1 Introduction

Problems of convective heat and mass transfer from a rotating disk for the Prandtl or Schmidt numbers much larger than unity may be encountered in many industrial and scientific applications. In electrochemistry, rotating disk electrode technique is widely used for the experimental determination of the diffusion coefficient [1–7]. For example, in experiments [1–5], Schmidt numbers were in the range of Sc = 34–10320. The second area of application actual for the present research is the naphthalene sublimation technique, where Sc = 2.28–2.5, often employed in experimental measurements of the mass transfer coefficients h m [8–13].

Based on experimental and theoretical studies, many authors developed empirical and semi-empirical equations for heat and mass transfer for the Prandtl and Schmidt numbers larger than unity. Resulting equations for the local and average Nusselt and Sherwood numbers can be written in the following generalized form [13–18]

$$ Nu = K_{1} Re_{\omega }^{{n_{\rm R} }} , \quad Nu_{\text{av}} = K_{2} Re_{\varphi }^{{n_{\rm R} }} , $$
(1)
$$ Sh = K_{1} Re_{\omega }^{{n_{\rm R} }} , \quad Sh_{\text{av}} = K_{2} Re_{\varphi }^{{n_{\rm R} }} , $$
(2)

where n R = 1/2 for laminar flow; values of n R for turbulent flow obtained by different authors are discussed below. Constants K 1 and K 2 depend on the boundary conditions, flow regime and Pr or Sc numbers. Energy and convective diffusion equation (or mass transfer equation), as well as their boundary conditions, are analogous, so that we will switch between theoretical solutions or experimental equations for the Nusselt and Sherwood numbers throughout the paper via substituting Sc, Sh and concentration C instead of Pr, Nu and temperature T (and vice versa), respectively.

The boundary condition for the disk’s surface temperature in many cases can be approximated by a power–law [14–18]

$$ T_{w} - T_{\infty } = c_{0} r^{{n_{*} }} , $$
(3)

where c 0 = const, n * = const. For the diffusion process, T being substituted with C, exponent n * is zero, which means constant concentration on the surface C w = const. Thus, heat/mass transfer analogy takes place, when both T w = const and C w = const.

An interrelation of constants in Eqs. 1, 2 is given in the following form [15]

$$ K_{2} = K_{1} {\frac{{2 + n_{*} }}{{2n_{\text{R}} + n_{*} + 1}}}. $$
(4)

According to numerous experimental investigations of different authors [1, 3–5, 8, 9, 12, 19–21], laminar flow over a rotating disk persists up to the values of the local Reynolds numbers Re ω = (2.0–2.3)105, transitional flow develops between Re ω = (2.0–2.3)105 and Re ω = (2.5–3.2)105, and the region of developed turbulent flow exists for the Reynolds number higher than Re ω = (2.5–3.2)105. In the present paper, only developed turbulent flow is considered.

Author [3] measured average Sherwood numbers for an entire disk. Transitional region existed at Re ω = (2.3–2.78)105. Data obtained for turbulent regime at Re φ = (0.278–1.8)106, Sc = 930–10320 were approximated with an empirical equation, whose asymptotical forms for the developed turbulent flow (also in view of Eq. 4) look as

$$ Sh_{\text{av}} = 1.08 \times 10^{ - 2} Re_{\varphi }^{0.87} Sc^{1/3} , $$
(5)
$$ Sh = 1.48 \times 10^{ - 2} Re_{\varphi }^{0.87} Sc^{1/3} . $$
(6)

Sherwood numbers for the entire disk were measured also in work [1] over the range of Re φ = 5 × 104–1.8 × 106, Sc = 345–6450. Transitional flow was observed at Re φ = (2.3–2.9)105. For the developed turbulent flow, i.e. for Re φ = 3 × 105–1.8 × 106, authors [1] obtained the following asymptotic correlation

$$ Sh_{\text{av}} = 0.0725Re_{\varphi }^{0.9} Sc^{0.33} . $$
(7)

Experimental data for Sh av for the entire disk as well as the data for the local Sherwood number at laminar, transitional and turbulent flow over the range Re ω = 4 × 104–2.2 × 106, Sc = 680–7200 were obtained by the authors [2]. Here transitional flow existed for Re ω = (2.2–3.0)105. For the developed turbulent flow at Re ω = 3.0·105 × 2.2 × 106, authors [2] obtained the following relations

$$ Sh = 1.09 \times 10^{ - 2} Re_{\omega }^{0.91} Sc^{1/3} , $$
(8)
$$ Sh_{\text{av}} = 7.67 \times 10^{ - 3} Re_{\varphi }^{0.91} Sc^{1/3} . $$
(9)

Experimental data [4] for Sh av for an entire disk were obtained for Re φ = 104–1.18 × 107, Sc = 34–1400. For the developed turbulent flow at Re φ = 8.9 × 105–1.18 × 107, authors [4] obtained the following empirical correlation

$$ Sh_{\text{av}} = 1.17 \times 10^{ - 2} Re_{\varphi }^{0.896} Sc^{0.249} . $$
(10)

One should point out significant deviations of the experimental data [4] from the approximating dependence (10).

