Fig. 1—Steady-state coupling of the hydraulic and thermal models of a pipeline segment. In the second step, which is similar to the approach adopted in Meziou et al. (2016), the steady-state-flow pattern, GVF level, and pressure gradient calculated in the first step are used to adjust the transient distributed-parameter model. A modal approximation of the hyperbolic transfer functions capturing the mixture-fluid transients is then performed to simulate the pipeline two-phase-flow dynamics (i.e., transient behavior) for each pipeline segment (Fig. 2). For causality, the pressure and flow rate at the same location cannot be used as inputs for the pipeline-mixture dynamics. Hence, either the inlet pressure and the outlet-flow rate or the outlet pressure and the inlet-flow rate are used as inputs for each considered pipe- line segment. The different segments are combined as shown in Fig. 3, when considering the outlet pressure and the inlet-flow rate as system boundary conditions. The same procedure can be applied if the inlet pressure and outlet-flow rate are used as system inputs. The modeling procedure discussed previously results in a steady-state thermal model coupled with a hydraulic dynamic model where the fluid properties for each pipe segment are updated depending on only steady-state temperature conditions. However, for some cases, such as pipeline-shutdown operations, transient heat-transfer phenomena are critical. For such cases, a fully transient hydraulic and thermal model is required. To achieve this goal, the steady-state hydraulic and thermal models in Step 1 (Fig. 1) are replaced by transient models. The estimated transient pressure and temperature are used to determine the fluid properties at each time- step by solving the loop presented in Fig. 1. The pipe segments are then interconnected, as shown in Fig. 3, to construct the overall pipe- line model. The modular nature of the developed model allows the additional transient thermal model to be easily disabled if only steady-state heat-transfer phenomena are of interest. pee 7 ae a ae UF oe 7. a oe RS ae ‘ei, oe Pry peo 7 a4 Pe) o (ge Fig. 4—Transient inlet pressure along the pipeline (four segments). where I is the propagation operator and Z, is the characteristic impedance. The derivation of the dissipative distributed-paramete1 model is detailed in Brown (1962). The hyperbolic models in Eq. 1 are highly nonlinear. A modal approximation of the fluid-line dynamics is therefore performed according to Oldenburger and Goodson (1964) and Hsue and Hullender (1983). The transfer functions TF (jk) in Eq. 1 will be written, using the Laplace operator s, as a finite sum of second-order rational transfer functions, as shown in Eq. 2, Table 1—System properties for approximation illustration. Fig. 6—Axial view of pipeline. Summarized in Fig. 7 are the different steps of the proposed method for determining the Nusselt number for two-phase distributive flow. This number is used to calculate the heat-transfer coefficient for distributive flow, using Eq. 25. It is shown that the procedure used to calculate the Nusselt number for distributive flow is similar to the method used for single-phase fluids. Segregated Flow. Segregated or separated flow comprises stratified and annular mist flows. In this flow pattern, the phases are sepa rated by a distinct interface. For laminar-flow conditions, Knott et al. (1959) and Kaminsky (1999) proposed that the liquid’s heat transfer coefficient, using the equivalent hydraulic diameter and the velocity of two-phase flow, can be used to estimate the two-phas« heat-transfer coefficient. The heat transfer caused by gas is considered negligible, and only heat transfer by means of the liquid to the surroundings is considered. According to Knott (1959) and Kaminsky (1999), this would estimate the entrance effects more efficiently than heat/momentum analogies and homogeneous models using equivalent gas and liquid properties. This is reasonable for segregatec flow where the interaction between the two phases is minimal and heat transfer is dependent on the amount of liquid making contac with the pipe walls. Furthermore, it has also been suggested that the ratio of the two-phase fluid and the superficial liquid heat-transfe coefficient using the Sieder and Tate (1936) correlation would provide a better estimate for the heat-transfer coefficient of the two phase fluid rather than using the Sieder and Tate (1936) equation directly (Kaminsky 1999). Thus, for fully wetted walls, as in the cas of annular flow, Kaminsky (1999) proposed Table 4—Hydraulic and thermal boundary conditions. Table 5—Effect of the GVF level on the equivalent fluid properties. Effect of the GVF Level. The GVF level is increased from 0 (single-phase-flow liquid) to 40% to study its effect on the mixture dynamics. Given in Table 5 are the equivalent fluid density, bulk modulus, dynamic viscosity, speed of sound, natural frequency, anc damping ratio for the considered GVF levels. The gas and liquid properties are determined for each pressure and temperature using the Property Integration Block. The derivation of the equivalent fluid properties used in the hydraulic model is detailed in Meziou et al. (2016). It is shown in Table 5 that adding gas into the two-phase-flow system results