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SYMPOSIUM PAPERS

Modeling Sensible and Latent Heat Transport in Crop and Residue Canopies

Decagon Devices, 950 NE Nelson Ct., Pullman, WA 99163 and Washington State Univ., Pullman, WA 99164

^* Corresponding author (gaylon{at}decagon.com).

Received for publication August 26, 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
Modeling of latent and sensible heat exchange in crop and residue canopies is complicated by the fact that the equations for the two processes are strongly coupled and require simultaneous solution. We derive a set of uncoupled equations where the new fluxes are enthalpy and isothermal latent heat and the new driving forces are equivalent temperature and vapor pressure deficit. The canopy network equations, using the new driving variables, are reduced, using Thevenin's theorem, to a single potential and a single resistance for each flux.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
MOST OF THE ENERGY EXCHANGE that occurs at the earth's surface is the result of absorption of radiation by plant and residue canopies and soil surfaces and the convective loss of this absorbed radiation as sensible and latent heat. The absorption of energy at the surfaces of canopy elements and the soil and its partitioning into sensible and latent heat involve a number of interacting processes. The processes themselves are not difficult to model, but their interactions make any model of the whole system complex and difficult to use. The purpose of this paper is to derive transformations that can be used to minimize interaction between fluxes in the canopy and use these transformations to produce equivalent, simple transport models that can be used as upper-boundary conditions for soil heat and water flow models. Two novel features of this derivation are a simpler set of driving variables for the decoupled equations and the use of Thevenin's theorem to reduce the complex canopy networks to a single resistance and potential.

	EQUATIONS FOR SENSIBLE AND LATENT HEAT FLUX

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
The transport of sensible heat, C (W m^-2), within the canopy (from canopy elements to the air or within the air spaces) is given by (Campbell and Norman, 1998):

[1]

where is the molar density of air (mol m^-3), c_p is the molar specific heat of air (J mol^-1 K^-1), K_H is an appropriate diffusivity (m² s^-1), T (°C) is temperature, and z (m) is distance.

The transport of latent heat is similarly described by

[2]

Here (J mol^-1) is the latent heat of vaporization for water, E (mol m^-2 s^-1) is the flux density of water vapor, K_v is a vapor diffusivity (molecular or eddy diffusivity, depending on location), p is atmospheric pressure (kPa), and e is the vapor pressure (kPa).

From the definition of relative humidity, h, the vapor pressure in Eq. [2] can be rewritten as the product e_sh, where e_s is the saturation vapor pressure at air temperature. Substituting this product for e in Eq. [2] and using the product rule from calculus gives:

[3]

The derivative in the first term on the right can be rewritten as dT/dz, where = de_s/dT is the slope of the saturation vapor pressure function (kPa °C^-1). The derivative in the second term in Eq. [3] can also be transformed by defining a vapor deficit as D = e_s(1 - h). The derivative of the vapor deficit is dD = -e_sdh. Substituting these into Eq. [4] gives

[4]

The second term on the right of Eq. [4] is defined as the isothermal latent heat flux density, E_i.

[5]

While it arises from mathematical expediency, isothermal latent heat flux is sometimes used to approximate actual latent heat flux, such as when evaporation is assumed proportional to a gradient in relative humidity. In an isothermal system, such an assumption would be correct, but in nonisothermal systems, an additional component of evaporation is due to the temperature difference (Eq. [4]).

Equations 1, 4, and 5 can now be combined to give

[6]

The apparent psychrometer constant * = K_H/K_v, where the thermodynamic psychrometer constant = c_p/ (K^-1). When K_H = K_v, * = . The slope of the saturation mole fraction function s = /p (K^-1).

The sum of the sensible and latent heat fluxes is the enthalpy flux:

[7]

Combining Eq. [6] and [7] to eliminate C gives

[8]

When H = R_n - G (R_n is net radiation; G is the rate of heat storage), Eq. [8] is the form of the Penman–Monteith equation given by Monteith (1980) for surfaces that are not saturated. A complementary equation for C is derived by eliminating E between Eq. [6] and [7]:

[9]

This equation is similar to one given by Monteith (1973), except here the humidity of the evaporating surface is included in the denominator while Monteith's original equation assumed evaporation from a saturated surface.

