http://people.csail.mit.edu/jaffer/SimRoof/Convection | |
Convection From a Rectangular Plate |
New research into convection yields practical formulas sufficient to calculate roof convection from roof parameters and meteorological data.
Through the use of effective-characteristic-length, the model is extended to natural convection for all orientations of a plate, not just those having a horizontal edge, off-axis forced (but in-plane) flows, and forced convection averaged over all in-plane directions of flow.
A related model for convection from symmetrical peaked roofs is also developed.
The most important thermal processes for a building's roof are absorption of solar radiation (insolation), convection, and thermal radiation.
Currently, established theory is not sufficient to compute convection from roof parameters and meteorological data. This work rectifies that situation.
Says Wikipedia:
Convective heat transfer, often referred to simply as convection, is the transfer of heat from one place to another by the movement of fluids.
To model the thermal dynamics of roofs we are interested in the convective flow of heat to or from the top surface of the roof. For a flat plate at a uniform temperature T, the rate of heat flow (in Watts) due to convection from one side of the plate is:
h ⋅ A ⋅ ΔT h W/(K⋅m^{2})_{ } convective surface conductance A m^{2}_{ } area of one side of plate ΔT K T − T_{env} T K plate temperature T_{env} K fluid temperature
The complication is that the value of h depends on temperatures, fluid-velocity, and the area, shape, orientation, and roughness of the plate surface. A value of h for a 1 m by 1 m plate will usually be larger (and never smaller) than h for a 2 m by 2 m plate under otherwise identical conditions. The larger plate will transfer more heat because it has four times the area of the smaller plate, but not more than 4 times the heat.
Convection is natural (free), forced, or mixed. In natural convection, fluid motion is driven by the difference in temperature between the plate and the fluid. In forced convection, fluid motion is driven by an external force such as wind. Mixed convection is a mixture of natural and forced convection occurring at low fluid speeds.
(Natural) upward convection is produced by an upward-facing plate which is warmer than the fluid, or a downward-facing plate which is colder than the fluid. (Natural) downward convection is produced by an upward-facing plate which is cooler than the fluid, or a downward-facing plate which is warmer than the fluid.
A correlation (approximate equation relating dimensionless quantities) can model an isothermal surface or a constant heat flux through the surface. The correlations for these two regimes usually differ only in coefficients and additive constants. This article addresses convection from isothermal plates.
A forced convection formula can be for local heat flow or for heat flow averaged across the surface. Section "Laminar-Turbulent Progression" gives the derivation for average heat flow from the local heat flow which appears in several texts. While it produces the customary averaged formulas for purely laminar or purely turbulent flow, the formula it produces for transitional flows requires a Reynolds-number threshold from measurement of the specific configuration under investigation, which limits its predictive ability.
Convection formulas can be derived from theory, numerical simulation, or experiment (empirical). The accepted formulas for natural convection are empirical.
At the plate surface, the fluid velocity is near zero. At some distance from the plate surface the fluid velocity approaches the bulk fluid velocity. In between is the boundary layer. Flow in the boundary layer is laminar or turbulent. The h values for turbulent regions of the plate (boundary layer) are larger than h values for laminar regions. Natural upward convection (from a horizontal plate) transitions from laminar to turbulent at Rayleigh numbers (Ra) near 10^{7} (Clear et al[82]), placing it between the ranges for formulas T9.7 and T9.8 (which intersect at 4.7×10^{6}). Convection from a vertical plate has a transition around Gr=Ra/Pr=10^{9}.
Natural convection is strongly affected by the inclination of a plate. The natural convective surface conductivity from a vertical plate is between the conductivities of upward facing and downward facing level plates of the same size. Upward convection has the largest heat flow. Forced convection is insensitive to inclination.
Forced convection is greater for rough surfaces than for smooth ones. Natural convection is insensitive to roughness whose mean height is much less than the dimensions of the plate.
symbol | units | description | ||
---|---|---|---|---|
T_{F} | K | Fluid (Air) Temperature | ||
T_{S} | K | Plate Surface Temperature | ||
T | K | Mean Temperature (T_{F}+T_{S})/2 | ||
P | Pa | Fluid (Atmospheric) Pressure | ||
V | m/s | Fluid Velocity | ||
R_{air} | J/(kg⋅K) | Gas Constant for Dry Air = 287.058 J/(kg⋅K) | ||
Φ | Pa/Pa | Relative Humidity |
From these parameters are derived density ρ, thermal conductivity k, specific heat c_{p}, and viscosity μ. For dry air, Kadoya, Matsunaga, and Nagashima[93] seems the authoritative source for viscosity and thermal conductivity.
symbol | forced | natural | formula | units | description | ||
---|---|---|---|---|---|---|---|
c_{p} | ✓ | ✓ | [see text] | J/(kg⋅K) | specific heat at constant pressure | ||
ρ | ✓ | ✓ | P/(R_{air} ⋅ T) | kg/m^{3} | density | ||
k | ✓ | ✓ | [see text] | W/(m⋅K) | thermal conductivity | ||
μ | ✓ | ✓ | [see text] | Pa⋅s | dynamic viscosity | ||
ν | ✓ | ✓ | μ/ρ | m^{2}/s | kinematic viscosity | ||
α | ✓ | k/(ρ⋅c_{p}) | m^{2}/s | thermal diffusivity | |||
β | ✓ | 1/T | K^{−1} | coefficient of thermal expansion |
The moist air values of these properties are computed by combining the values for dry air and water vapor in proportion to their presence in the moist air mixture, in some cases with correction factors. What is true of all the mixture formulas is that at 0% relative humidity the mixture values are identical with the dry air values and at 100% relative humidity at 100°C the mixture values are identical with the steam (water vapor) values.
These mixture formulas come from Tsilingiris[91] and Morvay and Gvozdenac[92] (not ASHRAE). Both sources contain errors; the obvious errors don't occur in the corresponding quantities. Wexler[94] seems the authoritative source for water-vapor (partial) pressure versus temperature.
At 100% relative humidity, the pressure and mass fraction of vapor increase with temperature, but remain less than 10% at 45°C. So humidity will not be a major influence on air's properties at outdoor temperatures.
Model moist air as a mixture of ideal gases. The relative-humidity, Φ, is the ratio of the partial-pressure of water vapor to P_{sat}. P_{sat} is the partial pressure of saturated water vapor at temperature T_{F} (Kelvins):
P_{sat} = 610.78 ⋅ 10^{7.5 (TF−273.15)/(TF−35.85)} |
Over the temperature range of interest, this formula for P_{sat} is well within 1% of the values returned by formula 16b in Wexler[94]:
P_{sat}^{ }= | exp( −0.63536311×10^{4}/T_{F} +0.3404926034×10^{2} −0.19509874×10^{−1}⋅T_{F} +0.12811805×10^{−4}⋅T_{F}^{2} ) |
Density ρ, thermal conductivity k, specific heat c_{p}, and viscosity μ are computed at temperature T, the average of T_{F} and T_{S}. But P_{sat} must be evaluated at the bulk fluid temperature T_{F} because the amount of water-vapor in the fluid doesn't change when heated to intermediate temperature T. If T_{S} is colder than T_{F}, then condensation may occur.
Simulation of roofs can side-step the issue by constraining the roof temperature to not drop below the ambient dew-point temperature. Because of water's high latent heat, this treatment should not result in large errors.
