Problem Of The Month

March 2001—Heat Exchanger IRIS Wall Thickness And Gumbel Smallest Distributions


A heat exchanger was placed into service 17 years ago.  The wall thickness of each tube in the exchanger was surveyed by eddy current inspection during a routine turnaround inspection at age 17 years.  Some potentially thin walls detected by the quick eddy current inspection suggested the use of an internal radial immersion technique using a more accurate ultrasonic inspection of 10% of the tube walls using IRIS inspection.  The IRIS tube wall thickness shows the following minimum wall thickness (inches)*quantity.

            0.050*1, 0.055*1, 0.056*2, 0.058*2, 0.059*1, 0.061*6, 0.063*9, 0.064*9, 0.065*4, 0.066*5, 0.067*2, 0.069*4


The 46 tubes are considered representative of the total heat exchanger with 460 tubes (although random locations were not drawn from a random number table for inspection and thus some would consider the inspection data to be haphazard although most tubes were inspected in the “bad zone” of the heat exchanger). 


The rule-of-thumb practice in this facility is:

1. Begin heat exchanger tubing inspection at turnarounds when the wall thickness has been reduced 1/3 (from 0.083” to 0.055”), and

2. Consider the heat exchanger for retubing when tube wall thickness has been reduced to ½ of the original wall thickness (from 0.083” to 0.0415”).

The minimum allowed wall thickness for this service (with environmental concerns and conditions) is 0.036”.  Starting wall thickness for the heat exchanger were not recorded when the heat exchanger was placed into service 17 years ago for the 1”OD x 14 BWG steel tubing.  Wall thickness tolerances for year zero are derived from the manufacturing tolerances assuming the minimum wall thickness is 0.083” and the maximum wall thickness is 0.101” based on manufacturing tolerances.


The cost to retube the exchanger is $50,000.  If the heat exchanger fails in service the unplanned failure cost (expedited retube cost and lost gross margin values from product not produced)  is seasonally driven at a cost of:

Summer =        $750,000,

Fall =               $500,000

Winter =          $100,000,  and

Spring -           $250,000. 

The key issue of tube wall failure is an environmental impact with high outage costs to facilitate repairs/retubing.


Should we:

Retube the heat exchanger now at age 17 years? 
Retube at the next turnaround scheduled for 3 years from now when the tubes are age 20 years? 
Retube at the second turnaround scheduled in 6 years from now when the tubes are 23 years of age?


Since the IRIS inspection records the minimum wall thickness in each tube, this represents a condition of minimum extremes typified by use of the Gumbel Lower (or smaller) distribution (a type I extreme value).  The Gumbel smallest extreme value distribution is from the family of extreme values best know by the Weibull distribution (a type III extreme value). 


The Gumbel smallest cumulative distributions are expressed in Figure 1 which are generically similar to equations for the smallest extreme value listed in Table 2 (page 24) of ”Statistical Theory Of Extreme Values And Some Practical Applications—A series Of Lectures by Emil J. Gumbel”., National Bureau of standards Applied Mathematics Series 33 Issued February 12, 1954 (Order PB175818 from the National Technical Information Service of the U.S. Department of Commerce.


This equation is often called the double exponential or Gumbel SMALLEST or Gumbel lower extreme value cumulative distribution function.  It explains what percent of the population will fail by time, t.

Figure 1:  Gumbel Smallest Distribution- x is a scale factor, d is a shape factor


A plot of the Y-axis of the Gumbel distribution is the same as for a Weibull distribution.  The X-axis for a Gumbel distribution is uniformly divided which makes it a little easier to describe the probability plots to novices as compared to the log scale for the Weibull distribution.  WinSMITH Weibull software can produce both Gumbel plots and Weibull plots with ease.


Figure 2:  Derivation of the Gumbel Smallest Extreme Value Straight Line Equation


The Weibull plot is well known for representing the “weakest link” (not to be confused with the tag line for a current television program where contestants are informed they are “the weakest link” because of their inferior performance). 


The Gumbel Smallest extreme value is considered a model for a system having n elements in a series and where the failure distributions of components are reasonably uniform and similar (See British Standard BS 5760). 


The conditions described in BS 5760 seems appropriate for the condition of a heat exchanger where the tubes are functionally in series as any tube failure causes the heat exchanger system to fail and the data for wall thickness represents the minimum value in each tube.


The heat exchanger minimum wall thickness inspection data is plotted in Figure 3.

Notice the stacks of data as shown above with the large number of data points all the same values.  This suggests the use of the “Inspection” option for analysis.  The Inspection option regresses the trend line through the top point in the data stacks.  This functions as if true data values actually were to the left of the stack (at smaller wall thickness) because they are rounded values from the inspection technique and exposure time.  The coefficient of determinations r^2 says this straight line explains 98.9% of scatter in the data.


