Experimental and Computational Modeling of Complex Fluid Dispensing Systems

Abstract

In this work, we present a series of experimental and computational modeling procedures related to precision controls of intermittent dispensing systems for complex fluids. With reductionist approaches, we have modeled key components of dispensing systems with different boundary conditions. With system approaches, we have also connected the pressure differential with the volume flow rate, as well as the characterization of the material properties. Finally, confirmed with a series of experiments and simulations, we demonstrate that traditional reductionist approaches and system modeling tools can be effectively and efficiently employed hand-in-hand with the help of the so-called inversed optimization approaches to derive useful information of relevance for industrial applications.

1. Introduction

With the availability of sophisticated Computational Fluid Dynamics (CFD) tools, engineering designs are becoming more simulation-based [1]. However, massive data derived from these simulation packages are sometimes counterproductive and pose challenges to engineers in the derivation of needed insights of sufficient relevance for fluid delivery and control. For continuous fluid delivery, engineers tend to utilize the so-called Moody chart or diagram in their respective design practice. The information based on a steady solution is adequate to predict the major friction loss coefficient f_m in relation to the pressure drop Δp, the pipe length L, and the diameter D for laminar or turbulent flow conditions, as well as the minor friction loss coefficient f_d for various flow components such as valves, elbows, diffusers, and nozzles [2]. In fact, the pressure losses calculated based on these coefficients are directly linked to the required pump power with consideration for the volume flow rate. Nevertheless, the precision control of the dispensing system in electrical vehicle (EV) manufacturing plants, as depicted in Figure 1, demands a better understanding of both transient or developing and steady flow conditions for non-Newtonian fluids with the complex and often time-dependent material properties [3].

In earlier studies, through a combination of theoretical and computational studies, we have demonstrated inner connections between well-established pressure drop and flow rate relationships through various computational models [4,5]. In this paper, we focus on the effects of time-dependent (thixotropic) non-Newtonian fluids with both shear thinning and thickening effects—namely, a so-called V-shape relationship between the dynamic viscosity μ and the shear rate $\dot{\gamma }$ [6,7]. Furthermore, confirmed by the experimental observations and numerous computational results, there exists a slight delay between the inlet pressure signal and the outlet pressure signal. Thus, a series of computational models with compressible or nearly incompressible non-Newtonian fluids are employed. In addition, different ramp sizes for the time-dependent inlet volume flow rate or average axial velocity conditions have been employed in the simulation, mimicking different powers of the servo motor that are used to squeeze the complex fluid through the needle valve for different volumes and volume flow rates. In traditional displacement pumps, the power input is equivalent to the product of the torque and the angular velocity. In fact, with a precise setting of the helical angle, the angular velocity is directly linked to the average axial velocity or the volume flow rate. Thus, for the precision delivery systems utilized in EV manufacturing settings, the servo motor power is established with the product of the pressure differential and the volume flow rate. Moreover, such a ramp size during the actuation of the servo motor, determined by the needle displacement and the actuation time, coupled with the required pressure, directly attribute to the required servo motor power and the total volume output, as well as the volume flow rate [8,9].

2. Thixotropic Non-Newtonian Rheological Properties

The rheological properties of the non-Newtonian fluid in question can be obtained through experiments as documented in Refs. [3,4,5]. Moreover, from the force equilibrium between the shear force F and the pressure drop Δp, it is obvious that the high pressure drop produces high shear stress. For this shear-thinning polymer melt, we can easily draw a conclusion that the effective viscosity for a high pressure drop or high shear stress is smaller; thus, the average axial velocity will be larger, and the relaxation time will be larger as well. For the typical power law distribution, $\mu =A{\left(\dot{\gamma }\right)}^a,$ where $\dot{\gamma }$ is the shear rate and μ stands for the dynamic viscosity, with the prescribed linear regression and normal equation, it is not difficult to come up with the constant A—in this case, 3307.0—and the power a—in this case, −0.6129. Of course, we can also use the regression analysis to come up with other non-Newtonian rheological properties, such as the Carreau model [6,7]. These parameters yield a very close relationship between the dynamic viscosity µ and the shear rate $\dot{\gamma }$ obtained through experiments.

Moreover, it is clear from

Table 1. A list of materials and their typical rheological property ranges.

Fluid	Dynamic Viscosity (Ns/m2)	Kinematic Viscosity (mm2/s)	Density (kg/m3)
Engine Oil	0.3	363.64	825
Glycerin	1.412	1121	1260
Vegetable Oil	3.680	4000	920
Honey	5	3530	1415
Window Putty	100,000	62,500,000	1600
Tar	30,000,000	26,000,000,000	1153

and

Table 2. Material property measurements for the non-Newtonian fluid.

