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Saint Petersburg Mining University, Russian Federation, 199106, St. Petersburg, 21st Line, 2, Russia

**Received date:** 02 May, 2017; **Accepted date:** 06 May, 2017

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The article describes the principle of nomographs building that make it possible to quickly determine the required power consumption of the electric heating systems for stationary thermal operating mode of a pipeline. For pipelines with standard outside diameters, dependency of the linear and power type of the required linear power consumption of heat tracing on the average oil temperature has been shown. Also, tabular and graphical forms that simplify the process of calculating the required thickness of pipeline thermal insulation have been shown. The obtained dependences, charts and nomographs make it possible to calculate the thermal mode of a "hot" trunk oil pipeline with the minimum amount of source data. The results shown in the paper have been obtained with the accuracy sufficient for calculation.

Heat tracing, Pipeline, Heat mode, Heat insulation, Nomograph

**Steady temperature condition of oil trunk pipelines**

The steady temperature condition is characterized by continuous movement of a viscous product in the pipeline. The temperature at the inlet and the outlet of the pipeline is considered to be the same. With that, the required linear power consumption of heat tracing (W/m) is defined as (Fonaryov, 1973):

(1)

where

*t _{o.av}*– the average temperature of the oil pumped
through the entire length of the pipeline;

t_{env}. – temperature of the environment, K (for
underground pipes laying, t_{0}: the temperature of the
soil in its natural thermal condition at the depth of
the pipeline axis makes sense);

λ_{ins} - coefficient of thermal insulation thermal
conductivity, W/(m∙K);

d_{ins} - the outer diameter thermal insulation, m;

d_{o} - the outer diameter of the pipeline, m;

α_{o} - the coefficient of heat transfer from the outer
surface of the insulation to the environment, W/
(m^{2}*K).

For the pipelines located in open areas, the coefficient kn of heat loss through the supports and valves is taken equal to 1.25, and for pipelines in confined areas - equal to 1.2. The unknown losses coefficient ko is taken equal to 1.1.

In the denominator of formula (1), the first summand is the thermal resistance of the heat insulating structure, and the second summand is the thermal resistance of heat transfer from the surface of the insulation structure to the environment. Here we will neglect the thermal resistance of the pipeline metal and thermal resistance of heat transfer from the oil to the inner wall of the pipe due to their small values.

The heat conduction coefficient of the main insulation
layer of the structur λ_{ins}, W/(m∙K) can be found using
formula (Kopko, 2002):

λ_{ins} = λk (2)

where λ in the heat conduction coefficient of dry material of the main layer, W/(m∙K) (taken according to (SNiP 41-03-2003, 2003);

k is the correction coefficient of increasing thermal conductivity from moisturizing (depending on the type of the insulation material and type of soil, kis taken according to (SNiP 41-03-2003, 2003).

For calculating the minimum required thickness of the insulation layer, density of the heat flow between the pipeline and the environment is to be found (SNiP 41-03-2003, 2003):

q=q_{L}K (3)

where q_{L} is the standard heat flux density, W/m,
(adopted according to (SNiP 41-03-2003, 2003);

K is the coefficient of changes of normal density of the heat flow, depending on the area of construction (adopted according to (SNiP 41-03-2003, 2003).

Thickness of the insulating layer δ_{ins}, depending
on the normal density of the heat flow, can be
determined by simplified formula (Kopko, 2002):

(4)

where coefficient B is found as:

(5)

For insulation structures made of compacting
materials, sealing of the main layer to the expected
values calculated with regard to the compaction
factor (Kopko, 2002) is provided. To determine the
ordered amount (volume) of compacting insulation
products, the volume of the insulating layer of
these products in the structure is multiplied by the
compaction factor K_{c} (SNiP 41-03-2003, 2003).

To build nomographs for quick selection of the required power consumption of the pipeline trace heating system, let us adopt the following values of unknown parameters:

- Thermal conductivity of the insulating material λ=0.05W/(m∙K) (based on the average values of the most common insulating materials according to (Vasiliev, 1971; Goova, 2002; Kammerer, 1965; Design and construction specifications 41-103-2000, 2001);

- The norm of the heat flow densityq will be defined by interpolating the tabular values of the norms of heat flow density for pipelines with positive temperatures located in the open air and the work number over 5,000 (SNiP 41-03-2003, 2003) (coefficient K is taken equal to 0.96);

- Coefficient k_{s} is taken equal to 1.25;

- The heat conduction coefficient from the surface
of insulation α_{o}=26 W/(m2*K) (recommended value
in the absence of specific information about wind
speed) (Design and construction specifications 41-
103-2000, 2001).

