http://people.csail.mit.edu/jaffer/SimRoof/Convection  
Convection From a Rectangular Plate 
This document as PDF.
A complete model for singlephase (noncondensing) convection from an isothermal rectangular plate is developed and tested against available published experimental data. It generalizes Fujii and Imura[76]'s natural convection model to the full range of inclinations (−90° to +90°) while eliminating the use of their of adhoc formula for the laminarturbulent threshold. The forced convection model finds a correlation for roughsurface convection by applying the Colburn analogy to flat plates. Through the use of effectivecharacteristiclength, the theory is extended to natural convection for all orientations of a plate, not just those having a horizontal edge, and offaxis forced (but inplane) flows. For mixed convection a correspondence is found between forced convection and the verticalplate mode of natural convection which allows them to be combined and used within the multimode natural convection solution.
A related model for convection from symmetrical peaked roofs is also developed.
The most important thermal processes for a building's roof are insolation, convection, and thermal radiation. The most complicated to compute is convection because it depends on so many variables: air and surface temperatures, humidity, wind, and roof size, shape, orientation, and (surface) roughness.
There are additional complications: total heat transfer is not equal to the sum of heat transfer from a shape's parts considered individually; wind usually varies both in speed and direction, but convective surface conductance is not a linear function of windspeed.
In 2011, established theory was not sufficient to compute convection from roof parameters and meteorological measurements. 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. Convection is usually the dominant form of heat transfer in liquids and gases.
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, fluidvelocity, 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 upwardfacing plate which is warmer than the fluid, or a downwardfacing plate which is colder than the fluid. (Natural) downward convection is produced by an upwardfacing plate which is cooler than the fluid, or a downwardfacing 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 correlation can be for local heat flow or for heat flow averaged across the surface. Section "LaminarTurbulent Progression" gives the derivation for average heat flow from the local heat flow which appears in several texts. While it produces the customary averaged correlations for purely laminar or purely turbulent flow, the correlation it produces for transitional flows requires a Reynoldsnumber threshold from measurement of the specific configuration under investigation, which limits its predictive ability.
Convection correlations can be derived from theory, numerical simulation, or experiment (empirical). The accepted correlations for natural convection are empircal.
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 correlations 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 downward convection (from a horizontal plate) is always laminar (its sole correlation having the exponent 1/5).
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.
The 1997 ASHRAE Fundamentals Handbook (SI) takes an engineering approach to convective heat transfer. Fig. 24.1 (reconstructed to the right) has a graph of surface conductance h as a function of wind velocity V for 6 building materials. That conductance is for a ΔT of 5.5°C at a mean temperature of −6.7°C and includes blackbody emissions, which add a constant amount to each curve. About Fig. 24.1, the Handbook also states:
Other tests on smooth surfaces show that the average value of the convection part of the surface conductance decreases as the length of the surface increases.
The graph is based on [84] which gives the dimensions of the plate as a 0.305 m square. Without a quantitative relation between surface conductance and plate size, application to surfaces as large as roofs is uncertain.
Two of the lines are bent: "Clear pine" and "Glass and white paint on pine". The formulas I fit for glass and pine are:
h(V) = (b^{2} − 4 a c + 4 a V)^{1/2} − b
2 ah_{g} Glass h_{p} Clear Pine a = 1/882 a = 4/3999 b = 257/882 b = 973/3999 c = −1060/441 c = −2680/1333
The first task is to separate the surface conductivity due to convection from that due to radiative transfer. Measurements of convective surface conductance take place in a large enclosure whose temperature matches the fluid (air) inside it. To the right is plotted the difference in blackbody radiance between the plate and enclosure divided by the temperature difference. Plots for temperature differences of 1°C, 5°C, and 25°C are superimposed and look identical at this scale.
The maximum surface conductivity due to radiation at −6.7°C±2.75°C is 4.29 W/(K⋅m^{2}). If the surface and test apparatus both have emissivity of 0.95, it is 3.87 W/(K⋅m^{2}). But the suite of materials tested do not necessarily have the same emissivity.
There are multiple books and webpages offering tables of emissivity for building materials. Of the five excerpted below, those that cite a source cite handbooks or textbooks, so they are not original data. [85] gives emissivities at specified temperatures. None of these five sources provided a value for stucco.
source  citing  

[E1] 
EMISSIVITY EXPLAINED IN LAYMAN'S TERMS, Fig 1.2 EMISSIVITY OF BUILDING MATERIALS, Horizon Energy Systems. 
Handbook of Chemistry 
[85] 
A Heat Transfer Textbook, Third Edition [PDF], by John H. Lienhard IV and John H. Lienhard V,, Cambridge, MA, Phlogiston Press, 2008 

[E3] 
Emissivity Coefficients of some common Materials, The Engineering Toolbox 

[E4] 
Emissivity Table, ThermoWorks Inc., 2012 

[70]  2009 ASHRAE Fundamentals Handbook (SI) American Society of Heating Refrigerating and Airconditioning Engineers Inc. ISBN: 9781931862707 (IP); 9781931862714 (SI) ISSN: 15237222 (IP); 15237230 (SI) 
Mills, A.F., 1999., Basic heat and mass transfer. Prentice Hall, Saddle River, NJ. 
glass  paint  wood  plaster  plaster  brick  concrete  

[E1]  .94  .94 avg 16 colors  .95  .91 rough coat  .92 common red  .95  
[85]  .94 smooth  .92.96 var. oils  .90 oak planed  .92 rough lime  .93 red rough  .94 rough  
[E3]  .92.94 smooth  .96  .95 pine  .98  .91 rough  .93 red rough  .94 rough  
[E4]  .92  .97 oil, grey, flat  .90 planed  .86.90  .91 rough coat  .93 common red  .92  
[70]  .91 smooth  .90 white acrylic  .89 rough  .90  .91 rough  
E1  E4 consensus values:  
.93  .95  .92  .91  .93  .94 
Source [70] gives emissivities which are less than those given by the other sources; so I didn't use it in finding consensus values. For six of the seven materials the spread in emissivity values is small enough for consensus values. Although nonmetallic paints are given emissivity values from 0.80 to 1.00, both 0.80 values are for enamel; the 1.00 value is only for black velvet coating 9560 series optical black, neither of which was used in the 1930 experiments.
In the figure to the right I have subtracted out the radiative conductances as moderated by the consensus emissivities and painted apparatus (ε=.95). The emissivity for plaster (.91) is used for stucco. Thus this is a plot of surface conductivities due only to convection, ie. ε=0.0.
Table 1 of chapter 24 gives two numbers for convection in moving air (any position) of a surface with emissivity ε=.9 at 21°C, but without specifying the surface material. These two points, with radiative conductance subtracted out, are marked with dots on the graph. They are closest to the curve for clear pine.
At V=0, the surface conductance is due to natural, not forced, convection.
Table 24.1 gives stillair conductance for flat plates in five positions with emissivities .90, .20, and .05 which, being collinear (chart at right), allow the stillair conductances to be extrapolated to zero emissivity, h_{0.0}.
Table 1 Surface Conductivity W/(m^{2}⋅K) θ orientation heatflow h_{0.0} −90° horizontal up 4.03 −45° inclined up 3.86 0° vertical up 3.06 +45° inclined down 2.27 +90° horizontal down 0.96
Surface Conductances as Affected by Air Velocity, Temperature and Character of Surface[84] by F.B. Rowley, A.B. Algren, and J.L. Blackshaw is the source for Fig. 24.1. Their test plate was inserted (flush with interior) into a .305 m by .305 m opening cut from one vertical side of a .152 m by .305 m rectangular duct over 7 m long.
Rowley at al take and report measurements with the flow shut off. In such a long duct with no forced airflow, equilibrium must involve conduction through the walls of the (insulated) duct. The equilibrium temperature range they report for 0 m/s flow is 9°C to 40°C, while for forced flow the equilibrium temperatures range is −16°C to 31°C. Fig. 24.1 specifies a temperature of −6.7°C; so they must have extrapolated the 0 m/s values.
ΔT  Mean Temperature  

10°C  50°C  90°C  
10 K  3.66 W/(m^{2}⋅K)  3.43 W/(m^{2}⋅K)  3.25 W/(m^{2}⋅K)  
20 K  4.49 W/(m^{2}⋅K)  4.20 W/(m^{2}⋅K)  3.97 W/(m^{2}⋅K)  
30 K  5.06 W/(m^{2}⋅K)  4.73 W/(m^{2}⋅K)  4.47 W/(m^{2}⋅K) 
As shown to the right, natural convective surface conductance is much more sensitive to temperature difference than mean temperature (average of surface and air temperatures); but only mean temperatures are revealed in their graphs. The mean temperature in this case uses the air temperature 25 mm from the center of the surface.
Temperature  

Test  Test Surface 
25 mm From Test Surface 
Mean  Difference  Surface Conductance 
257  30.3°C  1.6°C  15.9°C  28.7 K  10.90 W/(m^{2}⋅K) 
264  52.3°C  26.9°C  39.6°C  25.4 K  11.48 W/(m^{2}⋅K) 
Table 1 Test Data for Smooth Plaster Surface shows one page of their measurements. Two of its rows are for still air. Here they are converted to metric units. The surface conductance column includes both the convective and radiative surface conductances. The radiative surface conductance depends on the inner wall temperature of the duct, which is not provided. Rowley et al write:
In order to obtain average radiation conditions, the inside surface of the test duct was painted a dull gray, and all of the pipe outside the refrigerator was covered with a oneinch [25 mm] thick blanket insulating material. With this arrangement, the surfaces immediately around the test surface were at substantially the same temperature as the surrounding air, practically the same as that for the average wall...
Preliminary tests had shown that when the thermocouple was placed in contact with the test surface and gradually moved away from it, the temperature steadily dropped until the couple was about 1/2in [13 mm]. from the surface, after which this temperature remained uniform and equal to the air temperature, regardless of distance.
The air temperature they refer to is that of the refrigerated room connected through more than 5 m of level duct. It seems clear from the article that the "preliminary tests" were conducted with forced air, calling into question whether the duct inner surface temperature is the same as the air temperature 25 mm from the plate center when flow is unforced.
So how would convection from a vertical plate in a horizontal duct compare with convection from an openair vertical plate? Inside the duct heated rising air is obstructed and so accumulates in the duct. This reduces the heat transfer from the heated plate, but also raises the temperature of the air which is 25 mm from the plate. When the heat transfer is divided by ΔT, it increases the reported h values.
With processes driving h in opposite directions, these stillair measurements can be justified as neither upper nor lower bounds for natural convective surface conductance. Particularly disappointing is that the relationship between submillimeter surface roughness and natural convection from a flat vertical plate remains unquantified (the lack of measured emissivities also contributes uncertainty).
