CHAPTER-III CONVECTION IN A POROUS MEDIUM WITH EFFECT OF MAGNETIC FIELD, VARIABLE FLUID PROPERTIES AND VARYING WALL TEMPERATURE

Transcription

1 CHAPER-III CONVECION IN A POROUS MEDIUM WIH EFFEC OF MAGNEIC FIELD, VARIABLE FLUID PROPERIES AND VARYING WALL EMPERAURE 3.1. INRODUCION Heat transer studies in porous media ind applications in several Engineering and technological sstems, as pointed out in Cheng and Minowcz[0]. In view o the applications, several heat transer studies were carried out and a ew are quoted here. Minowcz and Cheng[43] and Cheng and Chang [19] have discussed ree convection in a porous medium adjacent to vertical and horizontal suraces when the surace temperature varies as a power unction o distance. Sobha and Ramarishna[60] studied the eect o magnetic ield and varing plate temperature on convective heat transer past a vertical plate in porous medium. As pointed out in Al. et.al. [5], in mied convection similarit eists onl i the ree stream velocit and the temperature o the plate var as the power unctions o the distance along the plate. Both assisting low and opposing low were discussed. Ranges o the values o or dierent values o were presented or which either a unique solution, dual solutions or no solution eist. he eects o and on the low and heat transer characteristics were discussed. Chin et al.[1] discussed mied convection boundar laer low over a vertical surace or the Darc model when viscosit varies inversel as a linear unction o temperature. Results o both assisting low and opposing low were presented which were discussed as unctions o the mied convection parameter and variable viscosit parameter. In the opposing low case, the eistence o c dual solutions and boundar laer separation were noticed. 75

2 here has been increasing attention to the stud o magnetic ield on convection lows in porous media as pointed out in Nield and Bejan [47]. Magnetic ield eects on the ree convection and mass transer low through a porous medium with constant suction and constant heat lu has been discussed in Achara et al.[1]. Magneto hdrodnamic mied convection low has been analzed in an annular region illed with a luid saturated porous medium in Barletta[10]. In that paper, a transverse magnetic ield which acts radiall is created b a stationar electric current that lows through a clindrical shaped electrical cable present in the annular region. he eect o non uniorm magnetic ield on the low and heat transer o the Darc model is discussed. Magneto hdrodnamic ree convection in a horizontal cavit illed with a luid saturated porous medium with internal heat generation has been studied in Grosan et al.[4]. Assuming that the magnetic ield is inclined at angle with the horizontal plane, the low and heat transer are discussed as unctions o inclination angle, Hartmann number Ha, Raleigh number Ra and aspect ratio a. When a magnetic ield is applied perpendicular to a plate, an additional orce, nown as Lorenz orce appears in the momentum equation. he orce is deined b F J B 0 where J is the current densit and 0 B is the magnetic induction vector. B the generalized Ohm s law, the current densit can be epressed as J E V where E B 0 represents the conduction current and V B 0 represents the induction current. We assume the induced magnetic and electric ields to be negligibl small so that the Mawell s equations need not be written or the analsis o luid low and heat transer. Also, when we neglect the aerodnamic and Joule heating, the energ equation remains the same as the one in the absence o magnetic ield. (reer Sparrow, Cess[6]). 76

3 In the present chapter the eects o magnetic ield, variable viscosit, variable thermal conductivit and varing plate temperature on ree and mied convection at a vertical plate in a porous medium are studied. he ecess o the plate temperature over the ambient is assumed to var as power unction o distance along the plate, viscosit and thermal conductivit are assumed to var as linear unctions o temperature and a magnetic ield is assumed to act normal to the plate. Similarit solutions are obtained or the problem when the plate is hot or cold in the case o ree convection and assisting low and opposing low are discussed or mied convection. In the opposing low case, dual solutions (reerred to as upper and lower solutions) are obtained or certain values o the mied convection parameter RP. Ranges o values o RP are also obtained or which either a unique solution, dual solutions or no solution eist. Signiicant dierences are noticed between the low and heat transer quantities related to the upper and lower solutions. 3. FORMULAION OF HE PROBLEM AND SOLUION Let a lat plate be embedded verticall in a porous medium saturated with a viscous incompressible homogeneous quiescent luid (see igures 3.1(a) and 3.1(b) or the phsical model and the coordinate sstem). he porous medium is assumed to be homogeneous and is in thermal equilibrium with the surrounding luid. Let a magnetic ield o uniorm strength be applied in a direction normal to the plate. emperature o the plate, variations o viscosit and thermal conductivit with temperature are taen as in Chapter-II. he boundar laer equations governing the Darc model or ree convection and mied convection are presented below GOVERNING EQUAIONS FREE CONVECION 77

