Turbulent Mixed Convection from an Isothermal Plate Aubrey G. Jaffer Digilant 2 Oliver Street, Suite 901 Boston MA US

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1 Turbulent Mixed Convection from an Isothermal Plate Aubrey G. Jaffer Digilant Oliver Street, Suite 901 Boston MA 09 US Abstract Mixed convection is the combined transfer of heat from a surface having simultaneous externally forced flow and buoyancy generated (natural) flow. For horizontal and vertical rectangular (smooth) plates and flows (parallel to the plate), there are 14 combinations of orientation and laminar and turbulent forced and natural flows. Including rough-turbulent forced flows raises the number of combinations to 3. By excluding laminar forced convection, mixed convection can be modeled as a function of the natural and forced convective surface conductances and orientation of the plate and flow. Furthermore, such mixed convection is bounded by the L 4 -norm and L -norm of the forced and natural convective surface conductances per each orientation. Systematic measurements at ynolds numbers from 500 to 0000 of the combinations of horizontal or vertical orientation and laminar natural flow mixed with turbulent and rough-turbulent forced flows were made in air using m square heated plates having 3.0 mm and 1.03 mm RMS height-of-roughness. The formula presented here matches nearly all the measurements within their expected uncertainties. In addition to the measurements made at horizontal and vertical orientations, measurements of buoyancy-aided and buoyancy-opposed mixed flows made at a plate angle of 84.5 (nearly face down) also match the generalized formula within expected uncertainties. Keywords: mixed convection; natural convection; forced convection; turbulent flow Table of Contents Overview Scope Prior Work Forced Convection Natural Convection Turbulent Mixed Convection Natural Convection from an Inclined Plate Mixed Convection from an Inclined Plate Conclusions ferences Appendix: Measurements of Mixed Convection in Horizontal and Vertical Orientations

2 1. Overview For a rectangular flat surface with at least one horizontal edge, the goal is a total convective surface conductance function of lengths, angles, and dimensionless quantities: h M = fn(pr,, Ra, θ, ψ,, ) Symbol Range Name Pr 0 <Pr Prandtl number 0 ynolds number of external flow Ra 0 Ra Rayleigh number θ 90 θ 90 Surface angle from vertical ( 90 is face up) ψ 0 ψ 180 Angle of in-plane fluid flow from vertical 0 < Length of plate 0 < Width of plate Table 1 Mixed Convection Parameters For the cases where the convection produced by the external flow alone would be turbulent or roughturbulent, the work presented here achieves a significant simplification: the mixed convective surface conductance depends only on the isolated forced and natural conductances and the plate orientation (θ, ψ). Moreover, fn is linear in h F and h N ; that is: h M = fn(h F, h N, θ, ψ) h M ζ = fn(h F ζ, h N ζ, θ, ψ) ζ 0 Which leads to (with U F = h F and U N = h N ): U M = fn(u F, U N, θ, ψ) Furthermore, this work finds that the (turbulent) mixed convection is bounded by the L 4 -norm and L -norm of the convective components: ( UF 4 + U N 4) 1/4 UM ( U F + U N ) 1/ These simplifications do not apply when the mixed convection is laminar. Jaffer[1] details some laminar flow measurements violating inequality (1). In the case of forced horizontal flow parallel to a vertical plate, their convection mixes as the L -norm as shown in Figure 9; this is not surprising as the magnitude of the sum of perpendicular vectors is the L -norm. Fig. 14(f) from Fujii and Imura[] shows that upward-facing convection draws fluid from the sides of the plate into a central plume. As horizontal forced flow increases, it will move the plume leeward; there will be more convection from the windward side and less from the leeward side. While not as intuitive as the vertical plate case, the L -norm also applies to this combination of perpendicular flows (Figure 7). Fig. 14(c) from Fujii and Imura[] shows that downward-facing convection has the fluid flowing out from opposite sides just below the convecting surface. A slow horizontal forced flow will oppose half of the natural flow and aid the rest. Downward flow adjacent to a heated vertical plate is also in conflict at low flow rates. The L 4 -norm applies over the entirety of the Nu-versus- curve for downward-facing (Figure 5), and the lower part of the vertical plate curve in Figure 11 (the rest is L -norm). In both cases the Nu-versus- curves are monotonically increasing. Velocity shear drives turbulence, but velocity shear for a vertical plate with forced upward flow is reduced when the aiding convections are close in magnitude, resulting in a small range of L 4 -norm mixing (with the rest L -norm) in Figure 13. Section 6 gives the formulas for horizontal and vertical plates. Section 7 gives the formula for all orientations (with one horizontal edge). (1)

