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Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

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Abstract

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.

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Fig. 3

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Abbreviations

a :

thermal diffusivity of liquid, m2 s−1

b :

coefficient in Eq. 16

c p :

specific heat, kJ kg−1 K−1

d :

coefficient in Eq. 16, K−1

D :

coefficient of molecular diffusion, m2 s−1

f(θ):

function, Eq. 3

F(φ):

function, Eq. 15

\(K_1 = \frac{{c_p }} {{dL}}\) :

dimensionless number

k c :

local mass transfer coefficient, m s−1

\(\overline{k}_{\text{c}}\) :

averaged mass transfer coefficient, m s− 1

L :

heat of dissolution, kJ kg−1

Le:

Lewis number (D/a)

N A :

mass flux density of the solute, kgm2 s−1

Pe1:

Peclet number for uniform-translational flow (RU/D)

Pe2:

Peclet number for axisymmetric shear flow (Ω R2/D)

Pe3:

Peclet number for a flow with parabolic velocity profile (Λ R3/D)

r :

radial coordinate, m

R :

sphere radius, m

R 0 :

initial radius of a sphere, m

Sh:

Sherwood number (Rkc/D)

\(\overline {{\text{Sh}}} \) :

\((R\overline{k} _{\text{c}} {\text{/}}D)\)

Sc:

Schmidt number (ν/D)

T :

temperature of liquid, K

T ′ s :

equilibrium temperature corresponding to bulk concentration, K

x A :

mass fraction of the solute

\(x_{A_s } \) :

mass fraction of the solute at solid–liquid interface

\(x_{A_0 } \) :

mass fraction of the solute in the bulk of liquid

\(x^\prime_{A_s}\) :

equilibrium mass fraction of solute at the bulk temperature

U :

uniform fluid flow velocity, m s−1

v y , vθ:

velocity components, m s−1

y :

distance from a surface of a solid sphere, m

Γ(z):

gamma function

γ:

correction factor for the effect of finite solute concentration level for isothermal dissolution

ζ c ,ζ T :

variables, Eq. 10, m9/2 s− 3/2

η c ,η T :

similarity variables, Eq. 11

θ:

angular coordinate, rad

θ0:

angle of accumulation, rad

Λ:

curvature of velocity profile on the axis of symmetry away from a particle, m−1 s−1

λ:

thermal conductivity of liquid, kJ m−1 s−1 K− 1

μ:

dynamic viscosity of liquid, Pa s

ν:

kinematic viscosity of liquid, m2 s−1

ρ:

bulk liquid density, kg m−3

ρ0:

saturated solution density, kg m−3

ρs:

solid density, kg m−3

φ:

dimensionless average mass velocity

Ω:

velocity gradient, s−1

ωs:

relative intensity of shear and translational motions (10ΩR/U)

0:

value in the bulk of liquid

A :

solute

θ:

tangential direction

is:

isothermal

0:

infinite dilution of the solute in a solvent

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Authors and Affiliations

  1. Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 84105, Israel

    T. Elperin & A. Fominykh

Authors
  1. T. Elperin
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  2. A. Fominykh
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Corresponding author

Correspondence to T. Elperin.

Appendix

Appendix

1.1 Derivation of Eqs. 5 and 6

Replacing variables (y, θ) by (ψ, θ) we obtain:

$$ \left( {\frac{{\partial x_A (\theta ,y)}} {{\partial \theta }}} \right)_y = \left( {\frac{{\partial x_A }} {{\partial \theta }}} \right)_\psi + \frac{{\partial x_A }} {{\partial \psi }}\frac{{\partial \psi }} {{\partial \theta }} = \left( {\frac{{\partial x_A }} {{\partial \theta }}} \right)_\psi + \frac{{\partial x_A }} {{\partial \psi }}v_y R^2 \sin \theta , $$
(26)
$$ \frac{{\partial x_A (\theta ,y)}} {{\partial y}} = \frac{{\partial x_A }} {{\partial \psi }}\frac{{\partial \psi }} {{\partial y}} = - v_\theta R\sin \theta \frac{{\partial x_A }} {{\partial \psi }}, $$
(27)
$$ \frac{{\partial ^2 x_A (\theta ,y)}} {{\partial y^2 }} = \frac{\partial } {{\partial y}}\left[ { - v_\theta R\sin \theta \frac{{\partial x_A }} {{\partial \psi }}} \right] = - v_\theta R\sin \theta \frac{\partial } {{\partial \psi }}\left[ { - v_\theta R\sin \theta \frac{{\partial x_A }} {{\partial \psi }}} \right]. $$
(28)

Substituting Eqs. 26, 27 and 28 into Eq. 1 we obtain:

$$ \frac{1} {R}\frac{{\partial x_A }} {{\partial \theta }} = DR\sin \theta \left\{ {\frac{\partial } {{\partial \psi }}\left[ {(v_\theta R\sin \theta )\frac{{\partial x_A }} {{\partial \psi }}} \right] + \frac{1} {{1 - x_{A_s } }}\left( {\frac{{\partial x_A }} {{\partial \psi }}} \right)_{\psi = 0} \frac{{\partial x_A }} {{\partial \psi }}} \right\}. $$
(29)

Expressing vθ in terms of ψ

$$ v_\theta = - 2\sqrt \psi \sqrt {f(\theta )} (R\sin \theta )^{ - 1} $$
(30)

After substitution of Eq. 30 to Eq. 29 we obtain Eq. 5. Following similar approach we obtain energy Eq. 6 from Eq. 2.

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Elperin, T., Fominykh, A. Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow. Heat Mass Transfer 41, 442–448 (2005). https://doi.org/10.1007/s00231-004-0555-z

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