# Similarity solutions

(Difference between revisions)
 Revision as of 01:55, 30 March 2010 (view source) (→Similarity Solution for Flow over a Wedge)← Older edit Current revision as of 13:49, 26 July 2010 (view source) (43 intermediate revisions not shown) Line 1: Line 1: - The form of the velocity, temperature, and concentration profiles for flow over a flat plate are presented in Figure 4.4, and are based on qualitative measurements.  These presentations indicate the possibility that these profiles are geometrically similar to each other along the flow direction for dependent variables such as velocity, temperature, and concentration, if the coordinates are stretched properly.  For example, the velocity profiles are geometrically similar along the flow in the x-direction, differing only by a stretching factor (similarity factor) if the coordinates are properly sketched. + [[Image:Fig4.4.png|thumb|400 px|altBoundary layer concept over a flat plate |Boundary layer concept over a flat plate.]] - As noted before, not all boundary layer flow configurations have similar geometric profiles (similarity solution), but some do [[#References|(Burmeister, 1993)]], especially for more conventional geometries and boundary conditions.  The steady, two-dimensional, laminar, momentum boundary layer equation is a second order, nonlinear, partial differential equation.  The two-dimensional, steady, laminar flow, energy, and species boundary layer equations are linear, second order, partial differential equations for most constant properties, and/or for decoupled heat and mass transfer problems. + The form of the velocity, temperature, and concentration profiles for flow over a flat plate are presented in the figure to the right, and are based on qualitative measurements.  These presentations indicate the possibility that these profiles are geometrically similar to each other along the flow direction for dependent variables such as velocity, temperature, and concentration, if the coordinates are stretched properly.  For example, the velocity profiles are geometrically similar along the flow in the x-direction, differing only by a stretching factor (similarity factor) if the coordinates are properly sketchedFaghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO.. + + As noted before, not all boundary layer flow configurations have similar geometric profiles (similarity solution), but some do Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ., especially for more conventional geometries and boundary conditions.  The steady, two-dimensional, laminar, momentum boundary layer equation is a second order, nonlinear, partial differential equation.  The two-dimensional, steady, laminar flow, energy, and species boundary layer equations are linear, second order, partial differential equations for most constant properties, and/or for decoupled heat and mass transfer problems. If a similarity solution exists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact.  A similarity technique converts conservation partial differential equations into ordinary differential equations and therefore, makes the solution much simpler.  However, the analytical solutions may still require numerical integrations. If a similarity solution exists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact.  A similarity technique converts conservation partial differential equations into ordinary differential equations and therefore, makes the solution much simpler.  However, the analytical solutions may still require numerical integrations. Line 11: Line 13: ===Similarity Solution for Flow over a Flat Plate=== ===Similarity Solution for Flow over a Flat Plate=== - We first apply the similarity solution to the classic problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled.  Next, similarity solutions of eqs. (4.25) to (4.27) with appropriate boundary conditions corresponding to Fig. 4.4 (a) and (b) are presented below. The viscous dissipation term is neglected for simplicity even though it is possible to obtain a similarity solution with viscous dissipation for this simple configuration. + We first apply the similarity solution to the classic problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled.  Next, similarity solutions of control equations with appropriate boundary conditions of above figure are presented below. The viscous dissipation term is neglected for simplicity even though it is possible to obtain a similarity solution with viscous dissipation for this simple configuration. - Assuming geometrically similar velocity profiles are possible along the flow over a flat plate using the following non-dimensional variables (Kays et al., 2004, Bejan, 2004, Burmeister ,1983): + Assuming geometrically similar velocity profiles are possible along the flow over a flat plate using the following non-dimensional variables Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NYBejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- $u=f(\eta )$ + $\begin{matrix}{}\\\end{matrix}u=f(\eta)$
| (1) | (1) Line 29: Line 31: |} |} - where $\eta$ is a non-dimensional independent variable (function of both x and y) that fulfills the similarity requirement. + where $\eta$ is a non-dimensional independent variable (function of both ''x'' and ''y'') that fulfills the similarity requirement. - Mass and momentum boundary layer equations for flow over a flat plate [eqs. (4.25) and (4.26)] can be converted to a new coordinate system $\eta$, f, and g by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$. + Mass and momentum boundary layer equations for flow over a flat plate (refer to [[Boundary layer approximations]]) can be converted to a new coordinate system $\eta$, ''f'', and ''g'' by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 54: Line 56: |} |} - Substituting eqs. (4.46)-(4.48) into (4.25) and (4.26) yields: + Substituting eqs. (3)-(5) into mass and momentum boundary layer equations for flow over a flat plate yields: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- $\nu {f}''g^{2}=f{f}'y{g}'+v{f}'g$ + $\begin{matrix}{}\\\end{matrix}\nu {f}''g^{2}=f{f}'y{g}'+v{f}'g$
| (6) | (6) Line 71: Line 73: |} |} - Combining eqs. (4.49) and (4.50) to eliminate $v$ and separating variables gives: + Combining eqs. (6) and (7) to eliminate $v$ and separating variables gives: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 80: Line 82: |} |} - Considering that both $\eta$ and x are independent variables, the left side of eq. (4.51) is a function of $\eta$, and the right side is a function of x, then each side must be a constant. + Considering that both $\eta$ and x are independent variables, the left side of eq. (8) is a function of $\eta$, and the right side is a function of ''x'', then each side must be a constant. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 89: Line 91: |} |} - Integrating (4.52) gives: + Integrating eq.(9) gives: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 98: Line 100: |} |} - At the leading edge of the flat plate ($x=0$), the boundary condition is $u=u_{\infty }$ which can be satisfied only at $\eta \to \infty$. In other words, $\eta$ must be infinite when $x=0$. It follows from eq. (4.45) that $g(0)$ must be infinite and consequently, c2 in eq. (4.53) must be zero. Therefore + At the leading edge of the flat plate ($x=0$), the boundary condition is $u=u_{\infty }$ which can be satisfied only at $\eta \to \infty$. In other words, $\eta$ must be infinite when $x=0$. It follows from eq. (2) that $g(0)$ must be infinite and consequently, ''c''2 in eq. (10) must be zero. Therefore {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 115: Line 117: |} |} - Since c1 is constant, the axial velocity is of the following form: + Since ''c''1 is constant, the axial velocity is of the following form: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 124: Line 126: |} |} - From eq. (4.51) + From eq. (8) {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 133: Line 135: |} |} - Integrating eq. (4.57) gives: + Integrating eq. (14) gives: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 142: Line 144: |} |} - The following boundary conditions are used to evaluate constants c1 and c3: + The following boundary conditions are used to evaluate constants ''c''1 and ''c''3: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- $\text{at }y=0,\text{ }u=v=0\text{ or at }\eta =0,\text{ }f=0$ + $\begin{matrix}{}\\\end{matrix}\text{at }y=0,\text{ }u=v=0\text{ or at }\eta =0,\text{ }f=0$
| (16) | (16) |} |} - Using eqs. (4.58) and (4.59): + Using eqs. (15) and (16): {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- ${f}''=0\text{ at }\eta =0$ + $\begin{matrix}{}\\\end{matrix}{f}''=0\text{ at }\eta =0$
| (17) | (17) |} |} - then, using eqs. (4.58) and (4.60): + then, using eqs. (15) and (4.17): - + {| class="wikitable" border="0" - $c_{3}=0$ + |- + | width="100%" |
+ $\begin{matrix}{}&{}\\\end{matrix}c_{3}=0$ +
+ | + |} therefore, therefore, Line 184: Line 191: |} |} - Then, using eq. (4.62) + Then, using eq. (19) {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 193: Line 200: |} |} - Substituting (4.63) in (4.61) gives: + Substituting eq.(20) in eq.(18) gives: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 211: Line 218: |} |} - From (4.65) we conclude that c1U∞ is a constant, non-dimensional variable. Let + From eq.(22) we conclude that ''c1U'' is a constant, non-dimensional variable. Let $c_{1}U_{\infty }=1/2$, eq. (22) becomes - $c_{1}U_{\infty }=1/2$ + - , eq. (4.65) becomes + {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 231: Line 236: |} |} - Equation (4.66) is known as the Blasius equation and is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions. + Equation (23) is known as the Blasius equation and is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 240: Line 245: |} |} - Equation (4.66) can be numerically integrated using boundary conditions given by eq. (4.68).  Blasius clearly integrated eq. (4.66) by hand without using a computer (Blasius, 1908).  The numerical results using the shooting method are presented in Fig. 4.7, which shows that both $u\text{ and }v$as functions of the independent variable $\eta$. + [[Image:Fig4.7.png|thumb|400 px|alt=Dimensionless velocity distribution in a laminar boundary layer |Dimensionless velocity distribution in a laminar boundary layer.]] - A number of conclusions can be made from the numerical results presented in Fig. 1. + - [[Image:Fig4.7.png|thumb|400 px|alt=Dimensionless velocity distribution in a laminar boundary layer |Figure 1: Dimensionless velocity distribution in a laminar boundary layer.]] + Equation (23) can be numerically integrated using boundary conditions given by eq. (25). Blasius clearly integrated eq. (23) by hand without using a computer Blasius, H., 1908, Grengschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phy., Vol. 56 p.4, Also NACA TM 1256.  The numerical results using the shooting method are presented in the figure to the right, which shows that both $u\text{ and }v$as functions of the independent variable $\eta$. + A number of conclusions can be made from the numerical results presented in the figure to the right. - 1. The edge of the momentum boundary layer is approximately at $\eta$=5. - 2. The vertical velocity component $v$ is non-zero at the edge of the boundary layer at $\eta$=5, even though the free stream flow is parallel to the plate. + 1. The edge of the momentum boundary layer is approximately at $\eta$=5.