The 2009 ASHRAE Fundamentals Handbook (SI) takes an entirely different approach, characterizing convective heattransfer using correlations (equations) of dimensionless quantities. Surface roughness is treated only for forced convection inside pipes and ducts. For external plates only horizontal and vertical orientations are covered. It has no example calculations of unobstructed convection from a plate.
It appears that with the move to a more theoretical treatment, 2009 Fundamentals lost practical applicability to roofs. The intent of this article is to develop a theory sufficient to predict convection from rough inclined rectangular roofs.
Formulas not leading to the final derivations have a gray background. 
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] is the authoritative source for viscosity and thermal conductivity.
The example calculations in this text were performed before the model incorporated humidity. Calculations with a relative humidity of 41% at sea level match the example calculations within 2%.
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] is the authoritative source for watervapor (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 relativehumidity, Φ, is the ratio of the partialpressure 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 watervapor 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 sidestep the issue by constraining the roof temperature to not drop below the ambient dewpoint 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 watervapor, 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 
The "trunc" traces are with the two higher order terms of c_{pa}(t) dropped.
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. 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} 
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. A linear fit to [93] data over the range of interest is:
μ_{a}^{ }=  16.74×10^{−6} Pa⋅s + (t − 10°C) ⋅ 49.25×10^{−9} Pa⋅s/K 
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} Pa⋅s
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). The simplified 99% RH curve, which is nearly identical to the 99% RH curve, uses the linear μ_{v} and μ_{a} models and substitutes 1.07 and 0.93 for Φ_{AV} and Φ_{VA}.
The authoritative formula for the thermalconductivity of dry air is from Kadoya, Matsunaga, and Nagashima[93]:
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 thermalconductivity (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, T9.3) 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 aspectratio for a rectangular plate with an area of 1 m^{2}. As the aspectratio r grows, L* tends to sqrt(A/r)/2.
The hydraulicdiameter, used as the characteristic length D_{H} in ducts, is related to L*, being 4 times the crosssection area divided by the crosssection 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 a subscripted V. Ranges for horizontal plates, where L_{c} is the ratio of area to perimeter, are marked with a subscripted H. The Prandtl number is insensitive to L_{c}, 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 rangeofinterest is 1 K rather than 0 K. Re will be zero when the windspeed is zero. In order to get an idea of the dynamic range of Re, the minimum V used for rangeofinterest 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 0.719≤Pr≤0.735 Prandtl number Gr β⋅ΔT⋅g⋅L_{c}^{3}
ν^{2}ΔT⋅g⋅L_{c}^{3}⋅ρ^{2}
T⋅μ^{2}9.35×10^{7}≤Gr_{V}≤2.02×10^{15}
1.17×10^{7}≤Gr_{H}≤2.79×10^{13}Grashof number for natural convection Ra Gr⋅Pr ΔT⋅g⋅L_{c}^{3}⋅ρ^{2}⋅c_{p}
T⋅μ⋅k6.72×10^{7}≤Ra_{V}≤1.49×10^{15}
8.40×10^{6}≤Ra_{H}≤2.05×10^{13}Rayleigh number for natural convection Re V⋅L_{c}/ν V⋅L_{c}⋅ρ/μ 2.46×10^{4}≤Re≤6.79×10^{7} Reynolds number for forced convection Nu SCAA
T8.9, T8.11(forced rough)
(forced smooth)1.25×10^{2}≤Nu≤3.46×10^{5}
1.08×10^{2}≤Nu≤6.13×10^{4}Nusselt number HHT*
T9.3(natural) 33.4≤Nu_{H}≤3.63×10^{3}
54.4≤Nu_{V}≤1.22×10^{4}h Nu⋅k/L_{c} (forced rough)
(forced smooth)2.38≤h≤202.
2.38≤h≤68.4convective surface conductance W/(m^{2}⋅K) Nu⋅k/L*
Nu⋅k/L_{c}(natural) 3.02≤h_{H}≤7.58
2.43≤h_{V}≤6.18
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.
The table below excerpts those sections of Table 9 from chapter 4 of 2009 ASHRAE Fundamentals Handbook (SI) which deal with natural convection from isothermal (uniform t_{s}) flat plates. HHT* is my generalization of T9.5 through T9.8.
Table 9 Natural Convection Correlations II. Vertical plate Properties at (t_{s}+t_{∞})/2 except β at t_{∞}. L=height 0.1<Ra<10^{9} Nu=0.68+ 0.67 Ra^{1/4}
[1+(0.492/Pr)^{9/16}]^{4/9}(T9.2)^{a} L=height 10^{9}<Ra<10^{12} Nu={.825+ .387 Ra^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} (T9.3)^{a} III. Horizontal plate Properties of fluid at (t_{s}+t_{∞})/2. hot top L=area/perimeter 1<Ra<200 Nu=0.96 Ra^{1/6} (T9.5)^{b} 200<Ra<10^{4} Nu=0.59 Ra^{1/4} (T9.6)^{b} 2.2×10^{4}<Ra<8×10^{6} Nu=0.54 Ra^{1/4} (T9.7)^{b} 8×10^{6}<Ra<1.5×10^{9} Nu=0.15 Ra^{1/3} (T9.8)^{b} hot top L=area/perimeter 1<Ra<1.5×10^{9} Nu={0.65 + 0.36 Ra^{1/6}}^{2} HHT* cold top L=area/perimeter 10^{5}<Ra<10^{10} Nu=0.27 Ra^{1/4} (T9.9)^{b}
Sources: ^{a}Churchill and Chu (1975a), ^{b}Lloyd and Moran (1974), Goldstein et al. (1973)
The range for T9.2 is supposed to cross over to T9.3 at Ra=10^{9}, but the two curves don't match at Ra=10^{9}. About T9.3, Churchill and Chu (1975a)[75] conclude that:
Equation (9) [T9.3] based on the model of Churchill and Usagi 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.
Roofs aren't vertical. But this formula is a component for modeling the cases of roofs which are neither horizontal nor vertical.
The stipulation: "Properties at (t_{s}+t_{∞})/2 except β at t_{∞}." is not in Churchill and Chu [75]. On this subject their paper says only:
For large temperature differences such that the physical properties vary significantly, Ede [2] recommends that the physical properties be evaluated at the mean of the surface and the bulk temperature. Wylie [18] provides more detailed theoretical guidance for the laminar boundarylayer regime.
The maximum relative error for Nu in T9.3 if β is 1/t_{∞} instead of 2/(t_{s}+t_{∞}) is
( t_{s}+t_{∞}
2 t_{∞}) ^{1/3}
−1 = ( Δt
2 t_{∞}+1 ) ^{1/3}
−1
At t_{∞}=20°C and t_{s}=70°C the maximum error is less than 3%.
Formulas T9.5 through T9.8 give the horizontal hot top correlations in 4 segments with a gap between 10^{4} and 2.2×10^{4}. A function (dashed line) constructed like T9.3 shows good agreement:
Nu*  range  

{0.65 + 0.36 Ra^{1/6}}^{2}  1<Ra<1.5×10^{9}  HHT* 
Despite being cited (in footnote ^{b}), there is no support for T9.9 in either Lloyd and Moran (1974)[74] or Goldstein et al. (1973)[73], which both state "The present results apply to either the heated upward facing heat transfer surface or the cooled downward facing heat transfer surface." Brucker and Majdalani (2005)[72] misattribute T9.9 to Lloyd and Moran (1974)[74]. Whitaker (2005) [71] also perpetuates T9.9.
With the L_{c} values which generate curves matching Fig. 24.1, T9.9 yields h values much larger than Table 24.1. Fujii and Imura[76] treat downward convection in their 1972 paper Freeconvection heat transfer from a plate with arbitrary inclination, analyzed next.
Fujii and Imura(1972)[76] provide convincing measurements that downward facing convection can be treated as scaling of Ra values by cosine of θ. Downward convection from a horizontal plate is thus the limit of θ approaching 90°. But having been written several years before the appearance of T9.2 and T9.3 in Churchill and Chu (1975a)[75], their model of the natural convection from a vertical plate does not include a minimum constant value.
Abstract An experimental study is described concerning freeconvection heat transfer from a plate with arbitrary inclination. The heat is transferred from one side surface of two plates of 30 cm height, 15 cm width and 5 cm height, 10 cm width. The main flow in the boundary layer is restricted twodimensionally.
width  height  area/perimeter  

L_{W}  L_{H}  L*  L_{H}/L*  
15 cm  30 cm  450/90 = 5 cm  6  
10 cm  5 cm  50/30 = 1.67 cm  3 
In their paper, L (L_{H} here) is the length of an edge of the plate which is parallel to the sidewalls which restrict the flow to two dimensions. For both plates, L_{H} is significantly larger than the L* used by Lloyd and Moran (1974)[74] and Goldstein et al. (1973)[73].
The straight line, equation (4) below, on Fujii and Imura's figure 6 is reproduced as the bottom trace. It is a good fit for the range of angles from 0° to 86.8°, but the measurements for an angle of 89° veer increasingly above the line as Gr⋅Pr⋅cosθ falls below 10^{7}. With Pr=5, the top trace is T9.2(Gr⋅Pr⋅cosθ), which has a minimum value of 0.68, causing a barely visible flare to the left. Fujii and Imura remark:
The fact that the experimental Nu value is smaller than the theoretical one seems to be caused by estimating the representative wall temperature too high and heat loss too large.
Another explanation is that the sidewalls responsible for the "twodimensionally" restriction reduce the flow, both by preventing flow around the sides and from drag with the sidewalls.
Nu range 0.56 (Gr Pr cos θ)^{1/4} 10^{5} < Gr Pr cos θ < 10^{11} 0° ≤ θ < 90° (4) 0.58 (Gr Pr)^{1/5} 10^{6} < Gr Pr < 10^{11} θ = 90° (5)
Fujii and Imura remark:
It seems the range of the inclination angle with the horizontal, within which expression (5) may be applied, becomes smaller with increase of GrPr.
Rather than limiting the range of correlations (4) and (5), using the (pointwise) maximum of expressions (4) and (5) is a model which matches the data points in Fujii and Imura's Figure 8 well.
Nu = max(0.56 (Gr Pr cos θ)^{1/4},0.58 (Gr Pr)^{1/5}) 0°≤θ≤90° UCT1
This formulation has a simple physical interpretation; convection will flow along the bottom of the plate unless Gr⋅Pr⋅cos θ is too small to support it. In that case convection will be as for a downward facing horizontal plate.
So far, we have a correlation which predicts natural convection from vertical to downward facing rectangular plates having sidewalls restricting flow to two dimensions. So what is the correlation without the sidewalls?
For a vertical plate we should expect higher Nu values. T9.2 providing 11% more than (4) fits the bill. It's also possible that losing the sidewalls enables a higher exponent for Ra > 10^{7}, in which case T9.3 is likely the right one. Churchill and Chu (1975a)[75] assert that the vertical isothermal plate behaves as T9.3. Because the convection at vertical (0°) must match between the downward and upward facing formulas, T9.3 must be used for both.
