Steady-state solutions in a nonlinear pool boiling model. (English) Zbl 1221.80010
Summary: We consider a relatively simple model for pool boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling fluid via a nonlinear boundary condition imposed on the fluid–heater interface. This results in a standard heat equation with a nonlinear Neumann boundary condition on part of the boundary. In this paper, we analyse the qualitative structure of steady-state solutions of this heat equation. It turns out that the model allows both multiple homogeneous and multiple heterogeneous solutions in certain regimes of the parameter space. The latter solutions originate from bifurcations on a certain branch of homogeneous solutions. We present a bifurcation analysis that reveals the multiple-solution structure in this mathematical model. In the numerical analysis a continuation algorithm is combined with the method of separation-of-variables and a Fourier collocation technique. For both the continuous and discrete problem a fundamental symmetry property is derived that implies multiplicity of heterogeneous solutions. Numerical simulations of this model problem predict phenomena that are consistent with laboratory observations for pool boiling processes.
MSC:
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 76E06 | Convection in hydrodynamic stability |
| 76R10 | Free convection |
References:
| [1] | Thome, J. R., Boiling, (Bejan, A.; Krause, A. D., Handbook of heat transfer (2003), Wiley and Sons: Wiley and Sons Hoboken), 635-717 |
| [2] | Mudawar, I., Assessment of high-heat flux thermal management schemes, IEEE Trans-CPMT: Compon Pack Technol, 24, 122-141 (2001) |
| [3] | Dhir, V. K., Boiling heat transfer, Ann Rev Fluid Mech, 30, 365-401 (1998) |
| [4] | Theofanous, T. G.; Tu, J. P.; Dinh, A. T.; Dinh, T. N., The boiling crisis phenomenon. Part I: nucleation and nucleate boiling heat transfer, Exp Therm Fluid Sci, 26, 775-792 (2002) |
| [5] | Dhir, V. K., Nucleate and transition boiling heat transfer, Int J Heat Fluid Flow, 12, 290-314 (1991) |
| [6] | Auracher, H.; Marquardt, W., Heat transfer characteristics and mechanisms along entire boiling curves under steady-state and transient conditions, J Heat Fluid Flow, 25, 223-242 (2004) |
| [7] | Auracher, H.; Marquardt, W., Experimental studies of boiling mechanisms in all boiling regimes under steady-state and transient conditions, Int J Therm Sci, 41, 586-598 (2002) |
| [8] | van Ouwekerk, H., Burnout in pool boiling. The stability of boiling mechanisms, Int J Heat Mass Transf, 15, 25-33 (1972) |
| [9] | Zhukov, S. A.; Barelko, V. V.; Merzhanov, A. G., Wave processes on heat generating surfaces in pool boiling, Int J Heat Mass Transf, 24, 47-55 (1980) |
| [10] | Blum, J.; Lüttich, T.; Marquardt, W., Temperature wave propagation as a route from nucleate to film boiling?, (Celata, G. P.; DiMarco, P.; Shah, R. K., Proceedings of the second international symposium on two-phase flow modelling and experimentation Rome, vol. 1 (1999), Edizioni ETS: Edizioni ETS Pisa) |
| [11] | Zhukov, S. A.; Barelko, V. V., Nonuniform steady states of the boiling process in the transition region between the nucleate and film regimes, Int J Heat Mass Transf, 26, 1121-1130 (1983) |
| [12] | Gabaraev, B. A.; Kovalev, S. A.; Molochnikov, Yu. S.; Solov’ev, S. L.; Usitakov, S. V., Rewetting and autowave change of boiling modes, High Temp, 39, 302-314 (2001) |
| [13] | Kovalev, S. A., An investigation of minimum heat fluxes in pool boiling of water, Int J Heat Mass Transf, 9, 1219-1226 (1966) |
| [14] | Kovalev, S. A.; Rybchinskaya, G. B., Prediction of the stability of pool boiling heat transfer to finite disturbances, Int J Heat Mass Transf, 21, 691-700 (1978) |
| [15] | Kovalev, S. A.; Usitakov, S. V., Analysis of the stability of boiling modes involving the use of stability diagrams, High Temp, 41, 68-78 (2003) |
| [16] | Gurevich, A. V.; Mints, R. G., Self-heating in normal metals and superconductors, Rev Mod Phys, 59, 941-999 (1987) |
| [17] | Blum J, Marquardt W. Objection to Haramura’s criterions for temperature uniformity across the surface in transition boiling, Internal report LPT-1998-14, Lehrstuhl für Prozesstechnik, RWTH Aachen; 1998.; Blum J, Marquardt W. Objection to Haramura’s criterions for temperature uniformity across the surface in transition boiling, Internal report LPT-1998-14, Lehrstuhl für Prozesstechnik, RWTH Aachen; 1998. |
| [18] | Speetjens MFM. Steady-state behaviour of 2D pool boiling problems. In: Nowak AJ, Bialecki RA, Wecel G, editors. In: Proceedings of the Eurotherm Seminar 82 Numerical Heat Transfer; 2005. p. 923-31.; Speetjens MFM. Steady-state behaviour of 2D pool boiling problems. In: Nowak AJ, Bialecki RA, Wecel G, editors. In: Proceedings of the Eurotherm Seminar 82 Numerical Heat Transfer; 2005. p. 923-31. |
| [19] | Kreyszig, E., Advanced engineering mathematics (1999), Wiley: Wiley Chichester |
| [20] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods in fluid dynamics (1987), Springer: Springer Berlin · Zbl 0636.76009 |
| [21] | Ott, E., Chaos in dynamical systems (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1006.37001 |
| [22] | Govaerts, W. J.F., Numerical methods for bifurcations of dynamical equilibria (2000), SIAM: SIAM Philadelphia · Zbl 0935.37054 |
| [23] | Deimling, K., Nonlinear functional analysis (1985), Springer: Springer Berlin · Zbl 0559.47040 |
| [24] | Blum, J.; Marquardt, W.; Auracher, H., Stability of boiling systems, Int J Heat Mass Transf, 39, 3021-3033 (1996) · Zbl 0964.76510 |
| [25] | Speetjens M, Reusken A, Marquardt W. Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, IGPM report 263, RWTH-Aachen; 2006.; Speetjens M, Reusken A, Marquardt W. Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, IGPM report 263, RWTH-Aachen; 2006. |
| [26] | Darabi, J.; Ohadi, M. M.; Fanni, M. A.; Dessiatoun, S. V.; Kedzierski, M. A., Effect of heating boundary conditions on pool boiling experiments, HVAC&R Res, 5, 1-14 (1999) |
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