4.6 Similarity Solutions

Transcription

1 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell s 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 dependant variables such as velocity, temperature, and concentration, if the coordinates are stretched properly. 1

2 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell For eample, the velocity profiles are geometrically similar along the flow in the direction, differing only by a stretching factor (similarity factor) if the coordinates are properly sketched.

3 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell As noted before, not all boundary layer flow configurations have similar geometric profiles, but some do (Burmeister, 1993); especially for more simple geometry and conventional boundary conditions. The steady, two dimensional, laminar, momentum boundary layer equation is a second order nonlinear partial differential equation. The two dimensional, steady, laminar boundary layer, energy, and species equations are linear second order partial differential equations for most constant properties, and for decoupled heat and mass transfer problems. 3

4 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell If a similarity solution eists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact. A similarity technique converts these partial differential equations to ordinary differential equations and therefore, make the solution much more simple. These analytical solutions still require numerical integrations. 4

5 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Uncoupled Mass, Momentum, and Heat Transfer Problems The conservation equations are uncoupled when each equation and its boundary condition can be solved independently of each other, ecept for continuity and momentum, which need to be solved simultaneously. Coupled transport phenomenon can also occur in some applications because of coupled conservation governing equations and/or boundary conditions for diffusion, momentum, or heat transfer. Similarity solution for flow and heat transfer over a plat plate. We first apply the similarity solution to the classical problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled. Similarity solution of equations (4.5) to (4.7) with appropriate boundary conditions corresponding to figures 4.4 a & b are presented. 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. (see problem 4.16) 5

6 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Assuming geometrically similar velocity profiles are possible along the direction then (Kays and Crawford, 004; Bejan, 004; Biermeister, 1983) (4.44) u = f( η) η = yg ( ) (4.45) where η is a non-dimensional independent variable (function of both and y) that fulfills the similarity requirement. 6

7 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Mass and momentum boundary condition equations for flow over a flat plate (eq 4.5 & eq 4.6) can be converted to a new coordinate system η, f, and g ' df ' dg by using the notation f = and Upon substitution of equations (4.46)-(4.48) into (4.5) and (4.6) we get: ν f g = ffyg + vfg Combining (5.48) & (5.49) to eliminate dη g = d u f f η = = = y y η y u f f η = = = η f ' g ' ' f yg y y y y η y y u f f f η = = = = f '' v f yg + = 0 y υ 1 d f 1 g = 3 f dη f ν g and separating variables gives: g (4.46) (4.47) (4.48) (4.49) (4.50) (4.51) 7

8 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Considering both η and are independent variables and the left side of eq. (4.51) is a function of η, and the right side is a function of, then each side must be a constant (c 1 ). 1 g 3 = c1 (4.5) ν g Integrating (4.5) gives: 1 = c (4.53) 1ν + c g c =0 since at =0, g = y η = cν (4.54) 1 cν The aial velocity is of the following form since c 1 is constant. y u = f From eq (4.51) g = 1 1 d f f dη f 1 '' ' = c 1 (4.55) (4.56) (4.57) 8

9 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Integrating eq (4.57) gives: f f '' = c fdη + c 1 3 Net, the following boundary conditions are used to evaluate constants c 1, c, and c3. (4.58) at y=0, u=v=0 or at η=0, f=0 (4.59) 9

10 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Using equations (4.58) and (4.59) '' f = 0 at η = 0 (4.60) then, using (4.58) and (4.60) Therefore, Let the function u velocity U c 3 = 0 f f '' ' d d = c η ' 1 0 fdη represent non-dimensional aial (4.61) ' u U f U (4.6) 10

11 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Then using equation (4.6) substituting (4.63) in (4.61) or (4.65) is a constant and non- From (4.65) we conclude that dimensional variable. Let where then η 1 f f '' ' ''' ς = '' ς ''' ς η dς = c1 U d η cu 0 = '' 1 ς ς dη ''' ς + cu ςς = cu = 1 '' 1 0 cu 1 ''' 1 '' ς + ςς = 0 y = ς ( η) = ν U (4.63) (4.64) u U (4.66) ' and (4.67) 11

12 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Equation (4.66) is known as the Blasius equation and therefore is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions. ς ' (0) = 0 ς ' ( ) = 1 ς (0) = 0 (4.68) Equation (4.66) can be numerically integrated using boundary conditions given by (4.68), however Blasius, in 1908, clearly integrated (4.66) by hand and not using a computer. The numerical results using shooting method are presented in figure (4.7), which shows both aial and radial velocity as a function of the independent variable η. 1

