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A solution to the turbulent Graetz problem by matched asymptotic expansions—II the case of uniform wall heat flux
Chemical Engineering Science, 197 1, Vol. 26, pp. 559-565:Pergamon
Press.
Printed in Great Britain.
A solution to the turbulent Graetz problem by matched asymptotic expansions-II The case of uniform wall heat flux ROBERT H. NOTTER and C. A. SLEICHER Department of Chemical Engineering, University of Washington, Seattle, Wash. 98105, U.S.A.
(Received 3 March 1970) Abstract-Consideration of the turbulent Graetz problem subject to a wall boundary condition of uniform heat flux leads to a Sturm-Liouville problem that can be solved for the higher eigenvalues by the method of matched asymptotic expansions. Analytical results are presented for the higher eigenvalues and eigenfunctions. The analytical results are compared with computer calculations of the fourth eigenvalues and constants for Prandtl numbers in the liquid metal range (Pr < 0.06). INTRODUCTION
IN PART I of this paper[2]
we developed an analytical solution to the turbulent Graetz problem, i.e., the problem of calculating the temperature distribution and.wall heat flux for a fluid of constant properties in fully developed turbulent flow in a pipe with a step change in wall temperature. The solution, found by the method of matched asymptotic expansions, is valid for intermediate and large values of the eigenvalues that characterize the solution. In the present paper we apply the same method to a similar problem, that of finding the temperature distribution and wall temperature in a pipe in which the wall heat flux has a step change, i.e., the wall heat flux is specified to be uniform for x > 0. This idealized condition is closely approximated by electrical heating of the pipe. The turbulent Graetz problem with uniform wall heat flux has been considered by Sparrow, Hallman, and Siegel [4]. In that paper the authors discuss a method of solution of the governing equation and then report numerical results for three Prandtl numbers (0.7, 10, and 100) and for three Reynolds numbers. The solutions appear in the form of a fully developed temperature profile plus a thermal-entry region term which is a summation of eigenfunctions. In the present paper we develop analytical expressions for these eigenfunctions which, like those for the case of uniform wall-temperature, are valid for
intermediate and large eigenvalues. These solutions will be useful for calculating heat transfer in the thermal entry region, especially for liquid metal flows. In Part III we will present calculations made with these solutions together with numerical calculations of the lower eigenvalues. STATEMENT
OF THE
PROBLEM
A fluid of constant properties is taken to be in fully-developed turbulent pipe flow. The pipe and fluid temperatures are uniform up to the axial position x = 0, at which point a uniform heat flux is imposed from the pipe wall to the fluid. The problem is to find the temperature distribution for all positions downstream If energy dissipation is small and if axial conduction can be neglected, then the energy equation is
(1) which is, of course, the same equation as for the case of uniform wall temperature[2,3]. Here x and r are the dimensionless axial and radial variables, and f and g are the dimensionless velocity and total diffusivity respectively. For the case of uniform wall heat flux, it is useful to define a dimensionless temperature
559
eJwTo) 4oro
(2)
R. H. NOTTER
and C. A. SLEICHER
where q,, is the uniform wall heat flux for x > 0. This definition is adopted from Sparrow et al. [4]. The other quantities written above are defined under Notation. Equation (1) is subject to the following boundary conditions on &x,r): e(o,r) = 0
$ (x,1)
(3a)
= 1
$ (x,0) = 0
(3c)
In addition to the boundary conditions (7) and (S), both e1 and & are subject to a third condition at x = 0, as yet unspecified. Solution for the fully distribution 8,
developed
temperature
The fully developed temperature profile, &, is specified by solving Eq. (5) subject to the boundary conditions (7). As noted by Sparrow et al. [4], a property of the fully-developed profile for the uniform heat flux case is that de,/ax = A, a constant for all values of r. This implies that the solution for e1 is of the form
and with these conditions the problem is specified.
13,(x,r) =
SOLUTION
The solution of this problem is in many ways analagous to the solution of the uniform wall temperature Graetz problem, although there are several features unique to the uniform wall flux case. Most of these are aptly pointed out by Sparrow, Hallman, and Siegel [4] who demonstrated that the relevant form of the energy equation could be solved if the temperature profile were written as 8 = C&+e2 (4)
;ix+h(r) .
(9)
Substitution of Eq. (9) into Eq. (5) gives the following ordinary differential equation for h(r): (10)
subject to the boundary conditions i;‘(l) = 1 h’(0) = 0.