For an ideal gas, density ρ is:
ρ=P M/(R T) | R=8.314 J/(kg⋅mol) |
Model moist air as a mixture of dry air and water vapor:
ρ = |
(P − Φ P_{sat}) M_{a}
+
Φ P_{sat} M_{v}
R T |
R = | 8.314 J/(kg⋅mol) |
M_{a} = | 0.028964 kg/mol |
M_{v} = | 0.018016 kg/mol |
The specific heat at constant pressure c_{p} for dry air and water vapor each vary little over our temperature range. But the mixture at a given relative humidity is sensitive to temperature. c_{pa} and c_{pv} are the specific heat of air and water-vapor, respectively. These formulas from Tsilingiris[91] take temperature t in degrees Celsius. x_{v}(T) takes temperature in Kelvins.
c_{pa}(t)^{ }= | 1034 −.2849⋅t +.7817×10^{−3}⋅t^{2} −.4971×10^{−6}⋅t^{3} +.1077×10^{−9}⋅t^{4} |
c_{pv}(t)^{ }= | 1869−2.578×10^{−1}⋅t +1.941×10^{−2}⋅t^{2} |
c_{p}= |
c_{pa}(t) (1−x_{v}) M_{a}
+
c_{pv}(t) x_{v} M_{v}
(1−x_{v}) M_{a} + x_{v} M_{v} | x_{v}(T) = Φ P_{sat}(T)/P |
Both [91] and [92] give formulas for the viscosity of dry air. [91] matches the data from [93] better; the values from [92] are about 1% lower; but result in (lower) Pr values closer to those from many sources. The formula from [91] is:
μ_{a}^{ }= | −9.8601×10^{−7} +9.080125×10^{−8}⋅T −1.17635575×10^{−10}⋅T^{2} +1.2349703×10^{−13}⋅T^{3} −5.7971299×10^{−17}⋅T^{4} |
The formula from [92] is:
μ_{a}^{ }= | −0.40401×10^{−6} +0.074582×10^{−6}⋅T −5.7171×10^{−11}⋅T^{2} +2.9928×10^{−14}⋅T^{3} −6.2524×10^{−18}⋅T^{4} |
A line for the viscosity at half air pressure (P=50 kPa) overlays [93]Dry air; so air pressure variations don't significantly affect viscosity at roof conditions.
The viscosity of saturated water vapor is less studied. Tsilingiris[91] gives a formula which returns a viscosity value for 100 C around half of that shown in his own graph of viscosity! Morvay and Gvozdenac[92] give a formula for the viscosity of water vapor which matches Tsilingiris' graph well. With γ=647.27/T:
μ_{v} = | γ^{−1/2}
018158.3 + 017762.4 γ + 010528.7 γ^{2} −003674.4 γ^{3} |
Over the temperature range of interest, μ_{v} is hardly different from:
μ_{v} = 9.2173×10^{−6} Pa⋅s + (T − 273.15 K) ⋅ 25.713×10^{−9} Pa⋅s/K |
Morvay and Gvozdenac introduce a parameter they call absolute humidity, the ratio of masses of water vapor and dry air:
χ= |
M_{v} P_{sat}
M_{a} (P−P_{sat}) |
The dynamic viscosity of the moist air mixture is:
μ = |
μ_{a}
1 + Φ_{AV} ⋅ χ |
+ |
μ_{v}
1 + Φ_{VA} / χ |
The terms Φ_{AV} and Φ_{VA} are complicated expressions having values ranging from 1.064 to 1.073 and .93 to .923 respectively over the roof range of interest (-25°C to 45°C).
Kadoya, Matsunaga, and Nagashima[93] seems an authoritative source for the thermal-conductivity of dry air:
ρ_{r} = | P / (287.058 ⋅ 314.3 ⋅ T) | ||||||||
T_{r}^{ }= | T / 132.5 | ||||||||
k_{a}^{ }= |
0.0259778 ⋅ (
0.239503⋅T_{r}
+0.00649768⋅T_{r}^{1/2}
+1.0
−1.92615⋅T_{r}^{−1}
+2.00383⋅T_{r}^{−2}
−1.07553⋅T_{r}^{−3}
+0.229414⋅T_{r}^{−4}
+0.402287⋅ρ_{r} +0.356603⋅ρ_{r}^{2} −0.163159⋅ρ_{r}^{3} +0.138059⋅ρ_{r}^{4} −0.0201725⋅ρ_{r}^{5} ) |
A line for the k at half air pressure (P=50 kPa) overlays [93]Dry air on the graph; so air pressure variations don't significantly affect k at roof conditions. Over the roof range of interest this is hardly different from:
k_{a} = | 0.02241 W/(m⋅K) + (T−250 K) ⋅ 76.46×10^{−6} W/(m⋅K^{2}) |
Tsilingiris[91] gives a formula for the thermal conductivity of water vapor:
k_{v}^{ }= | 1.761758242×10^{1} +5.558941059×10^{2} T +1.663336663×10^{4} T^{2} |
Morvay and Gvozdenac[92] also give a formula for the thermal conductivity of water vapor:
k_{v}^{ }= | 1.74822×10^{−2} +7.69127×10^{−5} t −3.23464×10^{−7} t^{2} +2.59524×10^{−9} t^{3} −3.17650×10^{−12} t^{4} |
They diverge mostly at the dry end of the curve, where k_{v} has insignificant effect on the moist air thermal-conductivity (k_{m}). Over the roof range of interest the latter curve is hardly different from:
k_{v}^{ }= | 0.0174822 W/(m⋅K) + (T−273.15 K) ⋅ 69.4587305×10^{−06} W/(m⋅K^{2}) |
Morvay and Gvozdenac[92]'s formula for moist air thermal conductivity is like the formula for viscosity, but with more complicated expressions for Φ_{AV} and Φ_{VA}. Tsilingiris[91]'s formula uses the same Φ_{AV} and Φ_{AV} as the viscosity formula:
k = |
k_{a}
1 + Φ_{AV} ⋅ χ |
+ |
k_{v}
1 + Φ_{VA} / χ |
The simplified 99% RH curve, which is nearly identical to the 99% RH curve, uses the linear k_{a} and k_{v} models and substitutes 1.07 and 0.93 for Φ_{AV} and Φ_{VA}.
For an ideal gas with pressure held constant, the volumetric thermal expansivity (i.e. relative change in volume due to temperature change) is the inverse of temperature. For natural convection from a horizontal plate it is:
β=1/T
For natural convection from a vertical plate (T9.2, Nu') it is:
β=1/T_{F}
For a flat rectangular plate, L_{c} is the length of the side parallel to the direction of flow.
For a flat plate, L* is the area (of one side) of the plate divided by its perimeter. Here are formulas for L* of four shapes. l is the length; w is the width; D is the diameter.
rectangle square infinite strip circular disk L*(w, l) = w⋅l
2 (w+l)L*(w, w) = w
4L*(w, ∞) = w
2π D^{2} / 4
π D= D
4
It is interesting that a square and the maximal circle inscribed within it have the same L*.
The plot to the right shows how the length of a side (sqrt(A/r)) and L* (area/perimeter) vary with aspect-ratio for a rectangular plate with an area of 1 m^{2}. As the aspect-ratio r grows, L* tends to sqrt(A/r)/2.
The hydraulic-diameter, used as the characteristic length D_{H} in ducts, is related to L*, being 4 times the cross-section area divided by the cross-section perimeter.
Below are formulas for dimensionless quantities governing convection along with ranges for air under the conditions:
Ranges for vertical plates, where L_{c} is the height, are marked with an apostrophe ('). Ranges for horizontal upward-facing plates, where L* is the ratio of area to perimeter, are marked with an asterisk (*). Ranges for horizontal downward-facing plates, where L_{R} is half the length of the shorter side, are marked with a subscript R. The Prandtl number is insensitive to L, depending only on fluid (air) properties. The L_{c} for Reynolds and Nusselt numbers in forced convection is the length of the plate in the direction of flow.
Both Gr and Ra have factors of |ΔT|; they will be zero when ΔT is zero. In order to get an idea of the dynamic range of Gr and Ra, the minimum ΔT used for range-of-interest is 1 K rather than 0 K. Re will be zero when the wind-speed is zero. In order to get an idea of the dynamic range of Re, the minimum V used for range-of-interest is 1 m/s rather than 0 m/s.
