Induction electromagnetic pump

3. Methods

3.1. Basic principles of electromagnetic pump

3.1.2. Induction electromagnetic pump

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Table 3.1.1. Loss mechanisms in an electromagnetic pump and their scales

.

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Figure 3.1.3. Structure and conceptual diagram of the ALIP

Maxwell’s equations, Ampere's law, Ohm's law, Gauss' law, continuity equation, and momentum equation are used to analyze the characteristics of the ALIP. In particular, magnetohydrodynamics, which combines electromagnetics and hydrodynamics, is utilized in the analysis of the theoretical aspects of the fluid equations, as shown in Figure 3.1.4 [57].

Figure 3.1.4. Analysis of electromagnetic pumps using magnetohydrodynamics

The equivalent circuit analysis method analyzes the pump characteristics by converting the electromagnetic pump into an electrical circuit diagram as shown in Figure 3.1.5 [58–59]. This figure shows that the primary side is set up as an electromagnet and the secondary side is the liquid metal.

The design variables are studied and analyzed using a MATLAB program that establishes a relationship between the developed pressure and efficiency based on the equivalent circuit interpretation of the electromagnetic principles. The equivalent circuit analysis method should be supplemented by an interpretation, considering minor and major losses caused by friction and flow, end effect, and edge effect.

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Figure 3.1.5. Equivalent circuit analysis method for designing an ALIP

The cross section of the ALIP has slots to insert coils for driving an external current. The real coil arrangement for generating a sinusoidal current can be converted into an equivalent sheet current [60].

The assumptions of the theoretical model are as follows [61]:

- An incompressible liquid metal exhibits stable laminar flow behavior in the narrow annular gap of the pump.

- The axial velocity of the flow changes along the radial coordinates.

- The ALIP is infinitely long and axially symmetric.

- The sheet current equivalent to the current by the real coil arrangement in the ALIP is sinusoidal, as seen in Equation (3.1.4) [62].

𝐽_𝑎(𝑟_𝑏, 𝑧, 𝑡) = 𝐽_𝑚cos(𝜔𝑡 − 𝑘𝑧) 𝜃̂ = 𝑅𝑒[𝐽_𝑚𝑒^{𝑖(𝜔𝑡−𝑘𝑧)}]𝜃̂ (3.1.4)

- The electric field intensity (E), magnetic field (B), and induced current density (J), are represented by sinusoidal functions, as presented in Equation (3.1.5).

E(r, z, t) = 𝑅𝑒[𝐸(𝑟)𝑒^{𝑖(𝜔𝑡−𝑘𝑧)}]𝜃̂

B(r, z, t) = 𝑅𝑒[{𝐵_𝑟(𝑟)𝑟̂ + 𝐵_𝑧(𝑟)𝑧̂}𝑒^{𝑖(𝜔𝑡−𝑘𝑧)}] (3.1.5) J(r, z, t) = 𝑅𝑒[𝐽(𝑟)𝑒^{𝑖(𝜔𝑡−𝑘𝑧)}]𝜃̂

The dimensionless MHD equations for the ALIP that is subjected to a Lorentz’ force 𝐉 × 𝐁 as an external force, is represented by Equation set (3.1.6) [61].

27 - Continuity

∇ ∙ 𝐕 = 0

- Momentum equation

𝜕𝐕

𝜕𝑡 + (𝐕 ∙ ∇)𝐕 = −𝛻𝑃 + 1

𝑅_𝑒∇²𝐕 +𝐻_𝑎² 𝑅_𝑒𝐉 × 𝐁

- Maxwell’s equations

∙ Ampere’s law: ∇ × 𝐁 = 𝑅_𝑚𝐉 (3.1.6)