Authors [5] presented their own approximation of the experimental data [1] for developed turbulent flow

$$ Sh_{\text{av}} = 7.8 \times 10^{ - 3} Re_{\varphi }^{0.9} Sc^{1/3} . $$
(11)

Theoretical correlations for the local Sherwood numbers for developed turbulent flow and high Schmidt numbers obtained with the help of different model assumptions in [22, 23] look as follows, respectively,

$$ Sh_{\text{av}} = 7.07 \times 10^{ - 3} Re_{\varphi }^{0.9} Sc^{1/3} , $$
(12)
$$ Sh_{\text{av}} = 5.93 \times 10^{ - 3} Re_{\varphi }^{0.91} Sc^{0.34} . $$
(13)

A solution for the average Sherwood number that coincides with experimental Eq. 11 was obtained in the theoretical investigation [24]. A similar solution was obtained using a theoretical model in [25] with the coefficient equal to \( 6.43 \times 10^{ - 3} \).

One can conclude from the material described above that the literature provides a variety of empirical and theoretical equations for Sherwood number for turbulent flow, which differ from each other rather significantly.

Therefore, the objectives of the present research were: (a) to validate the relations for the local and average mass transfer of a rotating disk at turbulent flow for Schmidt numbers significantly larger than unity and to provide recommendations for their further use; (b) to develop an original approach to modeling this problem via a further development of the integral method successfully used earlier by the author for the case of heat transfer at air cooling in rotating-disk systems.

2 Comparative analysis of the equations for the Sherwood number

Several empirical and theoretical equations for the Sherwood number agree well with each other. As seen from Fig. 1, empirical Eqs. 5, 7 and theoretical Eq. 12 practically coincide. It is easy to see that theoretical Eq. 13 also agrees well with Eqs. 5, 7 and 12. Approximation (11) [5] exceeds noticeably the original experimental data [1]. Theoretical curve [25] based on Eq. 11 with the coefficient \( 6.43 \times 10^{ - 3} \) would lie 9% lower than line 6 obtained using Eq. 12. Equation 9 exceeds significantly Eq. 5. Very good agreement of empirical equations 5 and 7 is thus evidence in favor of the reliability of these experimental data.

Fig. 1
figure 1

Average Sherwood number for the high Schmidt numbers. Approximation of experiments: 1 laminar flow, equation of Levich [6]; 2 Eq. 5 [3]; 3 Eq. 7 [1]; 4 Eq. 11 [5, 24]; 5 Eq. 9 [2]. Theoretical solution: 6 − Eq. 12 [22]

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Empirical equation 10 is in fact the only one where the power exponent 0.249 at the Schmidt number differs noticeably from 1/3. Such a significant disagreement with the results of all the other experimental and theoretical investigations known in the literature, together with the aforementioned deviation of the experimental data from the approximation curve [4], is most probably the evidence of the erroneous value of the exponent 0.249. As the data in Fig. 2 show, differences among the curves that one can obtain based on Eq. 10 in the coordinates \( Sh_{\text{av}} /Sc^{1/3} \) versus Re φ for the different Schmidt numbers (over the range of variation of Sc observed in experiments [4]) is rather significant. Therefore, Eq. 10 should be considered unreliable.

Fig. 2
figure 2

Average Sherwood number for the high Schmidt numbers, approximation of experiments: 1 laminar flow, equation of Levich [6]; 2 Eq. 5 [3]; 3 Eq. 10, Sc = 34 [4]; 4 Eq. 10, Sc = 109 [4]; 5 Eq. 10, Sc = 1300 [4]

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The range of the Schmidt numbers investigated in electrochemical experiments [1, 3], whose data we consider the most reliable, is Sc = 345–10320. Experimental data [8, 9] for the Sherwood numbers at naphthalene sublimation in air for Sc = 2.28 were published significantly later. A comprehensive review and analysis of experimental data and empirical equations for naphthalene sublimation in air is given in work [13]. It is therefore interesting to find out whether Eqs. 5–7 can be applied to the case of the moderate values Sc = 2.28–2.5.

For developed turbulent flow in the case of Sc = 2.28, authors [8, 9] obtained the following empirical equations, respectively

$$ Sh = 0.0512Re_{\omega }^{0.8} , $$
(14)
$$ Sh = 0.0518Re_{\omega }^{0.8} . $$
(15)

An analysis of the data for the naphthalene sublimation in air fulfilled in Fig. 3 shows that empirical Eq. 6 [3] used at Sc = 2.28 allows computing Sherwood numbers close enough to the experimental data [8, 9] and approximations (14) and (15). Equations 6 (curve 8) and (14) (curve 7) agree well with each other for the sufficiently high Reynolds numbers Re ω = (0.6–2.0)106. As could be expected, curve 9 plotted based on Eq. 8 exceeds significantly the experimental data in Fig. 3.

