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6.1 Laminar Flow

Convective heat and mass transfer over a single disk rotating in fluid with high Prandtl or Schmidt numbers can be found in many practical and research applications. For instance, in electrochemistry, where the Schmidt numbers are several orders of magnitude larger than unity, rotating disk electrode is involved in measurements of the convective diffusion coefficient [1–14]. Another example is naphthalene sublimation technique often used to measure mass transfer coefficients α m [15–29].

The differential Eq. (1.28) of convective diffusion, including the time-averaged fluctuating components, is analogous to the energy Eq. (2.5), provided that the temperature T and the thermal diffusivity a are replaced by the concentration C and the diffusion coefficient D m , respectively. The Navier–Stokes and continuity equations hold, if constant fluid properties are assumed.

If the Schmidt number Sc replaces the Prandtl number, and the nondimensional function θ is written as

$${\theta } = (C - C_{\infty } )/(C_w - C_{\infty } ),$$
(6.1)

then the self-similar Eqs. (2.32)–(2.36) for steady-state axisymmetric laminar flow become valid for convective mass transfer.

Surface concentration on the disk does not vary; thus \(C_w = {\text{const}}\). Therefore, rewritten convective diffusion Eq. (2.36) reduces to

$${\theta }^{\prime\prime} - {Sc}H{\theta }^{\prime} = 0.$$
(6.2)

The following equations (analogous to Eq. (3.4)) can be used for estimation of the Sherwood number

$${Sh} = K_{1} {Re}_{\omega}^{{n_{\text{R}} }} ,\quad {Sh}_{\text{av}} = K_{2} {Re}_{\varphi }^{{n_{\text{R}} }} .$$
(6.3)

The constants K 1 and K 2 in Eq. (6.3) are affected by the boundary conditions, flow type (laminar, transitional, or turbulent) and the Schmidt numbers. The exponent n R is affected by the flow type, whereas K 1 = K 2 and n R = 1/2 in a laminar flow regime.

Thus, the aforementioned analogy between convective heat and mass transfer, enables the use of theoretical solutions or empirical experimental equations simply via replacing C, Sc and Sh with of T, Pr and Nu (or vice versa), accordingly.

For laminar flow, Eqs. (2.32)–(2.36) for Pr > 1 and Sc > 1 at N = 0 and β = 0 were solved numerically using Mathcad [30]. Table 6.1 shows that the calculated coefficient K 1 is increasing with growing Pr or Sc numbers.

Table 6.1 Values of the constant K 1, exact solution of Eqs. (2.32)–(2.36) for Pr > 1 [30]
Full size table

For the same Prandtl number, the constant K 1 is an increasing function of the exponent n *: the value of K 1 at Pr = 0.71, 2.0 and 106 becomes 3.3, 2.73 and 2.2 times larger, respectively, if the constant n * changes from −1 to 3. Thus, at increased Prandtl numbers, the influence of the exponent n * on the constant K 1 gets less pronounced.

The approximate Eq. (3.6) for the coefficient K 1 for the boundary condition (2.30), Pr ≥ 1 and nonzero values n * was derived by Dorfman [31]. Values of K 1 by Eq. (3.6) surpass the exact solution. Equation (3.6) deviates from the exact solution at n * ≤ 0 by 16–40 % even for Pr = 1. For n * = 0 and Pr = 1–3, this deviation reaches 10–11 %. For larger exponents n * and Pr = 1–3, the deviation of Eq. (3.6) from the exact solution is smaller (1–6 %). However, at Pr → ∞, the inaccuracy of Eq. (3.6) abruptly increases [30].

The functional dependence of the constant K 1 on the Schmidt (or Prandtl) number according to Dorfman’s Eq. (3.6) and the exact solution for n * = 0 (T w  = const. or C w  = const.) is depicted in Fig. 6.1. The inaccuracies of Eq. (3.6) make it unusable already for Sc = 1–3.

Fig. 6.1
figure 1

Constant K 1 in Eq. (6.3), laminar flow at C w  = const. [30]. 1—Exact solution; 2—Eq. (3.6) for n * = 0; 3—Eqs. (6.4) and (6.5); 4—Eq. (6.6); 5—Eq. (6.7); 6—Eq. (6.8). Experiments: 7—K 1 = 0.59, Sc = 2.28 [15]; 8—K 1 = 0.604, Sc = 2.28 [17]; 9—K 1 = 0.625, Sc = 2.4 [20, 21, 26]; 10—K 1 = 0.636, Sc = 2.44 [19]; 11—K 1 = 0.625, Sc = 2.5 [25]; 12—K 1 = 0.69, Sc = 2.5 [24]; 13—K 1 = 0.628, Sc = 2.5 [22]

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Equations (3.7) and (3.8) can be rewritten for mass transfer for Sc = 0–∞, respectively

$$K_{1} = 0.6109{Sc}/(0.5301 + 0.3996{Sc}^{1/2} + {Sc})^{2/3} ,$$
(6.4)
$$K_{1} = 0.6{Sc}/(0.56 + 0.26{Sc}^{1/2} + {Sc})^{2/3} .$$
(6.5)

Equations (6.4) and (6.5) result nearly in the same values. Maximal deviation of them from the exact solution is 4 and 5 %, respectively, for Sc = 5–20 (Fig. 6.1). For higher Schmidt numbers, inaccuracies of Eqs. (6.4) and (6.5) tend to zero (Table 6.2).

Table 6.2 Constant K 1 by Eqs. (6.4)–(6.8), a rotating disk for C w  = const. or T w  = const. [30]
Full size table

Another expression was derived in the work [13]

$$K_{1} = 0.621{Sc}/(1 + 0.298{Sc}^{{{-}1/3}} + 0.14514{Sc}^{{{-}2/3}} ).$$
(6.6)

At the expense of a larger deviation from the exact solution for Sc = 1–2 (8 % at Sc = 1 and 4 % already at Sc = 2) Eq. (6.6) ensures only deviations of less than 1–3 % at higher Prandtl or Schmidt numbers (Fig. 6.1; Table 6.2).

For Sc → 0, Eqs. (6.4)–(6.6) reduce to the asymptotic relation K 1/Sc = 0.885 [32]. For Sc → ∞, they ensure agreement with another asymptotic \(K_{1} = 0.62{Sc}^{1/3}\) [11, 32].

One more relation for K 1 for Pr = 0−∞ was designed as a combination of asymptotic solutions for the cases Pr → 0 and Pr → ∞ [2]. Rewritten, using Sc number, this results in

$$K_{1} = \left[ {(0.88447{Sc})^{{{-}1.077}} + (0.62048{Sc}^{1/3} )^{{{-}1.077}} } \right]^{{{-}1/1.077}} .$$
(6.7)

For Sc = 2, Eq. (6.7) merges with the self-similar solution. Deviation of Eq. (6.7) from the exact solution grows up to 3.2 % at Sc = 2.5, exhibits a maximum of 5.6 % at Sc ≈ 20 and, for larger Schmidt numbers, diminishes to 2.7 % at Sc = 1000 and 0.6 % at Sc = 105.

Over the range of Sc < 2, deviation of Eq. (6.7) from the exact solution changes its sign, and increases in absolute values being 3.4 % at Sc = 1 and 8.2 % at Sc = 0.1 (Table 6.2).

To conclude, preference should be rendered to that of Eqs. (6.4)–(6.7) that ensures the lowest inaccuracy at the Schmidt numbers specific for the problem is to be solved.

Application to electrochemistry problems. Levich [11] derived an asymptotic solution for convective diffusion for very large Schmidt numbers \(Sc \gg 1\)

$$K_{1} = 0.62{Sc}^{1/3} .$$
(6.8)

It coincides with the asymptotic solution for heat transfer for \(Pr \gg 1\) given in [32].

Table 6.2 elucidates that Eq. (6.8) correlates well with the exact solution at Sc > 500 (deviation for Sc = 500 is 3.9 % and reduces to zero for Sc → ∞). Equation (6.8) overruns the exact solution by 7.1 % at Sc = 100 and by 56.7 % at Sc = 1. Levich’s Eq. (6.8) was successfully validated in experimental studies [4, 5, 7, 8, 12, 14] at high Schmidt numbers.