13 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell u U = f 1 v Re U 0 y 1 5 η = 7 Re Figure 4.7: Dimensionless velocity distribution in a laminar boundary layer over a flat plate 13

14 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell A number of conclusions can be made from the numerical results presented in Figure 4.7 The edge of the momentum boundary layer is at approimately η=5. The vertical velocity component ν is not zero at the edge of the boundary layer at η=5 even though the free stream flow is parallel to the plate. The momentum boundary layer thickness is approimately δ 5 = (4.69) Re U where Re = ν Using the definition of shear strength at the wall, τ and friction coefficient w c f, we obtain: du ρu τ w = µ = c f dy (4.70) y= 0 Rearranging equation (5.64) in terms of L gives: '' ς (4.71) c f ν ς '' y= = = U η = 0 Re c f 0.33 = Re 0 (4.7) 14

15 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell For the purposes of design and comparison with eperimental measurements, it is more practical to obtain the average value of, usually denoted by c f c U U 0 f ρ ρ total drag τ da w d c w (4.73) = A w = = Where w is the width of the plate, then from above c f f c 1 c = d = f f Re 1 (4.74) 15

16 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell It should be emphasized that the results presented above are for laminar flow parallel to a flat plate with constant free stream velocity. 16

17 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The approach presented above to obtain velocity distribution can be etended to obtain temperature distribution for laminar boundary layer flow with constant properties and constant free stream velocity and temperature over a flat plate. 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: With the notation that θ = ' θ = T T T dθ dη w T w and '' θ = d θ dη (4.75) 17

18 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The non-dimensional form of the energy equation (4.7) (neglecting viscous dissipation) and boundary conditions are as shown below. '' Pr ' (4.76) θ + ςθ = 0 θ (0) = 0 θ ( ) = 1 Equation (4.76) can be directly integrated: ' dθ Pr ςθ 0 dη + ' = ' dθ Pr ςdη 0 ' θ + = η Pr η θ = c ep ςdη dη+ c (4.77) (4.78) (4.79) (4.80) (4.81) 18

19 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Apply boundary conditions (4.77) and (4.78) c = 0 1 and c1 = (4.8) Pr η ep ς d η d η 0 0 Therefore, we obtain the following equation for the dimensionless temperature, θ: θη ( ) = η Pr ep Pr ep 0 0 η 0 0 η ςdη dη ςdη dη (4.83) 19

20 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Energy equation (4.76) is a second order linear ordinary differential equation. The dimensionless temperature θ is a function of η and Pr. Equation (4.83) can be integrated numerically for a given Pr. The numerical results of θ as a function of η for different Pr are presented in Figure 4.8 0

21 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell η = 1/ y Re θ = T T T w T w Figure 4.8 Dimensionless temperature distribution in a laminar boundary layer over a flat plate (Oosthuizen and Naylor, 1999) 1

22 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layers, as presented in figures 4.7 and 4.8. Temperature profile is identical to velocity for Pr=1 Temperature profile has strong dependence on the Prandtl number For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, δt > δ For Pr>1 the thermal boundary layer thickness is smaller than the momentum boundary layer thickness δ < δ T

23 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The local heat transfer coefficient can be easily obtained from the numerical results of equation (4.83) for a given Prandtl number by using Fourier s law of heat conduction. T k dθ η k '' y (4.84) q w y= 0 dη y w h = = = T T T T T T or local Nusselt number is presented in terms of dimensionless variables as follows: ' h θ 0 1 η = ' Nu = = = Re θ (0) K ν (4.85) U or using (4.83) Nu = w w w Re Pr ep η ς d η d η (4.86) 3

24 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The numerical results can be approimated over the range of Prandtl numbers of 0.5 to 0 by the following equation: 1 1 (4.87) Nu = 0.33 Pr Re The ratio of thermal boundary layer δ T to momentum boundary layer δ for flow over a flat plate can also be approimated over this range of Prandtl numbers according to the relation below. δ T δ = Pr Pr 10 (4.88) 4

