A way to explain the thermal boundary effects on laminar convection through a square duct

Abstract

A way using the reformulation of the energy conservation equation in terms of heat flux to explain the thermal boundary effects on laminar convective heat transfer through a square duct is presented. For a laminar convection through a square duct, it explains that on the wall surface, the velocity is zero, but convection occurs for uniform wall heat flux (UWHF) boundary in the developing region due to the velocity gradient term; for uniform wall temperature (UWT) boundary, only diffusion process occurs on the wall surface because both velocity and velocity gradient do not contribute to convection; for UWHF, the largest term of the gradient of velocity components (the main flow velocity) on the wall surface takes a role in the convection of the heat flux normal to the wall surface, and this role exists in the fully developed region. Therefore, a stronger convection process occurs for UWHF than for UWT on the wall surface. The thermal boundary effects on the laminar convection inside the flow are also detailed.

Keywords

convective transport / heat transfer / mass transfer / laminar flow / thermal boundary effects

Introduction

All results show that under the same flow condition, laminar convection will have a larger Nusselt number when uniform wall heat flux (UWHF) is applied on the wall than when uniform temperature (UWT) is enforced [1]. For example, at laminar fully developed fluid flow and under heat transfer condition, in square duct, the Nusselt number is 3.61 and 2.98 for UWHF and UWT boundary conditions [2], respectively. The boundary layer theory [3] is commonly used to explain the mechanism of convective heat transfer. One commonly accepted explanation of thermal boundary effects is that for UWHF and UWT boundary conditions, the boundary layers of temperature are different, and different heat transfer characteristics for these different boundary conditions can be obtained [2]; hence, the intensity of the convections are different. Another difference of UWHF and UWT boundary conditions for heat transfer is that, for the case of UWHF, there exists a temperature gradient on the surface of the wall, but for the case of UWT, such temperature gradient is vanished.

Recently, an additional description about convective heat transfer has been presented [4]. The main idea is that the energy conservation equation (ignoring the energy dissipation in flow)

(1){\displaystyle \frac{\partial T}{\partial t}}+(v\cdot \nabla )T=a{\nabla }^{\mathrm{2}}T,

can be written in the term of heat flux \mathit{q}=-\lambda \nabla T:

(2){\displaystyle \frac{\partial \mathit{q}}{\partial t}}+(v\cdot \nabla )\mathit{q}+e_{ij}\mathit{q}=a{\nabla }^{\mathrm{2}}\mathit{q}.

Displaying every term of Eq. (2), the following equations can be obtained:

(3){\displaystyle \frac{\partial q_x}{\partial t}}+\left(u{\displaystyle \frac{\partial q_x}{\partial x}}+v{\displaystyle \frac{\partial q_x}{\partial y}}+w{\displaystyle \frac{\partial q_x}{\partial z}}\right)+\left(q_x{\displaystyle \frac{\partial u}{\partial x}}+q_y{\displaystyle \frac{\partial v}{\partial x}}+q_z{\displaystyle \frac{\partial w}{\partial x}}\right)=a{\nabla }^2{q}_x,

(4){\displaystyle \frac{\partial q_y}{\partial t}}+\left(u{\displaystyle \frac{\partial q_y}{\partial x}}+v{\displaystyle \frac{\partial q_y}{\partial y}}+w{\displaystyle \frac{\partial q_y}{\partial z}}\right)+\left(q_x{\displaystyle \frac{\partial u}{\partial y}}+q_y{\displaystyle \frac{\partial v}{\partial y}}+q_z{\displaystyle \frac{\partial w}{\partial y}}\right)=a{\nabla }^2{q}_y,

(5){\displaystyle \frac{\partial q_z}{\partial t}}+\left(u{\displaystyle \frac{\partial q_z}{\partial x}}+v{\displaystyle \frac{\partial q_z}{\partial y}}+w{\displaystyle \frac{\partial q_z}{\partial z}}\right)+\left(q_x{\displaystyle \frac{\partial u}{\partial z}}+q_y{\displaystyle \frac{\partial v}{\partial z}}+q_z{\displaystyle \frac{\partial w}{\partial z}}\right)=a{\nabla }^2{q}_z.