In convection calculations, Nu is computed from the other dimensionless quantities; then solved for h, the surface conductance (having units W/(m^{2}⋅K)).
symbol formula expansion roof range of interest description Pr ν/α c_{p}⋅μ/k 0.697≤Pr≤0.729 Prandtl number Gr β⋅|ΔT|⋅g⋅L_{c}^{3}
ν^{2}|ΔT|⋅g⋅L_{c}^{3}⋅ρ^{2}
T⋅μ^{2}Grashof number for natural convection Ra Gr⋅Pr |ΔT|⋅g⋅L_{c}^{3}⋅ρ^{2}⋅c_{p}
T⋅μ⋅k9.34×10^{6}≤Ra*≤1.37×10^{13}
5.97×10^{8}≤Ra'≤8.74×10^{14}
7.47×10^{7}≤Ra_{R}≤1.09×10^{14}Rayleigh number for natural convection Re V⋅L_{c}/ν V⋅L_{c}⋅ρ/μ 4.40×10^{4}≤Re≤7.15×10^{7} Reynolds number for forced convection Nu RST
T8.9, T8.11(forced rough)
(forced smooth)2.00×10^{2}≤Nu_{r}≤3.28×10^{5}
1.70×10^{2}≤Nu_{s}≤6.36×10^{4}Nusselt number Nu*
Nu'
Nu_{R}(natural) 8.70≤Nu*≤3.45×10^{3}
1.05×10^{2}≤Nu'≤1.10×10^{4}
21.9≤Nu_{R}≤4.85×10^{3}h Nu⋅k/L_{c} (forced rough)
(forced smooth)5.87≤h_{r}≤4.78×10^{2}
2.90≤h_{s}≤1.59×10^{2}convective surface conductance W/(m^{2}⋅K) Nu⋅k/L*
Nu⋅k/L'
Nu⋅k/L_{R}(natural) 0.189≤h*≤7.62
2.46≤h'≤5.83
0.259≤h_{R}≤4.98
It is worth noting that a larger h value doesn't necessarily correspond to a larger Nu value because Nu gets divided by different L values.
From Thermodynamic Basis for Natural Convection from an Isothermal Plate the formulas for upward facing, vertical, and downward facing plates are:
Nu*(Ra*) = ( 0.671 + 0.370 Ra*^{1/6} )^{2} | (Nu*) | ||
Nu'(Ra') = 0.682 ( 1 + 0.469 [Ra' / Ξ(Pr)]^{1/6} )^{2} | (Nu') | ||
Nu_{R}(Ra_{R}) = 0.682 + 0.550 [Ra_{R} / Ξ(Pr)]^{1/5} | (Nu_{R}) | ||
Ξ(Pr) = ‖ 1, 0.5 / Pr ‖_{1/√3} | |||
‖ x, y ‖_{p} = ( x^{p} + y^{p} )^{1/p} |
These formulas are combined to yield natural convection from a flat plate inclined at angle θ:
h = k max( Nu'(Ra' |cos θ|) / L', Nu*(Ra* |sin θ|) / L* ), | if ΔT sinθ < 0; | ||
h = k max( Nu'(Ra' |cos θ|) / L', Nu_{R}(Ra_{R} |sin θ|) / L_{R} ), | otherwise. |
The upward characteristic-length L* is the area to perimeter ratio. The vertical L' is the length of the rectangle's vertical edge. The downward L_{R} is half of the rectangle's shortest edge.
The common formulas for forced convection from a smooth flat plate are due to Blasius' 1908 mathematical analysis.
Recalling Dimensionless Quantities, we introduce local versions of Nu and Re:
symbol formula expansion description Nu_{x} h_{x}⋅x/k local Nusselt number Nu h⋅L_{c}/k average Pr ν/α c_{p}⋅μ/k fluid Prandtl number Re_{x} V⋅x/ν V⋅x⋅ρ/μ local Reynolds number Re V⋅L_{c}/ν V⋅L_{c}⋅ρ/μ average
The table below excerpts those sections of Table 8 from chapter 4 of 2009 ASHRAE Fundamentals Handbook (SI) which deal with convection from flat plates. The plate orientation is not specified. L is the length of the plate [assumed in the direction of fluid flow]. No citations are given for these 5 formulas. Re_{c} does not appear in the text.
Chapter 4 of 2009 ASHRAE Fundamentals[70] states:
For a flat plate with a smooth leading edge, the turbulent boundary layer starts at distance x_{c} from the leading edge where the Reynolds number Re=V⋅x_{c}/ν is in the range 300000 to 500000 (in some cases, higher). In a plate with a blunt front edge or other irregularities, it can start at much smaller Reynolds numbers.
At 15°C, the kinetic viscosity of air is ν=15.7 ×10^{−6} m^{2}/s. At V=10 m/s and Re=4×10^{5}, x_{c}=Re⋅ν/V=.63 m. At V=2.5 m/s x_{c}=2.5 m. The higher wind speeds will induce a mixture of laminar and turbulent convection on smooth panels larger than 1 m.
Local Re changes with distance (x) from the leading edge of the plate. Casting T8.8 and T8.10 in terms of h and x:
h_{L} = 0.332 k⋅(x⋅V/ν)^{1/2} Pr^{1/3}
xlaminar x⋅V/ν<5×10^{5} h_{T} = 0.0296 k⋅(x⋅V/ν)^{4/5} Pr^{1/3}
xturbulent x⋅V/ν>5×10^{5}
The boundary position, x_{c}, is the minimum of L_{c} and 5×10^{5} ν/V.
x_{c} = min(L_{c}, 5×10^{5} ν/V)
Integrate along the direction of flow from leading edge to L_{c}; then divide by L_{c} to obtain the average value of h for the plate.
h = 1
L_{c}( x_{c}
∫
0h_{L}(x) dx + L_{c}
∫
x_{c}h_{T}(x) dx ) = k ⋅ Pr^{1/3}
L_{c}( x_{c}
∫
0.332 (V/ν)^{1/2} x^{−1/2} dx + L_{c}
∫
x_{c}.0296 (V/ν)^{4/5} x^{−1/5} dx ) = k ⋅ Pr^{1/3}
L_{c}( 0.664 (V/ν)^{1/2} ⋅ x_{c}^{1/2} + 0.037 (V/ν)^{4/5} ⋅ (L_{c}^{4/5}−x_{c}^{4/5}) ) (FH)
FH can be cast as a dimensionless formula by reversing the previous substitutions:
Nu = 0.664 Re_{c}^{1/2} Pr^{1/3} + 0.037 (Re^{4/5}−Re_{c}^{4/5}) Pr^{1/3} (FC)
When Re_{c}=5×10^{5}, T8.12 results. When the flow is entirely laminar, x_{c}=L_{c}, and T8.9 results. If the impinging flow is turbulent, then the boundary layer is all turbulent, x_{c}=0, and T8.11 results.
Skin-Friction and Forced Convection from an Isothermal Rough Plate gives a formula for forced convection from isothermal plates having self-similar or periodic roughness in terms of root-mean-squared (RMS) height-of-roughness ε > 0:
Nu = | Re Pr^{1/3}
6 ln^{2}(L/ε) |
L
ε |
≫ 1 | (RT) |
Dimensional analysis indicates that for periodic isotropic roughness with period L_{S} ≪ L, the least upper bounds for laminar and smooth-turbulent flow along the plate will be, respectively:
Re_{L} = 0.664^{2} | L L_{S}
2 ε^{2} |
Re_{S} = 0.036^{5} | L L_{S}^{4}
2^{4} ε^{5} |
When L_{S}/ε < 388, the flow should transition directly from laminar to rough-turbulence at Re_{L}. When L_{S}/ε > 388, the flow should transition from laminar to smooth-turbulence at critical Reynolds number Re_{L}, and to rough-turbulence at Re_{S}.
Not having a definitive laminar-turbulent threshold for the smooth plate substantially reduces predictive value. Every plate is a rough surface at some scale. If the profile roughness is weakly isotropic and has an effective roughness repeat length L_{S} ≪ L, then min(Re_{S},Re_{L}) can take the place of Re_{c}. L/L_{S}=15 and L/ε=5800 results in a critical Re_{L}=5×10^{5}.
Recasting Re_{L} and Re_{S} as functions of Re adapts formula (FC) to all three flow regimes:
Re_{L} = min(Re, 0.664^{2} | L L_{S}
2 ε^{2} |
) | Re_{S} = min(Re, 0.036^{5} | L L_{S}^{4}
2^{4} ε^{5} |
) |
When L_{S}/ε < 388:
Nu
Pr^{1/3} |
= 0.664 Re_{L}^{1/2} + | Re−Re_{L}
6 ln^{2}(L/ε) |
When L_{S}/ε > 388:
Nu
Pr^{1/3} |
= 0.664 Re_{L}^{1/2} + 0.037 (Re_{S}^{4/5}−Re_{L}^{4/5}) + | Re−Re_{S}
6 ln^{2}(L/ε) |
(FCC) |
Over flat plates, forced fluid flow is assumed to have straight, parallel streamlines. Thus convex plates with non-uniform characteristic-length can be analyzed by summing the results of the plate partitioned into strips which are parallel to the flow.
