Heat Transfer From a Small Isothermal Spanwise Strip on an Insulated Boundary

1963 ◽  
Vol 85 (3) ◽  
pp. 230-235 ◽  
Author(s):  
S. C. Ling
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Heat Conduction ◽  
Detailed Analysis ◽  
Similarity Solution ◽  
Leading Edge ◽  
Trailing Edge ◽  
Shear Field ◽  
Uniform Shear

Detailed analysis of heat transfer of an isothermal spanwise strip in a uniform shear field is presented. Solutions for the leading edge and the trailing edge are obtained by method of relaxation. These solutions are exact in that the streamwise heat conduction term, in the differential equation of energy, is not neglected. The results are compared with the Le´veˆque similarity solution, and the ranges where the solution is valid are defined. A similarity solution for the trailing wake is also presented.

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

scholarly journals Eigenvalues in trailing edge flows

1975 ◽  
Vol 19 (2) ◽  
pp. 210-221
Author(s):  
K. Capell
Keyword(s):  
Asymptotic Formula ◽  
Flat Plate ◽  
Similarity Solution ◽  
Shear Flows ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Uniform Shear

Abstract A wake similarity solution for symmetric uniform shear flows merging at the trailing edge of a flat plate has associated with it an eigenfunction problem which was overlooked by Hakkinen and O'Neil (1967). An asymptotic formula for large eigenvalues is obtained and compared with another such formula related to both the Goldstein (1930) inner wake solution and Tillett's (1968) similarity solution for a jet emerging from a two-dimensional channel.

Effect of Arbitrary Nonsteady Wall Temperature on Incompressible Heat Transfer

1962 ◽  
Vol 84 (4) ◽  
pp. 347-351 ◽  
Author(s):  
T. R. Goodman
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Wall Temperature ◽  
Integral Method ◽  
Nonsteady Heat ◽  
Uniform Heat ◽  
Special Cases ◽  
Title Problem

The title problem is solved using an integral method and ignoring viscous dissipation. A partial differential equation is derived which yields as special cases Lighthill’s non-uniform heat-transfer formula and the nonsteady heat conduction in a slab. The differential equation is then specialized to the nonsteady but uniform heat transfer on a flat plate. Comparisons with other solutions are made when available, and it is shown that the integral method produces accuracy of a few per cent in these limiting cases.

Analysis of Heat Distributing to Ensure Energy Saving

2013 ◽  
Vol 724-725 ◽  
pp. 880-884
Author(s):  
Xiao Chang ◽  
Tian Chi Li ◽  
Yu Bo Xian ◽  
Xin Chao Zhao
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Heat Conduction ◽  
Outer Edge ◽  
Heat Distribution ◽  
Order Partial Differential Equation ◽  
Conduction Model ◽  
Heat Transfer Theory ◽  
Different Shapes ◽  
Save Energy

In this paper, we present a mathematical model which aims to save energy of household appliances, take the brownie pan as example we take advantage of the heat conduction and draw the conclusion that the pan in shape with more edges has more even distribution of heat. In most cases, the shape of rectangle is the best choice to save energy when bake. In Heat conduction model, We raise a conception of doneness to accurately measure how hard the food is being baked, and we handle a second-order partial differential equation based on the heat transfer theory, considering both evenly heating vertically and heat transferring horizontally. We focus on the ratio of how well roasted the edges and the corners are shows the distribution of heat across the outer edge of a pan for pans of different shapes. The variance of all points heating level on the pan shows the evenness of the heat distribution.