Rotating disk electrodes are intensively employed in experimental electrochemical investigations [1, 11]. Convective diffusion, which displays itself as the diffusion of the electrical current on the electrode, is modeled by Eq. (1.28).

For this case, Eq. (6.3) for laminar flow in view of Eq. (6.8) is usually rewritten as [1, 11]

$$i_{\text{L}} = 0.62nFC_{\text{F}} C_{\infty } D_{m}^{2/3} {\nu }^{{{-}1/6}} {\omega }^{1/2} ,$$
(6.9)

where i L is the limiting diffusion current of electrons to the surface of a rotating disk electrode; n is the number of electrons that are involved in the current; F is the disk area; C F is the Faraday constant (96,485 C/mol); C ∞ is the concentration at infinity, mol/m3. Based on this, one can ascertain that the mass transfer coefficient can be written as \({\alpha }_{m} = i_{\text{L}} /(nFC_{\text{F}} C_{0} )\), while Eq. (6.9) translates into Eq. (6.8).

In practice, the following tasks are actual: (1) searching a functional dependence of i L on ω; (2) finding the diffusion coefficient D m , whereas the value of i L is measured; and (3) measurements of Volt–Ampere characteristics using a rotating disk electrode.

Naphthalene sublimation technique for experimental determination of the mass and heat transfer coefficients. Convective heat transfer from a surface to air is analogous to convective mass transfer in naphthalene sublimation to air. Naphthalene sublimation has been often employed to measure the average mass transfer of an entire disk weighted before and after the measurement to determine the amount of naphthalene lost by the disk as a result of experiments [19–21, 25–27]. Currently, accurate instrumentation is available for local pointwise scanning of the naphthalene layer thickness and subsequent calculation of local mass transfer coefficients for laminar, transitional, and turbulent flow [15–18, 22, 24, 28]. In frames of the analogy between the surface heat and mass transfer, constants K 1 in Eq. (3.4) for the Nusselt number and Eq. (6.3) for the Sherwood number can be expressed as [23]

$$K_{1} = C\;{Pr}^{{m_p}} ,$$
(6.10)
$$K_{1} = C\,{Sc}^{{m_p}} ,$$
(6.11)

where the coefficient C is identical in both equations. The effects of the Prandtl and Schmidt numbers are described by respective multipliers in Eqs. (6.10) and (6.11).

Equations (6.10) and (6.11) are used for the Prandtl and Schmidt numbers moderately diverging from unity: Pr = 0.7–0.74 for air, whereas Sc = 2.28–2.5 for naphthalene sublimation in air. Therefore, the constant C is assigned to be equal to the coefficient K 1 at Sc = 1 and Pr = 1 at T w  = const. or C w  = const. (see Table 6.1), i.e., C = 0.3963.

Authors [15–19, 24, 28] used the naphthalene sublimation technique to measure rotating disk mass transfer and set the exponent m p to be the same for all values of Pr and Sc, which yields a relation between the Nu and Sh numbers

$${Nu/Sh} = ({Pr/Sc})^{{m_p }} .$$
(6.12)

The scatter of the values of the exponent m p in the literature amounted up to 45 %: m p  = 1/3 [17], m p  = 0.4 [15, 17, 18, 20], m p  = 0.53 [19], and m p  = 0.58 [24].

Erroneous values m p entail fallacious results of post-processing of the experimental data from the naphthalene sublimation technique aimed at estimation of heat transfer in air. An analysis and recommendation of the proper value m p were made by the author [23].

Exponent m p can be detected from the self-similar solution of the problem (Tables 3.1 and 6.1). Table 6.3 lists exponents m p for the Prandtl/Schmidt numbers moderately deviating from unity [23]. It is evident from here that the function m p (Pr) exhibits a decreasing trend and varies from m p  = 0.5723 to m p  = 0.5024, if the Prandtl/Schmidt numbers grow from 0.7 up to 2.5. Consequently, the effective exponent m p  = 0.53 suggested in [19] is practically the average m p value weighted over the range Pr = 0.7–2.5.

Table 6.3 Value m p in Eqs. (6.10), (6.11) and (6.12) based on the exact solution of Eqs. (2.32)–(2.36) for laminar flow [23, 30]
Full size table

Figure 6.1 depicts different experimental data for the constant K 1 in naphthalene sublimation in air. These data agree well with the self-similar solution (see Table 6.2); only the too large value K 1 = 0.69 for Sc = 2.5 [24] falls out from the overall picture.

Post-processing [23] of the measured results using Eq. (6.12) at m p  = 0.53 to reduce them to conditions of heat transfer at Pr = 0.71 yields the values K 1 = 0.325 [17], K 1 = 0.328 [20, 21, 26], K 1 = 0.331 [19], K 1 = 0.321 [25], K 1 = 0.322 [22] that agree well with the exact value K 1 = 0.326 for Pr = 0.71 and T w  = const. (see Table 3.1) and reliable experimental data (see Chap. 3). Falling out of the overall good conformance are (a) the constant K 1 = 0.318 resulting from the low value K 1 = 0.59 in naphthalene sublimation at Sc = 2.28 measured in [15], and (b) the constant K 1 = 0.354 stemming from the high experimental value K 1 = 0.69 in naphthalene sublimation obtained in [24] (see Fig. 6.1).

The use of the value m p  = 0.4 suggested in [15, 17, 18, 20] and widely used throughout the literature brings for the heat transfer in air at Pr = 0.71 [23]: K 1 = 0.37 [15]; K 1 = 0.379 [17]; K 1 = 0.384 [20, 21, 26]; K 1 = 0.388 [19]; K 1 = 0.378 [25]; K 1 = 0.380 [22]; K 1 = 0.417 [24]. All these recalculated data are too large as compared to the exact value K 1 = 0.326.

Taking the exponent m p  = 1/3 [17], one can obtain for Pr = 0.71 [23] the constants K 1 = 0.4 [15]; K 1 = 0.409 [17]; K 1 = 0.416 [20, 21, 26]; K 1 = 0.421 [19]; K 1 = 0.411 [25]; K 1 = 0.413 [22]; K 1 = 0.454 [24]. They surpass the exact value K 1 = 0.326 to an even larger extent.

Involvement of the exponent m p  = 0.58 [24] yields for Pr = 0.71 the values K 1 = 0.3 [15], K 1 = 0.307 [17], K 1 = 0.308 [20, 21, 26], K 1 = 0.311 [19], K 1 = 0.301 [25], K 1 = 0.303 [22], that are too small [23]. Only the value K 1 = 0.332 [24] is acceptable, which is due to the high value m p  = 0.58 chosen by the authors [24] to agree with the exact solution K 1 = 0.326. However, it is clear that the too large exponent m p  = 0.58 results from the too large value K 1 = 0.69 in naphthalene sublimation measured in [24], which is discordant with the measurements of the other researchers.

Authors [25] rearranged Eq. (6.12) in the following way

$${Nu/Sh}_{{{Sc} = {{2{.}5}}}} = f(Pr){Pr}^{1/3} .$$
(6.13)

The value K 1 = 0.625 at Sc = 2.5 and function f(Pr) = 0.576, 0.634, 0.737, 0.842 and 0.926 at Pr = 0.1, 1, 2.5, 10 and 100, respectively, yield jointly the values K 1 = 0.321, 0.396, 0.625, 1.134 and 2.686 at the Prandtl numbers mentioned above. This is fully consistent with the self-similar solution at T w  = const. (Table 6.1, n * = 0). The correction function f(Pr) can be recast to incorporate the multiplier Pr 1/3. Use of Eq. (6.13) ensures higher accuracy than that conveyed by approaches operating with a single value of m p , though Eq. (6.13) is less practical as Eq. (6.12), because of the involvement of a tabulated function.

To conclude, Eq. (6.12) with the exponent m p  = 0.53 [19] can be suggested as the most accurate and practical one for post-processing of the measured laminar mass transfer coefficients of a rotating disk in naphthalene sublimation in air in order to recalculate it to laminar heat transfer in air. As an alternative, Eq. (6.13) (or its modification) can be used.