$\dot{\mathit{\gamma }}$ (1/s)	0.0096	0.02	0.05	0.1	0.2	0.5	1	5	10	50	100	1000
μ (Poise)	92,600	63,000	30,000	17,000	9120	4200	2370	540	350	260	260	660

that these non-Newtonian glues have fairly extreme dynamic and kinematic viscosities in comparison with other common fluids. In Refs. [4,5], parametric studies of inlet and outlet pressure drop transients for pipes with different volume flow rates and radii have been conducted for non-Newtonian fluids, as shown in Figure 2. These parametric studies will not be feasible without the initial phase of the project documenting the rheological properties through experimental means and detailed comparisons between full-fledged three-dimensional simulations with two-dimensional axisymmetric models [4,5]. In this paper, we will present further experimental results in comparison with time-dependent thixotropic non-Newtonian fluid models and their respective simulation results with consideration of the needle valve status [8,9,10]. Furthermore, the in-depth studies of the transient flow conditions with respect to different pressure and volume flow rate boundary conditions and different time scales, as well as different rheological properties, have been presented in Refs. [11,12,13].

Overall, two different inlet and outlet boundary conditions have been systematically evaluated [4,5]. In both cases, the outlet pressure is the atmospheric pressure. The first case is the force-controlled scenario in which a fixed force or force is imposed either as a Heaviside step function or with a finite ramp size. The second case is the displacement-controlled scenario in which a fixed displacement or volume flow rate is imposed either as a Heaviside step function or with a finite ramp size. In our preliminary studies, for the imposed force or pressure, one or multiple relaxation times can be used to model the transient pressure drop and volume flow rate relationships. For the imposed volume flow rate, we have observed, through experiments and simulations, similar exponential growth behaviors with one or multiple relaxation times, as well as more complicated transient behaviors [4,5,11]. Based on these preliminary results, it seems that the higher the ramp size, the smaller the pressure drop peak [12,13]. Furthermore, the simulation can provide important insights into the transient behavior of the fluid, which can be used to optimize the design and performance of the system for specific applications [4,13].

In this paper, we took a pragmatic approach with a simple compressible thixotropic non-Newtonian fluid model that exhibits an initial dynamic viscosity μ_i with a relaxation time τ_i and a final dynamic viscosity μ_f with a final relaxation time τ_f. Moreover, the shear rate is estimated as the average axial velocity, determined by the ratio between the volume flow rate $\dot{V}$ and the axial cross-sectional area A. Through a series of data analyses in comparison with the expected experimental measurements, material constants for the newly proposed thixotropic non-Newtonian model with both initial and final dynamic viscosities and their respective relaxation times have been identified. For example, at the flow rate of 1 cc/s, the average axial velocity $\overline{v}$ is around 0.0509 m/s for the pipe diameter D = 5 mm, which yields the equivalent shear rate as 20.372 1/s and the corresponding initial dynamic viscosity μ_i and the final dynamic viscosity μ_f as 535.12 Pa.s and 91.74 Pa.s, respectively. As the volume flow rate increases to 5 cc/s, the average axial velocity $\overline{v}$ is around 0.2546 m/s; for the same pipe diameter, the equivalent shear rate is 101.86 1/s, which yields the corresponding initial dynamic viscosity μ_i and the final dynamic viscosity μ_f as 446.79 Pa.s and 76.59 Pa.s, respectively.

Finally, for the thixotropic non-Newtonian fluid model, the time dependent dynamic viscosity μ(t) can be expressed as

$$\mu \left(t\right)={\mu }_f(1-e^{-t/{\tau }_f}) {+\mu }_i{e}^{-t/{\tau }_i}.$$

(1)

Note that, in Equation (1), the so-called initial dynamic viscosity μ_i and the final dynamic viscosity μ_f and their corresponding relaxation times τ_i and τ_f can be fine-tuned or optimized with the inverse optimization procedure, as reported in Ref. [1]. This important inverse analysis approach aided by data analysis tools can be very effective in directly linking computational, analytical, and experimental evidence with needed recommendations for design and manufacturing.

