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Minenergo-325. Massively trace backwards thermal-hydraulic regime for district heating network

Source: R/m325tracebw.R
m325tracebw.Rd

Trace values of thermal-hydraulic regime (temperature, pressure, flow rate, and other) in the bunched pipeline against the flow direction using norms of heat loss values prescribed by Minenergo Order 325.

Algorithm also suits for partially measurable district heating network with massive data lack conditions, when there are no temperature and pressure sensor readings on the majority of terminal nodes.

Usage

m325tracebw(
  sender = 6,
  acceptor = 7,
  temperature = 70,
  pressure = pipenostics::mpa_kgf(6),
  flow_rate = 20,
  a = 0,
  d = 100,
  wth = 6.5,
  len = 72.446,
  year = 1986,
  insulation = 0,
  laying = "tunnel",
  beta = FALSE,
  exp5k = TRUE,
  roughness = 0.001,
  inlet = 0.5,
  outlet = 1,
  method = "romeo",
  opinion = "median",
  strict_sizes = FALSE,
  verbose = TRUE,
  csv = FALSE,
  file = "m325tracebw.csv"
)

Arguments

sender

identifier of the node which heat carrier flows out. Type: any type that can be painlessly coerced to character by as.character.

acceptor

identifier of the node which heat carrier flows in. According to topology of test bench considered this identifier should be unique for every row. Type: any type that can be painlessly coerced to character by as.character.

temperature

sensor-measured temperature of heat carrier (water) sensor-measured on the terminal acceptor node, [°C]. Use NA_float_s for (terminal) nodes without temperature sensor. Type: assert_double.

pressure

sensor-measured absolute pressure of heat carrier (water) inside the pipe (i.e. acceptor's incoming edge), [MPa]. Type: assert_double.

flow_rate

sensor-measured amount of heat carrier (water) on terminal node that is transferred by pipe (i.e. acceptor's incoming edge) during a period, [ton/h]. Type: assert_double. Use NA_float_s for nodes without flow rate sensor.

a

heat carrier volume loss factor of pipe: 0 - no loss of heat carrier; 0.0025 - maximum possible loss of heat carrier allowed. Type: assert_double.

d

nominal (outside) diameter of pipe (i.e.diameter of acceptor's incoming edge), [mm]. Type: assert_double.

wth

nominal wall thickness of pipe, [mm]. Type: assert_double.

len

pipe length (i.e. length of acceptor's incoming edge), [m]. Type: assert_double.

year

year when the pipe (i.e. acceptor's incoming edge) is put in operation after laying or total overhaul. Type: assert_integerish.

insulation

identifier of insulation that covers the exterior of pipe (i.e. acceptor's incoming edge):

0

no insulation

1

foamed polyurethane or analogue

2

polymer concrete

Type: assert_subset.

laying

type of pipe laying depicting the position of pipe in space. Only five types of pipe laying are considered:

  • air,

  • channel,

  • room,

  • tunnel,

  • underground.

Type: assert_subset.

beta

logical indicator: should they consider additional heat loss of fittings located on this pipe (i.e. acceptor's incoming edge)? Type: assert_logical.

exp5k

logical indicator for regime of pipe (i.e. acceptor's incoming edge): if TRUE pipe is operated more that 5000 hours per year. Type: assert_logical.

roughness

roughness of internal wall of pipe (i.e. acceptor's incoming edge), [m]. Type: assert_double.

inlet

elevation of pipe inlet, [m]. Type: assert_double.

outlet

elevation of pipe outlet, [m]. Type: assert_double.

method

method of determining Darcy friction factor:

  • romeo

  • vatankhah

  • buzzelli

Type: assert_choice. For more details see dropp.

opinion

method for aggregating values of regime parameters on each node for the next tracing step:

mean

values of parameter are averaged before the next tracing step

median

median of parameter values are used for the next tracing step

Type: assert_choice.

strict_sizes

verify diameter and wall thickness with the actual pipe specifications produced. Type: assert_flag.

verbose

logical indicator: should they watch tracing process on console? Type: assert_flag.

csv

logical indicator: should they incrementally dump results to csv- file while tracing? Type: assert_flag.

file

name of csv-file which they dump results to. Type: assert_character of length 1 that can be used safely to create a file and write to it.