6.2 Transitional and Turbulent Flow for the Prandtl and Schmidt Numbers Moderately Different from Unity

Values of Pr ≤ 5 and Sc ≤ 5 are considered here as those moderately deviating from unity. The objective is again a validation of the experimental technique dealing with sublimation of naphthalene from a rotating disk in air at Sc = 2.28–2.5 [23].

Local Sherwood numbers in naphthalene sublimation experiments in air in transitional and turbulent flow obtained in the recent works [15, 18] together with the data for laminar flow and different empirical approximations are depicted in Fig. 6.2. Recast Eq. (3.13) [15] for transitional flow (corrected range of validity) and empirical equations [15, 18] for turbulent flow look as follows [15, 18]

$${Sh} = 2.0 \times 10^{{{-}19}} {Re}_{\omega }^{4} \quad {\text{for}}\quad {Re}_{\omega } = ({1.9}\text{--}{2.75}) \times {10}^{5}\;\text{(Ref. [15])} ,$$
(6.14)
$${Sh} = 0.0512{Re}_{\omega }^{0.8} \quad {\text{for}}\quad {Re}_{\omega } \ge {2.75} \times {10}^{5}\;\text{(Ref. [15])} ,$$
(6.15)
$${Sh} = 0.0518{Re}_{\omega }^{0.8} \quad {\text{for}}\quad {Re}_{\omega } \ge {2.5} \times {10}^{5} \;\text{(Ref. [18])}.$$
(6.16)
Fig. 6.2
figure 2

Local Sherwood numbers for naphthalene sublimation in air [23, 30]. Experiments: 1—Sc = 2.28 [15]; 2—Sc = 2.4 [21]; 3—Sc = 2.4 [26]; 4—Sc = 2.44 [19]; 5—Sc not mentioned [18]. Empirical approximations, Eq. (6.3): 6—laminar flow, n R = 1/2, K 1 = 0.625 [20, 21, 25, 26]; 7—laminar flow, n R = 1/2, K 1 = 0.604 [17]; 8—transitional flow, n R = 4, K 1 = 2 × 10−19, Eq. (6.14) [15]; 9—turbulent flow, n R = 0.8, K 1 = 0.0512, Eq. (6.15) [15]

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In practice, one often needs to estimate average Sherwood numbers Sh av (or average Nusselt numbers Nu av) of an entire disk, where areas occupied by laminar/transitional flow or laminar/transitional/turbulent flow emerge at the same time. For instance, only surface-averaged mass transfer coefficients of an entire disk were measured in [19–21, 25, 26].

Measurements of the average Sherwood number over an entire disk covered with areas of laminar, transitional and turbulent flow were performed by [19–21, 25, 26]. Reynolds analogy between mass transfer and fluid flow was involved to derive a quite inconvenient theoretical solution for Sh av for an entire disk [21, 26] incorporating parameters, which were rather difficult to determine by means of the used approach. More promising is the model for Sh av first used in the paper [7] and further generalized by the author of the present work [23, 30], which enables verifications of the recent measurements of the local Sherwood numbers by means of comparisons with the vast database for the average Sherwood numbers for an entire disk.

The author [7] assumed that laminar-turbulent transition sets on instantly at a radial coordinate r tr corresponding to the Reynolds number Re ω,tr. Subsequently, the value Sh av for an entire disk can be found using the following integral

$${Sh}_{\text{av}} = \frac{2}{b}\left[ {\int\limits_{0}^{{r_{\text{tr}} }} {{Sh}_{\text{lam}} \text{d}r + \int\limits_{{\text{r}_{\text{tr}} }}^{b} {{Sh}_{\text{turb}} \text{d}r} } } \right].$$
(6.17)

Sherwood numbers are presented by Eq. (6.3) accompanied with the constants K 1,lam and n R = 1/2 for laminar flow, and K 1,turb and n R = 0.8 for turbulent flow.

An integration of Eq. (6.17) yields

$${Sh}_{\text{av}} = K_{{1,{\text{lam}}}} {Re}_{{{\omega },{\text{tr}}}}^{1/2} \left( {\frac{{{Re}_{{{\omega },{\text{tr}}}} }}{{{Re}_{\varphi } }}} \right)^{1/2} + \frac{2}{{2n_{R} + 1}}K_{{1,{\text{turb}}}} {Re}_{\varphi }^{{n_{\text{R}} }} \left[ {1 - \left( {\frac{{{Re}_{{{\omega },{\text{tr}}}} }}{{{Re}_{\varphi } }}} \right)^{{n_{\text{R}} + {1/2}}} } \right].$$
(6.18)

If \({Re}_{\varphi } < {Re}_{{{\omega },{\text{tr}}}}\), the second summand in Eq. (6.18) must be discarded. Asymptotically at \({Re}_{\varphi } \gg {Re}_{{{\omega },{\text{tr}}}}\), Eq. (6.18) degenerates to Eq. (6.3) for turbulent flow with

$$K_{{2,{\text{turb}}}} = \frac{2}{{2n_{\text{R}} + 1}}K_{{1,{\text{turb}}}} .$$
(6.19)

Given n * = 0, which effectively means T w  = const. and C w  = const., Equation (6.18) translates into Eq. (3.25), while Eq. (6.19) turns to Eq. (3.35) in view of the relation \(2n_{\text{R}} = 1 + m\) resulting from Eqs. (2.78) and (3.31).

In [12] it is suggested taking into account regions of laminar, transitional and turbulent flow separately. If the transition sets on at the radial location r tr1 (or at Re ω,tr1) and ends at the radial location r tr2 (or Re ω,tr2), a definite integral for Sh av can be written as

$${Sh}_{\text{av}} = \frac{2}{b}\left[ {\int\limits_{0}^{{r_{\text{tr1}} }} {{Sh}_{\text{lam}} \text{d}r + \int\limits_{{\text{r}_{\text{tr1}} }}^{{r_{\text{tr2}} }} {{Sh}_{\text{tran}} \text{d}r} + \int\limits_{{\text{r}_{\text{tr2}} }}^{b} {{Sh}_{\text{turb}} \text{d}r} } } \right].$$
(6.20)

The transitional Sherwood number Sh tran is specified by the first of Eq. (6.3) complemented with experimental values of K 1,tran and n R,tran for transitional flow. Integration of Eq. (6.20) results in

$$\begin{aligned} {Sh}_{\text{av}} & = K_{{1,{\text{lam}}}} {Re}_{{{\omega },{\text{tr1}}}}^{1/2} \left( {\frac{{{Re}_{{{\omega },{\text{tr1}}}} }}{{{Re}_{\varphi } }}} \right)^{1/2} + \frac{2}{{2n_{{\text{R},{\text{tran}}}} + 1}}K_{{1,{\text{tran}}}} {Re}_{{{\omega },{\text{tr}}2}}^{{n_{{\text{R},{\text{tran}}}} }} \left( {\frac{{{Re}_{{{\omega },{\text{tr2}}}} }}{{{Re}_{\varphi } }}} \right)^{1/2} \\ & \quad \times \left[ {1 - \left( {\frac{{{Re}_{{{\omega },{\text{tr1}}}} }}{{{Re}_{{{\omega },{\text{tr}}2}} }}} \right)^{{n_{{\text{R},{\text{tran}}}} + {1/2}}} } \right] + \frac{2}{{2n_{\text{R}} + 1}}K_{{1,{\text{turb}}}} {Re}_{\varphi }^{{n_{\text{R}} }} \left[ {1 - \left( {\frac{{{Re}_{{{\omega },{\text{tr2}}}} }}{{{Re}_{\varphi } }}} \right)^{{n_{\text{R}} + {1/2}}} } \right]. \\ \end{aligned}$$
(6.21)

Equation (6.21) holds at \({Re}_{\varphi } \ge {Re}_{{{\omega },{\text{tr2}}}}\). If \({Re}_{\varphi } < {Re}_{{{\omega },{\text{tr2}}}}\), the last term in Eq. (6.21) is discarded, whereas the second summand turns to

$$\begin{aligned} Sh_{{{\text{av}}}} & = K_{{1,{\text{lam}}}} Re_{{\omega ,{\text{tr1}}}}^{{1/2}} \left( {\frac{{Re_{{\omega ,{\text{tr1}}}} }}{{Re_{\varphi } }}} \right)^{{1/2}} \\ & \quad + \frac{2}{{2n_{{\text{R},{\text{tran}}}} + 1}}K_{{1,{\text{tran}}}} Re_{\varphi }^{{n_{{\text{R},{\text{tran}}}} }} \left[ {1 - \left( {\frac{{Re_{{\omega ,{\text{tr1}}}} }}{{Re_{\varphi } }}} \right)^{{n_{{\text{R},{\text{tran}}}} + 1/2}} } \right]. \\ \end{aligned}$$
(6.22)

Asymptotically for \({Re}_{\varphi } \gg {Re}_{{{\omega },{\text{tr2}}}}\), Eq. (6.21) transforms to the second of Eq. (6.3), whereas the constant K 2,turb is given by Eq. (6.19). A solution derived in [12] is a particular case of Eq. (6.21), whose empirical constants resulting from experiments [12] at high Sc numbers are fixed numerical values. Hence, the solution [12] as it is can not be used to describe the experimental data for naphthalene sublimation.

