Forced heat convection in cylindrical channels: some problems involving potential and parabolic velocity distribution

Chemical

EnpineeringScience,

19.56, Yol.

S, pp.

18 to 19.

P~~OD~OII PIWELtd.

Forced heat convection Some problems

involving

in cylindrical

potential

and parabolic

LEONARD

channels

:

velocity

distribution

TOPPER*

The Johns Hopkins University,

Baltimore

19, Maryland

(RuXiosd 5 April, 1955) Abstract-Analytical solutions are presented for two kinds of problem in the convection of heat by a fluid. The llrat problem is tbat of the unifom and constant generation of heat wltbin the fluid ; the second is that of temperature equalization when the inlet temperature has one constant value for part of the radius and another value for the rest of the radius. Steady-state solutions are developed for both plug flow and laminar flow ; the wall of the tube is assumed to be isothermal. R&um&-L’auteur pr6sent.e des solutions analytiques de deux sortes de problbmes sur la convection thermique par un fluide : le premier relatif ir la production con&ante et uniforme de chaleur au sein du fluide ; le second concernant l’dgalisation des tempkratures quand la temp&ature internee a une valeur con&ante pour une partie du rayon et une autre valeur pour le reste. 11 donne des solutions ir l’dtat stationnaire soit pour l’tkoulement en piston soit pour l’koulement laminaire, en supposant que la paroi du tube est isotherme.

temperatures are calculated and presented analytically and also graphically. The solution of the first problem should be helpful in estimating temperatures developed in chemical and nuclear reactors, and that of the second problem in estimating the average life of non-uniformities of the temperature in such streams.

TEE merit of the analytical approach to problems in convective heat transfer has been demonstrated in the sixty-five years that have elapsed since the pioneer theoretical studies by GRAETZ The design of equipment, and by L. LORENZ. the prediction of its performance, and the planning and interpretation of experimental research are often conducted more effectively when the implications of the heat equation in the particular problem are clearly stated. Two different kinds of problem in forced convection are solved here. In both of them, a fluid flows through a cylindrical tube. The tube wall is isothermal, the properties of the fluid are independent of temperature, and we assume the fluid velocity to be entirely in the axial direction and to be steady. In the first problem, the fluid inlet temperature is uniform and there is a constant heat generation per unit volume ; in the second, the inlet temperature is uniform at one value for part of the radius and at another value for the rest of the radius. Both constant and parabolic velocity distributions are considered in The steady-state local each of the problems. l

Present Address

I.

UNIFORM

HEAT

GENERATION

The differential equation of heat conduction in a moving medium is

= q/pc + a A’T.

(1)

Assuming symmetry about the axis of the cylinder, neglecting axial conduction of heat 9 at the steady&ate (1) becomes

(2) The Auid enters at x = 0, at the uniform tem-

_: E&o Research and Engineering Co., P.O. Box 121, Linden, New Jersey. 18

LEONARD

TOPPER

perature T,,, and the wtall (I’ = a) is at the constant temperature T,. Define 8 as T - T,. Then the boundary conditions are :

1

at

x =

0, e =

at r = 8,

To - T,

eO =

s

Jo(/3,,w)Jo(&,w) w dzc = o,

(11)

1

(8)

e = 0.

m =/=IL

0

(4)

[JoOLw)]*

w dw =

(12)

f [J#,,]

0

A.

Potential

Flow

1

Here V’ is a constant,

V,, and (2) becomes

Jo(&w) w* dw = JAI%) - 4- JAB,) A

s 0

The homogeneous solution [references

e

e;=2 8”

equation 1, 21

(A = 0)

has

'9

the

JL%,

Jo (&w) exp

(

- g,” +

+$f+ m

Ati

---

ae0aP

and finally, from (7),

+

g(l

- w’). (7)

This automatically satisfies (4); @,, must computed from boundary condition (8) :

+$(I

1

1

The table below is for use in computing three terms of (‘16).

VI=1

2 N,J&?,,w)

[J,(/$,)]’

5 (9) (I) )

0 = E Nn J&&w) exp

m-1

;

and

The values of & are the positive roots of J&B,) = 9, w = (r/s) ; J,, and J, are Bessel functions of the first kind and zero and first order respectively. The solution of (5) has the form

e=e,=

=

B2

-rps)

n 1 2 8

be

(8)

B” 2 40 5.52 8.05

08) (14)

(15)

the hrst

J,uL)

0.520 - o&uJ 0.272

Fig. 1 is a plot of axial temperature (w = 0), from equation (16),using appropriate dimensionless groups. The average (or bulk) temperature at any axial station is another important quantity. It is defined as

Multiply (8) by J@;w) w dw where 8, need not be the same eigen value as /3,,, and integrate from w = 0 to w = 1. For each n, we have the result