4 78 he orientation o the plate is as shown in igure 3.1(a) or hot plate and igure 3.1(b) or cold plate. he equations governing the natural convection boundar laer low or the Darc model are written as 0 v u (3.3.1) 0 ) ( 0 u K u B g p p (3.3.) 0 v K p (3.3.3) v u c m p (3.3.4) where v u, are luid velocit components, is luid temperature, K is Permeabilit, m is thermal conductivit o the luid saturated porous medium, 0 B is the magnetic lu, is the electrical conductivit and other smbols have their usual meanings.

5 79 aing, )] ( 1 [, in the bod orce term in (3.3.), introducing a stream unction and eliminating luid pressure rom (3.3.) and (3.3.3), the governing equations are obtained as K g B K 0 (3.3.5) c m m p 1 (3.3.6) he boundar conditions on and are 0,, 0,, 0, 0 as at (3.3.7) Introducing Raleigh number ( Ra ), Hartman number ( M ), a magnetic interaction parameter C and non dimensional unctions, together with a similarit variable through the relations

6 * * * 0 0 ) ( ) ( m Ra Ra M K K C K L K L B M g K Ra (3.3.8) equations (3.3.5), (3.3.6) are rewritten as C C C 1 1 (3.3.9) 0 ) (1 )] 1/ ( [1 (3.3.10) he boundar conditions (3.3.7) become 0 0,, 0, 1, 0, as at (3.3.11) Equation (3.3.9) can be integrated once using the condition on at ininit to get 1 1 C C (3.3.1) Evaluation o this epression at 0 gives the slip velocit ) (0.

7 MIXED CONVECION he orientation o the plate is shown in Figure.3. (a) or hot plate assisting low and Figure.3. (b) or cold plate opposing low. he ambient luid lows with a constant velocit U parallel to the plate and 0 B, the magnetic lu is perpendicular to the plate. he governing equations o the mied convection boundar laer low or the Darc model are written as 0 v u (3.3.13) 0 ) ( ) ( ) ( 0 U u K U u B g p p (3.3.14) 0 v K p (3.3.15) v u c m p (3.3.16)

8 he boundar conditions will be at 0, 0 A, v 0, as,, u U (3.3.17) aing the ree stream velocit as U b where b is a constant, introducing Peclet number ( Pe ) and non dimensional unctions, together with a similarit variable through the relations 8

9 ) ( ) ( ) ( m m m b b U Pe (3.3.18) the equations (3.3.13) to (3.3.16) reduce to ) ( 1 1 RP C C C (3.3.19) 0 ) (1 (3.3.0) where M K K C,, 0 L B M, 0 K g Ra and Pe Ra RP. he boundar conditions in terms o and are

10 at 0, 1, as, 0, 0, 1 (3.3.1) Equation (3.3.19) can be integrated once using the condition on at ininit to get C 1 C ( RP) 1 1 C (3.3.) Evaluating at 0, we get the slip velocit (0). 3.4 SOLUION OF HE PROBLEM PARAMEERS OF HE PROBLEM he low and heat transer quantities are unctions o the parameters, C, the magnetic interaction parameter (parameter depending on the porosit o the medium and Hartmann number),, the viscosit variation coeicient,, the thermal conductivit variation coeicient and, the power o inde o plate temperature, constant A appearing in the epression or temperature o the plate and RP mied convection parameter. he parameter C taes smaller values (less than unit) when either the porous parameter taes smaller values or the Hartmann number taes larger values, that is, when porosit o the medium is high or the intensit o the magnetic ield is high. When there is no applied magnetic ield M taes zero value and as a result C taes the value unit. Solutions are ound or the values 0.1, 0.5 and 1 o C. Reduced low can be epected or smaller values o C or or increased intensit o the magnetic ield as the magnetic ield lines obstruct the low. he other parameters are as described in the previous chapter. 84