**Nomographs for selecting power consumption of
trace heating**

Based on the above dependencies, let us build a
nomograph for fast approximates selection of the
trace heating system power consumption for a
pipeline with nominal diameter 219 mm, with the
maximum possible temperature difference between
the pumped oil and the environment (**Fig. 1**).

The dashed line describes the variation of the required linear power consumption of trace heating at the maximum temperature difference between the oil product and the environment. Solid straight lines determine operation mode of the pipeline trace heating system for which the maximum difference between the temperature of oil and the environment is known, and help to define the linear power consumption of trace heating in case of increasing the temperature of the environment. Each of the modes described by the olid straight lines corresponds to a certain minimum thickness of the pipeline thermal insulation (indicated in the top part of the graph).

The ambient temperature is taken equal to the temperature of the coldest fivedays, based on the climate reference books (SNiP 23-01-99, 2008), with the probability of 0.92.

To determine the required linear power consumption
of trace heating, it is enough to know the maximum
difference of temperatures between the pumped fluid
and the environment, and the outer diameter of the
pipeline. For example, the maximum temperature
difference between oil and the environment is
50°C for a pipeline with the outer diameter equal
to 219 mm. In the nomograph (**Fig. 1**), let us draw
a vertical line from 50°C to the intersection with
the dashed line, and determine the required linear
power consumption of trace heating (49 W/m) and
insulation thickness (58 mm).

It is necessary to underline the fact that this method is suitable for calculating power consumption of trace heating in the stationary thermal mode of pipeline operation.

The nomographs data have been built with the assumption that the temperature of the pumped product is equal to the maximum temperature allowable for pumping oil, 70°C.

Nomographs may be built for all sizes of pipelines, including the pipelines of most common outer diameters: 32 mm, 57 mm, 83 mm, 108 mm, 133 mm, 159 mm, 219 mm, 245 mm, 273 mm, 325 mm, 426 mm, 530 mm, 630 mm, 720 mm, 820 mm, and 1,020 mm (GOST 30732-2006; Zhuravlev, 1954).

**The simplified method of determining thickness of
a trunk pipeline insulating layer**

The product temperature defines the value of the normal flux density, and, therefore, the values of the minimum allowable thickness of the insulation and the required trace heating power consumption.

Thus, in the next stage, the problem of obtaining
simplified linear dependencies of the required
power consumption of trace heating and thickness
of heat insulation on the temperature of pumped
product arises. Analysis of the initial data allowed
us to obtain dependencies of the required power
consumption of trace heating P_{d} on the temperature
of the product t_{o.av} like:

(6)

(7)

where A, B are coefficients.

The accuracy of describing the initial tabular data with the obtained equations was ensured by using the absolute σ and the relative σ values of the mean square approximation error:

(8)

where N – the number of sample points from the dataset;

*n* is the number of constants in the equation;

P_{i.theor} – P_{i.calc} is the discrepancy between the initial
values of trace heating power consumption and
those obtained with the use of the approximation
equation, W/m;.

(9)

**Table 1** shows the obtained dependences of heat
trace heating power consumption on the average
temperature of oil. The approximation error for these
equations is fairly small **Table 1** - it does not exceed
2 per cent – which bespeaks of the possibility of their
practical use.

Pipe diameter, mm | Linear power consumption, W/m | Root mean square approximation error, W/m | Relative mean square approximation error, % |
---|---|---|---|

32 | 0.23 | 1.37 | |

57 | 0.17 | 0.90 | |

83 | 0.20 | 1.10 | |

108 | 0.18 | 0.73 | |

133 | 0.23 | 1.12 | |

159 | 0.18 | 0.77 | |

219 | 0.23 | 0.80 | |

245 | 0.36 | 0.84 | |

273 | 0.23 | 0.57 | |

325 | 0.46 | 0.85 | |

426 | 0.66 | 0.97 | |

530 | 1.09 | 1.85 | |

630 | 0.56 | 0.66 | |

720 | 0.60 | 0.65 | |

820 | 0.74 | 0.74 | |

1,020 | 0.82 | 0.68 |

**Table 1.** Errors in calculation by obtained equations

Approximating equations for thermal insulation
thickness could not be obtained with the accuracy
sufficient for technical calculations, so the calculated
values of insulation thickness were presented in a
table. **Table 2** shows the values of heat insulation
thickness, depending on the average temperature
of the pumped product and the difference of
temperatures between the environment and the
pumped product. If the tabular data are presented
as a chart, a family of curves will be obtained (**Fig. 2**).