Fig. 3
figure 3

Local Sherwood number for the naphthalene sublimation in air. Experiments: 1 Sc = 2.28 [8]; 2 Sc = 2.4 [10]; 3 Sc = 2.4 [11]; 4 Sc = 2.44 [12]; 5 Sc not mentioned [9]. Approximation of experiments using Eqs. 2: 6 laminar flow, n R = 1/2, K 1 = 0.625 [10, 11]. Developed turbulent flow, Sc = 2.28: 7  Eq. 14 [8]; 8 Eq. 6 [3]; 9 Eq. 8 [2]

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Thus, Eqs. 5–7 are the most reliable for the computation of the Sherwood numbers for the cases of high Schmidt numbers at developed turbulent flow. As the analysis has shown, these equations with the maximal uncertainty less than 7% can allow estimating mass transfer intensity also at moderate Schmidt numbers of Sc = 2.28.

3 An integral method for modeling heat and mass transfer at Pr and Sc numbers much larger than unity

3.1 Basic equations and their solutions

An integral method developed by the author earlier has proved its validity for the air cooling problems in rotating-disk systems [17, 18, 26, 27]. It is therefore challenging to find out whether this method can be also valid for modeling heat and mass transfer for the Prandtl and Schmidt numbers much larger than unity.

The integral equations of the boundary layer are as follows [15, 18, 27]:

$$ {\frac{d}{dr}}\left[ {r\int\limits_{0}^{\delta } {v_{\rm r}^{2} dz} } \right] - \int\limits_{0}^{\delta } {v_{\varphi }^{2} dz} = - {{r\tau_{{{\text{w}}r}} } \mathord{\left/ {\vphantom {{r\tau_{{{\text{w}}r}} } \rho }} \right. \kern-\nulldelimiterspace} \rho }, $$
(16)
$$ {\frac{d}{dr}}\left[ {r^{2} \int\limits_{0}^{\delta } {v_{\rm r} v_{\varphi } dz} } \right] = - {{r^{2} \tau_{{{\text{w}}\varphi }} } \mathord{\left/ {\vphantom {{r^{2} \tau_{{{\text{w}}\varphi }} } \rho }} \right. \kern-\nulldelimiterspace} \rho }, $$
(17)
$$ {\frac{d}{dr}}\left[ {r\int\limits_{0}^{{\delta_{T} }} {v_{\rm r} (T - T_{\infty } )dz} } \right] = aNu(T_{\text{w}} - T_{\infty } ). $$
(18)

Velocity and temperature profiles throughout the boundary layer except for the immediate vicinity of the disk are approximated by the power–law functions [26, 27]

$$ \bar{v}_{\varphi } = \xi^{n} , $$
(19)
$$ \bar{v}_{\rm r} = \bar{v}_{\varphi } { \tan }\varphi , \quad { \tan }\varphi = \alpha (1 - \xi )^{2} , $$
(20)
$$ \Uptheta = \xi_{\text{T}}^{{n_{\text{T}} }} . $$
(21)

where \( \bar{v}_{\varphi } = (\omega \, r - v_{\varphi } )/(\omega \, r),\bar{v}_{\rm r} = v_{\rm r} /(\omega \, r). \) It is assumed in the present paper that n = n T. For turbulent flows, both n and n T can vary from 1/7 to 1/10.

Expressions for the shear stresses τwφ and τwr, as well as for the surface heat flux q w can be found from a two-layer model of the velocity and temperature profiles in the wall–law coordinates. In the outer region, Eqs. 19–21 can be rearranged as

$$ V^{ + } = \xi^{n} /\sqrt {c_{\text{f}} /2} , \quad T^{ + } = \xi_{\text{T}}^{{n_{\text{T}} }} \sqrt {c_{\text{f}} /2} /St. $$
(22)

In the viscous and heat conduction sub-layers near the disk surface, the total velocity and temperature profiles are linear

$$ V^{ + } = z^{ + } , \quad T^{ + } = Pr z^{ + } . $$
(23)

Splicing Eqs. 22 and 23 on the boundaries of the viscous (at the point \( z_{1}^{ + } \)) and heat conduction (at point \( z_{1T}^{ + } \)) sub-layers and performing further transformations, one can derive formulas for the friction coefficient and the Stanton number

$$ c_{\text{f}} /2 = \left( {z_{1}^{ + } } \right)^{2(n - 1)/(n + 1)} \times \mathop {Re}\nolimits_{{V_{*} }}^{ - 2n/(n + 1)} , $$
(24)
$$ St = \left( {z_{1}^{ + } } \right)^{{n_{\text{T}} - 1}} Re_{V*}^{{ - n_{\text{T}} }} ({{c_{\text{f}} } \mathord{\left/ {\vphantom {{c_{\text{f}} } 2}} \right. \kern-\nulldelimiterspace} 2})^{{{{\left( {1 - n_{\text{T}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - n_{\text{T}} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} \Updelta^{{ - n_{\text{T}} }} ({{z_{{1{\text{T}}}}^{ + } } \mathord{\left/ {\vphantom {{z_{{1{\text{T}}}}^{ + } } {z_{1}^{ + } }}} \right. \kern-\nulldelimiterspace} {z_{1}^{ + } }})^{{n_{\text{T}} - 1}} Pr^{{ - n_{\text{T}} }} . $$
(25)

