This paper investigates a problem on the steady-state, conduction-convection heat transfer process in cylindrical porous heat exchangers. The governing partial differential equations for the system are obtained using the energy conservation law. Solution of these equations and the concept of enthalpy lead to a new approach to prove a theorem that the sum of inverse squares of all the positive roots of the zero order Bessel function of the first kind equals to one-forth. As a corollary, it is shown that the sum of one over pth power (p greater than or equal to 2) of the roots converges to some constant

Agrawal, O. P.

Lin, X.-A.

English

NASA Technical Reports Server

N94-23667
A THEOREM REGARDING ROOTS OF THE ZERO-ORDER
BESSEL FUNCTION OF THE FIRST KIND
X.A. Lin and O.P. Agrawa/
Department of Mechanical Engineering and Energy Processes
Southern minois University at Carbondale
Carbondale, IL 62901
ABSTRACT
This paper investigates a problem on the steady-state, conduction-convection
heat transfer process in cylindrical porous heat exchangers. The governing partial
differential equations for the system are obtained using the energy conservation law.
Solution of these equations and the concept of enthalpy lead to a new approach to
prove a theorem that the sum of inverse squares of all the positive roots of the zero
order Bessel function of the first kind equals to one-forth. As a corollary, it is shown
that the sum of one over pth power (p > 2) of the roots converges to some constant.
NOMENCLATURE
PRKCliDiI'_G PAGE BLA;'_K NOT FR.MED
W specific mass flow rate , Kg/m2.s
Cp specific heat of fluid at constant pressure , J/Kg.°C
h convective heat transfer coefficient , W/m2.°C
R radius of cylindrical board, m
r radial coordinate, m
z axial coordinate , m
T, solid temperature, °C
T,o solid temperature at circumference , °C
7'/ fluid temperature , °C
Ts0 inlet fluid temperature, °C
A WCpR/A_ , dimensionless parameter
B i3hR/WCp , dimensionless parameter
t (T, - T/o)/(Tso - T]o), dimensionless temperature of solid media
489
https://ntrs.nasa.gov/search.jsp?R=19940019194 2020-06-16T15:22:44+00:00Z
TX
Y
( T 1 - Tfo ) / (T,o - Tfo ) , dimensionless temperature of fluid
z/R, dimensionless axial coordinate
r/R, dimensionless radial coordinate
Greek Letters
specific area of heat transfer , m2/m 3 : -: ....
7 _,/_ , ratio of the axial and radial thermal conductivity
Az axial thermal conductivity of solid, W/m.°C
AT radial thermal conductivity of solid, W/m.°C
INTRODUCTION
z 2z : -:
Porous heat exchangers play a significant role in many engineering appli-
cations, such as cryogenics, thermal storage systems, and chemical reactors. Such
systems lead to a set of Partial Differential Equations (PDEs) with a strong coupling
between equations for the solid and the fluid phases. Analytical schemes to solve
a general class of problems in this area include the method of separation of vari-
ables (refs. 1 and 2), Pdemarm method (ref. 3), orthogonal collocation techniques
(ref. 4) and collocation-perturbation scheme for packed beds (ref. 5). Siegwarth
and Radebough (ref. 6) present a numerical technique to solve the above problems
with variable physical properties. Lin, Guo, and Wang (ref. 7) present a com-
bined orthogonal collocation-perturbation method to solve the temperature field in
a cylindrical porous heat exchanger.
In this paper, the physical problem presented in (Ref. 7) is reconsidered.
Using the approach of separation of variables, the PDEs associated with the prob-
lem are reduced to two Boundary Value Problems (BVPs) of Ordinary Differential
Equations (ODEs). Solution of the resulting BVPs leads to a new method to prove
a theorem regarding roots of the zero-order Bessel function of the first kind. As
a corollary, it is shown that the sum of one over pth power (p > 2) of the roots
converges to some constant.
The theory of Bessel functions and related functions is well established and
the main results are summarized in textbooks and mathematical manuals (refs. 8
=
490
to 10). The properties of Bessel functions concerning operations of differentiation
and integration with respect to their order have also been studied by Apelblat and
Kravitsky (ref. 11). However, little is known about the convergence and summation
of series for roots of Bessel functions due to no explicit expressions for the roots
except for 4-1/2 order Bessel functions. Although in 1874 Rayleigh derived a gen-
eral form of the theorem discussed here by applying an Euler's formula (ref. 8),
his approach is suitable for even power series only. Furthermore, he did not study
the convergence of such series for a general case. Therefore, research is needed to