The streamlines which Fujii and Imura show for horizontal downflow (Fig.14 c) have two cylindrical rollers with axes perpendicular to the sidewalls. These will arise freely from a flow where heated air spreads from the center of the plate. For a rectangular plate without sidewalls, these two rollers should arise parallel to the longer edges. If a toroidal rolling Ra^{1/6} mode were possible for a square plate, then it would be even more likely for a circular plate. But Schulenberg [77] does not find it in correlation (47) below.
We need to incorporate the perpendicular 1/5 mode, which has a potentially different characteristic length than used in UCT1. How should the heat flow from perpendicular modes be compared? The h value (having units W/(m^{2}⋅K)) for each convection component is found using its L value through its computation.
h = k⋅max(  Nu_{9.3}(Ra(L_{H}) cos θ) L_{H} 
,  0.58 Ra(L_{H}/2)^{1/5} L_{H}/2 
,  0.58 Ra(L_{W}/2)^{1/5} L_{W}/2 
)  0°≤θ≤+90°  UCT2 
In Natural convection heat transfer below downward facing horizontal surfaces [77], Schulenberg develops correlations for downward convection for an infinite strip and circular plate (I added the equivalent T9.2style expressions):
In Schulenberg's derivation, the characteristiclength used for computing the Rayleighnumber (Ra) is R:
R halfwidth of the infinite strip or radius of the circular plate
Replacing Nu_{5} with Nu_{45} in UCT2 results in a formula for convective surface conductance (h) for a heated downwardfacing or a cooled upwardfacing L_{H} by L_{W} rectangular plate inclined θ from vertical around a level L_{W} edge, 90° (horizontal) to 0°(vertical). Because the derivative of h_{45}* is monotonically decreasing as a function of L, its two terms are collapsed to one term dependent on min(L_{H}, L_{W}).
R = min(L_{H}, L_{W})/2 h = k⋅max(Nu_{9.3}(Ra(L_{H}) cos θ)/L_{H} , Nu_{45}(Ra(R))/R) 0°≤θ≤+90° UCT3
Both the 5 cm and 30 cm plates (of FujiiImura) tilt around an axis (the θaxis) parallel to the fluid axes of rotation (as restricted by sidewalls perpendicular to the fluid axes). At the lowest edge of a heated plate, the flow due to the T9.3 mode of convection opposes the flow due to the Nu_{45} mode of convection. This explains why these modes are mutually exclusive when flow is restricted by sidewalls.
Without sidewalls, Nu_{45} flow can organize so its rotation axes are perpendicular to the θaxis, and coexist with T9.3 flow from the lower edge. Taking the maximum of h_{45} and h_{9.3}, which is equivalent to the L^{∞}norm, ignores mode cooperation. Finite norms blend the inputs. Putting L_{H} Nu_{45} in competition with L_{W} Nu_{45} blended with T9.3 results in a formula like:
‖x,y‖_{p} = (x^{p}+y^{p})^{1/p} x ≥ 0; y ≥ 0; p > 0 h^{ }= k⋅max ( Nu_{45}(Ra(L_{H}/2))
L_{H}/2, ‖ Nu_{45}(Ra(L_{W}/2) sin θ)
L_{W}/2, Nu_{9.3}(Ra(L_{H}) cos θ)
L_{H}‖_{p} )
The sin θ factor is included in order to make the formula continuous with T9.3 at θ=0. It will have negligible effect near θ=90°
There appears to be no published experimental data for slightly tilted downward convection without sidewalls. Not only is p unknown, the accuracy of Lnorms for this application is unknown. Additionally, the formula above will be difficult to extend to cases where the lower edge is not horizontal. So I return to a simpler formulation, even if might not be as accurate as the one above once p is known.
The graph to the right shows the natural convective surface conductance at angles near 90° (horizontal downfacing square plate) using five different Lnorms. The L^{4}norm looks to be midway between L^{2} and L^{∞}.
UCT4 is the formula resulting from replacing the L^{∞}norm (max) with the L^{4}norm in UCT3. The maximum relative error from using UCT4 where modes were actually exclusive is +19% at the point where the Nu_{9.3} and Nu_{45} terms are equal. Away from the crossover points the error is smaller.
R = min(L_{H}, L_{W})/2 ‖x,y‖_{p} = (x^{p}+y^{p})^{1/p} h^{ }= k⋅‖ Nu_{45}(Ra(R) sin θ)
R, Nu_{9.3}(Ra(L_{H}) cos θ)
L_{H}‖_{4} 0°≤θ≤+90° UCT4
The FujiiImura treatment of convection from upward facing plates is more difficult than the other orientations. While expression (7) is in line with the expressions from other sources, expression (6) is not. The problem they report with lower than expected Nu values seems to particularly affect their upwardfacing 30 cm plate.
The flow patterns in figure 14 (e) and (f) show plumes rising from the center of the (heated) 5 cm plate. Those plumes would draw from the longer sides of the plate; but such flow is impeded by the sidewalls of the long plate. Having its long sides obstructed could well explain the 30 cm plate's convection deficit.
Nu range 0.13 (Gr Pr)^{1/3} 5×10^{8} < Gr Pr −90°=θ (6) 0.16 (Gr Pr)^{1/3} Gr Pr < 2×10^{8} −90°=θ (7) 0.13 {(Gr Pr)^{1/3} − (Gr_{c} Pr)^{1/3}} + 0.56 (Gr_{c} Pr cos θ)^{1/4} Gr_{c} ≤ Gr 2×10^{8} ≤ Gr Pr −90°<θ≤0° (8) 0.16 {(Gr Pr)^{1/3} − (Gr_{c} Pr)^{1/3}} + 0.56 (Gr_{c} Pr cos θ)^{1/4} Gr_{c} ≤ Gr Gr Pr < 2×10^{8} −90°<θ≤0° (9) Implicit in this formulation is a reprise of expression (4): 0.56 (Gr Pr cos θ)^{1/4} Gr < Gr_{c} −90°<θ≤0°
The graphs in this section of the paper cover the range 3×10^{5} < Gr Pr < 10^{11}.
Expressions T9.8, (6), and (7) are interesting in that they are linear as functions of L_{c} and result in average h values which are independent of L_{c}.
h⋅L_{c}/k = 0.15 (Pr⋅β⋅ΔT⋅g⋅L_{c}^{3}/ν^{2})^{1/3} h = 0.15 k⋅(Pr⋅β⋅ΔT⋅g/ν^{2})^{1/3}
Expressions (8) and (9) introduce Gr_{c}, the Grashof number corresponding to the transition region from laminar to turbulent flow. Unfortunately, no expression for calculating Gr_{c} is given:
The data of the inclination angle between −60° and −90° for 30 cm plate agree well with expression (6).
... By the way. the values of Gr_{c}Pr at angles between −75° and −90° in Fig. 12 were decided by cut and try adjusting expression (9) to the experimental data for 5 cm heated plate.
Their figure 12 plots Gr_{c}⋅Pr values for 7 angles. As θ nears −90°, Gr_{c} should tend to zero. I created a curve (red) to the points, trying to get the best agreement with Figure 10 below.
Ra_{c} = Gr_{c} Pr  range  

10^{11}⋅(5×10^{5})^{(θ/90+.09 sin(6 θ))}−2×10^{5}  −60°<θ≤0°  
10^{11}⋅(5×10^{5})^{(θ/90)}−2×10^{5}  −90°≤θ≤−60° 
The color traces superimposed on Figure 10 show a good match overall. The trace for −15° is the poorest fit; but adjusting Ra_{c}(−15°) did not result in improvent.
Expressions (8) and (9) describe a regime combining verticalplate mode of convection from 0 to Ra_{c}(θ) with horizontalplate mode of convection from Ra_{c}(θ) to Ra. Expressing the equations in terms of h, x, and x_{c}. Let Ra(x)=C⋅x^{3}; then C=Pr⋅β⋅ΔT⋅g/ν^{2}.
Ra_{c}(θ) = Ra(x_{c}) = C⋅x_{c}^{3} x_{c} = { Ra_{c}(θ)
C}^{1/3}
x_{c} is independent of L_{c}. At the conditions of Fig.14(e), θ=85°, Ra=5.93×10^{7}, C=1.19×10^{9}, and L_{c}=50 mm, x_{c} is 7.7 mm, which is 15% of the plate height L_{c}.
When x_{c} ≤ L_{H}, the plate is considered in two pieces: L_{W} by L_{H}−x_{c} and L_{W} by x_{c}. Support for this treatment can be seen in comparing Fujii and Imura's Fig.14(e) and Fig.14(f); in (e) the plume of the inclined plate is shifted up the slope compared with the level plate (f). Less certain is the validity of this adhoc formula for Ra_{c}(θ) in convection which is not restricted to twodimensional flows.
Express correlation (9) as the sum of surface conductances for correlations (7) and (4):
h_{7}(x) = 0.16 k⋅(C⋅x^{3})^{1/3} / L_{H} h_{4}(x, θ) = 0.56 k⋅(C⋅x^{3}⋅cos θ)^{1/4} / L_{H} h = h_{7}(L_{H}) − h_{7}(x_{c}) + h_{4}(x_{c}, θ) = h_{7}(L_{H}−x_{c}) + h_{4}(x_{c}, θ) −90°≤θ≤0° H9
The most negative angle Fujii and Imura treat is −85°. The spike at −90° may be due to my adhoc function for Ra_{c}(θ). In any case, h_{θ} needs to be generalized to plates without sidewalls.
To generalize H9, substitute surfaceconductance expressions derived from HHT* and T9.3 for h_{7} and h_{4}, respectively:
L*(L_{W}, L_{H}) = L_{H}⋅L_{W} / (2 (L_{H}+L_{W})) h_{HHT*}(x)=k⋅HHT*(C⋅x^{3}) / L*(L_{W}, L_{H}) h_{9.3}(x, θ)=k⋅T9.3(C⋅x^{3}⋅cos θ) / L_{H} h = h_{9.3}(L_{H}, θ) L_{H} ≤ x_{c} −90°<θ≤0° UHT1 h = h_{HHT*}(L*(L_{W}, L_{H}−x_{c})) + h_{9.3}(x_{c}, θ) x_{c} < L_{H} −90°≤θ≤0° UHT2
The sharp transition near −58° is uncharacteristic of convex convective heat transfer functions. The problem with UHT2 is that any h_{T9.3} flow from the leading edge of plate prevents that area from having h_{HHT*} flow, even if h_{HHT*} flow would be greater. This seems plausible for a plate with sidewalls, but not for a plate with all edges open.
For downward convection, the mode with the highest convective heat transfer won; even when the flows of those modes were not mutually exclusive. Both HHT* and T9.3 have flow from the leading edge of the plate inward. Thus, h_{T9.3} flow will not prevent h_{HHT*} flow; the larger convective heat transfer should win.