Substitution of numerical values of the constants resulting from measurements at naphthalene sublimation in air [15] (see Eqs. (6.14), (6.15) and caption to Fig. 6.1) into the general Eqs. (6.18)–(6.22) yields

  1. (a)

    applied to Eq. (6.18)

    $${Sh}_{\text{av}} = 0.59{Re}_{{{\omega },{\text{tr}}}}^{1/2} \left( {\frac{{{Re}_{{{\omega },{\text{tr}}}} }}{{{Re}_{\varphi } }}} \right)^{1/2} + \frac{2}{2.6}0.512{Re}_{\varphi }^{0.8} \left[ {1 - \left( {\frac{{{Re}_{{{\omega },{\text{tr}}}} }}{{{Re}_{\varphi } }}} \right)^{1.3} } \right],$$
    (6.23)
  2. (b)

    applied to Eq. (6.19)

    $$K_{{2,{\text{turb}}}} = \frac{2}{2.6}K_{{1,{\text{turb}}}} = 0.0394,$$
    (6.24)
  3. (c)

    applied to Eq. (6.21)

    $$\begin{aligned} {Sh}_{\text{av}} & = 0.59 \times 1.9 \times 10^{5} \times {Re}_{\varphi }^{{{-}1/2}} + \frac{4}{9}10^{{{-}19}} (2.75 \times 10^{5} )^{4.5} \\ &\quad \times {Re}_{\varphi }^{{{-}1/2}} \left[ {1 - \left( {\frac{{1.9 \times 10^{5} }}{{2.75 \times 10^{5} }}} \right)^{{4.{5}}} } \right] \\ & \quad + 0.0394{Re}_{\varphi }^{0.8} \left[ {1 - \left( {\frac{{2.75 \times 10^{5} }}{{{Re}_{\varphi } }}} \right)^{1.3} } \right],\quad {Re}_{\varphi } \ge {2.75} \times {10}^{5} , \\ \end{aligned}$$
    (6.25)
  4. (d)

    applied to Eq. (6.22)

    $$\begin{aligned} Sh_{\text{av}} & = 0.59 \times 1.9 \times 10^{5} \times Re_{\varphi }^{ - 1/2} + \frac{4}{9}10^{ - 19} Re_{\varphi }^{4} \left[ {1 - \left( {\frac{{1.9 \times 10^{5} }}{{Re_{\varphi } }}} \right)^{4.5} } \right], \\ Re_{\varphi } & = \left( {1.9 \text{--} 2.75} \right) \times 10^{5} . \\ \end{aligned}$$
    (6.26)

The Reynolds number Re ω,tr (instant transition to turbulence) in Eq. (6.23) remains a free parameter to be tuned for a better agreement with particular experiments.

Figure 6.3 shows validations of Eqs. (6.23)–(6.28) by comparison with experimental data. Experimental data 1, 5 and curve 6 for Sh av for purely turbulent flow stem from the works [23, 30] and result from reprocessing of the measured data [15, 18] and Eq. (6.15) using Eq. (6.24). For laminar flow, we have K 2,lam = K 1,lam (curves 7 and 8). Curve 9 combining Eqs. (6.25), (6.26) and incorporating boundaries of transitional flow conforms to experiments [19, 21, 26] for Sh av for an entire disk depicted in Fig. 6.3.

Fig. 6.3
figure 3

Average Sherwood numbers, naphthalene sublimation in air [23, 30]. Experiments: 1—Sc = 2.28 [15]; 2—Sc = 2.4 [21]; 3—Sc = 2.4 [26]; 4—Sc = 2.44 [19]; 5—Sc not mentioned [18]. Calculation, Eq. (6.3): 6—turbulent flow, n R = 0.8, K 2 = 0.0394, Eq. (6.24) [15]; 7—laminar flow, n R = 1/2, K 1 = 0.625 [20, 21, 25, 26]; 8—laminar flow, n R = 1/2, K 1 = 0.59 [15]. Calculation of Sh av for the entire disk: 9—Eqs. (6.25) and (6.26); 10—Eq. (6.23) at Re ω,tr = 1.9 × 105; 11—Eq. (6.23) at Re ω,tr = 2.75 × 105; 12—Eq. (6.23) at Re ω,tr = 2.35 × 105

Full size image

In Fig. 6.3, experimental data 1 for Sh av for an entire disk were calculated in [23] using Eqs. (6.25), (6.26) and measurements [15] for laminar, transitional and turbulent flow. These data points go beyond curve 9 at respective values of the argument Re φ .

The replacement of the Reynolds number Re ω,tr in Eq. (6.23) (instant transition to turbulence) with its values at the onset and end of transition (i.e., 1.9 × 105 and 2.75 × 105) yields curves 10 and 11 lying above and below curve 9, respectively. Reynolds number of the instant transition to turbulence Re ω,tr = 2.35 × 105, an arithmetic mean of values Re ω,tr1 and Re ω,tr2, substituted into Eq. (6.23) conveyed curve 12, which agrees with curve 9.

In the asymptotic case of Re φ  → ∞, lines 9–12 coincide with curve 6 valid for purely turbulent flow.

Thus, Eqs. (6.21) and (6.22) incorporating terms accounting for the coexistence of laminar, transitional and turbulent flow areas ensure the best agreement with experiments for the Sh av number for an entire disk. Equation (6.18) resulting from a simpler model [7] ensures the efficiency similar to that of Eqs. (6.21) and (6.22), if an “effective” Reynolds number Re ω,tr of the instant transition to turbulent flow is chosen correctly.

Application to the naphthalene sublimation technique. Again, a recalculation of the mass transfer to heat transfer data is performed using Eq. (3.4) and (6.3), with the constants K 1 defined in Eqs. (6.10) and (6.11), accordingly [23]. The factor C is equal to the constant K 1 for Sc = 1, Pr = 1 under conditions T w  = const. or C w  = const.

Authors [15, 18] used the constant m p  = 0.4 for a turbulent flow regime and Pr = Sc = 0.7–2.5. Equation (6.12) at m p  = 0.4 yields the value K 1 = 0.0323 for heat transfer in air at T w  = const. and Pr = 0.72, starting from the values K 1 = 0.0512–0.0518 (see Eqs. (6.15) and (6.16)) for Sc = 2.28 as a base for the recalculation. But, in reality, experiments [35–37] (see Table 3.5) conveyed the value of the constant K 1 = 0.0188 at T w  = const. and Pr = 0.72. The theoretical model [38, 39] gave the value K 1 = 0.0187 for the same conditions. Thus, also for turbulent flow, an erroneous value m p leads to fallacious translation of the naphthalene sublimation data to heat transfer in air.

Equations (6.10), (6.11) are to be used for the Prandtl and Schmidt numbers moderately diverging from unity: Pr = 0.7–0.74 for air; Sc = 2.28–2.5 for naphthalene sublimation. Hence, the constant C in Eqs. (6.10), (6.11) must be equal to the coefficient \(K_{1} = 0.0232\) in turbulent flow at Sc = 1, Pr = 1 and conditions T w  = const., C w  = const. (Table 3.7).

