Heat transfer through the boundary layer of elliptical shaped leaves under free convection
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Graefe, Jan
The efficiency of heat and mass exchange between leaves and their environment under low wind speed is dominated by free convection. This is commonly quantified in terms of the Nusselt number (Nu) and the Rayleigh number (Ra). The currently available Nu = f(Ra) relations for inclined plates were mostly derived for infinite wide plates or from one-sided heat transfer studies. A comprehensive simulated data set of laminar free convection may be used to derive new Nu relations at any inclination and for both plate sides. The relevant equations for free convection in 3D are solved numerically using the computational fluid dynamic (CFD) software OpenFOAM. The simulated Nusselt numbers agree very good with previous measurements for vertical and horizontal circular plates having a diameter of 84 mm. Various finite thickness (0.5 mm) elliptical plates (i.e. leaves) having aspect ratios between 1 and 3, plate length ranging from 30 to 160 mm and a range of inclinations are simulated with plate to air differences set to 1?12 K. Simulated heat fluxes from each leaf side are used to parameterize a comprehensive set of Nu relations.
 I'm from a small plant science institute south of Berlin. You may have heard of it, Leibniz Institute of Vegetable and Ornamental Crops. And I want to talk today about a plant topic. So actually the transport efficiency of heat and mass through the boundary layer of elliptical shaped leaves. To give you a short overview of this topic, so may not, anybody of you is familiar with this problem. So we have here a very abstract scheme of a plant with two leaves and an underlying soil and an overlaying atmosphere and different energy fluxes and mass fluxes. So we have the short wave radiation fluxes from the sun and the sky and long wave radiation fluxes arising from the sky itself and from the leaves and from the soil. And those radiation fluxes are the driver of mass and heat flow from the leaves. And this is a fundamental problem in plant science to understand the mass and heat flow from the leaves because leaf temperature, transpiration, et cetera are determined by the leaf energy balance which is shortly shown here. The balance between the absorbed short wave and long wave radiation fluxes and the ingoing fluxes of a latent heat flux and a sensible heat flux and then emitted long wave radiation. And this is the base equation to start from to describe transpiration and leaf temperature and dew and so on. So, and here in my talk we want to focus on this H term, the sensible heat flow, that is the heat flow itself. So more why we focus on this problem in general. So heat and mass flow here, carbon dioxide and water vapor, are fundamental to proper plant function and survival in natural and environments also on greenhouses. And especially if we going to towards a quantification, then the equation describing the heat and mass flow are a central part of computation modules for leaf photosynthesis, transpiration, dew, thrust or even in thermographic vegetation analysis. So, this is a fundamental problem or even in thermographic vegetation analysis. And in those models and model system, I found that the effect of leaf shape and leaf inclination is somehow neglected or are under represented in describing the heat and mass flow. So I had a closer look on heat flow under the conditions of free convection. So what is meant with free convection, because not anybody might be familiar with this term, we usually we make the distinction between forced convection, which is driven by an air stream, which is generated external to the leaf scale. So say by winter or say by movement of a plant and we have on the other side, we have the natural or free convection, where density gradients close to the leaf surface generate a fluid flow close to the leaf. And usually this belief, it's known that in both cases for forced convection and for natural or free convection, there is a speed gradient close to the surface and this speed gradient induces a share and the share drives a transport of heat and mass molecules, scalars and so on. So we want to focus now on the right process of natural or free convection, so driven by temperature gradients between the leaf surface and the air and the bulk air, and the air and the bulk air volume. And very frequently this process is quantified by simple correlations, so-called nethered numbers, non-dimensional numbers or shareboard numbers for mass flow, and you may a simplified version of such a simple correlation would be the expression for mean heat flow per area, so we have a temperature gradient and we have an angle of our surface between the normal and the vertical and we have some characteristic lengths. So these are the main parameters, the temperature gradient, the characteristic lengths and the angle of the surface. And so we want to focus on elliptical leaves because this leaf shape wasn't, so usually only very simple shapes are usually considered as circular shapes or rectangular shapes. And so here we have a look on elliptical shapes and the surface may be inclined vertically and you see the characteristic length is a kind of mean, on a vertical surface we would have such streamlines of air flow and the characteristic length is a kind of a mean streamline length of this air flow. Of course, if our leaf is rotated