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Transient heat transfer in the laminar thermal entry region of a pipe: an analytical solution

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Abstract

Attention is directed toward the problem of unsteady convective heat transfer to a fluid flowing inside a pipe in a laminar, fully developed fashion when suddenly, an ambient fluid outside the pipe undergoes a step change in temperature. For the fastest portion of the resultant transient, time domain I, an analytical solution of the governing partial differential thermal energy equation is effected via the Laplace transformation. From this solution, response functions are found for the pipe wall temperature, surface heat flux, and fluid bulk mean temperature as a function of non-dimensional time for a range of values of a parameter which characterizes the heat transfer between the ambient and the pipe.

Comparison of results is made with a recent finite difference solution in the literature and with the standard quasi-steady type of analysis. It is found that the analytical solution presented herein extends and complements the finite difference solution and that the quasi-steady solution can be severely in error in this part of the transient.

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Abbreviations

â :

ρc p R/ρwcpwb Ratio of thermal energy storage capacity of fluid to wall material

b :

pipe wall thickness

C n :

defined by equation (24)

c p , c pw :

specific heat capacity of fluid and pipe wall, respectively

D n :

functions defined by equation (23)

erf, erfc:

error function and complimentary error function, respectively

F :

αt/R 2 Fourier number

g :

1−2S

h :

local surface coefficient of heat transfer between inside of pipe wall and inside flowing fluid

i n erfc:

n th repeated integral of the error function

k :

thermal conductivity of the inside fluid

N :

h(2R)/k Nusselt number

p :

Laplace transform parameter

q w :

local, instantaneous surface heat flux at inside of pipe wall

Q w :

2Rq w /k(T L −T i ) nondimensional surface heat flux

R :

pipe inside radius

S :

UR/k

t :

time

T :

local instantaneous fluid temperature

T B , T L , T i :

bulk mean, ambient, and initial, as well as inlet, temperature, respectively

u, u m :

local and mass average, fluid velocity, respectively

U :

overall heat transmission coefficient between ambient fluid outside of pipe and inside pipe wall

X, Y :

x/R, y/R nondimensional space coordinates along, and radially inward from, the pipe wall, respectively

α :

k/ρc p thermal diffusivity of inside fluid

ρ, ρ w :

mass density of inside fluid and wall, respectively

φ :

(T(x, y, t)−T i )/(T L −T i )

φ w , φ B :

wall, bulk mean value of φ, respectively

References

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Authors and Affiliations

  1. University of Maine, Room 219 Boardman Hall, 04 469, Orono, ME, USA

    James Sucec (Professor of Mechanical Engineering)

Authors
  1. James Sucec
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Sucec, J. Transient heat transfer in the laminar thermal entry region of a pipe: an analytical solution. Appl. Sci. Res. 43, 115–125 (1986). https://doi.org/10.1007/BF00386039

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  • DOI: https://doi.org/10.1007/BF00386039

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