Generalizing the analysis at the end of Laminar Convection finds that the ratio between h_{d} (diagonal) and h_{p} (axis-aligned) is 2^{(1+E)/2}/(1+E) where E is the exponent of Re. For forced turbulent convection (E=4/5), sensitivity to rotation of the square is less than 4%.
Most roofs aren't square. Consider a S by r⋅S rectangular plate with flow in the plane of the plate at angle φ from a length S side. For 0≤φ<π/2 and r≥tan φ, the strips which run between opposite r⋅S sides have length K=S/cos φ. The base of the parallelogram is P=r⋅S−K⋅sin φ. The width of the parallelogram perpendicular to the flow is P⋅cos φ.
To find the average surface conductance for the whole plate, sum the product of the area of each strip and its average h (using its length as L), then divide by the total area r⋅S^{2}:
( | h(K) K P cos φ + 2 | K cos φ sin φ ∫ 0 |
h( | w cos φ sin φ |
) | w cos φ sin φ |
dw | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | ( | K^{E} P cos φ + | 2 (cos φ sin φ)^{E} |
K cos φ sin φ ∫ 0 |
w^{E} | dw | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | ( | K^{E} P cos φ + 2 | K^{1+E} 1+E |
cos φ sin φ | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | (S/cos φ)^{E}
S/cos φ |
( | 1 + | 1−E 1+E |
⋅ | tan φ r |
) |
= .037 k Pr^{1/3} (V/ν)^{4/5} | (S/cos φ)^{−1/5} | ( | 1 + | tan φ 9 r |
) |
When 0<r≤tan φ (and 0≤φ<π/2), the strips which run between opposite S sides have length K=r⋅S/sin φ. The base of the parallelogram is P=S−K⋅cos φ. The width of the parallelogram perpendicular to the flow is P⋅sin φ. The average surface conductance for the whole plate is:
( | h(K) K P sin φ + 2 | K cos φ sin φ ∫ 0 |
h( | w cos φ sin φ |
) | w cos φ sin φ |
dw | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | ( | K^{E} P sin φ + | 2 (cos φ sin φ)^{E} |
K cos φ sin φ ∫ 0 |
w^{E} | dw | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | ( | K^{E} P sin φ + 2 | K^{1+E} 1+E |
cos φ sin φ | ) / (r S^{2}) |
= .037 k Pr^{1/3} (V/ν)^{E} | (r S/sin φ)^{E}
r S/sin φ |
( | 1 + | 1−E 1+E |
⋅ | r tan φ |
) |
= .037 k Pr^{1/3} (V/ν)^{4/5} | (r S/sin φ)^{−1/5} | ( | 1 + | r 9 tan φ |
) |
For every rectangle with r>1, there is an identical (but rotated 90°) rectangle with S'=r⋅S and r'=1/r. Taking the geometric mean of the side lengths r^{1/2}⋅S as the characteristic length L allows these expressions to be written as the standard forced convection surface conductances (derived from T8.11 and T8.9) times rotation factors F_{tur}(r, φ) and F_{lam}(r, φ):
h = .037 k Pr^{1/3} (V/ν)^{4/5}
L^{1/5}F_{tur}(r, φ) HTRR
F_{tur}(r, φ) = |cos φ|^{1/5} r^{1/10} ( 1 + |tan φ|
9 r) r>|tan φ| FTRR ( |sin φ|^{1/5} / r^{1/10} ) ( 1 + r
9 |tan φ|) 0<r≤|tan φ|
h = .664 k Pr^{1/3} (V/ν)^{1/2}
L^{1/2}F_{lam}(r, φ) HLRR
F_{lam}(r, φ) = |cos φ|^{1/2} r^{1/4} ( 1 + |tan φ|
3 r) r>|tan φ| FLRR ( |sin φ|^{1/2} / r^{1/4} ) ( 1 + r
3 |tan φ|) 0<r≤|tan φ|
To the right are shown the surface conductances for smooth (equal area) flat plates .35 m by .35 m, .5 m by .25 m, and .7 m by .175 m versus azimuth angle of the forced air flow (3.4 m/s at 21°C). At an azimuth of 0° the flow is parallel to a shortest side.
Both turbulent and laminar conductances are shown. The dips in the turbulent h will probably not be as sharp as shown due to the directional fluctuations in a turbulent flow.
The corresponding formulas are T8.11 and T8.9 times F_{tur}(r, φ) and F_{lam}(r, φ) (with L = r^{1/2}⋅S):
Nu = .037 Pr^{1/3} Re^{4/5} F_{tur}(r, φ) CTRR Nu = .664 Pr^{1/3} Re^{1/2} F_{lam}(r, φ) CLRR
This is a mathematical derivation; I know of no in-plane off-axis forced-convection experiments.
That F_{tur} and F_{lam} are independent of Re and L is a useful result. F_{tur} and F_{lam} will be used to find h averaged over all azimuth angles in the next section.
The shape factor for the rough-surface formula RS is more difficult to compute because it doesn't have a simple power dependency on L.
For r≥|tan φ|:
K = S / |cos φ| |
P = r S−K |sin φ| = S (r − |tan φ|) |
The average surface conductance for the whole plate is:
h_{av} = | ( | h(K) K P |cos φ| + 2 | K |cos φ sin φ| ∫ 0 |
h( | w |cos φ sin φ| |
) | w |cos φ sin φ| |
dw | ) / (r S^{2}) |
= | h(K) (1 − |tan φ| / r) + | 2 r S^{2} |
S |sin φ| ∫ 0 |
h( | w |cos φ sin φ| |
) | w |cos φ sin φ| |
dw |
When 0<r≤|tan φ|:
K = r S / |sin φ| |
P = S−K |cos φ| = S (1 − r / |tan φ|) |
The average surface conductance for the whole plate is:
h_{av} = | ( | h(K) K P |sin φ| + 2 | K |cos φ sin φ| ∫ 0 |
h( | w |cos φ sin φ| |
) | w |cos φ sin φ| |
dw | ) / (r S^{2}) |
= | h(K) (1 − r / |tan φ|) + | 2 r S^{2} |
r S |cos φ| ∫ 0 |
h( | w |cos φ sin φ| |
) | w |cos φ sin φ| |
dw |
The solid lines in the graph to the right show the rotation-factors for 1 m^{2} rectangular plates with ε=10^{−4} m and aspect-ratios of 1:1 (black), 2:1 (blue), and 4:1 (red).
Clear et al[82] give characteristic-length values of L=28.3 m and L*=10.2 m. The roof they measured was neither round nor rectangular, yet they find that meteorological wind direction doesn't correlate with turbulent convection from that flat horizontal surface:
None of the data showed any correlation with wind direction, so this was not included in any of the fits.
This could either be because surface condutivities of different wind directions on their roof are close in value; or because wind direction is so variable within their 15-minute sampling intervals that the variations mostly average out. The only information Clear et al give about the roof dimensions is area (2940 m^{2}), perimeter (287 m), and mean distance of the perimeter from the measurement point (28.3 m which they use for the local forced L). Because they are working to match the measurements from an instrument cluster on a roof, they calculated local surface conductances. For application, average surface conductances are more useful.
A rectangle (their roof wasn't) with that area and perimeter would have an aspect-ratio of 4.8. The ratio of surface conductivities for a rectangle with aspect-ratio r along its two axis is:
k ⋅ C ⋅ (V⋅S/ν)^{E} Pr^{1/3} / S
k ⋅ C ⋅ (V⋅r⋅S/ν)^{E} Pr^{1/3} / (r⋅S)= r_{ }^{1-E}
In the turbulent case, an aspect-ratio of 2 leads to a surface conductance ratio of 1.15. That probably (and larger ratios certainly) would have been detected if wind directions were stable; so the independence of wind direction is probably due to its short-term variability.
