loss test done by manufacturers
is 4300 seconds long, and we convert that to hours maintain
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We assumed the effect of the ambient temperature or change in ambient
temperature during a trip was negligible compared to the effect of
operation.  

The KeyON tank temperatures will be known by way of the calculations of
the tank temperatures from the previous soak.

Implementation in MOVES and use in RFS

Within the MOVES model, the algorithm described above is performed to
calculate real-time temperatures for hundreds of vehicles based on their
daily trip patterns.  MOVES then averages temperatures by hour of the
day and operating mode (cold soak, hot soak, and operating).  For the
RFS rule, we ran MOVES for Maricopa County, Arizona, for the month of
September (70-98 °F ambient temperature curve) to reflect a typical
summer day, and calculated the hourly weighted average tank temperature
from the mode-based temperatures using the hourly breakdown of activity
by mode (i.e. hours parked in cold soak, hours parked in hot soak, hour
operating) calculated by MOVES.

 Edwards, C. Henry, et. al.  Differential Equations – Computing and
Modeling.  3rd ed.  Pearson Custom Publishing, 2003

 Cam, Tam M, et al. SAE 930078 – Running Loss Temperature Profiles.