Equations (3) to (5) can be rewritten as

(6){\displaystyle \frac{\partial q_x}{\partial t}}+W_{\mathrm{c}-x}+W_{\mathrm{e}-x}=\alpha {\nabla }^{\mathrm{2}}{q}_x,

(7){\displaystyle \frac{\partial q_y}{\partial t}}+W_{\mathrm{c}-y}+W_{\mathrm{e}-y}=\alpha {\nabla }^{\mathrm{2}}{q}_y,

(8){\displaystyle \frac{\partial q_z}{\partial t}}+W_{\mathrm{c}-z}+W_{\mathrm{e}-z}=\alpha {\nabla }^{\mathrm{2}}{q}_z,

where

(9)W_{\mathrm{e}-x}=q_x{\displaystyle \frac{\partial u}{\partial x}}+q_y{\displaystyle \frac{\partial v}{\partial x}}+q_z{\displaystyle \frac{\partial w}{\partial x}},

(10)W_{\mathrm{e}-y}=q_x{\displaystyle \frac{\partial u}{\partial y}}+q_y{\displaystyle \frac{\partial v}{\partial y}}+q_z{\displaystyle \frac{\partial w}{\partial y}},

(11)W_{\mathrm{e}-z}=q_x{\displaystyle \frac{\partial u}{\partial z}}+q_y{\displaystyle \frac{\partial v}{\partial z}}+q_z{\displaystyle \frac{\partial w}{\partial z}},

(12)W_{\mathrm{c}-x}=u{\displaystyle \frac{\partial q_x}{\partial x}}+v{\displaystyle \frac{\partial q_x}{\partial y}}+w{\displaystyle \frac{\partial q_x}{\partial z}},

(13)W_{\mathrm{c}-y}=u{\displaystyle \frac{\partial q_y}{\partial x}}+v{\displaystyle \frac{\partial q_y}{\partial y}}+w{\displaystyle \frac{\partial q_y}{\partial z}},

(14)W_{\mathrm{c}-z}=u{\displaystyle \frac{\partial q_z}{\partial x}}+v{\displaystyle \frac{\partial q_z}{\partial y}}+w{\displaystyle \frac{\partial q_z}{\partial z}}.

These equations can be used to explain the thermal boundary effects on laminar convective heat transfer. In this paper, the difference of the convections under uniform wall temperature and heat flux conditions is presented in terms of W.

A way to explain thermal boundary effects on convection

Specifying the case considered

In order to explain the thermal boundary effects on convective heat transfer using Eqs. (6) to (8), every term in Eqs. (6) to (8) must be determined. Since it is not easy to find these terms by experimental methods, a numerical method is used in this paper instead. A steady-state developing convective heat transfer in a square duct with L_x × L_y × L_z = 0.3 m × 0.01 m × 0.01 m, the fully developed fluid flow, and the heat transfer in a square duct with size of L_x × L_y × L_z = 0.018 m × 0.01 m × 0.01 m are studied. The main flow is directed along the x-direction, and the fluid flow is limited as laminar. For the case of UWT, the temperature on the four walls of the duct is specified to be uniform; for the case of UWHF, a uniform normal wall heat flux is specified on the four walls of the duct. The numerical method [5,6] is used to conduct the simulation of the convective heat transfer. The local heat transfer coefficient is determined by

(15)h_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{{(-\lambda \partial T/\partial n)}_{\mathrm{w}}}{T_{\mathrm{w}}-T_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}}},

where

T_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}={\displaystyle \frac{{\displaystyle \iint u(x,y,z)T(x,y,z)\mathrm{d}y\mathrm{d}z}}{{\displaystyle \iint u(x,y,z)\mathrm{d}y\mathrm{d}z}}}.

The local and average Nusselt number are determined by

(16)N{u}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}=h_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}{d}_{\mathrm{h}}/\lambda ,N{u}_{\mathrm{S}}={\displaystyle \underset{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}{∳}(h_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}{d}_{\mathrm{h}}/\lambda )}dl/(\mathrm{4}{L}_y).