3 . Scope Convection Measurement Apparatus and Methodology [1] describes the Convection Machine apparatus constructed in 016 to measure forced convection from a rough plate[3]. Its small wind-tunnel chassis (1.3 m.65 m.65 m) afforded an opportunity to measure and characterize mixed convection at various orientations of the plate and flow. Table shows the combinations of flow types and orthogonal orientations. marks the combinations measurable by the apparatus with a m square plate; marks those combinations requiring a smaller smooth plate; marks those requiring a larger plate. 1 Natural Forced Down Up Vertical Opposing Aiding Laminar Laminar Laminar Turbulent Figure 6 Figure 8 Figure Figure 1 Figure 14 Laminar Rough Figure 5 Figure 7 Figure 9 Figure 11 Figure 13 Turbulent Turbulent Turbulent Rough Table Mixed Convective Modes In Correlating equations for laminar and turbulent free convection from a vertical plate [4] Churchill and Chu note that the correlation provides a good representation for the mean heat transfer for free convection from an isothermal vertical plate over a complete range of Ra and Pr from 0 to even though it fails to indicate a discrete transition from laminar to turbulent flow. This lack of transition may indicate that a single mixed convection formula will work for both laminar and turbulent natural convection from vertical plates. Current practice splits upward natural convection into four regions (and a gap) with indistinct transitions. Section 5 proposes a correlation uniting the four regions into a smooth curve. 3. Prior Work Much has been written about mixed convection, most of it about local laminar mixed convection, and most of that about aiding and opposing flows adjacent to a vertical object. The focus here is on total (not local) mixed convection where the forced component is turbulent. So most of this class of prior works is not germane to the topic at hand. The few turbulent mixed convection experiments which have been published have been limited to local convection and single orientations. In An Experimental Study of Mixed, Forced, and Free Convection Heat Transfer From a Horizontal Flat Plate to Air [5], X. A. Wang proposes local correlations for upward natural, mixed, and forced convection from a horizontal plate. But the integral for averaging the local convection fails to converge under some conditions of practical interest. Experimental Validation Data for CFD of Steady and Transient Mixed Convection on a Vertical Flat Plate [6] by Blake describes an apparatus for measuring local conditions throughout a vertical wind channel with one wall heated. Measurements of a channel are not applicable to plates. While the total mixed convection could be derived from a complete model for local mixed convection, no complete model for local mixed convection from an isothermal plate has been published. Fortunately, average convection formulas can sometimes be discovered, even when the local convection is intractable. In A general expression for the correlation of rates of transfer and other phenomena [7] Churchill and Usagi write that correlations of the form: Nu n (z) = Nu n 0 (z) + Nu n (z) () are remarkably successful in correlating rates of transfer for processes which vary uniformly between these limiting cases. 1 In order to measure laminar-laminar mixed convection in room temperature air, the forced characteristic length would need to be reduced by a factor of at least 3. Producing turbulent natural convection in air from a m (or smaller) square plate would require it to be heated to impractically high temperatures. 3

4 Lin, Yu, and Chen in Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates [8] state the form as: Nu n M = Nu n F ± Nu n N They report that n takes on the values of, 3, and 4 in prior work (n = 3 for uniform heat flux). However, they settle on n = 4 for local laminar correlations in horizontal and vertical orientations of uniform-walltemperature, with aiding being the sum of terms and opposing being the difference. The present work on average convection did not find any cases where the average natural convective component was subtracted, and casts the summing form equivalently in terms of the (vector-space) L p -norm function of the natural and forced convection components: U F, U N p = ( U F p + U N p ) 1/p 4. Forced Convection The correlations for laminar and turbulent convection from a smooth flat plate are: Nu F = / F Pr 1/3 (3) Nu F = /5 F Pr 1/3 (4) The transition between correlations (3) and (4) is dependent on c 5 5 which is very sensitive to the geometry of the surface and its leading edge: M = min ( F, c ) ( 4/5 F Nu F = / M Pr1/ /5 M ) Pr 1/3 Jaffer in Skin-Friction and Forced Convection from an Isothermal Rough Plate [3] gives a correlation (5) for a plate with (self-similar) root-mean-squared height-of-roughness ε: Nu F = F Pr 1/3 [ ] 0 < ε (5) ε 3 log The Convection Machine plates have roughness composed of equal areas at two elevations; such roughness is not self-similar. Jaffer[3] derives a correlation for this bi-level roughness which matches measurements of both plates (1 mm and 3 mm roughness) within their expected uncertainties up to F 80000: Nu F /Pr 1/3 = M = min ( M [ 3 log + ε pvl S S S F, ε pv L S ] + ε ( LF L S ( εpv L ) ) 5/4 t 0.37 t = 11 3 [ ( ) ] 1/5 S S L S + L S LF ( S S SS /5 F L S ) 1/ Pr 1/1 ( εpv 4/5 M ) 9/35 ( ) 149/175 F 149/175 M The rough convection of correlation (5) is in effect in the mixed convection region for the 3 mm roughness plate, while a combination of smooth and rough convection are active for the 1 mm roughness plate. ) (6) 4