Value

A data.frame containing results (detailed log) of tracing in narrow format:

node

Tracing job. Identifier of the node which regime parameters is calculated for. Values in this vector are identical to those in argument acceptor. Type: assert_character.

tracing

Tracing job. Identifiers of nodes from which regime parameters are traced for the given node. Identifier sensor is used when values of regime parameters for the node are sensor readings. Type: assert_character.

backward

Tracing job. Identifier of tracing direction. It constantly equals to TRUE. Type: assert_logical.

aggregation

Tracing job. Identifier of aggregation method: span, median, mean, or identity. Type: assert_character.

loss

Traced thermal hydraulic regime. Normative specific heat loss power of adjacent pipe, [kcal/m/h]. Type: assert_double.

flux

Traced thermal hydraulic regime. Normative heat flux of adjacent pipe, [W/m²]. Type: assert_double.

Q

Traced thermal hydraulic regime. Normative heat loss of adjacent pipe per day, [kcal]. Type: assert_character.

temperature

Traced thermal hydraulic regime. Traced temperature of heat carrier (water) that is associated with the node, [°C]. Type: assert_double.

pressure

Traced thermal hydraulic regime. Traced pressure of heat carrier (water) that is associated with the node, [MPa]. Type: assert_double.

flow_rate

Traced thermal hydraulic regime. Traced flow rate of heat carrier (water) that is associated with the node, [ton/h]. Type: assert_double.

job

Tracing job. Value of tracing job counter. Type: assert_count.

Type: assert_data_frame.

Details

They consider the topology of district heating network represented by m325nxdata:

The network may be partially sensor-equipped too:

In latter case no more than two nodes must be equipped with pressure and temperature sensors whereas for other nodes only flow rate sensors must be installed.

Tracing starts from sensor-equipped nodes and goes backwards, i.e against the flow direction.

Though some input arguments are natively vectorized their individual values all relate to common part of district heating network, i.e. associated with common object. It is due to isomorphism between vector representation and directed graph of this network. For more details of isomorphic topology description see m325nxdata.

Before tracing starts for the next node, previously calculated values of thermal-hydraulic parameters are aggregated by either averaging or by median. The latter seems more robust for avoiding strong influence of possible outliers which may come from actual heating transfer anomalies, erroneous sensor readings or wrong pipeline specifications.

Aggregation for values of flow rate at the node is always sum.

Optional verification of pipe diameters and wall thicknesses is performed against b36pipedata data.

See also

Other Regime tracing: dropg(), dropp(), dropt(), m325tracefw(), m325traceline(), tracebw(), tracefw(), traceline()

Examples

library(pipenostics)

## It is possible to run without specification of argument values:
m325tracebw()
#> 
#> 2025-09-19 11:38:08.550641 m325tracebw | start backward tracing; segments 1;
#> 2025-09-19 11:38:08.550641 m325tracebw | start job; job 0;
#> 2025-09-19 11:38:08.550641 m325tracebw | now process; 1 node(s); [7]
#> 2025-09-19 11:38:08.550641 m325tracebw | seen tracing; [1/1] are TP-sensor-equipped;
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing loss;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Specific heat loss power traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing heat flux;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Heat flux traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing heat loss per day;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Heat loss per day traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing temperature;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Temperature traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing pressure;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Pressure traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | tracing flow_rate;;
#> 2025-09-19 11:38:08.550641 m325tracebw | OK! Flow-rate traced from 1 nodes;[7];
#> 2025-09-19 11:38:08.550641 m325tracebw | finish job; job 0; processed node(s) 1
#> 2025-09-19 11:38:08.550641 m325tracebw | finish backward tracing;;
#>        node tracing backward aggregation loss     flux        Q temperature
#> 1         7  sensor     TRUE    identity 78.4 290.2324 136314.4          70
#> span      7  sensor     TRUE        span  0.0   0.0000      0.0           0
#> median    7  sensor     TRUE      median 78.4 290.2324 136314.4          70
#> mean      7  sensor     TRUE        mean 78.4 290.2324 136314.4          70
#>        pressure flow_rate job
#> 1      0.588399        20   0
#> span   0.000000        20   0
#> median 0.588399        20   0
#> mean   0.588399        20   0