In this paper, we employ τ_i = τ_f = 0.25 s. The final dynamic viscosity level μ_f is adjusted based on experimental evidence of the black glue. Furthermore, in addition to the thixotropic non-Newtonian properties in Equation (1), we employ a density of ρ = 1450 kg/m³ and a compressibility of α = 1.2 × 10⁻⁹ 1/Pa. Notice here the compressibility α is defined as

$$\alpha =-{\displaystyle \frac{1}{V}}{\displaystyle \frac{\mathit{\partial }V}{\partial p}},$$

(2)

with the volume V and the pressure p, which is, in fact, the reciprocal of the so-called bulk modulus κ.

Assuming the initial pressure p(t) and density ρ(t) for this newly proposed compressible thixotropic non-Newtonian model with the initial values p_o and ρ_o, we obtain

$$p\left(t\right)=p_o+{\int }_{{\rho }_o}^{\rho }{\displaystyle \frac{\alpha }{\rho }}d\rho ;$$

(3)

$$\rho \left(t\right)={\rho }_o+{\int }_{{\rho }_o}^{\rho }{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{\alpha }{\rho }}d\rho ,$$

(4)

where the wave speed c is assumed to be constant, defined as $c^2={\displaystyle \frac{dp}{d\rho }}.$

Therefore, this new time-dependent compressible non-Newtonian fluid model is based on the experimental test of the same material employed in the computational models reported in this paper. The computational results based on these material constants and the corresponding compressible thixotropic model match very well with the experimental data: with all the volume flow rates and total delivered fluid volumes [12,13]. In all these CFD simulations, the time ramp size t_i and time step size Δt are fixed—namely, t_i = 0.1875 s and Δt = 0.625 ms. The relaxation times τ_i and τ_f are also fixed—namely, τ_i = τ_f = 50 ms. For the range of the test volume flow rate, different sets of initial and final dynamic viscosities are chosen within the range of the actual material property measures. The simulation data based on thixotropic non-Newtonian fluid models yield a set of consistent results that match with the experimental measures of the pressure drops [12,13].

In this research project, through the solution of the partial differential equation in the cylindrical coordinate systems, it is possible for us to derive the relaxation time scales, which are important for the control of temperature and pressure drop [4,11]. However, in most engineering practices, it is still not feasible to simulate complex systems as a whole, such as the entire dispensing system, as illustrated in Figure 1, Figure 3 and Figure 4, even with the recent advent of both hardware and software developments. Therefore, it is important to validate the information derived from computational and analytical approaches with direct experimental measures and to combine traditional reductionist approaches with system approaches.

3. Computational Modeling of the T-Joint and Needle Valve

In this work, we have further explored the non-Newtonian rheological issues related to an intermittent polymer delivery system—in particular, the discharge from the T-joint and the needle valve. To tackle the modeling of the complex injection system as illustrated in Figure 1, Figure 3, and Figure 4, we opted to model, using Star CCM+ software, the T-joint injection system in order to address the needed annulus pipe length for the flow to resume uniformity, as discussed in Refs. [14,15]. In the 3D simulation, as depicted in Figure 5 and Figure 6, a volume flow rate of 20 cc per second was introduced in the T-joint inlet, and the other surfaces were walls, except for the outlet nozzle, which was set with an atmospheric pressure. Hence, the required pressure drop over the entire T-joint was modeled with a typical operational volume flow rate. Moreover, the velocity distributions, as depicted in Figure 5, suggest that the uniform annual flow will be established within five or six times the diameter of the transverse mixing pipe, which does match with the actual design, around 180 mm, of the long needle, as depicted in Figure 3 and Figure 4.

Again, the injection servo motor and the attached needle valve, as depicted in Figure 3 and Figure 4, have been studied in two different stages with different focuses. In the first stage, the T-injection model is introduced to illustrate the ensuring uniform annular flow, as discussed in Refs. [15,16]. The results reported in Figure 5 and Figure 6 demonstrate that a distance of five to six times the pipe diameter is sufficient for the injected flow from the side branch to establish a uniform flow field. A more detailed computational study on this mixing issue is recommended in Refs. [1,16]. Moreover, similar exponential development of the inlet and outlet pressure drop is established after the initial impulse, the relaxation time of which can be characterized with the inverse optimization methods [4,17].

Based on a series of computational results at different scales with typical volume flow rates, pipe geometries, and empirical formulas matching the available material properties, as well as the pressure drop or the load condition, as depicted in Figure 7 and Figure 8, a gap (gate) function was introduced near the tip in order to model the needle valve open and close mechanisms [18,19,20]. The simulation setups do quantitatively match with the operation conditions. Therefore, with a better understanding of the non-Newtonian rheological behaviors within the last stage of the dispensing system, we are finally ready to calibrate the operation timelines for various complex non-Newtonian fluids at different pressure differentials, volume flow rates, and total volume distributions.