somehow, those streamlines must not be aligned with the longest axis of the ellipse, so they can have any direction over the surface. Somewhat different is the situation if we have horizontal horizontal leaves than usually the air flow, here you see air flow from heated leaves, and comes from below and would leave the surface and abound in the center of the leaf. So some open questions for this problem would be, can we describe this mean streamline length for different scenarios, different aspect ratios of ellipses and different angles, including those horizontal scenarios. And actually how accurate is such a simple correlation compared to wind tunnel experiments or alternatively as here computational fluid dynamics simulations. And what about the different leaf sides? Do we need maybe different relations for the upper leaf side and for the lower leaf side? And this may have an influence for due occurrence on the upper leaf side or lower leaf side. So this talk is mostly about these open questions. So I had the following objectives to find analytical expressions for the characteristic length, to implement a virtual wind tunnel using an open source computational fluid dynamics tool and to use this virtual wind tunnel to generate data sets to calibrate new natural convection correlations for elliptical shaped leaves. So coming to the first point, those relations were found. So here we see the first step was to find the angle of the steepest S-kent, the angle with the smaller axis of the ellipses. So the steepest S-kent is denoted by the vector S. And depending on the rotation angles of the ellipse around the y-axes, this steepest S-kent angle can be derived. And having this angle, the mean streamline length over the ellipse can be estimated, which is where the equation is given here. This is not too hard and relatively simple as shown here. I used the equation for the ellipse and the equation for the line. And setting both angles, you get equations for the crosspoints, for the crossings of the line and the ellipse. And the distance between both crosspoints can easily be formulated. And then you can consider the mean distance between the origin of the ellipse and the farthest point. So here denoted by the intercept with the y-axis. So you integrate this distance or you average it out. So you get an analytic expression of the characteristic length of the ellipse given the angle of the steepest S-kent, which defines, of course, the slope of the line. So this done. There was left a problem for horizontal leaves. This is actually more difficult, but it is all more easy how you want. From stochastic geometry, it is known that the mean random chord length can be easily obtained by the surface area of our two-dimensional surface and the perimetre. And so this would be a good starting point. And besides that, it is known from true wind tunnel studies that the ratio between the surface area and the perimetre is a good length scale to describe a natural convection from different shaped surfaces. So we stayed, or I decided to stay on this expression for the case of horizontal inclination, for the limiting case of horizontal. A second objective, I wanted to implement a virtual wind tunnel for obtaining data sets on different inclined and dimensioned ellipses. And we used as CFT computational fluid dynamics software, we used Openform, which is maybe, I'm sure, quite well known to many of you. And there was a semi-unautomated workflow. So first of all, I created the 3D leaf models, which had a finite thickness. And this was done in Salome. Ellipses with different aspect ratios were generated with different sizes and rotations. And they were the different surfaces, so the upper surface, the edge and the lower surface. Those surface measures had to be marked in Salome to simplify the heat flux estimation later on with Openform tools. And inputting those surface measures of our 3D models into Openform tools, into Openform meshing tools, BlockMesh and SnappyHacksMesh, it was possible to generate an appropriate 3D mesh of the wind tunnel, including our elliptical leaf model here, here shown wide. And having generated this 3D mesh simulations were done, I had problems with the steady-state solver of Openform, so I used a transient solver called in this version of Openform, Bavarian-Boussinesque Pimpleform. And I had to simulate about, depending on the forcing of the temperature difference between the leaf and the bulk air of the wind tunnel, I had to apply different transient times until I got almost steady-state conditions around the leaf, almost steady-state heat flow conditions. Computing time was okay, one to five hours, and overall 350 hours of computation were done with two PCs and Windows, even with Openform you can use Windows very well. And then all those simulations had to be post-processed for heat flow computations for both leaf-side, streamlined-resolvation and the model fitting of our simplified model to describe natural convection from these surfaces. I used here also Openform tools, para-view for stream-man-resolation and mud-lub. There are not actually not many data to validate those CFD simulations, but one data set was obtained by Hassani and Hollens on an 8 cm diameter circular disk, which were inclined vertically or horizontally. And I simulated the bulk heat flow, the bulk from both sides, because measurements were from both sides. For those temperature gradients, we can see that simulations and real wind tunnel data