Fluid Flow Simulation in Pipes Using

Study Case Fluid Flow Simulation in Pipes Using a Linear Differential Equation System Approach Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract One of the complex problems in everyday life is the fluid flow system (water) in a multilevel container. This study develops a Linear Differential Equation System (LDES) model to represent water flow in a pipe installation system with a multi-level tank, then compares the numerical simulation results with actual experimental results. The LDE model was constructed based on direct observations in the field to determine the system constants (C₁, C₂, and C₃). Numerical simulations were performed using the Euler method with four variations of constant values. The results showed that the LDE model was able to describe changes in water volume in tanks A and B along with the rate of change. The constant values that provide results closest to the experimental data are C₁ = 0.01323 and C₂ = 0.01091, with a Root Mean Square Error (RMSE) of 0.2599. Keywords: numerical analysis; Euler method; differential equations; piping system. INTRODUCTION Dynamic systems that undergo changes every unit of time are generally difficult to solve using simple algebraic methods. One example of a fairly complex case is a fluid (water) flow system in a multi-level container or tank (Figure 1). In this system, the volume of water in tanks A and B will continue to change over time due to the input flow and output flow. Interestingly, the outflow rate from each tank is greatly influenced by the volume of water in the tank, in accordance with Torricelli's law [1]. Torricelli's law, or Torricelli's Theorem, explains that the velocity of fluid flowing out of a hole depends on the difference in height between the water surface and the outlet hole. This concept is visualized in Figure 2. MICHAEL KURNIAWAN 1 Study Case Theoretically, as shown in Figure 1, the volume and flow rate of water can change at any time due to various factors. For example, the volume of water in tank A depends on the inflow and outflow, while the outflow itself is also determined by the volume of water in the tank—making this system even more complex. Figure 1. Water flow installation system with multi-level tanks Figure 2. Illustration of the speed of water flowing out of a tank based on Torricelli's Law [1] To model the dynamics of this system, a time-dependent Linear Differential Equation System (LDES) approach is used. Differential equations are equations that involve derivatives of an unknown function [2] and have wide applications in engineering and science. Generally, such modeling is applied to study changes in physical variables such as temperature, pressure, displacement, velocity, voltage, electric current, and other quantities that depend on time or location [3]. The general LDS model for the tank system in Figure 1 can be expressed through the following equations (1)–(3): 𝑄 = 𝐶3 (1) 𝑑𝑉𝐴 = −𝐶1 × 𝐶3 (2) 𝑑𝑡 𝑑𝑉𝐵 = 𝐶1 × 𝑉𝐴 − 𝐶2 × 𝑉𝐵 𝑑𝑡 MICHAEL KURNIAWAN 2 (3) Study Case where: • Q is the water flow rate (liters/second), • VA and VB are the volumes of water in tanks A and B (liters), respectively, • t is the time (seconds), • C1and C2are flow constants. In this study, the SLDE model was solved using a numerical method—a technique formulated mathematically through iterative calculation operations [4]. The purpose of this study is to develop a linear differential equation model for water flow systems in multi-level pipe and tank installations as shown in Figure 1, and to compare the numerical simulation results with the experimental results. The results of this study are expected to assist in the application of SLDE in fluid installation system analysis so that changes in variables such as discharge, volume, and flow rate can be accurately predicted through a numerical approach. RESEARCH METHODOLOGY Equipment and Procedures The experimental system consists of three water tanks, four valves, one centrifugal pump, and one recording camera. Water is used as the test fluid. The pipe installation unit consists of tanks A, B, and C connected by pipes and valves as shown in Figure 3. The test procedure begins by filling tank C to capacity, then turning on the pump. The water flow rate entering tank A is measured to determine the value of Q or C3 (liters/second). The C3 value is calculated using equation (4): 𝐶3 = 𝑉𝐴(𝑛) − 𝑉𝐴(𝑛−1) 𝑡𝑛 − 𝑡𝑛−1 (4) Measurements are taken every 3 seconds until the volume of tank A reaches 20 liters. Next, to determine the constants C1 and C2, observations are made of the rate of change in water volume in tanks A and B. This process is carried out by fully opening valves 2 and 3 so that the water flows freely. The initial volume of tank A is set to 20 liters, and the change in volume is measured for 180 seconds (3 minutes). The rate of change in volume for each tank is calculated using equations (5) and (6): 𝑑𝑉𝐴 𝑉𝐴(𝑛) − 𝑉𝐴(𝑛−1) = 𝑑𝑡 𝑡𝑛 − 𝑡𝑛−1 MICHAEL KURNIAWAN 3 (5) Study Case 𝑑𝑉𝐵 𝑉𝐵(𝑛) − 𝑉𝐵(𝑛−1) = 𝑑𝑡 𝑡𝑛 − 𝑡𝑛−1 (6) Then the values of C1 and C2 are obtained through the following relationships: 𝑑𝑉𝐴 𝐶1 = 𝑑𝑡 𝐶3 (7) 𝑑𝑉𝐵 𝑑𝑡 𝐶2 = 𝑑𝑉𝐴 − 𝐶3 𝑑𝑡 (8) Figure 3. Isometric view of the water installation system Numerical Analysis Numerical analysis was performed using the obtained values of constants C1, C2, and C3. The model was simulated numerically using the Euler method, as used by Fajar et al. [5] in simulating the application of Torricelli's law in vessels, and by Wijaya et al. [6] in solving ordinary differential equations. Several numerical methods commonly used to solve differential equations include the Euler, Heun, Taylor series, and multistep Runge–Kutta methods [7]. Ng et al. [8] also used Torricelli's law in modeling numerical drainage systems. However, this study focused only on the Euler