Optimized design of thermal insulation and fluid drag reduction for circumferentially grooved annular seal with MMA and perturbation methods

Abstract

Circumferentially grooved seals have been widely used in pumps to eliminate outward leakage of rotating liquid. On many occasions, the turbulent flow enhances the drag force on the interface between the liquid and the stator as well as the interface between the liquid and the rotor, creating much higher heat exchange than the conventional thermal conduction in laminar flows. Attention must be paid in the seal design to prevent rapid heating by the seal liquid to the ambient stator. In this study, the geometry of a circumferentially grooved seal is optimized for a better design of thermal insulation as well as reduction in drag force of the seal fluid. For the forward problem, i.e., the hydraulic and thermal analysis of the seal, the theory of bulk flow is used to simplify the thin-film liquid to a two-dimensional field which preserves the average characteristics of the original flow. The method of three-control volume is adopted to partition the liquid into three types of cavity flows. The governing equations of continuity, momentum, and energy transportation are presented for each control volume, and are approximated by the perturbation method and the Fourier expansion. The fluid and thermal solutions by the present perturbation method are validated by a CFD simulation. For the seal optimization, the multi-objective optimization for thermal insulation and drag reduction is converted into an integrated optimization problem with key geometrical parameters of the seal. Response surfaces are generated through radial basis functions to make the constraint functions explicit for the efficiency of the optimization process. The method of moving asymptotes (MMA) is adopted to find the optimized design of the seal geometry with the best performance of thermal insulation and drag reduction of liquid. Examples are presented to demonstrate the effectiveness of the present optimization method.

Appendix

The nondimensional parameters for (14) are defined as (Yang et al. 1993):

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Coefficients for (24)and(26):

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(54)

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Coefficients for (19):

$$ {\displaystyle \begin{array}{c}{E}_0=\left({h}_0^2{\overline{T}}_0+2\varepsilon {h}_1{h}_0{\overline{T}}_0+\varepsilon {\overline{T}}_1{h}_0^2\right)\frac{\partial \overline{\rho}}{\partial \tau}\\ {}{E}_1={h}_0\frac{\partial \overline{\rho}}{\partial \theta}\left({h}_0{u}_0{T}_0+2\varepsilon {h}_1{u}_0{T}_0+\varepsilon {u}_1{h}_0{T}_0+\varepsilon {T}_1{u}_0{h}_0\right)\\ {}\kern1.1em +{h}_0\frac{\partial \overline{\rho}}{\partial \overline{z}}\left({h}_0{w}_0{T}_0+2\varepsilon {h}_1{w}_0{T}_0+\varepsilon {w}_1{h}_0{T}_0+\varepsilon {T}_1{w}_0{h}_0\right)\\ {}\kern1.4em +\varepsilon \overline{\rho}{h}_0\left(\begin{array}{c}\frac{\partial {h}_1}{\partial \theta }{u}_0{T}_0+\frac{\partial {u}_1}{\partial \theta }{h}_0{T}_0+\frac{\partial {T}_1}{\partial \theta }{h}_0{u}_0\\ {}+\frac{\partial {h}_1}{\partial \theta }{w}_0{T}_0+\frac{\partial {w}_1}{\partial \theta }{h}_0{T}_0+\frac{\partial {T}_1}{\partial \theta }{h}_0{w}_0\end{array}\right)\\ {}{E}_2=\varepsilon {T}_0{h}_1\left(\sigma \frac{\partial {p}_1}{\partial \tau }+{u}_0\frac{\partial {p}_1}{\partial \theta }+{w}_0\frac{\partial {p}_1}{\partial \overline{z}}\right)\\ {}{E}_3={k}_x\left({u}_0^2+2\varepsilon \left({u}_0{u}_1+{w}_0{w}_1\right)+{w}_0^2+\frac{1}{2}\left({u}_0+\varepsilon {u}_1\right)\varLambda \right)+{k}_r\left(\frac{1}{4}{\varLambda}^2-\left({u}_0+\varepsilon {u}_1\right)\varLambda \right)\end{array}} $$

