Method for calculating pressure loss in grouting pipe

Abstract

The application provides a method for calculating pressure loss in a grouting pipe, and belongs to the technical field of synchronous grouting. Firstly, based on the basic theory of fluid mechanics, a calculation formula of pressure loss of Newton fluid pipe laminar flow is derived; in view of rheological properties of synchronous grouting slurry, the calculation formula of pressure loss of non-Newton fluid pipe laminar flow is theoretically derived in combination with the actual flow law of the slurry; and according to the boundary layer theory, the theoretically derived calculation formula is corrected to improve the accuracy of the calculation formula; ANSYS Fluent numerical simulation software is used to simulate and calculate the pressure loss along the pipe, and the correctness of the theoretical calculation formula is verified by comparison; finally, the theoretically derived formula is verified by indoor tests, and the reliability of the theoretical calculation formula is further verified. The method can accurately predict the pressure loss, can avoid the influence of grouting construction caused by insufficient pumping capacity of the equipment, and can prevent the waste of equipment resources caused by blind selection of high-pressure pumps.

Description

Technical Field

[0001] This invention relates to the field of synchronous grouting technology, and more specifically to a method for calculating pressure loss in a grouting pipe. Background Technology

[0002] Synchronous grouting is a relatively advanced method of grouting behind the shield tunnel wall that has been developed in recent years. It can effectively ensure the stability of the strata during shield tunnel excavation and is widely used in modern shield tunnel construction. The grouting pump, as one of the core components of the synchronous grouting system, is usually located on the trolley behind the shield machine. The grouting location and the grouting pump are connected by pipelines. Pressure sensors are installed near the front shield, and synchronous grouting of the shield tunnel is achieved by controlling the grouting volume and pressure.

[0003] Pressure loss is a core technical parameter of synchronous grouting systems, and its calculation results directly affect the design of the grouting system, including pipe diameter selection, pipeline route design, and grouting pump selection. Currently, existing technologies, based on fundamental principles of fluid mechanics, have conducted extensive theoretical calculations and numerical simulation analyses on pressure loss during concrete pipeline transportation, establishing computational models and accumulating rich research experience. However, specific research on pressure loss during synchronous grouting in shield tunnels is still lacking. Accurate prediction of pressure loss can prevent grouting construction from being affected by insufficient pumping capacity and can also prevent the waste of equipment resources caused by blindly selecting high-pressure pumps, thus having significant guiding significance for improving the safety and economy of tunnel construction. Summary of the Invention

[0004] To address the problem of calculating pressure loss during synchronous grouting in shield tunnels, this invention proposes a method for calculating pressure loss in grouting pipes. This method accurately predicts pressure loss, which can both prevent grouting construction from being affected by insufficient pumping capacity and avoid wasting equipment resources due to blindly selecting high-pressure pumps.

[0005] To solve the above problems, the present invention adopts the following technical solution: A method for calculating pressure loss in a grouting pipe, the method comprising the following steps: S1: Based on the basic theory of fluid mechanics, derive the formula for calculating the pressure loss of Newtonian fluid laminar flow in a circular pipe; S2: Based on the rheological characteristics of synchronous grouting slurry and combined with the actual flow law of slurry, the calculation formula for pressure loss of non-Newtonian fluid in circular pipe laminar flow is derived. S3: Based on boundary layer theory, the calculation formula derived from the theory is modified to improve the accuracy of the calculation formula; S4: Use numerical simulation software to simulate and calculate the pressure loss along the friction line, and compare and verify the correctness of the theoretical calculation formula. S5: Through indoor experiments, the theoretically derived formulas are experimentally verified to further validate the reliability of the theoretical calculation formulas.

[0006] Furthermore, in S1, based on the Newtonian fluid assumption, a simplified expression for the friction loss of a fluid undergoing low-velocity laminar flow in a circular pipe of uniform diameter is derived using the Rayleigh method. An expression relating the friction loss to various parameters is established, and the volumetric flow rate Q of the slurry per unit time can be expressed as: ; In the formula, The plastic viscosity of the slurry. Where is the radius of the grouting pipe. For pressure loss along the process, For length; Converting equation (1) to exponential form, we get: ; In the formula, It is a dimensionless constant; For index; Solve according to the principle of dimensional harmony. The simplified expression for the laminar flow rate of the synchronously injected grout in a circular pipe of equal diameter is obtained. The pressure loss along the pipe is then: ; In the formula, The average flow velocity of the slurry.

