Extended heat-transfer relation for an impinging laminar flame jet to a flat plate

Abstract

Many industrial applications use flame impingement to obtain high heat-transfer rates. An analytical expression for the convective part of the heat transfer of a flame jet to a plate is derived. Therefore, the flame jet is approximated by a hot inert jet. In contradiction with existing convective heat-transfer relations, our analytical solution is applicable not only for large distances between the jet and the plate, but also for close spacings. Multiplying the convective heat transfer by a factor which takes chemical recombination in the cold boundary layer into account, results in an expression for the heat flux from a flame jet to the hot spot of a heated plate. Numerical and experimental validation show good agreement.

Introduction

Flame-impingement heating is a frequently employed method to enhance the heat transfer to a surface. Applications can for instance be found in the glass industry, where glass products are melted, cut and formed using impinging flame jets. It is well reckoned that these jets yield very high heat-transfer coefficients [1], [2], [3], [4]. Experiments have been performed concerning the heat-transfer characteristics of impinging flame jets as well [5], [6], [7], [8], [9], [10].

The flame jets are operated in a laminar as well as a turbulent configuration. Turbulent flame jets can particularly be found in glass furnaces. When the jets are used in smaller geometries to apply the heat locally, however, the flames will be laminar. These flows are highly viscous because of their high temperatures. Since these flames are laminar, mixing of ambient air causing a temperature decrease of the flame is suppressed.

The main heat-transfer mechanism for impinging flame jets is forced convection. Radiation from the flame is negligible because of the very low emissivity of a hot gas layer of small thickness [11], [1], [12]. In the last few decades it became more accepted to increase the amount of oxygen in the oxidizer stream. Enhancing the amount of oxygen in the flame jet results in a higher flame temperature and a higher burning velocity. Not only the convective heat transfer will now be increased, another heat-transfer mechanism called Thermochemical Heat Release (TCHR) will start to play an important role as well [12]. The oxygen-enhanced flames contain a lot of free radicals such as O, H and OH. When the flame impinges on the cold surface of the heated glass product, the radicals will recombine in the boundary layer resulting in an increased heat transfer. Cremers [13], [14], [15] found that the heat transfer coefficient of a methane–oxygen and a hydrogen–oxygen flame can be as much as doubled compared to a chemically frozen mixture. Baukal and Gebhart [16] experimentally found this result as well. Furthermore, they concluded that the peak heat flux for these flames occurs at the stagnation point. This is in contrast with fuel-air flames, where the peak heat flux is shifted away from the stagnation point.

The flow behaviour of flame jets and hot isothermal jets is comparable. According to Viskanta [1], the aerodynamics of a single flame jet is very similar to the aerodynamics of a single isothermal jet. Experiments by van der Meer [11] showed that the axial velocity decays slightly faster for the flame jets than for the isothermal jets, due to the axial temperature decay. The radial velocity gradients at the stagnation point are found to be equal.

Simple analytical expressions for the heat transfer of inert jets are very useful from an engineering point of view when the heat transfer needs to be estimated. Sibulkin derived a semi-analytical relation for the laminar heat transfer of an impinging flow to a body of revolution [17]. This relation has been the basis of most other experimental and theoretical results since [18], [19], [20], [21], [22]. An important limitation of this relation, however, is that it is only applicable for large nozzle-to-plate spacing. Nevertheless, smaller spacing becomes very interesting when the heat flux needs to be increased. Furthermore, for the limit of non-viscous flows an unrealistic heat transfer is predicted using this model. In an earlier study [23] we presented an analytically derived expression for the convective heat flux of a hot inert jet. With this expression it is possible to predict a realistic heat flux for the limit of non-viscous flows. On the other hand, it is only applicable for small nozzle-to-plate spacing. In this paper we will present an extension of the analytical expression, which will be valid for small as well as large nozzle-to-plate spacing. Because of the resemblance in flow behaviour between flame jets and hot isothermal jets, we will focus on hot inert flame jets in the rest of the paper.

First we will show how the analytical expression for the heat flux of a hot inert flame jet, applicable for small nozzle-to-plate distances, is derived. With this expression the local heat flux to the hot spot of the plate can be calculated. The solution is also applicable for a closely staggered array of jets. The derivation is performed for an axi-symmetrical case by solving the conservation equations taking only the most important contributions into account. Since it is not possible to find the full analytical solution, the results will be validated by numerical calculations. Using the numerical results, an extension of the analytically derived expression can be derived to make it valid for large nozzle-to-plate spacing as well. Finally we will show the results of temperature measurements of a quartz plate which was heated by a methane–oxygen as well as a hydrogen–oxygen flame. The experiments were carried out to validate the analytically and numerically obtained heat-transfer expressions.