Effects of Lewis number on coupled heat and mass transfer in a circular tube subjected to external convective heating

Abstract

Heat and mass transfer in a circular tube subject to the boundary condition of the third kind is investigated. The closed form of temperature and concentration distributions, the local Nusselt number based on the total external heat transfer and convective heat transfer inside the tube, as well as the Sherwood number were obtained. The effects of Lewis number and Biot number on heat and mass transfer were investigated.

Appendix

The general solution of equation θ (Eq. 6) can be obtained by separation of variables [14]

$$ \theta = F_{1} e^{{ - 2\beta_{\rm n}^{2} X}} J_{0} \left( {\beta \eta } \right) $$

(24)

where, F ₁ is an unspecified constant and β is the eigenvalue.

By using separation of variables method, the solution of φ can be written as

$$ \varphi = H(X)W(\eta ) $$

(25)

Substituting Eq. 25 into Eq. 7, we have

$$ \frac{1}{2}\frac{{H^{\prime} }}{H} = \frac{1}{{{\text{Lew}}W\eta }}\left( {\eta W^{\prime\prime} + W^{\prime} } \right) = - \lambda^{2} $$

(26)

where λ is the eigenvalue. Equation 26 can be rewritten as

$$ H^{\prime} + 2\lambda^{2} H = 0 $$

(27)

$$ W^{\prime\prime} + \frac{1}{\eta }W^{\prime} + \left( {\sqrt {\text{Lew}} \lambda } \right)^{2} W = 0 $$

(28)

The general solution of Eq. 27 is

$$ H = C_{1} e^{{ - 2\lambda^{2} X}} $$

(29)

Equation 28 is a zeroth order of Bessel equation, based on the boundary condition specified by Eq. 9, its solution can be expressed as

$$ W = C_{2} J_{0} \left( {\sqrt {\text{Lew}} \lambda \eta } \right) $$

(30)

Therefore, the dimensionless concentration solution becomes

$$ \varphi = F_{2} e^{{ - 2\lambda^{2} X}} J_{0} \left( {\sqrt {\text{Lew}} \lambda \eta } \right) $$

(31)

where, F ₂ is another unspecified constant.

Substituting Eqs. 24 and 31 into Eqs. 10 and 11, one obtains

$$ \lambda = \beta $$

(32)

$$ F_{1} J_{0} \left( \beta \right) = F_{2} J_{0} \left( {\sqrt {\text{Lew}} \beta } \right) $$

(33)

Obviously, F ₁ = F ₂ when Lew = 1, thus,

$$ BiJ_{0} \left( \beta \right) = \beta \left( {L + 1} \right)J_{1} \left( \beta \right) $$

(34)

When ${\text{Lew}} \ne 1,$ Eq. 34 should be replaced by

$$ J_{0} \left( {\sqrt {\text{Lew}} \beta } \right){{Bi}}J_{0} \left( \beta \right) = \beta \left[ {\frac{L}{{\sqrt {\text{Lew}} }}J_{1} \left( {\sqrt {\text{Lew}} \beta } \right)J_{0} \left( \beta \right) + J_{0} \left( {\sqrt {\text{Lew}} \beta } \right)J_{1} \left( \beta \right)} \right] $$

(35)

There will be many different eigenvalues which satisfy Eqs. 34 or 35, thus, eigenvalues β _m (m = 1, 2, 3,…) obtained by the eigen equations can be expressed by

$$ Bi J_{0} \left( {\beta_{\rm m} } \right) = \beta_{\rm m} \left( {L + 1} \right)J_{1} \left( {\beta_{\rm m} } \right)\; {\text{at}}\; {\text{Lew}} = 1 $$

(36)

$$ \begin{gathered} \beta_{\rm m} \left[ {\frac{L}{{\sqrt {\text{Lew}} }}J_{1} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)J_{0} \left( {\beta_{\rm m} } \right) + J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)J_{1} \left( {\beta_{\rm m} } \right)} \right] \hfill \\ \quad \quad = {{Bi}}J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)J_{0} \left( {\beta_{\rm m} } \right)\; {\text{at}}\;{\text{Lew}} \ne 1 \hfill \\ \end{gathered} $$

(37)

Thus, θ and φ can be written as

$$ \theta \left( {\eta ,X} \right) = \sum\limits_{m = 1}^{\infty } {A_{\rm m} e^{{ - 2\beta_{\rm m}^{2} X}} J_{0} \left( {\beta_{\rm m} \eta } \right)} $$

(38)

and

$$ \varphi \left( {\eta ,X} \right) = \sum\limits_{m = 1}^{\infty } {B_{\rm m} e^{{ - 2\beta_{\rm m}^{2} X}} J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} \eta } \right)} $$

(39)

The equation should satisfy the boundary conditions of Eq. 8, i.e.,

$$ 1 = \sum\limits_{m = 1}^{\infty } {A_{\rm m} J_{0} \left( {\beta_{\rm m} } \right)} $$

(40)

Thus,

$$ A_{\rm m} = \frac{{\int_{0}^{1} {\eta J_{0} \left( {\beta_{\rm m} \eta } \right)d\eta } }}{{\int_{0}^{1} {\eta \left( {J_{0} \left( {\beta_{\rm m} \eta } \right)} \right)^{2} d\eta } }} = \frac{{2J_{1} \left( {\beta_{\rm m} } \right)}}{{\beta_{\rm m} \left[ {\left( {J_{0}^{2} \left( {\beta_{\rm m} } \right)} \right) + \left( {J_{1}^{2} \left( {\beta_{\rm m} } \right)} \right)} \right]}} $$

(41)

Similarly:

$$ B_{\rm m} = \frac{{\int_{0}^{1} {\eta J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} \eta } \right)d\eta } }}{{\int_{0}^{1} {\eta \left( {J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} \eta } \right)} \right)^{2} d\eta } }} = \frac{{2J_{1} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)}}{{\sqrt {\text{Lew}} \beta_{\rm m} \left[ {J_{0}^{2} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right) + J_{1}^{2} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)} \right]}} $$

(42)

Therefore, Eqs. 38 and 39 can be rewritten as

$$ \theta \left( {\eta ,X} \right) = \sum\limits_{m = 1}^{\infty } {\frac{{2J_{1} \left( {\beta_{\rm m} } \right)J_{0} \left( {\beta_{\rm m} \eta } \right)}}{{\beta_{\rm m} \left[ {J_{0}^{2} \left( {\beta_{\rm m} } \right) + J_{1}^{2} \left( {\beta_{\rm m} } \right)} \right]}}e^{{ - 2\beta_{\rm m}^{2} X}} } $$

(43)

$$ \varphi \left( {\eta ,X} \right) = \sum\limits_{m = 1}^{\infty } {\frac{{2J_{1} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)J_{0} \left( {\sqrt {\text{Lew}} \beta_{\rm m} \eta } \right)}}{{\sqrt {\text{Lew}} \beta_{\rm m} \left[ {J_{0}^{2} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right) + J_{1}^{2} \left( {\sqrt {\text{Lew}} \beta_{\rm m} } \right)} \right]}}e^{{ - 2\beta_{\rm m}^{2} X}} } $$

(44)

where β _m can be obtained by Eqs. 36 and 37.