http://people.csail.mit.edu/jaffer/SimRoof/Convection Convection From a Rectangular Plate Introduction Sites Weather Net Radiative Transfer Lambertian Perfromance Limits CoolRoof Convection From a Rectangular Plate Abstract Introduction Motivation Convective Heat Transfer Types of Convection 1997 ASHRAE Fundamentals Thermal Emissivity Still Air 2009 ASHRAE Fundamentals Physical Parameters Thermodynamic and Transport Properties of Air Characteristic Length Dimensionless Quantities Natural Convection Vertical Plate Horizontal Plate T9.9 Inclined Plate Downward Convection Upward Convection Natural Convective Surface Conductance Double­Sided Natural Convection Natural Convection Summary Forced Convection Laminar­Turbulent Progression Laminar­Turbulent Transition Laminar Convection Convection in a Duct Surface Roughness Data Surface Roughness Forced Convection in a Pipe Boundary Layer Forced Convection from a Rough Plate Colebrook­White Equation Asymptotes Scaled Colburn Analogy Asymptote Forced Convection Summary Natural Convection from a Rough Plate Off­Axis Convection Forced Off­Axis Convection Effective Length Natural Off­Axis Convection Mixed Convection Local Mixed Convection Correlations From Field Measurements Vertical Plates Vertical Plates with Off­Axis Flow Inclined Plates Mixed Convection Summary Application to Roofs Wind Flat Roofs Peaked Roofs Roof Performance Conclusion Future Work Flat Convex Plates Convection with Phase­Change Experiments Bibliography Solar Screen Model Solar Screen Performance Thermal Infrared Optics (footnotes and supplemental information) Abstract 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. A plan for (an apparatus and) experiments to test the rough­surface correlations is presented. Introduction Motivation 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. Convective Heat Transfer 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 2 h W/(K⋅m ) convective surface conductance 2 A m area of one side of plate ΔT K T − Tenv T K plate temperature Tenv 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. Types of Convection 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 "Laminar­ Turbulent Progression" gives the derivation for average heat flow from the local heat flow which appears in several texts. While it produces the customary averaged 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 107 (Clear et al[82]), placing it between the ranges for correlations T9.7 and T9.8 (which intersect at 4.7×106). Convection from a vertical plate has a transition around Gr=Ra/Pr=109. 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. 1997 ASHRAE Fundamentals 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: hg Glass hp Clear Pine 2 1/2 (b − 4 a c + 4 a V) − b a = 1/882 a = 4/3999 h(V) = 2 a b = 257/882 b = 973/3999 c = −1060/441 c = −2680/1333 Thermal Emissivity 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⋅m2). If the surface and test apparatus both have emissivity of 0.95, it is 3.87 W/(K⋅m2). 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, Handbook of Chemistry Fig 1.2 EMISSIVITY OF BUILDING MATERIALS, Horizon Energy Systems. [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) Mills, A.F., 1999., American Society of Heating Refrigerating and Air­conditioning Engineers Inc.