The coordinate \( z_{1}^{ + } \) most often used in the form of the constant \( C_{n} = (z_{1}^{ + } )^{1 - n} \) depends only on the exponent n. Constant C n takes values 8.74; 9.71; 10.6 and 11.5 at n = 1/7; 1/8; 1/9 and 1/10 [15]. An approximation for C n was offered in the work [28]

$$ C_{n} = 2. 2 8+ 0. 9 2 4/n. $$
(26)

The relation of the shear stress components τwφ and τwr to the total value τ w is given by the following formulas [14, 15, 26, 27]

$$ \tau_{{{\text{w}}r}} = - \alpha \tau_{{{\text{w}}\varphi }} , \quad \tau_{{{\text{w}}\varphi }} = - \tau_{\text{w}} (1 + \alpha^{2} )^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} . $$
(27)

For the Prandtl numbers moderately different from unity (e.g. in air cooling systems), relation \( ({{z_{1T}^{ + } } \mathord{\left/ {\vphantom {{z_{1T}^{ + } } {z_{1}^{ + } }}} \right. \kern-\nulldelimiterspace} {z_{1}^{ + } }}) \) in Eq. 25 is a function of the Prandtl number only (see [17, 27]); parameter Δ (which is the unknown to be found from Eq. 18) depends on the boundary condition for T w(r) and Pr. Denoting further

$$ ({{z_{1T}^{ + } } \mathord{\left/ {\vphantom {{z_{1T}^{ + } } {z_{1}^{ + } }}} \right. \kern-\nulldelimiterspace} {z_{1}^{ + } }})^{{n_{T} - 1}} Pr^{{ - n_{T} }} = Pr^{{ - n_{\rm p} }} , $$
(28)

one can find the following relations for the Stanton and Nusselt number

$$ St = ({{c_{\text{f}} } \mathord{\left/ {\vphantom {{c_{\text{f}} } 2}} \right. \kern-\nulldelimiterspace} 2})\Updelta^{ - n} Pr^{{ - n_{\rm p} }} , $$
(29)
$$ Nu = St{\frac{{V_{*} r}}{\nu }}Pr = StRe_{\omega } Pr(1 + \alpha^{2} )^{1/2} , $$
(30)

with the constant n p to be found from the solution of Eq. 18.

The solution of the Eqs. 16 and 17 is as follows [17, 18, 26, 27]

$$ \delta = C_{\delta } r^{m} , \quad C_{\delta } = \gamma (\omega /\nu )^{ - 2n/(3n + 1)} , \quad {\delta \mathord{\left/ {\vphantom {\delta r}} \right. \kern-\nulldelimiterspace} r} = \gamma Re_{\omega }^{ - 2n/(3n + 1)} , $$
(31)
$$ \gamma = C_{n}^{ - 2/(3n + 1)} (1 + \alpha^{2} )^{0.5(1 - n)/(3n + 1)} H_{9}^{ - (n + 1)/(3n + 1)} , $$
(32)
$$ c_{\text{f}} /2 = A_{\text{c}} Re_{\omega }^{ - 2n/(3n + 1)} , $$
(33)
$$ A_{\text{c}} = C_{n}^{ - 2/(n + 1)} \gamma^{ - 2n/(n + 1)} (1 + \alpha^{2} )^{ - n/(n + 1)} , $$
(34)
$$ \alpha = \left[ {{\frac{{C_{1} }}{{(3 + m)B_{1} + (4 + m)D_{1} }}}} \right]^{1/2} , $$
(35)

where \( \begin{gathered} H_{9} = \alpha D_{1} (4 + m);B_{1} = 1/(2n + 1) - 2/(n + 1) + 6/(2n + 3) - 2/(n + 2) + {1 \mathord{\left/ {\vphantom {1 {\left( {2n + 5} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2n + 5} \right)}}; \hfill \\ \mathop D\nolimits_{1} = \mathop A\nolimits_{1} - \mathop D\nolimits_{2} ;C_{1} = 1 - 2/(n + 1) + 1/(2n + 1);A_{1} = 1/(n + 1) - 2/(n + 2) + 1/(n + 3) ; \hfill \\ m = (1 - n)/(3n + 1);D_{2} = 1/(2n + 1) - 1/(n + 1) + 1/(2n + 3). \hfill \\ \end{gathered} \)

Solution (31–35) was shown ([17, 27] and references) to agree well with known von Karman’s model [29] for the friction coefficient, however providing much better agreement for the radial velocity profile vr, parameters α and δ.

For particular values of the parameter n used further in the computations, one can obtain from Eq. 31: \( {\delta \mathord{\left/ {\vphantom {\delta r}} \right. \kern-\nulldelimiterspace} r} = 0.530Re_{\omega }^{ - 1/5} \) for n = 1/7, and \( {\delta \mathord{\left/ {\vphantom {\delta r}} \right. \kern-\nulldelimiterspace} r} = 0.477Re_{\omega }^{ - 1/6} \) for n = 1/9.