1 1

1

e2rw eO

I 0

JoWmz4 w dw =

3,

J‘ ”

JoU4mw) Job%@

1

--

As9 JoUL4 4a U

I

w dw

e,=

O1

1

I

wa dw -

0

The following properties (9) to produce (14) :

I

0

JoVmw)

w dw

I

of Jo me introduced

2vw

(17)

V,dw

0

(9)

Fig. 2 expresses the difference between the axial temperature and the mean temperature.

in

B. Laminar

1

Jo&w) w dw = J&U 849 u

v,dw

(10)

Flow

In laminar flow, V, is a function of w :

V,=ZV,(l 14

-my

(18)

Forced heat convection

in cylindrical

channeb

0.b

:;t~’

0 -0.0

02

04

0.8

0.6

IO

FIG. 1. Axial temperature in a tube. Potential uniform heat generation, isothermal wall.

A8a =o;

bs,

Asp (2) =l; 40

(8)e *0

The differential equation

=4;

eo

n-o

n(8) b *fO-1 lJ

(4)Aaa= 4I

(1):

6.

-

where B,, = 1, B2 = - ,I$ B,

1

=j;

(S)Ass= 0. &O

-zl?)

(22)

The N, of eq. (22)are calculated from eq. (8), since boundary condition (4) is automatically satisfied by (22). At t = 0, (20)

t9=e,=~~oN~R(w,&)+

‘;

(1 -

W2)

(28)

Multiply eq. (28) by R,,, zu (1 - ~0') dw, where m need not be equal to ?t. By R, or R,,, we will mean R (~0,&,,) or R (w, j3,) respectively. Integrate from w = 0 to w = 1, For each n,

=(&s [+%m-,

BZn_* and the &, are the roots of R

AP

=o; (a-&

+ g(l

R h As,) -P,’ - a -z 2 8V,8 )

10

= 0. This function was studied by GRAETZ, and a table of its numerical values follows. The first three zeros are (4): & = 2.704,j31= 6.68,/I2= 10.67. A solution of eq. (19) is

here is

exp (

0.8

0.6

x

lib. 2. Difference between axial and mean temperature. Potential flow, uniform heat generation, isothermal wall:

flow,

with the boundary conditions of eqs. (3) and (4). The homogeneous form of eq. (19) was solved by GRAETZ [2] discussed in the modern literature, [I], [8],[4]), who also presented equation (6). GRAETZfound the solution of the homogeneous form to be

e _=&$

0.4 a

s35-

-ST s”-

(1) -

0.2

(1, j3,J

14

LEOS-MID TOPPER 1

R,~w(I

-rc2)dw=;2

(27)

s

0

1 =&,

I 0

R,R,w(l

--w’)dw

1

s

R,w3(1

1

As’ R, w (1 -’ al*) dw +-la U U

-w*)dw

= w-1

0

IL.-I

I

w* d R,

1

(28)

CM-0 -

&w"(l

-

w2)dw

s IJ

Equations (25) to (28) are due to GR.AETZ; (28) follows from an integration by parts. These lead to

I

These properties of R are used in (24) :

IO-1 1

R,R,w(l

-w*)dw

= 0,

nt

F w*

*n

dR,

(25)

I

0

The first terms of the series solution of (22) are then

94

I

0.3

et- em

e.

0.0

0.2

0.4

0.6

04

I-0

1

I.2

Fro. 8. Axial temperature in a tube. Laminar flow, uniform

Ffo. 4. Dmrence betweenaxialandmefbnteln~~. Lamimu aow, uniform heat generation, imothermal w&

heat generation, isothermal wall: A9 (1) -

a@0

so;

(2) $

= 1:

(8) $

=4;

(4)-

o.2

A9

(1)$=0;

= 6.

A9

(2)118 ~4; 0

&O 16

(t)$

= 6.

Forced heat cow&ion

in

cylindriral channels

6 -=

,-ET2

4

3X

(

8 =

- o.aos+o'"~;f82) R,(w)exp( -22+) (

(

w 3w

1

(34)

Fig. 3, is a plot of axial temperature (W = 0), from equation (aO), using appropriate dimensionFig. 4 expresses the difference less groups. between the axial temperature and the mean temperature (eq. (17)). TEMPERATURE MIXIKG BETWEES CONCEXTRIC STREAMS

3_e= z ’ a*e P

38

1 (s+rosi

with the boundary

1

+

4,

(35)

s a/a

%(8, -

&a = ii [JkL,]'

zc J, (j3, zu) dw

1

(38)

eb)J, +

4 Jl(8”)

And finally,

1

1 (39)

Fig. 5 indicates the approach of the temperature of the fluid to that of the isothermal wall when

(31)

is 0.5.

