Heat transfer characteristics of internally heated liquid pools at high Rayleigh numbers A. Liaqat, A. C. Baytas Abstract Detailed numerical analysis is presented for buoyancy driven ¯ow of a Newtonian ¯uid contained in a square enclosure for high Rayleigh Ra) numbers. Natural convection is due to internal heating sources, which are assumed to be uniformly distributed within the enclosure. All walls of the cavity are maintained at constant tem- perature. Flow and heat transfer characteristics are in- vestigated for a Ra number range of 10 7 to 10 12 while Prandtl Pr) number is taken to be 7.0. Governing equa- tions in primitive variables) are discretised using control volume technique based on staggered grid formulation. These equations are solved using SIMPLER algorithm of Patankar. Flow and heat transfer characteristics, stream- lines, isotherms and average wall Nusselt Nu) number, are presented for whole range of Ra number considered. Finally, present results for average wall Nu numbers are compared with experimental observations obtained from open literature. It is concluded that both results are in very good agreement, which con®rmed the accuracy of the scaling used for present investigation. List of symbols D horizontal/vertical dimension of the cavity, m g gravitational acceleration, m/s 2 k thermal conductivity of the ¯uid, W/mK n direction normal to the wall Nu a average Nusselt number, Eq. 6) p dimensionless pressure, N/m 2 P dimensionless pressure  pD 2 qa 2 Ra Pr 4=5   Pr Prandtl number  m a   q 000 uniform volumetric heat source, W/m 3 Ra Rayleigh number  gbq 000 D 5 kam   S i spatial position, Eq. 7) t physical time, s T dimensional temperature, K DT reference temperature  q 000 D 2 kRa Pr 1=5   u; v velocity components in x; y directions, m/s U ; V dimensionless velocity components in X; Y directions  u;v a=DRa Pr 2=5   x; y Cartesian coordinates, m X; Y dimensionless Cartesian coordinates  x; y=D Greek symbols a thermal diffusivity of the ¯uid, m 2 /s a s rate of grid stretching, Eq. 7) b coef®cient of thermal expansion of the ¯uid, K 1 h dimensionless temperature  TT w DT   m kinematic viscosity of the ¯uid, m 2 /s q density of the ¯uid, kg/m 3 s dimensionless time  ta D 2 Ra Pr 2=5   Subscripts i; j grid points indices w value on the wall 1 Introduction Natural convection induced by internal heating is impor- tant in many practical problems such as nuclear insula- tion, cooling of radioactive waste, spent fuel storage, nuclear reactor safety analysis, post accident heat removal in nuclear reactors, low power operation of research reactors and reprocessing of spent fuel. Many experimental and numerical studies are available in open literature for buoyant ¯ows in different types of cavities containing volumetric heat sources. Steinberner et al. [1] reported an experimental study of natural con- vection heat transfer with internal heat sources for a Ra number varying from 5  10 10 to 3  10 13 . Kulacki et al. [2] investigated the natural convection in a horizontal ¯uid layer containing internal heat sources, for Ra num- bers from 114.0 to 1:8  10 6 . Emara et al. [3] performed a numerical analysis of a heat generating ¯uid layer for a Ra number range of 5:0  10 4 to 5:0  10 8 . Tzanos et al. [4] performed a numerical simulation of natural convection in a cylindrical pool of heat generating ¯uid, for Ra numbers from 1:33  10 9 to 8:69  10 11 . Bergholz [5] solved boundary layer equations analytically to study the natural convection in a heat generating ¯uid in a closed Cartesian cavity. Boundary layer analysis was used to obtain the equations valid near the walls and the corresponding system of equations valid in the core of the cavity. May [6] presented a detailed numerical investigation of buoyant ¯ow in a square enclosure having internal heating sources for a Ra number range of 10 4 to 1:5  10 5 . Baytas [7] Heat and Mass Transfer 36 2000) 401±405 Ó Springer-Verlag 2000 401 Received on 15 November 1999 A. Liaqat, A. C. Baytas &) Institute for Nuclear Energy Istanbul Technical University 80626 Maslak Istanbul, Turkey This paper has been supported by the Istanbul Technical University Research Fund, through grants 1190 and 843.