- 3. The momentum boundary layer thickness is approximately + 2. The vertical velocity component $v$ is non-zero at the edge of the boundary layer at $\eta$=5, even though the free stream flow is parallel to the plate.
+ + 3. The momentum boundary layer thickness is approximately {| class="wikitable" border="0" {| class="wikitable" border="0" Line 260: Line 266: where where - $\operatorname{Re}_{x}=\frac{U_{\infty }x}{\nu }$ + $\operatorname{Re}_{x}=\frac{U_{\infty }x}{\nu } - Using the definition of shear stress at the wall, τw, and the skin friction coefficient, [itex]c_{f}$, we obtain: + Using the definition of shear stress at the wall, ''τw'', and the skin friction coefficient, $c_{f}$, we obtain: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 271: Line 277: |} |} - Rearranging the above equation in terms of $\varsigma$ gives: + Rearranging the above equation in terms of $\varsigma$ gives: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 288: Line 294: |} |} - For the purposes of design and comparison with experimental measurements, it is more practical to obtain the average value of the skin friction, cf, which is usually determined by + For the purposes of design and comparison with experimental measurements, it is more practical to obtain the average value of the skin friction, ''cf'', which is usually determined by {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 297: Line 303: |} |} - where W is the width of the plate.  From eq. (4.72), + where ''W'' is the width of the plate.  From eq.(29), {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 317: Line 323: |} |} - The non-dimensional form of the energy equation (4.27) (neglecting viscous dissipation) and boundary conditions and using the variables $\eta$ and $\varsigma$ noted in eqs. (4.67) and (4.62) are as shown below. + The non-dimensional form of the energy equation (neglecting viscous dissipation) and boundary conditions and using the variables $\eta$ and $\varsigma$ noted in eqs. (24) and (19) are as shown below. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 329: Line 335: |- |- | width="100%" |
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- $\theta (0)=0$ + $\begin{matrix}{} & {}\\\end{matrix}\theta (0)=0$
| (34) | (34) Line 344: Line 350: with the notation that with the notation that ${\theta }'=\frac{d\theta }{d\eta }$ and ${\theta }''=\frac{d^{2}\theta }{d\eta ^{2}}$. ${\theta }'=\frac{d\theta }{d\eta }$ and ${\theta }''=\frac{d^{2}\theta }{d\eta ^{2}}$. - Equation (4.76) can be directly integrated: + Equation (33) can be directly integrated: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 369: Line 375: |} |} - Applying boundary conditions (4.77) and (4.78) gives + Applying boundary conditions (34) and (35) gives {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 387: Line 393: |} |} - [[Image:Fig4.8.png|thumb|400 px|alt=Dimensionless temperature distribution in a laminar boundary layer over a flat plate (Oosthuizen and Naylor, 1999; courtesy of Professors Oosthuizen and Naylor) |Figure 2: Dimensionless temperature distribution in a laminar boundary layer over a flat plate (Oosthuizen and Naylor, 1999; courtesy of Professors Oosthuizen and Naylor).]] + [[Image:Fig4.8.png|thumb|400 px|alt=Dimensionless temperature distribution in a laminar boundary layer over a flat plate |'''Dimensionless temperature distribution in a laminar boundary layer over a flat plate.''']] - The energy equation (4.76) is a second order, linear, ordinary differential equation.  The dimensionless temperature, $\theta$, is a function of $\eta$ and Pr.  Equation (4.83) can be integrated numerically for a given Pr.  The numerical results of $\theta$ as a function of $\eta$ for different Pr values are presented in Fig. 4.8. + The energy equation (33) is a second order, linear, ordinary differential equation.  The dimensionless temperature, $\theta$, is a function of $\eta$ and Pr.  Equation (40) can be integrated numerically for a given ''Pr''.  The numerical results of $\theta$ as a function of $\eta$ for different Pr values are presented in the figure to the right. - The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layer equations, as presented in Figs. 4.7 and 4.8. + The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layer equations, as presented in the above two figures.
- 1. Temperature profile is identical to velocity for Pr = 1. + 1. Temperature profile is identical to velocity for Pr = 1.
- 2. Temperature profile has strong dependence on the Prandtl number. + 2. Temperature profile has strong dependence on the Prandtl number.
- 3. For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, $\delta _{T}>\delta$. + 3. For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, $\delta _{T}>\delta$.
- 4. For Pr>1 the thermal boundary layer thickness is less than the momentum boundary layer thickness, $\delta _{T}<\delta$. + 4. For Pr>1 the thermal boundary layer thickness is less than the momentum boundary layer thickness, $\delta _{T}<\delta$.
- The local heat transfer coefficient can be easily obtained from the numerical results of eq. (4.83) for a given Prandtl number by using Fourier’s law of heat conduction. + The local heat transfer coefficient can be easily obtained from the numerical results of eq. (40) for a given Prandtl number by using Fourier’s law of heat conduction. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 418: Line 424: |} |} - or by using eq. (4.83): + or by using eq. (4.40): {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 431: Line 437: |- |- | width="100%" |
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- $Nu_{x}=0.332\Pr ^{1/3}\operatorname{Re}_{x}^{1/2}$ + $Nu_{x}=0.332\operatorname{Pr}^{1/3}\operatorname{Re}_{x}^{1/2}$
| (44) | (44) |} |} - The ratio of thermal boundary layer thickness, δT, to momentum boundary layer thickness, δ, for flow over a flat plate can also be approximated over this range of Prandtl numbers according to the relationship below: + The ratio of thermal boundary layer thickness, ''δT'', to momentum boundary layer thickness, ''δ'', for flow over a flat plate can also be approximated over this range of Prandtl numbers according to the relationship below: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- $\frac{\delta _{T}}{\delta }=\Pr ^{-1/3}\text{, }0.5\le \Pr <10$ + $\frac{\delta _{T}}{\delta }=\operatorname{Pr}^{-1/3}\text{, }0.5\le \Pr <10$
| (45) | (45) Line 446: Line 452: For both design purposes and comparison with experimental measurement it is more convenient to calculate the average or mean heat transfer coefficient, For both design purposes and comparison with experimental measurement it is more convenient to calculate the average or mean heat transfer coefficient, - $\bar{h}$ + $\bar{h}$. - . + {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 460: Line 465: |- |- | width="100%" |
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- $\overline{h}=\frac{1}{x}\int_{0}^{x}{h_{x}dx=0.664\frac{k}{x}\operatorname{Re}_{x}^{1/2}\Pr ^{1/3}}$ + $\overline{h}=\frac{1}{x}\int_{0}^{x}{h_{x}dx=0.664\frac{k}{x}\operatorname{Re}_{x}^{1/2}\operatorname{Pr}^{1/3}}$
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- $\overline{Nu}=\frac{\overline{h}x}{k}=2Nu_{x}=0.664\operatorname{Re}_{x}^{1/2}\Pr ^{1/3}$ + $\overline{Nu}=\frac{\overline{h}x}{k}=2Nu_{x}=0.664\operatorname{Re}_{x}^{1/2}\operatorname{Pr}^{1/3}$
| (48) | (48) Line 477: Line 482: ===Similarity Solution for Flow over a Wedge=== ===Similarity Solution for Flow over a Wedge=== Similarity solutions also exist for some other conventional geometries with simple boundary conditions.  Some of these cases are discussed below. Similarity solutions also exist for some other conventional geometries with simple boundary conditions.  Some of these cases are discussed below. - For flow over a wedge, as shown in Fig. 4.10, the free stream velocity varies according to potential (inviscid) flow theory: + For flow over a wedge, as shown in figure below, the free stream velocity varies according to potential (inviscid) flow theory: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- $U=cx^{m}$ + $\begin{matrix}{}\\\end{matrix}U=cx^{m}$
| (49) | (49) |} |} - where U is the free stream velocity at the outer wedge surface and m is related to the wedge angle $\beta$ by: + where ''U'' is the free stream velocity at the outer wedge surface and m is related to the wedge angle $\beta$ by: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 499: Line 504:     • $\beta =\pi$ corresponds to flow perpendicular to the walls.
• $\beta =\pi$ corresponds to flow perpendicular to the walls.
- U∞ is the oncoming free stream velocity, which is constant, as shown in Fig. 4.10 + ''U'' is the oncoming free stream velocity, which is constant, as shown in figure to the right. - [[Image:Fig4.10.png|thumb|400 px|alt=Wedge flow configuration |Figure 3: Wedge flow configuration.]] + [[Image:Fig4.10.png|thumb|400 px|alt=Wedge flow configuration |Wedge flow configuration.]] - Equation (4.97) can be also applied near the leading edge of a blunt object.  For example, the free stream velocity on the surface of a cylinder and a sphere near a stagnation point can be predicted by potential flow theory according to the equations below (Schlichting and Gersteu, 2000): + Equation (49) can be also applied near the leading edge of a blunt object.  For example, the free stream velocity on the surface of a cylinder and a sphere near a stagnation point can be predicted by potential flow theory according to the equations belowSchlichting, H. and Gersteu, K., 2000, Boundary layer theory, 8th enlarged and revised ed., Springer-Verlag New York: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
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- (cylinder)$U=2U_{\infty }\sin \left( \frac{x}{R} \right)\approx 2U_{\infty }\left( \frac{x}{R} \right)$ + (cylinder) $U=2U_{\infty }\sin \left( \frac{x}{R} \right)\approx 2U_{\infty }\left( \frac{x}{R} \right)$
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- (sphere)$U=\frac{3}{2}U_{\infty }\sin \left( \frac{x}{R} \right)\approx \frac{3}{2}U_{\infty }\frac{x}{R}$ + (sphere) $U=\frac{3}{2}U_{\infty }\sin \left( \frac{x}{R} \right)\approx \frac{3}{2}U_{\infty }\frac{x}{R}$
