Editing Boundary layer (section)

==Convective transfer constants from boundary layer analysis==

[[Paul Richard Heinrich Blasius]] derived an exact solution to the above [[Blasius boundary layer|laminar boundary layer]] equations.<ref>{{cite journal |last=Blasius |first=H. |year=1908 |title=Grenzschichten in Flüssigkeiten mit kleiner Reibung |url=https://archive.org/details/zeitschriftfrma07runggoog/page/n15/mode/2up |journal=Zeitschrift für Mathematik und Physik |volume=56 |pages=1–37}} ([http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf English translation])</ref>  The [[Boundary-layer thickness|thickness]] of the boundary layer <math>\delta</math> is a function of the [[Reynolds number]] for laminar flow.

:<math> \delta \approx 5.0{x \over \sqrt {Re}} </math>

:<math>\delta</math> = the thickness of the boundary layer: the region of flow where the velocity is less than 99% of the far field velocity <math>v_\infty</math>; <math>x</math> is position along the semi-infinite plate, and <math>Re</math> is the Reynolds Number given by <math>\rho v_\infty x / \mu</math> (<math>\rho =</math> density and <math>\mu =</math> dynamic viscosity).

The Blasius solution uses boundary conditions in a dimensionless form:

:<math> {v_x - v_S \over v_\infty - v_S} = {v_x \over v_\infty} = {v_y\over v_\infty}= 0</math>{{space|5}}at{{space|5}}<math>y=0</math>

:<math> {v_x - v_S \over v_\infty - v_S} = {v_x \over v_\infty} = 1</math>{{space|5}}at{{space|5}}<math>y=\infty</math> and <math>x=0</math>

[[File:Velocity and Temperature boundary layer similarity.png|thumb|alt=Velocity and Temperature boundary layers share functional form|Velocity Boundary Layer (Top, orange) and Temperature Boundary Layer (Bottom, green) share a functional form due to similarity in the Momentum/Energy Balances and boundary conditions.]]

Note that in many cases, the no-slip boundary condition holds that <math>v_S</math>, the fluid velocity at the surface of the plate equals the velocity of the plate at all locations. If the plate is not moving, then <math>v_S = 0</math>. A much more complicated derivation is required if fluid slip is allowed.<ref>{{Cite book|doi = 10.1063/1.1407604|chapter = Blasius boundary layer solution with slip flow conditions|title = AIP Conference Proceedings|year = 2001|last1 = Martin|first1 = Michael J.|volume = 585|pages = 518–523|hdl = 2027.42/87372}}</ref>

In fact, the Blasius solution for laminar velocity profile in the boundary layer above a semi-infinite plate can be easily extended to describe Thermal and Concentration boundary layers for heat and mass transfer respectively. Rather than the differential x-momentum balance (equation of motion), this uses a similarly derived Energy and Mass balance:

Energy:{{space|8}} <math> v_x {\partial T \over \partial x} + v_y {\partial T \over \partial y} = {k \over \rho C_p}{\partial^2 T \over \partial y^2} </math>

Mass:{{space|10}} <math> v_x {\partial c_A \over \partial x} + v_y {\partial c_A \over \partial y} = D_{AB}{\partial^2 c_A \over \partial y^2} </math>

For the momentum balance, kinematic viscosity <math>\nu</math> can be considered to be the ''momentum diffusivity''. In the energy balance this is replaced by thermal diffusivity <math>\alpha = {k / \rho C_P}</math>, and by mass diffusivity <math>D_{AB}</math> in the mass balance. In thermal diffusivity of a substance, <math>k</math> is its thermal conductivity, <math>\rho</math> is its density and <math>C_P</math> is its heat capacity. Subscript AB denotes diffusivity of species A diffusing into species B.

Under the assumption that <math>\alpha = D_{AB} = \nu</math>, these equations become equivalent to the momentum balance. Thus, for Prandtl number <math> Pr = \nu/\alpha = 1</math> and Schmidt number <math> Sc = \nu/D_{AB} = 1</math> the Blasius solution applies directly.

Accordingly, this derivation uses a related form of the boundary conditions, replacing <math>v</math> with <math>T</math> or <math>c_A</math> (absolute temperature or concentration of species A). The subscript S denotes a surface condition.