[0007] Furthermore, the mechanical equilibrium equations are established for the tiny cylinder: ; In the formula, , The pressure acting on both ends of the cylinder, The radius of the tiny cylinder; Based on Newton's law of internal friction of fluids, equation (7) is rearranged and the boundary conditions are substituted. The flow velocity equation is obtained as follows: ; In the formula, The above equation shows that Newtonian fluids undergo laminar flow in a circular tube of equal diameter. When the flow velocity is at its maximum, the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity can be obtained. ; In the formula, At the maximum fluid velocity, along the axis. At the pipe wall, The slurry at the wall of the circular pipe satisfies the no-slip condition, and its velocity at the wall is the same as the velocity at the wall surface. The circular pipe wall is stationary, and theoretically, the velocity at the wall surface is the Newtonian fluid velocity. The flow rate formula for laminar flow of fluid in a circular pipe of constant diameter is derived, and the average velocity of the Newtonian fluid in the pipe cross section is derived by performing variable integration. ; According to equation (21), the formula for the friction loss of a Newtonian fluid undergoing laminar flow in a circular pipe of equal diameter is obtained: ; Substituting equation (22) for friction loss into equation (6), we get: .

[0008] Furthermore, in S2, the Bingham model is selected to derive the theoretical derivation of pressure loss along the flow path for non-Newtonian fluid laminar flow in a circular pipe. The Bingham model formula is as follows: ; In the formula, For shear stress, For dynamic yield stress, The slurry shear rate; By combining Newton's law of internal friction of fluids, the expression for the internal friction force on the surface of a cylinder is derived; ; In the formula, The area of ​​a tiny cylinder; This represents the velocity gradient.

[0009] Furthermore, assuming that the fluid throughout the circular pipe undergoes shear flow, the mechanical equilibrium equations for laminar flow in the circular pipe are established as follows: ; Separate the variables from equation (27) and rearrange the differential relationship, let Integrating the variables on both sides yields: ; In the formula, For a dimensionless constant, the boundary conditions are: Substituting into equation (31), we obtain the flow velocity equation: ; The above equation shows that non-Newtonian fluids undergo laminar flow in a circular pipe of constant diameter. When the flow velocity is at its maximum, substituting the maximum flow velocity into equation (34), we obtain the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity: ; At the axis, At the pipe wall, Theoretically, the velocity of non-Newtonian fluid at the wall ; Based on the obtained velocity equation, the flow rate formula for a non-Newtonian fluid undergoing laminar flow in a circular pipe of constant diameter is derived: ; Integrating the variables on both sides of equation (37), we derive the average flow velocity across the cross-section of a non-Newtonian fluid pipe: ; Based on the above formula, the pressure loss formula along the flow path for a non-Newtonian fluid undergoing laminar flow in a circular pipe of constant diameter is obtained: .

[0010] Furthermore, in S3, the local Reynolds number is used. As a Stokes peristaltic flow criterion, the critical velocity threshold of the fluid is obtained through the Reynolds number calculation formula, the thickness of the stagnant layer is quantitatively analyzed, and a local Reynolds number is defined. At that time, the critical flow velocity threshold , Since the value is a constant, this region belongs to the retention layer. The thickness of the retention layer can be calculated. The analytical solution is: ; In the formula, combined with the friction loss formula obtained in S2, the corrected friction loss calculation formula is as follows: .

[0011] Furthermore, in S4, ANSYS Fluent software is used to establish a numerical model of the laminar flow motion of synchronous grouting slurry in a horizontally placed straight pipe, and numerical simulation calculations are performed on the pressure loss along the pipeline during the synchronous grouting process. The Bingham fluid model is constructed by user-defined functions to replace the software's built-in model.

[0012] Furthermore, in S5, the indoor test uses a synchronous grouting model device, which includes a grouting pump, a grout tank, and a grouting pipe. The grouting pump is connected to the grout tank, and a flexible rubber hose is provided between the grout tank and the grouting pipe. Pressure gauges are provided on both sides of the grouting pipe.

[0013] Furthermore, the grouting pump is a portable air compressor equipped with an air source processor for pressure stabilization, the grout tank has a volume greater than 30L, the grouting pipe is a steel pipe with an inner diameter of 50mm, the two ends of the steel pipe are equipped with ball valves for adjusting the flow rate, the pressure gauge is a flat diaphragm type pressure transmitter, and the pressure gauge spacing is 1m.

[0014] The beneficial effects of this invention are as follows: This invention patent proposes a method for calculating pressure loss in grouting pipes, which can accurately predict pressure loss. This can not only avoid the impact of insufficient pumping capacity of equipment on grouting construction, but also prevent the waste of equipment resources caused by blindly selecting high-pressure pumps. It has important guiding significance for improving the safety and economy of tunnel construction.