Detecting of the exponent m p for turbulent flow is performed using experimental data. Only experimental Eqs. (6.15) and (6.16) [15, 18] can serve for this purpose. Based on Eq. (6.15) as well as the values C = 0.0232, K 1 = 0.0188 (at T w  = const. and Pr = 0.72) [35–37], one can transform Eqs. (6.10) and (6.11) as follows

$$K_{1} = 0.0232{Pr}^{0.64} \quad {\text{for}}\;{Pr} \le {1},$$
(6.27)
$$K_{1} = 0.0232{Sc}^{0.96} \quad {\text{for}}\;{Sc} \ge {1}.$$
(6.28)

This means that the exponent m p for turbulent flow is not universal being a function of the Prandtl and Schmidt numbers, which apparently results from different effects of the Pr or Sc larger and smaller than unity. Equations (6.27) and (6.28) yield as a result

$${Nu/Sh} = {Pr}^{0.64}/{Sc}^{0.96}.$$
(6.29)

Using the idea of a correction function, Eq. (6.13), one can transform Eq. (6.29) as

$${Nu/Sh}_{{Sc} = {2.28}} = f({Pr}).$$
(6.30)

At Pr = 0.72, the correction function f(Pr) takes the value f(Pr) = 0.367.

As an alternative, one can use an effective value of the exponent m p so that

$${Nu/Sh} = ({Pr/Sc})^{0.87} .$$
(6.31)

The exponent m p  = 0.87 in Eq. (6.31) is more than twice larger than the value 0.4 mistakenly recommended in [15, 18]. Nevertheless, the value m p  = 0.87 must not be used in Eqs. (6.27) and (6.28) to avoid significant errors in predictions of the constant K 1.

The empirical Eq. (3.10) [37] for transitional flow at T w  = const. and Pr = 0.72 is the most appropriate to be used jointly with Eq. (6.14) for transitional flow at C w  = const. Thus, Eq. (6.12) should be recast in view of Eqs. (3.10) and (6.14) as follows

$${Nu/Sh} = ({Pr/Sc})^{0.6} .$$
(6.32)

Equation (3.10) is valid for Re ω  = 1.95 × 105–2.5 × 105, while Eq. (6.14) holds at Re ω  = 1.9 × 105–2.75 × 105. These differences are though rather insignificant.

To conclude, Eqs. (6.29)–(6.31) should be employed to recalculate the data for turbulent mass transfer for naphthalene sublimation in air to the conditions of heat transfer in air. Equation (6.32) should be applied for transitional flow for the same purpose [23].

6.3 Transitional and Turbulent Flow at High Schmidt Numbers

High values of the Schmidt numbers can be encountered in electrochemistry problems: Sc = 34–10,320 [4, 5, 7, 8, 12]. Main objectives of this section are validation and development of recommendations for the further use of the experimental and theoretical data of different authors [40].

Experimental data [7] for average Sherwood numbers for an entire disk at Re φ  = 0.278 × 106–1.8 × 106, Sc = 930–10,320 were described by a relation

$${Sh}_{\text{av}} = {Sc}^{1/3} {Re}_{\varphi }^{{{-}1/2}} [0.62{Re}_{{{\omega },{\text{tr}}}} + 1.08 \times 10^{{{-}2}} ({Re}_{\varphi }^{1.37} - {Re}_{{{\omega },{\text{tr}}}}^{{{1}{0.37}}} )].$$
(6.33)

Here, Eq. (6.18) at K 1,lam = 0.62Sc 1/3, K 1,turb = 0.0148Sc 1/3, Re ω,tr = 2.78 × 105 and n R = 0.87 was taken into account. In the transitional region at Re ω  = 2.3 × 105–2.9 × 105, Eq. (6.33) lies below the experimental data [7] (in analogy to curve 12 in Fig. 6.3), which results from simplifications incorporated in model (6.18) and mentioned in Sect. 6.2.

A reduced form of Eq. (6.33) for purely turbulent flow [7] and an equation for the local Sherwood numbers derived in [30, 40] have the following form

$${Sh}_{\text{av}} = 1.08 \times 10^{{{-}2}} {Re}_{\varphi }^{0.87} {Sc}^{1/3} ,$$
(6.34)
$${Sh} = 1.48 \times 10^{{{-}2}} {Re}_{\omega }^{0.87} {Sc}^{1/3} .$$
(6.35)

Measurements [4] of the average Sherwood numbers for an entire disk performed at Re φ  = 5 × 104–1.8 × 106, Sc = 345–6450 (transition at Re φ = 2.3 × 105–2.9 × 105) for the region of turbulent flow were described by the relation

$${Sh}_{\text{av}} = 0.0725{Re}_{\varphi }^{0.9} {Sc}^{0.33} .$$
(6.36)

The authors [5] measured local Sh and average Sh av numbers at laminar, transitional, and turbulent flow for Re ω  = 4 × 104–2.2 × 106, Sc = 680–7200 (transition at Re ω  = 2.2 × 105–3.0 × 105). Sherwood numbers for the turbulent flow [5] and average values for an entire disk (approximated in [30]) are given by the following relations, respectively

$${Sh} = 1.09 \times 10^{{{-}2}} {Re}_{\omega }^{0.91} {Sc}^{1/3} ,$$
(6.37)
$${Sh}_{\text{av}} = {7.67} \times {10}^{{{-}3}} {Re}_{\varphi }^{0.91} {Sc}^{1/3} ,$$
(6.38)
$${Sh}_{\text{av}} = {Sc}^{1/3} {Re}_{\varphi }^{{{-}{1/2}}} {[0.62Re}_{{{\omega },{\text{tr}}}} + 7.67 \times 10^{ - 3} ({Re}_{\varphi }^{1.41} - {Re}_{{{\omega },{\text{tr}}}}^{1.41} )],$$
(6.39)

where the Reynolds number of the abrupt transition was Re ω,tr = 2.78 × 105 [7].

Experiments [8] for Sh av for an entire disk were performed at Re φ  = 104–1.18 × 107, Sc = 34–1400. For purely turbulent flow at Re φ  = 8.9 × 105–1.18 × 107, authors [8] obtained

$${Sh}_{\text{av}} = {1.17} \times {10}^{{{-}2}} {Re}_{\varphi }^{0.896} {Sc}^{0.249} .$$
(6.40)

Experiments for the local Sherwood numbers in transitional flow at Re ω  = 2.0 × 105–3.0 × 105 and Sc = 1192–2465 were described by the empirical Eq. (3.14) [12].

The authors [12] deduced empirical equations for Sh av for turbulent flow (based on experiments [4]), simultaneous existence of laminar and transitional flow, as well as simultaneous existence of laminar, transitional and turbulent flow, respectively

$${Sh}_{\text{av}} = {7.8} \times {10}^{ - 3} {Re}_{\varphi }^{0.9} {Sc}^{1/3} ,$$
(6.41)
$${Sh}_{\text{av}} = {Sc}^{1/3} {Re}_{\varphi }^{{ - {1/2}}} {[0.89} \times {10}^{5} + {9.7} \times {10}^{ - {15}} {Re}_{\varphi }^{3.5} ],$$
(6.42)
$${Sh}_{\text{av}} = {Sc}^{1/3} {Re}_{\varphi }^{{ - {1/2}}} {[7.8} \times {10}^{{ - {3}}} {Re}_{\varphi }^{1.4} - 1.3 \times 10^{5} ].$$
(6.43)

Equations (6.42) and (6.43) are particular cases of Eqs. (6.22) and (6.21), accordingly, with Eq. (6.8) used for laminar, Eq. (3.14) for transitional and Eq. (6.41) for turbulent flow.

Theoretical solutions for the local turbulent Sherwood numbers at high Schmidt numbers derived in [9, 41] can be presented as follows, respectively,

$${Sh}_{\text{av}} = {7.07} \times {10}^{{ - {3}}} {Re}_{\varphi }^{0.9} {Sc}^{1/3} ,$$
(6.44)
$${Sh}_{\text{av}} = {5.93} \times {10}^{{ - {3}}} {Re}_{\varphi }^{0.91} {Sc}^{0.34} .$$
(6.45)

A theoretical solution obtained in [10] coincides with Eq. (6.41). The solution obtained in [42] has a form of Eq. (6.41) with the coefficient changed to 6.43 × 10−3.