(11)
Equation (10) is analogous to Eq. (5a) of the paper of Sparrow et al.[4] with the coefficients and boundary conditions slightly different because of a difference in the dimensionless variables used in the two studies. For example, the constant2 of-sparrow et al. is 4/Re Pr, while for this analysis A = 4. The quantity of interest here that can be determined from the solution of Eq. (10) is the value of the fully-developed Nusselt number. Sleicher (6) and Tribus[3] have shown that the fullydeveloped Nusselt number for the boundary condition of uniform wall heat flux can be found solely from the knowledge of the solutions for the analogous problem with the boundary con(7) dition of uniform wall temperature. The relation is
where 8, is the fully developed temperature profile and e2 is the entry region temperature profile. At large distances downstream of the thermal entrance f& must approach zero. Substitution of Eq. (4) into Eq. (1) results in two boundary value problems as follows:
with boundary conditions $(x,1)
= 1, J$ (x,0) =o
and
$
(x,1) = 0,
$
(x,0) = 0.
(8) 560
Nu, (uniform flux) =
1 16 C A ,,/h,4
(12)
A solution to the turbulent
where the A, and A, are found by solution of the uniform wall temperature case and are, of course, functions of Re and Pr. (The series converges rapidly; at most three terms suffice for its determination.) The local Nusselt number for the uniform flux case may then be written as Nu, (uniform flux) =
2 2/Nu, + C & e-‘nzr
Graetz
problem-
II
subject to the boundary conditions R;(O) = 0 R;(l) =0
and to the normalizing condition R,(O) = 1. The temperature distribution e2 is then given by
(13)
e2 = g C,R,(r) n=1
where Nu, is given by Eq. (12) and & and ia2 are found from the solution for the entry region temperature distribution ~,(x,v), Eq. (6). This result has been derived, in a* slightly different form, by Sparrow, Hallman, and Siegel. Since Nu, can be found from Eq. (I 2), it is not necessary for the purpose of the present analysis to solve Eq. (10) for i(r). If fully developed temperature profiles were necessary, I?(r) would have to be obtained by numerical integration of Eq. (IO) with appropriate eddy diffusivity and velocity profiles. Fortunately, to calculate the higher eigenvalues in2 and constants A;, necessary to obtain Nusselt numbers and heat transfer entry lengths, it is necessary to obtain only the solution for the entry region temperature distribution tV2(x,r).
exp (-in2x).
(17)
The c;, are constants that can be evaluated from Eqs. (14) and (17) and the orthogonality of the eigenfunctions R, as s 0’ -h(r)rfR,dr
En=
(18)
1
I 0 rf R,2dr
This equation can be cast in a form more convenient for analysis. The function i(r) is given by Eq. (10) and obeys the boundary conditions (11). Using these equations and the properties of the Sturm-Liouville system, it can be shown that
en=_,2
*
(19)
L;[R:(Ul n
Solution for the entry region distribution t12
Some of the analysis here is identical with the analysis of the uniform wall temperature case, and consequently is given little discussion. The part of the analysis that is unique to the case of uniform wall heat flux is described more thoroughly. The entry region profile g2 is specified by solving Eq. (6) subject to the boundary conditions (8). The third boundary condition, imposed at x = 0, gives f3,(O,r) =--h(r) (14)
(16)
The constants & that appear in the Nusselt number expression, Equation (13) are given by A-,= C,R,(l).
(20)
In order to apply an asymptotic solution procedure for the eigenfunctions R,(r), we transform Eq. (15) by the variable change used by Sleicher, Notter, and Crippen[2] so that it becomes
which follows from Eqs. (3a), (4), and (9). Equation (6) is separable and the problem reduces to finding solutions of
$+
[k2-w(a$)]y = 0.
(21)
The variables y and .$ are the redefined eigenfunctions and radial coordinate, and the k2 are the redefined eigenvalues. The boundary condititions on the transformed equation (2 1), y = 0 at 5 = 0 and n, are the same as those for the uni-
(15)
561
R. H. NOTTER
and C. A. SLEICHER
form wall temperature case [2]. This follows from the definition of the eigenfunction y and the fact that f, the dimensionless velocity distribution, vanishes at the wall, e = rr. Consequently, the result of applying the technique of matched asymptotic expansions to obtain expressions for the eigenfunctions for the uniform wall heat flux Graetz problem is the same as that for the uniform wall temperature case [2], but the eigenvalues and constants that appear in the solution will have different values because of the different boundary conditions on the eigenfunctions R,(r), Eq. (16). A brief description and results are given below. Solutions to Eq. (21) are sought in three regions of the pipe: in the pipe center, in the pipe middle, and near the pipe wall. The same asymptotic expansion for the eigenfunction y is employed in all three regions of the pipe: y = k-“2y,,+k-3~21n
k
y1+k-3'2y2+
parameter enters the problem through the expansion for the function w.(t) (see Eq. (21)) in the wall region, and is considered in more detail in Reference [2]. For the purposes of the present analysis it is sufficient to note that c is an unknown parameter, constant for a given Reynolds and Prandtl number. The remaining constants in Eq. (22) may be found by matching. The results are
E,,=
(krr-2n/3)
-2wsin
D, = -
c(~;1’2cos (kr--r/3) (23)
E = c(GgoY2 fl
1
cos
(kn-
2~13)
0(k-5'2)
where we use a three term series and neglect terms of 0(km5j2) and higher. Solutions are obtained for yc, ym, and Y,~, and they are given in Reference [2]. Only the solution for_yW is necessary to calculate the eigenvalues An2 and the constants A, for uniform heat flux. This solution is ylc = ~[Do.h(v)
(krr-n/3)
D,=wsin
The next step in the solution is the determination of the eigenvalues (the k's) and the constant c, and A-,. To determine k the boundary condition R’( 1) = 0 is applied. The procedure is to take Eq. (22) for yW, differentiate as indicated and let r = 1 (or equivalently u = 0). The result, after setting R’( 1) = 0 is
+EJ-,,3(v)]k-"2
+~[0151,3(2))+E1J_-1,3(27)]k-3'2
+& {[ I
In k
D
(22)
2,(cW1,3J-I,3
+c&J-u3J--1,3)d~'+~2
I lnkD 0
k
+lD ’
k
2
+32’3G2’3U+) 2H 1’3r(3)
1
1
GJ,,3(4
V&_,,,(v)
I
k-3'2.
