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The extended Graetz problem with piecewise constant wall temperature for laminar and turbulent flows through a concentric annulus
International Journal of Thermal Sciences 54 (2012) 89e97
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The extended Graetz problem with piecewise constant wall temperature for laminar and turbulent flows through a concentric annulus B. Weigand, K. Eisenschmidt* Institute of Aerospace Thermodynamics, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2011 Received in revised form 6 December 2011 Accepted 7 December 2011 Available online 11 January 2012
The effect of axial heat conduction on the heat transfer is of importance if the flow Peclet number is small or if the axial extension of the heating zone is small. This paper therefore presents an analytical solution to the extended Graetz problem in a concentric annulus with piecewise constant temperature at the outer wall. The solution is based on a selfadjoint formalism which results from the decomposition of the elliptic energy equation into two first-order partial differential equations. The gained solution is exact and simple. The need of regarding the axial heat conduction is displayed by evaluating the analytical solution. 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Graetz problem Axial heat conduction Concentric annulus
1. Introduction The concentric annulus is the simplest form of a two fluid heat exchanger. Understanding its flow and heat transfer characteristics therefore is of great theoretical interest and practical importance for the design of such technical applications. The heat transfer in the thermal development region of an annulus or a pipe is described by the classical Graetz problem, if the effects of axial heat conduction are negligible. The energy equation in this case is a parabolic partial differential equation. Overviews of published literature concerning the classical Graetz problem can be found e.g. in Ref. [1], and in Ref. [2]. There is also recent advancement of research on the extended Graetz problem in micro regime [3]. But the boundary conditions in micro regime (slip) differ from that of the problem considered here. For flows with small Peclet numbers the effects of axial heat conduction can be of importance, for example if a compact heat exchanger uses a liquid metal as working fluid. Furthermore the effects of axial heat conduction are not to be neglected if the length of a heat exchanger is small. The energy equation specifying this extended Graetz problem, as it is called, takes an elliptic form. Its solution presented in literature is often based on the classical Graetz problem’s series solution, which results in a non-selfadjoint eigenvalue problem and needs the construction of orthonormal
* Corresponding author. Tel.: þ49 711 685 62324; fax: þ49 711 685 62317. E-mail address: [email protected] (K. Eisenschmidt). 1290-0729/$ e see front matter 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.12.006
functions (see e.g. Ref. [4]). In accordance with Ref. [5] this is a cumbersome procedure. To overcome this problem, a solution for a laminar pipe flow had been developed by the authors in Ref. [6]. It is based on a selfadjoint formalism which results from the decomposition of the secondorder partial differential equation. This method had been extended for turbulent pipe and channel flows in Ref. [5]. A solution for the thermal entrance region of a concentric annulus was given by the authors in Ref. [7]. Here only the case of a step change in the constant external wall temperature had been contemplated. The authors of Ref. [8] considered the case of a piecewise constant wall heat flux on the external wall of a concentric annulus. The analytical solution showed the effects of axial heat conduction. In this paper we present an exact solution of the extended Graetz problem for laminar and turbulent flows through a concentric annulus. The temperature of the annulus’ outer wall is piecewise constant. That of the inner wall is adiabatic or also piecewise constant. The following section displays the analytical calculation of the velocity and temperature field. Afterwards we investigate the influence of the Peclet number on the heat transfer as well as that of the heating zone length. The semi-infinite heating zone is analysed as a special case. Finally, the error done by a parabolic calculation is estimated. It arises, that the axial heat conduction is significant for Peclet numbers less then 50. Furthermore it is shown that the effects of axial heat conduction have to be taken into account if the heated section is small.
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B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
Nomenclature
Latin symbols a function, defined by Eq. (23) A area, m2 Aj eigenconstant specific heat capacity at constant pressure, J/(kg K) cp hydraulic diameter Dh E axial energy flux ! F arbitrary function g function H Hilbert space Prandtl mixing length, m lm matrix operator L ¼ Nu Nusselt number p pressure, Pa Pe Peclet number Pr Prandtl number r radial coordinate, m R radius, m Re Reynolds number based on the hydraulic diameter ! S solution vector T temperature, K u axial velocity, m/s v radial velocity, m/s x axial coordinate, m length of the heating zone, m x1 Greek symbols c radii ratio k heat conductivity, W/(mK)
As it can be seen from the above given literature review and to the authors’ best knowledge the investigations presented in this paper had not been done before. 2. Analysis
l h F n r s Q
eigenvalue dynamic viscosity, kg/(ms) eigenfunction kinematic viscosity, m2/s density, kg/m3 shear stress s ¼ m(vu/vy), N/m2 dimensionless temperature
Subscripts b bulk ell elliptic i inner j counter m middle max maximum o outer para parabolic rel relative t turbulent w boundary condition inside the heating section wall wall 0 boundary condition before and after the heating section Superscripts 0 fluctuation averaged quantity w dimensionsless quantity þ dimensionsless quantity 0 derivative þ positive eigenvalues negative eigenvalues
0 ¼
2 vp v u 1 vu : þ þh vx vr 2 r vr
(1)
Integrating Eq. (1) and introducing the following dimensionless quantities
Ri ; Ro
u ; u
r Ri Ro Ri
2.1. Geometrical setup and boundary conditions
c ¼
The annulus under investigation has constant inner and outer radii, Ri and Ro. The flow enters the heating zone at x ¼ 0 with T0. The temperature at the outer wall is Tw in [x0,x1]. The temperature after the end of the heating zone at Ro is T0 again. The inner wall is either adiabatic (case 1) or its temperature varies in accordance with the outer wall (case 2). Fig. 1 illustrates the geometry and boundary conditions. We assume the flow to be steady-state, hydrodynamical fully developed and incompressible. The fluid is Newtonian with constant properties.
yields the velocity distribution for laminar flows
~ ¼ u
~r ¼
(2)
2.2. Velocity field The hydrodynamically fully developed velocity distribution is deduced from the NaviereStokes equations. There is a difference between laminar and turbulent flows. The flow for the present study is supposed to be laminar for Re 2000 and turbulent for Re 5000 (see e.g. Ref. [9]). 2.2.1. Laminar flow The NaviereStokes equation in axial direction for this problem is given by (see e.g. [1])
Fig. 1. Geometry and boundary conditions for both setups (case 1 and case 2).
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
h h i2 1 c2 h ii 2 1 ~r ð1 cÞ þ c ln ~r ð1 cÞ þ c ln c ~ ¼ u : 1 c2 1 þ c2 þ ln c
(3)
rv vp h v vu r 0 ¼ þ ðru0 v0 Þ: r vr vx r vr vr
(4)
can be modelled by introducing the turbulent viscosity nt as
vu vr
(5)
vu
nt ¼ l2m vr
þ þ for Rþ m r Ro [11]. The maximum velocity’s radius is unknown a priori and has to be calculated iteratively from the continuity equation in integral form
Z1 h i ~ ~r ð1 cÞ þ c d~r : u
1þc ¼ 2
(6)
The readers are referred to Refs. [10] and [11] for further details on the calculation of the velocity profile.