[0015] Indoor tests were conducted to test the pressure loss of samples with different air-entraining agents at different hydration times. The test results were compared with the theoretical calculation results. The test results of all groups of samples without air-entraining agents were greater than the theoretical calculation values, while the test results of most groups of samples with air-entraining agents were less than the theoretical calculation values, with the error within ±10%. Attached Figure Description

[0016] To more clearly illustrate the specific embodiments of the present invention, the accompanying drawings used in the description of the specific embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 This is a flowchart of the method of the present invention; Figure 2 A schematic diagram showing the laminar flow of grout in a circular pipe during synchronous grouting. Figure 3 for Pressure cloud map along the straight pipe; Figure 4 for Pressure cloud diagram of each cross section of the straight pipe; Figure 5 A graph showing the pressure variation with pipe length for circular pipes of different radii; Figure 6 This is a histogram showing the variation of pressure loss along the pipe radius in laminar flow within a circular pipe. Figure 7 A schematic diagram for comparing and verifying the results of theoretical calculations and numerical simulations of pressure loss along the friction line; Figure 8 This is a schematic diagram of a synchronous grouting model device.

[0018] In the diagram, 1-grouting pump; 2-grout tank; 3-grouting pipe; 4-flexible rubber hose; 5-pressure gauge. Detailed Implementation

[0019] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0020] This invention proposes a method for calculating pressure loss in grouting pipes. To accurately calculate the pressure loss along the pipe during synchronous grouting and provide guidance for engineering applications, the method first derives a formula for calculating pressure loss in laminar flow of Newtonian fluids in a circular pipe based on fundamental fluid mechanics theory. Then, considering the rheological characteristics of the grout and combining it with the actual flow patterns of the grout, a theoretical derivation of a formula for calculating pressure loss in laminar flow of non-Newtonian fluids in a circular pipe is performed. Furthermore, the derived formula is modified based on boundary layer theory to improve its accuracy. The pressure loss along the pipe is simulated using ANSYS Fluent numerical simulation software, and the correctness of the theoretical formula is verified by comparison. Finally, indoor experiments are conducted to verify the reliability of the theoretical formula. Figure 1 As shown, this method includes the following steps: S1: Based on the basic theory of fluid mechanics, derive the formula for calculating the pressure loss of Newtonian fluid laminar flow in a circular pipe; S2: Based on the rheological characteristics of synchronous grouting slurry and combined with the actual flow law of slurry, the calculation formula for pressure loss of non-Newtonian fluid in circular pipe laminar flow is derived. S3: Based on boundary layer theory, the calculation formula derived from the theory is modified to improve the accuracy of the calculation formula; S4: Use numerical simulation software to simulate and calculate the pressure loss along the friction line, and compare and verify the correctness of the theoretical calculation formula. S5: Through indoor experiments, the theoretically derived formulas are experimentally verified to further validate the reliability of the theoretical calculation formulas.

[0021] The method will be described in detail below with reference to specific embodiments.

[0022] A subway shield tunnel has an inner diameter of 5900mm, an outer diameter of 6600mm, a cutterhead diameter of 6880mm, and a ring width of 1500mm. Two synchronous grouting pipes are opened simultaneously, with an inner diameter of approximately 60mm. The calculated average flow velocity of the grout during synchronous grouting is approximately 0.6m / s, and the grout density is 1740~1920kg / m³. 3 The plastic viscosity fitted by the Bingham model is 10~30 Pa·s. Using this viscosity instead of the dynamic viscosity, then:

[0023] In the formula: It is the Reynolds number; The density of the slurry; The average flow velocity of the slurry; The radius of the grouting pipe; This refers to the plastic viscosity of the slurry.

[0024] Assuming the grout undergoes laminar flow within the grouting pipe, and inputting the calculation parameters, This indicates that the grout flows in a laminar flow within the grouting pipeline. Assuming this is correct, the grout is in a low-velocity laminar flow state during the grouting process.

[0025] Fluids possess a certain viscosity. When a fluid flows in a pipe, the particles in different flow layers undergo relative motion due to differences in flow velocity, resulting in tangential internal friction, or viscous force, between adjacent particles. This viscous force hinders the relative motion between flow layers; the greater the fluid viscosity and the greater the velocity difference, the stronger the viscous force. The flow resistance experienced by a fluid flowing along a pipe does not originate from the friction between the fluid and the pipe wall, but rather from the internal friction resistance generated by the relative motion between flow layers with different velocities. The presence of the pipe wall merely provides the necessary external constraint for this internal friction resistance, limiting the fluid flow range and creating a velocity gradient distribution. Fluid viscosity is the root cause of pressure loss along the flow path; the greater the fluid viscosity, the more energy is required to overcome the viscous force during flow, resulting in a greater pressure loss along the flow path.

