W~irrneund Stofffibertragung

W/irme- und Stofffibertragung 24, 349 352 (1989)

© Springer-Verlag 1989

A simple method for obtaining the fully developed heat transfer coefficient in turbulent pipe flow A. Campo, Miami, Florida, U S A and A. Salazar, Caracas, Venezuela

Abstract. This paper addresses a distinct and direct computational technique for calculating the characteristics of a thermally developed turbulent pipe flow in a circular pipe. The technique seeks to replace a partial differential energy equation into an equivalent ordinary differential energy equation. The latter is valid in the thermally developed region of the pipe. Numerical results show good agreement with experimental observations for gas and water flows over a wide range of Reynolds numbers. Eine einfaehe Methode zur Berechnung des Wiirmeiibertragungskoeffizienten fiir roll entwickelte turbulente Rohrstriimung

y+ Yi z Z~

dimensionless value of y, u~ y/v functions, Eq. (12) dimensionless value of x, x/D functions, Eq. (12)

Greek letters

c~ eH %t t/ 0 2 v

thermal diffusivity eddy diffusivity of heat eddy diffusivity of momentum dimensionless value of r, r/R dimensionless value of T, ( r - Tw)/(Te -- T~.) eigenvalue kinematic viscosity density shear stress

Zusammenfassung. Diese Untersuchung stellt eine direkte M6glichkeit zur Berechnung der Eigenschaften einer thermisch entwickelten turbulenten Rohrstr6mung vor. Die angewandte Technik basiert auf der Ersetzung einer partiellen Differentialgteichung ffir die Energie durch eine /iquivalente einfache Differentialgleichung ffir die Energie. Das beschriebene Verfahren gilt in dem thermisch entwickelten Bereich der R6hre. Die numerisch erhaltenen Ergebnisse zeigen fiber einen weiten Bereich der Reynolds-Zahl gute Ubereinstimmung mit den experimentell erhattenen Ergebnissen fiir Gas und Wasser.

b e w

Nomenclature

1 Introduction

al Cy D E h k

Turbulent pipe flow with heat transfer finds widespread use in practical applications of thermal engineering. Bhatti and Shah [1] have recently compiled a large n u m b e r of correlation equations for the fully developed Nusselt number in a smooth circular pipe. The recomended correlations are the Gnielinski correlation [2] for P r > 0.5 and the N o t t e r Sleicher correlations [3] for P r < 0.1. Alternatively, the above-mentioned problem m a y be also analyzed within the fi'amework of the thermal development of a h y d r o d y n a m i c developed turbulent flow in a pipe.' In this respect, finite-difference techniques are c o m m o n l y used wherein the solution progresses from the inlet to the far downstream region of the pipe. Regardless of the formulation employed (explicit or implicit), very small step sizes in both axial and radial directions of the pipe have to be utilized in order to guarantee satisfactory results. In light of the foregoing, step-by-step calculations of the temperature distribution in the thermal entrance region seems to be in-

rh

Nu p Pr Pr t r R R+ Re

T u u,~ u~ U x y

coefficients, Eq. (12) friction coefficient, Eq. (6) diameter diffusivity function, Eq. (8b) local convection coefficient thermal conductivity mass flow rate local Nusselt number, h D/k pressure Prandtl number, v/~ turbulent Prandtl number radial variable pipe radius dimensionless pipe radius Reynolds number, u,, D/v temperature axial velocity mean velocity shear velocity, (%/~)1/z dimensionless value of u, u/u m axial variable transverse variable

z

Subscripts

mean bulk entrance wall

350

W~irme- und Stoffi~bertragung24 (1989)

evitable in order to compute the fully developed Nusselt number. On the contrary, this paper deals with a computational methodology wherein the applicable energy equation in the region of thermal development (a partial differential equation) is transformed into an equivalent ordinary differential energy equation. The latter is solved at a particular station in the far downstream region of the pipe using a single large axial step. Discussion of the computed results for the fully developed Nusselt number is limited to gas flows (Pr = 0.7) and water flows (Pr = 2.5), although extensions to other fluids are rather straightforward.

