Heat transport coefficients for constant energy flux models of broad leaves

Experimental determinations of the local heat transfer by forced convection from model leaves heated by a constant energy flux were made in the laboratory under laminar and turbulent flow conditions.The results are expressed in a logarithmic dimensionless plot of the local Nusselt number, Nu _d , against the local Reynolds number, Re _d . For the laminar case, Nu _d was only a linear function of Re _d ^1/2 downwind from the leading edge regions, although this relationship departed from that predicted theoretically due to the finite size and thickness of the model. For the turbulent case, a simple relationship between Nu _d and Re _d was found over a wide range of Reynolds numbers. The enhancement of heat transfer in the turbulent case depends primarily on the scale of turbulence rather than on the turbulent intensity.Past workers have discussed their results in relation to a factor , defined as the ratio between the heat transfer predicted by the Polhausen equation, and that measured. The results suggest that is not a unique parameter and may not be useful in describing the overall turbulent transfer process.     

Simultaneous determinations of the sensible and latent heat transfer coefficients for tree leaves

The heat and mass transfer coefficients for exchange across the fluid dynamic boundary layer over tree leaves were simultaneously determined in a controlled environment chamber. The mass transfer coefficients were calculated from measured values of evaporation, air specific humidity and a value of leaf specific humidity at leaf temperature. The heat transfer coefficients were calculated from measured values of air temperature, leaf temperature and an estimate of the sensible heat flux density calculated as the measured net radiation at the leaf surfaces minus the latent heat flux density. The experiments described in this paper indicate that the equations based on laminar boundary-layer theory can give reasonable estimates of the transfer coefficients of real tree leaves for the velocities most commonly experienced in plant canopies, if they are adjusted by a constant multiplier greater than one. Calculations of local mass transfer coefficients based on temperature measurements at three locations at different distances from the leading edge of the leaves, indicate that the deviation from theory is probably the result of transition to turbulent boundary-layer flow at some distance from the leading edge.     

Heat transfer to and from vegetated surfaces: An analytical method for the bulk exchange coefficients

The semi-analytical model outlined in a previous study (Massman, 1987) to describe momentum exchange between the atmosphere and vegetated surfaces is extended to include the exchange of heat. The methods employed are based on one-dimensional turbulent diffusivities and use analytical solutions to the steady-state diffusion equation. The model is used to assess the influence that (1) the canopy foliage structure and density, (2) the wind profile structure within the canopy, and (3) the shelter factor can have upon the inverse surface Stanton number (kB ^–1) as well as to explore the consequences of introducing a scalar displacement height which can be different from the momentum displacement height. In general, the triangular foliage area density function gives results which agree more closely to observations than that for constant foliage area density. The intended application of this work and its predecessor (Massman, 1987) is for parameterizing the bulk aerodynamic resistances for heat and momentum exchange for use within large-scale models of plant-atmosphere exchanges.     

Mass transfer to bean leaves

A vapour of radio-lead (²¹²Pb) has been used to measure the Sherwood number, Sh, of model leaves at various angles of incidence,, to the airstream in a wind tunnel. The results for=0 are compared with Pohlhausen's formula and the results for 0, with Powell's experiments. The local values of Sh on the upwind and downwind sides of discs have been obtained. For leaves in the canopy, Sh was found to be about 25% greater than would be predicted by applying Pohlhausen's equation without correction for orientation.     

Air-sea bulk transfer coefficients in diabatic conditions

On the basis of recent data for the roughness Reynolds number of the sea surface, and using the Owen-Thomson theory on the transfers of heat and mass between a rough surface and the flow above it, the bulk transfer coefficients of the sea surface have been estimated. For a reference height of 10 m, the neutral-lapse transfer coefficient for water vapor is larger by only a few percent than that for sensible heat. When the wind speed at the 10-m height is u ₁₀>3 m s^–1, the coefficient for sensible heat C _H is larger by about 10% than that for momentum C _D . For u ₁₀<5 m s^–1, however, the value of C _D exceeds the value of C _H , and for u ₁₀=15 m s^–1 it is shown that C _H 0.8C _D . It may be also proposed that 10³ C _D =1.11 to 1.70, 10³ C _E =1.18 to 1.30, and 10³ C _H =1.15 to 1.26 for a range of u ₁₀=4 to 20 m s^–1. A plot of diabatic transfer coefficients versus wind speed is obtained by using a parameter of the sea-air temperature difference. For practical purposes, the coefficients are approximated by empirical formulae.     