:<math> {v_x - v_S \over v_\infty - v_S} = {T - T_S \over T_\infty - T_S} = {c_A - c_{AS} \over c_{A\infty} - c_{AS}}= 0</math>{{space|5}}at{{space|5}}<math>y=0</math>

:<math> {v_x - v_S \over v_\infty - v_S} = {T - T_S \over T_\infty - T_S} = {c_A - c_{AS} \over c_{A\infty} - c_{AS}} = 1</math>{{space|5}}at{{space|5}}<math>y=\infty</math> and <math>x=0</math>

Using the [[Stream function|streamline function]] Blasius obtained the following solution for the shear stress at the surface of the plate.

:<math>\tau_0 = \left( {\partial v_x \over \partial y} \right) _{y=0}=0.332 {v_\infty \over x} Re^{1/2}</math>

And via the boundary conditions, it is known that

:<math> {v_x - v_S \over v_\infty - v_S} = {T - T_S \over T_\infty - T_S} = {c_A - c_{AS} \over c_{A\infty} - c_{AS}}</math>

We are given the following relations for heat/mass flux out of the surface of the plate

:<math>\left( {\partial T \over \partial y} \right) _{y=0}=0.332 {T_\infty - T_S \over x} Re^{1/2}</math>

:<math>\left( {\partial c_A \over \partial y} \right) _{y=0}=0.332 {c_{A\infty} - c_{AS} \over x} Re^{1/2}</math>

So for <math>Pr=Sc=1</math>

:<math>\delta =\delta _T= \delta _c= {5.0 x\over\sqrt{Re}}</math>

where <math>\delta_T,\delta_c</math> are the regions of flow where <math>T</math> and <math>c_A</math> are less than 99% of their far field values.<ref name="citation 3">Geankoplis, Christie J. Transport Processes and Separation Process Principles: (includes Unit Operations). Fourth ed. Upper Saddle River, NJ: Prentice Hall Professional Technical Reference, 2003. Print.</ref>

Because the Prandtl number of a particular fluid is not often unity, German engineer E. Polhausen who worked with [[Ludwig Prandtl]] attempted to empirically extend these equations to apply for <math>Pr\ne 1</math>. His results can be applied to <math>Sc</math> as well.<ref>{{Cite journal|doi = 10.1002/zamm.19210010205|title = Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner reibung und kleiner Wärmeleitung|year = 1921|last1 = Pohlhausen|first1 = E.|journal = Zeitschrift für Angewandte Mathematik und Mechanik|volume = 1|issue = 2|pages = 115–121|bibcode = 1921ZaMM....1..115P|url = https://zenodo.org/record/1447401}}</ref>  He found that for Prandtl number greater than 0.6, the [[Thermal boundary layer thickness and shape|thermal boundary layer thickness]] was approximately given by:

[[File:Thermal Boundary Layer Thickness.png|thumb|alt=Prandtl number affects the thickness of the Thermal boundary layer. When the Prandtl is less than 1, the thermal layer is larger than the velocity. For Prandtl is greater than 1, the thermal is thinner than the velocity.|Plot showing the relative thickness in the Thermal boundary layer versus the Velocity boundary layer (in red) for various Prandtl Numbers. For <math>Pr=1</math>, the two are equal.]]

:<math>{\delta \over \delta_T} = Pr^{1/3}</math>{{space|10}}and therefore{{space|10}}<math>{\delta \over \delta_c} = Sc^{1/3}</math>

From this solution, it is possible to characterize the convective heat/mass transfer constants based on the region of boundary layer flow. [[Conduction (heat)#Fourier's law|Fourier's law of conduction]] and [[Convective heat transfer#Newton's law of cooling|Newton's Law of Cooling]] are combined with the flux term derived above and the boundary layer thickness.

:<math>{q\over A} = -k \left({\partial T \over \partial y} \right)_{y=0} = h_x(T_S-T_\infty)</math>

:<math>h_x = 0.332{k \over x} Re^{1/2}_x Pr^{1/3}</math>

This gives the local convective constant <math>h_x</math> at one point on the semi-infinite plane. Integrating over the length of the plate gives an average

:<math>h_L = 0.664{k \over x} Re^{1/2}_L Pr^{1/3}</math>

Following the derivation with mass transfer terms (<math>k</math> = convective mass transfer constant, <math>D_{AB}</math> = diffusivity of species A into species B, <math>Sc = \nu / D_{AB}</math> ), the following solutions are obtained:

:<math>k'_x = 0.332{D_{AB} \over x} Re^{1/2}_x Sc^{1/3}</math>

:<math>k'_L = 0.664{D_{AB} \over x} Re^{1/2}_L Sc^{1/3}</math>

These solutions apply for laminar flow with a Prandtl/Schmidt number greater than 0.6.<ref name="citation 3" />