25 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell For design purposes and comparison with eperimental measurement it is more convenient to calculate the average heat transfer coefficient A ( ) ( ) ( ) q = h T T da = W h T T d = h W T T w 0 w w From above, the mean heat transfer coefficient and mean Nusselt number become 1 k h = hd Re Pr = 0 1/ 1/3 h (4.89) (4.90) Nu h = = Nu = Re Pr k 1/ 1/3 (4.91) 5

26 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Thus the mean or average heat transfer coefficient is twice the local one similar to mean friction coefficient twice the local friction coefficient for flow over a flat plate. It should be noted that conclusions concerning the reduction between the local and average can not be etended to other configurations. 6

27 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Eample 4. Obtain the local Nusselt number for flow over a flat plate with constant free stream velocity and temperature for the limiting case of very low Prandtl numbers. 7

28 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Solution For very low Prantl numbers such as with liquid metals, the thermal boundary layer will develop much faster than the momentum boundary layer as shown in figure 4.9. δt δ 8

29 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell δ T U y Velocity profile T δ Figure 4.9 Laminar momentum and thermal boundary layer for a fluid with very low Prandtl number Pr<<1 over a flat plate. 9

30 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell As shown in Figure 4.9 one can safely assume the velocity inside the thermal boundary is constant and equal to U for very low Prandtl numbers. Equation (4.76) can be differentiated with respect ' u to η as well as using ς = = 1 to obtain the U following equation. '' ' dθ θ Pr 0 dη + = (4.9) 30

31 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The above equation needs to be integrated three times and requires three boundary conditions. The additional boundary condition is θ '' (0) = 0, which can be obtained from the energy equation. The numerical results of (4.9) yield: 1 1 Nu = Pr Re Pr 1 for (4.93) 31

32 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Eample 4.3 Obtain the Nusselt number for flow over a flat plate when the Prandtl number is much greater than 1. 3

33 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Solution When the Prandtl number is very high, the momentum boundary layer is much thicker than the thermal boundary layer, therefore, the thermal boundary layer is within the momentum boundary layer. We can assume a linear velocity near the wall within the thermal boundary layer. 33

34 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell From the numerical results presented in figure 4.7, one can conclude that ς (0)=0.33 Also, based on linear velocity: '' ς = 0.33 (4.94) Integrating the above equation twice with respect to η and applying ς = ς=0 gives: ς = 0.166η (4.95) Substitution of (4.95) into (4.83) and repeating numerical integration yields the Nusselt number from equation (5.85) Nu = Pr Re for Pr 1 (4.96) 34

35 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Similarity Solution for Flow Over a Wedge Similarity solutions also eist for some other conventional geometries with simple boundary conditions. Some of these cases are discussed below. Flow over a wedge, figure 4.10, when the free stream velocity varies according to potential (invisid) flow theory (4.97) Where U is the free stream velocity at the outer wedge surface and m is related to the wedge angle β by: β / π du m = = β / π U d (4.98) U When β 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 Fig = c m 35

36 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell y U β Figure 4.10: Wedge flow configuration 36

37 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Equation (4.97) can also apply near the leading edge of a blunt object. For eample, 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 respectively (Schlichting and Gersteu, 000) 37

38 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell U = U (4.99) sin U R R 3 3 U = U (4.100) sin U R R 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) 1 dp du m m 1 U m ρ = U = c c = d d (4.101) 38

39 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The two dimensional steady constant property boundary layer equation for mass, momentum, and energy for flow over a wedge and neglecting viscous dissipation are: u continuity equation + = 0 (4.10) momentum ( direction) u + v = ν (4.103) energy equation u + v = α (4.104) The above equations can be easily obtained by starting from the general equation from chapter and making appropriate boundary layer approimations using scaling and order of magnitude analysis. v y u u u 1 dp y y d y y T T T ρ 39

40 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Using equation (4.93) for the pressure and similar dimensionless similarity parameters as in that for 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. 40

41 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 1 ( ) ( ς + m+ 1 ςς + m 1 ς ) = 0 Pr ( m + 1) θ + ςθ = 0 (4.105) (4.106) The boundary condition for velocity and temperature are the same as in the case of flow over a flat plate. The ordinary differential equations (4.105) and (4.106) for dimensionless velocity and temperature can thus be solved for the given boundary condition. The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations: '' ς (0) c f = 1 (4.107) Re Nu where the constant (A) depends on m and Pr. = A Re 1 (4.108) 41