The variable in Eq. (22) is u = k(rr-t). The other quantities in this equation are defined under Notation. Equation (22) contains a constant, c. This
-2,3
E,=0(24)
where terms of O(kp513)are neglected. Inspection of Eq. (24) shows that Do must be of O(k-2’3), which in turn implies that sin (knn/3) is of 0(k-2’3). This is only possible if k=n+++6/r
+E2
k
(25)
where 6 is of 0(kw213).This equation may be used to generate approximations for the trigonometric functions which appear in the expressions for Do, D,, D2, and Eo, and the resulting expressions may be substituted in Eq. (24) to obtain k. The result is
562
A solution
to the turbulent
37’6G 2’31-($) 47rH1’3r(9) (n+9)213’
k=n++-
(26)
In Eq. (26) terms of O(k-‘) and higher are neglected, and (n+# has been substituted for k in the O(k-2’3) correction term. The c, are given by (19) and require an expression for R:(l). This expression was determined for the uniform wall temperature case and is given by 21’2H
dR $
1/6x
l/3
=-3%-F($)&
r=l
Graetz
problem
-
21/231/6~1/6~(~)G1/2r;n113
E +!!&V ’
k
k
2
Reynolds number
n
I0,000 50,000 100,000 500,000
10,000 50,000 500,000
llSp’3
(28) 10,000 50,000 100,000 500,000
0(k-4’3).
The determination of & requires an expression for R,(l). From the definition of the eigenfunction y, and the form off and g near the wall, it is possible to derive 21’2 =
31/6G1/6~1/6~(~)k1/3
& =
-37’6r (9) z-H l’3Gr (S)in513
10,000 50,000 100,000 500,000
32’3r(+) 1+4H”T(3)1;,2’3+ii2ti
l-
i
i
368 481 1746 number 402 656 967 3318 number 484 1045 1709 6620 number
337 867 1490 6230
340 865 1480 5950
calculations flux
(to
4H”T(f)h,2’3
-A,
-A;, asym.
= 0 0.0 147 0.0136 0.0134 0.0130
0.0125 0.0082 OX@68 oxtO
= 0.01 0.0143 0.0115 0+0.56
0.0122 oGt73 oGI25
= 0.02 0.0139 0.0098 0.0079 0.0036
0.0118 0.0066 0.0048 OW20
= 0.04 0.0130 00377 0.00567 0.00213
0.0111 0.0055 0.00388 0.00156
= 0.06 0.0190 0.0094 0.0064 0.0020
0.0159 0.0075 0+052 0.0021
_
---(clnk~+&-)+O(k-413)
32’3r(4)
563
number
478 1037 1704 6765 Prandtl
(29) Expressions for the Ei are substituted into the above equation and the results may be combined with Eqs. (20) and (28) to give
344 345 346 351
396 644 954 3342 Prandtl
10,000 50,000 I00,000 500,000
number
362 469 1737 Prandtl
In Eq. (28) we have retained the 0(k-2/3) correction term but neglected a correction term of
Xaz asym.
338 335 336 339 Prandtl
C = (-l)n+134’3G1’3F($)
R,(l)
x,2 Prandtl
Equation (27) may be differentiated with respect to h, (or equivalently k) to derive an expression for c, in terms of the constants Q and Ei. The trigonometric terms in the expressions for these constants are approximated as before, and the result for c, is 21/2rrgol/2H
Computer and asymptotic 0(k-2’:J)) for uniform wall
.
(27)
R
1.