## Consider isomorphic representation of District Heating Network graph:
DHN <- m325nxdata

## Distinct options for opinion aggregation lead to distinct traced
## temperature and pressure:

## * When aggregation is by mean:
output_mean <- do.call(
  "m325tracebw", c(as.list(DHN), verbose = FALSE, opinion = "mean")
)

## * When aggregation is by median:
output_median <- do.call(
  "m325tracebw", c(as.list(DHN), verbose = FALSE, opinion = "median")
)

## It is possible to process partially measurable District Heating Network:

## * Simulate lack of temperature and pressure sensors:
DHN[c(7, 10, 21, 24), c("temperature", "pressure")] <- NA_real_

## * Trace thermal-hydraulic regime:
do.call("m325tracebw", c(as.list(DHN), verbose = FALSE))
#> Warning: longer object length is not a multiple of shorter object length
#>           node  tracing backward aggregation      loss        flux            Q
#> 1            1   sensor     TRUE    identity  96.23600 237.5068537  92028.56208
#> 2            2   sensor     TRUE    identity  96.28800 237.6351877  92078.28864
#> 3            3   sensor     TRUE    identity  70.58400 326.6225170 130903.39238
#> 7            7   sensor     TRUE    identity        NA          NA           NA
#> 9            9   sensor     TRUE    identity  28.11520 130.1011191  20296.25042
#> 10          10   sensor     TRUE    identity        NA          NA           NA
#> 15          15   sensor     TRUE    identity  96.60000 238.4051921 184405.53600
#> 17          17   sensor     TRUE    identity  24.95960 184.7980817  13114.57255
#> 19          19   sensor     TRUE    identity  28.16580 130.3352671  35131.99098
#> 21          21   sensor     TRUE    identity        NA          NA           NA
#> 24          24   sensor     TRUE    identity        NA          NA           NA
#> 25          25   sensor     TRUE    identity  96.23600 237.5068537  58792.49712
#> 26          26   sensor     TRUE    identity  70.54000 326.4189101 130821.79104
#> span.1       1   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.10     10   sensor     TRUE        span        NA          NA           NA
#> span.15     15   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.17     17   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.19     19   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.2       2   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.21     21   sensor     TRUE        span        NA          NA           NA
#> span.24     24   sensor     TRUE        span        NA          NA           NA
#> span.25     25   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.26     26   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.3       3   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> span.7       7   sensor     TRUE        span        NA          NA           NA
#> span.9       9   sensor     TRUE        span   0.00000   0.0000000      0.00000
#> median.1     1   sensor     TRUE      median  96.23600 237.5068537  92028.56208
#> median.10   10   sensor     TRUE      median        NA          NA           NA
#> median.15   15   sensor     TRUE      median  96.60000 238.4051921 184405.53600
#> median.17   17   sensor     TRUE      median  24.95960 184.7980817  13114.57255
#> median.19   19   sensor     TRUE      median  28.16580 130.3352671  35131.99098
#> median.2     2   sensor     TRUE      median  96.28800 237.6351877  92078.28864
#> median.21   21   sensor     TRUE      median        NA          NA           NA
#> median.24   24   sensor     TRUE      median        NA          NA           NA
#> median.25   25   sensor     TRUE      median  96.23600 237.5068537  58792.49712
#> median.26   26   sensor     TRUE      median  70.54000 326.4189101 130821.79104
#> median.3     3   sensor     TRUE      median  70.58400 326.6225170 130903.39238
#> median.7     7   sensor     TRUE      median        NA          NA           NA
#> median.9     9   sensor     TRUE      median  28.11520 130.1011191  20296.25042
#> mean.1       1   sensor     TRUE        mean  96.23600 237.5068537  92028.56208