[0026] S1: Based on the basic theory of fluid mechanics, derive the formula for calculating the pressure loss of Newtonian fluid laminar flow in a circular pipe.

[0027] Based on Newton's fluid hypothesis, a simplified expression for the friction loss of a fluid undergoing low-velocity laminar flow in a circular pipe of constant diameter is derived using the Rayleigh method, establishing the relationship between friction loss and various parameters. Empirically, for a fluid undergoing laminar flow in a circular pipe of constant diameter, the volumetric flow rate is related to the fluid viscosity, pipe diameter, length, and friction loss, and is directly proportional to the friction loss and inversely proportional to the pipe length. Therefore:

[0028] In the formula: The volumetric flow rate of the slurry per unit time. For fluid viscosity, Where is the radius of the grouting pipe. This refers to pressure loss along the pipeline; For length.

[0029] Converting equation (1) from functional form to exponential form, we get:

[0030] In the formula, It is a dimensionless constant; Let be the exponent. Converting equation (2) from exponential form to a dimensional equation, we get:

[0031] In the formula: It has the dimension of length; Dimensioned in terms of time; It is a unit of mass.

[0032] According to the principle of dimensional harmony, the dimensional index of the foundation pit must satisfy:

[0033] Solve the system of equations (4). The simplified expression for the laminar flow rate of the synchronously injected grout in a circular pipe of equal diameter is obtained as follows:

[0034] Will Substituting into equation (5), we obtain a simplified expression for the pressure loss along the flow path of the synchronous grouting slurry in a circular pipe of equal diameter:

[0035] In the formula, The average flow velocity of the slurry.

[0036] In synchronous grouting, the grout undergoes laminar flow within a circular pipe of uniform diameter. Due to the grout's viscosity, tangential internal friction is generated between adjacent flow layers during flow, hindering the relative movement between the layers and thus consuming the fluid's mechanical energy. This continuous energy loss caused by internal friction due to fluid viscosity is defined as friction loss along the flow path. Figure 2 As shown, assuming the slurry is a Newtonian fluid, the expression is: ,in: Shear stress; The plastic viscosity of the slurry; Let be the slurry shear rate. The slurry at a radius of ... A horizontally placed circular tube undergoes laminar flow. A small cylindrical section of the tube, coinciding with its axis, is considered. The velocity of each particle inside; , The pressure acting on both ends of the cylinder; This refers to the internal friction force on the surface of the cylinder. Let be the radius of the cylinder.

[0037] Establish the mechanical equilibrium equations for a tiny cylinder:

[0038] According to Newton's law of internal friction of fluids:

[0039] In the formula: Let be the area of ​​a tiny cylinder. The velocity gradient is negative, so a negative sign is added to the formula to make it... It is a positive value.

[0040] make ,but:

[0041] Separate the variables from equation (10) and rearrange the differential relationship:

[0042] Integrating both sides of equation (11) with respect to the variables:

[0043] In the formula: It is a dimensionless constant.

[0044] Boundary conditions: Substitution (13):

[0045] Substituting equation (15) into equation (13), we obtain the flow velocity equation:

[0046] The above equation shows that Newtonian fluids undergo laminar flow in a circular tube of constant diameter. At that time, the flow velocity is at its maximum:

[0047] In the formula: This represents the maximum flow velocity of the fluid.

[0048] Substituting equation (17) into equation (16), we obtain the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity:

[0049] At the axis, , The maximum fluid velocity; at the pipe wall, The slurry at the wall of the circular pipe satisfies the no-slip condition, and its velocity at the wall is the same as the velocity at the wall surface. The circular pipe wall is stationary, and theoretically, the velocity at the wall surface is the Newtonian fluid velocity. .

[0050] Based on the obtained velocity equation, the flow rate formula for a fluid undergoing laminar flow in a circular pipe of constant diameter is derived:

[0051] Integrating both sides of equation (19) with respect to the variables:

[0052] The above formula is the flow rate formula for a Newtonian fluid undergoing laminar flow in a circular pipe of equal diameter, namely the Hagen-Poiseuille formula.

[0053] Based on equation (20), the average flow velocity of the cross section in a Newtonian fluid pipe is derived as follows:

[0054] In the formula: The average flow velocity of the fluid.

[0055] According to equation (21), the formula for the friction loss of a Newtonian fluid undergoing laminar flow in a circular pipe of equal diameter is obtained:

[0056] Substituting the friction loss formula (22) into equation (6), we get: .

