Heat and Mass Transfer

, Volume 31, Issue 3, pp 185–189

# 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

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

2

line-heat source of strength$$\dot Q_0 (t/t^ \star )\exp ( - \lambda t)$$

21

3

line-heat source of strength$$\dot Q_0 [1 + a\cos (\omega t)]$$

31

# 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.

## Preview

### References

1. 1.
Spraragen, W.;Claussen, G.E.: Temperature distribution during welding — A review of the literature to January 1, 1937. Welding J. Res. Sup. 16 (1937) 4–10Google Scholar
2. 2.
Rosenthal, D.: The theory of moving sources of heat and its application to metal treatments, Trans. Am. Soc. Mech. Engrs. 68 (1946) 849–866Google Scholar
3. 3.
Rohsenow, W.M.;Hartnett, J.P.;Ganic, E.N.: Handbook of Heat Transfer Fundamentals (2nd ed.), New York: McGraw-Hill 1985Google Scholar
4. 4.
Carslaw, H.S.;Jaeger, J.C.: Conduction of Heat in Solids, London: Oxford University Press 1959Google Scholar
5. 5.
Grigull, U.;Sandner, H.: Heat Conduction. (English Translation ed. by J. Kestin) Berlin Heidelberg New York: Springer 1984Google Scholar
6. 6.
Chaudhry, M.A.;Zubair, S.M.: Analytic study of temperature solutions due to gamma-type moving-point-heat sources. Int. J. Heat and Mass Transfer 36 (1993) 1633–1637Google Scholar
7. 7.
Zubair, S.M.;Chaudhry, M.A.: Temperature solutions due to steady, periodic-type, moving-point-heat sources in an infinite medium. Int. Comm. in Heat and Mass Transfer 21 (1994) 207–215Google Scholar
8. 8.
Chaudhry, M.A.;Zubair, S.M.: Generalized incomplete gamma functions with applications. J. of Comp. and Appl. Math. 55 (1994) 99–124
9. 9.
Chaudhry, M.A.;Zubair, S.M.: On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms. J. of Comp. and Appl. Math. 59 (1995) 253–284
10. 10.
Chaudhry, M.A.;Zubair, S.M.: Temperature and heat flux, solutions due to steady and non-steady periodic-type surface temperatures in a semi-infinite solid. Wärme-und Stoffübertragung 29 (1994) 205–210Google Scholar
11. 11.
Zubair, S.M.;Chaudhry, M.A.: Heat Conduction in a semi-infinite solid subject to steady and non-steady periodic-type surface-heat fluxes. Int. J. Heat and Mass Transfer 38 (1995) 3393–3399