11 he eects o simultaneous variation o the values o the parameters on the low and heat transer are presented in the discussion NUMERICAL SOLUION he equations or and i.e., (3.3.10), (3.3.1) are integrated numericall subject to appropriate boundar conditions or ree convection and (3.3.0), (3.3.) are integrated or mied convection b Runge-Kutta-Gill method, together with a shooting technique. Results o the present analsis when C=1, 0, 0, 0 (no magnetic ield, isothermal plate, constant luid properties) or ree convection and C 1, 0, 0, 0& RP 0 or mied convection agree well with appropriate results o Lai and Kulaci[36]. Results o our analsis when 0, 0 agree with corresponding results o Sobha and Ramarishna[60]. When the parameter C assumes the value unit, results o the present chapter reduce to those o Chapter-II DISCUSSION OF HE RESULS FREE CONVECION Interesting results related to the quantities o practical utilit, namel the shear stress, heat transer coeicient, velocit and temperature are presented, certain numerical results in the orm o tables 3.1, 3. and some other results in the orm o igures 3.3 to 3.9. Quantities lie the local Nusselt number and the local drag coeicient can readil be obtained rom the heat transer coeicient and the sin riction as in the previous chapter. 85

12 From tables 3.1, 3. we ma notice that thermal boundar laer thicness diminishes as as well as C tae increasing values. However variation in boundar laer thicness with changing values o is not quite signiicant. Plots o slip velocit ' (0)' and heat transer coeicient ' (0)' are presented in igures 3.3 and 3.4. he slip velocit as well as the heat transer coeicient tae larger values or 0 ( 0) than or 0 ( 0), this behaviour being more pronounced in the absence o magnetic ield than in its presence. he also diminish with increasing intensit o the magnetic ield (i.e., as C diminishes) and with diminishing values o. For all values o C, the heat transer coeicient increases with increasing values o. Plots o the stream unction or dierent values o C are shown in igure 3.5. Stream unction ' ( )' taes diminishing values with increasing intensit o the magnetic ield and also with increasing values o. 86

13 Variations in shear stress or dierent values o are shown in igure 3.6. Shear stress ' ( )' is observed to tae negative values and also its absolute value at the plate increases with diminishing values o. 87

14 Variation o sin riction ' (0 )' with or dierent values o C and are shown in igure 3.7. Absolute values o sin riction can be observed to increase with 88

15 increasing values C and. Absolute value o sin riction also assumes relativel larger values or negative values o, smaller values or zero value o and much smaller values or positive values o. Plots o non dimensional velocit are shown in igure Velocit at the plate ' ( )' taes larger values or 0 ( 1, 1), smaller values when variations in and are neglected ( 0, 0 ), and much smaller values or 0 ( 1, 1). Hdrodnamic boundar laer thicness can be observed to be more or 1, 1 than or 1, 1. Variations in heat transer coeicient with are shown in igure 3.9. he heat transer coeicient ' (0)' can be seen to tae increasing values with increasing values o and with diminishing values o, or all values o. Heat transer coeicient is also observed to tae increasing values as taes diminishing values. 89

16 3.6.. Mied convection Qualitativel interesting results related to the shear stress, heat transer coeicient, velocit and temperature are presented, some o them in the orm o tables 90

17 3.3, 3.4 and others in the orm o igures 3.10 to 3.7. Quantities such as the Nusselt number and drag coeicient can be readil obtained rom the heat transer coeicient and sin riction. Variations in slip velocit ' (0)', sin riction ' (0)' and heat transer coeicient ' (0)' or positive values o RP are presented in table 3.3. Sin riction ( 0 ) can be observed to be negative or positive values o RP or all values o the other parameters under consideration. Absolute values o ( 0 ) decrease with increasing values o both RP and while the increase with increasing values o C. Heat transer coeicient (0) taes increasing values with increasing values o RP and C while it taes decreasing values with increasing values o. In table 3.4 are presented the ranges o values o RP or which no solution, a single solution or dual solutions eist. he range o values can be seen to be more negative values o than or positive values o. he range can be seen to increase with increasing values o. he range decreases with increasing values o when taes positive values. 91