[0057] S2: Based on the rheological characteristics of synchronous grouting slurry and combined with the actual flow law of slurry, the formula for calculating the pressure loss of non-Newtonian fluid laminar flow in circular pipe is derived.

[0058] The Herschel-Bulkley model is a yield power-law nonlinear constitutive model, while the modified Bingham model is a nonlinear piecewise constitutive model; neither can be expressed analytically through elementary integration. The Bingham model, with its simple form and clearly defined physical parameters, is one of the most widely used and theoretically mature fluid models in engineering. Formulas can be derived based on Newtonian fluids, yielding explicit velocity and flow rate formulas, and subsequently, the friction loss formula. Considering the feasibility of theoretical formula derivation and the needs of practical engineering applications, this embodiment selects the Bingham model to conduct the theoretical derivation of friction loss in laminar flow of non-Newtonian fluids in circular pipes.

[0059] The Bingham model formula is:

[0060] In the formula: This refers to the dynamic yield stress.

[0061]

[0062] Substituting equation (24) into equation (23), we get:

[0063] Substituting equation (25) into equation (9), we obtain the expression for the internal friction force on the surface of the cylinder:

[0064] Assuming that the fluid inside the entire circular pipe undergoes shear flow, the mechanical equilibrium equations for laminar flow in the circular pipe are established as follows:

[0065] make ,but:

[0066] Separate the variables and rearrange the differential relationship of equation (28):

[0067] Integrating both sides of equation (29) with respect to the variables:

[0068] In the formula: It is a dimensionless constant.

[0069] Boundary conditions: Substitution (31):

[0070] Substituting equation (33) into equation (31), we obtain the flow velocity equation:

[0071] The above equation shows that non-Newtonian fluids undergo laminar flow in a circular pipe of constant diameter. At that time, the flow velocity is at its maximum:

[0072] Substituting equation (35) into equation (34), we obtain the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity:

[0073] At the axis, At the pipe wall, Theoretically, the velocity of non-Newtonian fluid at the wall .

[0074] Based on the obtained velocity equation, the flow rate formula for a non-Newtonian fluid undergoing laminar flow in a circular pipe of constant diameter is derived:

[0075] Integrating both sides of equation (37) with respect to the variables:

[0076] when At that time, the above flow formula degenerates into the Hagen-Poiseuille formula.

[0077] Based on equation (38), the average flow velocity of the cross section in a non-Newtonian fluid pipe is derived as follows:

[0078] According to equation (39), the formula for the friction loss of a non-Newtonian fluid undergoing laminar flow in a circular pipe of equal diameter is obtained: .

[0079] S3: Based on boundary layer theory, the calculation formula derived from the theory is modified to improve the accuracy of the calculation formula.

[0080] As shown by the derived velocity equation, the grout used in synchronous grouting of shield tunnels undergoes laminar flow in a circular pipe of equal diameter. Due to adhesion to the pipe wall, the grout velocity decreases sharply from the center to the wall radially. A low-velocity retention layer forms near the pipe wall under viscous action. In this region, due to the low velocity, the viscous shear force dominates, and the inertial force is negligible.

[0081] The retention layer has a certain radial geometric thickness, and is used This indicates that it will reduce the effective flow radius of the pipeline and change the flow resistance characteristics. The derivation of the Bingham fluid friction loss formula in this invention did not consider the influence of the stagnant layer, resulting in the calculated result of the friction loss theoretical derivation formula being less than the actual friction loss, and thus unable to accurately predict the friction loss.

[0082] The retention layer is extremely thin, and the fluid in the region is almost stagnant. Theoretically, the slurry velocity at the wall of the circular pipe is 0. It is difficult to accurately measure the radial geometric thickness of the retention layer using existing technology. Therefore, this section introduces a critical velocity threshold to quantitatively characterize the thickness of the retention layer.

[0083] Stokes creep flow is a typical viscous flow under low Reynolds number conditions. When the Reynolds number is sufficiently low, the viscous shear force of the fluid is much greater than the inertial force, and the influence of the inertial force can be ignored. This paper uses a local Reynolds number... As a Stokes peristaltic flow criterion, the critical velocity threshold of the fluid is obtained through the Reynolds number calculation formula, the thickness of the stagnant layer is quantitatively analyzed, and a local Reynolds number is defined. At that time, the critical flow velocity threshold , Since it is a constant, this region belongs to the retention layer.

[0084]

[0085] Will Substituting into equation (34), we get:

[0086] Substituting equation (42) into equation (43), we get:

[0087] Substituting equation (40) into equation (44), we get:

[0088] Solving equation (45) yields the analytical solution for the thickness of the retained layer:

[0089] Substituting equation (3.46) into (3.40), we obtain the corrected formula for calculating friction loss:

[0090] S4: Use numerical simulation software to simulate and calculate the pressure loss along the friction path, and compare and verify the correctness of the theoretical calculation formula.

