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Transient thermal entrance heat transfer in laminar pipe flows with step change in pumping pressure

Ausgleichsvorgang des Wärmeübergangs im thermischen Einlauf laminarer Rohrströmung bei plötzlicher Änderung des Pumpendrucks

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Abstract

Analysis is made for the transient heat transfer phenomena in the thermal entrance region of laminar pipe flows. The transient results from both the change in flow field, a step change in pressure gradient from zero to a fixed value, and the change in thermal field, a step change in the inlet temperature. An exponential scheme has been employed to solve the energy equation with the presence of axial heat conduction in the fluid. In order to demonstrate the results more clearly, a modified Nusselt number is introduced. The unsteady axial variations of conventional Nusselt number, modified Nusselt number, bulk fluid temperature and pipe wall temperature are presented for water and air over a wide range of outside heat transfer coefficients. It is observed that the outside heat transfer coefficient has a significant influences on the transient heat transfer processes. The results can be comprehensively interpreted by the interactions among the axial convection, axial diffusion, and radial diffusion.

Zusammenfassung

Ausgleichsvorgänge des Wärmeübergangs im thermischen Einlauf einer laminaren Rohrströmung werden analysiert. Die Änderung des Wärmeübergangs resultiert aus Änderungen der Strömlings- und Temperaturfelder. Erstere werden erzwungen durch Änderung des Druckgradienten von Null auf einen bestimmten Wert, letztere durch plötzliche Änderung der Eintrittstemperatur. Eine Exponentialmethode zur Lösung der Energiegleichung wird angewandt unter Berücksichtigung der axialen Wärmeleitung in der Flüssigkeit. Für die bessere Darstellung der Ergebnisse wird eine modifizierte Nusselt-Zahl eingeführt. Die instationare axiale Änderung der konventionellen und der modifizierten Nusselt-Zahlen, und der mittleren Flüssigkeits- und Wandtemperaturen werden für Wasser und Luft angegeben. Es wird gezeigt, daß der äußere Wärmeübergang die Ausgleichsvorgänge wesentlich beeinflußt. Die Ergebnisse werden erklärt durch das Zusammenwirken der axialen Konvektion, der axialen Wärmeleitung und des radialen Wärmeleittransports.

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Abbreviations

a ij :

coefficients; Eq. (13)

Aw :

cross sectional area of pipe wall

A f :

cross sectional area of fluid flow

c p :

specific heat of fluid

c pw :

specific heat of the pipe wall material

h :

local heat transfer coefficient inside the pipe; Eq. (15)

h e :

modified local heat transfer coefficient inside the pipe; Eq. (17)

ii,j, m :

indices for finite difference discretization

J :

total number of grid points in radial direction

J 0 :

Bessel function of the first kind of order zero

J 1 :

Bessel function of the first kind of order one

k :

thermal conductivity of the fluid in the pipe

k w :

thermal conductivity of the pipe wall material

Nu :

conventional local Nusselt number;h (2R)/k

Nu e :

modified local Nusselt number;h e (2R)/k

Nu 0 :

outside Nusselt number; equation (11)

p :

fluid pressure in the pipe

Pe :

Peclet number;u m (2R)/α.

Pr :

Prandtl number;ν/α

q″ w :

wall heat flux

r :

radial coordinate

R :

pipe radius

t :

time

T :

temperature

u :

axial velocity

U :

outside heat transfer coefficient; Eq. (5)

gv :

dimensionless axial velocity; Eq. (1)

x :

axial coordinate

α :

thermal diffusivity of the fluid in the pipe

Δη :

dimensionless radial interval

Δξ :

dimensionless axial interval

Δτ :

dimensionless time step

η :

dimensionless radial coordinate

θ :

dimensionless temperature difference

λ n :

eigenvalue; Eq. (3)

μ :

dynamic viscosity of the fluid in the pipe

ν :

kinematic viscosity of the fluid in the pipe

ϱ :

fluid density

ϱ w :

density of the pipe wall material

ξ :

dimensionless axial coordinate

ξ f :

coolant front due to convection only; Eq. (20)

τ:

dimensionless time

a :

ambient

b :

bulk

e :

entrance

m :

mean

o :

outside

w :

wall

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Author notes

  1. T. F. Lin

    Present address: Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, New York

Authors and Affiliations

  1. School of Mechanical Engineering, Purdue University, West Lafayette, Indiana

    T. F. Lin

  2. School of Mechanical Engineering, Purdue University, West Lafayette, Indiana

    K. H. Hawks &  W. Leidenfrost

Authors
  1. T. F. Lin
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  2. K. H. Hawks
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  3. W. Leidenfrost
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Lin, T.F., Hawks, K.H. & Leidenfrost, W. Transient thermal entrance heat transfer in laminar pipe flows with step change in pumping pressure. Wärme- und Stoffübertragung 17, 201–209 (1983). https://doi.org/10.1007/BF01002363

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