| (52) | (52) |} |} - where R is the radius of the cylinder or sphere, U∞ is the oncoming velocity and U is the free stream velocity just outside the boundary layer.  The pressure gradient term is related to free stream velocity by potential flow theory.  This can be done using the inviscid momentum equation ($\mu =0$) + where R is the radius of the cylinder or sphere, ''U'' is the oncoming velocity and ''U'' is the free stream velocity just outside the boundary layer.  The pressure gradient term is related to free stream velocity by potential flow theory.  This can be done using the inviscid momentum equation ($\mu =0$) {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 542: Line 547: |} |} - Momentum (x-direction): + Momentum (''x''-direction): {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 560: Line 565: |} |} - The above equations can be easily obtained by beginning with the general equations from Chapter 2 and making appropriate boundary layer approximations using scaling and order of magnitude analysis. + The above equations can be easily obtained by beginning with the general equations and making appropriate boundary layer approximations using scaling and order of magnitude analysis. - Using eq. (4.101) for the pressure and comparable dimensionless similarity parameters to flow over a flat plate, we get the following ordinary differential equation for momentum and energy for constant wall temperature with no blowing or suction at the wall. + Using eq. (53) for the pressure and comparable dimensionless similarity parameters to flow over a flat plate, we get the following ordinary differential equation for momentum and energy for constant wall temperature with no blowing or suction at the wall. {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 578: Line 583: |} |} - The boundary conditions for velocity and temperature are the same as in the case of flow over a flat plate.  Assuming that heating starts at the leading edge, the ordinary differential equations (4.105) and (4.106) for dimensionless velocity and temperature can be solved using the given boundary condition.  The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations: + The boundary conditions for velocity and temperature are the same as in the case of flow over a flat plate.  Assuming that heating starts at the leading edge, the ordinary differential equations (57) and (58) for dimensionless velocity and temperature can be solved using the given boundary condition.  The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 596: Line 601: where the constant A depends on m and Pr. where the constant A depends on m and Pr. - Solutions of eqs. (4.105) and (4.106) for m > 0 are unique.  For m < 0 two groups of solutions exist for a limited range of m values corresponding to negative $\beta$.  These two groups correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. Falkner and Skan (1931) originally developed the similarity solution for flow over a wedge with U∞ = cxm and an impermeable wall. The numerical results for the friction coefficient for selected values of m and β are presented in Table 4.1. m = β = 0 corresponds to the case of flow parallel to a flat plate with constant free stream values (Blasius Solution). + Solutions of eqs. (57) and (58) for ''m'' > 0 are unique.  For ''m'' < 0 two groups of solutions exist for a limited range of ''m'' values corresponding to negative $\beta$.  These two groups correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. Falkner and Skan Falkner, V.M., and Skan, S.W., 1931, Some Appropriate Solutions of the Boundary Layer Equations, Phil. May 12, pp. 865-896 originally developed the similarity solution for flow over a wedge with ''U'' = ''cxm'' and an impermeable wall. The numerical results for the friction coefficient for selected values of ''m'' and ''β'' are presented in the table below. ''m'' = ''β'' = 0 corresponds to the case of flow parallel to a flat plate with constant free stream values (Blasius Solution). - Eckert (1942) was the first to obtain the numerical results for eqs. (4.105) and (4.106) for constant free stream and wall temperature as presented in Table 4.2. + - In similarity solutions for both flow over a flat plate and over a wedge, it is assumed that there is no blowing or suction at the wall.  It can be shown (see Problem 4.7) that similarity solutions with blowing or suction exist only if the vertical velocity at the wall changes along the flow according to the following equation: + - {| class="wikitable" border="0" + - |- + - | width="100%" |
+ - $v_{w}\propto x^{(m-1)/2}$ + -
+ - | (61) + - |} + - + - where vw is the blowing or suction velocity.  Clearly this requirement restricts the use of the similarity solution to special cases that satisfy eq. (4.109). +
- '''Table 1''' The local friction coefficient for laminar boundary layer flow over a wedge, with U = cxm and an impermeable wall + '''Table''' The local friction coefficient for laminar boundary layer flow over a wedge, with U = cxm and an impermeable wall {| class="wikitable" border="1" {| class="wikitable" border="1" | align="center" style="background:#f0f0f0;" width="30%" |'''''β''''' | align="center" style="background:#f0f0f0;" width="30%" |'''''β''''' Line 623: Line 617:
- Table 2. Local values of NuxRex-1/2 for laminar boundary layer flow over a wedge with constant wall and free stream temperature and an impermeable wall with U∞ = cxm + Eckert Eckert, E.R.G., 1942, Die Berchnung des Wärmeüberganges in der Laminaren Grenzschicht um strömter körper,  VDI – Forschungsheft, vol. 416 pp. 1-24 was the first to obtain the numerical results for eqs. (57) and (58) for constant free stream and wall temperature as presented in the following table. + +
+
+ '''Table 2''' Local values of NuxRex-1/2 for laminar boundary layer flow over a wedge with constant wall and free stream temperature and an impermeable wall with U = cxm {| border="1" cellpadding="5" cellspacing="0" {| border="1" cellpadding="5" cellspacing="0" |- |- - | rowspan="2" style="text-align:center;"| A + | rowspan="2" align="center" style="background:#f0f0f0;"| A - | rowspan="2" style="text-align:center;"| B + | rowspan="2" align="center" style="background:#f0f0f0;"| B - | colspan="5" style="text-align:center;"| Pr + | colspan="5" align="center" style="background:#f0f0f0;"| Pr |- |- - | style="text-align:center;"| 1 + | align="center" style="background:#f0f0f0;"| 1 - | style="text-align:center;"| 2 + | align="center" style="background:#f0f0f0;"| 2 - | style="text-align:center;"| 3 + | align="center" style="background:#f0f0f0;"| 3 - | style="text-align:center;"| 4 + | align="center" style="background:#f0f0f0;"| 4 - | style="text-align:center;"| 5 + | align="center" style="background:#f0f0f0;"| 5 |- |- | -0.512
0
π/5
π/2
π
8π/5
| -0.512
0
π/5
π/2
π
8π/5
Line 644: Line 642: |0.570
0.730
0.851
1.013
1.344
2.236
|0.570
0.730
0.851
1.013
1.344
2.236
|} |} +
+ + In similarity solutions for both flow over a flat plate and over a wedge, it is assumed that there is no blowing or suction at the wall.  It can be shown that similarity solutions with blowing or suction exist only if the vertical velocity at the wall changes along the flow according to the following equation: + {| class="wikitable" border="0" + |- + | width="100%" |
+ $v_{w}\propto x^{(m-1)/2}$ +
+ | (61) + |} + + where vw is the blowing or suction velocity.  Clearly this requirement restricts the use of the similarity solution to special cases that satisfy eq. (61). == Coupled Mass, Momentum, and Heat Transfer Problems == == Coupled Mass, Momentum, and Heat Transfer Problems == + [[Image:Fig4.11.png|thumb|400 px|alt=Phase diagram for solid-liquid and solid-vapor phase change | Phase diagram for solid-liquid and solid-vapor phase change.]] Coupled transport phenomena occur frequently in problems associated with evaporation, sublimation, absorption, or combustion. Sublimation over a flat plate can find its application in an analogy between heat and mass transfer and will be used here as an example to show the methodology used in solving these coupled problems. Coupled transport phenomena occur frequently in problems associated with evaporation, sublimation, absorption, or combustion. Sublimation over a flat plate can find its application in an analogy between heat and mass transfer and will be used here as an example to show the methodology used in solving these coupled problems. - [[Image:Fig4.11.png|thumb|400 px|alt=Phase diagram for solid-liquid and solid-vapor phase change |Figure 1: Phase diagram for solid-liquid and solid-vapor phase change.]] + - When the pressure and temperature of ice are above the triple point and the ice is then heated, melting occurs. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through condensation. The phase-change processes related to  solids can be illustrated by a phase diagram in Fig. 4.11. Sublimation and  deposition will be the subjects of this sub-section. + When the pressure and temperature of ice are above the triple point and the ice is then heated, melting occurs. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through condensation. The phase-change processes related to  solids can be illustrated by a phase diagram in to the right. Sublimation and  deposition will be the subjects of this sub-section. - When a subcooled solid is exposed to its superheated vapor, as shown in Fig. 4.12(a), the vapor phase temperature is above the temperature of the solid-vapor interface and the temperature of the solid is below the interfacial temperature. The boundary condition at the solid-vapor interface is + When a subcooled solid is exposed to its superheated vapor, as shown in second figure (a), the vapor phase temperature is above the temperature of the solid-vapor interface and the temperature of the solid is below the interfacial temperature. The boundary condition at the solid-vapor interface is - $k_{s}\frac{\partial T_{s}}{\partial x}-h_{\delta }(T_{\infty }-T_{\delta })=\rho _{s}h_{sv}\frac{d\delta }{dt}$ (4.110) + {| class="wikitable" border="0" - where hδ is the convective heat transfer coefficient at the solid-vapor interface, hsv is the latent heat of sublimation, and δ is the thickness of the sublimated or deposited material. The interfacial velocity d/dt in eq. (4.110) can be either positive or negative, depending on the direction of the overall heat flux at the interface.  While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in Fig. 4.12(a), the solid-vapor interface is usually smooth and stable. + |- - [[Image:Fig4.12.png|thumb|400 px|alt=Temperature distribution in sublimation and deposition |Figure 1: Temperature distribution in sublimation and deposition.]] + | width="100%" |