, ° C | The temperature of pumped product, ° C | ||||||||

20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | |

10 | 26 | 19 | 14 | 11 | 10 | 8 | |||

15 | 43 | 30 | 23 | 19 | 16 | 13 | 12 | ||

20 | 62 | 43 | 33 | 26 | 22 | 19 | 17 | 15 | |

25 | 83 | 57 | 43 | 35 | 29 | 25 | 22 | 19 | 17 |

30 | 107 | 73 | 54 | 43 | 36 | 31 | 27 | 24 | 21 |

35 | 134 | 89 | 66 | 52 | 44 | 37 | 33 | 29 | 26 |

40 | 164 | 107 | 79 | 62 | 52 | 44 | 38 | 34 | 30 |

45 | 198 | 127 | 93 | 73 | 60 | 51 | 44 | 39 | 35 |

50 | 237 | 149 | 107 | 83 | 69 | 58 | 51 | 45 | 40 |

55 | 279 | 173 | 123 | 95 | 78 | 66 | 57 | 50 | 45 |

60 | 328 | 198 | 140 | 107 | 88 | 74 | 64 | 56 | 50 |

65 | 382 | 227 | 158 | 120 | 98 | 82 | 71 | 62 | 55 |

70 | 443 | 257 | 178 | 134 | 109 | 91 | 78 | 68 | 61 |

75 | 511 | 291 | 198 | 149 | 120 | 100 | 86 | 75 | 66 |

80 | 328 | 221 | 164 | 132 | 110 | 94 | 81 | 72 | |

85 | 368 | 245 | 181 | 144 | 120 | 102 | 88 | 78 | |

90 | 270 | 198 | 157 | 130 | 110 | 96 | 84 | ||

95 | 298 | 217 | 171 | 141 | 119 | 103 | 91 | ||

100 | 237 | 186 | 152 | 128 | 111 | 97 |

**Table 2.** Thickness of thermal insulation for a pipeline with outer diameter of 219 mm

These tabular and graphical forms of determining thermal insulation thickness can be built for the entire range of frequently used types and sizes of pipelines.

**Substantiation of applicability of the obtained
dependencies**

The Let us perform a comparative analysis of these
charts and the chartsobtained by Z. I. Fonariov
(Fonaryov, 1982; Fonaryov, 1984). (**Fig. 3**) shows,
for a ND 25 mm pipe, the charts of selecting power
consumption built according to the data of Z. I.
Fonariov and the above method in other equal
conditions (insulation thickness, heat conduction
coefficient of the insulation, etc.).

Thus, these dependencies are described by the same law with some deviation in the values.

**Table 3** shows the maximum possible relative
deviation of the obtained values with the use of the
two methods of calculating power consumption of
trace heating for DN 25-300 mm pipes in the range of
temperature differences of t_{o.av} – t_{i}=20...100°C.

Pipe DN, mm | The maximum absolute deviation of power consumption, W/m | The maximum relative deviation of power consumption, % | Standard deviation σ, W/m |
---|---|---|---|

25 | 2.62 | 41.23 | 1.90 |

50 | 2.24 | 5.22 | 1.76 |

75 | 2.27 | 8.80 | 1.23 |

100 | 3.83 | 15.24 | 1.90 |

150 | 12.87 | 18.04 | 8.76 |

200 | 16.63 | 10.15 | 10.48 |

250 | 30.86 | 17.59 | 21.53 |

300 | 37.15 | 19.34 | 26.81 |

**Table 3. **Comparing the data with the use of two methods

Large maximum relative deviations of power
consumption occur with small absolute values of
power consumption, when the deviation of the
charts by the y-axis is comparable to the absolute
values of power consumption at this point in (**Fig. 3**),
for example, at 20°C).

To check the compliance of the theoretical dependencies proposed by Z. I. Fonariov, let us use the analysis of the Fisher's F criterion: let us compare total variance to residual variance .

The total and residual variances are calculated as:

(10)

where P_{1} is the value of power consumption
obtained by the method of Z. I. Fonariov, W/m; n is
the number of analyzed points;

(11)

where P_{2} is the value of power consumption obtained
by the proposed method, W/m.

The stages of calculation for a DN 25 mm pipe are
shown in **Table 4**.