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Coefficients for (22):

$$ {\displaystyle \begin{array}{c}{E}_4={\overline{C}}_p{\operatorname{Re}}_s\frac{\partial \overline{\rho}}{\partial \tau }{h}_0{T}_0+{\overline{C}}_p{\operatorname{Re}}_p^{\ast}\left(\frac{\partial \overline{\rho}}{\partial \theta }{h}_0{u}_0{T}_0+\frac{\partial \overline{\rho}}{\partial \overline{z}}{h}_0{w}_0{T}_0\right)\\ {}{E}_5=\frac{\partial \overline{\rho}}{\partial \tau}\left({T}_0{h}_1+{h}_0{T}_1\right)+\overline{\rho}\frac{h_0\partial {T}_1+{T}_0\partial {h}_1}{\partial \tau}\\ {}{E}_6=\frac{\partial \overline{\rho}}{\partial \theta}\left({h}_0{u}_1{T}_0+{h}_0{u}_0{T}_1+{u}_0{T}_0{h}_1\right)+\overline{\rho}\frac{h_0{T}_0\partial {u}_1+{h}_0{u}_0\partial {T}_1+{u}_0{T}_0\partial {h}_1}{\partial \theta}\\ {}\kern1.6em +\frac{\partial \overline{\rho}}{\partial \overline{z}}\left({h}_0{w}_1{T}_0+{h}_0{w}_0{T}_1+{w}_0{T}_0{h}_1\right)+\overline{\rho}\frac{h_0{T}_0\partial {w}_1+{h}_0{w}_0\partial {T}_1+{w}_0{T}_0\partial {h}_1}{\partial \overline{z}}\\ {}{E}_7=\overline{\beta}{h_0}^2{T}_0\left(\sigma \frac{\partial {p}_1}{\partial \tau }+{u}_0\frac{\partial {p}_1}{\partial \theta }+{w}_0\frac{\partial {p}_1}{\partial \overline{z}}\right)+{h_0}^2\frac{\varLambda }{2}\cdot \frac{\partial {p}_1}{\partial \theta}\\ {}\kern1.2em +\overline{\mu}\left[{k}_x\left(\left(2{u}_0{u}_1+2{w}_0{w}_1\right)+\frac{1}{2}{u}_0{u}_1\varLambda \right)-{u}_1{k}_r\varLambda \right]\end{array}} $$

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Coefficients for (40)–(41):

$$ {\displaystyle \begin{array}{c}{F}_1=\frac{\partial \overline{\rho}}{\partial \tau}\left(-{T}_0\frac{\underline{y}}{\varepsilon }+{h}_0{T}_{II1s}(z)\sin \varOmega t\right)+\overline{\rho}{h}_0\left({T}_{II1s}(z)\cdot \varOmega \cdot \cos \varOmega t\right)-{T}_0\frac{\underline {\overset{\cdot }{y}}\kern0.1em }{\varepsilon}\\ {}{F}_2=\frac{\partial \overline{\rho}}{\partial \theta}\left({h}_0\sin \varOmega t\left({T}_0{u}_{II1s}(z)+{u}_0{T}_{II1s}(z)\right)-{u}_0{T}_0\frac{\underline{y}\kern0.1em }{\varepsilon}\right)\\ {}\kern1.3em -\overline{\rho}{h}_0 cos\varOmega t\left({T}_0\cdot {u}_{II1c}(z)-{u}_0\cdot {T}_{II1c}(z)\right)+{u}_0{T}_0\overline{\rho}\frac{\underline{x}\left(\tau \right)\kern0.1em }{\varepsilon}\\ {}\kern1.1em +\frac{\partial \overline{\rho}}{\partial \overline{z}}\left({h}_0\sin \varOmega t\left({T}_0{w}_{1s}(z)+{w}_0{T}_{II1s}(z)\right)-{w}_0{T}_0\frac{\underline{y}}{\varepsilon}\right)\\ {}\kern0.9000001em +\overline{\rho}{h}_0\sin \varOmega t\left({T}_0\frac{\partial {p}_{1s}(z)}{\partial z}+{w}_0\frac{\partial {T}_{II1s}(z)}{\partial z}\right)-\overline{\rho}{w}_0{T}_0\frac{\partial \underline{y}}{\varepsilon \cdot \partial z}\\ {}{F}_3=\overline{\beta}{h_0}^2{T}_0\left(\sigma {p}_{1s}(z)\cdot \varOmega \cos \varOmega t+{w}_0\frac{\partial {p}_{1s}(z)}{\partial z}\sin \varOmega t\Big)\right)\\ {}\kern1.3em -\left({h_0}^2\frac{\varLambda }{2}+{u}_0\overline{\beta}{h_0}^2{T}_0\right){p}_{1c}(z)\cdot \cos \varOmega t\\ {}{F}_4=\overline{\beta}{h_0}^2{T}_0\left(\sigma {p}_{1s}(z)\cdot \varOmega \cos \varOmega t+{w}_0\frac{\partial {p}_{1s}(z)}{\partial z}\sin \varOmega t\Big)\right)\\ {}\kern1.4em -\left({h_0}^2\frac{\varLambda }{2}+{u}_0\overline{\beta}{h_0}^2{T}_0\right){p}_{1c}(z)\cdot \cos \varOmega t\end{array}} $$