5 5. Natural Convection The correlation for natural convection from a vertical plate due to from Churchill and Chu[4] is: Nu V (Ra V ) = Ra 1/6 V [1 + (0.49/Pr) 9/16] 8/7 1 Ra V 1 (7) For a horizontal plate, if T > 0 and θ > 0 or If T < 0 and θ < 0, then it is downward convection and Schulenberg[9] teaches: Nu R (Ra R ) = Ra 1/5 R [1 + (0.785/Pr) 3/5] 1/3 (8) Otherwise it is upward convection. ASHRAE[] citing Lloyd and Moran[11] gives four formulas for upward convection: Range Correlation f. 1 < Ra < 00 Nu = 0.96 Ra 1/6 (T9.5) 00 < Ra < 4 Nu = 0.59 Ra 1/4 (T9.6). 4 < Ra < 8 6 Nu = 0.54 Ra 1/4 (T9.7) 8 6 < Ra < Nu = 0.15 Ra 1/3 (T9.8) Table 3 Upward Convection Correlations Correlation (9) is an approximation aggregating the four correlations for upward convection. It is plotted with the four segments in Figure 1. All Convection Machine Ra values are between and 6. 6, putting them in correlation (T9.7) of Table 3. Nu (Ra ) = ( Ra 1/6) 1 Ra (9) Natural Convective Heat Transfer Upward Facing Heated Plate log Nu log Ra Figure 1 As for natural convection from rough plates, the working hypothesis here is that under identical conditions, the average natural convection from a rough (0 < ε L) plate is the same as the average natural convection from a smooth plate. The agreement of rough plate measurements with theory over the 180 range in Figure means that if natural convection from rough plates is different from smooth plates, then it is a scale factor independent of angle. 5

6 6. Turbulent Mixed Convection Jaffer[1] details the mixed convection interactions between the sides and rough face of the plate assuming a mixing function U mix. This section discovers that mixing function for horizontal and vertical plates and flows. In the case of forced flow perpendicular to the natural flow, the hypothesized mixture model from Section 3 is a L p -norm function of the natural and forced convection components: U F, U N p = ( U F p + U N p ) 1/p Because the sides are perpendicular to the rough face, the p values for each orientation must be considered together. The mixed convection data-sets were run with various assignments of p to orientations. Vertical and upward-facing natural convections being larger than downward-facing convection, it quickly became clear that p = for upward-facing h in Figure 7 and vertical h V in Figure 9. In Figure 5 for the downward-facing plate (h R ), p. The L 3 -norm is closer than L -norm, but the L 4 -norm curve is closest. h = k h V = k h R = k Nu F Nu F Nu F, Nu L, Nu V, Nu R R () (11) (1) 4 The measurements leading to equations (), (11), and (1) were performed with horizontal forced flow, perpendicular to the natural convective flows from the plate. The horizontal-vertical mixing of equation (11) might be due to vector addition of the forced and natural flows or just the L -norm of the two magnitudes. In the vector case a drop in convection would be expected when a weak forced flow opposed the fluid rising due to convection from a vertical plate. The apparatus was modified to perform this experiment, producing Figure 11, which shows that neither was the case! It is L 4 -norm below =7500 and L -norm above. Equation (13) is a blending function implementing the two norms with a sharp transition between them: B (h F, h N ) = { [ h 4 F + h F h 16h 16 ] 1/4 N N 1 16h 16 N + + hn} 4 (13) h16 F h = k B ( NuF A vertical plate with forced upward convection was also surprising. In Figure 13 the convections aid each other as the L -norm except around =8500 where they mix as the L 4 -norm. This could be explained by the reduced velocity shear creating less turbulent mixing around =8500. The aiding transition (14) is much gentler than the opposing transition (13): B (h F, h N ) =, Nu V ) { [ h 4 F + h F h N 1 4h N h F 4h 4 N + h4 F h = k B ( NuF, Nu V ) ] + h 4 N} 1/4 (14) The ynolds numbers of the transitions between the L -norm and L 4 -norm depend only on the relative strength of the natural and forced convective components h N and h F. Instead of ynolds numbers, the blend functions operate directly on h N and h F, simplifying their form. As detailed in Jaffer[1], t = 35 3 when the wind-tunnel is vertical. It also details the velocity correction for the upward-facing plate configuration. 6