The Reynolds number is \mathit{R}\mathit{e}=\rho u_{\mathrm{m}}{d}_{\mathrm{h}}/\mu, and the hydraulic diameter is d_{\mathrm{h}}=L_y=L_z.

It is found that for different Reynolds numbers in a laminar region, the characteristics of the thermal boundary effects are the same. Only the results at Re=700 is presented here. The numerical method and its uncertainty are tested by the existing data in the fully developed region: Nu=2.98 for UWT, Nu=3.62 for UWHF, and fRe=57 [2], where f is the friction factor. The relative error of the numerical result and the former data is less than 1%. The grid independence is also tested. It is found that a grid system of 30 × 30 × 300 for developing region study or 30 × 30 × 30 for fully developed region study is big enough to obtain the reliable results not only for the fluid flow and heat transfer but also for the higher order difference terms appeared in Eqs. (6) to (8).

Comparisons under UWT and UWHF conditions should be performed with unit temperature difference. Any term related to heat flux is normalized by the local temperature difference DT=T_w-T_bulk for UWT, and DT=ST_w-T_bulk, where ST_w is the span averaged value of the wall temperature for UWHF conditions. The physical meaning of this normalization is that the intensity of the convection transport is evaluated in unit temperature difference generally. In the following sections, except the Nusselt number, with no special mentioning, all parameters defined in the above sections denote the normalized values.

Thermal boundary effects on convection in terms of normalized W

In the developing region on the wall surface

The transport characteristics of the heat flux normal to the wall surface y = 0 are presented in Fig. 1. On the wall surface, the velocity is zero, and the transportations of heat fluxes corresponding to three directions caused by velocity terms are all zero, which are represented by that the values of W_c-x, W_c-y, and W_c-z are zero (see Eqs. (12) to (14)). The span averaged SW_c-y is shown in Fig. 1(a). It is clearly seen that the velocity term has no contribution to the convection of q_y. The span averaged velocity gradient term SW_e-y presented in Fig. 1(a) discloses that on the wall surface, the velocity gradient has a contribution to the convection of q_y for UWHF condition. The SW_e-y of UWT has a positive value, but SW_e-y has a negative value for UWHF. In the inlet region, SW_e-y decreases along the main flow direction for UWHF. The tread of span averaged value of the summation of all terms S(W_e-y + W_c-y) corresponds to Sq_y, as shown in Fig. 1(b). The trend of Sq_y corresponds to the intensity of the convection of heat flux q_y on the wall surface, y=0. These figures indicate that the larger S(W_e-y + W_c-y), the stronger the convection occurs on wall surface for UWHF.

For the case of UWT, q_x and q_z are zero, W_e-y = q_y¶v/¶y, and it has zero value; for the case of UWHF, q_x and q_z exist, W_e-y = q_x¶u/¶y, and ¶u/¶y is the largest velocity gradient term. This detailed information declares that the largest velocity gradient term has no contribution to the convection of q_y for UWT. Contrary to the case of UWT, the largest velocity gradient term takes a major role in the convection of q_y on the wall surface for UWHF. From this, it can be found that different thermal boundary conditions have different transport mechanisms for the heat flux normal to the wall surface.

The local distributions of W_e-y on the wall surface y = 0 for UWT and UWHF are presented in Fig. 2. Figure 2(a) indicates that on the y = 0 surface, W_e-y is zero on lateral wall positions. For UWHF, near the lateral wall, there are two peak values for W_e-y. For UWT, the value of W_e-y is zero. The summation of all terms, W_e-y + W_c-y is shown in Fig. 2(b). The amplitude of the summation term of UWHF is significantly larger than its counterpart of UWT. Figures 2(c) and (d) indicate that the convection intensity of UWHF is stronger than that of UWT at the same position of x. It should be emphasized that the peak values of the summation of W_e-y and W_c-y do not need to correspond to the peak value of q_y or h, but in the region with these peak values, the gradient of q_y in the z direction changes rapidly, and the peak value corresponds to the turning point of q_y.

It is clear that with the promotion of the velocity gradient, at the inlet region, the convection of q_y takes place on the wall surface for UWHF. Therefore, for UWHF, powered by a large velocity gradient term, a stronger convection of q_y occurs on the wall surface. The strength of such promotion decreases along the main flow direction; hence, the intensity of the convection of q_y on the wall surface decreases downstream. For UWT, only diffusion takes place on the wall surface.

