 # Monte Carlo Simulation of a Down Hole Dewar Flask

A Dewar flask is used in the oil & gas industry to protect electronics from high temperatures experienced down hole.  As temperatures increase, reliability of electronic components decrease and ultimately fail.

We will analyze a Dewar flask design using Monte Carlo simulation of time down hole before an internal flask temperature of 435 Kelvin is reached.  We will also compare the simulation results to a more traditional single point estimate model.

The Dewar flask consists of two nested chambers that are separated by a vacuum space to limit heat transfer over most of the flask to radiation.  Therefore, it is a useful method for limiting the temperature of the electronics environment while performing operations in a well.

If the high temperature of a well bore wasn't enough, the electronics inside the flask generate heat as well.  A heat sink is used to absorb heat within the flask.  The heat sink absorbs both heat from the electronics and heat from the borehole fluid.

One end of the flask is open to insert the electronics and heat sink package.  The open end is then thermally isolated by using an insulator to close the flask.  Wires for the electronics are run through the insulator.

## Monte Carlo Simulation

We will use Monte Carlo simulation to analyze how long the flask can be down hole while keeping the internal temperature below 435 degrees Kelvin.  Radiation is the primary mode of external heat transfer and heat generated by the electronics is a source of internal heat.  A Dewar flask has heat transfer by conduction where the internal and external flasks are joined near the opening.  Conduction through the wires entering the flask, and insulating plug are also sources of heat into the flask.

For the purposes of this analysis, we will assume that any conduction into the flask is minor and will be disregarded for simplicity.  In reality heat transfer by conduction is minor, but not trivial, and should be accounted for in an actual analysis.

As with many properties of materials, published data on emissivities are usually given as a range of values.  As a result, we will treat emissivity of the surfaces inside the vacuum chamber as a random variable.  The emissivity ranges from .2 to .3.  We use use a normally distributed random variable with mean .025 and standard deviation of .002.  This will result in 99.7% of the values between .019 and .031.

The heat generated by the electronics will also be treated as a random variable.  A trapezoidal distribution is used with the following parameters:

Minimum: 5 watts
Mode 1: 8 watts
Mode 2: 12 watts
Maximum: 15 watts

By using the trapezoidal distribution, there is an equal probability of heat generation being from 8 to 12 watts (and the highest probability).  The further we go below 8 watts, the less likely this will occur.  The further we go above 12 watts, the less likely this will occur.

The model is shown below.  The temperature rise is modeled in one minute increments as an approximation.  As the temperature increments go to zero, we approach the true temperature rise.

During each time increment, the heat generated by the electronics is treated as a trapezoidal random variable.  Since there are 750 increments, there are 750 random variables for electronic heat generation.  The RVUSERDIST function is used to average the value of these random variables and record this average in the simulation data.  The electronic heat random variables are located on another sheet so they are not recorded on the simulation data sheet.

There are two emissivities that are needed, one for the inside of the outer flask, and one for the outside of the inner flask.  It is assumed that their surface conditions are the same so one value of emissivity is used for both surfaces.

The model was simulated for 25,000 iterations using Simulation Master, and the results are shown below.

Simulation Master has a correlation report tool so we can look at the sensitivity of down hole time to the random variables.  Cell B12 is emissivity, and from the chart below, it has a strong negative correlation with down hole time.  Cell B19 is average electronics heat generation and it is weakly correlated with time down hole.

Therefore, down hole time is most sensitive to variations in emissivity.

Often, when modeling a design, single point estimates are used.  In our case, we could use an emissivity of 0.025 and an average heat generation by the electronics of 10 watts which are the mid-points of the ranges used in the Monte Carlo simulation.

The next step might be to look at extremes and maybe some points in between.  Let's generate some values from a similar model with the random variables removed.

The table below shows time to reach 435 K for various values of emissivity and average heat generated by the electronics.

## Comparing Monte Carlo vs. Traditional Methods

So why do a Monte Carlo simulation?  Let's compare the results of the two analyses and try to see what information we can glean from them.

MC = Monte Carlo simulation
SP = Single point

### Comparing Extremes

MC Min: 447 minutes
SP Min: 384 minutes

MC Max: 620 minutes
SP Max: 804

SP has a far greater spread between minimum and maximum.  For the SP maximum, electronics heat is at the minimum value of 5 watts the entire time, and emissivity is at .02.  Running at minimum power the entire time is not a very realistic scenario.

At SP minimum,  electronics heat is at the maximum value of 15 watts the entire time, and emissivity is at .03.  Running at maximum power is probably more realistic than running at minimum, but still an unlikely scenario.

The MC simulation has a low probability of the electronics heat being at the extremes.  Therefore the spread of extremes is smaller.

### Central Tendencies

MC Mean: 520.65
MC 50th Percentile: 520
SP Mid-Point Value: 520

Note that 50% of MC simulations range from 506 to 534 minutes.  90% of MC simulations range from 488 to 556 minutes.

## Conclusion

By randomly varying heat generated by the electronics and emissivity, we get more realistic results than by picking values and plugging them into the single point model.  As a result, we can give a confidence level based on the percentile values from the simulation.

For this example, MC mean, MC 50th percentile, and SP mid-point value are all nearly identical.  If we used the SP mid-point value from a single point estimate, we would likely not make the down hole time in approximately 50% of the cases.  Stating down hole time with 95 or 99% certainty is a more desirable situation.

Let's say we want to state the down hole time with 95% confidence.  The 5th percentile of the MC is 488 minutes, or 95% of the simulations exceeded 488 minutes.  Therefore, we could rate the flask for 488 minutes with 95% confidence.

So, back to the question, why do a Monte Carlo simulation?  The simulation gives a more realistic picture of outcomes, and gives us the data needed to rate the Dewar flask to a desired confidence level.

There is a second article on this topic that uses Simulation Master's optimizer to optimize the heat sink volume given a minimum time down hole.  Please follow this link to read that article.

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