42 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Solutions of equations (4.105) and (4.106) for m>0 are unique. For m<0 two groups of solutions eist for a limited range of m values corresponding to negative β. These two groups are correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. 4

43 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Falkner and Skan (1931) originally developed the similarity solution for flow over a wedge with U = c m and impermeable wall equation (4.105). The numerical results for friction coefficient for selected values of m or 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). Table 4.1 The local friction coefficient for laminar boundary layer flow over a wedge with U = c m and impermeable wall. β π π π/ π / m ς "(0) = C Re 1/ f 1.33 Stagnation Flat plate Separation 43

44 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Eckert (194) obtained initially the numerical results for equation (4.105) and (4.106) for constant free stream and wall temperature as presented in table 4.. Table 4. Local values of Nu Re -1/ for laminar boundary layer flow over a wedge with constant wall and free stream temperature and impermeable wall with U = c m Pr β m π/ π/ π π/

45 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell In similarity solutions for both flow over a flat plate and over a wedge, it was assumed that there is no blowing or suction at the wall. It can be shown (see problem 4.8) that similarity solution with blowing or suction eist only if the vertical velocity at the wall changes along the flow according to the following equation. ( m 1) v w (4.109) v w where is the blowing or suction velocity. Clearly this requirement puts restrictions on using the similarity solution to special cases that meet the requirement of equation (4.109). 45

46 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 5.6. Coupled mass, momentum and heat transfer problems Coupled transport phenomenon occurs frequently in problems associated with evaporation, sublimation, absorption, or combustion. 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 eposed 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. 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

47 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell When a subcooled solid is eposed to its superheated vapor, as shown in Fig. 4.1(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 Ts dδ (4.110) ks hδ( T Tδ) = ρshsv dt where hδ is the convective heat transfer coefficient at the solidvapor 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. (4.110) can be either positive or negative, depending on the direction of the overall heat flu 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. 4.1(a), the solid-vapor interface is usually smooth and stable. 47

48 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Figure 4.11 Phase diagram for solid-liquid and solidvapor phase change. (a) Subcooled solid eposed to superheated vapor (b) Superheated solid eposed to supercooled vapor Figure 4.1 Temperature distribution in sublimation and deposition. 48

49 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell In another possible scenario, as shown in Fig. 4.1 (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 not stable, 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. 49

50 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Sublimation over a flat plate can find its application in analogy between heat and mass transfer (Zhang et al., 1996) and will be used here as an eample to show the methodology used in solving these problems. 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 flu heating underneath. A gas with the ambient temperature and mass fraction of sublimable material flows over the flat plate at a velocity of. The heat flu 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 another part is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow. 50

51 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The following assumptions are made in order to solve the problem: The flat plate is very thin, and so the thermal resistance of the flat plate can be neglected. The gas is incompressible, with no internal heat source in the gas. The sublimation problem is two-dimensional steady state. 51

52 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The governing equations for mass, momentum, energy and species of the problem are u u u v + = 0 y + v = ν y y u u u T T T + v = α y y u + v = D y y ω ω ω (4.111) (4.11) (4.113) (4.114) 5

53 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell v u Boundary layer y Nonslip condition at the surface of the flat plat require that u = 0, y = 0 For a binary miture that contains the vapor sublimable substance and gas, the molar flu of the sublimable substance at the surface of the flat plate is [see eq. (4.10)] ρd ω m =, y = 0 (4.115) 1 ω y Figure 4.13 Sublimation on a flat plate with constant heat flu. 53

54 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Since the mass fraction of the sublimable substance in the miture is very low, i.e.,, the mass flu at the wall can be simplified to m = ρd ω, y = 0 y (4.116) Sublimation at the surface causes a normal blowing velocity,, at the surface. The normal velocity at the surface of the flat plate is therefore ω v= v = ρd, y = 0 w y (4.117) y= 0 The energy balance at the surface of the flat plate is T ω k ρh, 0 svd = q w y = y y (4.118) 54

55 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 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, 1973, 1974): ω = at + b, y = 0 (4.119) where a and b are constants that depend on the sublimable material and its temperature. As y, the boundary conditions are u u, T T, ω ω (4.10) 55

56 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Introducing the stream function, ψ ψ u = v= (4.11) y the continuity equation is automatically satisfied, and the momentum equation in terms of the stream function becomes 3 ψ ψ ψ ψ ψ = ν (4.1) 3 y y y y Similarity solutions for eq. (4.1) do not eist unless the injection velocity vw is proportional to 1/, 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 non-similarity solution proposed by Zhang et al. (1996) will be presented here. 56

