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Heat-transfer in cylindrical channels involving potential velocity distributions
t3maid~sdeaea.1267,vols,pp.2l3sto266
Heat -transfer
PtagamoapraLtd.
in cylindrical
channels involving distributions
potential
velocity
SHIVA NANDAN SINGH Department of Applied Mathematics Indian Institute of Technology, Kharagpur, India (Rixeiued 1 &t&r
1956)
Abstract-Exact solution are presented for the problem of heat-transfer involving potential velocity distribution, in cylindrical channels, without making the usual assumption that 2 2 2 ( a$ + 5 $ - The temperature distribution is found to depend on the dimensionless Aa=
as in the approximate treatment of
the problem. RBsumB-Les auteurs pcedntent des solutions exactes du transfert de chaleur avec distribitution de vitesses, dons les canaux cylindriques sans faire I’hypothtse habituelle : 1 aT
$
11s trouvant que la distribution de la temp&ature depend des parametres sons dimensions : r x
Vmx As2
~3 is oL
-
(z
x
et
Aa2
1, -?-. -, asVm.9 Q
Comme dans le calcul approximatif du problbme. 1.
INTRoDUOTI~N
TOPPER [l] investigated the temperature distribution for plug flow, in cylindrical channels, kept at constant wall temperature, with uniform and constant generation of heat within the fluid, when the fluid enters the channel at a prescribed temperature. He also considered the problem 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. The solution of the first problem is helpful in estimating temperatures developed in chemical and nuclear reactors, and that of the second problem in estimating average life of non-uniformities of the temperature in such streams. His treatment of these problems is approximate and is based on the assumption that
The
solution
of
the
simplified energy equation reveals that the temperature distribution depends only on nondimensional quantities (r/s, As2/a) and a dimen0 a sionless group .- . sv, s In this paper, we give exact solution of the energy equation corresponding to TOPPER’S problem. We find that the temperature distribution depends on the dimensionless ouantities r il: Viz As2 A, . The dimensionless group ss a a
(
1
a
-_)
1
-,
does not appear in the exact solution. 2.
UNIFORM
HEAT
GENERATION
Consider a stream of fluid entering the channel of radius r = 8. At z = 0, the stream temperature is To, and the uniform wall temperature is T,. The energy equation for steady state, and in case of axial symmetry simplifies as
SEUVA NANDAN SINQH
3.
In this case, the temperature of the entering fluid (at ;I: = 0) is assumed to be T, for part of the radius, say from r = o to r = a, and T, for the rest of the radius (from r = a to r = s). The wall temperature remains constant T,. The energy equation (1) with q = 0 simplifies to
B is defined by the relation
T =8esbL!+ with the boundary
T,,
B = 0, = T,, -
(2)
conditions
9 = 0, for x > 0 and r = 8. T,, for x = 0 and r < s.
In case of potential and (1) becomes
flow, V, is constant
TEMPERATURE MIXING BETWEEN CONCENTRIC STREAMS
(3) (4)
aa8 p+g+s-
aa8
V,,
A’ e = 0,
(11)
where T = edh 8 + T,.
where h = -
(V,/2a).
(6)
The boundary conditions
are
at x = 0, ~9= 8, = T, -
T,, a > r > 0,
e = t$, =
The complete solution of equation (5) is
T, -
(12)
T, s > T > a,
at r = 8, 0 = 0.
(13)
The solution is (7)
Jo
(I$,),
and
(4)
,
n-1
where B,, and Is, are constants. J,, is the Bessel function of the first kind and of zero order. The condition (3) is satisfied if /3,, are zeros of
where 18, are the positive roots of Jo (&) = 0. B, are determined from (12).
gives
1 Bn
s 0
Jo(rs,~)Job’%n~)~~~~ 1
w
where w = (T/S). Multiplying (8) by J, (flm ~0))w c&o, where /3,,, need not be the same eigen value as j?,,, and integrating within the limits zero to 1, we have for each n, As2 1 l-__--, (9)
a00 A1.
