Radiative Heat Transfer in Finite Cylindrical Enclosures with Nonhomogeneous Participating Media

Results of a numerical solution for radiative heat transfer in homogeneous and nonhomogeneous participating media are presented. The geometry of interest is a finite axisymmetric cylindrical enclosure. The integral formulation for radiative transport is solved by the YIX method. A three-dimensional solution scheme is applied to two-dimensional axisymmetric geometry to simplify kernel calculations and to avoid difficulties associated with treating boundary conditions. As part of the effort to improve modeling capabilities for turbulent jet diffusion flames, predicted distributions for flame temperature and soot volume fraction are used to calculate radiative heat transfer from soot particles in such flames. It is shown that the nonhomogeneity of radiative property has very significant effects. The peak value of the divergence of radiative heat flux could be underestimated by 2 factor of 7 if a mean homogeneous radiative property is used. Since recent studies have shown that scattering by soot agglomerates is significant in flames, the effect of magnitude of scattering is also investigated and found to be nonnegligible

Scale/Analytical Analyses of Freezing and Convective Melting With Internal Heat Generation

Using a scale/analytical analysis approach, we model phase change (melting) for pure materials which generate constant internal heat generation for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. The analysis also consider constant heat flux (in a cylindrical geometry).We show the time scales in which conduction and convection heat transfer dominate

An adjusted analytical solution for thermal design in artificial ground freezing

Artificial ground freezing is a widely used, reliable method for excavation in water-bearing ground. The questions posed in the thermal design of ground freezing projects require solving moving boundary (Stefan) problems. Approximate analytical solutions, such as the ones by St¨ ander1 and Sanger and Sayles,2 have been developed for thermal engineering design and are used by practitioners across the industry. For instance, Sanger & Sayles’ solution is widely used for the single-freeze-pipe problem, but it has proven to be of limited accuracy.3 In the present paper, an adjustment to this formula is proposed based on the re-evaluation of their empirical assumption that the ratio between the temperature penetration depth and the phase-change radius equals a constant value of 3 regardless the conditions. A sensitivity study is performed using a verified numerical model as a benchmark to study several problems with different initial and boundary conditions (initial, phase change and freeze pipe temperatures) and thermal properties of the ground (water content, thermal conductivity and heat capacity). This is done for the freezing times of 10 and 365 days, in order to consider the potential change of the ratio with the freezing time. In this way, a calibrated formula is proposed to find appropriate values of this ratio and a suitable adjustment to Sanger & Sayles’ solution is determined. Adjusting Sanger & Sayles’ solution in this manner, a significantly higher and more consistent accuracy is achieved for different boundary and initial conditions. This accuracy improvement was checked for real conditions from an engineering project, which shows that the adjustment can be useful for thermal problems in engineering design of ground freezing

Linear approach to the orbiting spacecraft thermal problem

We develop a linear method for solving the nonlinear differential equations of a lumped-parameter thermal model of a spacecraft moving in a closed orbit. Our method, based on perturbation theory, is compared with heuristic linearizations of the same equations. The essential feature of the linear approach is that it provides a decomposition in thermal modes, like the decomposition of mechanical vibrations in normal modes. The stationary periodic solution of the linear equations can be alternately expressed as an explicit integral or as a Fourier series. We apply our method to a minimal thermal model of a satellite with ten isothermal parts (nodes) and we compare the method with direct numerical integration of the nonlinear equations. We briefly study the computational complexity of our method for general thermal models of orbiting spacecraft and conclude that it is certainly useful for reduced models and conceptual design but it can also be more efficient than the direct integration of the equations for large models. The results of the Fourier series computations for the ten-node satellite model show that the periodic solution at the second perturbative order is sufficiently accurate.Comment: 20 pages, 11 figures, accepted in Journal of Thermophysics and Heat Transfe

Improved finite element methodology for integrated thermal structural analysis

An integrated thermal-structural finite element approach for efficient coupling of thermal and structural analyses is presented. New thermal finite elements which yield exact nodal and element temperature for one dimensional linear steady state heat transfer problems are developed. A nodeless variable formulation is used to establish improved thermal finite elements for one dimensional nonlinear transient and two dimensional linear transient heat transfer problems. The thermal finite elements provide detailed temperature distributions without using additional element nodes and permit a common discretization with lower order congruent structural finite elements. The accuracy of the integrated approach is evaluated by comparisons with analytical solutions and conventional finite element thermal-structural analyses for a number of academic and more realistic problems. Results indicate that the approach provides a significant improvement in the accuracy and efficiency of thermal stress analysis for structures with complex temperature distributions