∙ Faraday’s law: ∇ × 𝐄 = −^𝜕𝐁_𝜕𝑡

∙ Gauss’ law: ∇ ∙ 𝐁 = 0

- Ohm’s law

𝐉 = 𝐄 + 𝐕 × 𝐁

where velocity (𝐕), electric field (𝐄), magnetic flux density (𝐁), current density (𝐉), pressure (P), time (t) and geometrical parameters (𝐑), are non-dimensionalized using the reference values, 𝑣_𝑠, 𝑣_𝑠𝐵_𝑜, 𝐵₀, σ𝑣_𝑠𝐵₀, ρ𝑣_𝑠², ^𝑅_𝑣⁰

𝑠, and 𝑅₀ respectively. As a result, the equation set includes dimensionless parameters, such as the Reynolds number (𝑅_𝑒), Hartmann number (𝐻_𝑎) and magnetic Reynolds number (𝑅_𝑚) [63–67].

Upon substituting the magnetic field, current density, and electric field intensity in the form of sinusoidal functions in Equation set (3.1.5); the Maxwell’s equations, and Ohm’s law in Equation set (3.1.6) are reduced to Equation set (3.1.7).

𝜕𝐵_𝑧

𝜕𝑟 + 𝑗𝑘𝑅₀𝐵_𝑟 = −𝑅_𝑚𝐽

𝐸 = −𝐵_𝑟 (3.1.7) 1

𝑟

𝜕

𝜕𝑟(𝑟𝐵_𝑟) − 𝑗𝑘𝑅₀𝐵_𝑧 = 0 J = −(1 − 𝑣)𝐵_𝑟

where 𝐵_𝑧 and 𝐵_𝑟 are, respectively, the axial and radial components of the magnetic field in the ALIP. Combining Equation set (3.1.7) and the momentum equation of Equation set (3.1.6), the Equation set for the velocity in the radial direction and the magnetic fields in the radial and axial directions can be derived, and are shown in Equation set (3.1.8).

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−𝜕𝑃

𝜕𝑧+ 1 𝑅_𝑒

1 𝑟

𝜕

𝜕𝑟(𝑟𝜕𝑣

𝜕𝑟) +𝐵_𝑟𝐵_𝑟^𝑐𝐻_𝑎²

2𝑅_𝑒 (1 − 𝑣) = 0

1 𝑟

𝜕

𝜕𝑟(𝑟^𝜕𝐵_𝜕𝑟^𝑟) − {_𝑟¹₂+ (𝑘𝑅₀)²+ 𝑗𝑘𝑅₀𝑅_𝑚(1 − 𝑣)}𝐵_𝑟 = 0 (3.1.8) 1

𝑟

𝜕

𝜕𝑟(𝑟𝐵_𝑟) − 𝑗𝑘𝑅₀𝐵_𝑧 = 0

As of the boundary conditions for velocity and magnetic field, the fluid is adhesive in the inner and outer wall of the duct annulus of the ALIP, and the axial magnetic field can be expressed by an equivalent sheet current of both inner and outer walls. Thus, the Equation set (3.1.9) represents the boundary conditions for velocity and magnetic field mathematically.

v(𝑟_𝑎) = 𝑣(𝑟_𝑏) = 0

𝐵_𝑧(𝑟_𝑎) = √2𝑘𝑅0 (3.1.9) 𝐵_𝑧(𝑟_𝑏) = 0

The solutions for the velocity and magnetic fields in Equation set (3.1.8) are given by Bessel functions of the 1st and 2nd kinds [61]. As a result, taking 𝐉 × 𝐁 and finding electromagnetic force called Lorenz force, it is expressed as Equation (3.1.10).