In [43] a theoretical solution claimed to be valid for Sc = 0.72–∞ has been proposed, however, as demonstrated in [30, 40], this relation is inaccurate.

Some of the theoretical and empirical relations for the Sherwood numbers for purely turbulent flow, as well as for average Sh av numbers for an entire disk simultaneously occupied by laminar, transitional, and turbulent flow areas agree well with each other. Curves by Eqs. (6.34), (6.36), (6.44), and (6.45) practically merge (see Fig. 6.4). Equation (6.41) significantly surpasses original experiments [4]; corrected coefficient 6.43 × 10−3 [42] shifts predictions by Eq. (6.41) 9 % below those by Eq. (6.44). Empirical Eqs. (6.34) and (6.36) practically coincide, which corroborates the reliability of these experiments.

Fig. 6.4
figure 4

Average Sherwood numbers at high Schmidt numbers [30]. Approximation of experiments: 1—laminar flow, Levich’s Eq. (6.8); 2—Eq. (6.33) for an entire disk [7]; 3—Eq. (6.34) [7]; 4—Eq. (6.36) [4]; 5—Eq. (6.41) [10, 12]. Theoretical solution: 6—Eq. (6.44) [9]

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Equation (6.38) for turbulent flow and Eq. (6.39) for an entire disk significantly surpass Eqs. (6.33) and (6.34), respectively (see Fig. 6.5). Only in Eq. (6.40) [8], exponent 0.249 at the Schmidt number is not equal to 1/3. The large scatter of experiments around the approximation curve [8] is rather an evidence that the exponent 0.249 is erroneous. As demonstrated in [30, 40], differences between the curves by Eq. (6.40) plotted in the relation \({Sh}_{\text{av}} {/Sc}^{1/3}\) versus Re φ for different Sc values revealed in experiments [8] is rather significant. Hence, Eq. (6.40) should be discarded as too inaccurate.

Fig. 6.5
figure 5

Average Sherwood numbers at high Schmidt numbers, approximation of experiments [30]. 1—laminar flow, Levich’s Eq. (6.8); 2—Eq. (6.33) for an entire disk [7]; 3—Eq. (6.39), entire disk; 4—Eq. (6.46), entire disk; 5—Eq. (6.34) [7]; 6—Eq. (6.38) [5]

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The exponent for the Reynolds number Re φ in Eq. (6.35) diverges from those in Eqs. (6.15) and (6.16). Equation (6.35) can be recast to make the exponent for Re φ equal to 0.8. This yields for the entire disk [30, 40]

$$Sh_{\text{av}} Sc^{ - 1/3} = 0.62{Re}_{{{\omega },tr}}^{1/2} \left( {\frac{{{Re}_{{{\omega },tr}} }}{{{Re}_{\varphi } }}} \right)^{1/2} + \frac{2}{20.6}0.0365{Re}_{\varphi }^{0.8} \left[ {1 - \left( {\frac{{{Re}_{{{\omega },tr}} }}{{{Re}_{\varphi } }}} \right)^{1.3} } \right].$$
(6.46)

Figure 6.5 depicts curves 2 and 4 plotted by Eqs. (6.33) and (6.46), respectively. Here again Re ω,tr = 2.78 × 105, like in Eq. (6.33). Curves 2 and 4 in fact merge for Re φ  ≤ 9.0 × 105; deviations start to become visible for Re φ  > 9.0 × 105.

For purely turbulent flow, Eq. (6.46) reduces asymptotically to the relations

$$Sh = 3.65 \times 10^{ - 2} {Re}_{\omega }^{0.8} Sc^{1/3} ,$$
(6.47)
$$Sh_{\text{av}} = 3.65 \times 10^{ - 2} \frac{2}{2.6}{Re}_{\varphi }^{0.8} Sc^{1/3} = 2.81 \times 10^{ - 2} {Re}_{\varphi }^{0.8} Sc^{1/3} .$$
(6.48)

Figure 6.6 demonstrates that Eq. (6.35) [7] used at Sc = 2.28 predicts Sherwood numbers close to the experiments [15, 18] for naphthalene sublimation in air and their approximation Eq. (6.15). Equations (6.35) (curve 10) and (6.15) (curve 9) correlate well at larger Reynolds numbers Re ω  = 0.6 × 106–2.0 × 106. Curve 11, Eq. (6.47), lies in the vicinity of curve 10 at smaller Reynolds numbers Re ω  ≤ 0.7 × 106. Equation (6.47) yields K 1 = 0.048 at Sc = 2.28, which is only 6.7 % below the value K 1 = 0.0512 in Eq. (6.15). Curve 12 by Eq. (6.37) goes noticeably beyond experiments and approximation curve 9 in Fig. 6.6.

Fig. 6.6
figure 6

Local Sherwood numbers for naphthalene sublimation [30]. Experiments: 1—Sc = 2.28 [15]; 2—Sc = 2.4 [21]; 3—Sc = 2.4 [26]; 4—Sc = 2.44 [19]; 5—Sc not mentioned [18]. Empirical Eq. (6.3): 6—laminar flow, n R = 1/2, K 1 = 0.625 [20, 25, 26]; 7—laminar flow, n R = 1/2, K 1 = 0.604 [17]; 8—transitional flow, n R = 4, K 1 = 2 × 10−19 [15]; 9—turbulent flow, n R = 0.8, K 1 = 0.0512 [15]. Developed turbulent flow, Sc = 2.28: 10—Eq. (6.35) [7]; 11—Eq. (6.47); 12—Eq. (6.37) [5]. Transitional flow, Sc = 2.28: 13—Eq. (3.14) [12]

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Dependence 13 in Fig. 6.6 plotted by experimental Eq. (3.14) [12] for transitional flow at Sc = 2.28 conforms well to Eq. (6.14) and experiments [15].

To conclude, the most reliable empirical relations for developed turbulent flow and an entire disk relying on the analysis made above are Eqs. (6.33)–(6.36).

6.4 An Integral Method for Pr and Sc Numbers Much Larger Than Unity

Model with a constant value \({\mathbf{\Delta}} \ll 1.\) The thickness of the thermal (or diffusion) boundary layer at very high Pr or Sc numbers is much smaller than the thickness of the velocity boundary layer (i.e., \({{\Delta}} \ll 1.\)). Hence, in Eq. (3.40) obtained for Δ = const. and \(T^{ + } \equiv T^{ + } (y^{ + } )\), all summands in the parentheses in its left-hand part but a * tend to zero

$$\Delta ^{2n + 1} a_{*} = \frac{4 + m}{{4 + m + n_{*} }}{(}a_{*} - 2b_{*} + c_{*} {)Pr}^{{ - n_{\text{p}} }} .$$
(6.49)

Relying on Eq. (6.49), one can derive analytical solutions for constants Δ and K 1

$$\Delta = \left[ {\frac{4 + m}{{4 + m + n_{*} }}\left( {1 - \frac{{2D_{3} }}{{C_{2} }}} \right)} \right]^{{\frac{1}{2n + 1}}} {Pr}^{{ - \frac{{n_{\text{p}} }}{2n + 1}}} ,$$
(6.50)
$$K_{1} = K_{3} \left[ {\frac{4 + m}{{4 + m + n_{*} }}\left( {1 - \frac{{2D_{3} }}{{C_{2} }}} \right)} \right]^{{\frac{ - n}{2n + 1}}} {Pr}^{{1 + n_{\text{p}} \left( {\frac{n}{2n + 1} - 1} \right)}} ,$$
(6.51)

where the coefficients C 2 and D 3 are described in the comments to Eqs. (2.68) and (2.69).