In the following the energy equation is analysed in order to obtain the temperature field and the Nusselt number. One has to distinguish between the two boundary conditions, case 1 and case 2. 2.3.1. Energy equation Following [5] the energy equation can be written as
vT 1v vT v vT r k k þ ¼ rcp v0 T 0 rcp u0 T 0 : vx r vr vr vx vx
rcp uðrÞ
The mixing length is given by [7]
2 4 Ro Ri r Ri r Ri 1 0:06 2 1 ½0:14 0:08 2 2 Ro Ri Ro Ri 0 1 sffiffiffiffiffiffiffiffiffiffiffiffi 1 0 jswall j B ðr Ri ÞCC B C B 1 r CC B B B1 exp B (7) CC: AC B @ 26 n @ A
(14) The turbulent heat fluxes rcp and rcp were modelled using Boussinesq’s hypothesis and the turbulent Prandtl number. The turbulent Prandtl number was calculated by an extended Kays and Crawford model u0 T 0
This well known approach is validated for our problem by comparing nt from our calculations with experimental results as can be seen in 3.1. Introducing further dimensionless quantities
r
¼ r
lþ m ¼
;
n
þ
u
sffiffiffiffiffiffiffiffiffiffiffiffi lm jswall j
n
rffiffiffiffiffiffiffiffiffiffiffiffi r ; ¼ u jswall j
þ
R
¼
R
n
sffiffiffiffiffiffiffiffiffiffiffiffi jswall j
r
;
(8)
vuþ vuþ s vuþ swall þ rn wall þ s ¼ 0: lþ2 m þ þ rn vr vr vr
(
0; < 0;
quadratic
equation
requires
(9) the
r þ Rþ Rþ m; i þ þ Rþ m < r Ro
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u þ þ2 þ2 þ2 u Ri Rm r þ2 Rþ i Rm r A @1 þ t1 þ 4lþ2 2 þ þ2 m þ þ2 þ2 r Rm R r Rm Rþ2 i i
for Rþ r þ Rþ m and i
(16)
(17)
The choice of the turbulent Prandtl number has a significant influence on the final temperature and Nusselt number distribution. The authors of Ref. [2] showed that the extended Kays and Crawford model works very well for the fluids with small Peclet numbers as liquid metals. Therewith the energy equation can be written as
rcp uðrÞ for for
100 PrRe0:888
nt Pr: n
following
nt vT nt vT vT 1v v r k þ rcp þ k þ rcp : ¼ vx r vr vx Prt vr Prt vx (18)
(10) It has to fulfil the following boundary conditions
where Rþ m is the radius where the maximum velocity occurs. Thus, the velocity distribution for the turbulent flow is given by
vuþ ¼ vr þ
(15)
and
Pet ¼
Eq. (4) can be written as
vuþ ¼ vr þ
1 exp
PrtN ¼ 0:85 þ
r
Solving this distinction
sffiffiffiffiffiffiffiffiffiffi 1 ð0:3Pet Þ2 PrtN !#!1 1 pffiffiffiffiffiffiffiffiffiffi 0:3Pet PrtN
where
sffiffiffiffiffiffiffiffiffiffiffiffi jswall j
r
v0 T 0
1 þ 0:3Pet 2PrtN "
Prt ¼
þ
(13)
0
2.3. Temperature field
which itself can be estimated from the turbulent mixing length lm
lm ¼
vuþ ¼ vr þ
(12)
2.2.2. Turbulent flow The velocity distribution for a hydrodynamically fully developed turbulent flow entering the annulus is calculated by using an approach similar to that of Ref. [10]. The turbulent fluxes in the NaviereStokes equation
u0 v0 ¼ nt
91
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 !,0 þ2 r þ2 þ2 þ2 Rþ R Rþ o m o Rm r A @1 þ 1 4lþ2 2 þ m þ þ2 þ2 r Rþ2 r Rþ2 m Ro m Ro
(11)
- case 1
r ¼ Ri : r ¼ Ro :
vT ¼ 0 vr ( T ¼ T0 ;
for
x<0
T ¼ Tw ;
for
0 x < x1
and
x x1
(19) ;
92
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
- case 2
r ¼ Ri ;
r ¼ Ro :
L¼
T ¼ T0 ; T ¼ Tw ;
for for
x < 0 and 0 x < x1
x x1
¼
:
Pe2 ~ u a i v h 4ð1 cÞ ~r ð1 cÞ þ c a v~r
Pe2 1 v i h a ð1 cÞ ð1 cÞ~r þ c v~r
Q¼
T Tw ; T0 Tw
and
~nt ¼
~ x ¼
nt n
x 1 ; Dh Pe
Dh ¼ 2Ro ð1 cÞ;
Pe ¼ PrRe
(29) In order to derive a series solution, Eq. (22) needs to be transformed into an eigenvalue problem. Since ¼ L is not selfadjoint, the eigenfunctions are not necessarily orthogonal [5]. Define an inner product between two eigenfunctions
! (21)
:
0
(20) Introducing the following dimensionless quantities
!
Fj ¼
!
Fj1 ~r
Fj2 ~r
and
!
!
Fk ¼
Fk1 ~r
Fk2 ~r
in addition to Eqs. (2) and (8) yields
~ u
as
vQi vQ 4 v h
~r ð1 cÞ þ c a ~r i ¼ h v~ x v~r ~r ð1 cÞ þ c v~r 1 v vQ a ~r v~ x x Pe2 v~
þ
! ! F j ; F k ¼ ð1 cÞ
0
(22)
1
C ~ i þ h A dr ; 2 4 ð1 cÞ~r þ c ð1 cÞ a
Pr ~nt : a ~r ¼ 1 þ Prt
(23)
The boundary conditions in dimensionless form are given by - case 1
~r ¼ 1 :
0h i c~ c B ð1 Þr þ a Fj1 Fk1 @ Pe2
Fj2 Fk2
where
~r ¼ 0 :
Z1
(30)
it can be shown (see e.g. [12]) that ¼ L is a symmetric operator for the inner product given by Eq. (30) in
Dð ¼ L Þ ¼ F˛H : d ¼ L F˛H ; F1 ð1Þ ¼ 0; F2 ð0Þ ¼ 0ðcase 1Þ
F1 ð1Þ ¼ 0; F1 ð0Þ ¼ 0ðcase 2Þg: (31)
vQ ¼ 0 v~r 8 < Q ¼ 1;
for
~ x<0
: Q ¼ 0;
for
0~ x<~ x1
and
~ x~ x1
(24) ;
Thus Eq. (22) results in the SturmeLiouville eigenvalue problem (see [12])
! ! L F j ¼ lj F j
(32)
¼
which is selfadjoint, semidefinite and in expanded form: - case 2
( ~r ¼ 0;
~r ¼ 1 :
Q ¼ 1; for ~x < 0 and ~x ~x1 : Q ¼ 0; for 0 ~x < ~x1
(25)
The second-order partial differential equation in elliptic form, Eq. (22), cannot be solved directly. Introducing the function Eð~ x; ~r Þ decomposes the equation into a system of two first-order partial differential equations, where Eð~ x; ~r Þ is given by
Z~r ih
h a vQi ~Q d~r E ~ x; ~r ¼ ð1 cÞ ð1 cÞ~r þ c u x Pe2 v~
(27)
where
Q E
:
The matrix operator ¼ L is given by
i vF h j1 4ð1 cÞ ð1 cÞ~r þ c a ¼ lj Fj2 : v~r
(34)
Substituting F2 from Eq. (34) into Eq. (33) yields an eigenvalue problem for F1
h
i
i vF i l a v h h ~ j ~r þ c a j1 ¼ 0 c 4 ð1 ð1 cÞ~r þ c lj Fj1 u Þ v~r v~r Pe2 (35)
with the homogenous boundary conditions
and represents the energy flux in the axial direction [5]. Inserting Eð~ x; ~r Þ into the energy Eq. (22) results in
! S ¼
(33)
(26)
0
! v! S ¼ ¼ L S v~ x
~ Pe2 vF u Pe2 1 i j2 ¼ lj Fj1 h Fj1 a a ð1 cÞ ð1 cÞ~r þ c v~r
(28)
Fj1 ð1Þ ¼ 0 and
F0j1 ð0Þ ¼ 0 ðcase 1Þ : Fj1 ð0Þ ¼ 0 ðcase 2Þ
! ! S defined in Eq. (28) is not an element of Dð ¼ L Þ because S ! does not fulfil the homogenous boundary condition. To write S in terms of the eigenfunctions the symmetry condition ! ! ! ! L F ki ¼ h ¼ L F j ; F k i therefore has to be modified as h F j; ¼
! ! D! ! E L S ; Fj S ; ¼ L Fj ¼ g ~ x Fj2 ð1Þ Fj2 ð0Þ
¼
where
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
g ~ x ¼
(
1 0
~ x < 0 and 0~ x<~ x1
for for
~ x~ x1
:
(36)
See e.g. [12] for detailed information. If this modification is taken into account, the temperature distribution is given by
! N ! X S ; Fj Q ¼ S1 ¼ ! 2 Fj1 j¼1
(45)
where Tb is the bulk temperature
Z TudA
(37)
Fj
A
Tb ¼ Z
:
(46)
udA
where
A
! 2 ! ! F j ¼ F j; F j :
(38)
! This scaling term k F j k2 can be derived from Eq. (32) as
! 2 dFj1 ð1Þ F j ¼ Fj2 ð1Þ dl
93
vT vr wall Nu ¼ ; Tb Twall
The Nusselt number at the inner wall is estimated by evaluating vT vT the temperature gradient and with for the Nusselt vr vr r¼Ri
r¼Ro
number at the outer wall. Inserting Eqs. (2), (8) and (21) one obtains l¼lj
:
(39)
vQ Nu ¼ v~r wall Qb Qwall 2
To determine the inner product in Eq. (37) one applies Eq. (30) to Eq. (27) resulting in
! ! v ! ! L S ; Fj : S ; Fj ¼ ¼ v~ x
(40)
where
(
Qwall ¼
With Eq. (36) the differential equation can be rewritten as
! ! v ! ! x Fj2 ð1Þ Fj2 ð0Þ : S ; F j lj S ; F j ¼ g ~ v~ x
(47)
1 0
~ x < 0 and 0~ x~ x1
for for
~ x>~ x1
(48)
and
(41)
This differential equation has to be solved by superposing the homogenous and a particular solution. Then we can specify the distribution of temperature for the three different regions as
Qb ¼
2 1þc
Z1
Qu~ ½ð1 cÞ þ cd~r:
(49)
0
h i 1: N < ~ x<0
h
i þ þ þ Fþ x 1 exp lj ~ x1 exp lj ~ N P j2 ð1Þ Fj2 ð0Þ Q¼ Fþ j1 !þ 2 þ j¼1 l F
Inserting the temperature distribution given above into Eq. (49) we get the bulk temperature for the three regions ½N < ~ x < 0, x1 ~ x < N. ½0 ~ x<~ x1 and ½~ Therewith Eq. (47) can be written as
þ1; (42) h i 2: 0 ~ x<~ x1
þ þ ~ ~ Fþ N P j2 ð1Þ Fj2 ð0Þ exp lj x x1 Q¼ Fþ j1 !þ 2 þ (43) j¼1 lj F j
~ F N P j2 ð1Þ Fj2 ð0Þ exp lj x F 2 j1 ; ! j¼1 lj F j i h x
~ 1 exp l ~ F N P j2 ð1Þ Fj2 ð0Þ exp lj x j x1 Q ¼ 1 F j1 : 2 j¼1 l F j j
