Asymptotic Solutions and Range of Validity for Heat Transfer in Packed-Bed and Capillary Tubes

DOI : 10.17577/IJERTV4IS080445

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Asymptotic Solutions and Range of Validity for Heat Transfer in Packed-Bed and Capillary Tubes

Chinyere Mbachu11 1Department of Chemical Engineering, Tennessee Technological University,

Prescott 214, 1020 Stadium Drive,

P. O. Box 5013, Cookeville, TN 38505, Tennessee, USA

Abstract:- In this paper, an asymptotic analytical solution was used to evaluate convective-diffusive heat transfer in packed- bed and capillary tubes. Asymptotic solution to the heat transfer problems found in packed-bed systems with axial convective, radial diffusive with constant heat flux at the wall, and in capillary tubes were studied and analyzed from the results of the analysis reported by Bird et al. [1] and Arce and Trigatti, [2]. The role of asymptotic solution and its ranges of the validity of both for the packed-bed and capillary tubes were assessed and ranges of validity reported.

Keywords: Asymptotic solution; heat transfer; convection; capillary tube; packed-bed tube.


while diffusion dominates the transport in the radial direction of the flow [5]. The fluid is assumed to exchange heat at a constant rate without a volumetric source present within the bulk domain.

B Mathematical Model Equation for Capillary Tube

The basic equations for the system under study and the boundary conditions were properly derived from the general heat transfer equation given by Bird et al. [1]. For the capillary tube, the mathematical model for the steady state convective diffusion equation under the assumptions stated is given by

Capillary and packed-bed tubes are used in

T 1


various industrial applications and are important to the

pC pVz r z

k r r r r


practical aspects of engineering processes such as reactors, heaters, evaporators, thermal, and energy storage units [1, 2]. Analysis of capillary tubes with laminar flow regime and packed-bed tubes for Newtonian fluids under incompressible conditions was presented with focus on


Vz r

r 2

Vmax 1


assessing the range of validity of an asymptotic solution for the heat transfer problem. A restriction on the asymptotic solutions is based on the fact that convective-diffusive transfer within the domain is controlled by a constant rate at the capillary walls. The assessment of the asymptotic solution is very important in the analysis of the design of reactors or heat exchangers [3]. Several studies have been done to correlate heat transfer by forced convection from circular tubes [4]. However, it appears that the validity ranges are not reported in the literature.


The boundary conditions for equation (2) and the parameters are introduced as follows:

at r = 0, T = finite or, alternatively, 0 (symmetry) (3)



A General Assumptions for the Mathematical Model

The geometry of the capillary and packed-bed tubes are assumed to be cylindrical with a length that

at r = R, k T


at z = 0, T = T1


(constant) (4)


satisfies the long channel assumption (R/L 1), see

By defining the following dimensionless variables [1]

figures 1 and 2. The energy equation for capillary

T T1 ;

r ; Z z


cylindrical coordinates is given by Bird et al. [1] and was described in section B. The fluid is assumed to be a Newtonian fluid and the flow is assumed to be pressure-

q0 R / k

R c

p vz,max

R2 / k

driven, fully developed flow, and incompressible. Convection is assumed to dominate in the axial direction

The capillary equation of heat transfer then reduces to



at 0 ,




) Z



at ,



The boundary conditions in equations (1) – (5) in non- dimensional terms are

at 0 , 0


An asymptotic solution for this model was explored in the sections below.

Figure 1: System description of a capillary tube.

C Mathematical Model Equation for Packed-Bed Tube

The convective-diffusive transport equation for the case of a Poiseuille flow in a packed-bed tube as reported by Arce and Trigatti [2] is given by

Where v is the area-averaged velocity of the fluid moving through the packed-bed, and with the same assumptions stated above, the wall of the packed-bed tube at Rt shows that the system (see figure 1) either gains or losses heat at a constant rate q0 and at

T k T

pm c pmv z


r r r


the center of the tube, symmetry is assumed by Arce and Trigatti [2]. The boundary conditions for equation 11 can be stated as

Then, by using eq. (12) and eq. (13), the following non- dimensionless forms are as follows

T 1

0 ;

  • k T

q ; and T x 0 T




r r 0

r r Rc



And if the dimensionless variables are given as

And the boundaries and entrance conditions are given by

r ;

z ; and

T T0




Z 0 0


R m c

vR2 / k









An asymptotic solution for this differential model was obtained below.

Figure 2: System description of a packed-bed tube.