_!E ’
II
Equations (30) and (26) give the asymptotic expressions for the constants A;, and the eigenvalues, k, needed to calculate entry region heat transfer rates from Eq. (13). In an effort to determine the lowest value of n for which the asymptotic results are valid, computer calculations of x, and I& were obtained by numerical integration of Eq. (15) for several Prandtl numbers in the liquid metal range. The computer results are compared with the asymptotic solutions in Table 1. The asymptotic values for the eigenvalues h, = k/G, calculated from Eq. (26), are in Table
1
-
I*
(30)
+ 0 ( k-4’3)
R. H. NOTTER
and C. A. SLEICHER
good agreement with the computer results for IZ= 4 for Prandtl numbers less than or equal to 0.04 and for IZ= 3 at a Prandtl number of 0.06. The asymptotic results for & are calculated from Eq. (30) with the O(k-‘) terms, containing the unknown parameter c, neglected. The agreement between the computer and asymptotic values of & for n = 4 is somewhat disappointing. As discussed in Reference [2], better agreement can be obtained by choosing the value of c (for a given Reynolds and Prandtl number) to match the asymptotic results for & with the computer results at some arbitrary n, say 12= 3 or 4. This procedure in effect forces the asymptotic results to hold at a lower value of n than might otherwise be the case. This is advantageous even though the intermediate eigenconstants may be slightly in error because the higher the value of II, the larger the error that can be tolerated. Clearly, the values for very large n will be correct irrespective of any error in c. The asymptotic results found here for the uniform wall heat flux Graetz problem, and those found in Reference [2] for the uniform wall temperature case, have been supplemented with computer calculations of the lower eigenvalues and constants for a wide range of Reynolds and Prandtl numbers in Reference [l] and will be the subject of Part III of this series of three papers. The results are used to predict fully developed and entry region heat transfer rates to fluids in turbulent flow pipes. Acknowledgment-The
authors gratefully acknowledge financial support of the National Science Foundation.
the
DiJ$ f fm
%l
go
= dimensionless total diffusivity constant value for g in pipe center region
G
i I1 mg
g
&-)
=
c, c D
564
constant
Ref,/32,
dimensionless
con-
stant Bessel function of order p G hn, redefined eigenvalue Nusselt number, hD/k fully-developed Nusselt number Prandtl number, Pr = v/a constant heat flux at the pipe wall 90 ? dimensional radial coordinate r/r,, dimensionless radial coordinate nth eigenfunction R: Reynolds number, Re = DuavgIv Re s k<, stretched radial coordinate, dimensionless To initial temperature, x < 0 k(n- - 5) = kn- - s = stretched distance V from the wall
JP k NU NU, Pr
WK)
dimensional axial variable 2f/RePrD, dimensionless
axial variable
R, redefined eigenfunction
Y
constants in Nusselt number expression for uniform wall temperature constants in Nusselt number expression for uniform wall heat flux constants in temperature distribution for uniform wall heat flux parameter, a function of Reynolds and Prandtl numbers 2r,, pipe diameter
dr, dimensionless
function in Eq. (9) for the fully developed temperature profile
H
P
NOTATION
in
1 + Pr (E&)
0
X
A,,
arbitrary constants in the expression for yw velocity/bulk average ~I&, = local average velocity Moody friction factor = APIA(LID)Iip
Greek symbols h, Jcl
nth eigenvalue, uniform T, nth eigenvalue, uniform Q, V/f2gd r, redefined
5 $ [ 0
radial variable
dimensionless
A solution to the turbulent Graetz problem-
0
dimensionless w
temperature
profile,
8=
for uniform T, and 19= (T 0 w To)lqoro/k for uniform Q, O1 fully-developed temperature profile for uniform Q,
II
O2 entry region temperature uniform QW Subscripts c m w
profile
center region middle region wall region
REFERENCES [l] NOTTER R. H., Ph.D. Dissertation, University of Washington 1969. [2] SLEICHER C. A., NOTTER R. H. and CRIPPEN M., Chem. Engng SC;. 1970 25 845 [3] SLEICHER C. A. and TRIBUS M., Trans. ASME 1957 79 789. [4] SPARROW E. M., HALLMAN T. M. and SIEGEL R., Appl. scient. Res. 1957 A7 37. R&urn&La consideration du probleme de turbulence Graetz, soumis a une condition limite a la paroi, d’un flux de chaleur uniforme. conduit a un probltme de Sturm-Liouville qui peut itre resolu pour les valeurs propres les plus elevees, par la methode des expansions asymptotiques assorties. Les resultats analytiques sont present& pour les valeurs propres les plus tlevees et les fonctions propres. Les resultats analytiques sont compares aux calculs des quatrieme valeurs propres et constantes pour des nombres de Prandtl dans la gamme metal liquide (Pr < 0.06). ZusammenfassungDie Betrachtung des turbulenten Graetz-Problems in Abhangigkeit von einer Wandgrenzbedingung eines greichformigen Wiirmeflusses fiihrt zu einem Sturm-Louiville Problem, das durch die Methode der angepassten asymptotischen Expansionen fur die hiiheren Eigenwerte gel&t werden kann. Es werden analytische Ergebnisse fiir die hiiheren Eigenwerte und Eigenfunktionen dargelegt. Die analytischen Ergebnisse werden mit Computerberechnungen der vierten Eigenwerte und Konstanten fir Prandtlsche Zahlen im Fliissigmetallbereich (Pr < 0,06) verglichen.