Under the model assumption Δ = const, the solution of Eq. 18 for the Nusselt numbers under the boundary condition (3) takes the form of Eqs. 1 with

$$ n_{\text{R}} = (n + 1)/(3n + 1), $$
(36)
$$ K_{1} = K_{3} \Updelta^{ - n} Pr^{{1 - n_{\rm p} }} , $$
(37)
$$ K_{2} = K_{1} \left( {n_{*} + 2} \right)/\left( {2 + n_{*} + m} \right), $$
(38)
$$ K_{3} = A_{\text{c}} (1 + \alpha^{2} )^{1/2} = C_{n}^{ - 2/(n + 1)} \gamma^{ - 2n/(n + 1)} (1 + \alpha^{2} )^{0.5(1 - n)/(n + 1)} , $$
(39)

where obviously Eq. 38 coincides with Eq. 4, as \( 2n_{\rm R} = 1 + m \). Here:

  1. (a)

    for Δ ≥ 1 [17, 26]

    $$ \Updelta^{ - n} = \left[ {{\frac{4 + m}{{2 + m + n_{*} }}}K_{\rm V} Pr^{{ - n_{\rm p} }} + (1 - K_{\rm V} )} \right]^{ - 1} , $$
    (40)
    $$ K_{1} = K_{3} Pr\left[ {{\frac{4 + m}{{2 + m + n_{*} }}}K_{\rm V} + (1 - K_{\rm V} )Pr^{{n_{\rm p} }} } \right]^{ - 1} , $$
    (41)
    $$ K_{\text{V}} = 1 - D_{2} /A_{1} , $$
    (42)
    $$ n_{\text{p}} = 0.4/(1 - K_{\text{V}} ), $$
    (43)
  2. (b)

    for Δ ≤ 1 [27]

    $$ \Updelta^{2n + 1} \left( {a_{*} - 2b_{*} \Updelta + c_{*} \Updelta^{2} } \right) = {\frac{4 + m}{{4 + m + n_{*} }}}\left( {a_{*} - 2b_{*} + c_{*} } \right)Pr^{{ - n_{\rm p} }} , $$
    (44)

    where \(a_{*}= 1/(1+2n)-1/(1+n), b_{*} = 1/(2+2n)-1/(2+n), c_{*} = 1/(3+2n)-1/(3+n).\)

Solution (36–39) with allowance for Eqs. 40–43 for Δ ≥ 0 proved to be efficient for the air cooling systems [17, 26, 27] providing better agreement with experimental data of different authors than known Dorfman’s solution [14, 15]. Transcendental equation (44) for Δ ≤ 0 has to be yet validated against experimental data. Thus, it looks challenging to undertake a validation of the solution (44) for the case where the Pr or Sc values are much larger than unity and, if necessary, to develop further the integral method (16–30) to take into account effects associated with the very thin thermal/diffusion boundary layers.

3.2 Model with a constant value of Δ

At very high Prandtl (or Schmidt) numbers, the ratio of the thicknesses of the thermal (or diffusion) and momentum boundary layers Δ becomes very small. Owing to this, Eq. 44 can be simplified, because all the terms in the parentheses in its left-hand side except for a * become negligibly small

$$ \Updelta^{2n + 1} a_{*} = {\frac{4 + m}{{4 + m + n_{*} }}}\left( {a_{*} - 2b_{*} + c_{*} } \right)Pr^{{ - n_{\rm p} }} . $$
(45)

From Eq. 45, one can obtain an analytical solution for Δ

$$ \Updelta = \left[ {{\frac{4 + m}{{4 + m + n_{*} }}}\left( {1 - {\frac{{2D_{3} }}{{C_{2} }}}} \right)} \right]^{{{\frac{1}{2n + 1}}}} Pr^{{ - {\frac{{n_{\rm p} }}{2n + 1}}}} , $$
(46)

where \( C_{2} = - 2\left( {1/(2n + 1) - 1/(n + 1)} \right);D_{3} = 2/(n + 2) - 1/(n + 3) - 1/(n + 1) + 1/(2n + 3). \) Based on Eqs. 37 and 46, the constant K 1 can be expressed as

$$ K_{1} = K_{3} \left[ {{\frac{4 + m}{{4 + m + n_{*} }}}\left( {1 - {\frac{{2D_{3} }}{{C_{2} }}}} \right)} \right]^{{{\frac{ - n}{2n + 1}}}} Pr^{{1 + n_{\rm p} \left( {{\frac{n}{2n + 1}} - 1} \right)}} . $$
(47)

As shown in Sect. 2, the exponent at the Prandtl number in Eq. (47) for very high values of Pr should be equal to 1/3. This leads to the following expression for n p

$$ n_{\rm p} = {\frac{2}{3}} \cdot {\frac{2n + 1}{n + 1}}. $$
(48)

Finally, the solution for the constant K 1 with account for Eq. 48 is

$$ K_{1} = K_{3} \left[ {{\frac{4 + m}{{4 + m + n_{*} }}}\left( {1 - {\frac{{2D_{3} }}{{C_{2} }}}} \right)} \right]^{{{\frac{ - n}{2n + 1}}}} Pr^{1/3} . $$
(49)

To make comparisons with electrochemical experiments, we will consider Sherwood numbers instead of the Nusselt numbers thus substituting also Pr with Sc.