further understand the properties of Bessel functions. The new method to prove the
theorem stated above is based on the mathematical model of a cylindrical porous
heat exchanger. A brief description of this model is presented next.
MATHEMATICAL MODELING
The model considered here is a semi-infinite cylindrical porous heat ex-
changer as shown in (fig. 1). Let r and z be the radial and the axial coordinates,
and R be the radius of the cylinder. A fluid flows through the porous media in axial
direction from left to right. The inlet temperature of the fluid is Tf0 which can be
higher or lower than the board circumference temperature T,o. Let W, Cv, and h
be the specific mass flow rate, the specific heat of fluid at constant pressure, and
the convective heat transfer coefficient respectively.
Porous materials used in such applications exhibit anisotropic behavior. It
is assumed that the thermal conductivity of the solid is symmetric about the axis
of the cylinder. Let _ be the specific area of heat transfer , kz and kr be the axial
and the radial thermal conductivities of the solid, and 7 be the ratio of ks and kr.
In the derivation to follow, we assume that: 1) the physical properties and
convective heat transfer coefficient between porous substance and fluid are con-
stants, 2) the dimensions of the porosity and solid particles are very small com-
pared to the overall dimension o{the Heat exchanger, and therefore, the porous
material can be treated as continuous media, 3) the fluid thermal conductivities
are negligible compared to the solid thermal conductivities, and 4) the inner wall
temperature is kept constant, and there are no thermal resistances between wall
491
and porous media or fluid. Finally, for simplicity, it is considered that T,0 > Tfo.
However, the formulation presented here equally holds for T,0 < T/o.
The differential equations of the system may be obtained by applying the
energy conservation law to a micro unit volume (ref. 7). Nondimensional forms of
these equations and the boundary conditions are given as follows:
Solid Phase :
c92t 10t c92t _ A cgT
- (i)
Fluid Phase : .....
0T
c9---_= B(t - T) (2)
=
Boundary Conditions :
t=l, y----1 (3a)
T = 0, x - 0 (3b)
t = 1, z _ +co (3c)
where t(= (T,- Tso)l(T,o-Ts0)) and T(= (T s -Tso)l(T,o-T/0)) are dimensionless
temperatures of the solid and the fluid, x(-- z/R) and y(- r/R) are dimension-
less radial and axial coordinates, and A(= WCpRIk,) and B(= _hRIWCp) are
dimensionless parameters. Here T, and T! are the solid and the fluid temperature
distributions. It should be noted that an additional boundary condition is required
to completely define the problem. However, it is not needed here and therefore it
is not considered.
Equations (1) to (3) can be used to find the temperature distributions of the
solid and the fluid phases. It will be shown that these equations can also be used
to prove the following theorem:
Theorem: Let c_,_denote the nth positive root of the zero-order Bessel function
of the first kind, then
Corollary: For p > 2, series
depends on p.
+_ 1 1
4 (4)
r_=l
Z+=_ _ converges to a constant C(p) which
492
Proof." In order to pro_e the above theorem, we begin by eliminating T from Eqs.
(1) and (2). This leads to
Let
_(02t l& 02t. _ 02_ l& 02t_Ox Oy 2 + y'_y + 7-Oz'iz2)= AB - B(._y 2 + y-_y + 7_-._.z2) (5)
t= I-XY
where X and Y are functions of z and y respectively. Using Eqs.
and the method of separation of variables, we obtain the following two equivalent
boundary value problems of ordinary differential equation:
(6)
(3) and (5)
and
y" + !y'_ AY- 0 (7a)
y
Y = o, u = 1 (7b)
[Y[< oo, u = 0 (?c)
f 7X'+TBX"+(A-AB)X'+ ABX=O (8a)
t X = 0, z ---* +oo (8b)
where a prime(') on X (Y) represents the derivative with respect to z (y), and
A is a separation constant. Equation (7c) suggests that Y is bounded at its center.
Equation (7) can be brought to standard Bessel equation form by a simple linear
transformation. This is a general Sturm - Liouville eigenvalue problem (ref. 14).
It has nontrivial solutions only when A,_ = -a_(a, _ 0). Using the properties of
Bessel functions, the solution of Eq. (7) for any o_,, may be written as
Y,, = c,,Jo(oe,_y) (9)
where Jo(z) is the zero order BesseI function of the first kind, a,, is the nth positive
root of Jo(Z), and c,_ is an undetermined constant.
Let the trial solution of Eq. (8) be given as
X : ale _x (10)
493
Substitution of Eq. (10) into Eq. (8) leads to the following characteristic equation,
7# 3 + 7B# 2 - (o_ + AB)/9 - c_B = 0 (11)
Equation (11) is cubic in _. The discrim!nant A-of thisequatlo n !s::given as
A = -4-_{4_(B27- a_) 2 + 18AB2a_7 + AB3(AB + 2_)+
4AB(A2B 2 + 3a_ + 3a_AB)} (12)
Equation (12) suggests that A < 0. Thus, using the theory of cubic equations, it