The rheology of the polymer material in question for the dispensing system as depicted in Table 1 and Table 2 is very much strain rate-dependent. In the first-year study, the largest relaxation times have been estimated and compared based on transient solutions with the zeroth Bessel function of the first kind for cylindrical coordinates and Fourier series for Cartesian coordinates, as reported in Refs. [4,5]. We recognize that the transient effects for the developing flow are as important as the steady flow conditions. Furthermore, the key difficulties for the precise and accurate dispensing of this complex non-Newtonian flow can also be attributed to the thixotropic behaviors of these glues with varying initial and final viscosities, as well as their respective relaxation times [12,13]. We must pay attention to how we change the rheology of the polymer through heating, a time-dependent process, or strain rate and how we ramp up the pressure difference, depending on the hydraulic or pneumatic systems.

In this paper, for the needle valve opening and closing settings, we employed the Computational Fluid Dynamics (CFD) feature of Bentley Systems ADINA to model the flow within annulus region [18,19,20]. We consider here the transient laminar flow of a homogeneous, viscous, compressible fluid with the same time-dependent non-Newtonian fluid properties as presented in Equations (1), (3) and (4). In order to focus on the opening and closing needles, following the T-joint modeling, we introduce a timeline mimicking the actual servo-motored driven needle valve, as shown in Figure 7. To establish an opening and closing mechanism, a gap (gate) function is introduced, along with a moving mesh, as depicted in Figure 8.

At the moment, the threshold for the needle valve to open and to close is within 3 to 3.5 mm. Of course, detailed studies of this threshold are still required with the consideration of the surface tension and specific nozzle geometries. In the needle valve modeling with Bentley Systems ADINA, the needle valve starts to retract—namely, open at 20 ms and reach the peak position at 80 ms, as illustrated in Figure 7 and Figure 8, which leaves around 60 ms for the flow to move from the top of the T-joint needle injection model, where the normal traction or pressure is applied within 2 ms. For a volume flow rate of 1 cc per second, the average axial velocity within the annulus region with an inner radius of 2.5 mm and an outer radius of 5 mm is 0.068 m/s. Thus, it takes approximately 735 ms for the fluid particles to cover the axial length of 50 mm for the needle valve model and to reach the tip of the needle valve before dispensing. The velocity fields, as shown in Figure 9 and Figure 10, do confirm that the uniform flow distribution in the negative z direction will be established around 1 s, as required in Figure 7. These specific details for different pressure differentials, volume flow rates, and total volume distributions are crucial to the timeline setting of the needle valve servo motor.

The gauge pressure within the needle valve gun applied at the top of the needle model as the normal traction is around 244.46 kPa; using the same thixotropic non-Newtonian model, the calculated total discharge of the polymer over the roughly two seconds is 1.344327 cc, which corresponds to the volume flow rate of 0.6722 cc/s or average axial velocity of 0.04564 m/s within the annulus region or the average axial velocity of 0.07514 m/s for the exit nozzle with a diameter of 2.1 mm. Notice that the actual design of the needle valve nozzle with an axial length over 60 mm and the annulus region does rend the average velocity to be nearly consistent. As a result, the fluid particle will travel 50 mm over the pipe length within roughly 1.0955 s. As depicted in Figure 11, the same level of gauge pressure is introduced before the needle valve is open or after the needle valve is closed. Note that this pressure is due to the injection pressure through the T-joint, marked as 3 in Figure 3, which is different from the servo motor actuation pressure for the engagement of the needle, as depicted in Figure 4.

Moreover, notice that, as the needle valve is closed, there will be a burst of fluid discharge within 60 ms, as depicted in Figure 12. In fact, as the needle valve is opened, a reverse flow, resemblance of a syringe needle suction, is observed in Figure 12. It is anticipated that this reversed flow can be properly controlled by the application of the proper amount of pressure ahead of the retraction of the needle, as well as the ramp size. Again, further study of this needle valve model for different pressure drops, ramp sizes, volume flow rates, and total volume distributions is very much required, assisted by inverse optimization methods. Furthermore, in the future, the same system can be easily expanded to different thixotropic non-Newtonian fluids with similar operation parameters. Overall, for this non-Newtonian fluid, the pressure drops over time behave very much like a vibration system with feedback control around the critical damping area. Thus, suitable control algorithms can be implemented for such a dispense system.