#> mean.10     10   sensor     TRUE        mean        NA          NA           NA
#> mean.15     15   sensor     TRUE        mean  96.60000 238.4051921 184405.53600
#> mean.17     17   sensor     TRUE        mean  24.95960 184.7980817  13114.57255
#> mean.19     19   sensor     TRUE        mean  28.16580 130.3352671  35131.99098
#> mean.2       2   sensor     TRUE        mean  96.28800 237.6351877  92078.28864
#> mean.21     21   sensor     TRUE        mean        NA          NA           NA
#> mean.24     24   sensor     TRUE        mean        NA          NA           NA
#> mean.25     25   sensor     TRUE        mean  96.23600 237.5068537  58792.49712
#> mean.26     26   sensor     TRUE        mean  70.54000 326.4189101 130821.79104
#> mean.3       3   sensor     TRUE        mean  70.58400 326.6225170 130903.39238
#> mean.7       7   sensor     TRUE        mean        NA          NA           NA
#> mean.9       9   sensor     TRUE        mean  28.11520 130.1011191  20296.25042
#> 14           4        1     TRUE    identity  96.23600 237.5068537  92028.56208
#> 22           4        2     TRUE    identity  96.28800 237.6351877  92078.28864
#> 31           5        3     TRUE    identity  70.58400 326.6225170 130903.39238
#> 4            6        7     TRUE    identity        NA          NA           NA
#> 5            6        9     TRUE    identity  28.11520 130.1011191  20296.25042
#> 6           11       10     TRUE    identity        NA          NA           NA
#> 71          14       15     TRUE    identity  96.60000 238.4051921 184405.53600
#> 8           16       17     TRUE    identity  24.95960 184.7980817  13114.57255
#> 91          20       19     TRUE    identity  28.16580 130.3352671  35131.99098
#> 101         20       21     TRUE    identity        NA          NA           NA
#> 11          22       24     TRUE    identity        NA          NA           NA
#> 12          22       25     TRUE    identity  96.23600 237.5068537  58792.49712
#> 13          23       26     TRUE    identity  70.54000 326.4189101 130821.79104
#> span.14     14       15     TRUE        span   0.00000   0.0000000      0.00000
#> span.22     22    24|25     TRUE        span   0.00000   0.0000000      0.00000
#> span.23     23       26     TRUE        span   0.00000   0.0000000      0.00000
#> span.4       4      1|2     TRUE        span   0.05200   0.1283341     49.72656
#> span.5       5        3     TRUE        span   0.00000   0.0000000      0.00000
#> median.14   14       15     TRUE      median  96.60000 238.4051921 184405.53600
#> median.22   22    24|25     TRUE      median  96.23600 237.5068537  58792.49712
#> median.23   23       26     TRUE      median  70.54000 326.4189101 130821.79104
#> median.4     4      1|2     TRUE      median  96.26200 237.5710207  92053.42536
#> median.5     5        3     TRUE      median  70.58400 326.6225170 130903.39238
#> mean.14     14       15     TRUE        mean  96.60000 238.4051921 184405.53600
#> mean.22     22    24|25     TRUE        mean  96.23600 237.5068537  58792.49712
#> mean.23     23       26     TRUE        mean  70.54000 326.4189101 130821.79104
#> mean.4       4      1|2     TRUE        mean  96.26200 237.5710207  92053.42536
#> mean.5       5        3     TRUE        mean  70.58400 326.6225170 130903.39238
#> 16           8        4     TRUE    identity 116.04494 214.7959391  39756.99751
#> 23           6        5     TRUE    identity  70.73402 327.3167083  61399.38989
#> 32          13       14     TRUE    identity  96.73317 238.7338618 158611.45156
#> 41          18       22     TRUE    identity 115.97953 214.6748595  85759.90298
#> 51          20       23     TRUE    identity  70.68993 327.1126787  88173.52308
#> span.20     20 19|21|23     TRUE        span  42.52413 196.7774117  53041.53210
#> span.6       6    7|9|5     TRUE        span  42.61882 197.2155892  41103.13947
#> median.20   20 19|21|23     TRUE      median  49.42786 228.7239729  61652.75703
#> median.6     6    7|9|5     TRUE      median  49.42461 228.7089137  40847.82015
#> mean.20     20 19|21|23     TRUE        mean  49.42786 228.7239729  61652.75703