But how does h_{HHT*} vary with angle? In H9 and UHT2 the term L_{H}−x_{c} reduced h with increasing angle. With h_{T9.3} flow not preventing h_{HHT*} flow, h_{HHT*} would be independent of angle. By analogy with the FujiiImura treatment of T9.3, multiply the Ra* argument to Nu_{HHT*} by sin θ.
L* = L_{H}⋅L_{W} / (2 (L_{H}+L_{W})) h = k⋅ max( Nu_{HHT*}(Ra(L*)⋅sin θ) / L*, Nu_{9.3}(Ra(L_{H})⋅cos θ) / L_{H} ) −90°≤θ≤0° UHT3
The competition between modes does not mean that they are strictly exclusive. When mode T9.3 is dominant, there can still be some HHT* type inflow from the nonhorizontal edges. Taking the maximum of multiple expressions is equivalent to the L^{∞}norm. Other norms blend the inputs. The L^{1} through L^{3}norms make the values for −45°>θ>−90° too large. The L^{4}norm (UHT4) yields a believable curve; as θ becomes more negative, T9.3 mode is assisted by HHT*, which eventually dominates the flow. The conductance is flat around −90° because small tilts (from level) won't much affect the HHT* mode of convection (see the Boundary Layer section).
h^{ }= k⋅‖ Nu_{HHT*}(Ra(L*) sin θ)
L*, Nu_{9.3}(Ra(L_{H}) cos θ)
L_{H}‖_{4} −90°≤θ≤0° UHT4
UHT4 has similarities with UCT4. Both use the L^{4}norm and T9.3. In both, the Ra arguments to correlations are multiplied by sin θ and cos θ. For an infinite strip, the characteristic lengths of the left terms are equal (R=L*).
In A heat transfer textbook[85], Lienhard and Lienhard attribute the use of g⋅cos θ for inclined laminar convection to:
B. R. Rich.
An investigation of heat transfer from an inclined flat plate in free convection.
Trans. ASME, 75:489–499, 1953.
They attribute the use of g⋅cos θ and g⋅sin θ for inclined turbulent convection to:
G. D. Raithby and K. G. T. Hollands.
Natural convection.
In W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, editors,
Handbook of Heat Transfer, chapter 4.
McGrawHill, New York, 3rd edition, 1998.
Here are the conditions for the Table 1 of chapter 24 stillair convection points, which are plotted in the lower right graph.
symbol value units or L description T 296.9 K mean of plate and ambient temperature ΔT 5.5 K temperature difference between plate and ambient P 101325 Pa air pressure (assumed, not specified) μ 1.86×10^{5} Pa⋅s dynamic viscosity k 0.0260 W/(m⋅K) thermal conductivity of air ρ 1.19 kg/m^{3} air density Pr 0.723 Prandtl number Ra 8.43×10^{6} L_{H} = 0.25 m Rayleigh numbers of smooth 0.25 m square plate Ra* 1.32×10^{5} L* = 0.0625 m Ra 2.31×10^{7} L_{H} = 0.35 m Rayleigh numbers of smooth 0.35 m square plate Ra* 3.61×10^{5} L* = 0.0875 m
The shape and size of the plate are not specified by Table 24.1. The graph to the left shows what the various correlations predict for a smooth square plate with side length from 0.24 m to 0.71 m. The top two traces are for horizontal upward convection using composite correlation HHT* and Table 9 correlation T9.7 respectively. The middle trace is for a vertical plate (h_{T9.3}). The bottom three traces are for a horizontal downward facing hot plate or upward facing cold plate. As discussed above, h_{T9.9} appears to have no basis in theory or published measurement; it is over 50% larger than surface conductances based on Fujii and Imura (Nu_{5}) and Schulenberg (Nu_{45}).
The h_{HHT*} curve matches the Table 24.1 −90° value at a squareside length of 0.35 m, while the h_{T9.3} curve matches the Table 24.1 vertical value at a squareside length of 0.37 m. The Nu_{45} and Nu_{5} curves match the +90° value around 0.7 m. The upper left plot shows that all of the conductances grow with decreasing length of the square's side; so simply adjusting the square's dimensions will not improve the fit.
Would nonsquare rectangular plates bring these points closer to their targets? We are looking to reduce h_{45}* while not reducing h_{HHT*} and h_{T9.3}. For the vertical plate, the width does not affect its surface conductance h_{T9.3}. So lets leave the height fixed and adjust the width. The conductance h_{45}* uses the length of whichever side is shorter. So h_{45}* does not change with increasing width. And h_{45}* increases with decreasing width, which degrades the match. Therefore downward facing surface conductance cannot be reduced by changing the aspectratio from 1 (square).
From this, we can conclude that the plateundertest in Table 24.1 was most likely square. In addition to coming within a few percent of the Table 24.1 values, a 0.35 m square is a practical size for measuring convection, requiring an enclosure no more than a few cubic meters in size. For the rest of this article, I will assume that the Table 24.1 plate was a 0.35 m square.
The plot of surface conductance is linked to a larger, highresolution version. The UHT2, UHT3, and UHT4 curves apply from −90° to 0°; the UCT3 and UCT4 curves apply from 0° to +90°.
The FujiiImura curve is not monotonic. This implies, and looking at Figure 10 confirms, that the lines of Nu values for 59.9° and 44.8° cross.
At θ values between −90° and −58°, the UHT2, UHT3 and UHT4 curves have significantly more conduction than the FujiiImura curve; this may be due to Fujii and Imura's device having sidewalls which restrict flow. Both FujiiImura and UHT2 reach a local minimum where x_{c}(θ)=L_{H}=.35 m at a θ value near −58°. At θ values greater than −58°, the T9.3 and (4) mode of convection flows along the full height of the plate, even though the HHT* mode has a higher surface conductance near θ=−58°. UHT3 uses whichever mode has higher convective heat transfer, and comes within 5% of the Table 24.1 value at −45°. UHT4 blends the modes using the L^{4}norm and comes within 1.2% of the the −45° value.
Small tilts around −90° should not change h much. And that is the behavior of h_{UHT3} and h_{UHT4}. h_{UCT3} and h_{UCT4} are very sensitive to tilts away from horizontal (+90°). Both curves exceed the Table 24.1 value by more than 16% at +45° (inclined facing down). The downward convection from a level plate exceeding the Table 24.1 value by more than 30% is unfortunate, but the natural convection from horizonatal and vertical plates is established by theory and experiment from multiple sources.
The asymmetry of the curve near vertical (0°) is interesting. As fluid heated at the bottom of a vertical plate rises, it stays near the plate, creating a thermal boundary layer which is wider at the top of the plate than at the bottom. Fig.14(a) from Fujii and Imura(1972)[76] shows that the hydraulic boundary layer is also thicker at the top than the bottom of heated vertical plate.
As the plate tilts up, buoyancy of heated fluid pulls parcels away from the plate, increasing convective transport and surface conductance. Further upwardtilting promotes more (HHT*) vertical mixing at the expense of (T9.3) fluid flowing the length of the plate.
Because the layer of heated fluid is thick at the upper end of a vertical plate, as the plate is tilted downward, the further thickening of the boundary layer does not much affect convection. Further downwardtilting reduces the buoyancy of fluid parcels traveling along the plate.
A correspondent wonders what the convection from both sides of a solarcell panel is. For natural convection, the doublesided panel shows much less variation with angle than the singlesided case.
Natural convection correlations for inclined rectangular plates involve characteristiclengths derived from L_{H} and L_{W}. The derivation of L_{eff} is in the section Natural OffAxis Convection Each correlation is computed using its own characteristiclength.
Proposed is the complete natural 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.
L* = L_{H}⋅L_{W} / (2 (L_{H}+L_{W})) R = min(L_{H}, L_{W})/2 L_{eff} = L_{H} L_{W}
cos φ L_{W} + sin φ L_{H}LVRR Pr = c_{p}⋅μ/k Ra(L) = ΔT⋅g⋅ρ^{2}⋅c_{p}⋅L^{3} / (T⋅μ⋅k) Nu_{HHT*}(Ra*) = {0.65 + 0.36 Ra*^{1/6}}^{2} 1≤Ra*≤1.5×10^{9} θ=−90° T9.3(Ra) = {.825 + .387 Ra^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} 1≤Ra≤10^{12} θ=0° Nu_{45}(Ra) = 0.544 Ra^{1/5}
[1+(0.785/Pr)^{3/5}]^{1/3}θ=+90° ‖x,y‖_{p} = (x^{p}+y^{p})^{1/p} x ≥ 0; y ≥ 0; p > 0
h^{ }= k⋅‖ Nu_{45}(Ra(R) sin θ)
R, Nu_{9.3}(Ra(L_{eff}) cos θ)
L_{eff}‖_{4} 0°≤θ≤+90° UCT4E h^{ }= k⋅‖ Nu_{HHT*}(Ra(L*) sin θ)
L*, Nu_{9.3}(Ra(L_{eff}) cos θ)
L_{eff}‖_{4} −90°≤θ≤0° UHT4E
These equations assume that T9.3 and Nu_{45} are independent of surface roughness (with meanheightofroughness much smaller than the characteristic length). Evidence from Clear et al supports the independence of HHT* from surface roughness. The section "Natural Convection from a Rough Plate" addresses the effects of surface roughness on T9.3 and Nu_{45}.
For a flat plate there is much less experimental data available for forced convection than for natural convection. To accurately measure forced convection requires all the equipment necessary for measuring natural convection incorporated in a very uniform, laminar windtunnel. Wind speed and direction add two independent variables, multiplying the number of measurements which must be taken.
The common correlations for forced convection 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 correlations. 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. But the smooth L=0.35 m panel posited for Fig. 24.1 would be entirely laminar.
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 correlation 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.
For a characteristiclength (L_{c}) of 0.305 m, FH yields a curve (lower red) which matches the Fig. 24.1 curves for glass and clearpine (black) only at windspeeds less than 2 m/s. If the impinging flow were laminar, the Reynolds numbers for this size smooth plate are low enough that the boundary layer would not progress to turbulence. If the data is valid, then turbulence must have been present (confirmed in Convection in a Duct)
The upper red curve is h_{8.11} (all turbulent).
The conventional Reynoldsnumber threshold Re_{c}=5×10^{5} is too high to affect a 0.305 m square smooth plate under these conditions. The description of Re_{c} in 2009 ASHRAE Fundamentals is quite tentative. Perhaps Re_{c} has a smaller value in this situation.
The graph to the right plots FH with Re_{c} held constant at seven values from 0 to 3×10^{5}. The top curve is for Re_{c}=0, which is the same as T8.11.
In the abstract for A comprehensive correlating equation for forced convection from flat plates[86], Stuart Churchill writes:
A correlating equation was developed which provides a continuous representation for all Pr and Re. Different constants are suggested for the local and mean Nusselt numbers and for uniform wall temperature and heating. These constants are based on the best available theoretical and experimental results.