56.5
for
Press.
Printed in Great Britain.
A solution to the turbulent Graetz problem by matched asymptotic expansions-II The case of uniform wall heat flux ROBERT H. NOTTER and C. A. SLEICHER Department of Chemical Engineering, University of Washington, Seattle, Wash. 98105, U.S.A.
(Received 3 March 1970) Abstract-Consideration of the turbulent Graetz problem subject to a wall boundary condition of uniform heat flux leads to a Sturm-Liouville problem that can be solved for the higher eigenvalues by the method of matched asymptotic expansions. Analytical results are presented for the higher eigenvalues and eigenfunctions. The analytical results are compared with computer calculations of the fourth eigenvalues and constants for Prandtl numbers in the liquid metal range (Pr < 0.06). INTRODUCTION
IN PART I of this paper[2]
we developed an analytical solution to the turbulent Graetz problem, i.e., the problem of calculating the temperature distribution and.wall heat flux for a fluid of constant properties in fully developed turbulent flow in a pipe with a step change in wall temperature. The solution, found by the method of matched asymptotic expansions, is valid for intermediate and large values of the eigenvalues that characterize the solution. In the present paper we apply the same method to a similar problem, that of finding the temperature distribution and wall temperature in a pipe in which the wall heat flux has a step change, i.e., the wall heat flux is specified to be uniform for x > 0. This idealized condition is closely approximated by electrical heating of the pipe. The turbulent Graetz problem with uniform wall heat flux has been considered by Sparrow, Hallman, and Siegel [4]. In that paper the authors discuss a method of solution of the governing equation and then report numerical results for three Prandtl numbers (0.7, 10, and 100) and for three Reynolds numbers. The solutions appear in the form of a fully developed temperature profile plus a thermal-entry region term which is a summation of eigenfunctions. In the present paper we develop analytical expressions for these eigenfunctions which, like those for the case of uniform wall-temperature, are valid for
intermediate and large eigenvalues. These solutions will be useful for calculating heat transfer in the thermal entry region, especially for liquid metal flows. In Part III we will present calculations made with these solutions together with numerical calculations of the lower eigenvalues. STATEMENT
OF THE
PROBLEM
A fluid of constant properties is taken to be in fully-developed turbulent pipe flow. The pipe and fluid temperatures are uniform up to the axial position x = 0, at which point a uniform heat flux is imposed from the pipe wall to the fluid. The problem is to find the temperature distribution for all positions downstream If energy dissipation is small and if axial conduction can be neglected, then the energy equation is
(1) which is, of course, the same equation as for the case of uniform wall temperature[2,3]. Here x and r are the dimensionless axial and radial variables, and f and g are the dimensionless velocity and total diffusivity respectively. For the case of uniform wall heat flux, it is useful to define a dimensionless temperature
559
eJwTo) 4oro
(2)
R. H. NOTTER
and C. A. SLEICHER
where q,, is the uniform wall heat flux for x > 0. This definition is adopted from Sparrow et al. [4]. The other quantities written above are defined under Notation. Equation (1) is subject to the following boundary conditions on &x,r): e(o,r) = 0
$ (x,1)
(3a)
= 1
$ (x,0) = 0
(3c)
In addition to the boundary conditions (7) and (S), both e1 and & are subject to a third condition at x = 0, as yet unspecified. Solution for the fully distribution 8,
developed
temperature
The fully developed temperature profile, &, is specified by solving Eq. (5) subject to the boundary conditions (7). As noted by Sparrow et al. [4], a property of the fully-developed profile for the uniform heat flux case is that de,/ax = A, a constant for all values of r. This implies that the solution for e1 is of the form
and with these conditions the problem is specified.
13,(x,r) =
SOLUTION
The solution of this problem is in many ways analagous to the solution of the uniform wall temperature Graetz problem, although there are several features unique to the uniform wall flux case. Most of these are aptly pointed out by Sparrow, Hallman, and Siegel [4] who demonstrated that the relevant form of the energy equation could be solved if the temperature profile were written as 8 = C&+e2 (4)
;ix+h(r) .
(9)
Substitution of Eq. (9) into Eq. (5) gives the following ordinary differential equation for h(r): (10)
subject to the boundary conditions i;‘(l) = 1 h’(0) = 0.
(11)
Equation (10) is analogous to Eq. (5a) of the paper of Sparrow et al.[4] with the coefficients and boundary conditions slightly different because of a difference in the dimensionless variables used in the two studies. For example, the constant2 of-sparrow et al. is 4/Re Pr, while for this analysis A = 4. The quantity of interest here that can be determined from the solution of Eq. (10) is the value of the fully-developed Nusselt number. Sleicher (6) and Tribus[3] have shown that the fullydeveloped Nusselt number for the boundary condition of uniform wall heat flux can be found solely from the knowledge of the solutions for the analogous problem with the boundary con(7) dition of uniform wall temperature. The relation is
where 8, is the fully developed temperature profile and e2 is the entry region temperature profile. At large distances downstream of the thermal entrance f& must approach zero. Substitution of Eq. (4) into Eq. (1) results in two boundary value problems as follows:
with boundary conditions $(x,1)
= 1, J$ (x,0) =o
and
$
(x,1) = 0,
$
(x,0) = 0.