- In another possible scenario, as shown in Fig. 4.12 (b), the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (4.110). Depending on the degrees of superheating in the solid phase and supercooling in the vapor phase (the relative magnitude of the first and second terms in eq. (4.110)) both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition,  on the other hand, the interface is dendritic and unstable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases. + $k_{s}\frac{\partial T_{s}}{\partial x}-h_{\delta }(T_{\infty }-T_{\delta })=\rho _{s}h_{sv}\frac{d\delta }{dt}$ - Naphthalene sublimation is also a case whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer (Eckert and Goldstein, 1976). The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen. +
- Figure 4.13 shows the physical model of a sublimation problem, where a flat plate is coated with a layer of sublimable material and is subject to constant heat flux underneath (Zhang et al., 1996). A gas with the ambient temperature  and mass fraction of sublimable material  flows over the flat plate at a velocity of U∞.  The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and the other is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow. + | (62) - [[Image:Fig4.13.png|thumb|400 px|alt=PSublimation on a flat plate with constant heat flux |Figure 1: Sublimation on a flat plate with constant heat flux.]] + |} + where ''hδ'' is the convective heat transfer coefficient at the solid-vapor interface, ''hsv'' is the latent heat of sublimation, and ''δ'' is the thickness of the sublimated or deposited material. The interfacial velocity ''dδ/dt'' in eq. (62) can be either positive or negative, depending on the direction of the overall heat flux at the interface.  While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in second figure(a), the solid-vapor interface is usually smooth and stable. + [[Image:Fig4.12.png|thumb|400 px|alt=Temperature distribution in sublimation and deposition |Temperature distribution in sublimation and deposition.]] + In another possible scenario, as shown in (b) of the figure to the right, the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (62). Depending on the degrees of superheating in the solid phase and supercooling in the vapor phase (the relative magnitude of the first and second terms in eq. (62) both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition,  on the other hand, the interface is dendritic and unstable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases. + Naphthalene sublimation is also a case whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer Eckert, E.R.G., and Goldstein, R.J., 1976, Measurement in Heat Transfer, McGraw-Hill, New York, NY. The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen. + [[Image:Fig4.13.png|thumb|400 px|alt=PSublimation on a flat plate with constant heat flux |Sublimation on a flat plate with constant heat flux.]] + Figure to the right shows the physical model of a sublimation problem, where a flat plate is coated with a layer of sublimable material and is subject to constant heat flux underneath Zhang, Y., Chen, Z.Q., and Chen, M., 1996, “Local Non-Similarity Solutions of Coupled Heat and Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of Thermal Science, Vol. 5, pp. 112-116. A gas with the ambient temperature  and mass fraction of sublimable material  flows over the flat plate at a velocity of ''U''.  The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and the other is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow. + The following assumptions are made in order to solve the problem: The following assumptions are made in order to solve the problem: - 1. The flat plate is very thin, so the thermal resistance of the flat plate can be neglected. + 1. The flat plate is very thin, so the thermal resistance of the flat plate can be neglected. - 2. The gas is incompressible, with no internal heat source in the gas. + 2. The gas is incompressible, with no internal heat source in the gas. - 3. The sublimation problem is two-dimensional steady state. + 3. The sublimation problem is two-dimensional steady state. The governing equations for mass, momentum, energy, and species of the problem are The governing equations for mass, momentum, energy, and species of the problem are - + {| class="wikitable" border="0" + |- + | width="100%" |
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ - (4.111) +
- + | (63) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}$ $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}$ - (4.112) +
- + | (64) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ - (4.113) +
- + | (65) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{\partial ^{2}\omega }{\partial y^{2}}$ $u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{\partial ^{2}\omega }{\partial y^{2}}$ - (4.114) +
- A no slip condition at the surface of the flat plat requires that + | (66) - $u=0$ + |} - at + - $y=0$ + A no slip condition at the surface of the flat plat requires that $u=0$ at $y=0$. - . + For a binary mixture that contains the vapor, sublimable substance, and gas, the molar flux of the sublimable substance at the surface of the flat plate is - For a binary mixture that contains the vapor, sublimable substance, and gas, the molar flux of the sublimable substance at the surface of the flat plate is [see eq. (1.109)] + {| class="wikitable" border="0" - + |- + | width="100%" |
${\dot{m}}''=-\frac{\rho D}{1-\omega }\frac{\partial \omega }{\partial y}\begin{matrix} [itex]{\dot{m}}''=-\frac{\rho D}{1-\omega }\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ , & y=0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.115) +
+ | (67) + |} + Since the mass fraction of the sublimable substance in the mixture is very low, i.e., ω$\ll$1, the mass flux at the wall can be simplified to Since the mass fraction of the sublimable substance in the mixture is very low, i.e., ω$\ll$1, the mass flux at the wall can be simplified to - + {| class="wikitable" border="0" + |- + | width="100%" |
${\dot{m}}''=-\rho D\frac{\partial \omega }{\partial y}\begin{matrix} [itex]{\dot{m}}''=-\rho D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ , & y=0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.116) +
- Sublimation at the surface causes a normal blowing velocity, + | (68) - $v_{w}=\dot{{m}''}/\rho$ + |} - . The normal velocity at the surface of the flat plate is therefore + - + Sublimation at the surface causes a normal blowing velocity,$v_{w}=\dot{{m}''}/\rho$. The normal velocity at the surface of the flat plate is therefore + {| class="wikitable" border="0" + |- + | width="100%" |
$v=v_{w}=-D\frac{\partial \omega }{\partial y}\begin{matrix} [itex]v=v_{w}=-D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ , & y=0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.117) +
+ | (69) + |} + The energy balance at the surface of the flat plate is The energy balance at the surface of the flat plate is - + {| class="wikitable" border="0" + |- + | width="100%" |
$-k\frac{\partial T}{\partial y}-\rho h_{sv}D\frac{\partial \omega }{\partial y}={q}''_{w}\begin{matrix} [itex]-k\frac{\partial T}{\partial y}-\rho h_{sv}D\frac{\partial \omega }{\partial y}={q}''_{w}\begin{matrix} , & y=0 \\ , & y=0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.118) +
- Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship (Kurosaki, 1974): + | (70) - + |} + + Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship Kurosaki, Y., 1974, “Coupled Heat and Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of the Japan Society of Mechanical Engineers, part B, Vol. 40, pp. 1066-1072: + {| class="wikitable" border="0" + |- + | width="100%" |
$\omega =aT+b\begin{matrix} [itex]\omega =aT+b\begin{matrix} , & y=0 \\ , & y=0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.119) +
+ | (71) + |} + where a and b are constants that depend on the sublimable material and its temperature. where a and b are constants that depend on the sublimable material and its temperature. - As + As $y\to \infty ,$ the boundary conditions are - $y\to \infty ,$ + {| class="wikitable" border="0" - the boundary conditions are + |- - + | width="100%" |
- $u\to U_{\infty }$ + $u\to U_{\infty }$, $T\to T_{\infty }$,$\omega \to \omega _{\infty }$ - , +
- $T\to T_{\infty }$ + | (72) - , + |} - $\omega \to \omega _{\infty }$ + - (4.120) + Introducing the stream function $\psi$, - Introducing the stream function $\psi$, + {| class="wikitable" border="0" - + |- + | width="100%" |
$u=\frac{\partial \psi }{\partial y}\begin{matrix} [itex]u=\frac{\partial \psi }{\partial y}\begin{matrix} {} & v=-\frac{\partial \psi }{\partial x} \\ {} & v=-\frac{\partial \psi }{\partial x} \\ \end{matrix}$ \end{matrix}[/itex] - (4.121) +
- the continuity equation (4.111) is automatically satisfied, and the momentum equation in terms of the stream function becomes + | (73) - + |} + + the continuity equation (63) is automatically satisfied, and the momentum equation in terms of the stream function becomes + {| class="wikitable" border="0" + |- + | width="100%" |
$\frac{\partial \psi }{\partial y}\frac{\partial ^{2}\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{\partial ^{2}\psi }{\partial y^{2}}=\nu \frac{\partial ^{3}\psi }{\partial y^{3}}$ $\frac{\partial \psi }{\partial y}\frac{\partial ^{2}\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{\partial ^{2}\psi }{\partial y^{2}}=\nu \frac{\partial ^{3}\psi }{\partial y^{3}}$ - (4.122) +
- Similarity solutions for eq. (4.122) do not exist unless the injection velocity vw is proportional to x-1/2, and the incoming mass fraction of the sublimable substance, ω∞, is equal to the saturation mass fraction corresponding to the incoming temperature, T∞ (Kurosaki, 1974; Zhang et al., 1996). The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. (1996) will be presented here. + | (74) + |} + + Similarity solutions for eq. (74) do not exist unless the injection velocity vw is proportional to ''x''-1/2, and the incoming mass fraction of the sublimable substance, ''ω'', is equal to the saturation mass fraction corresponding to the incoming temperature, ''T'' . The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. will be presented here. + Defining the following similarity variables: Defining the following similarity variables: - + {| class="wikitable" border="0" + |- + | width="100%" |
\begin{align} [itex]\begin{align} & \xi =\frac{x}{L},\text{ }\eta =y\sqrt{\frac{U_{\infty }}{2\nu L\xi }}\begin{matrix} & \xi =\frac{x}{L},\text{ }\eta =y\sqrt{\frac{U_{\infty }}{2\nu L\xi }}\begin{matrix} Line 745: Line 814: \end{matrix} \\ \end{matrix} \\ \end{align} \end{align}[/itex] - (4.123) +
- eqs. (4.122) and (4.113) – (4.114) become + | (75) - + |} + + eqs. (74) and (65) – (66) become + {| class="wikitable" border="0" + |- + | width="100%" |