In the developing flow region

As shown in Fig. 3(a), in the developing flow region, the velocity term and the velocity gradient term have contributions to the convection of q_x. CW_e-x and CW_c-x have different signs for both cases of UWT and UWHF. C (W_e-x + W_c-x) has the sign of CW_c-x for UWT but has the sign of CW_e-x for UWHF. This means that the velocity gradient term has a negative contribution to the transport of q_x, while the velocity has a positive contribution to the transport of q_x for UWT. However, for the case of UWHF, the velocity gradient term has a positive effect on the transport of q_x, whereas the velocity term has a negative effect. The trend of C(W_e-x + W_c-x) corresponds to the trend of cross-averaged heat flux in the x direction, Cq_x, presented in Fig. 3(b) at the inlet region. These tell that the convection of q_x is controlled by the velocity term and the velocity gradient term; inside the flow, the averaged contribution of the velocity gradient term to the transport of q_x is negative for UWT, but this changes direction for UWHF. The absolute value of C(W_e-x + W_c-x) for UWT is larger than that for UWHF, but the absolute value of Cq_x for UWT is smaller than that for UWHF. The fact that no good agreement is found between the trend of C (W_e-x + W_c-x) and Cq_x tells that diffusion process plays a major role in the transport of q_x for UWHF in the developing region.

The local information of W_e-x, and W_c-x, on the special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) at different stream wise locations is presented in Fig. 4. As shown in Figs. 4(a) and (b), W_e-x is negative, but W_c-x has a positive peak value for UWT; there are two other negative peak values of W_c-x immediately beside the regions with the peak values mentioned above; and all of the peak values of W_e-x and W_c-x decrease downstream in the UWT condition. W_e-x has four peak values, as shown in Fig. 4 (a), for UWHF. The peak values near the wall surfaces are positive; while the other two peak values are negative; W_c-x also has four peak values, as shown in Fig. 4(b). They have a similar distribution as W_e-x of UWT. On the specified line, near the wall surfaces, both W_e-x and W_c-x have the same sign and have contributions to the convection of q_x; all of the peak values of W_e-x and W_c-x decrease greatly downstream for UWHF. As shown in Figs. 4(a) and (b) for UWT, the amplitude of the largest peak value of W_e-x or W_c-x is larger than that for UWHF, but a stronger convection process for UWHF than for UWT is obtained because for the case of UWT, near the wall surfaces, W_e-x and W_c-x have different signs, and the combination of these terms has a small value.

The summation of W_e-x and W_c-x are presented in Fig. 4(c), where more peak values appear for UWT. The amplitude of these peak values decreases greatly compared with W_c-x. Every peak value corresponds to a turning point in the distribution of q_x shown in Fig. 4(d) for UWT and UWHF. On the wall surface, q_x exists for UWHF but does not for UWT. The region with a strong intensity of the convection of q_x is quite different for UWHF and UWT. For UWHF, the convection region with a strong intensity begins on the wall surfaces, but for UWT, this region is far away from the wall surfaces. The main reason for this is that on the wall surface of UWT, a zero flux of q_x is specified. As shown in Fig. 4 (d), the intensity of the convection of q_x for UWHF is stronger than that for UWT.

The information of the transport of the heat flux in the y direction, q_y, on special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) is presented in Fig. 5. The velocity gradient term and the velocity term have the same sign in near wall regions, as shown in Figs. 5(a) and (b), for UWT. The W_e-y of UWT has two small peak values compared with the large W_e-y of UWHF, but W_c-y has four peak values for UWT and UWHF. The absolute peak value of W_c-y is larger than that of W_e-y for UWT but is smaller than that of W_e-y for UWHF. W_e-y has a different sign with W_c-y. The summation of W_e-y + W_c-y is shown in Fig. 5(c). Four peak values appear for UWT, but the amplitude of these peak values does not decrease greatly compared with W_e-y and W_c-y of UWHF. Figure 5(c) also indicates that for the transport of q_y, the velocity term takes the major role for UWT, but the velocity gradient term takes the major role for UWHF. Every peak value corresponds to a turning point in the distribution of q_y shown in Fig. 5(d) if a turning point is noted on the wall surface for UWHF.