(8) 560
Nu, (uniform flux) =
1 16 C A ,,/h,4
(12)
A solution to the turbulent
where the A, and A, are found by solution of the uniform wall temperature case and are, of course, functions of Re and Pr. (The series converges rapidly; at most three terms suffice for its determination.) The local Nusselt number for the uniform flux case may then be written as Nu, (uniform flux) =
2 2/Nu, + C & e-‘nzr
Graetz
problem-
II
subject to the boundary conditions R;(O) = 0 R;(l) =0
and to the normalizing condition R,(O) = 1. The temperature distribution e2 is then given by
(13)
e2 = g C,R,(r) n=1
where Nu, is given by Eq. (12) and & and ia2 are found from the solution for the entry region temperature distribution ~,(x,v), Eq. (6). This result has been derived, in a* slightly different form, by Sparrow, Hallman, and Siegel. Since Nu, can be found from Eq. (I 2), it is not necessary for the purpose of the present analysis to solve Eq. (10) for i(r). If fully developed temperature profiles were necessary, I?(r) would have to be obtained by numerical integration of Eq. (IO) with appropriate eddy diffusivity and velocity profiles. Fortunately, to calculate the higher eigenvalues in2 and constants A;, necessary to obtain Nusselt numbers and heat transfer entry lengths, it is necessary to obtain only the solution for the entry region temperature distribution tV2(x,r).
exp (-in2x).
(17)
The c;, are constants that can be evaluated from Eqs. (14) and (17) and the orthogonality of the eigenfunctions R, as s 0’ -h(r)rfR,dr
En=
(18)
1
I 0 rf R,2dr
This equation can be cast in a form more convenient for analysis. The function i(r) is given by Eq. (10) and obeys the boundary conditions (11). Using these equations and the properties of the Sturm-Liouville system, it can be shown that
en=_,2
*
(19)
L;[R:(Ul n
Solution for the entry region distribution t12
Some of the analysis here is identical with the analysis of the uniform wall temperature case, and consequently is given little discussion. The part of the analysis that is unique to the case of uniform wall heat flux is described more thoroughly. The entry region profile g2 is specified by solving Eq. (6) subject to the boundary conditions (8). The third boundary condition, imposed at x = 0, gives f3,(O,r) =--h(r) (14)
(16)
The constants & that appear in the Nusselt number expression, Equation (13) are given by A-,= C,R,(l).
(20)
In order to apply an asymptotic solution procedure for the eigenfunctions R,(r), we transform Eq. (15) by the variable change used by Sleicher, Notter, and Crippen[2] so that it becomes
which follows from Eqs. (3a), (4), and (9). Equation (6) is separable and the problem reduces to finding solutions of
$+
[k2-w(a$)]y = 0.
(21)
The variables y and .$ are the redefined eigenfunctions and radial coordinate, and the k2 are the redefined eigenvalues. The boundary condititions on the transformed equation (2 1), y = 0 at 5 = 0 and n, are the same as those for the uni-
(15)
561
R. H. NOTTER
and C. A. SLEICHER
form wall temperature case [2]. This follows from the definition of the eigenfunction y and the fact that f, the dimensionless velocity distribution, vanishes at the wall, e = rr. Consequently, the result of applying the technique of matched asymptotic expansions to obtain expressions for the eigenfunctions for the uniform wall heat flux Graetz problem is the same as that for the uniform wall temperature case [2], but the eigenvalues and constants that appear in the solution will have different values because of the different boundary conditions on the eigenfunctions R,(r), Eq. (16). A brief description and results are given below. Solutions to Eq. (21) are sought in three regions of the pipe: in the pipe center, in the pipe middle, and near the pipe wall. The same asymptotic expansion for the eigenfunction y is employed in all three regions of the pipe: y = k-“2y,,+k-3~21n
k
y1+k-3'2y2+
parameter enters the problem through the expansion for the function w.(t) (see Eq. (21)) in the wall region, and is considered in more detail in Reference [2]. For the purposes of the present analysis it is sufficient to note that c is an unknown parameter, constant for a given Reynolds and Prandtl number. The remaining constants in Eq. (22) may be found by matching. The results are
E,,=
(krr-2n/3)
-2wsin
D, = -
c(~;1’2cos (kr--r/3) (23)
E = c(GgoY2 fl
1
cos
(kn-
2~13)
0(k-5'2)
where we use a three term series and neglect terms of 0(km5j2) and higher. Solutions are obtained for yc, ym, and Y,~, and they are given in Reference [2]. Only the solution for_yW is necessary to calculate the eigenvalues An2 and the constants A, for uniform heat flux. This solution is ylc = ~[Do.h(v)
(krr-n/3)
D,=wsin
The next step in the solution is the determination of the eigenvalues (the k's) and the constant c, and A-,. To determine k the boundary condition R’( 1) = 0 is applied. The procedure is to take Eq. (22) for yW, differentiate as indicated and let r = 1 (or equivalently u = 0). The result, after setting R’( 1) = 0 is
+EJ-,,3(v)]k-"2
+~[0151,3(2))+E1J_-1,3(27)]k-3'2
+& {[ I
In k
D
(22)
2,(cW1,3J-I,3
+c&J-u3J--1,3)d~'+~2
I lnkD 0
k
+lD ’
k
2
+32’3G2’3U+) 2H 1’3r(3)
1
1
GJ,,3(4
V&_,,,(v)
I
k-3'2.