${f}'''+f{f}''=2\xi \left( {f}'{F}'-{f}''F \right)$ ${f}'''+f{f}''=2\xi \left( {f}'{F}'-{f}''F \right)$ - (4.124) +
- ${\theta }''+\Pr (f{\theta }'-{f}'\theta )=2\Pr \xi \left( {f}'\Theta -{\theta }'F \right)$ (4.125) + | (76) - ${\phi }''+\operatorname{Sc}(f{\phi }'-{f}'\phi )=2\operatorname{Sc}\xi \left( {f}'\Phi -{\phi }'F \right)$ (4.126) + |} - where prime ' represents partial derivative with respect to η, and all upper case variables represent partial derivatives of primary similarity variables with respect to ξ. + {| class="wikitable" border="0" - + |- + | width="100%" |
+ ${\theta }''+\Pr (f{\theta }'-{f}'\theta )=2\Pr \xi \left( {f}'\Theta -{\theta }'F \right)$ +
+ | (77) + |} + {| class="wikitable" border="0" + |- + | width="100%" |
+ ${\phi }''+\operatorname{Sc}(f{\phi }'-{f}'\phi )=2\operatorname{Sc}\xi \left( {f}'\Phi -{\phi }'F \right)$ +
+ | (78) + |} + where prime ' represents partial derivative with respect to ''η'', and all upper case variables represent partial derivatives of primary similarity variables with respect to ''ξ''. + {| class="wikitable" border="0" + |- + | width="100%" |
$F=\frac{\partial f}{\partial \xi }\begin{matrix} [itex]F=\frac{\partial f}{\partial \xi }\begin{matrix} , & \Theta =\frac{\partial \theta }{\partial \xi }\begin{matrix} , & \Theta =\frac{\partial \theta }{\partial \xi }\begin{matrix} Line 759: Line 849: \end{matrix} \\ \end{matrix} \\ \end{matrix}$ \end{matrix}[/itex] - (4.127) +
- It can be seen from eqs. (4.124) – (4.126) that the similarity solution exists only if F = Θ = Φ = 0. In order to use eqs. (4.124) – (4.126) to obtain a solution for the sublimation problem, supplemental equations for F, Θ, and Φ must be obtained. Taking partial derivatives of eqs. (4.124) – (4.126) with respect to $\xi$ and neglecting the higher order term, one obtains + | (79) - + |} + + It can be seen from eqs. (76) – (78) that the similarity solution exists only if ''F'' = Θ = Φ = 0. In order to use eqs. (76) – (78) to obtain a solution for the sublimation problem, supplemental equations for ''F'', Θ, and Φ must be obtained. Taking partial derivatives of eqs. (76) – (78) with respect to $\xi$ and neglecting the higher order term, one obtains + {| class="wikitable" border="0" + |- + | width="100%" |
${F}'''+F{f}''+{F}''f=2\left( {f}'{F}'-{f}''F \right)$ ${F}'''+F{f}''+{F}''f=2\left( {f}'{F}'-{f}''F \right)$ - (4.128) +
- ${\Theta }''+\Pr (F{\theta }'+f{\Theta }'-{F}'\theta -{f}'\Theta )=2\Pr \left( {f}'\Theta -{\theta }'F \right)$ (4.129) + | (80) - ${\Phi }''+\operatorname{Sc}(F{\phi }'+f{\Phi }'-{F}'\phi -{f}'\Phi )=2\operatorname{Sc}\left( {f}'\Phi -{\phi }'F \right)$ (4.130) + |} - The boundary conditions of eqs. (4.124) – (4.126) and eqs. (4.128) – (4.130) are + {| class="wikitable" border="0" - + |- + | width="100%" |
+ ${\Theta }''+\Pr (F{\theta }'+f{\Theta }'-{F}'\theta -{f}'\Theta )=2\Pr \left( {f}'\Theta -{\theta }'F \right)$ +
+ | (81) + |} + {| class="wikitable" border="0" + |- + | width="100%" |
+ ${\Phi }''+\operatorname{Sc}(F{\phi }'+f{\Phi }'-{F}'\phi -{f}'\Phi )=2\operatorname{Sc}\left( {f}'\Phi -{\phi }'F \right)$ +
+ | (82) + |} + The boundary conditions of eqs. (76) – (78) and eqs. (79) – (81) are + {| class="wikitable" border="0" + |- + | width="100%" |
${f}'(\xi ,0)=0\begin{matrix} [itex]{f}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.131) +
- + | (83) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$f(\xi ,0)=-\frac{2}{3}B\left[ \xi ^{1/2}{\phi }'(\xi ,0)-\xi ^{3/2}{\Phi }'(\xi ,0) \right]\begin{matrix} [itex]f(\xi ,0)=-\frac{2}{3}B\left[ \xi ^{1/2}{\phi }'(\xi ,0)-\xi ^{3/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.132) +
- + | (84) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
${f}'(\xi ,\infty )=1\begin{matrix} [itex]{f}'(\xi ,\infty )=1\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.133) +
- + | (85) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
${F}'(\xi ,0)=0\begin{matrix} [itex]{F}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.134) +
- + | (86) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$F(\xi ,0)=-\frac{1}{3}B\left[ \frac{1}{2}\xi ^{-1/2}{\phi }'(\xi ,0)-\xi ^{1/2}{\Phi }'(\xi ,0) \right]\begin{matrix} [itex]F(\xi ,0)=-\frac{1}{3}B\left[ \frac{1}{2}\xi ^{-1/2}{\phi }'(\xi ,0)-\xi ^{1/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.135) +
- + | (87) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
${F}'(\xi ,\infty )=0\begin{matrix} [itex]{F}'(\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.136) +
- + | (88) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
${\theta }'(\xi ,0)+{\phi }'(\xi ,0)=-1\begin{matrix} [itex]{\theta }'(\xi ,0)+{\phi }'(\xi ,0)=-1\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.137) +
- + | (89) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\theta (\xi ,\infty )=0\begin{matrix} [itex]\theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.138) +
- + | (90) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
${\Theta }'(\xi ,0)+{\Phi }'(\xi ,0)=0\begin{matrix} [itex]{\Theta }'(\xi ,0)+{\Phi }'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.139) +
- + | (91) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\Theta (\xi ,\infty )=0\begin{matrix} [itex]\Theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.140) +
- + | (92) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\theta (\xi ,0)+\phi _{s}\xi ^{-1/2}\begin{matrix} [itex]\phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\theta (\xi ,0)+\phi _{s}\xi ^{-1/2}\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.141) +
- + | (93) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\phi (\xi ,\infty )=0\begin{matrix} [itex]\phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.142) +
- + | (94) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\Phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\Theta (\xi ,0)-\frac{\phi _{s}}{2\xi ^{3/2}}\begin{matrix} [itex]\Phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\Theta (\xi ,0)-\frac{\phi _{s}}{2\xi ^{3/2}}\begin{matrix} , & \eta =0 \\ , & \eta =0 \\ \end{matrix}$ \end{matrix}[/itex] - (4.143) +
- + | (95) + |} + + {| class="wikitable" border="0" + |- + | width="100%" |
$\Phi (\xi ,\infty )=0\begin{matrix} [itex]\Phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ , & \eta =\infty \\ \end{matrix}$ \end{matrix}[/itex] - (4.144) +
+ | (96) + |} + where where - + {| class="wikitable" border="0" + |- + | width="100%" |
$B=\frac{{q}''_{w}}{\rho h_{sv}\nu }\sqrt{\frac{2\nu L}{U_{\infty }}}$ $B=\frac{{q}''_{w}}{\rho h_{sv}\nu }\sqrt{\frac{2\nu L}{U_{\infty }}}$ - (4.145) +
+ | (97) + |} + reflects the effect of injection velocity at the surface due to sublimation, and reflects the effect of injection velocity at the surface due to sublimation, and - + {| class="wikitable" border="0" + |- + | width="100%" |
$\phi _{s}=\frac{\rho h_{sv}D(\omega _{sat,\infty }-\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L/U_{\infty }}}$ $\phi _{s}=\frac{\rho h_{sv}D(\omega _{sat,\infty }-\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L/U_{\infty }}}$ - (4.146) +
- represents the effect of the mass fraction of the sublimable substance in the incoming flow. + | (98) - $\omega _{sat,\infty }$ + |} - is saturation mass fraction corresponding to the incoming temperature: + - + represents the effect of the mass fraction of the sublimable substance in the incoming flow. $\omega _{sat,\infty }$ is saturation mass fraction corresponding to the incoming temperature: + {| class="wikitable" border="0" + |- + | width="100%" |
$\omega _{sat,\infty }=aT_{\infty }+b$ $\omega _{sat,\infty }=aT_{\infty }+b$ - (4.147) +
+ | (99) + |} - [[Image:Fig4.14.png|thumb|400 px|alt=Temperature and mass fraction distributions (Zhang et al. 1996). |Figure 1: Temperature and mass fraction distributions (Zhang et al. 1996).]] + The set of ordinary differential equations (76) – (78) and (79) – (81) with boundary conditions specified by eqs. (83) – (96) are boundary value problems that can be solved using a shooting method . - The set of ordinary differential equations (4.124) – (4.126) and (4.128) – (4.130) with boundary conditions specified by eqs. (4.131) – (4.144) are boundary value problems that can be solved using a shooting method (Zhang et al., 1996). Figure 4.14 shows typical dimensionless temperature and mass fraction profiles obtained by numerical solution. It can be seen that the dimensionless temperature and mass fraction at different $\xi are also different, which is further evidence that a similarity solution does not exist. + Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is - + {| class="wikitable" border="0" + |- + | width="100%" | [itex]Nu_{x}=\frac{h_{w}x}{k}=\frac{[{q}''_{w}/(T_{w}-T_{\infty })]x}{k}=\frac{\operatorname{Re}_{x}^{1/2}}{\sqrt{2}\theta (\xi ,0)}$ $Nu_{x}=\frac{h_{w}x}{k}=\frac{[{q}''_{w}/(T_{w}-T_{\infty })]x}{k}=\frac{\operatorname{Re}_{x}^{1/2}}{\sqrt{2}\theta (\xi ,0)}$ - (4.148) +
+ | (100) + |} + and the Nusselt number based on convective heat transfer is and the Nusselt number based on convective heat transfer is - + {| class="wikitable" border="0" + |- + | width="100%" |
$Nu_{x}^{*}=\frac{h_{x}x}{k}=-\frac{x}{T_{w}-T_{\infty }}\left( \frac{\partial T}{\partial y} \right)_{y=0}=-\frac{{\theta }'(\xi ,0)}{\sqrt{2}\theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ $Nu_{x}^{*}=\frac{h_{x}x}{k}=-\frac{x}{T_{w}-T_{\infty }}\left( \frac{\partial T}{\partial y} \right)_{y=0}=-\frac{{\theta }'(\xi ,0)}{\sqrt{2}\theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ - (4.149) +
+ | (101) + |} + The Sherwood number is The Sherwood number is - + {| class="wikitable" border="0" + |- + | width="100%" |
$Sh_{x}=\frac{h_{m}x}{D}=-\frac{x}{\omega _{w}-\omega _{\infty }}\left. \frac{\partial \omega }{\partial y} \right|_{y=0}=-\frac{{\phi }'(\xi ,0)}{\sqrt{2}\phi \theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ $Sh_{x}=\frac{h_{m}x}{D}=-\frac{x}{\omega _{w}-\omega _{\infty }}\left. \frac{\partial \omega }{\partial y} \right|_{y=0}=-\frac{{\phi }'(\xi ,0)}{\sqrt{2}\phi \theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ - (4.150) +
- [[Image:Fig4.15.png|thumb|400 px|alt=Nusselt number based on convection and Sherwood number (Zhang et al. 1996). |Figure 1: Nusselt number based on convection and Sherwood number (Zhang et al. 1996).]] + | (102) - Figure 4.15 shows the effect of blowing velocity on the Nusselt number based on convective heat transfer and the Sherwood number for $\phi _{sat,\infty }=0$, i.e., the mass fraction of sublimable substance is equal to the saturation mass fraction corresponding to the incoming temperature. It can be seen that the effect of blowing velocity on mass transfer is stronger than that on heat transfer. + |} ==References== ==References== - Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ. + {{Reflist}}