vQ Nu ¼ v~r wall ; 2 P 1þc
j
j
(44) It is remarkable, that the eigenvalues, that were taken into account in the region before the heating zone, are all positive, the eigenvalues after the heating zone are all negative and the eigenvalues in the heating zone are positive as well as negative. If the elliptic problem is transformed into a parabolic problem by increasing the Peclet number, only negative eigenvalues will remain, since the region before the heating zone looses its influence. The Nusselt number represents the dimensionless temperature gradient at the wall and describes the convective heat transfer. It is given by
h i 1: N < ~ x<0 (50)
2
where
þ N
Fþ X j2 ð1Þ Fj2 ð0Þ þ þ x 1 exp lj ~ x1 P ¼ exp lj ~ þ 2 ! j¼1 lþ F j
Z1
j
h i ~ ~ ~ Fþ j1 u ð1 cÞr þ c dr ;
0
(51) h i 2: 0 ~ x~ x1
Nu ¼
vQ v~r wall PþQ
ð1 þ cÞ
where
þ N h Z1 i
Fþ X j2 ð1Þ Fj2 ð0Þ þ þ ~ ð1 cÞ~r þ c d~r ~ ~ l F u x x P¼ exp 1 j j1 þ þ ! 2 j¼1
and
lj F j
0
94
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
N i Z1 h
F X j2 ð1Þ Fj2 ð0Þ ~ ~ ~ ~ l F x Q ¼ exp j1 u ð1 cÞr þ c dr ; j ! 2
lj F j
j¼1
0
(52) h i 3: ~ x1 < ~ x
(53)
2
where
N
F X j2 ð1Þ Fj2 ð0Þ x 1 exp lj ~ x1 P ¼ exp lj ~ 2 ! l F j¼1 j
j
Z1
h
i
~ ~ ~ F j1 u ð1 cÞr þ c dr
0
(54) A description of the semi-infinite heating zone is gained by letting ~ x1 /N. 3. Validation and results In the following we will display and interpret the results of the previous section by means of the local temperature and the Nusselt number. Thereby the influence of axial heat conduction effects comes to the fore. The presented data were calculated numerically. The iteration of the eigenvalues as well as the eigenfunctions of the eigenvalue problem given by Eq. (35) was carried out by using a four stage RungeeKutta method. The geometrical setup under consideration was an annulus with c ¼ 0.5. Grid independency is assured by using a numerical grid with n ¼ 10,000 points in radial direction. To solve the eigenvalue problem accurately 250 terms should be taken into account in the summations in Eqs. (42)e(44) and Eqs. (50)e(53). Then the temperature profile is smooth even for small Peclet numbers. 3.1. Validation As validation of the turbulent velocity field Fig. 2 compares our calculation’s turbulent viscosity nt with measurements from
Table 1 Comparison of positive eigenvalues and eigenconstants for Pe ¼ 5 between [7] and own calculation. j
lþ j
Aþ j
Own calculation
[7]
1 2 3 4 5 6 7 8 9 10
0.34939Eþ02 0.62435Eþ02 0.92858Eþ02 0.12378Eþ03 0.15493Eþ03 0.18617Eþ03 0.21747Eþ03 0.24880Eþ03 0.28014Eþ03 0.31151Eþ03
0.34940Eþ02 0.62435Eþ02 0.92858Eþ02 0.12378Eþ03 0.15493Eþ03 0.18617Eþ03 0.21747Eþ03 0.24880Eþ03 0.28014Eþ03 0.31151Eþ03
Own calculation 0.28355Eþ00 0.21311Eþ00 0.14878Eþ00 0.11300Eþ00 0.90672E01 0.75618E01 0.64817E01 0.56700E01 0.50382E01 0.45327E01
[7] 0.28355Eþ00 0.21311Eþ00 0.14878Eþ00 0.11300Eþ00 0.90670E01 0.75620E01 0.64820E01 0.56700E01 0.50380E01 0.45330E01
Ref. [13] for Re ¼ 350,000. The agreement is quite good. The temperature field can be validated with analytical data from Ref. [12] for infinite heating zones. These data itselves are compared þ to numerical data. The values of lþ j and Aj for Pe ¼ 5 for the geometrical setup case 1 (see Table 1) and the values of l j and Aj for ReDh ¼ 10,000 and Pr ¼ 0.001 for case 2 (see Table 2) are shown exemplary. The maximum deviation is in the order of 0.1%. 3.2. Results In the following we will analyse the results from calculations with different parameters. We look at both setups (case 1 and case 2), laminar and turbulent flows, different Peclet numbers and varying heating zone lengths. The semi-infinite heating zone is regarded as a special case of the finite heating zone. Finally we estimate the error done by a parabolic calculation. The Reynolds number of a laminar flow is fixed to Re ¼ 1000 while the Peclet number varies with Prandtl number. 3.2.1. Case 1 As it can be seen from Fig. 1, the wall temperature for 0 < ~ x<~ x1 is changed to Q ¼ 0 for case 1. Elsewhere Q is equal to one. The derived solution captures the effect of small heating sections. Thus ~ x1 tends to infinity the here presented x1 can vary arbitrarily. If ~ solutions approach the one given by the authors in Ref. [7]. In this section we represent the results for laminar flows only. 3.2.1.1. Variation of Peclet number. Fig. 3 shows the radial temperature profiles Qð~r Þ for Pe ¼ 1,5,10 and 50 at ~ x ¼ 0:1; 0:001; 0:001 and 0.1. The length of the heating zone x ¼ 0:1 the effect of axial heat conduction was ~ x1 ¼ 1. At ~ becomes only visible for a flow with Pe ¼ 1. At ~ x ¼ 0:001 the fluid with Pe ¼ 1 adapts to the boundary condition inside the heating zone uniformly along the radius of the Table 2 Comparison of negative eigenvalues and eigenconstants for Re ¼ 10,000 and Pr ¼ 0.001 between [7] and own calculations. j
Fig. 2. Comparison of turbulent viscosity between experimental data from Ref. [13] and own calculation for Re ¼ 350,000.
1 2 3 4 5 6 7 8 9 10
lj
A j
Own calc.
[7]
Own calc.
[7]
0.27601Eþ02 0.82875Eþ02 0.14294Eþ03 0.20454Eþ03 0.26666Eþ03 0.32905Eþ03 0.39158Eþ03 0.45420Eþ03 0.51687Eþ03 0.57957Eþ03
0.27650Eþ02 0.82988Eþ02 0.14304Eþ03 0.20464Eþ03 0.26673Eþ03 0.32913Eþ03 0.39165Eþ03 0.45427Eþ03 0.51695Eþ03 0.57966Eþ03
0.78839Eþ01 0.10378Eþ01 0.56323Eþ01 0.89974Eþ00 0.51651Eþ01 0.85911Eþ00 0.49929Eþ01 0.84263Eþ00 0.49138Eþ01 0.83498Eþ00
0.78826Eþ01 0.10389Eþ01 0.56474Eþ01 0.90273Eþ00 0.51836Eþ01 0.86200Eþ00 0.50098Eþ01 0.84509Eþ00 0.49280Eþ01 0.83690Eþ00
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
95
Fig. 3. Radial temperature distribution for different Peclet numbers at different axial positions (~x1 ¼ 1).
annulus due to the great influence of axial heat conduction. For fluids with higher Peclet number the effects of axial heat conduction gain in importance only in the near wall region. The flow velocity is reduced due to the no slip condition at the wall and forwards the upstream axial heat conduction. This can be seen from the energy Eq. (14). If convective heat transport looses influence due to reduced velocity, the terms describing heat conduction have to be of the same order of magnitude. Towards the middle of the annulus higher velocities are reached and axial heat conduction looses influence. Inside the heating zone at ~ x ¼ 0:001 the temperature level is still higher for higher Peclet numbers. The differences between the Peclet numbers concerning the temperature gradient and the thermal boundary layers thickness are pointed up here. Temperature decreases rapidly near the wall for fluids with small Peclet numbers while the gradient towards the middle of the annulus is small. The increase of temperature is more even for higher Peclet numbers and the maximum temperature is reached earlier. Therefore the thermal boundary layer thickness is larger for fluids with smaller Peclet numbers despite their higher temperature gradient near the wall. At ~ x ¼ 0:1 the influence of radial and axial heat conduction depending on the Peclet number is illustrated. The radial heat conduction gains in importance for flows with higher Peclet numbers while the heat transfer due to axial heat conduction declines. For flows with smaller Peclet numbers the axial heat conduction prevails the boundary condition’s influence. Consequently, at this point the temperature level of flows with smaller Peclet numbers exceeds that of higher Peclet number. Summing up, flows with smaller Peclet numbers begin more early to adapt themselves to a new boundary condition while the whole adaption foregoes more slowly.
