Mixed Convection
We already have discussed one form of mixed convection in a horizontal layer, namely, the onset of convection with throughflow when the heating is from below (see Sect 6.10). In this chapter, we discuss some more general aspects of mixed convection. Since
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Mixed Convection
8.1 8.1.1
External Flow Inclined or Vertical Plane Wall
We already have discussed one form of mixed convection in a horizontal layer, namely, the onset of convection with throughflow when the heating is from below (see Sect 6.10). In this chapter, we discuss some more general aspects of mixed convection. Since we have dealt with natural convection and forced convection in some detail, our treatment of mixed convection in a porous medium [first treated by Wooding (1960)] can be brief. It is guided by the surveys by Lai et al. (1991a) and Lai (2000). We endorse the statement by Lai (2000) that despite the increased volume of research in this field, experimental results are still very few. In particular experimental data on thermal dispersion are very scarce, and this is hindering the study of the functional relationship between effective thermal conductivity and thermal dispersion. We start with a treatment of boundary layer flow on heated plane walls inclined at some nonzero angle to the horizontal. The foundational study is that by Cheng (1977c). This configuration is illustrated in Fig. 8.1. The boundary layer equations [compare Eqs. (5.5) and (5.6)] for steady flow are @2C g bK @T ¼ x 2 @y n @y
(8.1)
@C @T @C @T @ @T ¼ am : @y @x @x @y @y @y
(8.2)
Here gx is the component of g in the positive x direction, i.e., the direction of the stream velocity U1 at infinity. The Plus sign corresponds to the case where the buoyancy force has a component “aiding” the general flow and the Minus
D.A. Nield and A. Bejan, Convection in Porous Media, 397 DOI 10.1007/978-1-4614-5541-7_8, # Springer Science+Business Media New York 2013
398
8 Mixed Convection vw
x y U∞
γπ
γπ
vw
x
x vw
2
2 y U∞ U∞
y
U∞ y
U∞ γπ
y
2 vw
γπ
2 x
vw
x
Fig. 8.1 Definition sketch for mixed convection over an inclined surface
sign to the “opposing” case. We seek a similarity solution and allow for suction/ injection at the wall. Hence we take as boundary conditions the set y¼0:
@C ¼ axn ; @x
(8.3)
@C ¼ U1 ¼ Bxm , @y
(8.4)
T ¼ T 1 Axl ; v ¼
y!1:
T ¼ T1, u ¼
where A, a, and B are constants. The exponent m is related to the angle of inclination gp/2 (to the incident free-stream velocity) by the relation g ¼ 2 m/(m + 1). We find that a similarity solution does exist if l ¼ m and n ¼ (m 1)/2. The range of possibilities includes the cases l ¼ m ¼ 0, n ¼ 1/2 (vertical isothermal wall, injection / x1/2) l ¼ m ¼ 1/3, n ¼ 1/3 (wall at 45 inclination, constant heat flux) l ¼ m ¼ 1, n ¼ 0 (stagnation flow normal to vertical wall (Fig. 1.1e), linear temperature variation, uniform injection) With the similarity variables U 1 x 1=2 y C T T1 ; f ðÞ ¼ ¼ ; yðÞ ¼ 1=2 am x T w T1 ðam U1 xÞ
(8.5)
and the wall suction parameter f w ¼ 2a=ðm þ 1Þðam BÞ1=2 ;
(8.6)
8.1 External Flow
399
Fig. 8.2 Nusselt numbers for aiding and opposing flow with injection and suction on a vertical flat plate (Lai and Kulacki 1990d)
we obtain the system Rax 0 y; Pex
(8.7)
lþ1 0 f y þ lf0 y 2
(8.8)
f 00 ¼
y0
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