## Current revision as of 13:49, 26 July 2010

Boundary layer concept over a flat plate.

The form of the velocity, temperature, and concentration profiles for flow over a flat plate are presented in the figure to the right, and are based on qualitative measurements. These presentations indicate the possibility that these profiles are geometrically similar to each other along the flow direction for dependent variables such as velocity, temperature, and concentration, if the coordinates are stretched properly. For example, the velocity profiles are geometrically similar along the flow in the x-direction, differing only by a stretching factor (similarity factor) if the coordinates are properly sketched[1].

As noted before, not all boundary layer flow configurations have similar geometric profiles (similarity solution), but some do [2], especially for more conventional geometries and boundary conditions. The steady, two-dimensional, laminar, momentum boundary layer equation is a second order, nonlinear, partial differential equation. The two-dimensional, steady, laminar flow, energy, and species boundary layer equations are linear, second order, partial differential equations for most constant properties, and/or for decoupled heat and mass transfer problems.

If a similarity solution exists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact. A similarity technique converts conservation partial differential equations into ordinary differential equations and therefore, makes the solution much simpler. However, the analytical solutions may still require numerical integrations.

The conservation equations are uncoupled when each equation and its boundary condition can be solved independently of one another, except for continuity and momentum, which must be solved simultaneously. Coupled transport phenomena can occur in some applications because of coupled conservation governing equations and/or boundary conditions for mass diffusion, momentum, or heat transfer.

## Uncoupled Mass, Momentum, and Heat Transfer Problems

### Similarity Solution for Flow over a Flat Plate

We first apply the similarity solution to the classic problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled. Next, similarity solutions of control equations with appropriate boundary conditions of above figure are presented below. The viscous dissipation term is neglected for simplicity even though it is possible to obtain a similarity solution with viscous dissipation for this simple configuration. Assuming geometrically similar velocity profiles are possible along the flow over a flat plate using the following non-dimensional variables [3][4][2][5]:

 $\begin{matrix}{}\\\end{matrix}u=f(\eta)$ (1)
 $\eta =y\cdot g(x)$ (2)

where η is a non-dimensional independent variable (function of both x and y) that fulfills the similarity requirement. Mass and momentum boundary layer equations for flow over a flat plate (refer to Boundary layer approximations) can be converted to a new coordinate system η, f, and g by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$.

 $\frac{\partial u}{\partial y}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial \eta }\frac{\partial \eta }{\partial y}={f}'g$ (3)
 $\frac{\partial u}{\partial x}=\frac{\partial f}{\partial x}=\frac{\partial f}{\partial \eta }\frac{\partial \eta }{\partial x}={f}'y{g}'$ (4)
 $\frac{\partial ^{2}u}{\partial y^{2}}=\frac{\partial ^{2}f}{\partial y^{2}}=\frac{\partial }{\partial y}\left( \frac{\partial f}{\partial y} \right)=\frac{\partial }{\partial \eta }\left( \frac{\partial f}{\partial y} \right)\left( \frac{\partial \eta }{\partial y} \right)={f}''g^{2}$ (5)

Substituting eqs. (3)-(5) into mass and momentum boundary layer equations for flow over a flat plate yields:

 $\begin{matrix}{}\\\end{matrix}\nu {f}''g^{2}=f{f}'y{g}'+v{f}'g$ (6)
 ${f}'y{g}'+\frac{\partial v}{\partial y}=0$ (7)

Combining eqs. (6) and (7) to eliminate v and separating variables gives:

 $\frac{1}{f}\frac{d}{d\eta }\left( \frac{{{f}''}}{{{f}'}} \right)=\frac{1}{\nu }\frac{{{g}'}}{g^{3}}$ (8)

Considering that both η and x are independent variables, the left side of eq. (8) is a function of η, and the right side is a function of x, then each side must be a constant.

 $\frac{1}{\nu }\frac{{{g}'}}{g^{3}}=-c_{1}$ (9)

Integrating eq.(9) gives:

 $-\frac{1}{2g^{2}}=-c_{1}\nu x+c_{2}$ (10)

At the leading edge of the flat plate (x = 0), the boundary condition is $u=u_{\infty }$ which can be satisfied only at $\eta \to \infty$. In other words, η must be infinite when x = 0. It follows from eq. (2) that g(0) must be infinite and consequently, c2 in eq. (10) must be zero. Therefore

 $\eta =\frac{y}{\sqrt{2c_{1}\nu x}}$ (11)
 $g=\frac{1}{\sqrt{2c_{1}\nu x}}$ (12)

Since c1 is constant, the axial velocity is of the following form:

 $u=f\left( \frac{y}{\sqrt{x}} \right)$ (13)

From eq. (8)

 $\frac{1}{f}\frac{d}{d\eta }\left( \frac{{{f}''}}{{{f}'}} \right)=-c_{1}$ (14)

Integrating eq. (14) gives:

 $\frac{{{f}''}}{{{f}'}}=-c_{1}\int{fd\eta +c_{3}}$ (15)

The following boundary conditions are used to evaluate constants c1 and c3:

 $\begin{matrix}{}\\\end{matrix}\text{at }y=0,\text{ }u=v=0\text{ or at }\eta =0,\text{ }f=0$ (16)

Using eqs. (15) and (16):

 $\begin{matrix}{}\\\end{matrix}{f}''=0\text{ at }\eta =0$ (17)

then, using eqs. (15) and (4.17):

 $\begin{matrix}{}&{}\\\end{matrix}c_{3}=0$

therefore,

 $\frac{{{f}''}}{{{f}'}}=-c_{1}\int_{0}^{\eta }{fd\eta }$ (18)

Let the function ${\varsigma }'=d\varsigma /d\eta$ represent the non-dimensional axial velocity, $u/U_{\infty }$:

 ${\varsigma }'=\frac{u}{U_{\infty }}=\frac{f}{U_{\infty }}$ (19)

Then, using eq. (19)

 $\frac{{{f}''}}{{{f}'}}=\frac{{{\varsigma }'''}}{{{\varsigma }''}}$ (20)

Substituting eq.(20) in eq.(18) gives:

 $\frac{{{\varsigma }'''}}{{{\varsigma }''}}=-c_{1}\int_{0}^{\eta }{U_{\infty }\frac{d\varsigma }{d\eta }}d\eta =-c_{1}U_{\infty }\varsigma$ (21)

or

 ${\varsigma }'''+c_{1}U_{\infty }\varsigma {\varsigma }''=0$ (22)

From eq.(22) we conclude that c1U is a constant, non-dimensional variable. Let $c_{1}U_{\infty }=1/2$, eq. (22) becomes

 ${\varsigma }'''+\frac{1}{2}\varsigma {\varsigma }''=0$ (23)

where

 $\eta =\frac{y}{\sqrt{\nu x/U_{\infty }}}\text{ and }{\varsigma }'(\eta )=\frac{u}{U_{\infty }}$ (24)

Equation (23) is known as the Blasius equation and is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions.

 ${\varsigma }'(0)=0,\text{ }{\varsigma }'(\infty )=1,\text{ }\varsigma (0)=0$ (25)
Dimensionless velocity distribution in a laminar boundary layer.

Equation (23) can be numerically integrated using boundary conditions given by eq. (25). Blasius clearly integrated eq. (23) by hand without using a computer [6]. The numerical results using the shooting method are presented in the figure to the right, which shows that both u and vas functions of the independent variable η. A number of conclusions can be made from the numerical results presented in the figure to the right.

1. The edge of the momentum boundary layer is approximately at η=5.

2. The vertical velocity component v is non-zero at the edge of the boundary layer at η=5, even though the free stream flow is parallel to the plate.

3. The momentum boundary layer thickness is approximately

 $\frac{\delta }{x}=\frac{5}{\sqrt{\operatorname{Re}_{x}}}$ (26)

where $\operatorname{Re}_{x}=\frac{U_{\infty }x}{\nu }$

Using the definition of shear stress at the wall, τw, and the skin friction coefficient, cf, we obtain:

 $\tau _{w}=\mu \left. \frac{\partial u}{\partial y} \right|_{y=0}=c_{f}\frac{\rho U_{\infty }^{2}}{2}$ (27)

Rearranging the above equation in terms of $\varsigma$ gives:

 $\frac{c_{f}}{2}=\sqrt{\frac{\nu }{U_{\infty }x}}\left. {{\varsigma }''} \right|_{\eta =0}=\frac{\left. {{\varsigma }''} \right|_{y=0}}{\sqrt{\operatorname{Re}_{x}}}$ (28)
 $\frac{c_{f}}{2}=\frac{0.332}{\sqrt{\operatorname{Re}_{x}}}$ (29)

For the purposes of design and comparison with experimental measurements, it is more practical to obtain the average value of the skin friction, cf, which is usually determined by

 $\text{total drag}=\int_{A}{\tau _{w}dA=W\int_{0}^{x}{c_{f}\frac{\rho U_{\infty }^{2}}{2}dx=\frac{\rho U_{\infty }^{2}}{2}\bar{c}_{f}xW}}$ (30)

where W is the width of the plate. From eq.(29),

 $\frac{\bar{c}_{f}}{2}=\frac{1}{x}\int_{0}^{x}{\frac{c_{f}}{2}dx=0.664\operatorname{Re}_{x}^{-1/2}}$ (31)

It should be emphasized that the results presented above are for laminar flow parallel to a flat plate with constant free stream velocity. The approach to obtain the velocity distribution presented above can be extended to obtain the temperature distribution for laminar boundary layer flow over a flat plate with constant properties, constant free stream velocity, and constant temperature. The constant property assumption is important to decouple the momentum and energy equations in this configuration in order to obtain velocity first and temperature second. The non-dimensional temperature, θ, is defined as:

 $\theta =\frac{T-T_{w}}{T_{\infty }-T_{w}}$ (32)

The non-dimensional form of the energy equation (neglecting viscous dissipation) and boundary conditions and using the variables η and $\varsigma$ noted in eqs. (24) and (19) are as shown below.