18 In the ollowing, more attention is paid to the discussion o the dual solutions o the opposing low case. As in the previous chapter the two solutions o the dual solution case are reerred to as upper and lower solutions respectivel. he changes in sin riction with negative values o the mied convection parameter RP are shown in igures 3.10, 3.11, 3.1 or dierent values o the parameters C,, and. he corresponding changes in heat transer coeicient are shown in igures 3.13, 3.14, 3.15 respectivel. One curve each corresponding to re. [5] are presented in igures 3.10, 3.11 and one curve corresponding to re.[1] in ig From the igures the range o values o RP over which solutions eist can be seen to be more in the presence o magnetic ield than in its absence. In the isothermal case, when viscosit is a constant as well as variable and in the presence as well as absence o magnetic ield, (0 ) is observed to be positive. For C 1.0, 0, 0 & 0 single solution eists or 1.0 RP 0, dual solutions eist or RP 1. 1 and no solution or RP Lie sin riction (0), heat transer coeicient also taes positive values when 0 (see 9

19 igures 3.10 and 3.13). Ecept or magnitude, behaviour o sin riction and heat transer coeicient when 1 are similar to the corresponding ones when = 0. Unlie in the isothermal case, when the plate temperature is variable (or eample 0.05), ( 0 ) is observed to tae both positive and negative values with changing negative values o RP, and dual solutions eist or a wide range o values o RP. For eample, when C =0.5, =0.5, =1 and 0 the range over which solutions eist is.1 RP 0. 1 while the range is.09 RP 1. 6 when C=0.5, 0.0, 1 and 0. Lie sin riction, heat transer coeicient also taes both positive and negative values with changing values o RP, when From Fig.3.1, the magnitude o (0) can be seen to change signiicantl with changing values o,. he range o values o RP over which eists can be seen to be maimum or 1, 1 and 0. From Fig.3.14, changes in ' (0)' can also be seen to be signiicant with changing values o., 93

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23 Plots o shear stress or the upper and lower solutions or dierent values o the parameters are shown in the igures 3.16, 3.17, 3.18 and Considerable dierences can be noticed in the behaviour o the shear stress or the upper and lower solutions (see igures 3.16 and 3.17). Curves o igure 3.16 correspond to those or 1, 0 while those o igure 3.17 correspond to 1, 0 when 0 and C=0.5. From igures 3.18 and 3.19 and also rom numerical results, it can be noticed that, or positive values o RP, the shear stress at the plate becomes negative thereb indicating separation o the boundar laer. 97

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25 Fluid velocit proiles or the two solutions o the opposing low case are presented in igures 3.0, 3.1 (or one tpe o luids) and in igures 3. and 3.3 (or another tpe o luids). It can be observed that hdrodnamic boundar laer thicness o the lower solution is much larger than that o the upper solution. Qualitative dierences between the two solutions can also be observed in the vicinit o the plate. Fluid temperature proiles corresponding to the upper and lower solutions are presented in igure 3.4 or certain negative values o RP. It can be noticed that thermal boundar laer thicness o the lower solution is much larger than that o the upper solution. Variations in the lower solutions with changing values o the parameters are signiicant than those in the other solution. Plots o sin riction are presented in Fig. 3.5 and plots o heat transer coeicient are presented in Fig.3.6 or positive values o RP. From igure 3.5, (0) can be seen to diminish as changes rom 0 to -1, and also when C taes increasing values. From igures 3.6 heat 99

26 transer coeicient ( (0) ) can be seen to increase as changes rom 0 to -1, and as C taes increasing values. Slip velocit proiles are presented in igures 3.7 or positive values o RP. Slip velocit can be seen to assume increasing values as changes rom o to -1 and as C changes rom 0.5 to

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29 As in the previous chapter, using the same deinitions, percentage variations in sin riction and heat transer coeicient in the ree convection case are computed and 103

30 are presented in the table.3.5. On a comparison o tables.1 and 3.5, it ma be noted that, percentage variations in both (0) and ' (0)' are signiicantl eected b the presence o the magnetic ield. Eect o magnetic ield can however be observed to be insigniicant onl when 0, 1 and 0,