  1. Capillary Tube Model Equation

    The fact that this solution is valid for case z 0 implies that the condition at z = 0 needs to be replaced by a global balance as stated by Bird et al. [1]. This is the reason to

    use , Z instead of simply , Z .

    2 R

    For distances that are away (z 0) from the entrance of the capillary tube, the temperature profile can be approximated by a linear relation in the z-direction as reported by Bird et al. [1]. This assumption yields the mathematical formulation of the asymptotic solution and in terms of the nomenclature used above as follows

    2Rzq0 pC p T T1 vz rdrd

    0 0

    Now, in dimensionless form, this equation leads to




    , Z C Z


    Z , Z 1 2 d 0


    As stated by the Bird et al. [1] function in equation

    (16) must satisfy

    Therefore, the final solution for the asymptotic solution for the cylindrical tube is obtained (see Figure 3) and given by

    1 d d 2

    , Z 4Z 2 1 4 7 (29)

    C 1

    d d o


    4 24

    B Packed-Bed Tube Model Equation

    And by applying the boundary conditions


    at 0 , = finite and (20)

    Similarly to the case of a capillary tube, the asymptotic solution for the Poiseuille flow with convective-diffusive heat transfer in a packed-bed tube can be written as

    at 1 , d 1



    , C


    By solving equation (19), the following result is obtained

    In order to determine function , this must satisfy the



    following differential equation

    C0 C1 ln C2


    d d

    4 16

    d d c0


    Where C1


    C2 are constants of integration that by

    applying the boundary condition given by equations (20) and (21) can be determined as

    As well as the boundary conditions

    C1 0

    C 7

    2 24




    d 0

    0 ; and


    d 1



    The equation (22) leads to the following general solution for

    The solution is reported by Arce and Trigatti [2] and only a summary of the key steps for the convenience of the reader was done. As before, see Equation (17), the condition at the inlet of the packed-bed must be replaced


    4 7

    by an integral condition valid also for the function



    16 24


    , Z . This equation can be derived using an integral energy balance in the system.



    [Energy due to transport by convection]

    Substituting equation (25) into Equation (16) and by applying the asymptotic solution, then the temperature


    0 rdrdpm c pmV (T T0 )


    profile is given by

    2 4

    [Energy loss at the wall of the tube] wRxq0




    , Z C Z C C ln C


    4 16 1 2

    Now, by using Equation (33) and Equation (34) and invoking the energy balance in the system gives


    ' ' '

    Or, alternatively,

    d ,




    4 7

    After using the non-dimensional variables, the solution for

    leads to

    , Z C0 Z C0


    16 24



    o 2 c ln c



    1 2

    Constant C0

    C0 4

    can now be obtained to yield


    And after using the boundary conditions in Equation (32), function is given by

    1 1


    2 2


    , Z changes sign within the domain of validity. This change of sign is a result that the asymptotic solution is no longer valid; therefore, the condition determines the

    The constant C0


    can be obtained from Equation (35) to

    values from which meaningful results.

    , Z

    yields physically

    C0 2


  2. Case 2- Packed-Bed Tube

For the packed-bed tube, the system is assumed to

Finally, the asymptotic function for , z is given by


be cooling. From the physical point of view, under Poiseuille flow conditions and in order to establish the ranges of validity for the packed-bed tube, the function

, , it is seen that when Z takes smaller values, the

, 2

1 2 2



function ,

tends to cross the axis

, 0 as it should be since the validity of

For the case of the packed-bed tube in cylindrical coordinates is shown in Figure 4.


, is for large. A condition for the validity can be obtained by simply stating that


A Case 1- Capillary Tube

The ranges of validity of the asymptotic solution in a capillary tube can be obtained using Equation (29) and

, 0

With this, Equation (39) can be written as

1 1 2 2



after imposing the condition following equation is obtained

, Z 0

, the


This equation is used to plot the graph as seen in




1 *4



for 0 1



Figure 4. From this figure 4, it is possible to see the following: first for 0 1 and Z 0.125 , the


0 and belongs to a physically meaningful

From the physical point of view, this system describes a heating fluid under Poiseuille flow conditions. This implies that according to the definition of used in here, the proper domain for this solution should be 0. Now, values that are located within the domain of 0 (see Figure 3) are not physically meaningful. Therefore, the boundary between the two domains for can be calculated from the condition 0 or alternatively

domain. However, for values of Z 0.125, the asymptotic solution is still valid (partially) for values of

0.5 1 . In other words, if the full domain of

0 1 is desired to be considered, the Z is restricted

to Z 0.125 in order for to be within the range of validity. On the other hand, if the full range of Z

0 Z is considered, then the partial values of

1 *2

1 *4 7

0.5 1

must be used in order for

to be


4 16



within the region of validity.