Churchill expresses his correlations in terms of Φ, which multiplies Re by a function of Pr:
Φ = Re Pr^{2/3}
[ 1 + (0.0468 / Pr)^{2/3} ]^{1/2}(3)
Too complicated to reproduce here, his formula (17) takes a parameter Φ_{um} to determine the transition from laminar to turbulent flow, but gives no insight on how to predict Φ_{um}, which varies more than an order of magnitude fitting the data sets plotted in his Fig. 2, shown here:
On Fig. 2 have been plotted his laminar and turbulent asymptotes (.667 Φ^{1/2} and .04 Φ^{4/5}), T8.12 (FC with Re_{c}=5×10^{5}), FC with Re_{c}=10^{5}, and the scaledColburnanalogy (developed later in Forced Convection from a Rough Plate) adapted to Φ: f_{D}(Φ/9.5,0)⋅Φ/8. Correlation T8.12 (in cyan) breaks sharply from .667 Φ^{1/2}, unlike the curves produced by Churchill's correlation. T8.12 has negative curvature while Churchill's curves have positive curvature at the bottom of the laminarturbulent transition regions.
It is troubling that, although there are many points on the turbulent asymptote, there are only nine measured points on the laminar asymptote. Those nine points are reported by only two of the sources: Parmelee and Huebscher; and Jakob and Dow. The lack of points on the laminar asymptote from the other two experimental sources (one has one close point) could indicate that their apparatus was producing only turbulent, not laminar flow. That could have been caused by turbulence in the impinging flow, surface roughness, dullness of the leading edge, or vibration[85]. With the small number of laminar points, there doesn't seem to be enough evidence in Fig. 2 to justify Churchill's formula (17) over correlation FC with adjustable Re_{c}.
The FC curves (cyan and green) stay under the turbulent asymptote because they model part of the plate being in laminar convection (contributing lower Nu values to the average). All but one of the points where Φ≥2×10^{6} also are less than the turbulent asymptote. This stands in support of FC over Churchill's formula (17). Note that many points are on or above the turbulent asymptote where Φ<2×10^{6}, indicating that those experiments may have produced turbulent flows over the entire plate.
In any case, we are left with no means to predict Re_{c} or Φ_{um}. Given how difficult it seems to be to produce high Re_{c} values, for engineering purposes the pointwise maximum of T8.9 and T8.11 is the simplest asymptotically correct correlation.
Nu = max( T8.9(Re, Pr), T8.11(Re, Pr) ) FA
Note that Churchill's data points are for smooth plates. The Scaled Colburn Analogy Asymptote puts Nu for rough surfaces on linear asymptotes of Φ above the (smooth) turbulent asymptote.
Transition to turbulence seems to be a phenomena like supercooling. When pure water lacks nucleation particles, it can be cooled below its freezing point. Its not that the freezing point is below 0°C; but that nucleation particles are required as well. In forced flatplate convection, transition to turbulence requires requires Φ greater than 10^{4} and some surface or leadingedge roughness.
Forced Convection Heat Transfer at an Inclined and Yawed Square Plate—Application to Solar Collectors[80] by Sparrow and Tien applies only to square (or nearly square) plates. This paper uses a different correlation regime for heat transfer than Chapter 4 of 2009 ASHRAE Fundamentals:
j = St_{ }Pr^{2/3} St = h / (ρ c_{p} V)^{ } (2a)
Sparrow and Tien write:
In view of the insensitivity of the results to the angles of attack and yaw, a single formula can be given which represents all the data to an accuracy of 2.5 percent. The formula is
j=0.931_{ }Re^{−1/2} (6)
L_{c} is the side of the square, 7.62 cm in the experiment. V ranged from 4.5 to 24 m/s; Re varied from 2×10^{4} to 10^{5}. Thus they tested only laminar flow.
The Stanton Number, St, can be expressed in terms of dimensionless quantities:
St=Nu / (Pr⋅Re)
Combining these equations and solving for Nu:
Nu=0.931 Re^{1/2} Pr^{1/3} FCLY
Although the exponents match, the coefficient of FCLY (top red trace) is 40% higher than the coefficient of T8.9 (Nu=0.664 Re^{1/2} Pr^{1/3}). h_{FCLY} (top red trace) and h_{T8.9} (lower red trace) are compared to Fig. 24.1 curves for pine and glass (black) in the graph to the right.
Correlation FCLY computes the same convective surface conductance as Sparrow and Tien do for their example application (1.22 m and 2.44 m square solarcells), assuming a temperature of 15°C. However, even with a temperature difference (ΔT) as low as 1 K, the Rayleigh numbers for plates this large (Ra=2×10^{8} and Ra=1.6×10^{9}) are far in excess of the natural convection turbulence threshold Ra=10^{7}. Clear et al[82] write:
In retrospect, it seems likely that any time natural convection is turbulent, then the mixed natural/forced convection should be turbulent also.
If that is correct, then FCLY won't apply to the large plates because natural convection from them is turbulent.
Given T8.9's poor match to Fig. 24.1, Sparrow and Tien's correlation not matching Fig. 24.1 is not troubling. But Sparrow and Tien's correlation coefficient being 40% higher than T8.9's is indeed troubling. Sparrow and Tien measured mass transfer, not heat transfer, and presented their data only in terms of dimensionless quantities. The information in their paper seems insufficient to resolve this discrepancy. An additional problem with this paper is detailed below.
In view of the insensitivity of the results to the angles of attack and yaw, a single formula can be given which represents all the data to an accuracy of ±2.5 percent.
Sparrow and Tien's claim that the sensitivity of laminar convection to flow angle is less than 2.5% raises the question of what variation the theory expects from rotation of the square plate. For flows parallel to the plate's surface (angleofattack=0°), this can be computed by summing the heat contributions from strips of the plate (with L equal to strip length) which are parallel to the impinging flow. The worst case variation should be in comparing an edgeparallel flow h_{p} to a diagonal flow h_{d}.
h_{p} = k ⋅ 0.664 (V⋅L/ν)^{1/2} Pr^{1/3} / L HP
For flow in the diagonal direction, integrate h times the strip length (2w), then divide by the area (L^{2}/2):
h_{d} = 2^{1/2}L
∫
0(k ⋅ 0.664 (2w⋅V/ν)^{1/2} Pr^{1/3} / 2w) 2 w dw / (L^{2} / 2)
h_{d} = k ⋅ 0.664 Pr^{1/3} (V/ν)^{1/2} 2^{3/2} 2^{1/2}L
∫
0w^{1/2} dw / L^{2} = k ⋅ 0.664 Pr^{1/3} (V/ν)^{1/2} (2^{3/2}) (2/3) (2^{1/2}L)^{3/2} / L^{2} = k ⋅ 0.664 Pr^{1/3} (V⋅L/ν)^{1/2} (2^{7/4}/3) / L = (2^{7/4}/3) h_{p} h_{d} ≈ 1.12 h_{p}
Theory expects the diagonal flow to convect 12% more heat than the axisaligned flow, more than double Sparrow and Tien's 5% range. Reviewing their figures, it turns out that the minimum angleofattack they measured was 25°. Nothing they report can be tested against established correlations. Neither can their results be applied to cases where the flow might be inclined less than 25° from the plate surface.
Surface Conductances as Affected by Air Velocity, Temperature and Character of Surface[84] by F.B. Rowley, A.B. Algren, and J.L. Blackshaw is the source for Fig. 24.1. Their test plate was inserted (flush with interior) into a .305 m by .305 m opening cut from one vertical side of a .152 m by .305 m rectangular duct fed by a fan through 5.18 m of this duct. They claim:
This arrangement provided a flow of air past the test surface without turbulence.
The pipe and duct correlations assume that a laminar flow enters the test section from open space. Their 5.18 m of duct preceding the test plate does not meet that criteria. The interior of this duct's hydraulic diameter (4 times crosssectional area divided by its perimeter) is: D_{H}=.2032 m. At the minimum nonzero flow they used for measurements, 3.6 m/s, the duct has a Rynolds number of 56985, well in excess of the laminar cutoff of 2300 at V=0.15 m/s.
So the flow is certainly turbulent. Moreover, flowing through 5.18 m of duct makes it a fullydeveloped turbulent flow (turbulent through whole crosssection), which enables modeling of convection using the Colburn analogy (see Surface Roughness).
The duct correlations assume that the test section is heated on all sides; but only one side is heated in the apparatus of Rowley et al. Unlike natural convection, forced flow is not much affected by heat convection. So the surface conductance for one heated side should be close to that for four sides.
However, forced flow is affected by the surface roughness. So the surfaceconductance values for rough foursided test segments may differ from rough onesided segments (with three smooth unheated sides).
Rowley et al's curve for glass (with radiative transfer removed) is very close to the Colburn analogy curve with ε=0 and the T8.5a curve above 3 m/s (see Surface Roughness). It shows the expected Re^{4/5} curvature. The ε values for the other (red) lines were picked to make them close to the Fig. 24.1 curves (in black). These meanheightofroughness (ε) values are plausible (see next section) for their respective materials. The three roughest surfaces have nearly straight trajectories, as expected from the Moody Chart. As cautioned earlier, the Fig. 24.1. curves diverge from those predicted by the Colburn analogy as the roughness increases.
Below Re=2300 the flow is laminar, Nu=3.66, and h=0.42 W/(m^{2}⋅K).
Here are mean height of roughness (ε) values from six sources for three materials called out in Fig. 24.1.
source  

[R1] 
[83]
S Beck and R Collins, University of Sheffield, File:Moody diagram.jpg, Wikipedia, the free encyclopedia, 2008, Linked from: http://en.wikipedia.org/wiki/Moody_chart 
[R2] 
Walter Benenson Handbook of Physics (2006) 
[R3] 
Chapter 8 Steady Incompressible Flow in Pressure
Conduits (Part B) Shandong University 
[R4] 
Fluid Flow in Pipes scribd.com, and CIVE2400 Fluid Mechanics, Section 1: Fluid Flow in Pipes University of Leeds 
[R5] 
Table 14–2* Yunus A. Çengel and Robert H. Turner Fundamentals of ThermoFluid Sciences McGrawHill Science/Engineering/Math; 3 edition (Jun 29 2007) also from University of Oslo. *The uncertainty in these values can be as much as ±60 percent. 
[R6] 
Moody, L. F., Friction factors for pipe flow, Transactions of the ASME 66 (8): 671–684 1994 
glass  wood  concrete  

[R1]  0.0025 mm  0.0250.25 mm  
[R2]  13 mm  
[R3]  0.0015 mm  0.180.6 mm  
[R4]  0.003 mm  0.03 mm  
[R5]  0 mm  0.5 mm  0.9–9 mm 
[R6]  0 mm  0.180.9 mm  0.33 mm 
There appears to be no published theory or measurements relating meanheightofroughness to forced convective heat transfer of open plates (Rowley et al[84] measured a duct). Someone trying to compute forced surface conductance of a rough plate has only the smooth plate conductance as a lower bound; the rest will be guesswork.