The Forced Off-Axis Convection section developed formulas F_{tur} and F_{lam} for turbulent and laminar rotation factors which are functions of aspect-ratio and azimuth angle in the plane of a rectangular plate using the geometric mean of the sides as characteristic length. To the right is a graph of those rotation factors (numerically) averaged over 360° of flow direction versus aspect-ratio. The laminar curve F_{lam} is solid black; turbulent F_{tur} is dashed black. The red curve is an approximation to F_{lam} as a function of area A and perimeter P of the rectangle:
0.615 + P
8.7 A^{1/2}= 0.615 + A^{1/2}
8.7 L*
Expressing the shape factor in terms of general shape metrics may make extension to other convex polygons possible. A regular polygon approaches the circle as its number of sides increases. The shape factors for a circular disk under laminar and turbulent flow are shown as solid and dashed blue circles in the figure. The turbulent disk shape factor is 4% higher than its respective square plate having the same area; the laminar disk is 10% higher.
Instead of multiplying h by a shape factor, Clear et al adjust the characteristic length to account for the effect of averaging over wind direction, calling it the effective length. The chart to the right shows factors which, when multiplied by the geometric mean of the sides (A^{1/2}) produces the effective-length L_{eff}. The effective-length shape factors are simply F_{tur}^{−5} and F_{lam}^{−2}
The spread between the shape factors grows from 3% at an aspect-ratio of 1:1 to 6% at an aspect-ratio of 10:1. Formulas with Re exponents between 1/2 and 4/5 will be contained in this band. Thus a single L_{eff} (shown in red), which stays within those bounds will approximate Re exponents between 1/2 and 4/5 well.
L_{eff} = A^{1/2} / ( 0.615 + P
8.7 A^{1/2})^{1/2} LEAA
Effective-length greatly simplifies treatment of uniformly distributed flow direction. Can it do the same for Forced Off-Axis Convection? To the right are plots of laminar and turbulent off-axis rotation factors.
Laminar and turbulent effective-lengths are very close, enabling a single formula for laminar and turbulent flows and use with the scaled-Colburn-analogy asymptote.
L_{eff} = A^{1/2} |cos φ|^{−1} r^{−1/2} ( 1 + |tan φ|
3 r)^{−2} r>|tan φ| LERR A^{1/2} |sin φ|^{−1} r^{+1/2} ( 1 + r
3 |tan φ|)^{−2} 0<r≤|tan φ|
The solid lines in the graph to the right show the rotation-factors for 1 m^{2} rectangular plates with ε=10^{−4} m and aspect-ratios of 1:1 (black), 2:1 (blue), and 4:1 (red) computed by integration. The dashed lines are the rotation factors computed using L_{eff}.
The graph to the right shows the rotation-factors by integration (solid) and using effective-length (dashed) for ε=10^{−3} m (higher) and ε=10^{−5} m (lower).
So the complexities of integrating RS turn out not to matter because the convective surface conductance of the rotated plate can easily be computed using effective-length LERR.
So far, the natural convective analysis has assumed a rectangular plate with at least one pair of horizontal edges. This section extends the analysis to horizontal and vertical rotated plates. Rotation of the plate out of the horizontal or vertical planes has already been covered.
The three modes of (convex flat plate) natural convection use different characteristic lengths, so they must be individually treated. The upward facing mode depends on L* (area/perimeter), which is already independent of rotation as evidenced by its working for a variety of convex shapes (Lloyd and Moran[74]).
The downward convection mode is only active when the plate is horizontal. Its characteristic-length is half of the shortest side, which is also independent of orientation.
The vertical plate convection mode corresponds to formula Nu':
Nu'(Ra) = ( 0.825 + 0.387 [Ra' / Ξ(Pr)]^{1/6} )^{2} |
Ra' is proportional to L^{3}. As a function of L, h' is an L^{1/2}-norm of 0.825 k/L and a constant term. The constant term is independent of L.
A rotation factor for the 0.825 k/L term can derived analogously to FTRR and FLRR, but which reduces to a simpler form:
F'(r, φ) = |cos φ| r^{1/2} + |sin φ| r^{−1/2} | FVRR |
The rotation factor for the constant term is 1.
The effective length L_{eff} is simply A^{1/2}/F'(r, φ). It can be further reduced:
L_{eff} = |
L_{H} L_{W}
|cos φ| L_{W} + |sin φ| L_{H} |
LVRR |
The graph on the left compares h'(L_{eff}) with h'(L)⋅F', both with mean side lengths of 10^{−4} m. For side lengths greater than 1 m, h' is much less sensitive to L_{eff} and rotation. The graph on the right compares h'(L_{eff}) with h'(L), both with mean side lengths of 1 m.
A^{1/2}=10^{−4} m | A^{1/2}=1 m |
In An Experimental Study of Mixed, Forced, and Free Convection Heat Transfer From a Horizontal Flat Plate to Air [78] X. A. Wang proposes local correlations for upward natural, mixed, and forced convection from a horizontal plate:
Nu_{xf}^{ } = 3+0.0253 Re^{0.8} for 0.068 Re^{2.2} > Gr (11) Nu_{x}^{ } = 2.7 ( Gr / Re^{2.2} )^{1/3} ( 3+0.0253 Re^{0.8} ) for 0.068 Re^{2.2} ≤ Gr ≤ 55.3 Re^{5/3} (12) Nu_{xe}^{ } = 0.14 Gr^{1/3} for Gr > 55.3 Re^{5/3} (13)
Fig.9 from the paper (recreated to the right) shows that the boundaries of the mixture zone (correlation 12) will cross. Equating the constraints finds that the intersection is at Re=2.86×10^{5} and Gr=6.87×10^{10}. The paper gives no guidance about how to treat Re and Gr values greater than this (which are in the roof range-of-interest).
The technique for finding average h in Laminar-Turbulent Progression only works when the correlation expression tends to zero as Re(x) goes to zero. The 3 added to 0.0253 Re_{x}^{4/5} prevents the integral from converging. At high Re values, formulas 11 and 12 can be approximated:
Nu_{xf}^{ } ≈ 0.0253 Re_{x}^{4/5} Nu_{x}^{ } ≈ 0.0683 Gr^{1/3} Re_{x}^{1/15}
The Wang formulas don't express a dependence on Pr. The working fluid is air with the plate temperature ranging from 19 K to 106 K hotter than ambient. The paper doesn't reveal the ambient temperature, and describes heaters but not coolers. Taking as ambient the mean temperature of Shanghai, 16.1°C, the Pr range for Wang's experiments is 0.715. to 0.723. Incorporating Pr back into formulas 11 and 13 yields formulas within 5% of T8.10 and T9.8:
Nu_{xf}^{ } ≈ 0.0282 Re_{x}^{4/5} Pr^{1/3} Nu_{xe}^{ } ≈ 0.156 Gr^{1/3} Pr^{1/3}
Why can the local Nu_{xe} be compared with the average Nu of T9.8? The dependence of Nu_{xe} on x is linear; so its average h is the same as its (constant) local h.
The measured data is shown only in Fig. 2, plotting local h versus distance from the leading edge. To the right, the values predicted by formulas 11, 12, and 13 are plotted in colors over Wang's graph. There are discontinuities at both transitions and, for x≥0.2 m, predicted h is significantly higher than measured.
Because Wang's (local) model doesn't work, trying to derive average h formulas from its divergent integrals would be pointless. Can a de novo analysis fit Wang's experiments?
The curves predicted by (forced laminar) T8.8, (forced turbulent) T8.10, and (natural turbulent) T9.8 for two cases are plotted on Fig. 2 to the right. For V=0.84, the transition from laminar to turbulent happens around x=0.075 m, at which Re=3548. This is much less than the Re=5×10^{5} threshold given by Table 8. The low Re threshold could be due to a rough leading edge or turbulence in the impinging flow.