Untersuchung von Gasstrahlung in Hochenthalpieströmungen

Radiative heat transfer is analyzed in rocket combustion chambers and in the flow around a re-entry vehicle. To do so, the governing equations of the P1 radiation transport model are derived, afterwards discretized using the Finite Volume Method and finally implemented in the CFD solver NSMB. For spectral integration, different models are combined with the P1 radiation model. For radiative heat transfer in rocket combustion chambers Weighted Sum of Gray Gases Models (WSGGM) are identified for spectral modeling and their governing equations with the P1 model are derived to implement them in NSMB. For radiative heat transfer in re-entry flows, a spectral model is developed based on a Full Spectrum k-Distribution (FSK) using the spectral database PARADE. The model is applicable to nonhomogeneous media with varying temperature and mole fractions. The governing equations of the P1 model in conjunction with this model are derived and the model is also implemented in NSMB. All models for radiative heat transfer are validated in several one-dimensional cases and show good agreement with analytical solutions. The sole P1 model yields an error below 5 %. The combination of the P1 model and the WSGGM gives satisfactory results. The FSK reproduces nearly exact results with errors below 1 % for homogeneous media. For nonhomogeneous media, the Multi Group Full Spectrum Correlated k-Distribution (MGFSCK) reduces devia-tions of the FSK from over 250 % to below 10 %. Radiative transfer in rocket combustion chambers is analyzed using the P1 model and several WSGGM for H2/O2 and CH4/O2 combustion. The results reveal that simple WSGG models yield nearly the same radiative wall heat flux (RWHF) with less computational efforts than more complex WSGGM. Using WSGGM appropriate for nonhomogeneous media decreases the RWHF. An enlarged chamber volume increases the RWHF. The influence of radiation on the flow is investigated in a loosely coupled simulation, revealing a negligible effect. For CH4/O2 combustion the maximum relative RWHF decreases compared to H2/O2 combustion. The maximum local ratio of the RWHF and total wall heat flux (TWHF) is between 8-10 % near the injector face plate while the integrated ratio is below 3 % for both propellant combi-nations. The analysis reveals a small influence of radiation on the heat loads in the combus-tion chambers investigated. The second system analyzed is the re-entry of the FIRE II capsule. Several models in NSMB for the simulation of the flow are improved and tested. With a final set of models, the convec-tive wall heat flux (CWHF) as well as the temperature and species number densities lie within 10 % deviation compared to former numerical investigations of the FIREII flight test. A one-dimensional Line-by-Line (LBL) radiative heat transfer analysis along the stagnation line is done afterwards with PARADE. The deviation of this analysis is below 2% in terms of RWHF at the stagnation point with regard to the flight experiment. The P1 model with the MGFSCK yields good accuracy compared to the LBL results with a reduction in computational effort by a factor of nearly 1000. Concerning RWHF at the stagna-tion point, the error is around 20 %. Concerning divergence of radiative heat flux the error is lower than 30 % over most of the stagnation line. The divergence of radiative heat flux predicted by the P1 model with the MGFSCK for the entire domain is coupled in the total energy equation of NSMB to examine the influence of radiation on the flow. It reveals that the CWHF decreases by a maximum of 10 % and the flow properties do not change by more than 5 %. This concludes a minor influence of radia-tion on the flow for the chosen trajectory point of the FIREII flight test.Der Wärmeübergang durch Strahlung wird in Raketenbrennkammern und in der Strömung um einen Wiedereintrittskörper untersucht. Dazu werden die Gleichungen des P1 Strahlungs-transportmodells mit Hilfe der Methode der Finiten Volumina diskretisiert und in den CFD Löser NSMB implementiert. Zur spektralen Integration werden verschiedene Modelle mit dem P1 Modell kombiniert: Für den Strahlungswärmetransport in Raketenbrennkammern werden geeignete Weighted Sum of Gray Gases Modelle (WSGGM) mit dem P1 Modell ge-koppelt und in NSMB eingebaut. Für den Strahlungswärmetransport in Wiedereintrittsströ-mungen