[0091] This embodiment uses ANSYS Fluent software to establish a numerical model of the laminar flow motion of synchronous grouting slurry in a horizontally placed straight pipe, and performs numerical simulation calculations of the pressure loss along the pipeline during the synchronous grouting process. The software's built-in Herschel-Bulkley model requires setting a critical shear rate, which is generally determined empirically and affects the pressure loss calculation results. To avoid the impact of the randomness of the critical shear rate setting on the accuracy of the calculation results, a Bingham fluid model is constructed using a user-defined function to replace the software's built-in model.

[0092] During grouting, the pressure loss in the inlet and outlet sections of the pipeline is relatively complex. To accurately study the pressure loss within the pipeline, a 2.2m horizontally placed straight pipe model (with the inlet as the starting point) was established, and the pressure difference in the middle section of the pipeline was used to calculate the pressure loss. To study the influence of pipeline diameter and flow velocity on the pressure loss along the pipeline, the grout density was set to 1900 kg / m³. 3 The viscosity is 20 Pa·s, and the dynamic yield stress is 500 Pa. Pipe models with radii of 0.02 m, 0.025 m, 0.03 m, 0.035 m, and 0.04 m were established. Boundary conditions were set as no-slip static walls, and the average inlet velocities were set to 0.2 m / s, 0.4 m / s, 0.6 m / s, 0.8 m / s, and 1.0 m / s, respectively. The outlet pressure was 200 kPa. For a 0.03 m radius pipe with an average inlet velocity of 0.6 m / s, the pressure calculation contour plots for each cross-section within the model pipe are shown below. Figure 3 , Figure 4 As shown in the diagram, analyzing the pressure cloud map reveals that at the pipe inlet section, the slurry impacts the pipe wall, resulting in a higher pressure distribution around the pipe wall and a relatively lower pressure inside the pipe. Entering the middle section, the slurry gradually develops into a more uniform laminar flow, characterized by lower pressure around the pipe wall and higher pressure closer to the center. At the outlet section, the slurry is no longer constrained by the pipe wall, and the pressure across the circular pipe cross-section appears discrete.

[0093] Extract the pressure at various cross-sections of circular pipes with different radii at an inlet average flow velocity of 0.6 m / s, and plot the laminar flow pressure as a function of pipe length for different radii. Figure 5 As shown, under pressure outlet conditions, circular pipes of different radii all exhibit a gradual decrease in pressure with increasing pipe length, which is consistent with the theoretical calculation formula. With increasing pipe radius, the slope of the pressure loss curve gradually decreases, meaning that the smaller the pipe radius, the greater the pressure loss per unit length. It is noteworthy that circular pipes of different radii all show a slightly larger slope in the inlet section, indicating a more pronounced pressure drop per unit length in the inlet section. Analysis suggests that this is because a stable, uniform laminar flow has not yet formed in the inlet section, leading to a more complex pressure loss.

[0094] Extract the pressure difference at the mid-section of a circular pipe with different radii and an average inlet velocity of 0.6 m / s, calculate the friction loss, and plot a histogram of laminar friction loss as a function of the pipe radius. Figure 6 As shown, this reflects the variation of the pressure loss per unit length of the slurry along the pipe radius. Figure 6 It is evident that the pipe diameter has a significant impact on the friction loss of the grout. Under the condition of a constant average flow velocity, the friction loss of the synchronous grout exhibits a significant non-linear decreasing trend with increasing pipe radius. The largest decrease occurs when the pipe radius increases from 0.02m to 0.025m, with the pressure loss decreasing from 386.0kPa / m to 247.9kPa / m. As the pipe diameter continues to increase, the rate of decrease in pressure loss gradually slows, and the decreasing trend tends to level off. In grouting design, rationally selecting the pipe diameter is one of the effective ways to control friction loss.

[0095] To verify the accuracy of the theoretical calculation formula of this method, a slurry density of 1900 kg / m³ was used. 3 With a viscosity of 20 Pa·s, a dynamic yield stress of 500 Pa, and a no-slip static wall boundary condition, an inlet velocity of 0.6 m / s, and an outlet pressure of 200 kPa, the pressure loss per meter of a circular pipe with different radii was calculated using both theoretical and modified theoretical calculation formulas. The results were then compared with those from numerical simulations, and the variation curves are shown below. Figure 7 As shown in the curve illustrating the change in pressure loss, the theoretical calculation, which does not consider the influence of the boundary layer, results in a larger value and a smaller radius, leading to a greater deviation from the corrected result. The numerical simulation analysis results are quite close to the results calculated using the corrected formula, demonstrating the rationality of the corrected formula.