follows that all roots of Eq. (ll) are real (ref. 12). Let these roots be given as _,,z,
]3_2, and $_a. From Eq. (11), it follows that
#nl .4- #n2 "4- _n3 = -B < 0 (13)
and
#.i" #.3 = > 0 (14)
Equation (13) suggests that atleast one of the roots is negative and Eq. (14) indi-
cates that negative roots appear in pair. Thus, it is concluded that Eq. (I1) has
two negative roots and one positive root for each a_. From physical consideration
(or Eq. (8b)), the positive root is disregarded. The general solution for X may now
be written as
2
X,_ : 7_ anie
i=1
Substituting Eqs. (9) and (15) into Eq. (6), we get
(15)
¢o 2
n=l i:1
(16)
where a,,i = c,_a'_,i. From Eqs. (1), (2) and (16), we obtain
1 oo 2
T = 1 - A-"B" _" Jo(any)" _., a,,,e _''::" (oL_ - 7#_, + AB)
n:l i:i
(17)
Differentiation of Eq. (17) with respect to z yields,
(18)
494 _::
Using Eq. (17), the inlet condition (Eq. (3b)), and the property of orthogonality
of Jo(vL,_y) in the interval [0, 1], we derive
2AB
a,, • J,(a,,) (19)
where Jl(z) is a Bessel function Of the first kind of order one.
ENTHALPY RELATIONS
Some enthaipy relations help prove the above theorem. Let H(z) and H(z +
dx) be quantities of enthalpy c_'ied by the fluid across the axial planes at z and
z + dx respectively (Figure 1). Expression for H(x) is given as
H(x) = WCp. _on{rdr • fo2_ d6. IT10 + T(T,o - T/0)]} (20)
A closed form expression for H(x) can be obtained by substituting Eq. (17) into
Eq. (20), and integrating the resulting equation. Expanding H(z +dx) in a Taylor's
series, and neglecting the second and the higher order terms, we obtain
dH(x) = H(=+ dx) - H(=) = OH(x)Ox • dx (21)
where dH(z) is the enthalpy variation of the fluid passing through the control
volume between z and z + dz (Figure 1). The total change in enthalpy of the fluid
is obtained by ir_tegrating Eq. (21) over the length of the heat exchanger. Thus,
• dz (22)
Using Eqs. (lS), (20), and (22), we obtain
2,._WC_(Tso- T_o) _ J,(o_.)AH +_
= AB -" z....,{_. _ a,_,(o_ - 7Z_, + AB)} (23)
n=, C_n i=1
AH +°° can also be evaluated directly by subtracting the enthalpy of the fluid at
the inlet (- H(0)) from that at the outlet (= H(c_)). Expressions for H(0) and
H(c_) are given as
H(O) = WC_,_rR _ • T1o
and
ar(_) = wc_,_R_. Ts0
495
Hence_
AH+== wcp,_m . (Too-T<o) ' (24)
Comparing Eqs. (23) and (24), we obtain
• _Jl(_-).{ a.,(_ - _, + AB)} = 1 (25)
n=l (T. "
Finally, substituting Eq. (19) into Eq. (25) and simplifying, we obtain
+oo 1 1
This proves the theorem. Since function J0(y) is symmetrical, it follows that
+_ __1 = 1 (26)
In order to prove the corollary, observe that all positive roots of Jo(x) are greater
than 1 (ref. 8). This implies that forp >_ 2
1 1
Equations (4) and (27) suggest that the series
c.@) =
V=I _i
(27)
increases monotonically and it is bounded. Thus, by monotone convergence theo-
rem (ref. 13), it follows that the series C,(p) (n _ oo) is convergent.
CONCLUSION
A mathematical m0del for a steady-State c0nduction-convection heat transfer
processes in a cylindrical porous heat exchanger has been investigated. It has been
shown that the equations resulting in this model may be used to prove a theorem
and a corollary regarding roots of the zero-order Bessel function of the first kind.
= Research :reported in this paper provides an alternate appr0acfi toprove a
theorem in this area. The advantages of this physical approach _are tl/at it-enriches
physical understanding of a theorem, and that we may avoid difficulties emerging
from rigid mathematical arguments.
E
496
ACKNOWLEDGMENT
The authors wish to thank Prof. M. Tarabek, Department of Mathematics,
SouthernIllinois University at Carbondalefor hisvaluablesuggestionsand comments.
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9. Abramowitz, A.; and Stegun, I.E.: Handbook of Mathematical Functions. U.S.
National Bureau of Standards, 1965.
10. Gary, A.; and Macrobert, T.M.: A Treatise on Bessel Functions and Their
Applications to Physics, Macmillan, 1922.
11. Apelblat, A.; and Kravitsky, N.: Integral Representations of Derivatives and
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Math., vol. 34, Mar. 1985, pp. 187-210.
12. Condon, E.U.; and Odishaw, H.: Handbook of Physics. McGraw-Hill, 1967.
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14. Haberman, R.: Elementary Applied PartiaI DifferentialEquations. pp. 134-135,
pp. 216-217, Prentice- Hall, 1987.
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Y
A
I
I
|
!
I
I
Wso
x
Porous mesa
x+dx
X
Figure 1. Porous heat exchanger model
498


A theorem regarding roots of the zero-order Bessel function of the first kind

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