Fig. 4. Nusselt number for different Peclet numbers for a finite heating zone (~x1 ¼ 1).
The behaviour of the Nusselt number as shown in Fig. 4 confirms these explanations. The Nusselt number decreases more slowly for smaller Peclet numbers and rises earlier due to the larger temperature gradient at the wall. If the Peclet number is high there is no consequent adaption to the boundary conditions after the heating section. In fact the Nusselt number jumps to infinity. This is explained by the fact that the problem is more and more parabolic in character with lower influence of the boundary conditions if the Peclet number is increased. 3.2.1.2. Variation of heating section’s length. The length of the heating zone influences the heat transfer significantly. Therefore different length of the heating zone have been investigated for a fluid with Pe ¼ 5. The Nusselt number in short heating zones does not decline as much as in longer heating zones because less heat can be transferred into the fluid. Therefrom a shorter heating zone has a longer thermal entrance region relatively to its length. Because the heat transfer is most effective in the thermal entrance region, the Nusselt number is higher for shorter heating zones (see Fig. 5). 3.2.1.3. Estimation of the error done by a parabolic calculation. The parabolic calculation neglects the effects of axial heat conduction. However, the previous chapter showed that this effect is important for flows with small Peclet numbers regarding the temperature as well as the Nusselt number. Now we will estimate the error, that is done if a laminar flow through a semi-infinite heating zone is calculated parabolically. On the basis of this error we will specify a limit, which allows a parabolic calculation. Fig. 6 shows the relative error
Fig. 5. Nusselt number for different heating sections lengths (Pe ¼ 5).
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B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
Fig. 6. Nusselt numbers relative error due to parabolic calculation (~x1 /N).
DNurel ¼
Nuell Nupara Nuell
(55)
due to the neglection of the axial heat conduction on the basis of the Nusselt number for different Peclet numbers. The discrepancy is large for small Peclet numbers. It is more than 90% for Pe ¼ 1. The axial heat conduction is neglegible only for fluids with Pe 50. This result can of course not been applied to very short heating zones (see Fig. 5). Here the errors can be enormous if the elliptic character of the problem is not considered.
Fig. 8. Nusselt number at the inner (i) and at the outer (o) wall for Pe ¼ 1 and Pe ¼ 5 (~x1 /N).
the Nusselt number increasing with the Reynolds number, because turbulence enforces heat transfer. The thermal developing region behaves similar to that of a laminar flow. The Nusselt number increases for smaller Peclet numbers due to its larger temperature gradient as Fig. 7 shows.
3.2.2.1. Variation of Peclet number for turbulent flows. In contrast to laminar flows, the heat transfer in turbulent flows is not influenced only by the Peclet number, but also by the Reynolds and the Prandtl number so that flows might behave differently although the Peclet number is the same. This difference is caused by turbulence. The term að~r Þ in Eq. (22) is not equal to 1 in turbulent flows as it is in a laminar flow. Therefore, the Prandtl number as well as the Reynolds number influence the temperature field and make statements about the flow more complicate. In order to elucidate this effect of turbulence a fluid with Pr ¼ 0.001 and varying Reynolds number (Re ¼ 5000; 10,000; 25,000) was considered. Regarding the fully developed flow we see
3.2.2.2. Comparison of Nusselt numbers at the inner and the outer wall. The Nusselt number (Eqs. (50)e(53)) can be evaluated for the inner wall (Nui) by computing the temperature gradient at ~r ¼ 0. If the Nusselt number is based on the temperature gradient at ~r ¼ 1, we call it Nuo. Fig. 8 shows the absolute values of both Nusselt numbers, jNui jand jNuo j, for a laminar flow with Pe ¼ 1 and Pe ¼ 5. Despite symmetric boundary conditions, the inner wall’s Nusselt number reaches higher values than the Nusselt number based on the temperature gradient of the outer wall. The enhanced heat transfer expressed by the higher Nusselt number is caused by larger temperature gradients due to a thinner thermal boundary layer at the inner wall. The smaller boundary layer thickness in turn can be ascribed to the velocities at the inner wall. They are higher due to the annulus’ asymmetric velocity distribution. (The velocity distribution in a concentric annulus depends on the radii ratio c as can be seen in chapter 2.2.) It should be noticed, that a comparison between the Nusselt numbers jNui j and jNuo j has to be seen as a rather qualitative one. From their definition (Eqs. (50)e(53)) one can see that both Nusselt numbers are related to the same temperature difference Qb Qwall . Because of the asymmetric temperature profile in the annulus each Nusselt number should be related to another temperature difference which is characteristic for the inner and the outer wall for quantitative statements.
Fig. 7. Nusselt number in the thermal developing region depending on the Reynolds number (~ x1 /N); Pr ¼ 0.001.
Fig. 9. Comparison of bulk temperatures for case 1 (1) and case 2 (2) boundary condition for Pe ¼ 1 and Pe ¼ 5 (~x1 /N).
3.2.2. Case 2 The semi-infinite heating zone as well as the differences between laminar and turbulent flow will be of particular interest in this section. Furthermore we will compare the Nusselt numbers on the inner and the outer wall and we will look at the differences between both boundary conditions.
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
3.2.2.3. Comparison of both boundary conditions (case 1 and case 2). We compare both boundary conditions (case 1 and case 2) by analysing the bulk temperatures and the Nusselt numbers for laminar flows with Pe ¼ 1 and Pe ¼ 5. The bulk temperature in an annulus with both walls heated (case 2) declines faster than for an annulus with one adiabatic wall (case 1), because the information about change in temperature reaches every radial point earlier (see Fig. 9). Therefore the thermal entrance length is shortened for case (2).
4. Conclusions Axial heat conduction has a strong impact on fluids with Peclet numbers smaller than 50. The neglection of axial heat transfer in our studies caused relative errors up to 90%. In flows with large Peclet numbers radial heat transfer prevailed the axial heat conduction. Consequently for flows with smaller Peclet numbers the adaption due to a modified boundary condition started earlier but it ran slower. Furthermore the heating zone’s length is an important parameter. We saw an effective heat transfer for short heating zones which results in large Nusselt number. However, the absolute heat transfer along the heating zone was small. In turbulent flows the influence of the Reynolds number and that of the Prandtl number was studied separately. Compared to laminar flows, the heat transfer was increased by turbulent transport. The flow’s behaviour was similar for both setups. The comparison showed that the Nusselt number reaches larger
97
values, if the inner wall is heated (case 2) instead of keeping it adiabatic (case 1). References [1] R.K. Shah, A.L. London, Laminar Flow Forced Convection, Academic Press, New York, 1978. [2] W.M. Kays, M.E. Crawford, B. Weigand, Convective heat and mass transfer, McGraw-Hill Education (2004). [3] S. Chen, Lattice Boltzmann method for slip flow heat transfer in circular microtubes: extended Graetz problem, Applied Mathematics and Computation 217 (2010) 3314e3320. [4] C.-J. Hsu, An exact analysis of low Peclet number thermal entry region heat transfer in transversely nonuniform velocity fields, AIChE Journal 17 (1971) 732e740. [5] B. Weigand, An exact analytical solution for the extended turbulent Graetz problem with Dirichlet wall boundary conditions for pipe and channel, International Journal of Heat and Mass Transfer 39 (1996) 1625e1637. [6] E. Papoutsakis, D. Ramkrishna, H.C. Lim, The extended Graetz problem with Dirichlet wall boundary conditions, Applied Scientific Research 36 (1980) 13e34. [7] B. Weigand, M. Wolf, H. Beer, Heat transfer in laminar and turbulent flows in the thermal entrance region of concentric annuli: axial heat conduction effects in the fluid, International Journal of Heat and Mass Transfer 33 (1997) 67e80. [8] B. Weigand, F. Wrona, The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli, International Journal of Heat and Mass Transfer 39 (2001) 313e320. [9] J.H. Spurk, Strömungslehre, Springer, 1996. [10] N. Wilson, J. Medwell, An analysis of heat transfer for fully developed turbulent flow in concentric annuli, International Journal of Heat Transfer 90 (1968) 43e50. [11] H. Pfitzer, Konvektiver Wärmetransport im axial durchströmten Ringspalt zwischen rotierenden Hohlwellen, Doctorial Thesis, TH Darmstadt, 1992. [12] B. Weigand, Analytical Methods for Heat Transfer and Fluid Flow Problems, Springer, 2004. [13] F. Durst, On Turbulent Flow Through Annular Passages with Smooth and Rough Cores, University of London, Dep. of Mech. Eng. Thermo-Fluids Sec., 1968.