matched quite well. We got some trust in our workflow, in our setup, and did all those mentioned simulations for different aspect ratios, sizes and rotations of ellipses, and used those data sets to estimate the heat flux for the upper and lower side of the leaves. And used this relative simple Nusselt number model to describe simulated data. In the end, all those equations are motivated by previous work in literature, but I decided to re-estimate six parameters from simulated data. This was the outcome. We see in graphical terms predicted heat fluxes by the simple model and simulated heat fluxes matched quite well. Overall, we simulated 356 leaves, and our parameterized or calibrated Nusselt number model had only an error of 4%. The previous model, which I obtained from the literature, had an error of about 30%. At the end, we obtained a very good, simplified description of this complex process. To show you some behavior, how simulated leaves behaved, and how the CFPT simulations and simplified model predictions matched each other, you see here simulations of heat flow from the lower leaf side and from the upper leaf side. Here, in the figure A, we have a circular shape, and in figures B, C, D, we have this elliptical shape. This is such how you can imagine the rotation around the x-axis, which is shown here as an x-axis of the coordinate system. We start here with a horizontal orientation, and 90 degrees is the vertical orientation of our circle or ellipse. Heat flow is increasing with rotation around the x-axis. We have a higher heat flow from the lower side than from the upper side, and both behaviors match very well. So CFT simulations and simplified model match here very well up. Graph C and D are a bit of special. In graph D, I changed the direction of heat flow. Here, from A and B, we have heated leaves, and in D, we have a cooled leaf. It turned out, what also literature says, you can use the same correlations, but just switch the sides of your simulated object. The switch equations used for the lower and the upper side, and switch the sides. The graph D is just a mirror image of the graph B. Those data were not used for calibration, and a verification of a known issue. Graph C is a little bit different. It was pre-rotated along the epsilon axis, and then it was rotated along the x-axis. We obtain a totally different pattern of heat flow with rotation, and almost an invariant. This y-axis can be quite well seen by visualization of the streamlines. Here we see a visualization of the streamlines for different rotations along the x-axis. It's almost horizontal, but it's rotated along the y-axis. That's why it's almost in the direction of the smaller axis of the ellipse. With increasing, if the ellipse is more rotated towards the vertical direction, the 50-degree streamlines are more aligned with the longer axis. The characteristic length is increasing, and this offsets somehow the promoting effect of the increasing angle. In that in sum, the response is a little bit leveled out here. We have now increased the heat flow with rotation, so it stays more or less horizontal in the response. That can be explained by the different. It can be simulated, and it can be explained graphically. Coming to a summary, in this work we derived equations for the characteristic length inclined and rotated ellipses of different aspect ratios. We re-parameterized known engineering equations for natural convection from those ellipses using CFD simulations. By this re-parameterization, we improved the prediction by 27%. Those new relations are not a large improvement, but it is an improvement in a topic which is quite heavily researched over the last 50 years, I would say. Those proposed equations may enhance forecast analysis, especially under low wind conditions. Say at night, they may have special importance. For dew and frost occurrence on leaves, they are useful for prediction of internal CO2, temperature, and stormwater conductance of leaves, which are part of calculation model roots for leaf photosynthesis and transpiration. We found that open foam and salome are quite powerful tools to solve almost any fluid heat flow problem and even complex or conjugated problems. Here I could only show one tiny part of this analysis. If you are interested in the further details, you may consult this publication. At this point, this must be not the end of the analysis. We did some simulation with ellipses which are folded along the mid-rip, which you can see in nature quite often. I am looking forward to use those data sets to parameterize, to extend the Nasset number model even for folded leaves, which you find quite frequently. Of course, it would be more valuable if one would have a true mixed convection model, because leaves are operating under wind. Usually we have a kind of mixture of natural and forced convection transport processes and natural ecosystems. This is possible with open foam, especially if you can consider laminar flow. I forget to say that all simulations were laminar here, because we consider turbulence as too rare to occur. I am a bit like an outlaw in a plant science institute with CFT simulations. This problem is a little bit out of our opportunities in terms of setup and available computational resources. Of course, the most challenging analysis would combine radiation, heat and mass transport through pores. All happens together. An optimal case should be simulated together. Thank you for your attention.