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$$ {\displaystyle \begin{array}{c}{F}_5=\frac{\partial \overline{\rho}}{\partial \tau}\left(-{T}_0\frac{\underline{x}}{\varepsilon }+{h}_0{T}_{II1c}(z)\cos \varOmega t\right)-\overline{\rho}{h}_0{T}_{II1\mathrm{c}}(z)\cdot \varOmega \cdot \sin \varOmega t-{T}_0\frac{\underline {\overset{\cdot }{x}}\kern0.1em }{\varepsilon}\\ {}{F}_6=\frac{\partial \overline{\rho}}{\partial \theta}\left({h}_0\cos \varOmega t\left({T}_0{u}_{II1c}(z)+{u}_0{T}_{II1c}(z)\right)-{u}_0{T}_0\frac{\underline{x}}{\varepsilon}\right)\\ {}+\overline{\rho}{h}_0\sin \varOmega t\left({T}_0\cdot {u}_{II1s}(z)\sin \varOmega t+{u}_0{T}_{II1s}(z)\right)-{u}_0{T}_0\overline{\rho}\frac{\underline{y}\kern0.1em }{\varepsilon}\\ {}\kern1em +\frac{\partial \overline{\rho}}{\partial \overline{z}}\left({h}_0\cos \varOmega t\left({T}_0\cdot {w}_{1c}(z)+{w}_0{T}_{II1c}(z)\right)-{w}_0{T}_0\frac{\underline{x}}{\varepsilon}\right)\\ {}\kern1em +\overline{\rho}{h}_0\cos \varOmega t\left({T}_0\frac{\partial {p}_{1c}(z)}{\partial z}+{w}_0\frac{\partial {T}_{II1c}(z)}{\partial z}\right)-\overline{\rho}{w}_0{T}_0\frac{\partial \underline{x}}{\varepsilon \cdot \partial z}\\ {}{F}_7=-\sigma {p}_{1c}(z)\cdot \varOmega \sin \varOmega t+{w}_0\frac{\partial {p}_{1c}(z)}{\partial z}\cos \varOmega \\ {}{F}_8=\left({h_0}^2\frac{\varLambda }{2}+{u}_0\overline{\beta}{h_0}^2{T}_0\right){p}_{1s}(z)\cdot \sin \varOmega t\\ {}{F}_9={k}_x\left(\left(2{u}_0+\frac{1}{2}{u}_0\varLambda \right){u}_{II1c}(z)+2{w}_0{w}_{1c}(z)\right)-{k}_r\varLambda {u}_{II1c}(z)\end{array}} $$

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Derivation procedure from (24) and (26) to (36)–(39):

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