Contents lists available at SciVerse ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
The extended Graetz problem with piecewise constant wall temperature for laminar and turbulent flows through a concentric annulus B. Weigand, K. Eisenschmidt* Institute of Aerospace Thermodynamics, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2011 Received in revised form 6 December 2011 Accepted 7 December 2011 Available online 11 January 2012
The effect of axial heat conduction on the heat transfer is of importance if the flow Peclet number is small or if the axial extension of the heating zone is small. This paper therefore presents an analytical solution to the extended Graetz problem in a concentric annulus with piecewise constant temperature at the outer wall. The solution is based on a selfadjoint formalism which results from the decomposition of the elliptic energy equation into two first-order partial differential equations. The gained solution is exact and simple. The need of regarding the axial heat conduction is displayed by evaluating the analytical solution. 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Graetz problem Axial heat conduction Concentric annulus
1. Introduction The concentric annulus is the simplest form of a two fluid heat exchanger. Understanding its flow and heat transfer characteristics therefore is of great theoretical interest and practical importance for the design of such technical applications. The heat transfer in the thermal development region of an annulus or a pipe is described by the classical Graetz problem, if the effects of axial heat conduction are negligible. The energy equation in this case is a parabolic partial differential equation. Overviews of published literature concerning the classical Graetz problem can be found e.g. in Ref. [1], and in Ref. [2]. There is also recent advancement of research on the extended Graetz problem in micro regime [3]. But the boundary conditions in micro regime (slip) differ from that of the problem considered here. For flows with small Peclet numbers the effects of axial heat conduction can be of importance, for example if a compact heat exchanger uses a liquid metal as working fluid. Furthermore the effects of axial heat conduction are not to be neglected if the length of a heat exchanger is small. The energy equation specifying this extended Graetz problem, as it is called, takes an elliptic form. Its solution presented in literature is often based on the classical Graetz problem’s series solution, which results in a non-selfadjoint eigenvalue problem and needs the construction of orthonormal
* Corresponding author. Tel.: þ49 711 685 62324; fax: þ49 711 685 62317. E-mail address: [email protected] (K. Eisenschmidt). 1290-0729/$ e see front matter 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.12.006
functions (see e.g. Ref. [4]). In accordance with Ref. [5] this is a cumbersome procedure. To overcome this problem, a solution for a laminar pipe flow had been developed by the authors in Ref. [6]. It is based on a selfadjoint formalism which results from the decomposition of the secondorder partial differential equation. This method had been extended for turbulent pipe and channel flows in Ref. [5]. A solution for the thermal entrance region of a concentric annulus was given by the authors in Ref. [7]. Here only the case of a step change in the constant external wall temperature had been contemplated. The authors of Ref. [8] considered the case of a piecewise constant wall heat flux on the external wall of a concentric annulus. The analytical solution showed the effects of axial heat conduction. In this paper we present an exact solution of the extended Graetz problem for laminar and turbulent flows through a concentric annulus. The temperature of the annulus’ outer wall is piecewise constant. That of the inner wall is adiabatic or also piecewise constant. The following section displays the analytical calculation of the velocity and temperature field. Afterwards we investigate the influence of the Peclet number on the heat transfer as well as that of the heating zone length. The semi-infinite heating zone is analysed as a special case. Finally, the error done by a parabolic calculation is estimated. It arises, that the axial heat conduction is significant for Peclet numbers less then 50. Furthermore it is shown that the effects of axial heat conduction have to be taken into account if the heated section is small.
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B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
Nomenclature
Latin symbols a function, defined by Eq. (23) A area, m2 Aj eigenconstant specific heat capacity at constant pressure, J/(kg K) cp hydraulic diameter Dh E axial energy flux ! F arbitrary function g function H Hilbert space Prandtl mixing length, m lm matrix operator L ¼ Nu Nusselt number p pressure, Pa Pe Peclet number Pr Prandtl number r radial coordinate, m R radius, m Re Reynolds number based on the hydraulic diameter ! S solution vector T temperature, K u axial velocity, m/s v radial velocity, m/s x axial coordinate, m length of the heating zone, m x1 Greek symbols c radii ratio k heat conductivity, W/(mK)
As it can be seen from the above given literature review and to the authors’ best knowledge the investigations presented in this paper had not been done before. 2. Analysis
l h F n r s Q
eigenvalue dynamic viscosity, kg/(ms) eigenfunction kinematic viscosity, m2/s density, kg/m3 shear stress s ¼ m(vu/vy), N/m2 dimensionless temperature
Subscripts b bulk ell elliptic i inner j counter m middle max maximum o outer para parabolic rel relative t turbulent w boundary condition inside the heating section wall wall 0 boundary condition before and after the heating section Superscripts 0 fluctuation averaged quantity w dimensionsless quantity þ dimensionsless quantity 0 derivative þ positive eigenvalues negative eigenvalues
0 ¼
2 vp v u 1 vu : þ þh vx vr 2 r vr
(1)
Integrating Eq. (1) and introducing the following dimensionless quantities
Ri ; Ro
u ; u
r Ri Ro Ri
2.1. Geometrical setup and boundary conditions
c ¼
The annulus under investigation has constant inner and outer radii, Ri and Ro. The flow enters the heating zone at x ¼ 0 with T0. The temperature at the outer wall is Tw in [x0,x1]. The temperature after the end of the heating zone at Ro is T0 again. The inner wall is either adiabatic (case 1) or its temperature varies in accordance with the outer wall (case 2). Fig. 1 illustrates the geometry and boundary conditions. We assume the flow to be steady-state, hydrodynamical fully developed and incompressible. The fluid is Newtonian with constant properties.
yields the velocity distribution for laminar flows
~ ¼ u
~r ¼
(2)
2.2. Velocity field The hydrodynamically fully developed velocity distribution is deduced from the NaviereStokes equations. There is a difference between laminar and turbulent flows. The flow for the present study is supposed to be laminar for Re 2000 and turbulent for Re 5000 (see e.g. Ref. [9]). 2.2.1. Laminar flow The NaviereStokes equation in axial direction for this problem is given by (see e.g. [1])
Fig. 1. Geometry and boundary conditions for both setups (case 1 and case 2).
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
h h i2 1 c2 h ii 2 1 ~r ð1 cÞ þ c ln ~r ð1 cÞ þ c ln c ~ ¼ u : 1 c2 1 þ c2 þ ln c
(3)
rv vp h v vu r 0 ¼ þ ðru0 v0 Þ: r vr vx r vr vr
(4)
can be modelled by introducing the turbulent viscosity nt as
vu vr
(5)
vu
nt ¼ l2m vr
þ þ for Rþ m r Ro [11]. The maximum velocity’s radius is unknown a priori and has to be calculated iteratively from the continuity equation in integral form
Z1 h i ~ ~r ð1 cÞ þ c d~r : u
1þc ¼ 2
(6)
The readers are referred to Refs. [10] and [11] for further details on the calculation of the velocity profile.