Comparisons of the solution for Sh av based on Eqs. 2, 38 and 49 (at n * = 0) with empirical Eq. 5 [3] and theoretical formula (12) [22] are performed in Fig. 4. As may be concluded, curves 4 and 5 computed based on Eq. 49 at n = 1/7 and 1/9 lie 20 to 30% lower than curves 2 and 3 plotted based on Eqs. 5 and 12, respectively. Such deviations of the theory from experiments are unacceptable. Besides, the slope angle of curves 4 and 5 (the exponent at Re φ is equal to 0.8 and 0.833, with the constant K 2 being equal to 0.0207 and 0.0126, respectively) is visibly different from the slope angle of curves 2 and 3 (the exponent at Re φ is 0.87 and 0.9, respectively). This is the evidence of the partial inadequacy of the model assumptions used in the considered integral method as applied to the conditions at high Pr and Sc values. Therefore, it is necessary to revise these assumptions to improve the agreement with the empirical equations and solutions obtained using other theoretical models.

Fig. 4
figure 4

Average Sherwood number for the high Schmidt numbers, approximation of experiments: 1 laminar flow, equation of Levich [6]; 2 Eq. 5 [3]. Theoretical solutions: 3 Eq. 12 [22]; 4 Eq. 49 for n = 1/7; 5 Eq. 49 for n = 1/9; 6 Eq. 68

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3.3 Model with a variable value of Δ

When Prandtl and Schmidt numbers are moderately different from unity, the thickness of the thermal/diffusion boundary layer is larger or moderately less than that of the momentum boundary layer. Hence, to obtain solutions (40–44), the integral in the right-hand side of Eq. 18 has been taken using profile (20) for the radial velocity component vr across the entire momentum boundary layer. Viscous and heat conduction sub-layers are ignored at the integration, since their thickness is negligibly small and does not change the value of the definite integral. Integration in Eqs. 16 and 17 is usually done in the very same way [14, 15, 17, 26–29].

The situation is principally different for very high Prandtl and Schmidt numbers. Here thermal/diffusion boundary layers are very thin and develop within the viscous sub-layer of the momentum boundary layer, where profile of vr is a linear function of the coordinate z orthogonal to the disk [2, 6, 16, 22, 24, 25].

Linear profile of vr in the vicinity of the wall can be written down as

$$ {\text{v}}_{\rm r} = {\frac{{\tau_{{{\text{w}}r}} }}{\mu }}\,z = {\frac{{\tau_{\text{w}} \alpha }}{{\mu (1 + \alpha^{2} )^{1/2} }}}\,z = {\frac{{\rho V_{*}^{2} \alpha }}{{\mu (1 + \alpha^{2} )^{1/2} }}}{\frac{{c_{\text{f}} }}{2}}\,z = \alpha (1 + \alpha^{2} )^{1/2} \omega A_{c} Re_{\omega }^{{n_{\rm R} }} z, $$
(50)

where exponent n R is given in Eq. 36.

Based on the definition of \( z_{1}^{ + } \), one can derive the following formula

$$ {\frac{{z_{1} }}{\delta }} = {\frac{{z_{1}^{ + } }}{{\gamma (1 + \alpha^{2} )^{1/2} A_{\text{c}}^{1/2} Re_{\omega }^{1/(1 + 3n)} }}}. $$
(51)

According to Eq. 51, linear model (50) holds up to the values from z/δ ≤ 0.01 to 0.02. As an analysis in Fig. 5 confirms, linear model (50) approximately up to \( z/\delta_{\varphi }^{**} \le 0.2 \) or z/δ ≤ 0.02 agrees well with experiments and power–law profiles of v r. Thus, value Δ = 0.02 outlines the limit up to which the linear model (50) can be used.

Fig. 5
figure 5

Radial velocity profiles in the turbulent boundary layer over a free rotating disk. Model (20): 1 n = 1/7, 2 1/8, 3 1/9. Line 4 Eq. 50 for Re ω = 1.0 × 106. Experiments: 5 Re ω = 0.4 × 106, 6 0.65 × 106, 7 0.94 × 106, 8 1.6 × 106 [30], 9 0.6 × 106, 10 1.0 × 106 [31]

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Assumption (28) is justifiable for the Prandtl and Schmidt numbers moderately deviating from unity. At high values of Pr or Sc, when the entire temperature/concentration profile develops within the viscous sub-layer, one can assume that Eq. 28 has a more complicated form

$$ ({{z_{1T}^{ + } } \mathord{\left/ {\vphantom {{z_{1T}^{ + } } {z_{1}^{ + } }}} \right. \kern-\nulldelimiterspace} {z_{1}^{ + } }})^{{n_{T} - 1}} Pr^{{ - n_{T} }} = K_{\alpha } Pr^{{ - n_{\rm p} }} . $$
(52)