 I'm from a small plant science institute south of Berlin.  You may have heard of it,  Leibniz Institute of Vegetable and Ornamental Crops.  And I want to talk today about a plant topic.  So actually the transport efficiency of heat and mass  through the boundary layer of elliptical shaped leaves.  To give you a short overview of this topic,  so may not, anybody of you is familiar with this problem.  So we have here a very abstract scheme of a plant  with two leaves and an underlying soil  and an overlaying atmosphere  and different energy fluxes and mass fluxes.  So we have the short wave radiation fluxes  from the sun and the sky  and long wave radiation fluxes arising from the sky itself  and from the leaves and from the soil.  And those radiation fluxes are the driver of mass  and heat flow from the leaves.  And this is a fundamental problem in plant science  to understand the mass and heat flow from the leaves  because leaf temperature, transpiration, et cetera  are determined by the leaf energy balance  which is shortly shown here.  The balance between the absorbed short wave  and long wave radiation fluxes  and the ingoing fluxes of a latent heat flux  and a sensible heat flux  and then emitted long wave radiation.  And this is the base equation to start from  to describe transpiration and leaf temperature and dew  and so on.  So, and here in my talk we want to focus on this H term,  the sensible heat flow, that is the heat flow itself.  So more why we focus on this problem in general.  So heat and mass flow here, carbon dioxide and water vapor,  are fundamental to proper plant function and survival  in natural and environments also on greenhouses.  And especially if we going to towards a quantification,  then the equation describing the heat and mass flow  are a central part of computation modules  for leaf photosynthesis, transpiration, dew, thrust  or even in thermographic vegetation analysis.  So, this is a fundamental problem  or even in thermographic vegetation analysis.  And in those models and model system,  I found that the effect of leaf shape and leaf inclination  is somehow neglected or are under represented  in describing the heat and mass flow.  So I had a closer look on heat flow  under the conditions of free convection.  So what is meant with free convection,  because not anybody might be familiar with this term,  we usually we make the distinction between forced convection,  which is driven by an air stream,  which is generated external to the leaf scale.  So say by winter or say by movement of a plant  and we have on the other side,  we have the natural or free convection,  where density gradients close to the leaf surface  generate a fluid flow close to the leaf.  And usually this belief,  it's known that in both cases for forced convection  and for natural or free convection,  there is a speed gradient close to the surface  and this speed gradient induces a share  and the share drives a transport of heat  and mass molecules,  scalars and so on.  So we want to focus now on the right process  of natural or free convection,  so driven by temperature gradients  between the leaf surface and the air  and the bulk air,  and the air and the bulk air volume.  And very frequently this process is quantified  by simple correlations,  so-called nethered numbers,  non-dimensional numbers or shareboard numbers  for mass flow,  and you may a simplified version  of such a simple correlation  would be the expression for mean heat flow per area,  so we have a temperature gradient  and we have an angle of our surface  between the normal and the vertical  and we have some characteristic lengths.  So these are the main parameters,  the temperature gradient,  the characteristic lengths and the angle of the surface.  And so we want to focus on elliptical leaves  because this leaf shape wasn't,  so usually only very simple shapes  are usually considered as  circular shapes or rectangular shapes.  And so here we have a look on elliptical shapes  and the surface may be inclined vertically  and you see the characteristic length  is a kind of mean,  on a vertical surface we would have such streamlines  of air flow  and the characteristic length  is a kind of a mean streamline length of this air flow.  Of course, if our leaf is rotated somehow,  those streamlines must not be aligned  with the longest axis of the ellipse,  so they can have any direction over the surface.  Somewhat different is the situation  if we have horizontal horizontal leaves  than usually the air flow,  here you see air flow from heated leaves,  and comes from below  and would leave the surface  and abound in the center of the leaf.  So some open questions for this problem would be,  can we describe this mean streamline length  for different scenarios,  different aspect ratios of ellipses  and different angles,  including those horizontal scenarios.  And actually how accurate is such a simple correlation  compared to wind tunnel experiments  or alternatively as here  computational fluid dynamics simulations.  And what about the different leaf sides?  Do we need maybe different relations  for the upper leaf side and for the lower leaf side?  