 ${\theta }''+\frac{\Pr }{2}\varsigma {\theta }'=0$ (33)
 $\begin{matrix}{} & {}\\\end{matrix}\theta (0)=0$ (34)
 $\theta (\infty )=1$ (35)

with the notation that ${\theta }'=\frac{d\theta }{d\eta }$ and ${\theta }''=\frac{d^{2}\theta }{d\eta ^{2}}$. Equation (33) can be directly integrated:

 $\frac{d{\theta }'}{d\eta }+\frac{\Pr }{2}\varsigma {\theta }'=0$ (36)
 $\frac{d{\theta }'}{{{\theta }'}}+\frac{\Pr }{2}\varsigma d\eta =0$ (37)
 $\theta =c_{1}\int_{0}^{\eta }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta +c_{2}}$ (38)

Applying boundary conditions (34) and (35) gives

 $c_{2}=0\text{ and }c_{1}=\frac{1}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (39)

Therefore, we obtain the following equation for the dimensionless temperature, θ:

 $\theta (\eta )=\frac{\int_{0}^{\eta }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (40)
Dimensionless temperature distribution in a laminar boundary layer over a flat plate.

The energy equation (33) is a second order, linear, ordinary differential equation. The dimensionless temperature, θ, is a function of η and Pr. Equation (40) can be integrated numerically for a given Pr. The numerical results of θ as a function of η for different Pr values are presented in the figure to the right. The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layer equations, as presented in the above two figures.

1. Temperature profile is identical to velocity for Pr = 1.

2. Temperature profile has strong dependence on the Prandtl number.

3. For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, δT > δ.

4. For Pr>1 the thermal boundary layer thickness is less than the momentum boundary layer thickness, δT < δ.

The local heat transfer coefficient can be easily obtained from the numerical results of eq. (40) for a given Prandtl number by using Fourier’s law of heat conduction.

 $h_{x}=\frac{{q}''_{w}}{T_{w}-T_{\infty }}=\frac{-k\left. \frac{\partial T}{\partial y} \right|_{y=0}}{T_{w}-T_{\infty }}=\frac{-k\frac{d\theta }{d\eta }\left. \frac{\partial \eta }{\partial y} \right|_{w}}{T_{w}-T_{\infty }}$ (41)

The local Nusselt number is presented in terms of dimensionless variables as follows:

 $Nu_{x}=\frac{h_{x}x}{k}=\frac{x\left. {{\theta }'} \right|_{\eta =0}}{\sqrt{\nu x/U_{\infty }}}=\operatorname{Re}_{x}^{1/2}{\theta }'\left( 0 \right)$ (42)

or by using eq. (4.40):

 $Nu_{x}=\frac{\operatorname{Re}_{x}^{1/2}}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (43)

The numerical results can be approximated for the range of Prandtl numbers from 0.5 to 15 by the following equation:

 $Nu_{x}=0.332\operatorname{Pr}^{1/3}\operatorname{Re}_{x}^{1/2}$ (44)

The ratio of thermal boundary layer thickness, δT, to momentum boundary layer thickness, δ, for flow over a flat plate can also be approximated over this range of Prandtl numbers according to the relationship below:

 $\frac{\delta _{T}}{\delta }=\operatorname{Pr}^{-1/3}\text{, }0.5\le \Pr <10$ (45)

For both design purposes and comparison with experimental measurement it is more convenient to calculate the average or mean heat transfer coefficient, $\bar{h}$.

 $q=\int_{A}^{{}}{h\left( T_{w}-T_{\infty } \right)dA}=W\int_{0}^{x}{h\left( T_{w}-T_{\infty } \right)}dx=\overline{h}\text{ }W\left( T_{w}-T_{\infty } \right)x$ (46)

From above, the mean heat transfer coefficient and mean Nusselt number become:

 $\overline{h}=\frac{1}{x}\int_{0}^{x}{h_{x}dx=0.664\frac{k}{x}\operatorname{Re}_{x}^{1/2}\operatorname{Pr}^{1/3}}$ (47)
 $\overline{Nu}=\frac{\overline{h}x}{k}=2Nu_{x}=0.664\operatorname{Re}_{x}^{1/2}\operatorname{Pr}^{1/3}$ (48)

This shows that the average heat transfer coefficient is twice the local heat transfer coefficient, just as the mean skin friction coefficient is twice the local skin friction coefficient for flow over a flat plate. It should be noted that these conclusions concerning the relation between the local and average transfer coefficients can not be extended to other configurations.

### Similarity Solution for Flow over a Wedge

Similarity solutions also exist for some other conventional geometries with simple boundary conditions. Some of these cases are discussed below. For flow over a wedge, as shown in figure below, the free stream velocity varies according to potential (inviscid) flow theory:

 $\begin{matrix}{}\\\end{matrix}U=cx^{m}$ (49)

where U is the free stream velocity at the outer wedge surface and m is related to the wedge angle β by:

 $m=\frac{\beta /\pi }{2-(\beta /\pi )}=\frac{x}{U}\frac{dU}{dx}$ (50)

• When β (measured in radians) is positive, the free stream velocity increases along the wedge surface.
• When β is negative, the free stream velocity decreases along the wedge surface.
• β = 0 corresponds to flow parallel to a flat plate.
• β = π corresponds to flow perpendicular to the walls.

U is the oncoming free stream velocity, which is constant, as shown in figure to the right.

Wedge flow configuration.

Equation (49) can be also applied near the leading edge of a blunt object. For example, the free stream velocity on the surface of a cylinder and a sphere near a stagnation point can be predicted by potential flow theory according to the equations below[7]:

 (cylinder) $U=2U_{\infty }\sin \left( \frac{x}{R} \right)\approx 2U_{\infty }\left( \frac{x}{R} \right)$ (51)

 (sphere) $U=\frac{3}{2}U_{\infty }\sin \left( \frac{x}{R} \right)\approx \frac{3}{2}U_{\infty }\frac{x}{R}$ (52)

where R is the radius of the cylinder or sphere, U is the oncoming velocity and U is the free stream velocity just outside the boundary layer. The pressure gradient term is related to free stream velocity by potential flow theory. This can be done using the inviscid momentum equation (μ = 0)

 $\frac{1}{\rho _{\infty }}\frac{dp}{dx}=-U\frac{dU}{dx}=-cx^{m}cmx^{m-1}=-\frac{U^{2}m}{x}$ (53)

The two dimensional, steady, constant property boundary layer equations for mass, momentum, and energy of flow over a wedge neglecting viscous dissipation are:

Continuity equation:

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (54)

Momentum (x-direction):

 $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}-\frac{1}{\rho }\frac{dP}{dx}$ (55)

Energy equation:

 $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ (56)

The above equations can be easily obtained by beginning with the general equations and making appropriate boundary layer approximations using scaling and order of magnitude analysis. Using eq. (53) for the pressure and comparable dimensionless similarity parameters to flow over a flat plate, we get the following ordinary differential equation for momentum and energy for constant wall temperature with no blowing or suction at the wall.

 ${\varsigma }'''+\frac{1}{2}\left( m+1 \right)\varsigma {\varsigma }''+m\left( 1-{\xi }'^{2} \right)=0$ (57)
 ${\theta }''+\frac{\Pr \left( m+1 \right)}{2}\varsigma {\theta }'=0$ (58)

The boundary conditions for velocity and temperature are the same as in the case of flow over a flat plate. Assuming that heating starts at the leading edge, the ordinary differential equations (57) and (58) for dimensionless velocity and temperature can be solved using the given boundary condition. The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations:

 $c_{f,x}=\frac{2{\varsigma }''(0)}{\operatorname{Re}_{x}^{1/2}}$ (59)
 $Nu_{x}=A\operatorname{Re}_{x}^{1/2}$ (60)

where the constant A depends on m and Pr. Solutions of eqs. (57) and (58) for m > 0 are unique. For m < 0 two groups of solutions exist for a limited range of m values corresponding to negative β. These two groups correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. Falkner and Skan [8] originally developed the similarity solution for flow over a wedge with U = cxm and an impermeable wall. The numerical results for the friction coefficient for selected values of m and β are presented in the table below. m = β = 0 corresponds to the case of flow parallel to a flat plate with constant free stream values (Blasius Solution).

Table The local friction coefficient for laminar boundary layer flow over a wedge, with U = cxm and an impermeable wall

 β m ${\varsigma }''(0)=\frac{1}{2}C_{f}\operatorname{Re}_{x}^{1/2}$ 2πππ/2π/50-0.14-0.18-0.199 ∞1.00.3330.1110-0.0654-0.0826-0.0904 ∞1.233 Stagnation0.7570.5120.332 Flat plate0.1640.0820 Separation

Eckert [9] was the first to obtain the numerical results for eqs. (57) and (58) for constant free stream and wall temperature as presented in the following table.

Table 2 Local values of NuxRex-1/2 for laminar boundary layer flow over a wedge with constant wall and free stream temperature and an impermeable wall with U = cxm

 A B Pr 1 2 3 4 5 -0.5120π/5π/2π8π/5 -0.075300.1110.3331.04 0.2420.2920.3310.3840.4960.813 0.2530.3070.3480.4030.5230.858 0.2720.3320.3780.4400.5700.938 0.450.5850.6690.7921.0431.736 0.5700.7300.8511.0131.3442.236

In similarity solutions for both flow over a flat plate and over a wedge, it is assumed that there is no blowing or suction at the wall. It can be shown that similarity solutions with blowing or suction exist only if the vertical velocity at the wall changes along the flow according to the following equation:

 $v_{w}\propto x^{(m-1)/2}$ (61)

where vw is the blowing or suction velocity. Clearly this requirement restricts the use of the similarity solution to special cases that satisfy eq. (61).

## Coupled Mass, Momentum, and Heat Transfer Problems

Phase diagram for solid-liquid and solid-vapor phase change.

Coupled transport phenomena occur frequently in problems associated with evaporation, sublimation, absorption, or combustion. Sublimation over a flat plate can find its application in an analogy between heat and mass transfer and will be used here as an example to show the methodology used in solving these coupled problems.