The two horizontal T9.8 lines are close in height to the right side of Wang's corresponding curves. But there were several problems: T9.8 should use L*=area/perimeter=0.05⋅L/(0.1+L), not the linear L that T8.8 and T8.10 use. In Wang's 0.1×1 m apparatus, L* ranges between 0 and 0.045 m. At the ΔT values from Fig.2A, Ra is bounded by 4.8×10^{5}, an order of magnitude below the range of applicability of T9.8.
Forced convection is parametrized by distance from the leading edge because the flow controls the boundary layer near the edge. Natural upward convection from a horizontal plate has flow from all edges; that is why it uses L*. The heated strip in Wang's experiment is 10 times longer than it is wide. The forced flow at the leading edge will not affect most of that length, instead behaving as an infinite strip, L*=w/2. Unfortunately, the heated strip is flanked by insulated strips 0.075 m wide on each side. These reduce the natural convective flow across the edge of the heated strip, perhaps explaining why the measured h is lower than Nu* predicts, both for 0.1 m and 0.25 m widths. The graph to the right shows Nu* at two temperatures for (fully heated) widths of 0.1 m and 0.25 m.
An Empirical Correlation for the Outside Convective Air Film Coefficient for Horizontal Roofs[82] by R.D. Clear, L. Gartland and F.C. Winkelmann takes a quite different approach. The roofs of two commercial buildings were instrumented with thermal and meteorological sensors (at their centers) and sampled at 15 minute intervals for over a year. Because of the poor predictive power of the sky emissivity models, they exclued cloudy days. Along with exclusions for condensation and other reasons, they used less than 1/6 of the data collected.
Because of the size of the roofs, Ra at the measurement point always exceeded the range for laminar natural convection for ΔT > 0 and, in fact, often exceeded by a factor of 100 or more the recommended range of the equation for turbulent natural convection. Re is proportional to wind speed, and there were a substantial number of low wind speed points (< 0.1 m/s) that gave Re values that were nominally in the laminar flow region. However, the fits to the data were almost always better if the flow was assumed to be turbulent. In retrospect, it seems likely that any time natural convection is turbulent, then the mixed natural/forced convection should be turbulent also. All of our fits are based on turbulent flow at the measurement point for both the natural and forced convection conditions.
C and D are the fitted constants in:
f_{2}(h_{n},h_{f} ) ΔT = [ η C h_{n} + D h_{f} ] ΔT (4b)
where η is 1 for still air and tends to 0 as wind-speed increases:
η = ln(1 + Gr_{x} / Re_{x}^{2})
1 + ln(1 + Gr_{x} / Re_{x}^{2})
Re_{x} is the local Reynolds number because their wind-speed was measured at one point on the roof. The subscript of Gr_{x} implies that the Grashof Number is also local, but the formulas they used (T9.7 and T9.8) are for average surface conductance. It is possible that η requires adjustment for use with average Reynolds numbers.
Their fitted values are C=1.03 and D=1.66 with standard-error of 0.03 and 0.02, respectively.
With L*=10.2 m and ΔT=25 K, Ra is larger than 2×10^{12}, exercising a scale of operation which is difficult to test in the laboratory. The two graphs below are of:
... a representative sample of data (July 1996, Davis) with cloudy days and condensation conditions removed, and corrected for conduction time lag between sensor location and roof surface.
We removed all data with ΔT < 0, i.e., roof temperature less than outside air temperature. This exclusion was, like the non-clear sky exclusion, due to our reliance on a Q_{sky} estimation.
(In Figs. 8, 9 and 10, a small number of points with h < -20 or > 40 were not plotted to avoid losing detail in the remaining data.) During the day h generally increases during the morning to a mid-afternoon peak, then declines. This pattern reflects the ΔT and wind speed patterns at the site.
Figure 9 shows h vs. ΔT. The dependence on ΔT is fairly weak and we see the loss in precision as ΔT approaches zero.
Figure 10 shows h vs. wind speed. The dependence on wind speed is slightly sub-linear.
Over Fig. 9 I have plotted the convective surface conductances predicted by formulas T9.8 and Nu* (natural convection above a hot horizontal plate). Above surface-to-air temperature differences of 20 K, the measured values are greater than or equal to those predicted by Nu*. This is consistent with the natural convection being aided by forced convection at low wind speeds. The forced convection contribution appears to diminish for larger temperature differences; this could be explained by the larger temperature differences being acheived only at low wind speeds (higher wind speeds cool the roof too much).
When the temperature difference is small:
... the long-wave radiative loss from the roof dominates: it is typically 10 times or more higher than the convective and conductive heat transfer. Under these conditions a small percentage error in Q_{sky} can lead to large and systematic errors in hΔT ...
This could explain the flaring of h values below temperature differences of 10°C.
Because of the large Rayleigh numbers, Nu* is very close to T9.8. (Nu in) T9.8 is a linear function of L*; so h has no dependence on L*. Thus h at the center of the roof (where they measured it) should be the same as the average for the whole roof. The situation for forced convection is more complicated. The forced convection component at the center of the roof is local convection with characteristic length from the edge to center, rather than across the whole roof.
The characteristic length for forced convection is defined as the average distance from the roof perimeter to the heat transfer measurement point.
Their characteristic length for forced convection at the Davis site is 28.3 m. While the curve drawn on Fig. 9 is the curve for the average over the roof, the curves drawn on Fig. 10 are intended to reproduce the measurements taken at the center of the roof. The roof average will use a different L and involve T8.11 instead of T8.10 (whose ratio is 5:4).
Over Fig. 10 I have plotted f_{2}(h_{n},h_{f}) from equation (4b) for D values of 1.66 and 1.33 with ΔT values of 22 K and 4.4 K, respectively.
There are only 20 points at wind-speeds greater than 4 m/s. Considered in isolation they are a strong signal for a D value of 1.33. Because there are so few points at higher wind-speeds, one might conclude that h values at wind-speeds above 4 m/s are unimportant. But looking at the histogram of wind speeds for Honolulu shows that the yearly average wind speed is greater than 4 m/s at that locale.
In discarding 86% of the points, the authors raised the average surface-to-air-temperature-difference from 4.4 K to 22 K, a fivefold increase! The removal of low ΔT points introduced a large bias into the D value that the authors fit (1.66).
Can anything be salvaged from Fig. 10? While forced convection at low wind-speed aids natural convection, natural convection does not aid forced convection at higher wind-speeds. Thus the h values for wind-speeds greater than 4 m/s should be much less affected by the culling of low ΔT points than those at lower wind-speeds.
Clear et al model the effect of surface roughness as a scale factor of Nu (and h). The h versus V curves from Surface Roughness are similar enough to support fitting a scale factor. But the authors didn't measure the roughness of the roof; someone trying to apply their results has insufficient information for predicting h without reproducing their experiment to fit the h_{T8.10} coefficient D.
What about mixed convection from plates which are not level upward facing? Under the heading Limitations and Applicability, Clear et al write:
Our expressions for h are applicable in the following situations:
Caveat 1 rules out application of their work to vertical plates; caveat 2 rules out most downward convection.
The function η compares Gr=Ra/Pr and Re^{2}. But in Natural Convection Summary, Nu', Nu*, and Nu_{R} all use different characteristic-lengths, and Ra being proportional to L^{3} magnifies these differences. Adding to the difficulty of finding an effective Gr is that for downward convection from an inclined plate one of three characteristic-lengths applies depending on the h values computed with them. Furthermore, the Ra arguments to formulas Nu' and Nu* are multiplied by cos θ and |sin θ|, respectively. Should η include these factors with Gr? The published convection literature provides no experimental basis for resolving any of these issues.
Returning to horizontal plates, the graph to the right gives an expanded view of the green trace from Figure 10 above (local mixed convection). The red trace has double the forced characteric length L and uses T8.11 instead of T8.10 to model the average convective surface conductance from that roof. The dip at V=.25 m/s is due to the doubling of L (28.3 m to 56.6 m), not the switch from T8.10 to T8.11.
This plot compares the red trace from the previous graph with scaled-Colburn-analogy curves for various surface roughness values (ε). Because of surface roughness, all but the lowest scaled-Colburn-analogy trace are nearly straight above 1 m/s; the lowest trace and the scaled T8.11 trace in red are proportional to Re^{4/5} above 1 m/s.