=
Jo (&P’)
~a
W&u +
eb
Jo M,,
4
W&L
(15)
s a/'
s 0
-jq&-32 ; (0, - ‘%,)J,(i-6 ;) + ObJl (&)] ’
Hence4, =
(q
And finally
Finally
T-T*_ m,-
2 e-”
co c “=,
&
(10) +--
A s2
(17)
+J~M,)
4 a80 284
{Jl MI))”
I
e
Heat-tmneferin cylindrid channelsinvolving potenti8l velocity diatributiona 4.
DISCUSSION
OF THE
From equations (10) and (17), tbat the temperature
B,, = coeflticient of characteristicfunction
RESULTS
in v8rioueequations.
it is obvious
T, T,, Tb, To, T, = temperature,initial temperatureat
of the fluid at any point
ra!VaAsa
r -z a.
whereas in 1 TOPPER’S example, the temperature depends on
is 8 function of r
a
l
-, -,
K9
8s
a
-
initial
with those of Topper’s, cannot be plotted as
* 80,B8tW=0,~do
NOTATIONS
x = -
A = (q/p), uniformheat generation.
=(T-T,)@. (V,/Za). ....
In conclusion, I thank
Jo, .7, = Besselfunctionof the lhst hind, and
Dr. S. D. NIGAM for
suggesting tbis problem to me. REFEI~ENCE
[l]
8t
a = r8di8l coordin8te8t discontinuityof tempemture. c = heat capacity. q = heat generationper unit volumeand time. r 3: r8di81COOldill8te. 8 = tube radius. w = (r/s). 33= 8Xi8l COOldhl8te. OL= thermaldiffueivity. ~n=QzeroofJ~ p = den&y.
Here it is not possible
of zero and &et order.
1
temper&we.
1
4.
temuemture
V,, V, = mean velocity, loablaxial velocity.
. He plots the temperature distri!, @! ISV, 8 a bution on the axis of the channel (7 = 0), against
because the temperature
initial
a < 7 < 9, initi8l temperature,wall
--
TOPPERLe-onardChem.En@& Sci. 19565 1 18-19.
285
Heat -transfer
PtagamoapraLtd.
in cylindrical
channels involving distributions
potential
velocity
SHIVA NANDAN SINGH Department of Applied Mathematics Indian Institute of Technology, Kharagpur, India (Rixeiued 1 &t&r
1956)
Abstract-Exact solution are presented for the problem of heat-transfer involving potential velocity distribution, in cylindrical channels, without making the usual assumption that 2 2 2 ( a$ + 5 $ - The temperature distribution is found to depend on the dimensionless Aa=
as in the approximate treatment of
the problem. RBsumB-Les auteurs pcedntent des solutions exactes du transfert de chaleur avec distribitution de vitesses, dons les canaux cylindriques sans faire I’hypothtse habituelle : 1 aT
$
11s trouvant que la distribution de la temp&ature depend des parametres sons dimensions : r x
Vmx As2
~3 is oL
-
(z
x
et
Aa2
1, -?-. -, asVm.9 Q
Comme dans le calcul approximatif du problbme. 1.
INTRoDUOTI~N
TOPPER [l] investigated the temperature distribution for plug flow, in cylindrical channels, kept at constant wall temperature, with uniform and constant generation of heat within the fluid, when the fluid enters the channel at a prescribed temperature. He also considered the problem 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. The solution of the first problem is helpful in estimating temperatures developed in chemical and nuclear reactors, and that of the second problem in estimating average life of non-uniformities of the temperature in such streams. His treatment of these problems is approximate and is based on the assumption that
The
solution
of
the
simplified energy equation reveals that the temperature distribution depends only on nondimensional quantities (r/s, As2/a) and a dimen0 a sionless group .- . sv, s In this paper, we give exact solution of the energy equation corresponding to TOPPER’S problem. We find that the temperature distribution depends on the dimensionless ouantities r il: Viz As2 A, . The dimensionless group ss a a
(
1
a
-_)
1
-,
does not appear in the exact solution. 2.