The plot of this boundary is shown in Figure 3 and it is observed that the upper region for the boundary between the two domains for is the physical meaningful domain. From Figure 3, this is clearly when Z 0.073

and 0 1 . In other words, for all values


Based on the prediction suggested by the validity of the capillary tube, Figure 5 is presented. All values of Z0.073 (i.e., Z=1, Z= 0.075, Z=0.5) comply with the

solution , being within the physically

0 1 , the smallest value of Z , when the solution

meaningful domain. For example, for Z= 0.25, Z= 0.1, Z=

Z 0.073

Z 0.05 and Z= 0.01, the values of , fall outside the

, is valid is . Moreover, if values of are

such that Z 0Z Z 0.073, the solution is only valid for * 1 ; where * = 0.29. The reason

behind this condition as indicated above is the fact that

region of physically meaningful domain. Similarly, Figure 6 illustrates the results for the case of the packed-bed tube- for example, Z=0.25, Z=0.5, Z=0.75, and Z=1 all have

values of , being inside the physically

meaningful domain for this case. However, the values for

Z=0.1, Z=0.05, and Z=0.01 do not belong to this region. As a final remark, the results of these figures effectively

illustrate the useful predictions of the ranges of validity.

Figure 3: Sketch of the range of validity of asymptotic solution in a cylindrical capillary tube.

(No physically meaningful

(Physically meaningful

Figure 4: Sketch of the range of validity of asymptotic solution in a packed-bed tube.

Figure 5: Heating process in a capillary tube.

Figure 6: Cooling process in a packed-bed tube.


Differential models for two cases of heat transfer (i.e. a capillary system under heating conditions and a packed-bed system under cooling conditions) were developed and their boundary conditioned formulated. Asymptotic solutions for both models were proposed by following the suggestions of Bird et al. [1] that are consistent with a case of constant rate of cooling (or heating) at the external walls [6]. Moreover, to the best of

the researchers knowledge, the ranges of validity of both asymptotic solutions were determined and illustrations of the behavior of the systems (within the ranges) were presented. The analysis suggests that for these cases, the asymptotic solutions present limitations for both independent variables as opposed to only one as suggested before. Finally, these solutions and the range of validity are useful to improve the understanding of the behavior of the systems described.


Financial support of the Schlumberger Foundation Faculty for the Future Fellowship is greatly acknowledged. Thanks to Offices of Research and Graduate Studies, Tennessee Technological University, Cookeville, Tenessee, USA.

c pm


c1 , c2 DH


Heat capacity of the cooling/heating fluid Proportionality constant

Integration constants



  1. R. B., Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, New York, 2002, pp. 311-316.

  2. P. E. Arce and I. M. B. Trigatti, Convective-diffusive heat transfer in packed-bed with constant heat flux at the wall: An asymptotic solution for large, International Communications in Heat

    and Mass Transfer. 21(3), 1994: 435-446.

  3. M. Mansour, A. Sheboul, and A. N. Hussein, An analytical solution for diffusion and reaction in a laminar flow tubular reactor, International Communications in Heat and Mass Transfer. 16, 1989: 603-608.

  4. S. M. Ross, A mathematical model of mass transport in a long permeable tube with radial convection, Journal of Fluid Mechanics. 63, 1974: 157-175.

  5. G. R. Ahmed and M. M. Yovanovich, Analytical method for forced convection from flat plates, circular cylinders, and spheres, Journal of Thermophysics and Heat Transfer. 9 (3), 1995: 516-523.

  6. P. E. Arce, A. E. Cassano, and H.A. Irazoqui, The tubular reactor with laminar flow regime: An integral equation approach homogeneous reaction with arbitrary kinetics. Computers and Chemical Engineering. 12, 1988: 1103-1113.

h Convective heat transfer coefficient

H Width of the packed-bed

  1. Thermal diffusivity

    kg Global mass constant

  2. Length


, m Density m Convective heat transfer coefficient

q0 Heat flux at the wall of the system

r Radial position of the packed-bed tube R Radius of the packed-bed tube

T Temperature profile

T0 Non-dimensional temperature profile v, V Area Averaged Velocity of the fluid

z,max Maximum Velocity in the z-direction

vz (r) Average velocity in the z- direction

Greek symbols

Dynamic viscosity (Pa s)






Radial component

Non-dimensional axial position

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