And this may have an influence  for due occurrence on the upper leaf side  or lower leaf side.  So this talk is mostly about these open questions.  So I had the following objectives  to find analytical expressions  for the characteristic length,  to implement a virtual wind tunnel  using an open source computational fluid dynamics tool  and to use this virtual wind tunnel  to generate data sets  to calibrate new natural convection correlations  for elliptical shaped leaves.  So coming to the first point,  those relations were found.  So here we see the first step was  to find the angle of the steepest S-kent,  the angle with the smaller axis of the ellipses.  So the steepest S-kent is denoted by the vector S.  And depending on the rotation angles  of the ellipse around the y-axes,  this steepest S-kent angle can be derived.  And having this angle,  the mean streamline length over the ellipse  can be estimated,  which is where the equation is given here.  This is not too hard  and relatively simple as shown here.  I used the equation for the ellipse  and the equation for the line.  And setting both angles,  you get equations for the crosspoints,  for the crossings of the line and the ellipse.  And the distance between both crosspoints  can easily be formulated.  And then you can consider the mean distance  between the origin of the ellipse  and the farthest point.  So here denoted by the intercept with the y-axis.  So you integrate this distance  or you average it out.  So you get an analytic expression  of the characteristic length of the ellipse  given the angle of the steepest S-kent,  which defines, of course, the slope of the line.  So this done.  There was left a problem for horizontal leaves.  This is actually more difficult,  but it is all more easy how you want.  From stochastic geometry,  it is known that the mean random chord length  can be easily obtained by the surface area  of our two-dimensional surface and the perimetre.  And so this would be a good starting point.  And besides that, it is known from true wind tunnel studies  that the ratio between the surface area and the perimetre  is a good length scale to describe a natural convection  from different shaped surfaces.  So we stayed, or I decided to stay on this expression  for the case of horizontal inclination,  for the limiting case of horizontal.  A second objective, I wanted to implement a virtual wind tunnel  for obtaining data sets on different inclined and dimensioned ellipses.  And we used as CFT computational fluid dynamics software,  we used Openform, which is maybe, I'm sure, quite well known to many of you.  And there was a semi-unautomated workflow.  So first of all, I created the 3D leaf models,  which had a finite thickness.  And this was done in Salome.  Ellipses with different aspect ratios were generated  with different sizes and rotations.  And they were the different surfaces, so the upper surface,  the edge and the lower surface.  Those surface measures had to be marked in Salome  to simplify the heat flux estimation later on with Openform tools.  And inputting those surface measures of our 3D models  into Openform tools,  into Openform meshing tools, BlockMesh and SnappyHacksMesh,  it was possible to generate an appropriate 3D mesh  of the wind tunnel, including our elliptical leaf model here,  here shown wide.  And having generated this 3D mesh simulations were done,  I had problems with the steady-state solver of Openform,  so I used a transient solver called in this version of Openform,  Bavarian-Boussinesque Pimpleform.  And I had to simulate about, depending on the forcing  of the temperature difference between the leaf and the bulk air  of the wind tunnel, I had to apply different transient times  until I got almost steady-state conditions around the leaf,  almost steady-state heat flow conditions.  Computing time was okay, one to five hours,  and overall 350 hours of computation were done with two PCs  and Windows, even with Openform you can use Windows very well.  And then all those simulations had to be post-processed  for heat flow computations for both leaf-side,  streamlined-resolvation and the model fitting of our simplified model  to describe natural convection from these surfaces.  I used here also Openform tools,  para-view for stream-man-resolation and mud-lub.  There are not actually not many data to validate those CFD simulations,  but one data set was obtained by Hassani and Hollens  on an 8 cm diameter circular disk,  which were inclined vertically or horizontally.  And I simulated the bulk heat flow, the bulk from both sides,  because measurements were from both sides.  For those temperature gradients, we can see that simulations  and real wind tunnel data matched quite well.  We got some trust in our workflow, in our setup,  and did all those mentioned simulations for different aspect ratios,  sizes and rotations of ellipses,  and used those data sets to estimate the heat flux  for the upper and lower side of the leaves.  And used this relative simple Nusselt number model  to describe simulated data.  In the end, all those equations are motivated by previous work in literature,  but I decided to re-estimate six parameters from simulated data.  