If the value of W_e-y + W_c-y is taken as reference, near wall surfaces the velocity term has a negative contribution to the convection of q_y for UWHF, but both of W_e-y and W_c-y contribute to the convection of q_y for UWT. The velocity term has a negative contribution to the convection of q_y in the region with peak values near the flow core for UWT and UWHF. As shown in Fig. 5(d), the peak values with large amplitudes near the wall surface correspond to a very strong convection of q_y near wall regions. The decrease of this amplitude corresponds to a decrease of q_y downstream. The small value of the summation in the region of flow core corresponds to a small absolute value of q_y for both UWT and UWHF.

The transport of q_z on special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) takes the similar characteristics as q_y special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y).

In the fully developed region on the wall surface

When the fluid flow and the heat transfer are fully developed, on the wall surface, not only are W_c-x, W_c-y, and W_c-z zero, but also W_e-x, W_e-y, and W_e-z are zero (see Eqs. (9) to (11)) for UWT. For UWHF, W_e-y exists on the wall surface of y=0. Among them, the span averaged value, SW_c-y and SW_e-y on y=0 surface are shown in Fig. 1(c). It is clearly seen that neither the velocity term nor the velocity gradient term has any contribution to the convection of q_y in the case of UWT. Since UWHF SW_e-y has a constant value, on the wall surface, the promotion for the convection of q_y is the velocity gradient term, SW_e-y; hence, there exists a stronger intensity of convection of q_y on the wall surface, as shown in Fig. 1(d). Sq_y has the same trend as that of span averaged W_e-y.

In the fully developed region, on the wall surface local distributions of W_e-y are shown in Fig. 6. W_e-y=0 for UWT, but W_e-y has a stream wise independent value for UWHF, as indicated by Fig. 6(a). The summation of the velocity term and the velocity gradient term is presented in Fig. 6(b). Such characteristics correspond to the trend of stream wise independent q_y, as shown in Fig. 6(c), and the heat transfer coefficient, as shown in Fig. 6(d). UWHF has a larger h than UWT has.

It is clear that for UWT W_c-x, W_c-y, W_c-z, W_e-x, W_e-y, and W_e-z are zero on the surface y=0. Therefore, no convection process occurs on the wall surface. For UWHF, although the velocity term has no contribution to the convection of q_y, the velocity gradient term takes the role for the convection of q_y.

In the fully developed flow region

The cross-averaged values of W_e-x, W_c-x, and their summation are presented in Fig. 3(c). As shown in Fig 3, CW_e-x is zero, but CW_c-x is not zero. The cross-averaged summation of W_e-x and W_c-x takes the shape of W_c-x. Along the main flow direction, CW_c-x is independent of x for UWT. The cross-averaged summation of W_e-x and W_c-x is zero for UWHF. The distribution of Cq_x is shown in Fig. 3(d). For UWHF, q_x is larger than that for UWT, and it is independent of x. For UWT, it is found that v and w are zero for the fully developed flow, and ¶q_x/¶x is not zero, as shown in Fig. 7(a). In Eq. (12), u¶q_x/¶x is just W_c-x and has no zero value for UWT. For UWHF, the velocity gradient terms along x are all zero for the fully developed flow, and hence, W_e-x has zero value. In Eq. (12), v and w are zero for the fully developed flow, and ¶q_x/¶x is zero, as indicated in Fig. 7(b); hence, W_c-x = u¶q_x/¶x has zero value for UWHF.

It is found that for UWHF, Eq. (6) becomes

(17)a{\nabla }^{\mathrm{2}}{q}_x=\mathrm{0}

From energy conservation law described by Eq. (1), it is found that the velocity has really carried the heat flux, q_x. From this point of view along the main flow direction, the convection of q_x exists, but because u(¶T/¶x) is constant along x, only no net convection of q_x exists. Where there is no net convection, it means that changing the temperature of fluid cannot be done.