𝑓_𝑧 = (1 − 𝑣)^{𝐵𝑟𝐵𝑟𝑐}

2 𝐻𝑎2

𝑅𝑒 (3.1.10) Considering the small skin effect at 60 Hz, the change in the radial magnetic field across the narrow

annular flow gap is negligible. Hence, ^{𝐵𝑟𝐵𝑟}

𝑐

2 in Equation (3.1.10) is assumed to be constant [68]. The average slip in the flow gap is defined in Equation (3.1.11) and the average electromagnetic force generated along the axial direction in the flow gap is expressed as Equation (3.1.12).

s ≡ ∫ (1 − 𝑣)𝑑𝑟_𝑟^𝑟𝑏

𝑎 (3.1.11) 𝑓_𝑧= ∫ (1 − 𝑣)^𝐵𝑟₂^{𝐵𝑟𝑐𝐻}_𝑅^𝑎2

𝑒𝑑𝑟

𝑟_𝑏

𝑟𝑎 (3.1.12)

= s𝐵_𝑟𝐵_𝑟^𝑐 2

𝐻_𝑎² 𝑅_𝑒

Substituting the hydrodynamic, geometric, and electromagnetic variables into Equation (3.1.12), the driving pressure in the ALIP is obtained, as shown in Equation (3.1.13),

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∆P̅̅̅̅ =_𝑝𝑅^{36𝜎𝑠𝑓𝜏2(𝜇0𝑘𝑤𝑁𝐼)2}

02{𝜋²+(2𝜇₀𝜎𝑠𝑓𝜏²)² (3.1.13) which is expressed as a function of the electrical conductivity of the liquid metal, slip, input frequency, pole pitch, turns of coil, input current, flow gap, and number of pole pairs.

The efficiency of the ALIP is calculated by taking the ratio of the hydraulic output to the electrical input of the pump. The relation between the electrical input and the hydraulic output can be found by using an electrical equivalent circuit for the ALIP, as seen in Figure 3.1.5 [59, 69]. In Figure 3.1.5, the ALIP is divided into two parts, namely the primary and secondary. The primary part is the one with the electromagnetic core and exciting coils, and the secondary part contains the liquid sodium flow [69]. 𝑅₁ and 𝑋₁ are primary resistance and leakage reactance in the primary part, respectively, and 𝑋_𝑚 and 𝑅₂ are the magnetization reactance and secondary resistance in the secondary part, respectively. Those equivalent resistance and reactance are expressed as functions of the pump geometric and electromagnetic variables, which are represented in Equations (3.1.14) to (3.1.17) using Laithwaite’s standard design formulae [69–71].

𝑅₁ =^{𝜋𝜌𝑐𝑞𝑘𝑝2𝑚2𝐷0𝑁2}

𝑘𝑓𝑘𝑑𝑝𝜏² (3.1.14) 𝑋₁ ≅^{2𝜋𝜇0𝜔𝐷0𝜆𝑐𝑁2}

𝑝𝑞 (3.1.15) 𝑋_𝑚 =^{6𝜇0𝜔𝜏𝜋𝐷}_{𝜋2𝑝𝐺}^{0(𝑘𝑤𝑁)2}

𝑒 (3.1.16) 𝑅₂=^{6𝜋𝐷𝑚𝜌𝑠(𝑘𝑤𝑁)2}

𝜏𝑝 (3.1.17) Balancing the input and output in the equivalent electric circuit of Figure 3.1.5 and using Equations (3.1.14)–(3.1.17), the efficiency is expressed in Equation (3.1.18) [14].

𝜀 = ^{∆𝑃𝑑∙𝑄}

√3𝑉𝐼𝑐𝑜𝑠𝜑= ^{(2𝑝−1)6𝑘𝑤2(1−𝑠)}

(2𝑝+1){𝜌𝑐𝑞𝑘𝑝2𝑚2𝜎𝑔𝑒

𝑘𝑓𝑘𝑑𝜏 {1+( ^𝜋

2𝜇0𝑓𝑠𝜎𝜏2)²}+^6𝑘𝑤_𝑠²}𝑐𝑜𝑠𝜑

−^{𝑓𝐷

𝜌𝐿𝑣2

2𝐷ℎ+𝐾_𝑚^𝜌𝑣2₂ }∙𝑄

√3 𝑉𝐼𝑐𝑜𝑠𝜑 (3.1.18)

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