2 Basic equations 2.1 Velocity subproblem

Under the assumption of hydrodynamically developed flow, the equation for conservation of momentum in a circular pipe is expressed as dp

dz

dx

dy

....

(1)

where the pressure gradient dp/dx is constant and the shear stress is given by the relation

The numerical integration of Eq. (4) may be performed by a Runge-Kutta technique providing the velocity field U (t/) in the circular tube. This field is a unique function of the Reynolds number because the friction coefficient is C f~2 = 0.023

(6)

Re -°'2

Note that the preceding expression has been adopted here for simplicity, although other more complex expressions could be also implemented. 2.2 Temperature subproblem

Although the aim of this paper is to study the fully developed temperature field, we will start analyzing the thermal entrance region of the pipe. In this broad sense, the equation for conservation of energy assuming constant properties may be written as 8T - _ l ; I r ( c ~ + @ ) S r T l

U 8X

(7)

r

where e and en designate the molecular and the eddy diffusivity of heat, respectively. Now, defining the dimensionless variables 0 -

T--T w

x z =

Te--T,v

-

(8a)

D

du

v/O = (v + eu) dy

(2)

In this equation, eM denotes the eddy diffusivity for momentum and y refers to the wall-based radial coordinate. Upon introducing the dimensionless variables U = u/u,,

(3)

~ = r/R

and a diffusivity function E Pr

E = 1 + Pr(eH/v ) = 1 + ~

(¢u/V)

(8b)

Eq. (7) may be rewritten in dimensionless form as follows 80

1 8 07E80)

U~z-t/St/

at/

(9)

and the watt coordinates y+ _ u ~ y _ U m Y ( C f / 2 ) 1/2

v

The imposed boundary conditions are

R+ __ R ( C f / 2 ) 1/2

v

2

the combination of Eqs. (1)-(3) yields the momentum equation in dimensionless form, namely (1 +

-

dU

eM/v)~ -

=

(4)

~/

This equation is to be solved subject to the condition U = 0 at t/= 1. The eddy diffusivity ei appearing in Eq. (4) varies with y+ and will be modeled by a two-region equation proposed originally by Reichardt and recommended by Petukhov [4]: ~2vx= 0.4 [y+ - 11 tanh(y+/ll)];

0 _< y+ _< 50

(5a)

e~( = (0.4/3) y+ (0.5 + ~72)(1 + t/);

y+>50

(5b)

V

where y+ = ( I - - r / ) R +.

0=l,

z=0

(9a)

80 - - = 0, 8~

t/= 0

(9 b)

0 = 0,

t/= 1.

(9c)

Note that the eM/v distribution appearing in the diffusivity function E has already been described in connection with the velocity subproblem. Furthermore, for heat transfer calculations of fluids having Prandtl numbers in the vicinity of one, it is common practice to use P5 = 0.9 [5]. Correspondingly, the solution of the temperature field 0 (z, q) is parameterized by the Reynolds number Re and the Prandtl number Pr only. This will be explained in detail in the next section. Upon calculation of the velocity and temperature fields, the local Nusselt number constitutes the most important heat transfer result for engineering purposes. This quantity may be determined from its definition Nu . .hD. .

k

2 ~ 0 / ~ In= 1 0b

(10)

A. Campo and A. Salazar: A simple method for obtaining the fully developed heat transfer coefficientin turbulent pipe flow

351

where the dimensionless bulk temperature distribution 0b is given by z*

1

0b 2~ UOrldrl

(11)

=

0

-I ThermQily developed region

qt.._~ O z --

Fig. 1. Axial station in the thermally developed region 3 Coupling of the method of separation of variables and the transversal method of lines

cretization procedure leads to

Invoking the classical method of separation of variables [6], a product solution

0 = ~ aiZ,(z)

Y~(t/)

(12)

i=0

transforms Eq. (9) into a system of ordinary differential equations: Z', + 2~ Z -- 0 Y/'+