method, which is categorized as a first-order method because it only uses first-order derivatives. Simulations were performed using Microsoft Excel software to calculate the water volume in tanks A and B with several variations in the values of constants C 1 and C2, in order to obtain model results that most closely approximated the experimental data. The Euler method calculations are expressed by equations (9) and (10): MICHAEL KURNIAWAN 4 Study Case 𝑑𝑉𝐴 × Δ𝑡 𝑑𝑡 𝑑𝑉𝐵 = 𝑉𝐵(𝑖) + × Δ𝑡 𝑑𝑡 𝑉𝐴(𝑖+1) = 𝑉𝐴(𝑖) + (9) 𝑉𝐵(𝑖+1) (10) RESULTS AND DISCUSSION Value of Constant C₃ (Pump Flow Rate) Based on the procedure described in the Research Method section, the constant C₃ is determined through observation results. The value of C₃ is constant because the water flow rate produced by the pump remains constant as long as the pump rotation speed does not change. The results of the pump water flow rate measurements are presented in Table 1. Table 1. Results of volume observations in tank A Observations were made every three seconds to obtain a more accurate scale reading. By entering the values of VA and time t into equation (4), the value of C3 was obtained. The average value of C3 was calculated from 0 to 30 seconds, resulting in a value of 0.175 liters/second. MICHAEL KURNIAWAN 5 Study Case Values of Constants C₁ and C₂ The constants C₁ and C₂ were obtained through direct observation of the water flow installation system. The system was operated according to the previous procedure, in which water was continuously pumped at a constant flow rate (C3) to tank A. The water then flowed from tank A to tank B through valve 2, and from tank B to tank C through valve 3. The calculation of the values of C1 and C2 was performed using equations (5)–(8). Volume changes were observed every 10 seconds in both tanks. Based on the measurement results shown in Table 2, the average values of C1=0.01312 and C2=0.01093 were obtained. Table 2. Results of observations of changes in water volume in tanks A and B MICHAEL KURNIAWAN 6 Study Case SLDE Modeling and Numerical Simulation Results The values of constants C₁, C₂, and C₃ obtained were then used in equations (2) and (3) to construct an initial model of a system of linear differential equations (SLDE) in the case of multi-level water flow. The model equation can be written as: 𝑑𝑉𝐴 = 0.175 − 𝐶1 𝑑𝑡 𝑑𝑉𝐵 = 𝐶1 𝑉𝐴 − 𝐶2 𝑉𝐵 𝑑𝑡 (11) (12) Table 3. Constant value scenarios in numerical simulation The values of C1 and C2 used were taken from the measurement results (Table 2), with Scenario 1 as the average value, while Scenarios 2–4 were determined randomly based on observation data. Comparison of Numerical Simulation and Experimental Results The numerical simulation results of the four scenarios, along with the experimental results for the water volume in tanks A and B, are shown in Figure 3. Figure 3. Changes in water volume from experiments and numerical simulations: (a) tank A; (b) tank B MICHAEL KURNIAWAN 7 Study Case Figure 3 shows that Scenarios 1, 2, and 4 provide results that are closest to the experimental curve. To determine the best scenario, the Root Mean Square Error (RMSE) calculation was used, where a smaller RMSE value indicates a more accurate model [10]. The RMSE calculation results are presented in Table 4. Table 4. Results of RMSE calculations for each scenario Based on Table 4, the model with the smallest RMSE is Scenario 1 with a value of 0.2599, so uet considered the most representative model. The simulation results using the Euler method show good performance, similar to the findings of Fajar et al. [6], who reported an error rate of less than 2%. In general, the numerical simulation results in this study were able to describe the changes in water volume in tanks A and B realistically. Although the experimental results curve appears rough uet o measurement inaccuracies (such as limitations in volume measuring instruments, time inaccuracies, and manual observation), the pattern between the experimental data and simulation results shows a high degree of conformity. The best model, Scenario 1, has values of C1=0.01323 and C2=0.01091, with the final equation as follows: 𝑑𝑉𝐴 = 0.175 − 0.01323 (13) 𝑑𝑡 𝑑𝑉𝐵 = 0.01323𝑉𝐴 − 0.01091𝑉𝐵 𝑑𝑡 MICHAEL KURNIAWAN 8 (14) Study Case Figure 4. Comparison of the rate of change in the volume of tanks A and B between the experimental results and numerical simulation From Figure 4, it can be observed that the experimental results show fluctuations caused by the limitations of the measuring instruments and visual observation. However, in general, the simulation results have successfully represented the rate of change in the volume of water in both tanks accurately, and Scenario 1 proved to be the most consistent with the experimental data. CONCLUSION From the comparison between the experimental method and the numerical method using the Linear Differential Equation System (LDES) approach, it can be concluded that the changes in water volume and their rates in tanks A and B show very similar trends. The best scenario was obtained in Scenario 1, with values of C1=0.01323 and C2=0.01091, resulting in the smallest RMSE value of 0.2599. The small inaccuracy in the experimental results was due to the limitations of the volume measuring device and manual observation. The Euler method applied in numerical simulation successfully modeled the ideal conditions of a multi-level water flow system, enabling predictions of volume changes and flow rates that closely approximate actual conditions. RECOMMENDATIONS To improve the accuracy of experimental results, it is recommended to use more precise measuring instruments and utilize digital data loggers so that data can be recorded simultaneously and human observation errors can be reduced. MICHAEL KURNIAWAN 9 Study Case REFERENCE MICHAEL KURNIAWAN 10