A procedure for the formulation of bulk transfer coefficients over water

A method suitable for predicting bulk transfer coefficients appropriate to any reasonable height of measurement in the atmospheric surface boundary layer and incorporating the effects of atmospheric stability is based on the assumption that eddy and molecular diffusivities are additive near a water surface. This assumption is supported in the case of sensible heat, by results obtained over Lake Michigan and over an industrial cooling pond at Dresden, Illinois, as well as by published measurements made over Lake Flevo, Holland. The verification appears to extend to wind speeds in the range 10–15 m s^–1. The results permit evaluation of transfer coefficients applicable in the demanding situations of inland lakes and artificial cooling ponds.Work performed under the auspices of the U.S. Energy Research and Development Administration.     

Mass transfer from discrete water drops on leaves in a cereal canopy

The boundary-layer resistance, r _d , for water vapour transfer from single drops on a wheat leaf was derived from field measurements of the evaporation rate, drop temperature and air humidity. Parameters are estimated in an equation to calculate r _d from drop diameter and wind speed. The relationship between resistance and wind speed is compared with that from other sources, and possible systematic errors in temperature measurements are examined using a model of the drop energy balance.On secondment from Department of Agricultural Sciences, University of Bristol, IACR, Long Ashton Research Station, Long Ashton, Bristol BS18 9AF, U.K.     

Eddy flux measurements over the ocean and related transfer coefficients

Eddy correlation measurements of vertical turbulent fluxes made during AMTEX 1975 are used to assess the reliability of flux prediction from established bulk transfer relations, using both surface-layer and planetary boundary-layer formulations. The surface-layer formulae predict momentum and latent heat fluxes to an accuracy comparable to the direct eddy correlation method, using transfer coefficients of C _DN (at 10m and in neutral conditions) increasing with wind speed, and a constant C _EN - 1.5 × 10 ^–3 . The data suggest C _CHN , for sensible heat, increases significantly with wind speed and is on average 30% lower than C _CEN The boundary-layer drag coefficient, C _GD , agrees within about 40% of recently published values using a vertically averaged geostrophic wind to the height of the lowest temperature inversion, corrected for trajectory curvature. Values of _* / from which C _CGH is derived, are in excellent agreement if the published values are modified to account for inappropriate surface temperatures used in their derivation. Preliminary values of C _GE are also presented.     

Momentum and scalar transfer coefficients over aerodynamically smooth antarctic surfaces

Vertical profiles of wind speed, temperature and humidity were used to estimate the roughness lengths for momentum (z ₀), heat (z _H ) and moisture (z _Q) over smooth ice and snow surfaces. The profile-measurements were performed in the vicinity of a blue ice field in Queen Maud Land, East Antarctica. The values ofz ₀ over ice (3·10^–6 m) seem to be the smallest ever obtained over permanent, natural surfaces. The settling of snow on the ice and the loss of momentum at saltating snow particles serve as momentum dissipating processes during snow-drift events, expressed as a strong dependence ofz ₀ on u_#.The scalar roughness lengths and surface temperature can be evaluated from the temperature and humidity profile measurements if the ratioz _H /z _Q is specified. This new method circumvents the difficult measurement of surface temperature. The scalar roughness lengths seem to be approximately equal toz0 for a large range of low roughness Reynolds numbers, despite the frequent occurrence of drifting snow. Possible reasons for this agreement with theory of non-saltating flow are discussed.     