	DECOUPLING THE HEAT AND VAPOR TRANSPORT EQUATIONS

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
It is clear that the partitioning of heat (enthalpy, or in our system, net radiation minus heat storage) into sensible and latent components depends on the diffusivities for heat and water, the temperature, the vapor deficit, and the strength of the heat source. Each flux therefore depends on the other for its value. A simpler system, where the fluxes are independent, can be derived from linear combinations of C and E. One combination is given in Eq. [7] and is the enthalpy flux. When Eq. [8] and [9] are added, the terms dependent on D add to zero, leaving just the enthalpy flux term. The driving force for this flux is shown by Monteith (1973) to be the equivalent temperature:

[10]

It is derived by adding Eq. [1] and [2] and combining the derivatives. When K_v = K_H, as is the case for eddy transport within and above the canopy, * = and * = .

A second combination is E - (hs/*)C. When Eq. [8] and [9] are combined in this way, the enthalpy terms add to zero, and only the isothermal latent heat flux, E_i, remains. The driving force for this flux is the vapor deficit gradient, dD, as shown in Eq. [5]. Chen (1984) chose a different combination of H and E. He defined a saturation heat flux density as J = C - (/s)E. This choice does decouple the equations but does not eliminate H from the second set of equations, resulting in a more complicated analysis and a flux that is less intuitive than the isothermal latent heat flux density. There are, of course, many linear combinations of C and E that result in sets of decoupled equations for canopy transport. The set chosen here is thought preferable to Chen's because of its simplicity. As previously mentioned, it has some physical meaning in that it is the evaporation rate in systems with negligible temperature gradients.

	APPLICATION TO TRANSPORT IN PLANT AND RESIDUE CANOPIES

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
Soil–plant–atmosphere models, such as Cupid (Norman and Campbell, 1983), arbitrarily divide the canopy and soil into layers and then use numerical solutions to the differential equations for flow to determine the temperature and moisture distribution and the fluxes of sensible and latent heat in the system. An electrical analog of the canopy and soil surface is shown in Fig. 1 . The coupling between the two circuits is indicated by the fact that fluxes of both sensible and latent heat depend on canopy properties. Resistances to transport between layers within the canopy are obtained by integrating Eq. [1] and [2] over the layer depth, giving r_v = z/K_v and r_H = z/K_H. Since K_H and K_v are assumed equal for turbulent transport within the canopy (Campbell and Norman, 1998), the resistances for heat and vapor transport between canopy layers are equal. Resistances shown as r' in Fig. 1 are the boundary-layer resistances for canopy elements. The resistances between canopy elements and the air for vapor can be quite different from those for heat. When water evaporates from substomatal cavities, then r^'_v is the sum of a stomatal resistance and a boundary-layer resistance for the leaf. When the leaf surface is wet, the resistance is just the vapor boundary-layer resistance of the canopy element, which is about 10% smaller than the corresponding value for r^'_H (Campbell and Norman, 1998).

View larger version (18K): [in this window] [in a new window]   Fig. 1. Electrical analog of sensible and latent heat transport in a plant canopy with four layers. The symbol for a current source (flux) is shown as an arrow within a circle.

Figure 2 shows the decoupled electrical analogs for the flux of enthalpy and isothermal latent heat. Boundary conditions are _a and D_a for the air above the canopy and _o and D_o at the soil surface. Values for _a, D_a, _o, and D_o are computed from known temperatures and relative humidities of the atmosphere and the soil surface. If boundary conditions and resistances are known, then all other values of and D can be computed. Once these are known, T and e can be found from the definitions of equivalent temperature and vapor deficit.

View larger version (19K): [in this window] [in a new window]   Fig. 2. Equivalent uncoupled circuits for enthalpy (left) and isothermal latent heat (right) fluxes.

	CALCULATION OF THE THEVENIN EQUIVALENT POTENTIALS AND RESISTANCES

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

[11]

and

[12]

View larger version (7K): [in this window] [in a new window]   Fig. 3. Thevenin equivalent circuits of the canopy analogs shown in Fig. 2.

The method for calculating the Thevenin equivalent potentials and resistances is straightforward. The circuit is broken at points AA' (Fig. 2 and 3). The Thevenin resistance is the resistance of the circuit with batteries shorted and current sources open-circuited. The Thevenin potential is the open circuit potential, or the potential measured at AA' when nothing is connected. For the enthalpy circuit

[13]

The Thevenin potential is

[14]

where n is the number of layers and _a is the equivalent temperature of the air above the canopy.