This was the outcome.  We see in graphical terms predicted heat fluxes  by the simple model and simulated heat fluxes matched quite well.  Overall, we simulated 356 leaves,  and our parameterized or calibrated Nusselt number model  had only an error of 4%.  The previous model, which I obtained from the literature,  had an error of about 30%.  At the end, we obtained a very good, simplified description  of this complex process.  To show you some behavior, how simulated leaves behaved,  and how the CFPT simulations and simplified model predictions matched each other,  you see here simulations of heat flow from the lower leaf side  and from the upper leaf side.  Here, in the figure A, we have a circular shape,  and in figures B, C, D, we have this elliptical shape.  This is such how you can imagine the rotation around the x-axis,  which is shown here as an x-axis of the coordinate system.  We start here with a horizontal orientation,  and 90 degrees is the vertical orientation of our circle or ellipse.  Heat flow is increasing with rotation around the x-axis.  We have a higher heat flow from the lower side than from the upper side,  and both behaviors match very well.  So CFT simulations and simplified model match here very well up.  Graph C and D are a bit of special.  In graph D, I changed the direction of heat flow.  Here, from A and B, we have heated leaves,  and in D, we have a cooled leaf.  It turned out, what also literature says,  you can use the same correlations,  but just switch the sides of your simulated object.  The switch equations used for the lower and the upper side,  and switch the sides.  The graph D is just a mirror image of the graph B.  Those data were not used for calibration, and a verification of a known issue.  Graph C is a little bit different.  It was pre-rotated along the epsilon axis,  and then it was rotated along the x-axis.  We obtain a totally different pattern of heat flow with rotation,  and almost an invariant.  This y-axis can be quite well seen by visualization of the streamlines.  Here we see a visualization of the streamlines for different rotations along the x-axis.  It's almost horizontal, but it's rotated along the y-axis.  That's why it's almost in the direction of the smaller axis of the ellipse.  With increasing, if the ellipse is more rotated towards the vertical direction,  the 50-degree streamlines are more aligned with the longer axis.  The characteristic length is increasing,  and this offsets somehow the promoting effect of the increasing angle.  In that in sum, the response is a little bit leveled out here.  We have now increased the heat flow with rotation,  so it stays more or less horizontal in the response.  That can be explained by the different. It can be simulated,  and it can be explained graphically.  Coming to a summary,  in this work we derived equations for the characteristic length  inclined and rotated ellipses of different aspect ratios.  We re-parameterized known engineering equations  for natural convection from those ellipses using CFD simulations.  By this re-parameterization, we improved the prediction by 27%.  Those new relations are not a large improvement,  but it is an improvement in a topic which is quite heavily researched  over the last 50 years, I would say.  Those proposed equations may enhance forecast analysis,  especially under low wind conditions.  Say at night, they may have special importance.  For dew and frost occurrence on leaves,  they are useful for prediction of internal CO2, temperature,  and stormwater conductance of leaves,  which are part of calculation model roots for leaf photosynthesis and transpiration.  We found that open foam and salome are quite powerful tools  to solve almost any fluid heat flow problem  and even complex or conjugated problems.  Here I could only show one tiny part of this analysis.  If you are interested in the further details, you may consult this publication.  At this point, this must be not the end of the analysis.  We did some simulation with ellipses which are folded along the mid-rip,  which you can see in nature quite often.  I am looking forward to use those data sets to parameterize,  to extend the Nasset number model even for folded leaves,  which you find quite frequently.  Of course, it would be more valuable if one would have a true mixed convection model,  because leaves are operating under wind.  Usually we have a kind of mixture of natural and forced convection transport processes  and natural ecosystems.  This is possible with open foam, especially if you can consider laminar flow.  I forget to say that all simulations were laminar here,  because we consider turbulence as too rare to occur.  I am a bit like an outlaw in a plant science institute with CFT simulations.  This problem is a little bit out of our opportunities in terms of setup and available computational resources.  Of course, the most challenging analysis would combine radiation, heat and mass transport through pores.  All happens together.  An optimal case should be simulated together.  Thank you for your attention.
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