The variable in Eq. (22) is u = k(rr-t). The other quantities in this equation are defined under Notation. Equation (22) contains a constant, c. This
-2,3
E,=0(24)
where terms of O(kp513)are neglected. Inspection of Eq. (24) shows that Do must be of O(k-2’3), which in turn implies that sin (knn/3) is of 0(k-2’3). This is only possible if k=n+++6/r
+E2
k
(25)
where 6 is of 0(kw213).This equation may be used to generate approximations for the trigonometric functions which appear in the expressions for Do, D,, D2, and Eo, and the resulting expressions may be substituted in Eq. (24) to obtain k. The result is
562
A solution
to the turbulent
37’6G 2’31-($) 47rH1’3r(9) (n+9)213’
k=n++-
(26)
In Eq. (26) terms of O(k-‘) and higher are neglected, and (n+# has been substituted for k in the O(k-2’3) correction term. The c, are given by (19) and require an expression for R:(l). This expression was determined for the uniform wall temperature case and is given by 21’2H
dR $
1/6x
l/3
=-3%-F($)&
r=l
Graetz
problem
-
21/231/6~1/6~(~)G1/2r;n113
E +!!&V ’
k
k
2
Reynolds number
n
I0,000 50,000 100,000 500,000
10,000 50,000 500,000
llSp’3
(28) 10,000 50,000 100,000 500,000
0(k-4’3).
The determination of & requires an expression for R,(l). From the definition of the eigenfunction y, and the form off and g near the wall, it is possible to derive 21’2 =
31/6G1/6~1/6~(~)k1/3
& =
-37’6r (9) z-H l’3Gr (S)in513
10,000 50,000 100,000 500,000
32’3r(+) 1+4H”T(3)1;,2’3+ii2ti
l-
i
i
368 481 1746 number 402 656 967 3318 number 484 1045 1709 6620 number
337 867 1490 6230
340 865 1480 5950
calculations flux
(to
4H”T(f)h,2’3
-A,
-A;, asym.
= 0 0.0 147 0.0136 0.0134 0.0130
0.0125 0.0082 OX@68 oxtO
= 0.01 0.0143 0.0115 0+0.56
0.0122 oGt73 oGI25
= 0.02 0.0139 0.0098 0.0079 0.0036
0.0118 0.0066 0.0048 OW20
= 0.04 0.0130 00377 0.00567 0.00213
0.0111 0.0055 0.00388 0.00156
= 0.06 0.0190 0.0094 0.0064 0.0020
0.0159 0.0075 0+052 0.0021
_
---(clnk~+&-)+O(k-413)
32’3r(4)
563
number
478 1037 1704 6765 Prandtl
(29) Expressions for the Ei are substituted into the above equation and the results may be combined with Eqs. (20) and (28) to give
344 345 346 351
396 644 954 3342 Prandtl
10,000 50,000 I00,000 500,000
number
362 469 1737 Prandtl
In Eq. (28) we have retained the 0(k-2/3) correction term but neglected a correction term of
Xaz asym.
338 335 336 339 Prandtl
C = (-l)n+134’3G1’3F($)
R,(l)
x,2 Prandtl
Equation (27) may be differentiated with respect to h, (or equivalently k) to derive an expression for c, in terms of the constants Q and Ei. The trigonometric terms in the expressions for these constants are approximated as before, and the result for c, is 21/2rrgol/2H
Computer and asymptotic 0(k-2’:J)) for uniform wall
.
(27)
R
1.