In the fully developed flow region, the local information of W_e-x, W_c-x on the special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) are presented in Fig. 8. As shown in Figs. 8(a) and (b), for UWT and UWHF, W_e-x is zero, but W_c-x has a positive peak value for UWT. Because all of the velocity gradient terms are zero along the x direction, it is not difficult to find that W_e-x is zero from Eq. (12). The summation of W_e-x and W_c-x is presented in Fig. 8(c). This value is zero for UWHF but not for UWT. The transport of q_x is a no net diffusion process with no net convection process for UWHF. As shown in Fig. 8(d), the transport intensity per unit temperature difference of q_x is uniform along y and independent of x on the specified line for UWHF. The transport intensity per unit temperature difference of q_x is not uniform along y but is also independent of x on the specified line for UWT.

The information of transport of q_y on special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) is presented in Fig. 9. For UWHF, the velocity gradient term exists, as shown in Fig. 9(a), but the velocity term is zero, as shown in Fig. 9(b), which means that the convection of q_y is powered by the velocity gradient term. For UWT, both the velocity term and the velocity gradient term exist, having the same order and the same sign in the corresponding region. The velocity gradient term and the velocity term have the same effects on the convection of q_y for UWT. The summation of all terms is presented in Fig. 9(c) and is not zero for UWT and UWHF, which tells that on specified line, the transport of q_y is carried out by a convection process for UWT and UWHF, but for UWHF, the velocity term has no contribution to the convection of q_y. The q_y on the specified line for UWT is larger than that for UWHF, as shown in Fig. 9(d), which is also indicated in Fig. 6(c). Although at some region on the wall surface, the value of q_y is larger for UWT than for UWHF, and the averaged value is smaller for UWT than for UWHF. On the specified line, q_y is independent of x for UWT and UWHF.

Considering the symmetrical characteristics in square duct case, q_z on special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) has similar characteristics as q_y on special lines (intersection lines of the plane z = 0.5L_z and the planes x = 1.2L_y, x = 3.0L_y, and x = 5.7L_y) .

Conclusion

On the wall surface, the velocity is zero but convection occurs for UWHF in the developing region. The main promotion of the convection of the heat flux normal to the wall surface is the velocity gradient term. For UWT, the velocity gradient term is zero, and a diffusion process takes place on the wall surface. For UWHF, the largest term of the gradient of velocity components (the main flow velocity) on the wall surface takes a role in the convection of the heat flux normal to the wall surface, and this role exists in the fully developed region. Therefore, a strong convection process occurs for UWHF on the wall surface.

In the developing flow region, the velocity term and the velocity gradient term contribute to the transport of the heat flux in the main flow direction for UWHT and UWT cases, but the convection promotion is weak, and the diffusion process takes a very important role in transport of heat flux along the main flow.

In the fully developed flow region, for UWT, the promotion of the convection of the flux along the main flow direction is produced by the velocity term of the main flow, but for UWHF, there is no promotion of this convection, which tells that in the fully developed region, the heat flux along the main flow transports through no net diffusion and no net convection processes for UWHF but through convection process for UWT.

In the fully developed flow region, the velocity term and the velocity gradient term contribute to the convection of the heat flux normal to the wall surface for UWT. The velocity gradient term contributes only to the convection of the heat flux normal to the wall surface for UWHF.

References

[1] Shah R K, London A L. Laminar flow forced convection in ducts. In: Hartnett J P, Irvine T F. eds. Advances in Heat Transfer, New York: Academic Press, 1978, 78-384

[2] Kays W M, Crawford M E, Weigand B. Convection Heat and Mass Transfer. 4th ed. New York: Mc Graw Hill, 2005

[3] Schlichting H. Boundary Layer Theory. 6th ed. New York: McGraw-Hill, 1968

[4] Wang L B, Li X X, Lin Z M, Yang X, Wang X H. Additional description of laminar heat convection in tube with uniform wall temperature. Journal of Thermalphysics and Heat Transfer, 2009, 23: 200-205

[5] Li Z Y, Tao W Q. A new stability-guaranteed second-order difference scheme. Numerical Heat Transfer, Part B, 2002, 42: 349-365

[6] Patankar S V. Numerical Heat Transfer and Fluid Flow. Washington: Hemisphere, 1981

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