Finally, in order to further understand the complex non-Newtonian fluid rheology, we can also introduce the so-called inverse optimization approach as reported in Refs. [1,16] and search with the Newton-Raphson iterative procedures for the optimal match of intrinsic system parameters or missing operation conditions by comparing simulation results with experimental measurements. Although a direct three-dimensional simulation based on a similar needle valve and servo motor design does demonstrate that an inlet and outlet transient pressure drop is similar to a series of axisymmetric simulation results for pipe flows with the same non-Newtonian fluid properties at various volume flow rates, a series of systematic studies with different focuses will provide much more relevant information with variable operation conditions. Again, it is very important to recognize that such a full-fledged, three-dimensional simulation does take more resources and more computation time for each variation, and the cost of such systematic studies with numerous design and operation variations could be prohibitive [20,21].

4. Experimental and Computational Modeling of a Set of Short and Long Pipes

Also shown in Figure 13, the yellow colored pressure transducers are from the GM Technical Center, which is connected with National Instrument (NI) LabView Data Acquisition system and Matlab Simulink. In this paper, a sampling frequency of 1000 Hz the was introduced for the GM data acquisition system. The special adaptors connecting Echometer Total Asset Monitor (TAM) pressure measurement systems with the GM dispensing systems are depicted in Figure 14.

In order to maintain the same operation conditions during the switch of different sets of pressure transducers, special connectors have been designed and manufactured by Echometer Company (Wichita Falls, TX, USA). As shown in Figure 14, these special connectors utilize specific threads suitable for operations in high fluid pressure conditions and can be easily exchanged for different pressure transducers and corresponding data acquisition systems. Moreover, the sampling frequency for the Echometer TAM pressure measurement system is selected at 30 Hz. Since there is no synchronization between these two data acquisition systems, a set of time shifts has been introduced. Separate pressure signals have been presented in Echometer TAM software for the inlet (upward location) and outlet (downward location), as shown in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28. Due to the complex non-Newtonian rheological properties of the test fluid, the so-called black glue, a black-colored glue employed in actual EV manufacturing plants with a finite bulk modulus and compressibility, a clear time delay between the inlet pressure signal and the outlet pressure signal does exist, which will transpire as an upward blip in the inlet and outlet pressure drop, as depicted in Figure 15 and Figure 16.

The ramp sizes are directly related to the initial inlet pressure elevation for a prescribed volume flow rate or an average axial velocity. Also discussed in Refs. [12,13], the ramp size has larger effects on the pressure blip than the compressibility. In all the simulation cases conducted in this paper, based on computational and experimental evidence, we choose the time step Δt = 0.625 ms, and the leading time ramp size t_i = t_f = 0.1875 s, which is equivalent to 300 steps with the prescribed time step Δt.

Note that the time registered in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28 is Central Standard Time (CST) using the TAM software installed on an MSU Texas Laptop, whereas the time recorded in the GM pressure transducers is Eastern Standard Time (EST), which is one hour ahead of CST utilized in the state of Texas. As expected, higher sampling frequencies tend to have high noise levels. Moreover, mimicking the actual operation conditions of the servo motors, we take the trapezoidal time function with a leading slope defined by the leading ramp size or time t_i and a trailing slope defined by the trailing ramp size or time t_o. Therefore, the overall volume delivery V_o, with the consideration of the steady state volume flow rate $\dot{V}$ and the actual operation setting, yield the total delivered volume, calculated as

$$V_o=({\displaystyle \frac{t_i+t_o}{2}}+t_m)\dot{V,}$$

where the servo motor maintains the average volume flow rate $\dot{V}$ for a time duration t_m.

A series of tests was conducted in the Materials and Manufacturing Research Laboratory at GM Technical Center in Warren, MI, USA, on 4 April 2024. Two sets of pressure transducers were implemented—namely, a GM pressure transducer set connected with LabView Data Acquisition system and Matlab Simulink with a sampling rate of 1000 Hz and an Echometer pressure transducer set connected through TAM software with a sampling rate of 30 Hz. Various volume flow rates and total fluid volumes related to the actual production operations—namely, 1, 2, 5, and 10 cc/s, as well as 1, 10, 15, 30, and 60 cc, respectively, as shown in

Table 3. Flow rate and total volume for the non-Newtonian fluid delivery.