(13a)

;i+

Y/+2 i 2~Y~=0

2-2 0

(14)

+

~t/2"

~

=

- -

(15)

2E

Here, it should be mentioned that Z constitutes a weighted average value of the squares of the eigenvalues [7]. At this point, let us assume that the dependence of Zwith z may be neglected in Eqs. (14) and (15) for purposes of integration. Therefore, the solution of Eq. (14) simply becomes 0 = e -Z~~

(16)

once the boundary condition at the entrance, Eq. 0 a) has been used. Consequently, 2-2 may be written explicitly as a function of 0 as 2-2 =

ln[0l

(17)

Z

In view of this, attention may now be turned to the combination of Eqs. (15) and (17). Accordingly, after rearranging terms, it results in

d20 (~ ---

+

E_~)dO [U +

-

2

+

+

-

z* l

0= 0

(19)

In fact, since z* is a fixed number, from a mathematical point of view, the preceding ordinary differential equation governs the temperature profile O(z*, tl) at a pre-determined downstream axial station in the region where the temperature is fully developed. These asymptotic temperature profiles depend solely on the controlling parameters, namely Re and

Pr.

(13b)

Next, multiplying Eqs. (13 a) and (13 b) by ~ and Z i respectively, results in a new system of equations in terms of the dependent variable 0: ~0 Oz

--

ln~0 ]] o =

o

(18)

Let z* designate a dimensionless distance between the entrance z = 0 and any axial station z = z placed in the thermally developed region of the tube (Fig. 1). Thus, by virtue of the transversal method of lines [8], the dimensionless axial coordinate z may be replaced by this fixed quantity z* in the source term of Eq. (18). Consequently, this dis-

4 Numerical procedure

The numerical integration of the equation of conservation of momentum, Eq. (4) may be easily carried out by a RungeKutta algorithm giving the corresponding fully developed velocity profile U(t/) in terms of Re. This velocity profile constitutes the input data for the transformed energy equation eq. (19), which is subject to the radial boundary conditions Eqs. (9b) and (9 c). Hence, the numerical solution of Eq. (19) provides the fully developed temperature distribution 0 (z*, 0) parameterized by Re and Pr, respectively. In spite of the fact that the numerical integration of Eq. (4) could be easily done using a Runge-Kutta algorithm, it is advantageous to reconsider the problem in its overall pattern formulated by a system of ordinary differential equations comprising Eq. (4) for the velocity field and Eq. (19) for the temperature field at a certain station z* in the thermMly developed region. Correspondingly, this coupled system may be readily integrated numerically on a personal computer using a variety of subroutines for boundary value problems. In this paper, the PASVA3 algorithm [9] has been adopted to carry out the numerical calculations. Accordingly, Eqs. (4) and (19) are rewritten in finite-difference form for a pre-selected value of z* and the resulting system of algebraic equations is iteratively solved until an adequate convergence criterion is achieved. In this particular algorithm, the nodal points are clustered closer to the wall in order to resolve the larger velocity and temperature gradients associated to turbulent pipe flows accounting for heat transfer. Ultimately; numerical determination of the fully developed Nusselt number Nu~ as a function of both Reynolds and Prandtl numbers, may be readily accomplished utilizing gqs. (10) and (11).

352

W~irme- und Stofft~bertragung 24 (1989)

5 Additional considerations

5.1 Velocity subprobtem The degree of difficulty in solving the system of ordinary differential equations may be alleviated considerably using an analytical expression for the fully developed velocity in wall coordinates u + and y+. As seen in [1], all authors except Reichardt [10], Spalding [11] and van Driest [12] were forced to employ at least two piecewise expressions in y+ to describe the velocity profile adequately for all values of y +. In this sense, on the basis of convenience, the preferred direct expressions are those of [10, 11], which when inserted into Eq. (9) provides the variable coefficient multiplying the axial temperature gradient 80/8z.