wird ein eigenes Spektralmodell auf Basis der Full Spectrum k-Distribution (FSK) entwickelt und in NSMB implementiert. Alle Strahlungsmodelle werden anhand eindimensionaler Fälle validiert und ergeben eine gute Übereinstimmung mit den analytischen Lösungen. Das P1 Modell weist Abweichungen von unter 5 % auf und auch die Kombination aus P1 Modell und WSGGM ergibt gute Resul-tate. Das FSK Modell reproduziert die nahezu exakten Ergebnisse für homogene Medien mit einem Fehler von weniger als 1 %. Für inhomogene Medien reduziert die Multi Group Full Spectrum Correlated k-Distribution (MGFSCK) die Abweichungen des FSK von über 250 % auf weniger als 10 %. Der Strahlungstransfer in Raketenbrennkammern wird mit dem P1 Modell und mehreren WSGGM für H2/O2 und CH4/O2 Verbrennung analysiert. Die Ergebnisse zeigen, dass einfa-che WSGGM mit weniger Rechenaufwand nahezu denselben Strahlungswandwärmestrom vorhersagen wie komplexere WSGGM. Die Verwendung von WSGGM für inhomogene Me-dien verringert den Strahlungswandwärmestrom, wohingegen eine Vergrößerung des Brenn-kammervolumens ihn erhöht. Der Einfluss der Strahlung auf die Strömung wird im Rahmen einer lose gekoppelten Simulation untersucht und ergibt einen geringen Effekt. In der CH4/O2 Verbrennung verringert sich der maximale relative Strahlungswandwärmestrom im Vergleich zur H2/O2 Verbrennung. Das maximale Verhältnis aus Strahlungswandwärmestrom zu kon-vektivem Wandwärmestrom liegt lokal zwischen 8 und 10 % nahe dem Injektor und integral bei unter 3 % für beide Brennstoffkombinationen. Die Untersuchung ergibt einen geringen Einfluss des Strahlungswandwärmestroms auf die Wärmelasten der Brennkammerwände. Das zweite untersuchte System ist der Wiedereintritt der FIRE II Kapsel in die Erdatmosphä-re. Zur Simulation der Strömung werden verschiedene Modelle in NSMB verbessert und ge-testet. Mit diesen Modellen liegen der konvektive Wandwärmestrom sowie die Temperatur und Teilchendichten nahe an den Ergebnissen voriger Simulationen des FIRE II Wiederein-tritts, mit Abweichungen von unter 10 %. Der Strahlungswärmetransport wird zunächst eindimensional entlang der Staupunktstromlinie mit Hilfe von sehr genauen Line-by-Line (LBL) Spektraldaten untersucht. Die Abweichung des Strahlungswandwärmestroms am Staupunkt zu den Ergebnissen des realen Wiedereintritts beträgt weniger als 2 %. Die Kombination aus P1 Modell und MGFSCK liefert gute Ergeb-nisse im Vergleich zur LBL Untersuchung, bei einer Verringerung des Rechenaufwandes um nahezu den Faktor 1000. Die Abweichung des Strahlungswandwärmestroms zur LBL Rech-nung beträgt ca. 20 %, während die Divergenz des Strahlungswärmestroms im Feld größten-teils Abweichungen von unter 30 % aufweist. Die Divergenz des Strahlungswärmestroms, basierend auf dem P1 Modell und dem MGFSCK, wird in die Erhaltungsgleichung der Totalenergie in NSMB gekoppelt. Dies ver-ringert den konvektiven Wandwärmestrom um 10 % und verändert die Strömungsgrößen um weniger als 5 %, was einen geringen Einfluss der Strahlung auf die Strömung für den gewähl-ten Punkt der FIRE II Wiedereintrittstrajektorie ergibt

STUDIES ON ABLATION OF OBJECTS TRAVERSING AN ATMOSPHERE

Ablation-type thermal protection of objects traversing an atmosphere - earth and mar

Improved finite element methodology for integrated thermal structural analysis

An integrated thermal-structural finite element approach for efficient coupling of thermal and structural analysis is presented. New thermal finite elements which yield exact nodal and element temperatures for one dimensional linear steady state heat transfer problems are developed. A nodeless variable formulation is used to establish improved thermal finite elements for one dimensional nonlinear transient and two dimensional linear transient heat transfer problems. The thermal finite elements provide detailed temperature distributions without using additional element nodes and permit a common discretization with lower order congruent structural finite elements. The accuracy of the integrated approach is evaluated by comparisons with analytical solutions and conventional finite element thermal structural analyses for a number of academic and more realistic problems. Results indicate that the approach provides a significant improvement in the accuracy and efficiency of thermal stress analysis for structures with complex temperature distributions