(a;.~) is 0.5, and

The radial

temperature distribution is plotted at z = 0 and at two values of the dimensionless distance down-

*4t x = 0, e = 8, = T, - T,, (q’s) > I(' > 0

stream,

8 = eb = T, - T,, 1 < z(’< (ah) (32) W)

x

( ) 2.

*

xv,

B. Laminar Flow The problem

-4. Potedal Flow

G is a constant,

(- 8: ;+;j

1

conditions

.4tw=l,e=o

N, J, (8, W) esp

Here

The rate of “ mixing-out ” of non-uniformities of temperature or of composition in a flowing stream is often important to the analysis of processes. The calculations that follow are equivalent to the rate of equalization of temperature due to the diffusion of heat alone. In most practical situations, the initial non-uniformity of temperature induces non-uniform velocities that expedite the temperature equalktion. For this reason, the actual rate of temperature mixing should always be at least as great as indicated here. Consider a stream of fluid entering a cylindrical tube of radius S. At x = 0,the stream temperature is T,, from the axis (r = 0)to r = a (8 > a); it is T, from r = a to I = 8. The wall temperature remains constant at T,, and the assumptions of eq. (1) apply here as well. The differential equation is

“3x

2

n=1

where the 8, arc the positive roots of J, (p,,) = O. The &Vm are determined from eq. (32). Subject to certain restrictions on f(zc) that arc satisfied here, when

(30)

v

a’e I i-- 38

3wz

which has the solution

1476 -

2.

s*

2

and (31) becomes 17

is

38 G (1 -- w2)TX =

3

(

(41)

3”8 + ;,gw b

w2

1

LEOXARD TOPPER

Table 1. W

R, Function

&adz’s

%

Rl

O-0

I

o-1 0.2 0.8 0.4 0.5 0.0 0.7 0.8 0.0 1.0

O*D02 0929 0+40 om8 0*615 0488 0351 0.224 @WY 0

f(w) =

1 O*W6 O.158 - 0*815 - 0392 - 0.142 0.170 0.882 o*aos 0,168 0

Sv,

to eq. (85)

%

(47)

R, (w) exp

8

(

+ 0.588 [1 _ ; (!?)

- 22.8 0:

8 v,

f

81

(d2;@‘D-a/n

R, (w) exp

of eqs. (82) and (88).

B=n~oN,R(w,8.)exp(-fl~~!) where R (1, ##,) = 0.

---

1 om2 Odo6 om4 - 0.110 -O+Ml - 0482 -0ma - om4 0.141 0

with boundary conditions The solution is

The first three terms of the solution are

[4]

Fig. 6 expresses the temperature equalisation in laminar flow in a tube having an isothermal when

(42)

The N, are from eq. (82) :

(a/s)

is 0.5 (the same con-

is 0.5 and

ditions as for Fig. 5).

n;o X R, (4

(48)

Multiply eq. (48) by R, w (1 - w”) dw, where tn need not be equal to n. Integrate from w = 0 to zc = 1. For each 12, using eq. (25) to (28),

+

R,w(

eb

- wS) dw

(44)

a/=

is most conveniently

CO-C+ graphical differentiation graphical integration :

--

of R,,

calculated or else by

0

by a

Fra. 6.

JSqualization of (O/8)

s

R, w (1 -

w*) dw (46)

IJ

18

0.4

0.2

=

0.5,

0.6 W-

0.8

temperature.

I.0

Potential flow,

Foreed heat ronvwtion

in rylindrical

chnnn& NOTATION .7o, J, = Bessel functions of the first kind and zero and Hrst order _V, = coef3cient of rhararteristic various equations

R (w, &) = Graetz’s

function,

I’,,

0.6

0.4

1; = mean velocity,

by eq. (21)

local axial velocity

(I = radial coordinate initial temperature

1 !

I 0

0.2

8 =

0.6 W-

Equalization

of temperature.

of

per unit volume

and

7 = radial coordinate

ix

0.4

at discontinuity

c = heat capacity

\I

q = heat generation time

6.

in

T, T,. Tb. To. T, = temperatute, initial temperature at r < a, initial temperature at u < r
eb

Fro.

defined

fundion

Ltuninar

How,

tube radius

ru =

(r/u)

P =

axial

coordinate

OL= thermal

diffusivity

&=azeroofJoorofH, p = density 8, Bc = T -

‘I;, 0 at zu = 0

7 = time

REFERENCES [I]

DREW,

[9]

Crwrrz,

T. B.;

Tr. A. 1. C’h. E. 1931 26 49.

[3]

JAKOB, hf.; Heof Tran+r,

[4]

SCAENK, J. and DcwonE,

L.; 2. Moth.

P/Q&.

1880 25 318.

Vol. 1, John Wiley, J. II.; Appl. Sci.

Re8.

1949. 1953 A4 39.