7 7. Natural Convection from an Inclined Plate Following the approach of Fujii and Imura[], the Ra V argument to the vertical correlation (7) is scaled by cos θ in order to model the reduced vertical mode of convection on a tilted plate. Similarly, the Ra argument to the upward correlation (9) is scaled by sin θ. Although it has negligible effect on the correlation, the Ra R argument to the downward correlation (8) is also scaled by sin θ so that the combination of formulas (15) and (16) is continuous around θ = 0. When V = 0, if T > 0 and θ > 0 or if T < 0 and θ < 0 (downward convection), then: ( NuV (Ra V cos θ ) h = k max, Nu ) R (Ra R sin θ ) (15) R Otherwise (upward convection): ( NuV (Ra V cos θ ) h = k max, Nu (Ra ) sin θ ) L G. D. Raithby and K. G. T. Hollands writing in chapter 4 of Handbook of heat transfer [1] recommend ( NuV (Ra V cos θ ) h = k max, Nu (Ra sin θ ) L, Nu ) R(Ra R ) L R where Nu R uses an unscaled Ra R. Currently achievable measurement uncertainties are much larger than the difference expected between these two approaches. Given the many uncertainties of natural convection measurements and the coarse assumptions of the sides model equation developed in [1], a graph of natural convection h versus angle θ is a speculative venture. But points in Figure are surprisingly close to the curves predicted by formulas (15) and (16). Three Ra values are involved in generating each of the curves labeled Theory in Figure. Ra R and Ra are taken from the Rayleigh numbers during measurements at θ = +90 and θ = 90, respectively. Ra V is derived from the geometric mean of Ra R and Ra. Natural Convection vs Angle - 3mm Roughness (16) 5 Convective Surface Conductance, h, W/(m K) Theory T=11K Expected Uncertainty T=11K T=11K Theory T=3.8K Expected Uncertainty T=3.8K T=3.8K Angle θ in Degrees Figure 7

8 00 Opposing Mixed Convection From Inclined Plate θ=+84.5, 1mm Roughness, T=K +75 Mixed Mixed Exp. Uncertainty +90 Mixed 0 t =11x 3 = Figure 3 00 Aiding Mixed Convection From Inclined Plate θ=+84.5, 1mm Roughness, T=11K +75 Mixed +84 Mixed Exp. Uncertainty +90 Mixed 0 t =11x 3 = Figure 4 8

9 8. Mixed Convection from an Inclined Plate Combining the natural and mixed horizontal flow equations so that the mixed equations (), (11), and (1) result when θ is an integer multiple of 90, and the natural equations (15) and (16) result when V = 0, leads to formula (17) for mixing with horizontal flow: ( Nu F h = k max ( Nu F h = k max, Nu V (Ra V cos θ ), Nu V (Ra V cos θ ),, Nu F Nu F, Nu ) R (Ra R sin θ ) R 4, Nu (Ra ) sin θ ) L 0 θ θ 0 (17) Let 0 ψ 180 be the angle of the forced flow from the zenith. The blend function for the vertical convective mode (18) is the L -norm except for the vertical downward component of the forced flow which transitions from the L 4 -norm to the L -norm as the forced flow increases, and the upward component of the forced flow which is the L -norm except where the convections are close in value, which uses the L 4 -norm: M(h F, h N, ψ) = max(0, cos ψ) 4h N h F 4h 4 N + h4 F 16h 16 N min(0, cos ψ) 16h 16 N + h16 F { [h B(h F, h N, ψ) = F + h } (18) 1/4 N] h N h F M(h F, h N, ψ) ( ( NuF h = k max B ( h = k max B, Nu ) V (Ra V cos θ ), ψ, ( NuF, Nu ) V (Ra V cos θ ), ψ, Nu F Nu F, Nu ) R (Ra R sin θ ) R 4, Nu (Ra ) sin θ ) L 0 θ θ 0 (19) While equation (19) collapses to validated equations (), (11), (1), (13), and (14) when θ is an integer multiple of 90 and to (15) and (16) when V = 0, that does not prove that (19) is valid for mixed convection from an inclined plate. Figure 3 and Figure 4 show mixed convection measured at a plate angle of was chosen because it is in the range of orientations where convection is most sensitive to θ. The curves for θ = +90 and θ = +75 are shown for comparison. Figure 3 has forced flow opposing its natural convection, ψ = The obstruction of the plume by the wind-tunnel is made worse at low flow rates, resulting in lower than expected convection. Figure 4 has forced flow aiding its natural convection, ψ = Conclusions When mixed convection is turbulent, the formulas presented here provide a straightforward framework for predicting convection in terms of the natural and forced convective surface conductances. The natural and forced convective modes cooperate as the L -norm or L 4 -norm depending on orientation and their relative strengths. In the absence of laminar forced flow, mixed convection is never less than the least of the natural and the forced components and is bounded by their L -norm and L 4 -norm. Horizontal and vertical convective regimes compete when θ is not an integer multiple of 90. The resulting convection is the maximum (which is the L -norm) of each natural-forced mixture. The winner is not always the regime with the greatest natural convection because the downward regime mixes as L 4 -norm while others mix as L -norm. Acknowledgments Thanks to John Cox and Doug Ruuska for machining the plates. Thanks to Martin Jaffer and Roberta Jaffer for their assistance and problem-solving suggestions. The forced flow is parallel to the plane of the test surface. 9