#> mean.6       6    7|9|5     TRUE        mean  49.42461 228.7089137  40847.82015
#> 18           8        6     TRUE    identity  96.17197 237.3488224 139602.45780
#> 27          18       20     TRUE    identity  96.20009 237.4182284 154140.24934
#> span.18     18    22|20     TRUE        span  19.77944  22.7433689  68380.34636
#> span.8       8      4|6     TRUE        span  19.87298  22.5528833  99845.46029
#> median.18   18    22|20     TRUE      median 106.08981 226.0465440 119950.07616
#> median.8     8      4|6     TRUE      median 106.10846 226.0723808  89679.72766
#> mean.18     18    22|20     TRUE        mean 106.08981 226.0465440 119950.07616
#> mean.8       8      4|6     TRUE        mean 106.10846 226.0723808  89679.72766
#> 110         11        8     TRUE    identity 115.99003 214.6942911  83512.81952
#> 28          16       18     TRUE    identity 115.99053 214.6952242  85768.03843
#> span.11     11     10|8     TRUE        span   0.00000   0.0000000      0.00000
#> span.16     16    17|18     TRUE        span  91.03093  29.8971425  72653.46589
#> median.11   11     10|8     TRUE      median 115.99003 214.6942911  83512.81952
#> median.16   16    17|18     TRUE      median  70.47507 199.7466530  49441.30549
#> mean.11     11     10|8     TRUE        mean 115.99003 214.6942911  83512.81952
#> mean.16     16    17|18     TRUE        mean  70.47507 199.7466530  49441.30549
#> 111         13       11     TRUE    identity 116.01115 214.7333980  83528.03152
#> 29          13       16     TRUE    identity 116.70069 216.0097058  84024.49588
#> span        13 14|11|16     TRUE        span  19.96751  24.0004637  75083.42004
#> median      13 14|11|16     TRUE      median 116.01115 216.0097058  84024.49588
#> mean        13 14|11|16     TRUE        mean 109.81501 223.1589885 108721.32632
#> 112         12       13     TRUE    identity 152.81349 189.1996643 362285.34328
#> span1       12       13     TRUE        span   0.00000   0.0000000      0.00000
#> median1     12       13     TRUE      median 152.81349 189.1996643 362285.34328
#> mean1       12       13     TRUE        mean 152.81349 189.1996643 362285.34328
#>           temperature     pressure flow_rate job
#> 1         69.30000000 5.883990e-01        30   0
#> 2         69.40000000 5.883990e-01        30   0
#> 3         68.60000000 5.883990e-01        16   0
#> 7                  NA           NA        20   0
#> 9         69.20000000 5.883990e-01        16   0
#> 10                 NA           NA        10   0
#> 15        70.00000000 5.883990e-01        30   0
#> 17        71.40000000 5.883990e-01        10   0
#> 19        69.30000000 5.883990e-01        16   0
#> 21                 NA           NA        20   0
#> 24                 NA           NA        30   0
#> 25        69.30000000 5.883990e-01        30   0
#> 26        68.50000000 5.883990e-01        16   0
#> span.1     0.00000000 0.000000e+00        30   0
#> span.10            NA           NA        10   0
#> span.15    0.00000000 0.000000e+00        30   0
#> span.17    0.00000000 0.000000e+00        10   0
#> span.19    0.00000000 0.000000e+00        16   0
#> span.2     0.00000000 0.000000e+00        30   0
#> span.21            NA           NA        20   0
#> span.24            NA           NA        30   0
#> span.25    0.00000000 0.000000e+00        30   0
#> span.26    0.00000000 0.000000e+00        16   0
#> span.3     0.00000000 0.000000e+00        16   0
#> span.7             NA           NA        20   0
#> span.9     0.00000000 0.000000e+00        16   0
#> median.1  69.30000000 5.883990e-01        30   0
#> median.10          NA           NA        10   0
#> median.15 70.00000000 5.883990e-01        30   0
#> median.17 71.40000000 5.883990e-01        10   0
#> median.19 69.30000000 5.883990e-01        16   0
#> median.2  69.40000000 5.883990e-01        30   0
#> median.21          NA           NA        20   0
#> median.24          NA           NA        30   0
#> median.25 69.30000000 5.883990e-01        30   0
#> median.26 68.50000000 5.883990e-01        16   0
#> median.3  68.60000000 5.883990e-01        16   0