	Short Pipe (139.1 mm)						Long Pipe (220.7 mm)
Flow Rate (cc/s)	1	1	1	1	2	2	5	5	5	5	5	5
Volume (cc)	1	1	10	10	10	10	15	15	30	30	60	60

, have been tested. For the experimental test case performed on 4 April 9:06 CST or 10:06 EST, 2024, the actual fluid, a black-colored glue, was a very complicated material. Based on the experimental observations, this particular glue has a V-shape relationship between the dynamic viscosity and the shear rate—namely, shear thinning for a low to moderately high shear rate and shear thickening for an extremely high shear rate. Moreover, the material also demonstrates a combination of the Bingham plastic and thixotropic properties [3,8].

Note that it is impossible to implement the precise material properties at all spatial points and all temporal stations. The computational and experimental solutions presented in Refs. [12,13] demonstrated that there is a need to accommodate the compressibility. Again, the average axial velocity $\overline{v}$ can be expressed as

$$\overline{v}={\displaystyle \frac{\dot{V}}{A}}.$$

In this paper, we adopt simple inlet and outlet pressure measurements for both long and short dispensing tubes, as shown in Figure 15 and Figure 16. As depicted in Figure 13, the gray-colored pressure transducers are from Echometer Company, a world leader in underground liquid level detection in petroleum industries. The data from these gray pressure transducers were collected and analyzed with Echometer’s Total Asset Monitor (TAM) software, as shown in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28, for both short and long pipe test cases. In this study, a sampling frequency of 30 Hz was introduced for the Echometer data acquisition system. Note that, due to the dispensing system safety threshold around 150 bar, which is equivalent to 2183.17 psi, when the inlet pressure approaches that level, the dispensing system automatically reduces the inlet pressure, which explains the wave-like pressure measurements for the cases with high volume flow rates, such as 10 cc/s [18].

The thixotropic compressible material properties are similar to the basics of thixotropy determined by Anton Paar Instruments [8]. In fact, the adopted parameters actually yield a shear stress and shear rate relationship within the ranges of the snapshots of experimental measurement data for the black glue, as shown in Figure 29 and Table 2. Note the slight reversal of the dynamic viscosity when the shear rate is above 100 1/s in Figure 29 and Table 2. Of course, in the near future, inverse optimization approaches can be employed to fine tune the search for the optimum material constants [16]. Also confirmed in Figure 29, within the shear rate according to the operation conditions, the thixotropic material does exhibit so-called shear-thinning effects. Also suggested by the experimental evidence in Echometer’s pressure measures depicted in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28, compressible models must be considered.

This set of detailed comparisons between the experimental and computational results has directly validated mathematical models introduced for thixotropic compressible non-Newtonian fluids utilized in EV manufacturing plants. In the experimental tests on 4 April 2024, the chosen sampling rate for the GM pressure transducers was 1000 Hz, whereas the employed sampling rate for the Echometer pressure transducers and TAM Data Acquisition System was 30 Hz. Thus, they do have different noise levels. In the computational modeling, we proposed a new thixotropic compressible non-Newtonian fluid that has two distinct dynamic viscosities, each of which has a relaxation point as illustrated in Equations (1), (3) and (4). The selected leading and trailing ramp sizes yielded a comparable blip of the inlet pressure, matching what was recorded in the experiment. Ultimately, such a ramp size must be connected with the power of the servo motor actuated in coordination with the opening and closing of the needle valve.

The selected material constants in the newly proposed compressible thixotropic non-Newtonian fluid are based on the direct experimental data and the comparisons between the computational solutions and the experimental measurements, as depicted in Figure 29. The ramp sizes, as well as the material constants, directly provide the operation configurations such that the precise control of the dispensing system can be accomplished for different fluids and operation conditions. In the near future, better understanding of the compressible thixotropic non-Newtonian rheological behaviors of specific polymers at different temperature ranges and the reliable and stable delivery of a dispensing system with consideration of the pump, doser/meter, hose, and nozzle selections currently utilized in General Motors EV manufacturing plants can be improved and optimized [22,23].

5. Conclusions

In this paper, a new slightly compressible non-Newtonian thixotropic material model is proposed for the modeling of a precise delivery system for complex fluids. The selected material constants are confirmed to be within the experimental data range of the material tests. Moreover, the computational simulation based on the proposed time-dependent non-Newtonian fluid model produces results matching well with the corresponding experimental measurements for both long and short dispensing tubes. Moreover, the in-depth modeling of the T-joint and the needle valve with the actuation conditions—namely, the selected ramp size and opening duration—yield pressure and velocity profiles consistent with the experimental observations. This type of model analysis assisted by experimental and computational approaches, along with inversed optimization methods, is promising in providing direct links to the precision control design and manufacture settings of dispensing systems for complex fluids.