As mentioned earlier, measuring convective heat transfer from flat plates is difficult. Measuring (forced) convective heat transfer from flow inside pipes and ducts is comparatively easy; theory and experimental data are abundant. Can forced convection pipe correlations be adapted to flat plates?
The ratio of the mean height of roughness in a pipe to the pipe diameter (its characteristic length) affects the friction factor used to compute head loss in pipes. The Chilton and Colburn Jfactor analogy relates friction factors to forced convective heat transfer. Citing Dittus and Boelter (1930), the ASHRAE Fundamentals 2009 Table 4.8 gives correlations for a smooth pipe which are similar to the correlations resulting from Colburns's analogy:
II. Internal Flows for Pipes and Ducts: Characteristic length = D, pipe diameter, or D_{h}, hydraulic diameter. Nu = (f/2) Re Pr^{1/3} Colburn's analogy (T8.1) Turbulent:
Fully developedNu = 0.023 Re^{4/5}Pr^{0.4} Heating fluid
Re ≥ 10000(T8.5a)^{b} Nu = 0.023 Re^{4/5}Pr^{0.3} Cooling fluid
Re ≥ 10000(T8.5b)^{b}
Combining equation T8.1 with T8.5 and solving for the Darcy friction factor (which is 4 times the Fanning friction factor) yields:
f_{D} = 4⋅2⋅0.023 Re^{−1/5} DF5
The Moody chart (from Wikipedia) shown below is for the Darcy friction factor, which is 4 times the Fanning friction factor. The Moody chart consists of two segments: 64/Re for Re values less than 2300, and solutions of the ColebrookWhite equation:
1
f_{D}^{1/2}= −2 log_{10} ( ε
3.7 D_{H}+ 2.51
Re ⋅ f_{D}^{1/2})
In the Colburn analogy, Nu is proportional to the product of the friction factor and Re. Thus horizontal sections of the friction factor curve produce Nu and h values which increase linearly with Re. DF5 is shown in red on the Moody chart. It is a good match for the middle of the smooth pipe curve.
Lienhard and Lienhard[85] write:
Turbulence near walls
... In a turbulent boundary layer, the gradients are very steep near the wall and weaker farther from the wall where the eddies are larger and turbulent mixing is more efficient. This is in contrast to the gradual variation of velocity and temperature in a laminar boundary layer, where heat and momentum are transferred by molecular diffusion rather than the vertical motion of vortices. In fact,the most important processes in turbulent convection occur very close to walls, perhaps within only a fraction of a millimeter. The outer part of the b.l. is less significant.
The viscous sublayer
... The [viscous] sublayer is on the order of tens to hundreds of micrometers thick, depending upon the fluid and the shear stress. Because turbulent mixing is ineffective in the sublayer, the sublayer is responsible for a major fraction of the thermal resistance of a turbulent boundary layer. Even a small wall roughness can disrupt this thin sublayer, causing a large decrease in the thermal resistance (but also a large increase in the wall shear stress).
Conversely, when the wall roughness is too small to disrupt the viscous sublayer, the convective surface conductance should be the same as for a smooth plate under the same conditions.
At the scale of the viscous sublayer, the inner radius curvature (being perpendicular to the flow) in a large diameter pipe should be indistinguishable from a flat plate. Because most of the convective thermal resistance occurs close to the surface of a pipe or plate, it is reasonable to expect the turbulent correlations for pipe and plate to be related.
Consider the similarity of Colburns's analogy (T8.1) to T8.9 and T8.11 for forced convection on plates:
Nu = (f/2) Re Pr^{1/3} Colburn's analogy (T8.1) Laminar boundary layer:
Re < 5 × 10^{5}Nu = 0.664 Re^{1/2} Pr^{1/3} Average value of h (T8.9) Turbulent boundary layer
beginning at leading edge:
All ReNu = 0.037 Re^{4/5} Pr^{1/3} Average value of h (T8.11)
Combining equation T8.1 with each correlation and solving for Darcy friction factors yields:
f_{D} = 4⋅2⋅0.664 Re^{−1/2} DF9 f_{D} = 4⋅2⋅0.037 Re^{−1/5} DF11
DF11 in blue is parallel to DF5 in red because Re has the same exponent in both. DF9 in cyan looks like it may be parallel to the asymptote of the low Re end of the solutions of the ColebrookWhite equation.
If D (and Re) in the ColebrookWhite equation were scaled to move D5 to D11 then this scaled ColebrookWhite equation would yield smooth flat plate correlations using the Colburn analogy.
Although it must have the correct relation to dimensions, the scale of characteristiclength is otherwise a matter of convention. The forcedconvection correlations are usually monomials in Re and scale easily. With their characteristiclengths being perpendicular, the geometry of pipes and flat plates are so different from each other that it is not unreasonable that scaling may be necessary in order to make correspondences between them.
Scaling the characteristiclength and Reynolds number by 9.5 in the ColebrookWhite equation moves the friction curves to nestle into the flat plate asymptotics (D_{H}=L/9.5):
1
f_{P}^{1/2}= −2 log_{10} ( 9.5 ⋅ ε
3.7 ⋅ L+ 9.5 ⋅ 2.51
Re ⋅ f_{P}^{1/2}) SCW
The result of combining T8.1 with SCW can be expressed in terms of the unscaled ColebrookWhite equation:
Nu = 1
8f_{D}(Re/9.5, 9.5 ε/L) Re Pr^{1/3} ScaledColburnanalogy SCA
Correlation SCA transitions gradually between T8.9 and T8.11 around Re=17000.
The graph to the right is of convective surface conductance versus (turbulent) fluidspeed for a smooth plate having length (in the direction of flow) of .305 m, 1 m, 10 m, and 100 m comparing T8.11 with the scaledColburnanalogy (SCA). The curves of each pair are close to each other. The scaledColburnanalogy curve is higher than the T8.11 curve for L=100 m because the Darcy friction factor levels out at high Reynolds numbers, raising the effective exponent from 4/5 (to 1).
The graph to the right shows that h becomes a nearly linear function of V when ε>0.
Inside a closed duct there is a limited volume of fluid to receive the convected heat; the air above a flat plate is unlimited. Thus the open plate's forced convective surface conductance should be larger than the same plate in a duct.
That is the case when comparing the Fig. 24.1 curves (from Rowley et al) with the scaled Colburn analogy for the same roughness heights posited for forced convection in a duct. If this analysis is correct, then Fig. 24.1 underestimates forced turbulent convection from a plate.
At low Reynolds numbers f_{D} has little sensitivity to relative roughness.
As Re tends to infinity, the ColebrookWhite equation becomes (solid red in graph to the right):
f_{D} = 1
4log_{10} ( ε / D
3.7)^{−2}
As Re tends to infinity the scaledColebrookWhite equation becomes:
f_{P} = 1
4log_{10} ( 9.5 ε
3.7 L)^{−2} SCWA
Note that the graph of the ColebrookWhite equation and the scaled ColebrookWhite equation have different abscissa ranges.
The ColebrookWhite equation assumes that the mean height of roughness, ε, is much smaller than the characteristiclength. As ε approaches 0.39 L, the denominator of SCWA goes to 0. So the range of application can be tightened to ε≪0.4 L.
Although the (friction factor) solutions of the ColebrookWhite equation are monotonic as a function of Re, the Nusselt numbers resulting from their use in the Colburn analogy are not monotonic at low Reynolds numbers. These values are out of range for the Colburn analogy because the Colburn analogy assumes a laminar boundary layer depth dependent on the Reynoldsnumber [Prof. J. Lienhard, private communication]. So it behooves us to examine the correlation produced by asymptotics of the analogy. Combining the Colburn analogy (T8.1) with SCWA in the preceding section we find that for large Reynolds numbers over rough surfaces:
Nu = 1
4 ⋅ 8log_{10} ( 9.5 ε
3.7 L)^{−2} Re Pr^{1/3} ≈ 0.031 Re Pr^{1/3}
log_{10}( 2.57 ε / L )^{2}ε≪0.4 L RS
The black dashed lines for RS in the graph to the right indeed tend to the scaledColburnanalogy correlations for rough plates at large Reynolds numbers.
The scaledColburnanalogyasymptote correlation (RS) can be combined with the correlation FA developed for smooth plates. This is accomplished by the SCAA correlation, which takes the pointwise maximum (Nu) of the T8.9, T8.11, RS correlations:
Nu = max( T8.9(Re, Pr), T8.11(Re, Pr), RS(Re, Pr, ε/L) ) SCAA
The established theory for forced convection on plates gives correlations for the laminar and turbulent asymptotes, but not for the transition zone. Working with asymptotes means that the proposed correlation for the rough plate convection asymptote can be replaced if the data from experiments turns out to significantly differ with it.
Turbulent flow over both flat smooth plates and inside smooth pipes has Nu proportional to Re^{4/5}. For the pipe, surface roughness which disrupts the viscous sublayer has Nu proportional to Re. Scaling the pipe correlations to match the plate results in correlation RS (which is proportional to Re) for rough plates.
Proposed is the convective surface conductance for one side of a flat rectangular isothermal plate having mean height of roughness ε(≪L) with flow at velocity V parallel to one edge which is L long.
Re = V⋅L / ν
Laminar flow:
Nu_{8.9} = 0.664 Re^{1/2} Pr^{1/3} (T8.9)
Turbulent flow:
Nu_{8.11} = 0.037 Re^{4/5} Pr^{1/3} (T8.11)
Turbulent Flow over rough surface:
Nu_{RS} = 0.031 Re Pr^{1/3}
log_{10}( 2.57 ε / L )^{2}ε≪0.4 L RS
Combined:
Nu_{SCAA} = max( Nu_{8.9}, Nu_{8.11}, Nu_{RS} ) SCAA
The forcedconvection surface conductance is:
h = Nu_{SCAA} k / L
The effect of meanheightofroughness on surface conductance predicted by correlation RS looks reasonable but hasn't been tested experimentally.
The Boundary Layer section asserts that laminar flow over a rough surface has the same convective surface conductance as laminar flow over a smooth plate. Note that when the meanheighofroughness is larger than the thickness of the viscoussublayer (but still much less than L_{c}), the flow becomes turbulent and the assertion about laminar flow doesn't apply.
The boundary layer for downward convection (from a horizontal plate) is always laminar, its sole correlation having the Re exponent 1/5. It has the thickest boundary layer among orientations. If turbulence does appear, it will be in plumes from the edges of the plate, well away from the nearly all of the boundary layer (and not affect h).
The upward convection (from horizontal plate) measured by Clear et al was turbulent because of the large roof size. Their Rayleigh number of 10^{15} dwarfs the T9.8 upper bound of 1.5×10^{9}, yet they fit a coefficient of 1.01 to T9.8 for a horizontal roof (which wasn't smooth), but a significantly larger coefficient fitting T8.8 (local forced convection). They report the standarderror as 0.03 and 0.02, respectively. The coefficient of 1.01 indicates that natural convection from the rough roof was hardly different from that expected for a smooth roof.