in a lower equivalent fluid density and a lowe equivalent bulk modulus. However, the effect of the reduction in the bulk modulus is predominant, especially at low GVF levels. because of the use of the GVF-weighted parallel combination. This results in a significantly lower equivalent speed of sound, and con- sequently a lower system natural frequency and higher damping ratio. In addition, the introduction of a higher gas-flow rate results in a lower equivalent dynamic viscosity, specifically at higher GVF levels, leading to a lower steady-state pressure drop. Table 7—Effect of the outlet pressure on the equivalent fluid properties. Fig. 9—Inlet-pressure time response to inlet-flow-rate step for different inlet temperatures. Table 6—Effect of the inlet temperature on the equivalent fluid properties. Fig. 8—Inlet-pressure time response to inlet-flow-rate step for different GVFs. Fig. 11—Inlet-pressure time response to inlet-flow-rate step for different number of modes. Effect of the Hyperbolic-Function Approximation Order. Shown in Fig. 11 is the normalized inlet transient pressure for differen truncation orders of the transfer functions in Eq. 1. Increasing the number of modes in Eq. 1 enables a more-accurate approximation o the hyperbolic transfer functions by including the higher-order pipeline-mixture dynamics. This will translate, in the time domain, int a more-oscillating transient pressure. Although using a greater number of modes results in a more-accurate and -realistic estimation o the pressure and flow dynamic response, it will lead to a higher computational time. Meziou et al. (2016) discussed the effect of th truncation order on the model accuracy and computational time and provided a guideline on the selection of the appropriate number o modes to be used in the model approximation. Fig. 12—(a) Steady-state pressure profile; (b) steady-state fluid-temperature profile. density (Fig. 13a). It is shown in Fig. 13b that the equivalent fluid viscosity will increase considerably along the pipeline, driven by the reduction in the fluid temperature. As demonstrated in Tables 6 and 7, it is confirmed by analyzing Fig. 13c that the decrease in the pres- sure and temperature along the pipeline leads to a lower equivalent bulk modulus, and hence a higher fluid compressibility. Finally, as previously discussed, the effect of the bulk modulus reduction overcomes the reduction in the equivalent fluid density, leading to a slightly lower equivalent speed of sound (Fig. 13d). Fig. 13—Equivalent-fluid-properties profile: (a) equivalent density, (b) equivalent dynamic viscosity, (c) equivalent bulk modulus, and (d) equivalent speed of sound. To evaluate the effect of the pipe segmentation on its dynamic response, the baseline case simulation was performed using one, twe and four segments. Presented in Fig. 14 is the normalized transient pressure at the pipeline inlet. Note that a one-mode approximatio of the exact transfer functions in Eq. | is adopted for all the cases simulated in Fig. 14. The increase in the number of pipeline segment used in the simulations results in a more-dynamic pressure response. Although a one-mode approximation of the transfer functions i Eq. | was considered, the model was able to capture the higher-order dynamics similarly to using a higher truncation order because c the coupling of the smaller segments and a more-accurate estimation of the fluid-characteristics distribution along the pipeline. The normalized pressure is monitored at different locations along the pipeline using four segments (Fig. 4). It is shown that the pres- sure at the pipeline inlet (location of the step in the flow rate) is characterized by higher frequency oscillations and a higher overshoot. Those oscillations will be damped as we move closer to the pipeline outlet, characterized by a constant-pressure boundary condition. It is also noticed that the model was able to capture the time delay in the pressure response as the distance from the pipeline inlet increases. The pipeline segmentation enables a considerable improvement in the model-estimation capabilities. However, similar to using a higher hyperbolic-function truncation order, the increase in the number of segments will result in a higher computational cost. aioe a Fig. 15—Computation time vs. mean-squared error. Moills a Sa a ia aay UVTI MLTULIV ELUTE LLULIVALIUT UU T, UI TINT doOw Lil Lily LUO UE RVetiwiite Will tout tii a ADL MARAE ERR EEE XLVUdSL. Presented in Fig. 15 are the computation time and the mean-squared error as a function of the number of pipeline segments for th base case. Note that a one-mode approximation is considered for the studied cases, whereas a 20-mode approximation with 10 segment is used as a baseline. As the number of pipelines segments is increased, the mean-squared error drops quickly at the beginning while th computation time undergoes a low increase. This tendency is inverted when considering the higher number of segments where the com putation time increases dramatically without a noticeable improvement in the model accuracy. Hence, it is crucial to select the suitabl number of pipe segments depending on the desired model application and available computing power. This type of study can be per formed to evaluate the maximum number of segments that can be used not to exceed a required simulation time for the real-time moni toring of the pipeline’s dynamics or the minimum number of segments that should be considered to achieve a certain degree o accuracy and reliability in the design of safety equipment (Meziou et al. 2014). Effect of the Transient Thermal Module. The baseline-case parameters are used to evaluate the effect of the transient thermal model on the pipeline’s dynamic response. To simulate startup conditions, the inlet-flow rate is stepped at t= 200 seconds from 0 to 5x 1077 m?