Development - Development and analysis of electromagnetic pump design program

3. Methods

3.2. Development and analysis of electromagnetic pump design program

3.2.1. Development

Figure 3.2.1. Logical flowchart of the electromagnetic pump design program

The design variable input phase can be largely divided into one for the electromagnetic design variables and another for the geometric design variables. The electromagnetic design variables include data involving input power, number of windings in the coil, input frequency, number of pole pairs,

ratio of slots/poles/phase, number of phases, and electromagnetic properties, such as permeability and electrical resistance. The geometric design variables include diameter and length of the cores, size of flow gap, thickness of the inner and outer ducts, width and thickness of coils, and ratio of slot width/slot pitch. Based on these variables, the developed pressure is derived using the relevant equations inside the design program. To derive a relatively accurate developed pressure, the following items are further considered:

A. Effective magnetic gap

Basically, the ALIP forms an induced magnetic field in a radial direction around the axis through the inner and outer cores. The magnetic reactance value is required to calculate the developed pressure, and the value of the magnetic gap between the cores is used. The magnetic gap between the inner and outer cores can be easily calculated by adding the thickness of the inner ducts, outer ducts, and flow gap in the structure of the ALIP. However, in the case of the outer cores, the E-shaped core, where the tooth exists and the coil is inserted in the slot in between, is continuously connected up and down.

This geometry changes the value of the magnetic gap, applied to the magnetic reactance value, when the actual annular induction electromagnetic pump is operated [70]. The value of the magnetic gap considering these geometric structures is called the effective magnetic gap, which is calculated using Equation (3.2.1).

𝐺_𝑒𝑓𝑓 =_(𝑤^𝑝(𝑤^𝑡^+𝑤)𝐺

𝑡+𝑤(1−𝐶)) (3.2.1) In the Equation (3.2.1), the Carter coefficient 𝐶 is used to calibrate the effective magnetic gap value, which is calculated as shown in Equation (3.2.2) [70].

𝐶 =²

𝜋(arctan (^𝑤

2𝐺) −^𝑤

𝐺ln (1 + (^𝑤

2𝐺)²)) (3.2.2)

B. Interpretation of electrical equivalent circuits

The electrical equivalent circuit analysis and MHD analysis for the ALIP were demonstrated to produce the same results, and the verification of the electrical equivalent circuit analysis was carried out in a previous study [76]. However, in the case of an analysis of the electrical equivalent circuits, there is a partial simplification of the developed pressure relationship equation. The electrical equivalent circuits in Figure 3.1.5 show only the electrical resistance to the conductive liquid flux in the secondary side of the electrical resistance; however, in the case of actual operation of the ALIP, the electrical resistance by the internal and external stainless steel ducts covering the flow path shall

be considered. Therefore, if an electrical equivalent circuit is drawn close to reality, it appears as shown in Figure 3.2.2 [70].

Figure 3.2.2 Electrical equivalent circuit (1 phase) of an ALIP

If the above equivalent circuit is expressed in the equation for the developed pressure by geometric and electromagnetic design variables of the linear induction electromagnetic pump, the simplification fails to proceed and becomes very complicated [77]. As an alternative to this, multiple replacements can be used to simplify the equations. Firstly, the electrical resistance of the internal and external ducts connected in parallel with the resistance to the conductive liquid is expressed as shown in Equation (3.2.3).

𝑅_𝑐 = ^𝑅^𝑖𝑑^𝑅^𝑜𝑑

𝑅_𝑖𝑑+𝑅_𝑜𝑑 (3.2.3) The magnetic reactance of the secondary side, the electrical resistance of the duct, and the electrical resistance of the conductive liquid can be combined to be expressed as an impedance of the magnetic gap, as shown in Equation (3.2.4).

𝑍₂= 1 ¹ 𝑋𝑚𝑖+_𝑅𝑐¹+^𝑠

𝑅2

(3.2.4)

In this case, the equation for the developed pressure of the electromagnetic pump is calculated as shown in Equation (3.2.5), which is similar to Equation (3.1.13).

∆𝑃_𝑑=^3𝐼_𝑅²^𝑠𝑍²²

2𝑄_𝑠 (3.2.5) When calculating the developed pressure relationship equation of the linear induction electromagnetic pump in the same manner as in Equation (3.2.5), more accurate developed pressure can be derived by considering the electrical resistance of ducts surrounding the flow path. In the electrical equivalent circuit analysis presented in Section 3.1, the effective magnetic gap was defined to be same as the thickness of the flow path to simplify the expression, or the diameter of the internal core was defined as the thickness of the fluid. When calculating by substitution, as shown in Equation (3.2.5) of this section, it is possible to calculate the developed pressure more accurately because there is no error caused by the simplification process.

C. Magnetic reactance correction

In the analysis of the electrical equivalent circuits, presented in Section 3.1, the formula deduced on the basis of the linear induction motor may be somewhat different from the formula for the ALIP. The magnetic reactance formula can be deduced in a different way based on the magnetic current, and this relationship is considered to be more accurate [70].

