It is often the cases that the temperature of the piping system is not constant throughout the operating cycle, or there is potentially more than one operating cycle (i.e., pump A on, pump B on, both pumps on, both pumps off).

In these cases it is common to find that the temperature rises on some occasion to a maximum value, say Te; then, as events occur during the normal course of operation the temperature varies between Te and other lower temperature states, say T1, T2,..., Tn. In these cases the piping codes have provided a simplified method by which the cumulative damage due to the various thermal cycles may be evaluated by converting reduced thermal expansion cycles into equivalent full temperature cycles. The user will find that cumulative damage rules usually become important only either the number of thermal variations is large, or when the magnitude of the temperature variation is a large percentage of the maximum design temperature expected. The following rules should be followed when evaluating systems with multiple temperature variation cycles:

Te should be selected as the highest operating temperature of the piping system, even if the startup cycle does not go directly to this temperature.

The expansion allowable stress should be based on Te, i.e. Sa should be calculated from Sh for temperature Te.

The range dTe is determined as the difference between Te and the ambient temperature. Ne should be estimated as the total number of times during the life of the unit that the temperature will be expected to vary from ambient to Te.

The temperature ranges between Te and each of the other reduced temperature states should be calculated, i.e.:

dTi = Te - Tb dT2 = Te - T2,... dTn = Te - Tn

The number of cycles associated with each operating mode should be estimated as:

Temperature change dTi occur Ni times, Temperature change dT2 occur N2 times,... , Temperature change dTn occurs Nn times

The total number of equivalent full temperature cycles that these partial cycles represent can be calculated as:

The cyclic reduction factor f should be selected based upon the number of equivalent cycles, N, while other components of Sa and Se shouldbe based upon temperature Te.

Example: A particular process line varies in temperature as the quality of the catalyst varies. The particulars of the operation are outlined below:

Ambient = 70°F

Startup goes to 560°F

It is estimated that the maximum temperature ever required will be 650°F and the minimum temperature required during operation will be 430°F. The temperature will fluctuate between 560°F and 650°F perhaps 10 times per day, and between 560°F and 450°F perhaps 5 times per day. The design life of the unit is 12 years, and it is estimated that the unit will be shut down at least once each month for maintenance.

Te should be selected as the highest operating temperature of the piping system. In this case, it is equal to 650°F.

The range dTe is determined as the difference between 650° and the ambient temperature of 70°F, so dTe = 580°F. The estimate of Ne, the total number of times that the temperature will be expected to vary through this range is:

Ne = 1 shutdown/month x 12 months/yr x 12 yr = 144

The temperature ranges between Te and each of the other reduced temperature states are:

Ti = 560°F; so dTx = 650<>F - 560°F = 90°F T2 = 450°F; so dT2 = 650°F - 450°F = 200°F

The number of cycles associated with each operating mode are:

Ni = 10 times/day x 365 days/yr x 12 yr = 43800 N2 = 5 times/day x 365 days/yr x 12 yr = 21900

The total number of equivalent full temperature cycles is:

N = 144 + (90/580)^5 x 43800 + (200/580)^5 x 21900 = 255

The cyclic reduction factor f is selected based upon 255 cycles, so f = 1.0 (for fewer than 7000 cycles). As noted, the material allowable stresses Sa and Sh are based upon 650°F, and the expansion stress, Sg, is calculated for the system operating at 650°F.

Warning: These cumulative damage rules don't fully address those cases where one part of the piping system stays at Te while another part of the piping system undergoes a temperature fluctuation. In these cases it is common to simply analyze each case separately. The ASME Section III, Subsection NB (Nuclear Class 1 Piping) Code provides rules which may be followed should the user be concerned about the cumulative damage where different parts of the piping system cycle through different temperature states. The requirements are described below:

** Cumulative Damage**: If there are two or more types of stress cycles which produce significant stresses, their cumulative effect shall be evaluated as stipulated in Steps 1 through 6 below:

Designate the specified number of times each type of stress cycle of types 1,2,3, ..., n, will be repeated during the life of the component as ni, n2, n3, ..., nn, respectively. In determining ni, n2, n3..., nn consideration shall be given to the superposition of cycles of various origins which produce the greatest total alternating stress range. For example, if one type of stress cycle produces 1000 cycles of a stress variation from zero to +60,000 psi and another type of stress cycle produces 10,000 cycles of a stress variation from zero to -50,000 psi, the two cycles to be considered are shown below: (a) Cycle type 1: ni=1000; and Saiti=(60000+50000)/2 (b) Cycle type 2, n2=9000; and Sait2=(50000+0)/2

For each type of stress cycle, determine the alternating stress intensity Salt, which for our application is one half of the range between the expansion stress cycles (as shown above). These alternating stress intensities are designated as Salt1, Salt2, ... Saitn.

On the applicable design fatigue curve find the permissible number of cycles for each Salt computed. These are designated as Ni, N2,..., Nn.

For each stress cycle calculate the usage factors U1, U2,Un, where U1 = n/N1, U2 = n2/N2,..., Un = nn/Nn.

Calculate the cumulative usage factor U as U = U1 + U2 + ... + Un.

The cumulative usage factor shall not exceed 1.0.

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