/s at a constant temperature (350 K) while keeping a fixed outlet pressure (1x 10’ Pa). A steady-state coupling of the hydrau- lic and thermal module is first used to generate the inlet pressure and outlet temperature. The proposed transient thermal module is then enabled to assess the effect of transient heat transfer on the pressure and temperature flow conditions. Given in Fig. 16 are the pressure and temperature predicted using the steady-state and transient heat-transfer models. It can be noticed from Fig. 16 that both the transient and steady-state thermal models lead to the same steady-state pressure and tem- perature conditions, However, unlike the model with steady-state heat transfer, characterized by very fast pressure transients (settling time of 50 seconds), the full transient model is characterized by significantly slower pressure and temperature transients (settling time of 1,500 seconds). This feature is very important, especially when designing pumps for startup conditions or calculating the cool-down time for shutdown conditions. In Fig. 16a, the transient inlet pressure predicted by the full transient model is considerably higher than the one simulated using the steady-state heat-transfer model. To understand in greater detail the effect of the transient temperature var- iations on the pipeline dynamic response, the equivalent fluid properties calculated by the steady-state and transient heat-transfer models are given in Fig. 17. Fig. 17—Equivalent fluid properties for the startup condition: (a) equivalent density, (b) equivalent dynamic viscosity, (c) equiva- lent bulk modulus, and (d) equivalent speed of sound. considering the entrance length is critical for obtaining an acceptable accuracy of the two-phase-flow heat-transfer coefficient. As seen in Figs. 19 and 20, the proposed segregated model is able to predict the overall heat-transfer coefficient with a good accuracy for hori- zontal flows. Thus, the approximation in this model (that most of the heat transfer occurs across the liquid’s boundary) is valid for hori- zontal pipes. For vertical segregated flows (annular flow), the proposed model shows a higher absolute error. This can be explained by the fact that the model assumes that the liquid-film thickness across the entire pipeline’s length is constant, which leads to a constant heat-transfer coefficient along the entire length. This assumption can introduce an error for vertical pipes where gravity is prevalent. A better estimate of the liquid-film thickness in annular flows might increase the accuracy of the proposed thermal model. This can be achieved through computational-fluid-dynamics (CFD) modeling of two-phase flow in vertical pipelines. The proposed intermittent model heavily relies on the distributed and segregated models. For horizontal flow, where the accuracy of the segregated model is rela- tively higher than that for vertical flow, the accuracy of the intermittent model is also higher. Fig. 18—Superficial liquid and gas velocities of the data used for validation of the steady-state thermal model. Table 8—Experimental data used for validation of thermal modeling. Fig. 19—Comparison of predicted and experimental heat-transfer coefficients for different flow patterns. Fig. 22—Fractional error for different flow patterns with respect to pipe diameters. Fig. 21—Comparison of predicted and experimental heat-transfer coefficients for different fluid combinations. Fig. 20—Fractional error for different flow patterns with respect to GVFs. Fig. 25—Fractional error for different flow patterns with respect to liquid viscosities. Fig. 24—Fractional error for different flow patterns with respect to gas viscosities. Fig. 23—Fractional error for different flow patterns with respect to pipe lengths. Table 9—Steady-state thermal mechanistic-model accuracy. Fig. 26—Fractional error for different flow patterns with respect to the ratio of superficial liquid velocity to superficial gas velocity Fig. 27—Shutdown condition: (a) inlet pressure, (b) outlet temperature (0% GVF). predicts a faster transient temperature response for the case of a 40% GVF level (Fig. 29b), leading to a lower cool-down tim (0.79 hours for the Low-D model and 1.44 hours for OLGA). This discrepancy can lead to different hydrate-prevention strategies. It : also shown in Figs. 27b, 28b, and 29b that both models predict that increasing the amount of gas in the system will result in more important heat losses in the pipeline, and therefore a lower cool-down time. Fig. 28—Shutdown condition: (a) inlet pressure, (b) outlet temperature (20% GVF). Fig. 29—Shutdown condition: (a) inlet pressure, (b) outlet temperature (40% GVF). Table 10—Characteristic parameters of a pipeline-shutdown scenario.