[0096] S5: Through indoor experiments, the theoretically derived formulas are experimentally verified to further validate the reliability of the theoretical calculation formulas.

[0097] like Figure 8As shown, to verify the accuracy of the calculation formula for pressure loss along the flow path in laminar flow through a circular pipe derived from the Bingham model, this study independently developed a test device system for grout pressure loss during synchronous grouting in shield tunnels. The grouting pump 1 is a portable air compressor equipped with an air source processor for pressure stabilization. The grout tank 2 is a self-made modified pressure tank with a volume greater than 30L. The grouting pipe 3 is a steel pipe with an inner diameter of 50mm, and a ball valve is installed at the end to regulate the flow rate. For ease of installation, a flexible rubber hose 4 connects the grout tank 2 and the grouting pipe 3. Two pressure gauges 5 are installed at both ends of the steel pipe, spaced 1m apart. To prevent clogging of the pressure gauges during the grouting test, the pressure gauges 5 are flat diaphragm type pressure transmitters.

[0098] During the experiment, the pressure tank was set and pressurization began. Once the set pressure was reached, the ball valve was opened, allowing the grout to flow out through the grouting pipe, simulating the grouting process. Based on the mass of grout pressed out per unit time, the grout volume was calculated, and then the average flow velocity of the grout in the grouting pipe was calculated. Substituting these values ​​into the formula, the pressure loss was calculated, and the results were compared with the pressure difference readings of the two pressure gauges. The test results are shown in Table 1.

[0099] Table 1. Pressure Loss of Grouting with Different Air-Entraining Agent Doses

[0100]

[0101] Analysis of the above data shows that the measured pressure loss of all samples without air-entraining agent was greater than the corrected theoretical calculation value, while the measured pressure loss of most samples with air-entraining agent was less than the calculated value, with the error generally within ±10%. The reasons for this are as follows: Samples with air-entraining agent contain a large number of air bubbles in the slurry, which act as a lubricant during slurry flow. This effect was not considered in the theoretical calculations, hence the observed phenomenon. Additionally, the limited volume of the slurry tank in the experiment resulted in a short simulated grouting process, preventing prolonged observation of the pressure gauge readings during slurry flow. The resulting fluctuations in pressure gauge readings were also a major contributing factor to the error.

[0102] The present invention has been described in detail above through embodiments, but the content is only a preferred embodiment of the present invention and should not be considered as limiting the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the patent coverage of the present invention.

Claims

1. A method for calculating pressure loss in a grouting pipe, characterized in that: The method includes the following steps: S1: Based on the basic theory of fluid mechanics, derive the formula for calculating the pressure loss of Newtonian fluid laminar flow in a circular pipe; S2: Based on the rheological characteristics of synchronous grouting slurry and combined with the actual flow law of slurry, the calculation formula for pressure loss of non-Newtonian fluid in circular pipe laminar flow is derived. S3: Based on boundary layer theory, the calculation formula derived from the theory is modified to improve the accuracy of the calculation formula; S4: Use numerical simulation software to simulate and calculate the pressure loss along the friction line, and compare and verify the correctness of the theoretical calculation formula. S5: Through indoor experiments, the theoretically derived formulas are experimentally verified to further validate the reliability of the theoretical calculation formulas.

2. The method for calculating pressure loss in a grouting pipe according to claim 1, characterized in that: In S1, based on the Newtonian fluid assumption, a simplified expression for the friction loss of a fluid undergoing low-velocity laminar flow in a circular pipe of uniform diameter is derived using the Rayleigh method. The relationship between friction loss and various parameters is established, and the volumetric flow rate Q of the slurry per unit time can be expressed as: ； In the formula, The plastic viscosity of the slurry. Where is the radius of the grouting pipe. For pressure loss along the process, For length; Converting equation (1) to exponential form, we get: ； In the formula, It is a dimensionless constant; For index; Solve according to the principle of dimensional harmony. The simplified expression for the laminar flow rate of the synchronously injected grout in a circular pipe of equal diameter is obtained. The pressure loss along the pipe is then: ； In the formula, The average flow velocity of the slurry.