Constants K α and n p in Eq. 52 should be found from the comparison with experimental data for Nu or Sh numbers. Therefore, Eqs. 29 and 30 for the Stanton and Nusselt numbers can be written as

$$ St = ({{c_{\text{f}} } \mathord{\left/ {\vphantom {{c_{\text{f}} } 2}} \right. \kern-\nulldelimiterspace} 2})\Updelta^{ - n} Pr^{{ - n_{\rm p} }} K_{\alpha } = A_{\text{c}} Re_{\omega }^{ - 2n/(3n + 1)} \Updelta^{ - n} Pr^{{ - n_{\text{p}} }} K_{\alpha } , $$
(53)
$$ Nu = StRe_{\omega } Pr(1 + \alpha^{2} )^{1/2} = A_{\text{c}} (1 + \alpha^{2} )^{1/2} Re_{\omega }^{{n_{\text{R}} }} \Updelta^{ - n} Pr^{{1 - n_{\text{p}} }} K_{\alpha } . $$
(54)

Finding the definite integral in the left-hand side of Eq. 18 with allowance for Eq. 50 for vr and Eq. 21 for Θ, and substituting Eqs. 53 and 54 into the right-hand side of Eq. 18, one can reduce Eq. 18 to the following form

$$ {\frac{n}{2(n + 2)}}\,\alpha \omega\,{\frac{d}{dr}}\left[ {r\delta^{2} \Updelta^{2} Re_{\omega }^{{n_{\rm R} }} \Updelta T} \right] = K_{\alpha } \Updelta^{ - n} Pr^{{ - n_{\text{p}} }} Re_{\omega }^{{n_{\text{R}} }} \nu \Updelta T. $$
(55)

In the integral method described in Sects. 3.1 and 3.2, it was assumed that Δ = const. This condition is inapplicable to Eq. (55), because at Δ = const exponents at the variable r in both sides of Eq. 55 do not coincide.

One can assume by analogy to Eqs. 31 that

$$ \Updelta = C_{\Updelta } r^{k} . $$
(56)

Substituting Eq. 56 into 55, one can derive with account for Eqs. 3 and 31

$$ \Updelta = C_{{\Updelta}^*} Re{}_{\omega }^{k/2} , $$
(57)
$$ C_{{\Updelta}^{*}} = C_{{\Updelta}{^*}{^*}}Pr^{{ - n_{\rm p} /(2 + n)}} , $$
(58)
$$ C_{\Updelta^{**}} = \left[ {{\frac{{K_{\alpha } 2(n + 2)/n}}{{\alpha \gamma^{2} (1 - nk + n_{*} + 2n_{\text{R}} )}}}} \right]^{1/(n + 2)} , $$
(59)
$$ k = - 2m/(2 + n). $$
(60)

As a result, Eqs. 1, 37, 38 and 43 undergo the following transformations

$$ Nu = K_{1} Re{}_{\omega }^{{n_{{{\text{R}}^*}} }} , $$
(61)
$$ Nu_{\text{av}} = K_{2} Re{}_{\varphi }^{{n_{{{\text{R}}^*}} }} , $$
(62)
$$ n_{{{\text{R}}^*}} = n_{\text{R}} + mn/(2 + n), $$
(63)
$$ K_{1} = K_{\alpha } K_{3} C_{{\Updelta}{^*}{^*}}^{-n}Pr^{1/3} , $$
(64)
$$ K_{2} = K_{1} {\frac{{2 + n_{*} }}{{2n_{{{\text{R}}^*}} + n_{*} + 1}}}, $$
(65)
$$ n_{\text{p}} = (2 + n)/3. $$
(66)

Relation (66) for the exponent n p was obtained in view of the requirement that the cumulative exponent at the Prandtl number in Eq. 64 should be equal to 1/3.

Thus, in Eqs. 61 and 62, the cumulative exponent n R* at the Reynolds number given in Eq. 63 increases in comparison with Eq. 36 by the additional term mn/(2 + n) owing to that parameter Δ becomes variable decreasing with the increasing local Reynolds number (i.e. local coordinate r) in accordance with Eq. 57.

At n = 1/7 one has n R*  = 0.84, and at n = 1/9 one obtains n R*  = 0.868, which practically coincides with the exponent 0.87 at Re φ in Eq. 6 [3]. In order for Eqs. 61 and 62 at n = 1/9 and n R*  = 0.868 agree with Eqs.  6 and 5, respectively, one should choose the value K α = 1.254, that finally entails for n *  = 0

$$ Nu = 1.52 \times 10^{ - 2} Re_{\omega }^{0.868} Pr^{1/3} , $$
(67)
$$ Nu_{\text{av}} = 1.11 \times 10^{ - 2} Re_{\varphi }^{0.868} Pr^{1/3} . $$
(68)
$$ \Updelta = 18.31Re_{{{\upomega}}}^{ - 0.3158} Pr^{ - 1/3} . $$
(69)

Curve 6 plotted in Fig. 4 based on Eq. 68 coincides with curve 2, depicted based on Eq. 5 [3]. Equations 67 and 6 also practically coincide.