In the following the energy equation is analysed in order to obtain the temperature field and the Nusselt number. One has to distinguish between the two boundary conditions, case 1 and case 2. 2.3.1. Energy equation Following [5] the energy equation can be written as
vT 1v vT v vT r k k þ ¼ rcp v0 T 0 rcp u0 T 0 : vx r vr vr vx vx
rcp uðrÞ
The mixing length is given by [7]
2 4 Ro Ri r Ri r Ri 1 0:06 2 1 ½0:14 0:08 2 2 Ro Ri Ro Ri 0 1 sffiffiffiffiffiffiffiffiffiffiffiffi 1 0 jswall j B ðr Ri ÞCC B C B 1 r CC B B B1 exp B (7) CC: AC B @ 26 n @ A
(14) The turbulent heat fluxes rcp and rcp were modelled using Boussinesq’s hypothesis and the turbulent Prandtl number. The turbulent Prandtl number was calculated by an extended Kays and Crawford model u0 T 0
This well known approach is validated for our problem by comparing nt from our calculations with experimental results as can be seen in 3.1. Introducing further dimensionless quantities
r
¼ r
lþ m ¼
;
n
þ
u
sffiffiffiffiffiffiffiffiffiffiffiffi lm jswall j
n
rffiffiffiffiffiffiffiffiffiffiffiffi r ; ¼ u jswall j
þ
R
¼
R
n
sffiffiffiffiffiffiffiffiffiffiffiffi jswall j
r
;
(8)
vuþ vuþ s vuþ swall þ rn wall þ s ¼ 0: lþ2 m þ þ rn vr vr vr
(
0; < 0;
quadratic
equation
requires
(9) the
r þ Rþ Rþ m; i þ þ Rþ m < r Ro
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u þ þ2 þ2 þ2 u Ri Rm r þ2 Rþ i Rm r A @1 þ t1 þ 4lþ2 2 þ þ2 m þ þ2 þ2 r Rm R r Rm Rþ2 i i
for Rþ r þ Rþ m and i
(16)
(17)
The choice of the turbulent Prandtl number has a significant influence on the final temperature and Nusselt number distribution. The authors of Ref. [2] showed that the extended Kays and Crawford model works very well for the fluids with small Peclet numbers as liquid metals. Therewith the energy equation can be written as
rcp uðrÞ for for
100 PrRe0:888
nt Pr: n
following
nt vT nt vT vT 1v v r k þ rcp þ k þ rcp : ¼ vx r vr vx Prt vr Prt vx (18)
(10) It has to fulfil the following boundary conditions
where Rþ m is the radius where the maximum velocity occurs. Thus, the velocity distribution for the turbulent flow is given by
vuþ ¼ vr þ
(15)
and
Pet ¼
Eq. (4) can be written as
vuþ ¼ vr þ
1 exp
PrtN ¼ 0:85 þ
r
Solving this distinction
sffiffiffiffiffiffiffiffiffiffi 1 ð0:3Pet Þ2 PrtN !#!1 1 pffiffiffiffiffiffiffiffiffiffi 0:3Pet PrtN
where
sffiffiffiffiffiffiffiffiffiffiffiffi jswall j
r
v0 T 0
1 þ 0:3Pet 2PrtN "
Prt ¼
þ
(13)
0
2.3. Temperature field
which itself can be estimated from the turbulent mixing length lm
lm ¼
vuþ ¼ vr þ
(12)
2.2.2. Turbulent flow The velocity distribution for a hydrodynamically fully developed turbulent flow entering the annulus is calculated by using an approach similar to that of Ref. [10]. The turbulent fluxes in the NaviereStokes equation
u0 v0 ¼ nt
91
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 !,0 þ2 r þ2 þ2 þ2 Rþ R Rþ o m o Rm r A @1 þ 1 4lþ2 2 þ m þ þ2 þ2 r Rþ2 r Rþ2 m Ro m Ro
(11)
- case 1
r ¼ Ri : r ¼ Ro :
vT ¼ 0 vr ( T ¼ T0 ;
for
x<0
T ¼ Tw ;
for
0 x < x1
and
x x1
(19) ;
92
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
- case 2
r ¼ Ri ;
r ¼ Ro :
L¼
T ¼ T0 ; T ¼ Tw ;
for for
x < 0 and 0 x < x1
x x1
¼
:
Pe2 ~ u a i v h 4ð1 cÞ ~r ð1 cÞ þ c a v~r
Pe2 1 v i h a ð1 cÞ ð1 cÞ~r þ c v~r
Q¼
T Tw ; T0 Tw
and
~nt ¼
~ x ¼
nt n
x 1 ; Dh Pe
Dh ¼ 2Ro ð1 cÞ;
Pe ¼ PrRe
(29) In order to derive a series solution, Eq. (22) needs to be transformed into an eigenvalue problem. Since ¼ L is not selfadjoint, the eigenfunctions are not necessarily orthogonal [5]. Define an inner product between two eigenfunctions
! (21)
:
0
(20) Introducing the following dimensionless quantities
!
Fj ¼
!
Fj1 ~r
Fj2 ~r
and
!
!
Fk ¼
Fk1 ~r
Fk2 ~r
in addition to Eqs. (2) and (8) yields
~ u
as
vQi vQ 4 v h
~r ð1 cÞ þ c a ~r i ¼ h v~ x v~r ~r ð1 cÞ þ c v~r 1 v vQ a ~r v~ x x Pe2 v~
þ
! ! F j ; F k ¼ ð1 cÞ
0
(22)
1
C ~ i þ h A dr ; 2 4 ð1 cÞ~r þ c ð1 cÞ a
Pr ~nt : a ~r ¼ 1 þ Prt
(23)
The boundary conditions in dimensionless form are given by - case 1
~r ¼ 1 :
0h i c~ c B ð1 Þr þ a Fj1 Fk1 @ Pe2
Fj2 Fk2
where
~r ¼ 0 :
Z1
(30)
it can be shown (see e.g. [12]) that ¼ L is a symmetric operator for the inner product given by Eq. (30) in
Dð ¼ L Þ ¼ F˛H : d ¼ L F˛H ; F1 ð1Þ ¼ 0; F2 ð0Þ ¼ 0ðcase 1Þ
F1 ð1Þ ¼ 0; F1 ð0Þ ¼ 0ðcase 2Þg: (31)
vQ ¼ 0 v~r 8 < Q ¼ 1;
for
~ x<0
: Q ¼ 0;
for
0~ x<~ x1
and
~ x~ x1
(24) ;
Thus Eq. (22) results in the SturmeLiouville eigenvalue problem (see [12])
! ! L F j ¼ lj F j
(32)
¼
which is selfadjoint, semidefinite and in expanded form: - case 2
( ~r ¼ 0;
~r ¼ 1 :
Q ¼ 1; for ~x < 0 and ~x ~x1 : Q ¼ 0; for 0 ~x < ~x1
(25)
The second-order partial differential equation in elliptic form, Eq. (22), cannot be solved directly. Introducing the function Eð~ x; ~r Þ decomposes the equation into a system of two first-order partial differential equations, where Eð~ x; ~r Þ is given by
Z~r ih
h a vQi ~Q d~r E ~ x; ~r ¼ ð1 cÞ ð1 cÞ~r þ c u x Pe2 v~
(27)
where
Q E
:
The matrix operator ¼ L is given by
i vF h j1 4ð1 cÞ ð1 cÞ~r þ c a ¼ lj Fj2 : v~r
(34)
Substituting F2 from Eq. (34) into Eq. (33) yields an eigenvalue problem for F1
h
i
i vF i l a v h h ~ j ~r þ c a j1 ¼ 0 c 4 ð1 ð1 cÞ~r þ c lj Fj1 u Þ v~r v~r Pe2 (35)
with the homogenous boundary conditions
and represents the energy flux in the axial direction [5]. Inserting Eð~ x; ~r Þ into the energy Eq. (22) results in
! S ¼
(33)
(26)
0
! v! S ¼ ¼ L S v~ x
~ Pe2 vF u Pe2 1 i j2 ¼ lj Fj1 h Fj1 a a ð1 cÞ ð1 cÞ~r þ c v~r
(28)
Fj1 ð1Þ ¼ 0 and
F0j1 ð0Þ ¼ 0 ðcase 1Þ : Fj1 ð0Þ ¼ 0 ðcase 2Þ
! ! S defined in Eq. (28) is not an element of Dð ¼ L Þ because S ! does not fulfil the homogenous boundary condition. To write S in terms of the eigenfunctions the symmetry condition ! ! ! ! L F ki ¼ h ¼ L F j ; F k i therefore has to be modified as h F j; ¼
! ! D! ! E L S ; Fj S ; ¼ L Fj ¼ g ~ x Fj2 ð1Þ Fj2 ð0Þ
¼
where
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
g ~ x ¼
(
1 0
~ x < 0 and 0~ x<~ x1
for for
~ x~ x1
:
(36)
See e.g. [12] for detailed information. If this modification is taken into account, the temperature distribution is given by
! N ! X S ; Fj Q ¼ S1 ¼ ! 2 Fj1 j¼1
(45)
where Tb is the bulk temperature
Z TudA
(37)
Fj
A
Tb ¼ Z
:
(46)
udA
where
A
! 2 ! ! F j ¼ F j; F j :
(38)
! This scaling term k F j k2 can be derived from Eq. (32) as
! 2 dFj1 ð1Þ F j ¼ Fj2 ð1Þ dl
93
vT vr wall Nu ¼ ; Tb Twall
The Nusselt number at the inner wall is estimated by evaluating vT vT the temperature gradient and with for the Nusselt vr vr r¼Ri
r¼Ro
number at the outer wall. Inserting Eqs. (2), (8) and (21) one obtains l¼lj
:
(39)
vQ Nu ¼ v~r wall Qb Qwall 2
To determine the inner product in Eq. (37) one applies Eq. (30) to Eq. (27) resulting in
! ! v ! ! L S ; Fj : S ; Fj ¼ ¼ v~ x
(40)
where
(
Qwall ¼
With Eq. (36) the differential equation can be rewritten as
! ! v ! ! x Fj2 ð1Þ Fj2 ð0Þ : S ; F j lj S ; F j ¼ g ~ v~ x
(47)
1 0
~ x < 0 and 0~ x~ x1
for for
~ x>~ x1
(48)
and
(41)
This differential equation has to be solved by superposing the homogenous and a particular solution. Then we can specify the distribution of temperature for the three different regions as
Qb ¼
2 1þc
Z1
Qu~ ½ð1 cÞ þ cd~r:
(49)
0
h i 1: N < ~ x<0
h
i þ þ þ Fþ x 1 exp lj ~ x1 exp lj ~ N P j2 ð1Þ Fj2 ð0Þ Q¼ Fþ j1 !þ 2 þ j¼1 l F
Inserting the temperature distribution given above into Eq. (49) we get the bulk temperature for the three regions ½N < ~ x < 0, x1 ~ x < N. ½0 ~ x<~ x1 and ½~ Therewith Eq. (47) can be written as
þ1; (42) h i 2: 0 ~ x<~ x1
þ þ ~ ~ Fþ N P j2 ð1Þ Fj2 ð0Þ exp lj x x1 Q¼ Fþ j1 !þ 2 þ (43) j¼1 lj F j
~ F N P j2 ð1Þ Fj2 ð0Þ exp lj x F 2 j1 ; ! j¼1 lj F j i h x
~ 1 exp l ~ F N P j2 ð1Þ Fj2 ð0Þ exp lj x j x1 Q ¼ 1 F j1 : 2 j¼1 l F j j
