Abstract
A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-line-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i)\(\dot Q_1 (t) = \dot Q_0 \exp ( - \lambda t)\), (ii)\(\dot Q_2 (t) = \dot Q_0 (t/t^ \star )\exp ( - \lambda t)\), and\(\dot Q_3 (t) = \dot Q_0 [1 + a\cos (\omega t)]\), whereλ andω are real parameters andt⋆ characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ (α,x;b) and its decompositionsC Γ andS Γ. It is also demonstrated that the present analysis covers the classical temperature solution of a constant strength source under quasi-steady-state situations.
Zusammenfassung
Es wird ein in geschlossener Form beschreibbares Modell zur Berechnung der Temperaturverteilung in einem unendlich ausgedehnten, isotropen Körper mit zeitabhängiger, wandernder, linienförmiger Wärmequelle untersucht, wobei sich die Lösungen auf folgende Zeitfunktionen für die Wärmequelle beziehen: (1)\(\dot Q_1 (t) = \dot Q_0 \exp ( - \lambda t)\); (2)\(\dot Q_2 (t) = \dot Q_0 (t/t^ \star )\exp ( - \lambda t)\) und (3)\(\dot Q_3 (t) = \dot Q_0 [1 + a\cos (\omega t)]\). Hierin sindλ undω reelle Parameter;t⋆ charakterisiert eine Grenzzeit. Die normierten Temperaturfeldlösungen werden als Funktionen einer unvollständigen Gamma-Funktion Γ(α,x;b) und hirer DekomposiertenC Γ undS Γ angegeben. Es läßt sich zeigen, daß die mitgeteilten Lösungen das bekannte Ergebnis für eine Quelle konstanter Energielieferung im quasistationären Fall einschließen.
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Abbreviations
- a :
-
controlling factor of steady-periodic oscillations
- C p :
-
specific heat at constant pressure, [kJ/(kg K)]
- C Γ :
-
decomposition function
- Fo :
-
Fourier number (Fo=αt/r 2)
- k :
-
thermal conductivity, [W/(mK)]
- S Γ :
-
decomposition function
- t :
-
time, [s]
- T :
-
temperature, [K]
- r :
-
distance from the line-heat source, [m]
- u :
-
source velocity, [m/s]
- V :
-
reduced velocity (V=ut/r)
- α :
-
thermal diffusivity (α=k/ρC p ), [m2/s]
- β :
-
dimensionless parameter [β=(V/4Fo)2-τλ/4Fo]
- β 0 :
-
dimensionless parameter [β 0=(V/4Fo)2]
- Γ :
-
generalized incomplete gamma function
- θ :
-
reduced (or dimensionless) temperature
- ρ :
-
density, [kg/m3]
- τλ :
-
reduced (or dimensionless) time (τλ=λt)
- τ ω :
-
reduced (or dimensionless) time (τ ω =ωt)
- 1:
-
line-heat source of strength\(\dot Q_0 \exp ( - \lambda t)\)
- 11:
-
constant strength, quasi-steady case
- 2:
-
line-heat source of strength\(\dot Q_0 (t/t^ \star )\exp ( - \lambda t)\)
- 21:
-
pulse-type strength, quasi-steady case
- 3:
-
line-heat source of strength\(\dot Q_0 [1 + a\cos (\omega t)]\)
- 31:
-
quasi-steady case
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The authors acknowledge the support provided by King Fahd University of Petroleum and Minerals under Research Project MS/GAMMA/171.
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Zubair, S.M., Chaudhry, M.A. Temperature solutions due to time-dependent moving-line-heat sources. Heat and Mass Transfer 31, 185–189 (1996). https://doi.org/10.1007/BF02333318
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DOI: https://doi.org/10.1007/BF02333318