Heat and Mass Transfer

, Volume 31, Issue 3, pp 185–189 | Cite as

Temperature solutions due to time-dependent moving-line-heat sources

  • S. M. Zubair
  • M. A. Chaudhry
Originals

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 decompositionsCΓ andSΓ. It is also demonstrated that the present analysis covers the classical temperature solution of a constant strength source under quasi-steady-state situations.

Keywords

Temperature Distribution Fluid Dynamics Generalize Representation Apply Physic Transport Phenomenon 

Nomenclature

a

controlling factor of steady-periodic oscillations

Cp

specific heat at constant pressure, [kJ/(kg K)]

CΓ

decomposition function

Fo

Fourier number (Fo=αt/r2)

k

thermal conductivity, [W/(mK)]

SΓ

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)

Greek symbols

α

thermal diffusivity (α=k/ρC p ), [m2/s]

β

dimensionless parameter [β=(V/4Fo)2λ/4Fo]

β0

dimensionless parameter [β0=(V/4Fo)2]

Γ

generalized incomplete gamma function

θ

reduced (or dimensionless) temperature

ρ

density, [kg/m3]

τλ

reduced (or dimensionless) time (τλt)

τω

reduced (or dimensionless) time (τ ω =ωt)

Subscripts

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

Temperaturfeldermittlung für zeitabhängige, wandernde, linienförmige Wärmequellen

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 DekomposiertenCΓ undSΓ 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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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • S. M. Zubair
    • 1
  • M. A. Chaudhry
    • 2
  1. 1.Mechanical Engineering DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Mathematical Sciences DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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