10 . ferences [1] A. Jaffer. Convection measurement apparatus and methodology, 018. [Online; accessed 30-August- 018]. [] Tetsu Fujii and Hideaki Imura. Natural-convection heat transfer from a plate with arbitrary inclination. International Journal of Heat and Mass Transfer, 15(4): , 197. [3] A. Jaffer. Skin-friction and forced convection from an isothermal rough plate, 017. [Online; accessed 0-December-017]. [4] Stuart W Churchill and Humbert HS Chu. Correlating equations for laminar and turbulent free convection from a vertical plate. International journal of heat and mass transfer, 18(11): , [5] X. A. Wang. An experimental study of mixed, forced, and free convection heat transfer from a horizontal flat plate to air. Journal of Heat Transfer, 4(1): , 198. [6] Blake Lance. Experimental Validation Data for CFD of Steady and Transient Mixed Convection on a Vertical Flat Plate. Utah State University Graduate Theses and Dissertations, 015. [7] S. W. Churchill and R. Usagi. A general expression for the correlation of rates of transfer and other phenomena. AIChE Journal, 18(6): , 197. [8] HT Lin, WS Yu, and CC Chen. Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates. Wärme-und Stoffübertragung, 5(6): , [9] T Schulenberg. Natural convection heat transfer below downward facing horizontal surfaces. International journal of heat and mass transfer, 8(): , [] Mark S. Owen, editor. Handbook, ASHRAE Fundamentals (SI). American society of heating, refrigerating and air-conditioning engineers, 009. [11] JR Lloyd and WR Moran. Natural convection adjacent to horizontal surface of various planforms. Journal of Heat Transfer, 96(4): , [1] W.M. Rohsenow, J.P. Hartnett, and Y.I. Cho. Handbook of heat transfer. McGraw-Hill handbooks. McGraw-Hill, Appendix: Measurements of Mixed Convection in Horizontal and Vertical Orientations

11 00 Mixed Convection from Downward Facing Plate, 3mm Roughness, T=K L 3 Mixed L 4 Mixed Exp. Uncertainty 0 t =11x 3 = Figure 5 00 Mixed Convection from Downward Facing Plate, 1mm Roughness, T=11K L 4 Mixed Exp. Uncertainty 0 t =11x 3 = Figure 6 11

12 00 Mixed Convection from Upward Facing Plate, 3mm Roughness, T=K L Mixed Exp. Uncertainty 0 t =11x 3 = Figure 7 00 Mixed Convection from Upward Facing Plate, 1mm Roughness, T=.3K L Mixed Exp. Uncertainty 0 t =11x 3 = Figure 8 1

13 00 Mixed Convection from Vertical Plate, 3mm Roughness, T=K L Mixed Exp. Uncertainty 0 t =11x 3 = Figure 9 00 Mixed Convection from Vertical Plate, 1mm Roughness, T=11K L Mixed Exp. Uncertainty 0 t =11x 3 = Figure 13

14 00 Opposing Mixed Convection Vertical Plate, 3mm Roughness, T=K L Mixed Mixed Exp. Uncertainty L 4 Mixed 0 t =35x 3 = Figure Opposing Mixed Convection Vertical Plate, 1mm Roughness, T=.5K L Mixed Mixed Exp. Uncertainty L 4 Mixed 0 t =35x Figure 1 14

15 00 Aiding Mixed Convection Vertical Plate, 3mm Roughness, T=K L Mixed Mixed Exp. Uncertainty L 4 Mixed 0 t =35x 3 = Figure Aiding Mixed Convection Vertical Plate, 1mm Roughness, T=.5K L Mixed Mixed Exp. Uncertainty L 4 Mixed 0 t =35x Figure 14 15