According to Eq. 69, Δ = 0.036 at minimal values Sc = 930 and Re ω = 0.278 × 106 in experiments [3]. At Sc = 10320 and Re ω = 0.278 × 106, one has Δ = 0.015; at Sc = 930 and Re ω = 1.8 × 106 one has Δ = 0.02. These values Δ agree well with the aforementioned limit Δ ≤ 0.02, at which the linearity of the radial velocity profile remains in force.

3.4 Model with a variable value Δ and profile T + depending on Re ω

It was obtained in works [6, 22, 24, 25] based on the analysis of models for the eddy-viscosity in the near-wall region at high Prandtl numbers that

$$ Nu = K_{\text{N}} (1 + \alpha^{2} )^{1/2} (c_{\text{f}} /2)^{1/2} Re_{\omega } Pr^{1/3} , $$
(70)
$$ St = K_{\rm N} (c_{\rm f} /2)^{1/2} Pr^{ - 2/3} , $$
(71)

where constant K N takes different values depending on the model assumptions.

In the above-mentioned works, Eq. 33 was used for the friction coefficient with n = 1/7. Then the relations for the Nusselt numbers with allowance for Eq. 70 reduce to the form of Eqs. 61 and 62, where

$$ K_{1} = K_{\text{N}} (1 + \alpha^{2} )^{1/2} A_{\text{c}}^{1/2} Pr^{1/3} , $$
(72)
$$ n_{{{\text{R}}{^*}}} = (2n + 1)/(3n + 1), $$
(73)

and K 2 is given in Eq. 65. For n = 1/7, Eq. 73 results in n R*  = 0.9.

As shown in Sect. 2, Eq. 12 is the most accurate among the correlations obtained in [6, 22, 24, 25] based on model (70). Then, agreeing Eqs. 72 and 12 with allowance for Eq. 65, once can obtain the value K N = 0.05986.

Solving the thermal boundary layer equation using assumption (56), like it was done in Sect. 3.3, one can come again to the solution (57) for Δ, where

$$ C_{{\Updelta}{^*}} = C_{{\Updelta}{^*}{^*}} Pr^{ - 1/3} , $$
(74)
$$ C_{\Updelta^{**}} = \left[ {{\frac{{K_{\text{N}} 2(n + 2)/n}}{{\alpha \gamma^{2} (1 + 2m + 2k + n_{*} + 2n_{\text{R}} )A_{\text{c}}^{1/2} }}}} \right]^{1/(n + 2)} , $$
(75)
$$ k = (2n - 1)/(3n + 1). $$
(76)

Substituting values n = 1/7 and n *  = 0 into Eqs. 57, 74–76 results in

$$ \Updelta = 12.54Re_{{{\upomega}}}^{ - 1/4} Pr^{ - 1/3} . $$
(77)

For the minimal values Sc = 930 and Re ω = 0.278 × 106 [3], parameter Δ in accordance with Eq. 77 is equal to Δ = 0.037. At Sc = 10320 and Re ω = 0.278 × 106, one can obtain Δ = 0.016, while at Sc = 930 and Re ω = 1.8 × 106 one has Δ = 0.023. These values Δ exceed insignificantly the data obtained based on Eq. 69, and they also agree rather well with the value Δ ≤ 0.02, at which the linear model (50) holds.

In frames of the model described in Sect. 3.3, profile T + in the region where the power–law (21) is valid remains independent of the Reynolds number

$$ T^{ + } = (z^{ + } )^{n} (1 + \alpha^{2} )^{ - n/2} \gamma^{ - n} A_{\text{c}}^{ - (n + 1)/2} Pr^{{n_{\rm p} }} K_{\alpha }^{ - 1} . $$
(78)

This agrees with suggestions given in works [32, 33]. Model based on Eqs. 70 and 71 leads to the profile of T + dependent on Re ω

$$ T^{ + } = (z^{ + } )^{n} (1 + \alpha^{2} )^{ - n/2} \gamma^{ - n} A_{\text{c}}^{ - n/2} C_{{\Updelta}{^*}}{ - n} K_{N}^{ - 1} Pr^{2/3} Re_{\omega }^{{ - 0.5(2n^{2} + n)/(3n + 1)}} . $$
(79)

4 Conclusions

The investigations performed above resulted in the following accomplishments:

  1. 1.

    Available in the literature empirical/theoretical relations for Sherwood numbers were thoroughly validated, and Eqs. 5–7 were distinguished as the most reliable.

  2. 2.

    A novel approach to modeling temperature/concentration profiles for the values of Pr and Sc significantly larger than unity was proposed. An original integral method developed based on this allows estimating relative thickness Δ of the thermal/diffusion boundary layers that has not been accomplished by the other theoretical models. It was shown that namely decrease in Δ with the increasing local radius entails the additional increase of the exponent at the Reynolds number in the expressions for the Nu or Sh numbers in comparison with the air flows. As a result, solutions obtained for the surface heat and mass transfer agree well with the selected empirical formulas.