_!E ’
II
Equations (30) and (26) give the asymptotic expressions for the constants A;, and the eigenvalues, k, needed to calculate entry region heat transfer rates from Eq. (13). In an effort to determine the lowest value of n for which the asymptotic results are valid, computer calculations of x, and I& were obtained by numerical integration of Eq. (15) for several Prandtl numbers in the liquid metal range. The computer results are compared with the asymptotic solutions in Table 1. The asymptotic values for the eigenvalues h, = k/G, calculated from Eq. (26), are in Table
1
-
I*
(30)
+ 0 ( k-4’3)
R. H. NOTTER
and C. A. SLEICHER
good agreement with the computer results for IZ= 4 for Prandtl numbers less than or equal to 0.04 and for IZ= 3 at a Prandtl number of 0.06. The asymptotic results for & are calculated from Eq. (30) with the O(k-‘) terms, containing the unknown parameter c, neglected. The agreement between the computer and asymptotic values of & for n = 4 is somewhat disappointing. As discussed in Reference [2], better agreement can be obtained by choosing the value of c (for a given Reynolds and Prandtl number) to match the asymptotic results for & with the computer results at some arbitrary n, say 12= 3 or 4. This procedure in effect forces the asymptotic results to hold at a lower value of n than might otherwise be the case. This is advantageous even though the intermediate eigenconstants may be slightly in error because the higher the value of II, the larger the error that can be tolerated. Clearly, the values for very large n will be correct irrespective of any error in c. The asymptotic results found here for the uniform wall heat flux Graetz problem, and those found in Reference [2] for the uniform wall temperature case, have been supplemented with computer calculations of the lower eigenvalues and constants for a wide range of Reynolds and Prandtl numbers in Reference [l] and will be the subject of Part III of this series of three papers. The results are used to predict fully developed and entry region heat transfer rates to fluids in turbulent flow pipes. Acknowledgment-The
authors gratefully acknowledge financial support of the National Science Foundation.
the
DiJ$ f fm
%l
go
= dimensionless total diffusivity constant value for g in pipe center region
G
i I1 mg
g
&-)
=
c, c D
564
constant
Ref,/32,
dimensionless
con-
stant Bessel function of order p G hn, redefined eigenvalue Nusselt number, hD/k fully-developed Nusselt number Prandtl number, Pr = v/a constant heat flux at the pipe wall 90 ? dimensional radial coordinate r/r,, dimensionless radial coordinate nth eigenfunction R: Reynolds number, Re = DuavgIv Re s k<, stretched radial coordinate, dimensionless To initial temperature, x < 0 k(n- - 5) = kn- - s = stretched distance V from the wall
JP k NU NU, Pr
WK)
dimensional axial variable 2f/RePrD, dimensionless
axial variable
R, redefined eigenfunction
Y
constants in Nusselt number expression for uniform wall temperature constants in Nusselt number expression for uniform wall heat flux constants in temperature distribution for uniform wall heat flux parameter, a function of Reynolds and Prandtl numbers 2r,, pipe diameter
dr, dimensionless
function in Eq. (9) for the fully developed temperature profile
H
P
NOTATION
in
1 + Pr (E&)
0
X
A,,
arbitrary constants in the expression for yw velocity/bulk average ~I&, = local average velocity Moody friction factor = APIA(LID)Iip
Greek symbols h, Jcl
nth eigenvalue, uniform T, nth eigenvalue, uniform Q, V/f2gd r, redefined
5 $ [ 0
radial variable
dimensionless
A solution to the turbulent Graetz problem-
0
dimensionless w
temperature
profile,
8=
for uniform T, and 19= (T 0 w To)lqoro/k for uniform Q, O1 fully-developed temperature profile for uniform Q,
II
O2 entry region temperature uniform QW Subscripts c m w
profile
center region middle region wall region
REFERENCES [l] NOTTER R. H., Ph.D. Dissertation, University of Washington 1969. [2] SLEICHER C. A., NOTTER R. H. and CRIPPEN M., Chem. Engng SC;. 1970 25 845 [3] SLEICHER C. A. and TRIBUS M., Trans. ASME 1957 79 789. [4] SPARROW E. M., HALLMAN T. M. and SIEGEL R., Appl. scient. Res. 1957 A7 37. R&urn&La consideration du probleme de turbulence Graetz, soumis a une condition limite a la paroi, d’un flux de chaleur uniforme. conduit a un probltme de Sturm-Liouville qui peut itre resolu pour les valeurs propres les plus elevees, par la methode des expansions asymptotiques assorties. Les resultats analytiques sont present& pour les valeurs propres les plus tlevees et les fonctions propres. Les resultats analytiques sont compares aux calculs des quatrieme valeurs propres et constantes pour des nombres de Prandtl dans la gamme metal liquide (Pr < 0.06). ZusammenfassungDie Betrachtung des turbulenten Graetz-Problems in Abhangigkeit von einer Wandgrenzbedingung eines greichformigen Wiirmeflusses fiihrt zu einem Sturm-Louiville Problem, das durch die Methode der angepassten asymptotischen Expansionen fur die hiiheren Eigenwerte gel&t werden kann. Es werden analytische Ergebnisse fiir die hiiheren Eigenwerte und Eigenfunktionen dargelegt. Die analytischen Ergebnisse werden mit Computerberechnungen der vierten Eigenwerte und Konstanten fir Prandtlsche Zahlen im Fliissigmetallbereich (Pr < 0,06) verglichen.
56.5
for