5.2 Temperature subproblem In addition to gas and water flows, the hybrid methodology developed here applies equally well for very high as well as very low Prandtl number fluids. For these extreme cases, appropriate expressions describing the variation of the turbulent Prandtl number P4 with Pr need to be introduced in Eq. (9). 6 Results and discussion In industrial applications, the quality of most practical interest for heat transfer calculations is the asymptotic Nusselt number Nuo~. Therefore, to asses the validity of the direct computational procedure it is appropriate to compare the fully developed results obtained numerically with established experimental data reported in the heat transfer literature. For the comparison, the reliable observations of Volkov and Svanova [13] and Petukhov and Roisen [14] cited in [4] are being used in Fig. 2. The abcissa of this figure ,

400

,

200

o P

t

100 %

o

o

60

8

40

20 o Experimen'Io[ dab [13, 14,] I

"lO 5000

I

I

I I

10000

__

I

ZOO00 Re

I

I

I

50000

I

,I

I,I[

...........

100000

Fig. 2. Comparison of Nuoo with experimental results

ZOO000

corresponds to Re ranging between 5 x 103 and 2 x 105, while the parameter is Pr = 0.7 (gas) and 2.5 (water), respectively. Since correlation equations for this kind of problems have a stated range of applicability of z* >_ 60 to insure fully developed thermal conditions, the numerical calculations have been performed in Eq. (19) for z* = 60. This extremely large quantity, indeed corresponds to the magnitude of the axial interval Az chosen for the numerical calculations of the fully developed turbulent temperature profiles. F r o m the figure, it is clearly seen that the deviations between the experimental data and the numerical predictions are very small for both values of Pr. The excellent agreement serves to support the direct numerical methodology which is especialized to the thermally developed region incorporating large axial steps.

References l. Bhatti, M. S.; Shah, R. K.: Turbulent and transition flow convective heat transfer in ducts. In Handbook of Single-Phase Convective Heat Transfer. (Eds: Kakac, S.; Shah, R. K. and Aung W.) New York: Wiley 1989 2. Gnielinski, V.: New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng. 16 (t976) 359-368 3. Notter, R. H.; Sleicher, C. A.: A solution of the turbulent Graetz Problem III. Fully developed and entry region heat transfer rates. Chem. Eng. Sci. 24 (1972) 1179-1189 4. Petukhov, B. S.: Heat transfer and friction in turbulent pipe flow with variable physical properties. Adv. Heat Transfer 6 (1970) 503-563 5. Kays, W.: Convection heat transfer. New York: McGraw-Hill 1968 6. Churchill, R. V.; Brown, J. W.: Fourier series and boundary value problems. New York: McGraw-Hill 1978 7. Salazar, A.; Campo, A.; Schuler, C.: On a solution of the GraetzNusselt problem and its extension with uniform wall heating. W/irme-Stoffiibertrag, 22 (1988) 141-146 8. Liskovets, O. A.: The method of lines (review). Differential equations 1 (1965) 1308-1323 9. IMSL library reference manual, computer library for FORTRAN subroutines, IMSL Inc., Houston, Texas, USA, 1982 10. Reichardt, H.: Vollstfindige Darsteltung der turbutenten Geschwindigkeitsverteilung in glatten Leitungen, Z. Angew. Math. Mech. 31 (1951) 208-219 1I. Spalding, D. B.: A single formula for the law of the wall. J. Appl. Mech. 28 (1961) 455-458 12. Van Driest, E. R.: On turbulent flow near a wall. J. Aero. Sci. 23 (t956) t007-101t 13. Volkov, P. M.; Svanova, A. V.: Heat transfer and hydrodynamics in elements of power equipment. Tr. Ts KTI 73, 1966 14. Petukhov, B. S.; Roisen, L. I.: Teplofiz. Vysoc. Temperatur 1 (1963) Dr. Antonio Campo Mechanical Engineering Department Florida International University Miami, Florida 33199, USA Abraham Salazar INTEVEP, S.A. Caracas, Venezuela Received November 4, 1988