vQ Nu ¼ v~r wall ; 2 P 1þc
j
j
(44) It is remarkable, that the eigenvalues, that were taken into account in the region before the heating zone, are all positive, the eigenvalues after the heating zone are all negative and the eigenvalues in the heating zone are positive as well as negative. If the elliptic problem is transformed into a parabolic problem by increasing the Peclet number, only negative eigenvalues will remain, since the region before the heating zone looses its influence. The Nusselt number represents the dimensionless temperature gradient at the wall and describes the convective heat transfer. It is given by
h i 1: N < ~ x<0 (50)
2
where
þ N
Fþ X j2 ð1Þ Fj2 ð0Þ þ þ x 1 exp lj ~ x1 P ¼ exp lj ~ þ 2 ! j¼1 lþ F j
Z1
j
h i ~ ~ ~ Fþ j1 u ð1 cÞr þ c dr ;
0
(51) h i 2: 0 ~ x~ x1
Nu ¼
vQ v~r wall PþQ
ð1 þ cÞ
where
þ N h Z1 i
Fþ X j2 ð1Þ Fj2 ð0Þ þ þ ~ ð1 cÞ~r þ c d~r ~ ~ l F u x x P¼ exp 1 j j1 þ þ ! 2 j¼1
and
lj F j
0
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B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
N i Z1 h
F X j2 ð1Þ Fj2 ð0Þ ~ ~ ~ ~ l F x Q ¼ exp j1 u ð1 cÞr þ c dr ; j ! 2
lj F j
j¼1
0
(52) h i 3: ~ x1 < ~ x
(53)
2
where
N
F X j2 ð1Þ Fj2 ð0Þ x 1 exp lj ~ x1 P ¼ exp lj ~ 2 ! l F j¼1 j
j
Z1
h
i
~ ~ ~ F j1 u ð1 cÞr þ c dr
0
(54) A description of the semi-infinite heating zone is gained by letting ~ x1 /N. 3. Validation and results In the following we will display and interpret the results of the previous section by means of the local temperature and the Nusselt number. Thereby the influence of axial heat conduction effects comes to the fore. The presented data were calculated numerically. The iteration of the eigenvalues as well as the eigenfunctions of the eigenvalue problem given by Eq. (35) was carried out by using a four stage RungeeKutta method. The geometrical setup under consideration was an annulus with c ¼ 0.5. Grid independency is assured by using a numerical grid with n ¼ 10,000 points in radial direction. To solve the eigenvalue problem accurately 250 terms should be taken into account in the summations in Eqs. (42)e(44) and Eqs. (50)e(53). Then the temperature profile is smooth even for small Peclet numbers. 3.1. Validation As validation of the turbulent velocity field Fig. 2 compares our calculation’s turbulent viscosity nt with measurements from
Table 1 Comparison of positive eigenvalues and eigenconstants for Pe ¼ 5 between [7] and own calculation. j
lþ j
Aþ j
Own calculation
[7]
1 2 3 4 5 6 7 8 9 10
0.34939Eþ02 0.62435Eþ02 0.92858Eþ02 0.12378Eþ03 0.15493Eþ03 0.18617Eþ03 0.21747Eþ03 0.24880Eþ03 0.28014Eþ03 0.31151Eþ03
0.34940Eþ02 0.62435Eþ02 0.92858Eþ02 0.12378Eþ03 0.15493Eþ03 0.18617Eþ03 0.21747Eþ03 0.24880Eþ03 0.28014Eþ03 0.31151Eþ03
Own calculation 0.28355Eþ00 0.21311Eþ00 0.14878Eþ00 0.11300Eþ00 0.90672E01 0.75618E01 0.64817E01 0.56700E01 0.50382E01 0.45327E01
[7] 0.28355Eþ00 0.21311Eþ00 0.14878Eþ00 0.11300Eþ00 0.90670E01 0.75620E01 0.64820E01 0.56700E01 0.50380E01 0.45330E01
Ref. [13] for Re ¼ 350,000. The agreement is quite good. The temperature field can be validated with analytical data from Ref. [12] for infinite heating zones. These data itselves are compared þ to numerical data. The values of lþ j and Aj for Pe ¼ 5 for the geometrical setup case 1 (see Table 1) and the values of l j and Aj for ReDh ¼ 10,000 and Pr ¼ 0.001 for case 2 (see Table 2) are shown exemplary. The maximum deviation is in the order of 0.1%. 3.2. Results In the following we will analyse the results from calculations with different parameters. We look at both setups (case 1 and case 2), laminar and turbulent flows, different Peclet numbers and varying heating zone lengths. The semi-infinite heating zone is regarded as a special case of the finite heating zone. Finally we estimate the error done by a parabolic calculation. The Reynolds number of a laminar flow is fixed to Re ¼ 1000 while the Peclet number varies with Prandtl number. 3.2.1. Case 1 As it can be seen from Fig. 1, the wall temperature for 0 < ~ x<~ x1 is changed to Q ¼ 0 for case 1. Elsewhere Q is equal to one. The derived solution captures the effect of small heating sections. Thus ~ x1 tends to infinity the here presented x1 can vary arbitrarily. If ~ solutions approach the one given by the authors in Ref. [7]. In this section we represent the results for laminar flows only. 3.2.1.1. Variation of Peclet number. Fig. 3 shows the radial temperature profiles Qð~r Þ for Pe ¼ 1,5,10 and 50 at ~ x ¼ 0:1; 0:001; 0:001 and 0.1. The length of the heating zone x ¼ 0:1 the effect of axial heat conduction was ~ x1 ¼ 1. At ~ becomes only visible for a flow with Pe ¼ 1. At ~ x ¼ 0:001 the fluid with Pe ¼ 1 adapts to the boundary condition inside the heating zone uniformly along the radius of the Table 2 Comparison of negative eigenvalues and eigenconstants for Re ¼ 10,000 and Pr ¼ 0.001 between [7] and own calculations. j
Fig. 2. Comparison of turbulent viscosity between experimental data from Ref. [13] and own calculation for Re ¼ 350,000.
1 2 3 4 5 6 7 8 9 10
lj
A j
Own calc.
[7]
Own calc.
[7]
0.27601Eþ02 0.82875Eþ02 0.14294Eþ03 0.20454Eþ03 0.26666Eþ03 0.32905Eþ03 0.39158Eþ03 0.45420Eþ03 0.51687Eþ03 0.57957Eþ03
0.27650Eþ02 0.82988Eþ02 0.14304Eþ03 0.20464Eþ03 0.26673Eþ03 0.32913Eþ03 0.39165Eþ03 0.45427Eþ03 0.51695Eþ03 0.57966Eþ03
0.78839Eþ01 0.10378Eþ01 0.56323Eþ01 0.89974Eþ00 0.51651Eþ01 0.85911Eþ00 0.49929Eþ01 0.84263Eþ00 0.49138Eþ01 0.83498Eþ00
0.78826Eþ01 0.10389Eþ01 0.56474Eþ01 0.90273Eþ00 0.51836Eþ01 0.86200Eþ00 0.50098Eþ01 0.84509Eþ00 0.49280Eþ01 0.83690Eþ00
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95
Fig. 3. Radial temperature distribution for different Peclet numbers at different axial positions (~x1 ¼ 1).
annulus due to the great influence of axial heat conduction. For fluids with higher Peclet number the effects of axial heat conduction gain in importance only in the near wall region. The flow velocity is reduced due to the no slip condition at the wall and forwards the upstream axial heat conduction. This can be seen from the energy Eq. (14). If convective heat transport looses influence due to reduced velocity, the terms describing heat conduction have to be of the same order of magnitude. Towards the middle of the annulus higher velocities are reached and axial heat conduction looses influence. Inside the heating zone at ~ x ¼ 0:001 the temperature level is still higher for higher Peclet numbers. The differences between the Peclet numbers concerning the temperature gradient and the thermal boundary layers thickness are pointed up here. Temperature decreases rapidly near the wall for fluids with small Peclet numbers while the gradient towards the middle of the annulus is small. The increase of temperature is more even for higher Peclet numbers and the maximum temperature is reached earlier. Therefore the thermal boundary layer thickness is larger for fluids with smaller Peclet numbers despite their higher temperature gradient near the wall. At ~ x ¼ 0:1 the influence of radial and axial heat conduction depending on the Peclet number is illustrated. The radial heat conduction gains in importance for flows with higher Peclet numbers while the heat transfer due to axial heat conduction declines. For flows with smaller Peclet numbers the axial heat conduction prevails the boundary condition’s influence. Consequently, at this point the temperature level of flows with smaller Peclet numbers exceeds that of higher Peclet number. Summing up, flows with smaller Peclet numbers begin more early to adapt themselves to a new boundary condition while the whole adaption foregoes more slowly.