No. | ()^{2} |
|||||||

1 | 0 | 6.19 | 38.305 | 8.74 | -2.55 | 6.511 | 156.295 | 4.081 |

5 | 8.32 | 69.168 | 10.93 | -2.61 | 6.809 | |||

3 | 30 | 10.49 | 110.051 | 13.11 | -2.62 | 6.869 | ||

4 | 35 | 12.71 | 161.555 | 15.30 | -2.59 | 6.688 | ||

5 | 40 | 14.98 | 24.295 | 17.48 | -2.51 | 6.276 | ||

6 | 45 | 17.29 | 98.898 | 19.67 | -2.38 | 5.656 | ||

7 | 50 | 19.65 | 386.005 | 1.85 | -2.21 | 4.863 | ||

8 | 55 | 2.05 | 486.267 | 4.04 | -1.99 | 3.944 | ||

9 | 60 | 4.50 | 600.352 | 6.22 | -1.72 | .960 | ||

10 | 65 | 7.00 | 728.937 | 8.41 | -1.41 | 1.985 | ||

11 | 70 | 9.54 | 872.713 | 30.59 | -1.05 | 1.105 | ||

12 | 75 | 32.13 | 1,032.385 | 32.78 | -0.65 | 0.419 | ||

13 | 80 | 34.77 | 1,208.669 | 34.96 | -0.20 | 0.039 | ||

14 | 85 | 37.45 | 1,402.295 | 37.15 | 0.30 | 0.089 | ||

15 | 90 | 40.17 | 1,614.005 | 39.33 | 0.84 | 0.707 | ||

16 | 95 | 42.95 | 1,844.554 | 41.52 | 1.43 | .042 | ||

17 | 100 | 45.77 | ,094.710 | 43.70 | .06 | 4.259 | ||

∑ | 425.95 | 13,173.165 | 61.222 |

**Table 4. **Calculation of variances.

Fisher's F criterion is calculated as relation (Lvovsky, 1982):

(12)

The obtained theoretical dependence for power consumption with the 5% significance level adequately describes the values of power consumption obtained according to the method of Z. I. Fonariov, due to the fact that the calculated Fisher's criterion was greater than the tabular one (Lvovsky, 1982):

F = 38,30 > F_{(16;15;5%)} = 2,39

**Table 5** shows the values of the Fisher's F criterion
obtained with other conventional tube diameters.

DN, mm | 25 | 50 | 75 | 100 | 150 | 200 | 250 | 300 |

F |
38.30 | 70.87 | 247.5 | 155.16 | 11.26 | 15.58 | 4.95 | 4.39 |

2.39 | 2.39 | 2.39 | 2.39 | 2.39 | 2.39 | 2.39 | 2.39 |

**Table 5. **Fisher's F criteria

As the table shows, the obtained theoretical dependencies for all pipe diameters with the 5% significance level adequately describe the values of power consumption obtained according to an alternative method of Z. I. Fonariov.

Air Thus, nomographs were built and developed that make it possible to determine the required power consumption of trace heating for a pipeline with certain outer diameter, and the thickness of the insulation layer provided that certain temperature of the pumped fluid is maintained. It should be noted that the proposed nomographs do not have the shortcomings present in the work of (Z. I. Fonariov, 1984). In addition, with the accuracy sufficient for technical calculations, the approximating formulas were obtained for determining the required power consumption of trace heating, depending on the average temperature of oil flow. The obtained formulas make it possible to quickly and accurately enough obtain values of the required power consumption of the pipelines trace heating.

The results of the research make it possible:

1. Knowing the current readings of oil flow temperature measuring instruments and the actual thickness of the insulating layer, to define either the electrical power consumption for maintaining a stable temperature of the pipeline (for automated trace heating system), or the required frequency of enabling/disabling the trace heating systems (timer operation mode) in the conditions of temperature changes in the environment (daily/seasonal temperature fluctuations).

2. Knowing the current readings of oil flow temperature measuring instruments and the ranges of the ambient temperature, to assess the cost of the trace heating system at the stage of project feasibility study: the required amount of the heating cable, heat insulation, cost of electrical energy for operation of the trace heating system, etc.

The results show that the method of determining power consumption of the heat tracing system and the thickness of heat insulation for maintaining stable temperature of a non-isothermal pipeline should be described as a special method. From the point of view of practical significance, the method will be convenient after specialized software (a computer application) is used as its basis.

The obtained results are useful for a wide range of persons engaged in applied research and performing engineering tasks in the area of studying and choosing thermal conditions of pipelines.

With regard to the fundamental differences between the processes of oil flow transfer and heat exchange in the near-wall layer of the pipeline and in its central part (the so-called "flow core"), the obtained results may be used for further studies, making it possible to obtain similar dependencies for shorter pipelines (technological pipelines).

During the research, an integrated approach was used, which combined theoretical and experimental methods of research: planning experiments, testing in specialized laboratory installations, and numerical experiments on computer mathematical models of the pipeline built with the use of modern computer programs.

The authors express their deep gratitude to I. A. Vishnyakov, Ph. D. for help in formulating the overall idea of the research, for directing the research and for the provided materials.

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