Model experiments on free-convection heat and mass transfer of leaves and plant elements

Effects of orientation, shape, and surface structure on the free convection mass transfer of plates, leaf- and plant-models were studied in an electrolytic system. The results were extrapolated to the transfer of heat and mass in air, in an effort to predict realistic boundary-layer transfer coefficients for leaves and plant-like surfaces. Flow visualization complemented the investigation.The results indicate that increases in transfer of 50% over that of a vertical plate are possible for non-vertical rough leaves, but that a 25% increase is a more reasonable estimate for an average plant. Evidence is also presented that more reliable caluclations of the transfer properties of pine needles are obtained if the needles are approximated by vertical cylinders, rather than by horizontal ones.     

Drag and bulk transfer coefficients associated with a shallow water surface

During September, 1971, measurements of Reynolds stress and sensible heat flux were made over a shallow coral reef off the coast of Papua. The water varied (with the tide) from 0 to 2.5 m in depth. Two independent measurements of the momentum flux confirmed that the surface deviated only slightly from aerodynamically smooth. The independent determinations of the sensible heat flux are consistent with this result, indicating a much lower Stanton number than would be expected over deeper water (viz., 1.1 × 10^–3 rather than about 1.5 × 10^–3).The near smoothness of the surface appears to be associated with either the attenuation of small surface waves resulting from the presence of a surface film or their inability to grow in the absence of longer wave components. Although it is not clear which of these effects was predominant, the data show that great care should be taken in extrapolating shallow water results to open ocean conditions, particularly when it is conceded that surface films may exist in either situation.Now with Argonne National Laboratory, Argonne, Illinois, U.S.A.Now with CSIRO Division of Mineral Physics, Sydney, N.S.W., Australia.     

A theory for the scalar roughness and the scalar transfer coefficients over snow and sea ice

Although the bulk aerodynamic transfer coefficients for sensible (C _H ) and latent (C _E ) heat over snow and sea ice surfaces are necessary for accurately modeling the surface energy budget, they have been measured rarely. This paper, therefore, presents a theoretical model that predicts neutral-stability values of C _H and C _E as functions of the wind speed and a surface roughness parameter. The crux of the model is establishing the interfacial sublayer profiles of the scalars, temperature and water vapor, over aerodynamically smooth and rough surfaces on the basis of a surface-renewal model in which turbulent eddies continually scour the surface, transferring scalar contaminants across the interface by molecular diffusion. Matching these interfacial sublayer profiles with the semi-logarithmic inertial sublayer profiles yields the roughness lengths for temperature and water vapor. When coupled with a model for the drag coefficient over snow and sea ice based on actual measurements, these roughness lengths lead to the transfer coefficients. C _E is always a few percent larger than CH. Both decrease monotonically with increasing wind speed for speeds above 1 m s^–1, and both increase at all wind speeds as the surface gets rougher. Both, nevertheless, are almost always between 1.0 × 10^–3 and 1.5 × 10^–3.     

Inequality of eddy transfer coefficients for vertical transport of sensible and latent heats during advective inversions

Similarity of transport of water vapour and sensible heat was investigated within an advective inversion layer by measuring eddy fluxes together with gradients of temperature and humidity. The experimental site was a field of rice, grown under flood irrigation, which was situated in a semi-arid region. The fetch was about 300 m and local stabilities (z/L) over the rice ranged from 0 to 0.1. Results were expressed as the ratio of eddy transfer coefficients for sensible heat (K _H) and water vapour (K _w). Near neutral stability, K _H/K_W approached 1, but the ratio decreased to about 0.65 with increasing stability. Existing theory predicts the result qualitatively but accounts for only about one fifth of the decrease in K _H/K_W with stability.     

Characteristic heat transfer coefficients near the ground at night during strong winds

Based on theoretical radiative cooling values and observed temperature changes with time near the surface during the night, the bulk heat transfer coefficient C _H is estimated from standard meteorological observations obtained from stations representative of open rural, small town and large urban areas, for nights with clear skies and relatively strong winds. It is shown that C _H is smaller than the drag coefficient C _M, and that C _H/C _M over urban areas is smaller than that over open countryside.     