UNIFORM
HEAT
GENERATION
Consider a stream of fluid entering the channel of radius r = 8. At z = 0, the stream temperature is To, and the uniform wall temperature is T,. The energy equation for steady state, and in case of axial symmetry simplifies as
SEUVA NANDAN SINQH
3.
In this case, the temperature of the entering fluid (at ;I: = 0) is assumed to be T, for part of the radius, say from r = o to r = a, and T, for the rest of the radius (from r = a to r = s). The wall temperature remains constant T,. The energy equation (1) with q = 0 simplifies to
B is defined by the relation
T =8esbL!+ with the boundary
T,,
B = 0, = T,, -
(2)
conditions
9 = 0, for x > 0 and r = 8. T,, for x = 0 and r < s.
In case of potential and (1) becomes
flow, V, is constant
TEMPERATURE MIXING BETWEEN CONCENTRIC STREAMS
(3) (4)
aa8 p+g+s-
aa8
V,,
A’ e = 0,
(11)
where T = edh 8 + T,.
where h = -
(V,/2a).
(6)
The boundary conditions
are
at x = 0, ~9= 8, = T, -
T,, a > r > 0,
e = t$, =
The complete solution of equation (5) is
T, -
(12)
T, s > T > a,
at r = 8, 0 = 0.
(13)
The solution is (7)
Jo
(I$,),
and
(4)
,
n-1
where B,, and Is, are constants. J,, is the Bessel function of the first kind and of zero order. The condition (3) is satisfied if /3,, are zeros of
where 18, are the positive roots of Jo (&) = 0. B, are determined from (12).
gives
1 Bn
s 0
Jo(rs,~)Job’%n~)~~~~ 1
w
where w = (T/S). Multiplying (8) by J, (flm ~0))w c&o, where /3,,, need not be the same eigen value as j?,,, and integrating within the limits zero to 1, we have for each n, As2 1 l-__--, (9)
a00 A1.
=
Jo (&P’)
~a
W&u +
eb
Jo M,,
4
W&L
(15)
s a/'
s 0
-jq&-32 ; (0, - ‘%,)J,(i-6 ;) + ObJl (&)] ’
Hence4, =
(q
And finally
Finally
T-T*_ m,-
2 e-”
co c “=,
&
(10) +--
A s2
(17)
+J~M,)
4 a80 284
{Jl MI))”
I
e
Heat-tmneferin cylindrid channelsinvolving potenti8l velocity diatributiona 4.
DISCUSSION
OF THE
From equations (10) and (17), tbat the temperature
B,, = coeflticient of characteristicfunction
RESULTS
in v8rioueequations.
it is obvious
T, T,, Tb, To, T, = temperature,initial temperatureat
of the fluid at any point
ra!VaAsa
r -z a.
whereas in 1 TOPPER’S example, the temperature depends on
is 8 function of r
a
l
-, -,
K9
8s
a
-
initial
with those of Topper’s, cannot be plotted as
* 80,B8tW=0,~do
NOTATIONS
x = -
A = (q/p), uniformheat generation.
=(T-T,)@. (V,/Za). ....
In conclusion, I thank
Jo, .7, = Besselfunctionof the lhst hind, and
Dr. S. D. NIGAM for
suggesting tbis problem to me. REFEI~ENCE
[l]
8t
a = r8di8l coordin8te8t discontinuityof tempemture. c = heat capacity. q = heat generationper unit volumeand time. r 3: r8di81COOldill8te. 8 = tube radius. w = (r/s). 33= 8Xi8l COOldhl8te. OL= thermaldiffueivity. ~n=QzeroofJ~ p = den&y.
Here it is not possible
of zero and &et order.
1
temper&we.
1
4.
temuemture
V,, V, = mean velocity, loablaxial velocity.
. He plots the temperature distri!, @! ISV, 8 a bution on the axis of the channel (7 = 0), against
because the temperature
initial
a < 7 < 9, initi8l temperature,wall
--
TOPPERLe-onardChem.En@& Sci. 19565 1 18-19.
285