The cumulative exponent at the Prandtl number in Eq. (6.51) for Pr ≫ 1 must be equal to 1/3 (see Sect. 6.3), which yields the following expression for п p

$$n_{\text{p}} = \frac{2}{3} \cdot \frac{2n + 1}{n + 1}.$$
(6.52)

As a result, the constants K 1 and K 2 in view of Eq. (3.35) can be written as

$$K_{1} = K_{3} \left[ {\frac{4 + m}{{4 + m + n_{*} }}\left( {1 - \frac{{2D_{3} }}{{C_{2} }}} \right)} \right]^{{\frac{ - n}{2n + 1}}} {Pr}^{1/3} ,$$
(6.53)
$$K_{2} = K_{3} \left[ {\frac{4 + m}{{4 + m + n_{*} }}\left( {1 - \frac{{2D_{3} }}{{C_{2} }}} \right)} \right]^{{\frac{ - n}{2n + 1}}} \frac{{n_{*} + 2}}{{2 + n_{*} + m}}{Pr}^{1/3} .$$
(6.54)

To enable validations against electrochemical experiments, let us further treat the Sherwood numbers rather than the Nusselt numbers and replace Pr with Sc.

In Fig. 6.7, Eq. (6.54) for Sh av (at n * = 0) is validated against the empirical Eq. (6.34) [7] and theoretical Eq. (6.44) [9]. Curves 5 and 6 predicted by Eq. (6.54) at n = 1/7 and 1/9 lie 20–30 % below the curves 3 and 4 suggested by Eqs. (6.34) and (6.44), accordingly. Such a discrepancy between theory and measurements is too high. In addition, the slope of the curves 5 and 6 (exponents at Re φ being 0.8 and 0.833, constants K 2 being 0.0207 and 0.0126, accordingly) distinctly deviates from the slope of curves 3 and 4 (exponents at Re φ being 0.87 and 0.9, constants K 2 being 0.0207 and 0.126, accordingly). Therefore, some model approaches incorporated in the present integral method partially fail at high Pr and Sc numbers and need to be improved. In Eq. (3.32) for the coefficient K 1, the total exponent at the Reynolds number can be increased, provided that the relative thickness Δ is assigned to be a decreasing function of the local Reynolds number Re ω .

Fig. 6.7
figure 7

Average Sherwood numbers for high Schmidt numbers [30]. Approximation of experiments: 1—laminar flow, Levich’s Eq. (6.8) [11]; 2—Eq. (6.33) for an entire disk [7]; 3—Eq. (6.34) [7] Theoretical solutions: 4—Eq. (6.44) [9]; 5—Eq. (6.54) for n = 1/7; 6—Eq. (6.54) for n = 1/9; 7—Eq. (6.73)

Full size image

Model with a variable value of Δ. The present integral method incorporates a model, in frames of which a boundary layer consists two parts. In the vicinity of the wall, a viscous and heat conduction sub-layers emerge, where the velocity and temperature profiles are described by Eq. (2.62). In the main part of the boundary layer (outside of the viscous sub-layer), velocity components are described by the power-law functions (see Chaps. 2 and 3). If Prandtl and Schmidt numbers are slightly different from unity, the thermal/diffusion and velocity boundary layers have a thickness of the same order of magnitude [30, 40]. Hence, integration of Eq. (2.23) for the thermal boundary layer has been performed over the entire velocity boundary layer. Viscous and heat conduction sub-layers are not taken into account in this integration, because they are negligibly thin in comparison with the overall boundary layer thickness. Velocity profiles in Eqs. (2.17)–(2.19) are integrated in the same way [31, 38, 39, 44, 45].

At very high Prandtl and Schmidt numbers, the boundary layer structure changes drastically. A very thin thermal/diffusion boundary layer is fully incorporated inside the viscous sub-layer of the velocity boundary layer; here the radial velocity profile varies linearly depending on the coordinate z (curve 4 in Fig. 6.8). This fact is taken into account in theoretical models [5, 9–11, 32, 42] for large Pr and Sc numbers.

Fig. 6.8
figure 8

Radial velocity profiles in the turbulent boundary layer over a free rotating disk [30]. 1—n = 1/7; 2—1/8; 3—1/9. Equation (2.41), [44]: 4—Eq. (6.55) at Re ω  = 1.0 × 106; 5—Eq. (3.27) at σ = 0, n = 1/7 (see Fig. 3.7). Experiments: 6—Re ω  = 0.4 × 106; 7—0.65 × 106; 8—0.94 × 106; 9—1.6 × 106 [46]; 10—0.6 × 106; 11—1.0 × 106 [47]

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Next to the wall, the radial velocity v r varies as a linear function

$${v}_{r} = \frac{{{\tau_{w{r}}}}}{\mu }z = \frac{{{\tau }_w {\alpha }}}{{{\mu }(1 + {\alpha }^{2} )^{1/2} }}z = \frac{{{\rho }\,V_{*}^{2} {\alpha }}}{{{\mu }(1 + {\alpha }^{2} )^{1/2} }}\frac{{c_{\text{f}} }}{2}z = {\alpha }(1 + {\alpha }^{2} )^{1/2} {\omega }\,A_{\text{c}} {Re}_{\omega }^{{n_{\text{R}} }} z.$$
(6.55)

Here the constant п R is defined in Eq. (3.31).

The coordinate of the boundary of the viscous sub-layer \(z_{1}^{ + }\), where the linear model (6.55) holds, can be written as

$$\frac{{z_{1} }}{\delta } = \frac{{z_{1}^{ + } }}{{{\gamma }(1 + {\alpha }^{2} )^{1/2} A_{\text{c}}^{1/2} {Re}_{\omega }^{1/(1 + 3n)} }},$$
(6.56)

where \(z_{1}^{ + }\) = 12.54; 13.44; 14.23 and 15.09 for n = 1/7; 1/8; 1/9 and 1/10, respectively (see Chap. 2). According to Eq. (6.56), this corresponds to z 1/δ = 0.01–0.02. Figure 6.8 confirms the validity of the model (6.55) up to \(z/{\delta }_{\varphi }^{**} = 0.2\), \(z/{\delta } = 0.02\), or \(\Delta = {\delta }_{\text{T}} / {\delta } = 0.02\).

In the power-law model, the Stanton number is given by Eq. (2.64). In Sect. 2.4.3, the model assumption \(\left( {{{z_{1\text{T}}^{ + } } \mathord{\left/ {\vphantom {{z_{1\text{T}}^{ + } } {z_{1}^{ + } }}} \right. \kern-0pt} {z_{1}^{ + } }}} \right)^{{n_{\text{T}} - 1}} Pr^{{ - n_{\text{T}} }} = Pr^{{ - n_{\text{p}} }}\) completes Eq. (2.64), whereas validations of this model against experiments deliver the value of the exponent п p.

At high Prandtl or Schmidt numbers, the entire thermal/diffusion boundary layer is included inside the viscous sub-layer of the velocity boundary layer. Hence, one can assume that the relation between the coordinates \(z_{1}^{ + }\) and \(z_{1\text{T}}^{ + }\) (viscous and heat conduction sub-layer) can be recast as

$$(z_{1\text{T}}^{ + } /z_{1}^{ + } )^{{n_{\text{T}} - 1}} {Pr}^{{ - n_{\text{T}} }} = K_{\alpha } {Pr}^{{ - n_{\text{p}} }} .$$
(6.57)

Validations of the model (6.57) against experiments for the Nu or Sh numbers enable finding the coefficient K α and exponent п p. Consequently, given n = n T, Eqs. (2.66) and (2.67) turn to

$${St} = (c_{\text{f}} /2)\Delta^{ - n} {Pr}^{{ - n_{\text{p}} }} K_{\alpha } = A_{\text{c}} {Re}_{\omega }^{ - 2n/(3n + 1)}\Delta^{ - n} {Pr}^{{ - n_{\text{p}} }} K_{\alpha } ,$$
(6.58)
$$Nu = St{Re}_{\omega } Pr (1 + {\alpha }^{2} )^{1/2} = A_{\text{c}} (1 + {\alpha }^{2} )^{1/2} {Re}_{\omega }^{{n_{\text{R}} }}\Delta ^{ - n} Pr^{{1 - n_{\text{p}} }} K_{\alpha } .$$
(6.59)

Substituting Eqs. (2.53), (6.55), (6.58) and (6.59) into Eq. (2.20), one can transform the latter to the following notation:

$$\frac{n}{2(n + 2)} {\alpha \omega }\frac{\text{d}}{{\text{d}r}}\left[ {r{\delta }^{2}\Delta^{2} {Re}_{\omega }^{{n_{\text{R}} }}\Delta T} \right] = K_{\alpha }\Delta^{ - n} {Pr}^{{ - n_{\text{p}} }} {Re}_{\omega }^{{n_{\text{R}} }} \nu\Delta T,$$
(6.60)

where Eq. (2.77) determines the boundary layer thickness δ, while \(\Delta T = T_{w} - T_{\infty }\).