#> median.7           NA           NA        20   0
#> median.9  69.20000000 5.883990e-01        16   0
#> mean.1    69.30000000 5.883990e-01        30   0
#> mean.10            NA           NA        10   0
#> mean.15   70.00000000 5.883990e-01        30   0
#> mean.17   71.40000000 5.883990e-01        10   0
#> mean.19   69.30000000 5.883990e-01        16   0
#> mean.2    69.40000000 5.883990e-01        30   0
#> mean.21            NA           NA        20   0
#> mean.24            NA           NA        30   0
#> mean.25   69.30000000 5.883990e-01        30   0
#> mean.26   68.50000000 5.883990e-01        16   0
#> mean.3    68.60000000 5.883990e-01        16   0
#> mean.7             NA           NA        20   0
#> mean.9    69.20000000 5.883990e-01        16   0
#> 14        69.42782396 5.903436e-01        30   1
#> 22        69.52789112 5.903437e-01        30   1
#> 31        68.94094679 6.234568e-01        16   1
#> 4                  NA           NA        20   1
#> 5         69.25285830 6.020493e-01        16   1
#> 6                  NA           NA        10   1
#> 71        70.25610481 5.922822e-01        30   1
#> 8         71.45462914 6.624527e-01        10   1
#> 91        69.39149422 6.119858e-01        16   1
#> 101                NA           NA        20   1
#> 11                 NA           NA        30   1
#> 12        69.38166041 5.896413e-01        30   1
#> 13        68.84073921 6.234551e-01        16   1
#> span.14    0.00000000 0.000000e+00        30   1
#> span.22    0.00000000 0.000000e+00        60   1
#> span.23    0.00000000 0.000000e+00        16   1
#> span.4     0.10006716 9.263514e-08        60   1
#> span.5     0.00000000 0.000000e+00        16   1
#> median.14 70.25610481 5.922822e-01        30   1
#> median.22 69.38166041 5.896413e-01        60   1
#> median.23 68.84073921 6.234551e-01        16   1
#> median.4  69.47785754 5.903436e-01        60   1
#> median.5  68.94094679 6.234568e-01        16   1
#> mean.14   70.25610481 5.922822e-01        30   1
#> mean.22   69.38166041 5.896413e-01        60   1
#> mean.23   68.84073921 6.234551e-01        16   1
#> mean.4    69.47785754 5.903436e-01        60   1
#> mean.5    68.94094679 6.234568e-01        16   1
#> 16        69.50546726 5.909539e-01        60   2
#> 23        69.10086062 6.417874e-01        16   2
#> 32        70.47637826 5.956180e-01        30   2
#> 41        69.44121824 5.909584e-01        60   2
#> 51        69.07038927 6.518350e-01        16   2
#> span.20    0.32110495 3.984920e-02        52   2
#> span.6     0.15199768 3.973813e-02        52   2
#> median.20 69.23094174 6.319104e-01        52   2
#> median.6  69.17685946 6.219183e-01        52   2
#> mean.20   69.23094174 6.319104e-01        52   2
#> mean.6    69.17685946 6.219183e-01        52   2
#> 18        69.28873011 6.336366e-01        52   3
#> 27        69.35446193 6.416679e-01        52   3
#> span.18    0.08675631 5.070943e-02       112   3
#> span.8     0.21673715 4.268267e-02       112   3
#> median.18 69.39784008 6.163131e-01       112   3
#> median.8  69.39709869 6.122953e-01       112   3
#> mean.18   69.39784008 6.163131e-01       112   3
#> mean.8    69.39709869 6.122953e-01       112   3
#> 110       69.42816896 6.167532e-01       112   4
#> 28        69.42974946 6.208915e-01       112   4
#> span.11    0.00000000 0.000000e+00       122   4
#> span.16    2.02487967 4.156121e-02       122   4
#> median.11 69.42816896 6.167532e-01       122   4
#> median.16 70.44218930 6.416721e-01       122   4
#> mean.11   69.42816896 6.167532e-01       122   4
#> mean.16   70.44218930 6.416721e-01       122   4
#> 111       69.45669763 6.220417e-01       122   5
#> 29        70.47088347 6.469634e-01       122   5
#> span       1.01968064 5.134543e-02       274   5
#> median    70.47088347 6.220417e-01       274   5
#> mean      70.13465312 6.215410e-01       274   5
#> 112       70.52596949 6.376271e-01       274   6
#> span1      0.00000000 0.000000e+00       274   6
#> median1   70.52596949 6.376271e-01       274   6
#> mean1     70.52596949 6.376271e-01       274   6