57 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Defining the following similarity variables: eqs. (4.1) and (4.113) (4.114) become f + ff = ξ ( f F f F ) θ + Pr( fθ f θ) = Prξ( f Θ θ F) u ψ ξ =, η = y, f = L νlξ νu Lξ kt ( T ) ρh v D( ω ω ) θ =, ϕ = q νlξ / u q νlξ / u w (4.13) (4.14) (4.15) (4.16) where prime ' represents partial derivative with respect to η, and all upper case variables represent partial derivative of primary similarity variable with respect to ξ. w ( ) ϕ + Sc( fϕ f ϕ) = Scξ f Φ ϕ F 57

58 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell F f θ ϕ =, Θ=, Φ= ξ ξ ξ (4.17) It can be seen from eqs. (4.14) (4.16) that the similarity solution eists only if F=Θ=Φ=0. In order to use eqs. (4.14) (4.16) to obtain a solution for the sublimation problem, the supplemental equations about F, Θ, and Φ must be obtained. Taking partial derivatives of eqs. (4.14) (4.16) with respect to and neglecting the higher order term, one obtains (4.18) ( ) F + Ff + F f = f F f F Θ + Pr( Fθ + fθ F θ f Θ ) = Pr( f Θ θ F) Φ + Sc( Fϕ + fφ F ϕ f Φ ) = Sc( f Φ ϕ F) (4.19) (4.130) 58

59 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The boundary conditions of eqs. (4.14) (4.16) and eqs. (4.18) (4.130) are f f ( ξ,0) = 0, η = 0 ξ = ξ ϕ ξ ξ Φ ξ η = 3 1/ 3/ (,0) B (,0) (,0), 0 f ( ξ, ) = 1, η = F ( ξ,0) = 0, η = / 1/ F( ξ,0) = B ξ ϕ ( ξ,0) ξ ( ξ,0), η 0 3 Φ = F ( ξ, ) = 0, η = θ ( ξ,0) + ϕ ( ξ,0) = 1, η = 0 θξ (, ) = 0, η= Θ ( ξ,0) +Φ ( ξ,0) = 0, η = 0 Θ( ξ, ) = 0, η = (4.131) (4.13) (4.133) (4.134) (4.135) (4.136) (4.137) (4.138) (4.139) (4.140) 59

60 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell ϕξ ah 1 = θ ξ + ϕξ η = sv 1/ (,0) (,0) s, 0 cp Le ϕξ (, ) = 0, η= ahsv 1 ϕs Φ ( ξ,0) = Θ( ξ,0), η = 0 3/ c Le ξ p Φ( ξ, ) = 0, η = (4.141) (4.14) (4.143) (4.144) 60

61 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Where B= q w ν L ρhsvν u (5.145) reflects the effect of injection velocity at the surface due to sublimation, and ρhsvd( ωsat, ω ) ϕs = q ν L/ u w (4.146) represents the effect of the mass fraction of the sublimable substance in the incoming flow. is saturation mass fraction corresponding to the incoming temperature: ωsat, = at + b (4.147) 61

62 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The set of ordinary differential equations (4.14) to (4.16) and (4.18) to (4.130) with boundary conditions specified by eqs. (4.131) to (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 ξ are also different, which is further evidence that a similarity solution does not eist. 6

63 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell θ Figure 4.14 Temperature and mass fraction distributions (Zhang et al. 1996). 63

64 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Figure 4.15 Nusselt number based on convection and Sherwood number (Zhang et al. 1996). 64

65 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Once the converged solution is obtained, the local Nusselt number based on the total heat flu at the bottom of the flat plate is 1/ h w [ q w/( Tw T )] Re Nu (4.148) = = = k k θξ (,0) and the Nusselt number based on convective heat transfer is * h T θ ( ξ,0) 1/ Nu = = = Re k T T y θξ The Sherwood number is h m ω ϕ ( ξ,0) Sh = = = Re D ω ω y ϕθ ( ξ,0) w y= 0 w y= 0 (,0) (4.149) (4.150) Figure 4.15 shows the effect of blowing velocity on the Nusselt number based on convective heat transfer and the Sherwood number for, 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. 1/ 65