The condition Δ = const. is inapplicable to Eq. (6.60), otherwise the exponents at the variable r on the left- and right-hand sides of Eq. (6.60) are not equal to each other.

Let us assume the parameter Δ to be a power-law function

$$\Delta (r) = C_{\Delta } r^{k} .$$
(6.61)

Substituting Eq. (6.61) into Eq. (6.60) and keeping in mind Eqs. (2.30), (2.77), (2.78) and (3.31), one can finally obtain

$$\Delta = C_{{\Delta *}} {Re}_{\omega }^{k/2} ,$$
(6.62)
$$C_{{\Delta *}} = C_{{\Delta **}} {Pr}^{{ - n_{\text{p}} /(2 + n)}} ,$$
(6.63)
$$C_{{\Delta **}} = \left[ {\frac{{K_{\alpha } 2(n + 2)/n}}{{{\alpha \gamma }^{2} (1 - nk + n_{*} + 2n_{\text{R}} )}}} \right]^{1/(n + 2)} ,$$
(6.64)
$$k = - 2m/(2 + n).$$
(6.65)

Equation (6.59) for the Nu number and the expression for Nu av can be written as follows:

$${Nu} = K_{1} {Re}_{\omega }^{{n_{\text{R}*} }} ,$$
(6.66)
$${Nu}_{\text{av}} = K_{2} {Re}_{\varphi }^{{n_{\text{R}*} }} ,$$
(6.67)
$$n_{\text{R}*} = n_{\text{R}} + mn/(2 + n),$$
(6.68)
$$K_{1} = K_{\alpha } K_{3} C_{{\Delta **}}^{ - n} {Pr}^{1/3} ,$$
(6.69)
$$K_{2} = 2K_{1} /(2n_{\text{R}*} + 1),$$
(6.70)
$$n_{\text{p}} = (2 + n)/3.$$
(6.71)

Equation (6.71) takes into account the fact that the total exponent at the Pr number in Eq. (6.69) must be equal to 1/3.

Thus, in Eqs. (6.66) and (6.67), the total exponent n R* at the Reynolds number is larger than that in Eq. (3.31) due to the additional term \(mn/(2 + n)\) (see Eq. (6.68)). This summand emerges as a result of the model with the variable parameter Δ being a subsiding function of the coordinate r or, in other words, local Re ω (see Eq. (6.62)).

The values of the exponent n R* are: n R*  = 0.84 at n = 1/7, and n R*  = 0.868 at n = 1/9. The latter agrees well with the exponent 0.87 at the Re φ number in the experiment-based Eq. (6.35) [7]. To bring Eq. (8.73) at n R*  = 0.868 into agreement with Eq. (6.35), the constant K α must be equal to K α  = 1.254, which yields at n *  = 0

$${Nu} = 1.52 \times 10^{ - 2} {Re}_{\omega }^{0.868} {Pr}^{1/3} ,$$
(6.72)
$${Nu}_{\text{av}} = {1.11} \times {10}^{{ - {2}}} {Re}_{\varphi }^{0.868} {Pr}^{1/3} ,$$
(6.73)
$$\Delta = 18.31{Re}_{\omega }^{ - 0.3158} {Pr}^{ - 1/3} .$$
(6.74)

In Fig. 6.7, curve 7 by Eq. (6.73) and curve 3 by Eq. (6.34) merge. Equations (6.72) and (6.35) are also practically identical.

The parameter range in experiments [7] is Re φ  = 0.278 × 106–1.8 × 106, Sc = 930–10,320. At minimal values Sc = 930 and Re ω  = 0.278 × 106 [7], Eq. (6.74) yields Δ = 0.036. Parameter Δ is a decreasing function of the Schmidt and Reynolds numbers. Thus Δ = 0.015 at Sc = 10,320 and Re ω  = 0.278 × 106, whereas Δ = 0.02 at Sc = 930 and Re ω  = 1.8 × 106. This conforms to the limit \(\Delta \le 0.02\) restricting validity of the linear model of the radial velocity profile.

Model with variable Δ and profile T + depending on Re ω . In the theoretical works [9–11, 42], the Nusselt number at high Pr values is described by a relation

$${Nu} = K_{\text{N}} (1 + {\alpha }^{2} )^{1/2} (c_{\text{f}} /2)^{1/2} {Re}_{\omega } {Pr}^{1/3} ,$$
(6.75)

where K N is an empirical constant; Eq. (2.82) at n = 1/7 was used for c f/2. Local and average Nusselt numbers take a form of Eqs. (6.66) and (6.67), respectively, with

$$K_{1} = K_{\text{N}} (1 + {\alpha }^{2} )^{1/2} A_{\text{c}}^{1/2} {Pr}^{1/3} ,$$
(6.76)
$$n_{\text{R}*} = (2n + 1)/(3n + 1),$$
(6.77)

while the constants K 1 and K 2 are related with Eq. (6.70). At n = 1/7, Eq. (6.77) brings n R*  = 0.9.

Matching Eqs. (6.44) and (6.75) in view of Eq. (6.70), one can obtain K N = 0.05986.

Substitution of Eq. (6.61) into the thermal boundary layer equation yields again Eq. (6.62) for Δ with

$$C_{{\Delta *}} = C_{{\Delta **}} {Pr}^{ - 1/3} ,$$
(6.78)
$$C_{{\Delta **}} = \left[ {\frac{{K_{\text{N}} 2(n + 2)/n}}{{{\alpha \gamma }^{2} (1 + 2m + 2k + n_{*} + 2n_{\text{R}} )A_{\text{c}}^{1/2} }}} \right]^{1/(n + 2)} ,$$
(6.79)
$$k = (2n - 1)/(3n + 1).$$
(6.80)

Setting the values n = 1/7 and n *  = 0 into Eqs. (6.62), (6.78)–(6.80) one can obtain

$$\Delta = 12.54{Re}_{\omega }^{ - 1/4} {Pr}^{ - 1/3} .$$
(6.81)

At the lower experimental limit of Sc = 930 and Re ω  = 0.278 × 106 [7], the value of Δ in view of Eq. (6.81) reduces to Δ = 0.037. For the conditions Sc = 10,320 and Re ω  = 0.278 × 106: Δ = 0.016. For Sc = 930 and Re ω  = 1.8 × 106: Δ = 0.023. These values for Δ conform to the data obtained by Eq. (6.74) and the upper limiting boundary \(\Delta \le 0.02\) of the validity of the linear model for the radial velocity.

According to the models with a constant and variable value of Δ, the function T + in wall coordinates, defined by the power-law Eq. (2.23) at n = n T does not depend on the Reynolds number, which is consistent with the results presented in [48, 49].

Model incorporating Eq. (6.75) results in the profile of T + being a function of Re ω

$$T^{ + } = (z^{ + } )^{n} (1 + {\alpha }^{2} )^{ - n/2} \gamma^{ - n} A_{\text{c}}^{ - n/2} C_{{\Delta *}}^{ - n} K_{\text{N}}^{ - 1} {Pr}^{2/3} {Re}_{\omega }^{{ - 0.5(2n^{2} + n)/(3n + 1)}}.$$
(6.82)

To conclude, a novel methodology for simulations of temperature/concentration profiles for the values of Pr and Sc much larger than unity was outlined in this section. An original integral method enabled evaluating a relative thickness Δ of the thermal/diffusion boundary layers that has not been attained by the other investigators. It was demonstrated that the model with a subsiding function Δ( r ) yields a new summand in the expression for the exponent at the Reynolds number, which determines functional dependence of Nu or Sh numbers on the local radius r. Consequently, theoretical relations obtained for Nusselt and Sherwood numbers are in a good consistency with the selected empirical equations.