Consider the natural upward convection from a heated horizontal plate with a forced laminar flow parallel to the plate. Each (laminar) layer is shifted in the direction of the flow, the shift increasing with the layer's height above the plate. The relative shifting of the layers slightly increases the mixing of hotter fluid with cool fluid, increasing the heat transferred from plate to fluid. Therefore the surface conductance should be monotonically increasing with air velocity; so the η function, which produces a local minimum in the graphs above, doesn't appear to correctly model average surface conductance in laminar flows.
Forced turbulent flow erodes the laminar boundar layer, which also increases the surface conductance. Thus mixed convective surface conductance should increase monotonically with fluid speed.
My article Mixed Convection from a Rough Plate develops a theory of convection for rough plates at any inclination and tests it against measurements from the Convection Machine. The Convection Machine plate with 3 mm of roughness is so rough that it doesn't explore the transition between turbulent flow from a smooth plate (Re^{4/5}) and turbulent flow from the rough plate (linear Re). It will be modified to do so. In the meantime, I will assume that the two modes are in simple competition:
Nu_{8.11}= 0.037 Re^{4/5} Pr^{1/3} (T8.11) Nu_{RT} = Re Pr^{1/3}
32 log_{10}( ε / L )^{2}ε
L_{F}< 0.085
Re^{1/5}RT
h_{F} = k⋅max(Nu_{RT}(Re), Nu_{8.11}(Re))/L_{F}
Allowing for the plate to be rotated φ around its center adds more complication, which is detailed below. R_{eff} is the effective-length for natural-convection from the plate.
Proposed is the complete convective surface conductance for one side of a flat L_{H} by L_{W} rectangular isothermal plate rotated φ around the center of the plate (in its plane) and inclined θ from vertical (θ<0 is facing upward) and with (forced) bulk fluid velocity is V_{f} at angle ψ from the normal projection of vertical in the plane of the plate.
Pr = c_{p}⋅μ/k
Shape Parameters
A = L_{W} L_{H} r = L_{W} / L_{H} L* = A
2 (L_{W}+L_{H})R = min(L_{H}, L_{W})/2 R_{eff} = L_{H} L_{W}
|cos φ| L_{W} + |sin φ| L_{H}‖x,y‖_{p} = (x^{p}+y^{p})^{1/p} x ≥ 0; y ≥ 0; p > 0 θ_{e} = θ T_{S}≥T_{F} −θ T_{S}<T_{F}
Natural Convection
Ra* = β⋅|ΔT|⋅g⋅L*^{3}
ν^{2}Pr |sin θ_{e}| Ra_{V} = β⋅|ΔT|⋅g⋅R_{eff}^{3}
ν^{2}Pr |cos θ_{e}| Ra_{R} = β⋅|ΔT|⋅g⋅R^{3}
ν^{2}Pr |sin θ_{e}|
Nu_{Nu*} = {0.671 + 0.370 Ra*^{1/6}}^{2} −90°≤θ_{e}≤0° 1≤Ra*≤10^{12} Nu' = {.825 + .387 Ra_{V}^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} 1≤Ra_{V}≤10^{12} Nu_{R} = 0.544 Ra_{R}^{1/5}
[1+(0.785/Pr)^{3/5}]^{1/3}0°≤θ_{e}≤+90°
The (forced) bulk fluid velocity is V_{f} at angle ψ from the normal projection of vertical in the plane of the plate.
The effective (characteristic) length for flow at angle ψ from the rectangle orientation is:
L_{eff} = A^{1/2} |cos ψ|^{−1} r^{−1/2} ( 1 + |tan ψ|
3 r)^{−2} r>|tan ψ| LERR A^{1/2} |sin ψ|^{−1} r^{+1/2} ( 1 + r
3 |tan ψ|)^{−2} 0<r≤|tan ψ|
Reynolds number for forced-convection:
Re = V⋅L_{eff} / ν
The forced component Nusselt number:
Nu_{8.11}= 0.037 Re^{4/5} Pr^{1/3} (T8.11) Nu_{RT} = Re Pr^{1/3}
32 log_{10}( ε / L_{eff} )^{2}ε
L_{eff}< 0.085
Re^{1/5}RT
Nu_{F} = max(Nu_{RT}(Re), Nu_{8.11}(Re))
The mixed convective surface conductance is:
M(h_{F}, h_{N}, ψ)^{ }= max(0, cos ψ) 4h_{F}^{2}h_{N}^{2}
h_{F}^{4}+4h_{N}^{4}−min(0, cos ψ) 16h_{N}^{16}
h_{F}^{16}+16h_{N}^{16}
B(h_{F}, h_{N}, ψ)^{ }= {[h_{F}^{2}+h_{N}^{2}]^{2}−2h_{F}^{2}h_{N}^{2}M(h_{F}, h_{N}, ψ)}^{1/4}
h = k⋅max( B( Nu_{F}
L_{eff}, Nu_{R}
R, ψ ) , ‖ Nu_{V}
L_{V}, Nu_{F}
L_{eff}‖_{2}) sin θ_{e}≥0 h = k⋅max( B( Nu_{F}
L_{eff}, Nu_{Nu*}
L*, ψ ) , ‖ Nu_{V}
L_{V}, Nu_{F}
L_{eff}‖_{2}) sin θ_{e}≤0
The heat flow is h⋅A⋅ΔT.
A rotating cup anemometer at 10 m height is the standard wind-speed measuring instrument for the Automated Surface Observing System (ASOS); only the horizontal component of wind velocity is reported. There are formulas for estimating wind at lower heights. The important wind properties for convection are those near the roof surface. Wind speed and direction over the roof will be affected the shape and openings of the building; the most accurate modeling will require simulating airflow around the building.
For the low-pitch roofs examined here, the wind-speed, scaled for its height above terrain, should be a good estimate for the airspeed at the roof surface.
The range table for h predicts (using correlations RS and T8.11) that, for a 1 m square plate, 2 mm surface roughness can produce a forced convective surface conductance three times that of a smooth plate. In locations where ambient temperatures can exceed 45°C, a smooth roof may be preferable to a rough one in order to reduce wind-driven roof heating on very hot days.
The simplest roofs are flat, either horizontal or slightly inclined. These are handled directly by the model. θ will be (slightly) greater than or equal to −90°.
Some horizontal flat roofs have low walls around their perimeters. This will reduce the forced convective surface conductance. As noted in the section Natural Convection from a Rough Plate, most of the inflow comes from above the plate, not along its surface. So the reduction of natural convective surface conductance may not be large.
Four Java methods in in Flat_roof.java calculate the mixed convection from a rough or smooth surfaced flat (rectangular) roof at given orientation, wind-direction and speed, temperature, air-pressure, and relative-humidity.
data-type name return-units description double flatRoofConvect
W heat flow double flatRoofHeatFlux
W/m^{2} heat flux double flatRoofH
W/(K⋅m^{2}) convective surface conductance double flatRoofHa
W/K heat flow per degree of temperature difference
They all take the same arguments:
data-type argument units description double h__m m length double w__m m width double eps__m m mean height of surface roughness double phi__deg ° rotation around center in plane of roof double theta__deg ° inclination from vertical around width edge; negative is upward facing double psi__deg ° forced flow angle from width edge in plane of roof double v_f__m_per_s m/s forced bulk flow velocity double t_a__k K ambient (bulk) air temperature double t_s__k K roof surface temperature double p__pa Pa bulk air pressure double rh relative humidity
Consider a symmetrical peaked roof composed of two rectangular plates joind at the ridge. In the example to the right (top), each section has an 8 m span, a 2 m rise, and is 12 m long.
No rotation in the plane of the plates (which are not necessarily coplanar) is allowed, so φ=0. One of the constraints on the peaked roof model is that it should become identical to the Mixed Convection Summary model when θ=−90 (because the roof would then be flat).
−90≤θ<−60 is the inclination from vertical; the roof pitch is 90+θ (between 0° and 30°). L_{W} is the length of the roof ridge and L_{H} is the distance between the lower (parallel) edges of the roof.