Analysis of Flow and Heat Transfer in the End-Wall Region of a Turbine Blade Passage

2000 ◽  
Author(s):  
Yumin Xiao ◽  
R. S. Amano
Keyword(s):  
Heat Transfer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Horseshoe Vortex ◽  
High Heat ◽  
Wall Region ◽  
Trailing Edge ◽  
High Heat Transfer ◽  
Transfer Region ◽  
End Wall

A numerical study has been performed to predict a three-dimensional turbulent flow and end-wall heat transfer in a blade passage. The complex three-dimensional flow in the end-wall region has an important impact on the local heat transfer. The leading edge horseshoe vortex, the leading edge corner vortices, the passage vortex, and the trailing edge wake cause large variations in the entire end-wall region. The heat transfer distributions in the end-wall region are calculated and the mechanism for the high heat transfer region has been revealed. The calculations show that the algebraic turbulence model lacks the ability to predict the heat transfer in the transition region, but it is valid in other flow region. The local high heat transfer downstream of the trailing edge is enhanced by the wake downstream of the trailing edge. The horseshoe vortex results a high heat transfer region near the leading edge and induces the leading edge corner vortices which cause high heat transfer on the end-wall at both sides of blade end-wall corner.

Turbine Heat Flux Measurements: Influence of Slot Injection on Vane Trailing Edge Heat Transfer and Influence of Rotor on Vane Heat Transfer

1985 ◽  
Vol 107 (1) ◽  
pp. 76-83 ◽  
Author(s):  
M. G. Dunn
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Leading Edge ◽  
Temperature Ratio ◽  
Trailing Edge ◽  
Heated Air ◽  
Flux Measurements ◽  
Nozzle Guide ◽  
Nozzle Guide Vanes ◽  
Blowing Parameter

This paper describes the measurement of heat flux distributions obtained for a Garrett TFE 731-2 hp turbine. Measurements were obtained for a full turbine both with and without injection and for the nozzle guide vanes with and without a rotor. A shock tube is used as a short-duration source of heated air and miniature thin-film gages are used to obtain the heat flux measurements. Results are presented for values of the blowing parameter (ρcVc/ρ∞V∞)at SLOT, in the range of 0.8–1.3. The injection gas (air) as a percentage of turbine weight flow, Wc/Wo, was in the range of 2.1–3.5 percent. A comparison is presented between results obtained with the rotor operating at 100 percent of corrected speed and those obtained with the rotor replaced by a row of flow straighteners. The results suggest that: (i) the reduction of heat flux due to injection is a function of the blowing parameter, the temperature ratio, and the physical location relative to the tip or hub endwall and (ii) the presence of the rotor has a significant affect on the vane trailing edge Stanton number, increasing it by 15 to 25 percent. The vane leading edge and midchord regions were generally unaffected.

Heat Transfer and Film Cooling of Blade Tips and Endwalls

2011 ◽  
Vol 134 (4) ◽  
Author(s):  
S. Naik ◽  
C. Georgakis ◽  
T. Hofer ◽  
D. Lengani
Keyword(s):  
Heat Transfer ◽  
High Pressure ◽  
Film Cooling ◽  
Leading Edge ◽  
Pressure Side ◽  
High Pressure Turbine ◽  
Trailing Edge ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Blade Tip

This paper investigates the flow, heat transfer, and film cooling effectiveness of advanced high pressure turbine blade tips and endwalls. Two blade tip configurations have been studied, including a full rim squealer and a partial squealer with leading edge and trailing edge cutouts. Both blade tip configurations have pressure side film cooling and cooling air extraction through dust holes, which are positioned along the airfoil camber line on the tip cavity floor. The investigated clearance gap and the blade tip geometry are typical of that commonly found in the high pressure turbine blades of heavy-duty gas turbines. Numerical studies and experimental investigations in a linear cascade have been conducted at a blade exit isentropic Mach number of 0.8 and a Reynolds number of 9×105. The influence of the coolant flow ejected from the tip dust holes and the tip pressure side film holes has also been investigated. Both the numerical and experimental results showed that there is a complex aerothermal interaction within the tip cavity and along the endwall. This was evident for both tip configurations. Although the global heat transfer and film cooling characteristics of both blade tip configurations were similar, there were distinct local differences. The partial squealer exhibited higher local film cooling effectiveness at the trailing edge but also low values at the leading edge. For both tip configurations, the highest heat transfer coefficients were located on the suction side rim within the midchord region. However, on the endwall, the highest heat transfer rates were located close to the pressure side rim and along most of the blade chord. Additionally, the numerical results also showed that the coolant ejected from the blade tip dust holes partially impinges onto the endwall.