Water vapor transfer from a vegetated surface: A numberical study of bulk transfer coefficients and canopy resistances

The semi-analytical model outlined in previous studies (Massman, 1987a, b) to describe momentum and heat exchange between the atmosphere and vegetated surfaces is extended to include water vapor exchange. The methods employed are based on one-dimensional turbulent diffusivities and use numerical solutions to the steady-state diffusion equation. The model formulates stomatal response as a function of vapor pressure deficit and the within-canopy profile of mean photosynthetically-active radiation (PAR). It is then used to assess the influence that foliage structure, density, and sheltering can have upon the bulk transfer coefficient, kB _v ^-1, and the canopy resistance. A general analytical formulation of the canopy resistance based on the mean within-canopy profile of PAR is proposed and found to agree with the model's solutions for canopy resistance to within a few percent.     

Electrochemical simulations of mass transfer from isolated wet spots and droplets on realistic fluttering leaves

This paper reports on forced-convection mass transfer from isolated discs on rectangular plates as well as hemispheres on realistic fluttering leaves. An electrochemical method was used where the convective transfer of ions to the test electrode (the droplet or the wet spot) in an electrolytic flow system was measured as a function of flow rates, sizes of discs and hemispheres. Measurements showed that the local transfer coefficient for uniformly transferring plates varied as expected while the transfer from isolated discs on plates was much less a function of the distance from the leading edge. An expression to describe the transfer coefficient for an isolated disc as a function of distance from the leading edge was determined. An expression describing the transfer from hemispherical drops on fluttering leaves was derived and compared with the predictions from transfer theory for a sphere in free space.     

Turbulent characteristics and bulk transfer coefficients over the desert in the HEIFE area

Observations of surface-layer turbulence and turbulent fluxes were made over a desert in northwestern China as a part of HEIFE (HEIhe river Field Experiment). These show that the normalized variations of the vertical wind component and of the air temperature obey Monin-Obukhov similarity well, especially in free convective conditions. However, the variations of specific humidity do not obey Monin-Obukhov similarity.Mean bulk transfer coefficients of sensible heat and momentum flux are obtained as functions of stability over a wide stability range from the observed data of turbulent fluxes and mast profiles. However, the bulk transfer coefficient for water vapor could not be obtained because of the large scatter of the data. In free convective conditions, the sensible heat flux was found to be approximately proportional to the 1.4 power of temperature difference between the surface and 20m. The bulk transfer coefficient of sensible heat is also obtained as a function of the bulk Richardson number for practical convenience.     

Stability dependence of fluxes and bulk transfer coefficients in a tropical boundary layer

Turbulence data obtained aboard a NOAA P-3 research aircraft during flights over the eastern equatorial Pacific Ocean are used to obtain mean vertical profiles of water vapor density, potential temperature, wind speed and fluxes of latent heat, buoyancy and momentum. The variation of eddy fluxes and bulk transfer coefficients as a function of atmospheric stability are plotted for two of the flights. The observed transfer coefficients generally agree with those obtained from parameterizations based on surface-layer similarity theory (Deardorff, 1968; Kondo, 1975).     

Stability dependence of the drag and bulk transfer coefficients over a coastal sea surface

Bulk transfer coefficients were evaluated from eddy correlation flux measurements on a fixed pier during onshore winds. The mean values are C _D = 1.69 × 10^-3, C _H = 2.58 × 10^-3 and C _E = 1.51 × 10^-3. The drag coefficient, C _D, gradually increases with wind speed but C _H and C _E are independent of wind speed. According to theory and empirical formulas based on experimental results over flat grassland, the transfer coefficients should gradually increase with increasing instability. This is confirmed experimentally in the stable region in our case. However, the drag coefficient appears to decrease with increasing instability, which is against the theoretical result. A stability dependence is not clearly observed for C _H or C _E.