Year 17 results show the probability of having a wall thickness less than 0.036 is very small from Figure 2.   Using the “Predict” feature of WinSMITH Weibull, the trend line predicts 1.29E-04 chance of occurrence, which is also the probability of failure.  Our risk today using the summer failure conditions shows the $Risk = POF*$Consequence = (1.29E-04*$750,000) = $96.75 which is an inconsequential amount of financial exposure.


To predict tube wall thickness at year 20 requires knowing wall thickness at year 0 for determining progression of the wall thickness to smaller values as time progresses.  Use the maximum/minimum wall thickness as 0.083” and 0.101” based on manufacturing tolerances and consider these span ~6*s conditions at 0.1% and 99.9% which covers 99.8% of the wall thickness expected from new materials.  Use the “Line Only” feature of WinSMITH Weibull and draw a line through the min/max wall thickness at (0.083”, 0.1%) and (0.101”, 99.9%) to get Figure 4.



Notice that the line slope (Del) for year 0 is about the same as year 17.  Similar line slopes gives clues that uniform corrosion is at work.  If the line slope at year 17 were much flatter, then expect both uniform corrosion and accelerated corrosion.  Look at the Xi values for the lines where (0.09706 – 0.06427)/17 = 0.03279/17 = 0.0019288” per year for the characteristic wall thickness which says at year 20 to expect the characteristic wall thickness is forecast to be Xi = (0.09706 - 0.0019288*20) = (0.09706 - 0.0385765) = 0.0584835” with a slope of 0.0031573 (assuming corrosion mechanisms remain unchanged.  Using the “Line Only” and the “Predict” features the predicted line will show a 8.075E-04 chance for a wall thickness of 0.036” which takes the $risk = 8.075E-04*$750,000 = $606 which is a very manageable risk for reaching year 20 without a leak using the existing tubes.  Therefore do not retubeat year 17—take the risk and proceed with operations.


What’s the risk for proceeding to year 23 without a retube?  Using the same methodology (and update with inspection data from year 20), forecast Xi = (0.09706 - 0.0019288*23) = (0.9706 - 0.0443624) = 0.0526976” with a slope of 0.0031573 (assuming corrosion mechanisms remain unchanged) to predict the probability of having a wall thickness smaller than 0.036” is 5.03625E-03 which takes the $risk = 5.03625E-03*$750000 = $3,777 which may be too high for an environmental incident.  Thus seriously consider a retube at year 20 since the thin wall tubes will incur a high risk for reaching year 23 without a failure.


You can also use WinSMITH Weibull’s Accelerated Test/Degradation icons to forecast the march of wall thickness to smaller values.  Instead of using the “Line Only” feature for the assumed wall distribution, you must draw a precise Monte Carlo simulation of 46 data points from Xi = 0.09076 and Del = 0.0020362 (you can toggle the Monte Carlo icon from scatter around the trend line to a precise draw of data with no variation from the line. 


The Accelerated Test/Degradation icon allows selection of the same line slopes or allowing for the program to calculate the expected line slope.  Of course for this simple problem, computer assist is overkill but available if you need it for more complicated analysis.


This problem of the month has shown how to use the smallest value of data acquired.  An IRIS tube inspection for heat exchangers will report only one value for each tube and it fits the Gumbel smallest value distribution.  Heat exchanger life is terminated with thin wall—not thick walls and the Gumbel distribution helps with the decision making process.  You don’t want to waste money by retubing heat exchangers too soon, however, you also do not want to be foolish and wait too long with increasing risks by blindly ignoring the details from inspections. 


Can the decision process for this heat exchanger be improved by use of confidence limits for the data set?  Yes.  In this case you should consider the use of left hand limits to provide guidance for the expected lower wall thickness limits from thin tubes. 


Please be aware, that inclusion of confidence limits will subject you to endless arguments about the subject matter!!  For presentation purposes, you may decide to use the trend line and keep the $risks low rather than being so technically correct that decisions are deferred by endless (and non productive) conversations/arguments by the well intended by technical neophytes!


Are others using the Gumbel distribution for heat exchanger decisions?  Yes.  Reports from Singapore describe their activities—unfortunately, their reports are in Japanese and perhaps readers of this Problem Of The Month can send PDF files to the mail box shown below.


Related Gumbel distributions:

See the Gumbel largest value distribution for another example of how to use extremes in data.  The largest value example shows how to use the largest data acquired to make important decisions.


Refer to the caveats on the Problem Of The Month Page about the limitations of the following solution. Maybe you have a better idea on how to solve the problem. Maybe you find where I've screwed-up the solution and you can point out my errors as you check my calculations. E-mail your comments, criticism, and corrections to: Paul Barringer by     clicking here.   Return to top of page by clicking here.  You can download a copy of this Gumble Lower paper as a PDF file.

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Last revised 01/07/2007
© Barringer & Associates, Inc. 2001

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