When the pressure and temperature of ice are above the triple point and the ice is then heated, melting occurs. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through condensation. The phase-change processes related to solids can be illustrated by a phase diagram in to the right. Sublimation and deposition will be the subjects of this sub-section. When a subcooled solid is exposed to its superheated vapor, as shown in second figure (a), the vapor phase temperature is above the temperature of the solid-vapor interface and the temperature of the solid is below the interfacial temperature. The boundary condition at the solid-vapor interface is

 $k_{s}\frac{\partial T_{s}}{\partial x}-h_{\delta }(T_{\infty }-T_{\delta })=\rho _{s}h_{sv}\frac{d\delta }{dt}$ (62)

where hδ is the convective heat transfer coefficient at the solid-vapor interface, hsv is the latent heat of sublimation, and δ is the thickness of the sublimated or deposited material. The interfacial velocity dδ/dt in eq. (62) can be either positive or negative, depending on the direction of the overall heat flux at the interface. While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in second figure(a), the solid-vapor interface is usually smooth and stable.

Temperature distribution in sublimation and deposition.

In another possible scenario, as shown in (b) of the figure to the right, the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (62). Depending on the degrees of superheating in the solid phase and supercooling in the vapor phase (the relative magnitude of the first and second terms in eq. (62) both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition, on the other hand, the interface is dendritic and unstable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases. Naphthalene sublimation is also a case whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer [10]. The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen.

Sublimation on a flat plate with constant heat flux.

Figure to the right shows the physical model of a sublimation problem, where a flat plate is coated with a layer of sublimable material and is subject to constant heat flux underneath [11]. A gas with the ambient temperature and mass fraction of sublimable material flows over the flat plate at a velocity of U. The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and the other is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow.

The following assumptions are made in order to solve the problem: 1. The flat plate is very thin, so the thermal resistance of the flat plate can be neglected. 2. The gas is incompressible, with no internal heat source in the gas. 3. The sublimation problem is two-dimensional steady state.

The governing equations for mass, momentum, energy, and species of the problem are

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (63)
 $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}$ (64)
 $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ (65)
 $u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{\partial ^{2}\omega }{\partial y^{2}}$ (66)

A no slip condition at the surface of the flat plat requires that u = 0 at y = 0. For a binary mixture that contains the vapor, sublimable substance, and gas, the molar flux of the sublimable substance at the surface of the flat plate is

 ${\dot{m}}''=-\frac{\rho D}{1-\omega }\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (67)

Since the mass fraction of the sublimable substance in the mixture is very low, i.e., ω$\ll$1, the mass flux at the wall can be simplified to

 ${\dot{m}}''=-\rho D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (68)

Sublimation at the surface causes a normal blowing velocity,$v_{w}=\dot{{m}''}/\rho$. The normal velocity at the surface of the flat plate is therefore

 $v=v_{w}=-D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (69)

The energy balance at the surface of the flat plate is

 $-k\frac{\partial T}{\partial y}-\rho h_{sv}D\frac{\partial \omega }{\partial y}={q}''_{w}\begin{matrix} , & y=0 \\ \end{matrix}$ (70)

Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship [12]:

 $\omega =aT+b\begin{matrix} , & y=0 \\ \end{matrix}$ (71)

where a and b are constants that depend on the sublimable material and its temperature. As $y\to \infty ,$ the boundary conditions are

 $u\to U_{\infty }$, $T\to T_{\infty }$,$\omega \to \omega _{\infty }$ (72)

Introducing the stream function ψ,

 $u=\frac{\partial \psi }{\partial y}\begin{matrix} {} & v=-\frac{\partial \psi }{\partial x} \\ \end{matrix}$ (73)

the continuity equation (63) is automatically satisfied, and the momentum equation in terms of the stream function becomes

 $\frac{\partial \psi }{\partial y}\frac{\partial ^{2}\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{\partial ^{2}\psi }{\partial y^{2}}=\nu \frac{\partial ^{3}\psi }{\partial y^{3}}$ (74)

Similarity solutions for eq. (74) do not exist unless the injection velocity vw is proportional to x-1/2, and the incoming mass fraction of the sublimable substance, ω, is equal to the saturation mass fraction corresponding to the incoming temperature, T [12][11]. The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. [11] will be presented here.

Defining the following similarity variables:

 \begin{align} & \xi =\frac{x}{L},\text{ }\eta =y\sqrt{\frac{U_{\infty }}{2\nu L\xi }}\begin{matrix} , & f=\frac{\psi }{\sqrt{2\nu U_{\infty }L\xi }} \\ \end{matrix} \\ & \theta =\frac{k(T-T_{\infty })}{{q}''_{w}\sqrt{2\nu L\xi /U_{\infty }}}\begin{matrix} , & \phi =\frac{\rho h_{sv}D(\omega -\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L\xi /U_{\infty }}} \\ \end{matrix} \\ \end{align} (75)

eqs. (74) and (65) – (66) become

 ${f}'''+f{f}''=2\xi \left( {f}'{F}'-{f}''F \right)$ (76)
 ${\theta }''+\Pr (f{\theta }'-{f}'\theta )=2\Pr \xi \left( {f}'\Theta -{\theta }'F \right)$ (77)
 ${\phi }''+\operatorname{Sc}(f{\phi }'-{f}'\phi )=2\operatorname{Sc}\xi \left( {f}'\Phi -{\phi }'F \right)$ (78)

where prime ' represents partial derivative with respect to η, and all upper case variables represent partial derivatives of primary similarity variables with respect to ξ.

 $F=\frac{\partial f}{\partial \xi }\begin{matrix} , & \Theta =\frac{\partial \theta }{\partial \xi }\begin{matrix} , & \Phi =\frac{\partial \phi }{\partial \xi } \\ \end{matrix} \\ \end{matrix}$ (79)

It can be seen from eqs. (76) – (78) that the similarity solution exists only if F = Θ = Φ = 0. In order to use eqs. (76) – (78) to obtain a solution for the sublimation problem, supplemental equations for F, Θ, and Φ must be obtained. Taking partial derivatives of eqs. (76) – (78) with respect to ξ and neglecting the higher order term, one obtains

 ${F}'''+F{f}''+{F}''f=2\left( {f}'{F}'-{f}''F \right)$ (80)
 ${\Theta }''+\Pr (F{\theta }'+f{\Theta }'-{F}'\theta -{f}'\Theta )=2\Pr \left( {f}'\Theta -{\theta }'F \right)$ (81)
 ${\Phi }''+\operatorname{Sc}(F{\phi }'+f{\Phi }'-{F}'\phi -{f}'\Phi )=2\operatorname{Sc}\left( {f}'\Phi -{\phi }'F \right)$ (82)

The boundary conditions of eqs. (76) – (78) and eqs. (79) – (81) are

 ${f}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (83)
 $f(\xi ,0)=-\frac{2}{3}B\left[ \xi ^{1/2}{\phi }'(\xi ,0)-\xi ^{3/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix}$ (84)
 ${f}'(\xi ,\infty )=1\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (85)
 ${F}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (86)
 $F(\xi ,0)=-\frac{1}{3}B\left[ \frac{1}{2}\xi ^{-1/2}{\phi }'(\xi ,0)-\xi ^{1/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix}$ (87)
 ${F}'(\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (88)
 ${\theta }'(\xi ,0)+{\phi }'(\xi ,0)=-1\begin{matrix} , & \eta =0 \\ \end{matrix}$ (89)
 $\theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (90)
 ${\Theta }'(\xi ,0)+{\Phi }'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (91)
 $\Theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (92)
 $\phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\theta (\xi ,0)+\phi _{s}\xi ^{-1/2}\begin{matrix} , & \eta =0 \\ \end{matrix}$ (93)
 $\phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (94)
 $\Phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\Theta (\xi ,0)-\frac{\phi _{s}}{2\xi ^{3/2}}\begin{matrix} , & \eta =0 \\ \end{matrix}$ (95)
 $\Phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (96)

where

 $B=\frac{{q}''_{w}}{\rho h_{sv}\nu }\sqrt{\frac{2\nu L}{U_{\infty }}}$ (97)

reflects the effect of injection velocity at the surface due to sublimation, and

 $\phi _{s}=\frac{\rho h_{sv}D(\omega _{sat,\infty }-\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L/U_{\infty }}}$ (98)

represents the effect of the mass fraction of the sublimable substance in the incoming flow. $\omega _{sat,\infty }$ is saturation mass fraction corresponding to the incoming temperature:

 $\omega _{sat,\infty }=aT_{\infty }+b$ (99)

The set of ordinary differential equations (76) – (78) and (79) – (81) with boundary conditions specified by eqs. (83) – (96) are boundary value problems that can be solved using a shooting method [11].

Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is

 $Nu_{x}=\frac{h_{w}x}{k}=\frac{[{q}''_{w}/(T_{w}-T_{\infty })]x}{k}=\frac{\operatorname{Re}_{x}^{1/2}}{\sqrt{2}\theta (\xi ,0)}$ (100)

and the Nusselt number based on convective heat transfer is

 $Nu_{x}^{*}=\frac{h_{x}x}{k}=-\frac{x}{T_{w}-T_{\infty }}\left( \frac{\partial T}{\partial y} \right)_{y=0}=-\frac{{\theta }'(\xi ,0)}{\sqrt{2}\theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ (101)

The Sherwood number is

 $Sh_{x}=\frac{h_{m}x}{D}=-\frac{x}{\omega _{w}-\omega _{\infty }}\left. \frac{\partial \omega }{\partial y} \right|_{y=0}=-\frac{{\phi }'(\xi ,0)}{\sqrt{2}\phi \theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ (102)

## References

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2. 2.0 2.1 Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ.
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4. Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ.
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10. Eckert, E.R.G., and Goldstein, R.J., 1976, Measurement in Heat Transfer, McGraw-Hill, New York, NY
11. 11.0 11.1 11.2 11.3 Zhang, Y., Chen, Z.Q., and Chen, M., 1996, “Local Non-Similarity Solutions of Coupled Heat and Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of Thermal Science, Vol. 5, pp. 112-116
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