Fig. 4. Nusselt number for different Peclet numbers for a finite heating zone (~x1 ¼ 1).
The behaviour of the Nusselt number as shown in Fig. 4 confirms these explanations. The Nusselt number decreases more slowly for smaller Peclet numbers and rises earlier due to the larger temperature gradient at the wall. If the Peclet number is high there is no consequent adaption to the boundary conditions after the heating section. In fact the Nusselt number jumps to infinity. This is explained by the fact that the problem is more and more parabolic in character with lower influence of the boundary conditions if the Peclet number is increased. 3.2.1.2. Variation of heating section’s length. The length of the heating zone influences the heat transfer significantly. Therefore different length of the heating zone have been investigated for a fluid with Pe ¼ 5. The Nusselt number in short heating zones does not decline as much as in longer heating zones because less heat can be transferred into the fluid. Therefrom a shorter heating zone has a longer thermal entrance region relatively to its length. Because the heat transfer is most effective in the thermal entrance region, the Nusselt number is higher for shorter heating zones (see Fig. 5). 3.2.1.3. Estimation of the error done by a parabolic calculation. The parabolic calculation neglects the effects of axial heat conduction. However, the previous chapter showed that this effect is important for flows with small Peclet numbers regarding the temperature as well as the Nusselt number. Now we will estimate the error, that is done if a laminar flow through a semi-infinite heating zone is calculated parabolically. On the basis of this error we will specify a limit, which allows a parabolic calculation. Fig. 6 shows the relative error
Fig. 5. Nusselt number for different heating sections lengths (Pe ¼ 5).
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B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
Fig. 6. Nusselt numbers relative error due to parabolic calculation (~x1 /N).
DNurel ¼
Nuell Nupara Nuell
(55)
due to the neglection of the axial heat conduction on the basis of the Nusselt number for different Peclet numbers. The discrepancy is large for small Peclet numbers. It is more than 90% for Pe ¼ 1. The axial heat conduction is neglegible only for fluids with Pe 50. This result can of course not been applied to very short heating zones (see Fig. 5). Here the errors can be enormous if the elliptic character of the problem is not considered.
Fig. 8. Nusselt number at the inner (i) and at the outer (o) wall for Pe ¼ 1 and Pe ¼ 5 (~x1 /N).
the Nusselt number increasing with the Reynolds number, because turbulence enforces heat transfer. The thermal developing region behaves similar to that of a laminar flow. The Nusselt number increases for smaller Peclet numbers due to its larger temperature gradient as Fig. 7 shows.
3.2.2.1. Variation of Peclet number for turbulent flows. In contrast to laminar flows, the heat transfer in turbulent flows is not influenced only by the Peclet number, but also by the Reynolds and the Prandtl number so that flows might behave differently although the Peclet number is the same. This difference is caused by turbulence. The term að~r Þ in Eq. (22) is not equal to 1 in turbulent flows as it is in a laminar flow. Therefore, the Prandtl number as well as the Reynolds number influence the temperature field and make statements about the flow more complicate. In order to elucidate this effect of turbulence a fluid with Pr ¼ 0.001 and varying Reynolds number (Re ¼ 5000; 10,000; 25,000) was considered. Regarding the fully developed flow we see
3.2.2.2. Comparison of Nusselt numbers at the inner and the outer wall. The Nusselt number (Eqs. (50)e(53)) can be evaluated for the inner wall (Nui) by computing the temperature gradient at ~r ¼ 0. If the Nusselt number is based on the temperature gradient at ~r ¼ 1, we call it Nuo. Fig. 8 shows the absolute values of both Nusselt numbers, jNui jand jNuo j, for a laminar flow with Pe ¼ 1 and Pe ¼ 5. Despite symmetric boundary conditions, the inner wall’s Nusselt number reaches higher values than the Nusselt number based on the temperature gradient of the outer wall. The enhanced heat transfer expressed by the higher Nusselt number is caused by larger temperature gradients due to a thinner thermal boundary layer at the inner wall. The smaller boundary layer thickness in turn can be ascribed to the velocities at the inner wall. They are higher due to the annulus’ asymmetric velocity distribution. (The velocity distribution in a concentric annulus depends on the radii ratio c as can be seen in chapter 2.2.) It should be noticed, that a comparison between the Nusselt numbers jNui j and jNuo j has to be seen as a rather qualitative one. From their definition (Eqs. (50)e(53)) one can see that both Nusselt numbers are related to the same temperature difference Qb Qwall . Because of the asymmetric temperature profile in the annulus each Nusselt number should be related to another temperature difference which is characteristic for the inner and the outer wall for quantitative statements.
Fig. 7. Nusselt number in the thermal developing region depending on the Reynolds number (~ x1 /N); Pr ¼ 0.001.
Fig. 9. Comparison of bulk temperatures for case 1 (1) and case 2 (2) boundary condition for Pe ¼ 1 and Pe ¼ 5 (~x1 /N).
3.2.2. Case 2 The semi-infinite heating zone as well as the differences between laminar and turbulent flow will be of particular interest in this section. Furthermore we will compare the Nusselt numbers on the inner and the outer wall and we will look at the differences between both boundary conditions.
B. Weigand, K. Eisenschmidt / International Journal of Thermal Sciences 54 (2012) 89e97
3.2.2.3. Comparison of both boundary conditions (case 1 and case 2). We compare both boundary conditions (case 1 and case 2) by analysing the bulk temperatures and the Nusselt numbers for laminar flows with Pe ¼ 1 and Pe ¼ 5. The bulk temperature in an annulus with both walls heated (case 2) declines faster than for an annulus with one adiabatic wall (case 1), because the information about change in temperature reaches every radial point earlier (see Fig. 9). Therefore the thermal entrance length is shortened for case (2).
4. Conclusions Axial heat conduction has a strong impact on fluids with Peclet numbers smaller than 50. The neglection of axial heat transfer in our studies caused relative errors up to 90%. In flows with large Peclet numbers radial heat transfer prevailed the axial heat conduction. Consequently for flows with smaller Peclet numbers the adaption due to a modified boundary condition started earlier but it ran slower. Furthermore the heating zone’s length is an important parameter. We saw an effective heat transfer for short heating zones which results in large Nusselt number. However, the absolute heat transfer along the heating zone was small. In turbulent flows the influence of the Reynolds number and that of the Prandtl number was studied separately. Compared to laminar flows, the heat transfer was increased by turbulent transport. The flow’s behaviour was similar for both setups. The comparison showed that the Nusselt number reaches larger
97
values, if the inner wall is heated (case 2) instead of keeping it adiabatic (case 1). References [1] R.K. Shah, A.L. London, Laminar Flow Forced Convection, Academic Press, New York, 1978. [2] W.M. Kays, M.E. Crawford, B. Weigand, Convective heat and mass transfer, McGraw-Hill Education (2004). [3] S. Chen, Lattice Boltzmann method for slip flow heat transfer in circular microtubes: extended Graetz problem, Applied Mathematics and Computation 217 (2010) 3314e3320. [4] C.-J. Hsu, An exact analysis of low Peclet number thermal entry region heat transfer in transversely nonuniform velocity fields, AIChE Journal 17 (1971) 732e740. [5] B. Weigand, An exact analytical solution for the extended turbulent Graetz problem with Dirichlet wall boundary conditions for pipe and channel, International Journal of Heat and Mass Transfer 39 (1996) 1625e1637. [6] E. Papoutsakis, D. Ramkrishna, H.C. Lim, The extended Graetz problem with Dirichlet wall boundary conditions, Applied Scientific Research 36 (1980) 13e34. [7] B. Weigand, M. Wolf, H. Beer, Heat transfer in laminar and turbulent flows in the thermal entrance region of concentric annuli: axial heat conduction effects in the fluid, International Journal of Heat and Mass Transfer 33 (1997) 67e80. [8] B. Weigand, F. Wrona, The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli, International Journal of Heat and Mass Transfer 39 (2001) 313e320. [9] J.H. Spurk, Strömungslehre, Springer, 1996. [10] N. Wilson, J. Medwell, An analysis of heat transfer for fully developed turbulent flow in concentric annuli, International Journal of Heat Transfer 90 (1968) 43e50. [11] H. Pfitzer, Konvektiver Wärmetransport im axial durchströmten Ringspalt zwischen rotierenden Hohlwellen, Doctorial Thesis, TH Darmstadt, 1992. [12] B. Weigand, Analytical Methods for Heat Transfer and Fluid Flow Problems, Springer, 2004. [13] F. Durst, On Turbulent Flow Through Annular Passages with Smooth and Rough Cores, University of London, Dep. of Mech. Eng. Thermo-Fluids Sec., 1968.