Unlike plates in other orientations, buoyancy drives parcels of fluid warmed by an upward facing plate to change places with cooler fluid above them, directly away from the plate. Fluid flowing away from the plate would not experience friction with surface roughness. Is unheated fluid flow along the surface of plate affected by surface roughness? The streamlines in Fujii and Imura[76]'s figure 14(f) show most of the unheated fluid approaching the plate at angles between than 30° and 45° from horizontal. Without much fluid flowing along the plate's surface, it makes sense that turbulent upward convection would be unaffected by surface roughness.
The longer a parcel of fluid is in contact with the plate, the less additional heat is transferred to it. The most heat is transferred when the most fluid contacts the plate per unit time. Because the parcels immediately move away from the plate in T9.8 mode, it has the highest convective surface conductance among orientations.
Forced flow in a pipe experiences head (pressure) loss. The pressure being a force that acts through a distance (along the pipe), means that the flow does mechanical work on the system. The same should be true for flow along a vertical plate. The (convective) conversion of the heat difference to mechanical energy necessarily has poor efficiency. Thus the fluid velocity produced by natural convection must be much less than the forced fluid velocity producing the same surface conductance, all else being equal. The low turbulent shear stress resulting from low fluid speeds results in a thick viscus sublayer (described by Lienhard and Lienhard[85]) where laminar flow dominates. As asserted by the Boundary Layer section, when the wall roughness is too small to disrupt the viscous sublayer, the convective surface conductance should be the same as for a smooth plate under the same conditions.
With the possible exception of very high Rayleighnumbers on nearly vertical plates, natural convection from a rectangular plate is insensitive to surface roughness (ε≪L).
Over flat plates, forced fluid flow is assumed to have straight, parallel streamlines. Thus convex plates with nonuniform characteristiclength 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} (axisaligned) 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 correlations 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 inplane offaxis forcedconvection 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 roughsurface correlation 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 rotationfactors for 1 m^{2} rectangular plates with ε=10^{4} m and aspectratios of 1:1 (black), 2:1 (blue), and 4:1 (red).
Clear et al[82] give characteristiclength 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 15minute 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 aspectratio of 4.8. The ratio of surface conductivities for a rectangle with aspectratio 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_{ }^{1E}
In the turbulent case, an aspectratio 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 shortterm variability.
The Forced OffAxis Convection section developed formulas F_{tur} and F_{lam} for turbulent and laminar rotation factors which are functions of aspectratio 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 aspectratio. 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 effectivelength L_{eff}. The effectivelength shape factors are simply F_{tur}^{−5} and F_{lam}^{−2}
The spread between the shape factors grows from 3% at an aspectratio of 1:1 to 6% at an aspectratio of 10:1. Correlations 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
Effectivelength greatly simplifies treatment of uniformly distributed flow direction. Can it do the same for Forced OffAxis Convection? To the right are plots of laminar and turbulent offaxis rotation factors.
Laminar and turbulent effectivelengths are very close, enabling a single formula for laminar and turbulent flows and use with the scaledColburnanalogy 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 rotationfactors for 1 m^{2} rectangular plates with ε=10^{4} m and aspectratios 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 rotationfactors by integration (solid) and using effectivelength (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 effectivelength 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 characteristiclength is half of the shortest side, which is also independent of orientation.
The vertical plate convection mode corresponds to correlation T9.3 (1≤Ra≤10^{12}):
T9.3(Ra) =  {.825 +  .387 Ra^{1/6} [1+(.492/Pr)^{9/16}]^{8/27} 
}^{2} 
Ra is proportional to L^{3}. As a function of L, h_{9.3} 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_{9.3}(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_{9.3}(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_{9.3}(L_{eff}) with h_{9.3}(L)⋅F_{9.3}, both with mean side lengths of 10^{4} m. For side lengths greater than 1 m, h_{9.3} is much less sensitive to L_{eff} and rotation. The graph on the right compares h_{9.3}(L_{eff}) with h_{9.3}(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 rangeofinterest).
The technique for finding average h in LaminarTurbulent 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, correlations 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 correlations 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 correlations 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 correlations 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 HHT* predicts, both for 0.1 m and 0.25 m widths. The graph to the right shows HHT* 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 windspeed 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 windspeed was measured at one point on the roof. The subscript of Gr_{x} implies that the Grashof Number is also local, but the correlations 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 standarderror 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 nonclear 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 midafternoon 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 sublinear.
Over Fig. 9 I have plotted the convective surface conductances predicted by correlations T9.8 and HHT* (natural convection above a hot horizontal plate). Above surfacetoair temperature differences of 20 K, the measured values are greater than or equal to those predicted by HHT*. 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 longwave 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, HHT* 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 windspeeds 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 windspeeds, one might conclude that h values at windspeeds 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 surfacetoairtemperaturedifference 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 windspeed aids natural convection, natural convection does not aid forced convection at higher windspeeds. Thus the h values for windspeeds greater than 4 m/s should be much less affected by the culling of low ΔT points than those at lower windspeeds.
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, T9.3, HHT*, and Nu_{45} all use different characteristiclengths, 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 characteristiclengths applies depending on the h values computed with them. Furthermore, the Ra arguments to correlations T9.3 and HHT* 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 scaledColburnanalogy curves for various surface roughness values (ε). Because of surface roughness, all but the lowest scaledColburnanalogy 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.
Among orientations, a vertical plate produces the natural convection which is the most like its (vertical) forced convection. The natural and forced convection correlations for vertical plates both have characteristiclength equal to the plate height. In both cases, the flows are vertical. In both, the boundary layer is thin at the leading edge and thickens toward the leeward edge.
The plots to the right are of Lienhard and Lienhard[85]'s approximations to the normalized natural and forced laminar boundarylayer profiles. They show that the natural and forced temperature profiles are quite similar. Because of the similarity of the temperature profiles, the overall surface conductance should be similar.
If the characteristic velocity u_{c} is scaled down by 1.6 (62.5%), then the curves match near the origin. In the plot to the right, the trace of the superposition of opposing forced and natural flows is zero up to y/δ=0.05. The fluid in this region is static, so heat transfer across it is conduction, not convection.
Although Lienhard and Lienhard also analyze the turbulent forced boundary layer, a similar analysis of the turbulent natural boundary layer is proving difficult to find. We proceed with the assumption of turbulent natural/forced similarity and justify it later.
The natural/forced similarity suggests that we can find a bulk fluid velocity V_{fe} whose forced convective surface conductance is equal to the natural convective surface conductance. That is, find a functional inverse for h_{SCAA}(Re(V)). The first step is to compute Re_{fe}. For laminar flows:
Re_{fe} = (Nu/Pr^{1/3})^{2}
0.664^{2}V_{fe} = ν (Nu/Pr^{1/3})^{2}
0.664^{2} L
For turbulent flows:
Re_{fe} = (Nu/Pr^{1/3})^{5/4}
0.037^{5/4}V_{fe} = ν (Nu/Pr^{1/3})^{5/4}
0.037^{5/4} L
For flow over a rough plate, use the scaledColburnanalogyasymptote.
Re_{fe}^{ }= 32 log_{10}( 2.57 ε / L )^{2} Nu/Pr^{1/3} V_{fe} = 32 log_{10}( 2.57 ε / L )^{2} ν Nu/Pr^{1/3}
L
These expressions can be combined:
Re_{fe} = min ( (Nu/Pr^{1/3})^{2}
0.664^{2}, (Nu/Pr^{1/3})^{5/4}
0.037^{5/4}, 32 log_{10}( 2.57 ε / L )^{2} Nu/Pr^{1/3} ) V_{fe} = ν min ( (Nu/Pr^{1/3})^{2}
0.664^{2} L, (Nu/Pr^{1/3})^{5/4}
0.037^{5/4} L, 32 log_{10}( 2.57 ε / L )^{2} Nu/Pr^{1/3}
L) V_{fe} = ν min ( L (h/Pr^{1/3})^{2}
0.664^{2} k^{2}, L^{1/4} (h/Pr^{1/3})^{5/4}
0.037^{5/4} k^{5/4}, 32 log_{10}( 2.57 ε / L )^{2} h/Pr^{1/3}
k)
Re_{fe} is shown in the graph to the right, along with the scaledColburnanalogy for a smooth plate.
How does the natural laminarturbulent transition compare with the forcedequivalent laminarturbulent transition? The following chart shows the Reynolds, Nusselt, and Grashof numbers (with Pr=.728) for several values used as laminarturbulent transitions for convection from a flat vertical plate:
Re  Nu  Gr=Ra/Pr  source 

1.51×10^{4}  73.5  2.52×10^{8}  intersection of T8.9 and T8.11 
2.56×10^{4}  112  10^{9}  Latif M. Jiji[89] 
5×10^{5}  1210  1.77×10^{12}  2009 ASHRAE Fundamentals 
The values for a Re=5×10^{5} transition point are more than an orderofmagnitude larger than those from the other sources. Transition numbers that high are reached with smooth plates with sharp leading edges.
A number of secondary sources give the natural laminarturbulent transition as 10^{8}≤Gr≤10^{9}. The T8.9T8.11 intersection Grashof number is in that range and fairly close to 10^{9}. This transition agreement gives confidence that the forcedequivalent flow can stand in for natural flow in interactions with other forcing.
A forced vertical flow will either aid or oppose the natural convection. So the mixed velocity will be either the sum or difference of V and V_{fe}. If V and V_{fe} are opposing and of equal magnitude, then the mixed velocity near the plate will be zero. The acutal minimum thermal conductance would not be less than that resulting from thermal conduction or molecular diffusion through static fluid.
Because the natural and forced characteristiclengths are the same in this case, calculations can be done in terms of dimensionless quantities. For laminar flows:
Nu = 0.664 Re ± Re_{fe}^{1/2} Pr^{1/3}
Nu = 0.664  Nu_{f}^{2}
0.664^{2} Pr^{2/3}± Nu_{9.3}^{2}
0.664^{2} Pr^{2/3}^{1/2} Pr^{1/3}
Nu = Nu_{f}^{2} ± Nu_{9.3}^{2}^{1/2}
The chart to the right plots laminar vertical mixed convection of Nu_{f} from 1 to 40 with Nu_{9.3} values of 1, 2, 5, and 10.
The chart to the right plots turbulent vertical mixed convection of Nu_{f} from 250 to 10000 with Nu_{9.3} values of 250, 500, 1250, and 2500.