Effect of Adiabatic Co-Planar Extension Surfaces on Wind-Related Solar-Collector Heat Transfer Coefficients

1981 ◽  
Vol 103 (2) ◽  
pp. 268-271 ◽  
Author(s):  
E. M. Sparrow ◽  
S. C. Lau
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Solar Collector ◽  
Leading Edge ◽  
Transfer Coefficients ◽  
Trailing Edge ◽  
Heat Transfer Coefficients ◽  
Planar Extension ◽  
Flat Plate Solar Collector

The heat transfer response to framing the thermally active cover surface of a flat plate solar collector with adiabatic co-planar extension surfaces has been investigated by wind tunnel experiments. Various framing patterns were employed (leading edge and/or trailing edge and/or side edge framing), along with frames of different width. The experiments were performed for various angles of inclination of the plate surface relative to the oncoming airstream and for a range of Reynolds numbers. It was found that the wind-related heat transfer coefficients can be substantially lower when the collector is framed than when it is unframed. An estimate of the possible reduction of the average heat transfer coefficient can be obtained from the equation h/h* = (Lc/Lf)1/2, where h and h* respectively denote the coefficients in the presence and in the absence of the frame. The quantity Lc is a dimension that is characteristic of the thermally active area of the cover surface, while Lf is a characteristic dimension of the outer edges of the frame. With respect to the reduction of the heat transfer coefficient, framing along the side edges appears to be more beneficial than framing along the leading and trailing edges, as is framing along the trailing edge compared with framing along the leading edge.

Heat Transfer and Film Cooling of Blade Tips and Endwalls

2010 ◽  
Author(s):  
S. Naik ◽  
C. Georgakis ◽  
T. Hofer ◽  
D. Lengani
Keyword(s):  
Heat Transfer ◽  
High Pressure ◽  
Film Cooling ◽  
Leading Edge ◽  
Pressure Side ◽  
High Pressure Turbine ◽  
Trailing Edge ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Blade Tip

This paper investigates the flow, heat transfer and film cooling effectiveness of advanced high-pressure turbine blade tips and endwall. Two blade tip configurations have been studied, including a full rim squealer and a partial squealer with a leading edge and trailing edge cut-out. Both blade tip configurations have pressure side film cooling, and cooling air extraction through dust holes which are positioned along the airfoil camber line on the tip cavity floor. The investigated clearance gap and the blade tip geometry are typical of that commonly found in the high pressure turbine blades of heavy-duty gas turbines. Numerical studies and experimental investigations in a linear cascade have been conducted at a blade exit isentropic Mach number of 0.8 and a Reynolds number of 9 × 105. The influence of the coolant flow ejected from the tip dust holes and the tip pressure side film holes has also been investigated. Both the numerical and experimental results showed that there is a complex aero-thermal interaction within the tip cavity and along the endwall. This was evident for both tip configurations. Although, the global heat transfer and film cooling characteristics of both blade tip configurations were similar, there were distinct local differences. The partial squealer exhibited higher local film cooling effectiveness at the trailing edge but also low values at the leading edge. For both tip configurations, the highest heat transfer coefficients were located on the suction side rim within the mid-chord region. However on the endwall, the highest heat transfer rates were located close to the pressure side rim and along most of the blade chord. Additionally, the numerical results also showed that the coolant ejected from the blade tip dust holes partially impinges onto the endwall.

Two-Dimensional Vibrating And Thermal Response of Skin Tissue To Different Types of Heat Sources Considering The Hyperbolic Heat Transfer Equation

2021 ◽  
Author(s):  
Mina Ghanbari ◽  
Ghader Rezazadeh
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Skin Tissue ◽  
Transient Temperature ◽  
Hyperbolic Heat Conduction ◽  
Two Dimensional ◽  
Thermomechanical Response ◽  
Different Types

Abstract Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation of tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of skin displacement and heat transfer with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along the depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

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