Classical Graetz Problem Research Articles

Laminar mixed convection in the thermal entrance region of horizontal isothermal rectangular channels

AbstractLaminar mixed convection in the thermal entrance region of horizontal isothermal rectangular channels for moderate and small Prandtl number (air) is investigated using the vorticity‐velocity formulation of the Navier‐Stokes equation. The numerical results, including the cross‐stream velocity vectors, local Nusselt numbers and local friction factor ratios are presented for the aspect (width/height) ratios 0.5, 1.0 and 2.0, Rayleigh numbers 0 ∼ 2 × 105 and Prandtl numbers 0.7, 5.0 and 100. The strength and pattern of the secondary flow induced by buoyancy effects is found to depend on the magnitude of Rayleigh number and aspect ratio, and the secondary flow leads to a significant enhancement of heat transfer in the entrance region. The classical Graetz problem is shown to be a limiting case which applies only when Ra ≤ 10.3 The behavior of the local Nusselt number for Pr = 100 compares well with the existing results for Pr → ∞.

Numerical solution to the Graetz problem: Achieving accuracy in the entrance region

Numerical solutions to the classical Graetz problem are often incapable of predicting correct values of Nusselt number in the entrance region. Parameters affecting the accuracy of a numerical solution employing the control volume approach were investigated and reasons for the discrepancy between previous numerical and analytical solutions were revealed.

Application of the general purpose collocation software, PDECOL, to the graetz problem

The software package PDECOL is capable of solving coupled systems of parabolic, nonlinear, partial differential equations. Its use of B-spline collocation for spatial discretization makes it valuable for obtaining solutions in which extreme gradients are present. A severe limitation of this package is the requirement that the boundary and initial conditions be consistent. The package can, however, be modified to overcome this limitation. To demonstrate the utility of the modified PDECOL package, the solution of the classical Graetz problem is obtained for different flow geometries and for some common boundary conditions. The results indicate that excellent agreement with the analytical results can be obtained provided that a suitable distribution of a sufficient number of spatial grid points is used and an error criterion compatible with the desired accuracy is chosen for the time integration. For this steady-state problem this integration is with respect to the spatial dimension which appears in the first derivative.

An assessment of the accuracy of numerical solutions to the graetz problem

It has been well documented in prior studies that numerical solutions to the classical Graetz problem in the entrance region of pipes produce estimates of Nusselt number that are greatly in error. In this report, we examine this problem and attempt to provide plausible explanation for the discrepancy.

Crystallization on a cooled wall with forced convection in the melt

The local heat transfer in a liquid, which flows in a laminar stream through a duct, towards a crystal layer, formed from that liquid, has been studied. This situation resembles the classical Graetz problem, however, the occurrence of a crystal layer on the wall of the duct results in a substantial difference in heat transfer. The relevant equations for momentum and heat transfer in the duct have been solved by numerical integration. The solutions have been tested against experimental data. Two cases have been considered, i.e. the stationary state (no growth or melting by the crystal layer) and the non-stationary state (growing or melting crystal layer). In the stationary state it has been found that near the leading edge of the crystal layer the local Nusselt number decreases more rapidly than in the situation without crystal layer. Further downstream the local Nusselt number approaches an asymptotic limit which depends on the crystal layer thickness. The theoretical solution given agrees satisfactorily with the experimental results. In the non-stationary state the local Nusselt number with growth or melting ( Nu) has been compared with the local Nusselt number ( Nu 0) in the stationary state. The experimental Nu/Nu 0-ratio expresses a more pronounced influence of the growth rate than the theoretical solution. For melting a broad agreement between theory and experiments has been found.

Natural convection effects on graetz problem in horizontal isothermal tubes

The classical Graetz problem with natural convection effect in isothermally cooled or heated horizontal tubes is approached by a numerical method using large Prandtl number assumption. Numerical solutions are obtained for a range of Rayleigh numbers Ra = 0–10 6. The developing secondary flow and temperature fields, bulk temperature, local and average Nusselt numbers are presented to study the natural convection effect. The Nusselt number results are compared against the experimental data and the agreement is found to be satisfactory.

Natural convection effects on graetz problem in horizontal rectangular channels with uniform wall temperature for large Pr

The effect of buoyancy forces on laminar forced convective heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall temperature is studied by a numerical method for the case of large Prandtl number fluids. The numerical results are presented for the aspect ratios (width/height) 0·5, 1, 2 and Rayleigh numbers 0 ~ 5 × 10 5. The correlation equations for the prediction on the onset of significant free convection effect are developed. The asymptotic behavior of local Nusselt number is compared against the known asymptote for the uniform wall temperature boundary condition. The entrance region where the free convection effect is significant is clearly established and the classical Graetz problem is shown to be a limiting case and is applicable only when Ra ⩽ 10 3.

Laminar forced convection in the thermal entrance region of curved pipes with uniform wall temperature

AbstractThe thermal entrance region heat transfer problem for fully developed laminar flow in curved pipes with uniform wall temperature is approached by an alternating direction implicit method for the parabolic energy equation for a flow regime with Dean number ranging from 0 to an order of 100. This work represents an extension of the classical Graetz problem in straight tubes to curved pipes. The graphical results for temperature developments in the form of temperature profiles through the horizontal and vertical planes, isothermals and local Nusselt number variations in the thermal entrance region are presented in such a way as to illustrate clearly the interaction between the secondary flow and the developing temperature field for Prandtl numbers of 0.1, 0.7, 10 and 500. For a given Dean number, the effect of Prandtl number is to shorten the thermal entrance length (I/Gz) and the temperature field develops rather rapidly with large Prandtl number. The effect of Dean number is similar to that of Prandtl number with Dean number effect becoming much more appreciable at high Prantdl numbers than at low Prandtl number.

Heat Transfer to Flow in a Round Tube With Arbitrary Velocity Distribution

Abstract An extension of the classical Graetz problem is presented to cover various velocity profiles; and the variation of Nusselt modulus with respect to these profiles is shown.

Heat Transfer to Laminar Flow in a Round Tube or Flat Conduit—The Graetz Problem Extended

Abstract The complete set of eigenvalues and eigenfunctions for the classical Graetz problem is presented and the solution is extended to cover arbitrary wall temperature or heat-flux variations.