With the decrease in the pipeline temperature and the precipitation of the wax crystals in the oil, the crude oil may be transformed from a Newtonian fluid to a non-Newtonian fluid. With the reduction of pipeline transportation throughput and the decrease of the pipeline transport temperature, the tube segments of the non-Newtonian flow pattern will lengthen [
14]. Considering the different rheological properties of crude oil in different temperature ranges, a method is proposed to divide the oil pipeline into transportation intervals according to characteristic temperatures, such as wax appearance point, critical transition temperature and anomalous point. A formula for calculating the unavoidable destroyed exergy of crude oil under different flow conditions is derived.
3.1. Crude Oil Critical Transition Temperature
Considering the stable operation of an oil pipeline, the Reynolds number of the same crude oil can be expressed as a function of temperature. The critical transformation temperature of the flow pattern and flow regime can be calculated by the critical Reynolds number inverse calculation. In this method, traditional flow regime judgment (based on Reynolds number) and flow pattern judgment (using the anomalous point
as the standard) are unified to use the pipeline temperature as the basis for judgment [
15,
16].
For Newtonian waxy crude oil, the critical Reynolds number is 2000 for internal flow [
17] when the flow of the Newtonian fluid is converted from laminar to turbulent. Reynolds number is defined as:
where,
—Inner diameter of oil pipeline, m;
—Crude oil flow velocity, m/s;
—Oil flow viscosity, m
2/s.
It should be noted that when calculating the transition temperature, the viscosity of the oil flow for Newtonian flow is calculated based on the critical Reynolds number . According to its viscosity temperature curve, is obtained.
For non-Newtonian waxy crude oil, the specification provides that the critical Reynolds number is 2000, which marks the change from laminar flow to turbulent flow, and the Reynolds number is defined as follows [
17]:
where,
—Density of crude oil, kg/m
3;
—Consistency coefficient of crude oil, Pa·s;
—Rheological index of crude oil.
When waxy crude oil becomes a non-Newtonian fluid, it is usually a pseudoplastic fluid. In the temperature range of pseudoplastic fluid, the definition
and its rheological parameters
and
can be expressed as a function of temperature. Using Formulas (5) and (6), the flow transition temperature can be back calculated. That is [
18]:
where,
,
,
,
—constant.
3.2. Determination of the Specific Heat Capacity of Waxy Crude Oil
It should be noted that wax deposition may occur during the transportation of waxy crude oil. The total heat transfer coefficient of the pipeline gradually decreases with the increase of the thickness of the wax layer. There will be a wax-free section, an initial paraffin section and a wax deposition tail section in the oil pipeline, and the specific heat capacities of the oil are different in each section:
(1) When the oil temperature is higher than the wax appearance temperature of waxy crude oil, there is no wax phenomenon in the crude oil pipeline (this is called the wax-free section). In this temperature range, the specific heat capacity of crude oil increases slowly with the rise of temperature. The relationship is [
19]:
where,
—The crude oil specific heat capacity, kJ/(kg·°C);
—The crude oil relative density at 15 °C, dimensionless;
T— the crude oil temperature, °C.
(2) When the temperature of the oil is lower than the wax appearance point, the wax will precipitate. In particular, after the oil temperature drops to the wax appearance point of crude oil, the wax in the pipeline will increase, and the wax layer will gradually thicken along the oil delivery direction until a peak is reached—that is, the maximum thickness of wax deposition. This section is called the initial paraffin section. Due to the release heat from the wax crystal in the above temperature range, the specific heat capacity of crude oil increases with decreasing temperature, and the relationship between crude oil and temperature fits the following formula [
19]:
where,
—constant, different from crude oil, kJ/(kg·°C);
a—constant, different from crude oil, 1/°C.
(3) From the peak point of wax deposition to the interval of crude oil intakes, it is called the wax deposition tail section. With the increase of wax deposit thickness, the operation conditions of oil pipelines correspondingly change. In this temperature range, the specific heat capacity of crude oil decreases with decreasing oil temperature. The relationship between specific heat and crude oil temperature in this interval can be expressed as [
19]:
where,
B—constant, different from crude oil, kJ/(kg·°C);
m—constant, different from crude oil, 1/°C.
The parameters of several waxy crude oils are shown in
Table 1.
3.3. Temperature Range of Pipeline Transportation
In the process of pipeline transportation, the specific heat capacity of oil and viscous frictional resistance vary with temperature; thus, the oil pipeline is divided into temperature intervals according to the wax appearance point
, anomalous point
and flow transition temperature
,
. Specifically [
20,
21,
22]:
(1) If the temperature range is , there is only a single temperature range in the pipeline transportation, and the crude oil will remain in the Newtonian turbulence state;
(2) If the temperature range is and , the pipeline transportation process is divided into two temperature intervals and ; thus, the pipeline crude oil is in the Newtonian flow state;
(3) If the temperature range is and , the pipeline transportation process is divided into three temperature intervals , and ; thus, the pipeline crude oil is in Newtonian flow and there may be laminar flow state;
(4) If the temperature range is and , pipeline transportation is divided into four temperature intervals , , and ; thus, non-Newtonian fluid characteristics begin to appear;
(5) If the temperature range is and , pipeline transportation is divided into five temperature intervals , , , and ; thus, the pipeline crude oil presents non-Newtonian laminar flow in some pipe sections.
The division of the temperature range for a waxy crude oil pipeline is shown in
Figure 1.
During crude oil transportation,
is the ambient temperature around the pipeline,
G is the throughput of oil transported by the pipeline and
i is the hydraulic gradient. When fluid flows through pipe section d
L, the corresponding temperature drop is d
T. The energy balance of elementary section d
L under stable conditions for a waxy crude oil pipeline is [
23]:
For Formula (10), the left is the heat dissipation from the dL pipe to the surrounding medium in unit time. The first item at the right is the heat release from the temperature drop dT of the oil flow in the pipe. The second item is the heat transformed by friction loss of oil flow in dL section.
The total length of the wax oil pipeline is the sum of the temperature range lengths for each oil transportation, and the temperature intervals lengths of the pipeline are calculated. The length of each temperature range
can be obtained according to the Formula (11).
where,
—Initial point temperature of the crude oil pipeline transportation temperature range, °C;
—Terminal point temperature of the crude oil pipeline transportation temperature range, °C.