In an ALIP, the magnetic current can be defined by Equation (3.2.6). In this case, the number of amps per magnetic gap is defined in Equation (3.2.7).

𝐼_𝑚 =^2.22𝐴𝑇_𝑚𝑁𝑘 ^𝑔^𝑝

𝑝𝑘_𝑑 (3.2.6) 𝐴𝑇_𝑔= 0.313𝐵_𝐺𝛿𝑘_𝑠𝑘_𝑟 (3.2.7)

Thus, when Equation (3.2.7) is substituted into Equation (3.2.6) and expressed as an ALIP design variable, it appears as Equation (3.2.8). The magnetic reactance, the induced voltage, and the magnetic current have a relation as given by Equation (3.2.9), and 𝐸_𝐴 can be defined through Equation (3.2.10).

𝐼_𝑚=^1.765×10⁶^𝐺^𝑒^𝐵^𝐺

𝑚𝑁𝑘_𝑝𝑘_𝑤𝑞 (3.2.8) 𝑋_𝑚 =^𝐸^𝐴

𝐼_𝑚 (3.2.9) 𝐸_𝐴 =^{2√2𝑓𝑁𝑘}^𝑝^𝑘^𝑤^𝑞

𝜋 𝛷_𝑇 (3.2.10) If Equation (3.2.8) and (3.2.10) are substituted into Equation (3.2.9), they are developed as shown in Equation (3.2.11). In this case, 𝐵_𝐺 is defined in Equation (3.2.11) for the ALIP.

𝑋_𝑚 =^3.56×10⁻⁶^{×2√2𝑓𝑚𝛷}^𝑇^(2𝑁𝑘^𝑝^𝑘^𝑤^𝑝𝑞)²

4𝜋𝑝²𝐵_𝐺𝐺_𝑒 (3.2.11) 𝐵_𝐺= ^𝛷^𝑇

2𝜋𝑟𝐷_𝑝 (3.2.12)

Therefore, Equation (3.2.12) is substituted to Equation (3.2.11), the and magnetic reactance at the ALIP can be defined by Equation (3.2.13). This is applied to the magnetic reactance of the developed pressure relationship in the ALIP design program.

𝑋_𝑚=^2.52×10⁻⁶×√2𝑓𝑚𝜏𝐷(2𝑁𝑘𝑝𝑘_𝑤𝑝𝑞)²

𝜋𝑝𝐺_𝑒 (3.2.13) If the values calculated through the developed pressure equation are assessed after the error correction of the above developed pressure relation equation, the major designs classified as hydrodynamic, geometric, and electromagnetic design values are printed in a tabular form, and the program is terminated.

D. Analysis of major and minor pressure losses

A conduit in which, the flow is without a free surface regardless of the shape of the cross section is called a pipeline. The energy losses need to be considered when designing a system with pipes.

Therefore, it is necessary to consider additional pressure loss in the developed pressure equation of the electromagnetic pump.

In terms of conventional hydrodynamics, Bernoulli’s theorem states that the reference lines and energy lines in the pipeline remain parallel resulting in the conservation of energy; however, the loss of hydraulic head is shown as Equation (3.2.14) and Figure 3.2.3. The hydraulic head loss indicates the quantity of energy, out of which the mechanical energy is converted into heat energy by friction.

𝑣₁²

2𝑔 +^𝑝_𝑟¹+ 𝑧₁ =^𝑣_2𝑔²²+^𝑝_𝑟²+ 𝑧₂+ ℎ_𝐿 (3.2.14)

Figure 3.2.3. Flow and energy relationship in a pipe channel

The energy loss caused by the system can be largely divided into two components. The first is the energy loss caused by friction, which is classified as a major loss, and the second is the minor loss.

The head loss for a fully developed flow within a duct with a constant cross section area is determined only by the detailed characteristics of the flow through the duct, which means that there is no minor loss.

On the other hand, if the flow in the duct has to pass through various types of components, bends, and rapid area changes, additional head loss occurs, mainly due to the separation of the flow. In other words, energy is dissipated by intense mixing in the separation area. If the flow path in the duct is predominantly of a certain cross-sectional area, the minor loss may not be significant; however, depending on the shape of the flow path, the minor loss may be greater than the major loss in certain cases [78].

The pressure loss due to friction within the pump flow path and on the walls of pipes connected to the pump during the operation of the electromagnetic pump is shown in Equation (3.2.15) by Darcy- Weisbach formula [79].