Nu = Nu_{f}^{5/4} ± Nu_{9.3}^{5/4}^{4/5}
Because h∝Nu and Re∝V, for laminar and turbulent flows:
h = h_{f}^{2} ± h_{9.3}^{2}^{1/2} h = h_{f}^{5/4} ± h_{9.3}^{5/4}^{4/5}
in the plane of a vertical plate, consider forced convection at bulk velocity V_{f} inclined at angle ψ from vertical with a forced effective bulk velocity V_{fe}. the combined velocity vector is:
( V_{f} sin ψ, V_{fe} + V_{f} cos ψ )
its magnitude and angle are:
V = ( V_{f}^{2} + 2 V_{fe} V_{f} cos ψ + V_{fe}^{2} )^{1/2} ζ = arctan V_{f} sin ψ
V_{fe} + V_{f} cos ψ
using ζ and LERR, the characteristiclength L_{eff} can be computed. L_{H} is the height of the plate; A=L_{H}⋅L_{W} and r=L_{W}/L_{H}. Using V and L_{eff} with SCAA, the mixed convective surface conductance h can be computed.
The polar plot to the right shows the surface conductance of a 1 m by 1 m plate having 2 m/s forcedequivalent natural convection combined with .5 m/s, 2 m/s, or 8 m/s inplane flow at all angles. Below are polar plots of the surface conductance resulting from equal area plates with 4:1 and 1:4 aspect ratios.
While natural convection from a vertical plate is akin to forced convection, the modes of natural convection for horizontal plates are not. Both are symmetrical, having no net flow parallel to the plane of the plate. To first order, adding a uniform fluid velocity to these modes strengthens the convective surface conductance for one half while reducing it for the other.
The theory developed for natural convection from an inclined plate teaches that for downward inclined plates the interaction between h_{9.3}(Ra⋅cos θ) and h_{45}(Ra) is competitive; whichever would produce the highest h wins. This is unsurprising because h_{45} is very weak compared with h_{9.3}. Less intuitive (perhaps) is that for upward inclined plates the interaction between h_{9.3}(Ra⋅cos θ) and h_{HHT*}(Ra⋅sin θ) is also competitive. For the plate with sidewalls used by Fujii and Imura[76], multiple modes can be present simultaneously. Because they are not convex, plates with sidewalls are not covered by this analysis.
Because natural convection from a vertical plate is so much like forced convection, it is logical to expect that h_{9.3} forcedequivalent convection will also be competitive with h_{45} and h_{HHT*}. Physically this means that verticalmode convection and external forcing combine through addition of their fluid velocity vectors, but do not exist simultaneously with the horizontalplate modes of convection.
The procedure to compute the (mixed) convective surface conductance of a rectangular plate at angle θ from vertical is then:
Allowing for the plate to be rotated φ around its center adds more complication, which is detailed below. R_{eff} is the effectivelength for naturalconvection 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
Nu_{HHT*} = {0.65 + 0.36 Ra*^{1/6}}^{2} −90°≤θ_{e}≤0° 1≤Ra*≤1.5×10^{9} Nu_{9.3} = {.825 + .387 Ra_{V}^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} 1≤Ra_{V}≤10^{12} Nu_{45} = 0.544 Ra_{R}^{1/5}
[1+(0.785/Pr)^{3/5}]^{1/3}0°≤θ_{e}≤+90°
The bulk fluid velocity which would have produced the same Nu as Nu_{9.3} is:
V_{fe} = ν min ( (Nu_{9.3}/Pr^{1/3})^{2}
0.664^{2} R_{eff}, (Nu_{9.3}/Pr^{1/3})^{5/4}
0.037^{5/4} R_{eff}, 32 log_{10}( 2.57 ε / R_{eff} )^{2} Nu_{9.3}/Pr^{1/3}
R_{eff})
The (forced) bulk fluid velocity is V_{f} at angle ψ from the normal projection of vertical in the plane of the plate. The magnitude and angle relative to the plate of the mixed velocity vector ( V_{f} sin ψ, V_{fe} + V_{f} cos ψ ) is:
V = ( V_{f}^{2} + 2 V_{fe} V_{f} cos ψ + V_{fe}^{2} )^{1/2} ζ = arctan ( V_{f} sin ψ
V_{fe} + V_{f} cos ψ) − φ
The effective (characteristic) length for flow at angle ζ from the rectangle orientation is:
L_{eff} = A^{1/2} cos ζ^{−1} r^{−1/2} ( 1 + tan ζ
3 r)^{−2} r>tan ζ LERR A^{1/2} sin ζ^{−1} r^{+1/2} ( 1 + r
3 tan ζ)^{−2} 0<r≤tan ζ
Reynolds number for forcedconvection:
Re = V⋅L_{eff} / ν
The forced/T9.3 Nusselt number where ε≪0.4 L_{eff}:
Nu_{SCAA} = max( 0.664 Re^{1/2} Pr^{1/3}, 0.037 Re^{4/5} Pr^{1/3}, 0.031 Re Pr^{1/3}
log_{10}( 2.57 ε / L_{eff} )^{2}) SCAA
The mixed convective surface conductance is:
h^{ }= k⋅‖ Nu_{45}
R, Nu_{SCAA}
L_{eff}‖_{4} sin θ_{e}≥0 UCT4E h^{ }= k⋅‖ Nu_{HHT*}
L*, Nu_{SCAA}
L_{eff}‖_{4} sin θ_{e}≤0 UHT4E
The heat flow is h⋅A⋅ΔT.
A rotating cup anemometer at 10 m height is the standard windspeed 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 lowpitch roofs examined here, the windspeed, 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 winddriven 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, winddirection and speed, temperature, airpressure, and relativehumidity.
datatype name returnunits 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:
datatype 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 θA = 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_{9.3} flow from the ridgeline will dominate Nu_{45} flow when the roof is not flat. The effective plate dimensions for Nu_{45} will be L_{W} by L_{H} when θ=−90 and are uncritical otherwise. The characteristic length for Nu_{9.3} 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_{HHT*} = {0.65 + 0.36 Ra*^{1/6}}^{2} −90°≤θ_{e}≤0° 1≤Ra*≤1.5×10^{9} Nu_{9.3} = {.825 + .387 Ra_{V}^{1/6}
[1+(.492/Pr)^{9/16}]^{8/27}}^{2} 1≤Ra_{V}≤10^{12} Nu_{45} = 0.544 Ra_{R}^{1/5}
[1+(0.785/Pr)^{3/5}]^{1/3}0°≤θ_{e}≤+90°
The bulk fluid velocity which would have produced the same Nu as Nu_{9.3} is:
V_{fe} = ν min ( (Nu_{9.3}/Pr^{1/3})^{2}
0.664^{2} R_{eff}, (Nu_{9.3}/Pr^{1/3})^{5/4}
0.037^{5/4} R_{eff}, 32 log_{10}( 2.57 ε / R_{eff} )^{2} Nu_{9.3}/Pr^{1/3}
R_{eff})
The (forced) bulk fluid velocity is V_{f} at angle ψ from the normal projection of vertical in the plane of the left plate. The left and right mixed velocity vectors are ( V_{f} sin ψ, V_{fe} + V_{f} cos ψ ) and (−V_{f} sin ψ, V_{fe} + V_{f} cos ψ ). Their magnitudes are:
V_{l} = ( V_{f}^{2} + 2 V_{fe} V_{f} cos ψ + V_{fe}^{2} )^{1/2} V_{r} = ( V_{f}^{2} − 2 V_{fe} V_{f} cos ψ + V_{fe}^{2} )^{1/2}
Their angles are:
ζ = ±arctan ( V_{f} sin ψ
V_{fe} + V_{f} cos ψ)
The effective (characteristic) length for the mixed flow is independent of the sign of ζ. So in both cases it is:
L_{eff} = A^{1/2} cos ζ^{−1} r^{−1/2} ( 1 + tan ζ
3 r)^{−2} r>tan ζ LERR A^{1/2} sin ζ^{−1} r^{+1/2} ( 1 + r
3 tan ζ)^{−2} 0<r≤tan ζ
The Reynolds numbers for the left and right plates:
Re_{l} = V_{l} L_{eff} / ν Re_{r} = V_{r} L_{eff} / ν
For the left and right plates with ε≪0.4 L_{eff}:
Nu_{l} = max( 0.664 Re_{l}^{1/2} Pr^{1/3}, 0.037 Re_{l}^{4/5} Pr^{1/3}, 0.031 Re_{l} Pr^{1/3}
log_{10}( 2.57 ε / L_{eff} )^{2}) Nu_{r} = max( 0.664 Re_{r}^{1/2} Pr^{1/3}, 0.037 Re_{r}^{4/5} Pr^{1/3}, 0.031 Re_{r} Pr^{1/3}
log_{10}( 2.57 ε / L_{eff} )^{2})
The average of the left and right Nusselt numbers is for the combined forced/T9.3 flow. The mixed convective surface conductance is:
h^{ }= k⋅‖ Nu_{45}
R, Nu_{l} + Nu_{r}
2 L_{eff}‖_{4} sin θ_{e}≥0 h^{ }= k⋅‖ Nu_{HHT*}
L*, Nu_{l} + Nu_{r}
2 L_{eff}‖_{4} 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 winddirection and velocity, temperature, airpressure, and relativehumidity.
datatype name returnunits 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:
datatype 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 T9.3 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 1 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 windspeeds greater than 4 m/s the surface roughness has greater effect on heatflow 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_{45} mode is overwhelmed by T9.3 in still air, so its downward natural convection does not get as close to 0 as the flat plate does.
A complete mixed singlephase convection model has been presented for flat rectangular plates. Its novel features are:
natural convection from a rectangular plate in any orientation,
a correlation for forced convection over a rough flat surface,
correspondence between forced convection and the verticalplate mode of natural convection,
vector sum of forcedequivalent 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 shapefactor for both the forced and verticalplate modes can be computed by integration as was done in Forced OffAxis Convection and Natural OffAxis Convection.
The upward convective mode (HHT*) 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 characteristiclength for downward convection from a horizontal plate. Downward convection from a rectangular plate selforganizes into two rollers parallel to a longest edge and has characteristiclength, 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 characteristiclength R, can these two correlations be unified?
Convection in fluids which can undergo phase changes is of great practical interest in the case of watervapor condensation on roofs. While there is a body of theory and experiment dealing with condensation from a pure phasechange fluid (eg. steam), there is less available dealing with condensation from a mixture of condensing and noncondensing 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 scaledColburnanalogyasymptote model predicts for plates with relative roughnesses of 0.01, 0.0025, 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} 2.47×10^{2} 4.00×10^{4} 4.94×10^{2} 8.00×10^{4} 9.89×10^{2} 0.0025 2.00×10^{4} 1.30×10^{2} 4.00×10^{4} 2.60×10^{2} 8.00×10^{4} 5.20×10^{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}
The effort to perform this experiment is documented in Measurements of Convection From a Rectangular Plate.
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 lowspeed flow in the wind tunnel, which can be difficult to achieve and measure accurately.
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SimRoof  
agj @ alum.mit.edu  Go Figure! 