3. The method for calculating pressure loss in a grouting pipe according to claim 2, characterized in that: Establish the mechanical equilibrium equations for a tiny cylinder: ； In the formula, , The pressure acting on both ends of the cylinder, The radius of the tiny cylinder; Based on Newton's law of internal friction of fluids, equation (7) is rearranged and the boundary conditions are substituted. The flow velocity equation is obtained as follows: ； In the formula, The above equation shows that Newtonian fluids undergo laminar flow in a circular tube of equal diameter. At this point, the flow velocity is at its maximum, thus yielding the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity: ； In the formula, At the maximum fluid velocity, along the axis. At the pipe wall, The slurry at the wall of the circular pipe satisfies the no-slip condition, and its velocity at the wall is the same as the velocity at the wall surface. The circular pipe wall is stationary, and theoretically, the velocity at the wall surface is the Newtonian fluid velocity. The flow rate formula for laminar flow of fluid in a circular pipe of constant diameter is derived, and the average velocity of the Newtonian fluid in the pipe cross section is derived by performing variable integration. ； According to equation (21), the formula for the friction loss of a Newtonian fluid undergoing laminar flow in a circular pipe of equal diameter is obtained: ； Substituting equation (22) for friction loss into equation (6), we get: .

4. The method for calculating pressure loss in a grouting pipe according to claim 1, characterized in that: In S2, the Bingham model is selected to derive the theoretical derivation of pressure loss along the flow path for non-Newtonian fluid laminar flow in a circular pipe. The formula for the Bingham model is as follows: ； In the formula, For shear stress, For dynamic yield stress, The slurry shear rate; By combining Newton's law of internal friction of fluids, the expression for the internal friction force on the surface of a cylinder is derived; ； In the formula, The area of ​​a tiny cylinder; This represents the velocity gradient.

5. The method for calculating pressure loss in a grouting pipe according to claim 4, characterized in that: Assuming that the fluid inside the entire circular pipe undergoes shear flow, the mechanical equilibrium equations for laminar flow in the circular pipe are established as follows: ； Separate the variables from equation (27) and rearrange the differential relationship, let Integrating the variables on both sides yields: ； In the formula, For a dimensionless constant, the boundary conditions are: Substituting into equation (31), we obtain the flow velocity equation: ； The above equation shows that non-Newtonian fluids undergo laminar flow in a circular pipe of constant diameter. When the flow velocity is at its maximum, substituting the maximum flow velocity into equation (34), we obtain the relationship between the flow velocity at any point inside the circular pipe and the maximum flow velocity: ； At the axis, At the pipe wall, Theoretically, the velocity of non-Newtonian fluid at the wall ; Based on the obtained velocity equation, the flow rate formula for a non-Newtonian fluid undergoing laminar flow in a circular pipe of constant diameter is derived: ； Integrating the variables on both sides of equation (37), we derive the average flow velocity across the cross-section of a non-Newtonian fluid pipe: ； Based on the above formula, the pressure loss formula along the flow path for a non-Newtonian fluid undergoing laminar flow in a circular pipe of constant diameter is obtained: 。 6. The method for calculating pressure loss in a grouting pipe according to claim 5, characterized in that: In S3, the local Reynolds number is used. As a Stokes peristaltic flow criterion, the critical velocity threshold of the fluid is obtained through the Reynolds number calculation formula, the thickness of the stagnant layer is quantitatively analyzed, and a local Reynolds number is defined. At that time, the critical flow velocity threshold , Since the value is a constant, this region belongs to the retention layer. The thickness of the retention layer can be calculated. The analytical solution is: ； In the formula, combined with the friction loss formula obtained in S2, the corrected friction loss calculation formula is as follows: 。 7. The method for calculating pressure loss in a grouting pipe according to claim 1, characterized in that: In S4, ANSYS Fluent software is used to establish a numerical model of the laminar flow motion of synchronous grouting slurry in a horizontally placed straight pipe. Numerical simulation calculations are performed on the pressure loss along the pipeline during the synchronous grouting process. The Bingham fluid model is constructed by user-defined functions to replace the software's built-in model.

8. The method for calculating pressure loss in a grouting pipe according to claim 1, characterized in that: In S5, the indoor test adopts a synchronous grouting model device, which includes a grouting pump (1), a grout tank (2) and a grouting pipe (3). The grouting pump (1) is connected to the grout tank (2), and a flexible rubber hose (4) is provided between the grout tank (2) and the grouting pipe (3). Pressure gauges (5) are provided on both sides of the grouting pipe (3).

9. The method for calculating pressure loss in a grouting pipe according to claim 8, characterized in that: The grouting pump (1) is a portable air compressor, which is equipped with an air source processor for pressure stabilization. The grout tank (2) has a volume of >30L. The grouting pipe (3) is a steel pipe with an inner diameter of 50mm. Ball valves for adjusting flow are provided at both ends of the steel pipe. The pressure gauge (5) is a flat diaphragm type pressure transmitter. The spacing between the pressure gauges (5) is 1m.

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