∆𝑃_𝐿=^{𝜌𝜂(𝐿+2𝐿}^𝑟^)𝑣²

2𝐷ℎ +^𝜌𝜂^𝑙^𝐿^𝑙^𝑣^𝑙²

2𝐷_ℎ𝑙 +¹

2𝜌𝐾𝑣_𝑙²+ 𝜌𝑔ℎ (3.2.15)

The right side of Equation (3.2.15) is divided into four components; the first is the pressure drop caused by friction in the flow path inside the electromagnetic pump; the second is the pressure drop caused by friction in the flow path inside the loop; the third is the pressure drop caused by a change in

the shape of the geometric flow path, such as valves, reducers inside the loop; and the fourth is the pressure drop caused by height difference and gravity. Darcy-Weisbach formula in Equation (3.2.15) means that the conditions of the pump operation are determined when the developed pressure of the pump is equal to the sum of the pressure loss. However, this section does not consider the rest of the pressure drop formula except the first component. In other words, a characteristic analysis of the developed pressure and flow rate of only the electromagnetic pump (i.e., excluding the loop) is conducted.

To calculate the major loss in accordance with Darcy-Weisbach formula, it is essential to identify the Darcy friction factor. Because the formulae are different for laminar flow and turbulent flow, firstly, Reynolds number in the laminar flow is considered as shown in Equation (3.2.16) [78].

𝜂 =⁶⁴_𝑅

𝑒 (3.2.16) In other words, the friction coefficient in a laminar flow is a function of only the Reynolds number, and it is not related to the roughness of the flow path, which is also proved in the Moody’s experimental results as shown in Figure 3.2.4.

Figure 3.2.4. Friction coefficient for a fully developed flow in circular pipes

When the flow reaches the critical Reynolds number, a transition to turbulence begins, and the laminar flow becomes a turbulent flow. Therefore, it begins to be affected by the relative roughness.

In Figure 3.2.4, when the relative roughness is less than or equal to 0.001, the friction coefficient tends to follow the smooth pipe curve. However, as the Reynolds number increases, the viscous sublayer thickness around the ducts decreases. The effect of roughness becomes important when the roughness- induced projections begin to penetrate the viscous sublayer, and the friction coefficient changes to a function of the Reynolds number and relative roughness. When the relative roughness is greater than 0.001, the friction coefficient is also influenced by the relative roughness and the Reynolds number, even in a small Reynolds number. Table 3.2.1 shows the roughness of commonly used duct materials [78].

Table 3.2.1. Roughness of the flow paths made from typical industrial materials

Surface material Roughness [mm]

Rivet steel

Pipe

0.9 – 9

Concrete 0.3 – 3

Wood 0.2 – 0.9

Cast iron 0.26

Galvanized steel 0.15

Asphalt cast iron 0.12

Commercial steel or soft iron 0.046

Invalent 0.0015

Polyvinyl chloride and plastics 0.0015 – 0.007

Copper, lead, brass, aluminum 0.001 – 0.002

Stainless steel 0.015

Stretched steel 0.015

Welding steel 0.045

Rusty steel 0.15 – 4

New cast iron 0.25 – 0.8

Worn cast iron 0.8 – 1.5

Rusty cast iron 1.5 – 2.5

Soft cement 0.3

Concrete 0.3 – 1

Coarse concrete 0.3 – 5

Processed wood 0.18 – 0.9

Wood 5

For a turbulent flow, a theoretical calculation of the fluid flow is more involved; therefore, the analysis usually relies on the experimental data. Colebrook Equation (3.2.17) is deduced from the data analysis of accumulated experiment results. It takes the form of a negative function, in which the friction coefficient exists on both sides [80].

√𝜂= −2𝑙𝑜𝑔(^𝑒/𝐷_3.72^ℎ+_𝑅^2.51

𝑒√𝜂) (3.2.17)

As an alternative to Colebrook equation, Haaland developed an equation that would approximate the Colebrook equation, and appears in the form of a positive function, as shown in Equation (3.2.18). A feature of Equation (3.2.18) is that if the Reynolds number is greater than 3,000, the result may be within 2% of that obtained by Colebrook equation without a recalculation [81].

√𝜂= −1.8𝑙𝑜𝑔(^𝑒/𝐷^ℎ

3.72)^1.11+^6.9

𝑅𝑒 (3.2.18)

The minor loss is usually calculated using Equation (3.2.19), or Equation (3.2.20), depending on the type of the device [78].

ℎ_𝑙𝑚= 𝐾^𝑉

2 (3.2.19) ℎ_𝑙𝑚= 𝑓^𝐿^𝑐

𝐷 𝑉²

2 (3.2.20) The loss coefficient of Equation (3.2.19) is determined experimentally for each case and expression 𝐿_𝑐 in Equation (3.2.20) is the equivalent length. There is a significant amount of experimental data for the minor loss; however, there are differences in the results depending on the source of the data even for the same flow geometry. Therefore, it is important to reduce the error in the results by considering the major and minor losses when conducting an analysis of the developed pressure and flow rate of an electromagnetic pump.

Dalam dokumen Electromagnetic Pump for Liquid Metal Cooled Micro Reactor (Halaman 45-55)