Constant Heat Flux

At constant heat flux, from theory the Nusselt numbers are predicted as follows [41]  

For asymptotic values of Re Pr di/L, the mean Nu and friction factors are listed in Table 9.2, summarized from [41, 63], for constant wall temperature (T) and constant heat flux (H) at all four charmel walls. Solutions for different boundary conditions can be found in [41], 

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Constant wall temperature Constant heat flux... 

Limiting Nusselt numbers for slug-flow annub may be predicted (for constant heat flux) from Trefethen (General Discu.s.sion.s on Heat Tran.sfer, London, ASME, New York, 1951, p. 436) ... 

For 0.003 Heat Mass Transfer, 18, 677 (1975)] obtained the correlation... 

At constant heat flux, CHF occurs at lower steam qualities as pressure rises, thus reinforcing the need to provide the very highest quality of waterside conditions in HP industrial and power boilers. 

The studies of Hicks and Baer were based on a constant heat flux to the propellant surface. Price (P8) considers the more important complex cases which have been analyzed, and the reader is referred to his work for the details. 

The condition of constant heat flux at the surface, as opposed to constant surface temperature, is then considered in a later section. 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

The data analysis procedure for the case of constant heat flux is based on the theory describing the response of an infinite line source model (Ingersoll and Plass, 1948 Mogeson, 1983). Although this model is a simplification of the actual experiment, it can successfully be used to derive the geothermal properties (e.g. Kavenaugh, 1984 Austin,. 1998 Gehlin, 1998). 

The complete data series is used to calculate the temperature response, but only certain parts of the experimental data are used to calculate the error. An example of a calibration run is given in Figure 53, the final calibrated TRNSYS model run is shown in Figure 54. Using the first part of the data (with constant heat flux) an estimate of ground thermal conductivity of 2.15 was obtained. Yavatzturk s method yielded an estimate of 2.18, while the estimate obtained with the TRNSYS parameter estimation method was 2.10. 

The first hours were run with a constant heat flux, allowing an estimate with the line source model. Results of the line source model are an estimated ground thermal conductivity of 2.05 W/m K and a borehole resistance of 0.12 K/(W/m), taking into account fluid properties, flow conditions and average shank spacing this borehole resistance equates to a conductivity of the borehole material of about 2.0 W/m K. 

The Geothermal Response Test as developed by us and others has proven important to obtain accurate information on ground thermal properties for Borehole Heat Exchanger design. In addition to the classical line source approach used for the analysis of the response data, parameter estimation techniques employing a numerical model to calculate the temperature response of the borehole have been developed. The main use of these models has been to obtain estimates in the case of non-constant heat flux. Also, the parameter estimation approach allows the inclusion of additional parameters such as heat capacity or shank spacing, to be estimated as well. 

Measurement based on heat flux effects This approach uses local probing devices such as hot-wire anemometers and microthermocouples. The hot-wire anemometer can be either a constant-temperature system or a constant-heat-flux system. Because of the difference in heat transfer between the exposed fluid (liquid or gas)... 

As explained above, one long side of the compartment wall was split into a large number of thin strips and the heat flux to the center of each strip calculated. For a constant heat flux, assuming the wall material to be semi—infinite, the wall surface temperature... 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

More recently, Boley (B9) shows that the method can be extended to find minimum and maximum bounds for ablation rates. The proof is based on a uniqueness theorem deriving from a theorem due to Picone (P3) and leads to the physically obvious result that the higher the heat input rate, the higher the melting rate. The procedure consists of forming a lower bound for the known arbitrary heat input function in terms of a sequence of constant heat flux periods, for which, as noted above, the solution can be written in terms of integrals of the error function. Upper bounds are constructed in a similar manner. 

This is an expression of the fact that at constant heat flux the melt thickness initially grows linearly with time. For small times an exact solution of this problem has been given by Evans et al. (E3), who expanded Ai (tji) in a Maclaurin series and found the first five coefficients by direct substitution ... 

The ignition time for each test, in which a constant heat flux was impinging on the sample, was obtained by examining the second derivate of the mass loss data or the first derivative of the HRR data. Both methods yield similar results after the time delay for transporting the hot gas to the HRR analyzer in the hood is accounted for. A summary of the ignition time for all the tests conducted is... 

The radiation from the ambient is negligible, whereas the absorption coefficient a and the emis-sivity e shown in Figure 19.27 are taken equal to one. Note that for a given material exposed to a constant heat flux 7net 0 remains constant during pyrolysis. 

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

A constant-property fluid flows in a laminar manner in the x direction between two large parallel plates. The same constant heat flux qw is maintained from the plates to the fluid for all x> 0. The fluid temperature is Tin at x = 0. Find an expression of the local Nusselt number by the integral method. What is this expression if Pr= 1 ... 

One operating concern for a rich combustor is the occurrence of high combustor wall temperatures. In a fuel-rich combustor, air cannot be used to film-cool the walls and other techniques (e.g., fin cooling) must be employed. The temperature rise of the primary combustor coolant was measured and normalized to form a heat flux coefficient which included both convective and radiative heat loads. Figure 7 displays the dependence of this heat flux coefficient on primary combustor equivalence ratio. These data were acquired in tests in which the combustor airflow was kept constant. If convective heat transfer were the dominant mechanism a constant heat flux coefficient of approximately 25 Btu/ft -hr-deg F would be expected. The higher values of heat flux and its convex character indicate that radiative heat transfer was an important mechanism. 

Water flows at the rate of 0.5 kg/s in a 2.5-cm-diameter tube having a length of 3 m. A constant heat flux is imposed at the tube wall so that the tube wall temperature is 40°C higher than the water temperature. Calculate the heat transfer and estimate the temperature rise in the water. The water is pressurized so that boiling can not occur. 

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