R_{H} = L_{H}
|sin θ|A = L_{W} R_{H} r = L_{W} / R_{H} R = min(R_{H}, L_{W})/2 R_{eff} = R_{H}/2 L* = A
2 (L_{W}+R_{H})θ_{e} = θ T_{S}≥T_{F} −θ T_{S}<T_{F} ‖x,y‖_{p} = (x^{p}+y^{p})^{1/p} x ≥ 0; y ≥ 0; p > 0
A=L_{W}⋅L_{H}/|sin θ| is the surface area of the roof and is the area by which h is multiplied.
For downward convection (roof cooler than air) Nu' flow from the ridgeline will dominate Nu_{R} flow when the roof is not flat. The effective plate dimensions for Nu_{R} will be L_{W} by L_{H} when θ=−90 and are uncritical otherwise. The characteristic length for Nu' flow is R_{eff}=R_{H}/2.
Natural Convection
Ra* = β⋅|ΔT|⋅g⋅L*^{3}
ν^{2}Pr |sin θ_{e}| Ra_{V} = β⋅|ΔT|⋅g⋅R_{eff}^{3}
ν^{2}Pr |cos θ_{e}| Ra_{R} = β⋅|ΔT|⋅g⋅R^{3}
ν^{2}Pr
Nu_{Nu*} = {0.671 + 0.370 Ra*^{1/6}}^{2} −90°≤θ_{e}≤0° 1≤Ra*≤10^{12} Nu' = {.825 + .387 Ra_{V}^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} 1≤Ra_{V}≤10^{12} Nu_{R} = 0.544 Ra_{R}^{1/5}
[1+(0.785/Pr)^{3/5}]^{1/3}0°≤θ_{e}≤+90°
The effective (characteristic) length for the mixed flow is independent of the sign of ψ. So in both cases it is:
L_{eff} = A^{1/2} |cos ψ|^{−1} r^{−1/2} ( 1 + |tan ψ|
3 r)^{−2} r>|tan ψ| LERR A^{1/2} |sin ψ|^{−1} r^{+1/2} ( 1 + r
3 |tan ψ|)^{−2} 0<r≤|tan ψ|
Reynolds number for forced-convection:
Re = V⋅L_{eff} / ν
The forced component Nusselt number:
Nu_{8.11}= 0.037 Re^{4/5} Pr^{1/3} (T8.11) Nu_{RT} = Re Pr^{1/3}
32 log_{10}( ε / L_{eff} )^{2}ε
L_{eff}< 0.085
Re^{1/5}RT
Nu_{F} = max(Nu_{RT}(Re), Nu_{8.11}(Re))
The mixed convective surface conductance is:
h^{ }= k⋅max(‖ Nu_{R}
R, Nu_{F}
L_{eff}‖_{4} , ‖ Nu_{V}
L_{V}, Nu_{F}
L_{eff}‖_{2}) sin θ_{e}≥0 h^{ }= k⋅max(‖ Nu_{Nu*}
L*, Nu_{F}
L_{eff}‖_{2} , ‖ Nu_{V}
L_{V}, Nu_{F}
L_{eff}‖_{2}) sin θ_{e}≤0
The heat flow is h⋅A⋅ΔT =h⋅ΔT⋅L_{W}⋅L_{H}/|sin θ|.
Four Java methods in in Peaked_roof.java calculate the mixed convection from a rough or smooth surfaced symmetrical peaked roof at given wind-direction and velocity, temperature, air-pressure, and relative-humidity.
data-type name return-units description double peakedRoofConvect
W heat flow double peakedRoofHeatFlux
W/m^{2} heat flux double peakedRoofH
W/(K⋅m^{2}) convective surface conductance double peakedRoofHa
W/K heat flow per degree of temperature difference
They all take the same arguments:
data-type argument units description double h__m m length double w__m m width double eps__m m mean height of surface roughness double theta__deg ° inclination from vertical; −90≤θ<−60 double psi__deg ° forced flow angle from ridge in horizontal plane double v_f__m_per_s m/s forced bulk flow velocity double t_a__k K ambient (bulk) air temperature double t_s__k K roof surface temperature double p__pa Pa bulk air pressure double rh relative humidity
The convecting surfaces of the peaked roof are convex when considered in combination. Two rectangular plates meeting at a lower height than their centers is not convex and their analysis is not addressed by current theory. For a valley roof, unless the joined edge is short compared with the other dimension, the Nu' mode will be significantly reduced because of the extra distance air must travel to enter or exit near the joined edge.
The graph to the right compares heat flux from a flat level roof (black) with that from a symmetrical peaked roof with 30° pitch (red). The upper two traces are with 5 mm mean height of roughness; the lower two traces for a smooth surface. All are for roofs covering a 10 m by 10 m building with the roof 5°C hotter than ambient.
The wind direction is diagonal across the roof. At wind-speeds greater than 4 m/s the surface roughness has greater effect on heat-flow than the roof configuration.
The graph to the right shows the heat flux from the same roofs, but with the roofs 5°C cooler than ambient. The peaked roof's Nu_{R} mode is overwhelmed by Nu' in still air, so its downward natural convection does not get as close to 0 as the flat plate does.
A complete mixed single-phase convection model has been presented for flat rectangular plates. Its novel features are:
natural convection from a rectangular plate in any orientation,
a formula for forced convection over a rough flat surface,
correspondence between forced convection and the vertical-plate mode of natural convection,
vector sum of forced-equivalent velocity and forced velocity vectors,
competition between convective modes, and
L^{4}-norm combination of natural convective modes.
The model could be extended to arbitrary convex shapes by extending each of the convective component modes to arbitrary shapes. The shape-factor for both the forced and vertical-plate modes can be computed by integration as was done in Forced Off-Axis Convection and Natural Off-Axis Convection.
The upward convective mode (Nu*) is a function of L*, which is defined for all flat convex shapes. Lloyd and Moran[74] find the formula accurate for a variety of convex shapes.
The challenge is finding the characteristic-length for downward convection from a horizontal plate. Downward convection from a rectangular plate self-organizes into two rollers parallel to a longest edge and has characteristic-length, R, which is half of the shorter edge. Schulenberg gives a separate correlation (47) for downward convection from a disk which has the same exponents as correlation (45) for an infinite strip. Through appropriate choice of characteristic-length R, can these two correlations be unified?
Convection in fluids which can undergo phase changes is of great practical interest in the case of water-vapor condensation on roofs. While there is a body of theory and experiment dealing with condensation from a pure phase-change fluid (eg. steam), there is less available dealing with condensation from a mixture of condensing and non-condensing gases (eg. air).
We know that for small Reynolds number flow over low surface roughness, the convective surface conductance is close to that of a smooth plate. So only strong turbulent flow over a rough surface needs to be measured. Not needing laminar flow simplifies the wind tunnel requirements considerably.
The table below shows the dimensionless quantities that the scaled-Colburn-analogy-asymptote model predicts for plates with relative roughnesses of 0.01, 0.005, and 0 (smooth).
ε/L Re Nu/Pr^{1/3} Re Nu/Pr^{1/3} Re Nu/Pr^{1/3} 0.0100 2.00×10^{4} 1.56×10^{2} 4.00×10^{4} 3.12×10^{2} 8.00×10^{4} 6.25×10^{2} 0.0050 2.00×10^{4} 1.18×10^{2} 4.00×10^{4} 2.36×10^{2} 8.00×10^{4} 4.72×10^{2} 0.0000 2.00×10^{4} 1.02×10^{2} 4.00×10^{4} 1.78×10^{2} 8.00×10^{4} 3.09×10^{2}
This measurement has been performed by the Convection Machine for ε/L=0.01 between Re=5000 and Re=100000; the results appear to confirm R8.11.
The reasoning that natural vertical convection is insensitive to surface roughness might not hold at high Rayleigh numbers.
The plate described in Convection Measurements is not large enough to reach turbulent natural convection in air with a practical temperature difference (<50K).
Because the natural convection arising from the the plate described in Convection Measurements is expected to be laminar, it can be used to do this measurement. However, the apparatus designed for testing the higher heat transfer rates of forced convection will suffer from long settling times measuring natural convection.
This would require stable laminar low-speed flow in the wind tunnel, which can be difficult to achieve and measure accurately.
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