The potential and resistance for the vapor and isothermal latent heat flux calculation are obtained similarly. The Thevenin potential is

[15]

where is the product operator and R_i is calculated from the recursive relationship

[16]

starting with layer n, where R_n = r_v,n.

The Thevenin resistance is

[17]

where R₂ is computed from Eq. [16].

	THEVENIN EQUIVALENT CIRCUITS AS BOUNDARY CONDITIONS FOR THE SOIL PROBLEM

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
The soil models in Cupid (Norman and Campbell, 1983) and in Campbell (1985) find temperature and water potential at each soil layer using a Newton–Raphson iterative technique. Values for the surface heat and water vapor fluxes are needed as well as their derivatives with respect to surface temperature and water potential or humidity. Both fluxes depend on both variables, as is clear from the forgoing equations, but linked transport models generally hold surface humidity constant while the heat flow equations are solved and surface temperature constant while the water flow equations are solved (Bristow et al., 1986). The surface temperature and surface humidity, which are needed for the lower boundary condition of the canopy model, are solved for as part of the Newton–Raphson iteration. The derivatives that are needed, dE_o/dh_o and dC_o/dT_o, are

[18]

and

[19]

These are obtained by substituting Eq. [11] and [12] into [8] and [9] and taking derivatives.

	IMPLEMENTATION

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
The details of how to implement this model are beyond the scope of this paper but are well documented (Bristow et al., 1986; Norman and Campbell, 1983; Campbell, 1985). While other models have not used these methods to decouple heat and water flow, the computation of resistances, enthalpy fluxes, etc., are similar. An outline of the approach for applying this model is as follows:

1. Divide the canopy into layers and compute the net radiation, wind speed, resistance to transport between layers, boundary-layer resistances, and layer stomatal resistances for each layer.
   
2. From the resistances and net radiation values, compute the Thevenin equivalent potentials and resistances for the canopy.
   
3. Use the Thevenin resistances and potentials as boundary conditions to the soil temperature and moisture simulation and solve for the soil temperatures and water potentials.
   
4. From the known soil surface conditions, solve the canopy equations for and D and then for T and e.
   
5. Solve for leaf temperatures and transpiration rates.
   

Iteration may be required at several points, especially for long time steps. For example, as soil water is depleted, stomatal resistance increases due to decreasing leaf water potential. There is therefore a link between steps 1 and 3. Also, the temperature profile influences atmospheric stability, which influences transport resistance. Steps 4 and 1 are therefore linked.

	GAPS AND UNRESOLVED ISSUES

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
The canopy model presented here is a steady-state model since no storage of heat or water occurs within the canopy. It works well for plant canopies where the vapor deficit in substomatal cavities is nearly zero, and stomatal resistances remain fairly constant over model time steps. A steady-state model like this one fails, however, when the water content of canopy elements changes markedly. Canopy storage becomes important when dew condenses on or evaporates from leaves and when water condenses in or evaporates from crop residues. In the case of leaves, water condensed on the surface drastically changes the surface resistance for vapor transport, so the model needs to include both a storage capacity and a storage-dependent surface resistance. In the case of crop residues, the humidity depends on the amount of water stored, so the model needs to include a storage-dependent humidity term (moisture release curve). It is not clear that the uncoupled equations can be applied in these cases. More work is needed to address these two important cases.

Once the canopy saturates, the rate of evaporation or condensation of layer j can be computed from Eq. [8], modified to account for evaporation from a saturated surface:

[20]

If the value is negative, condensation is occurring. The value for H is known for each layer (R_{n j} - G_j), but a value for D_i is needed to make the calculation.

A serious shortcoming of the present model is an implicit use of K-theory to describe turbulent transport within the canopy. New solutions to the Lagrangian description of scalar transport in canopies have recently become available (Raupach, 1989; Warland and Thurtell, 2000). These typically are used for heat, water vapor, or CO₂, but presumably would also work for enthalpy and vapor deficit. This is a clear opportunity for further development.

	APPENDIX:

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 
DEFINITION OF SYMBOLS
C, W m^-2 sensible heat flux density

D, kPa water vapor pressure deficit, e_s (1 - h) or e_s - e

E, mol m^-2 s^-1 flux density of water vapor

E_i, mol m^-2 s^-1isothermal flux density of water vapor

G, W m^-2 soil heat flux density or rate of heat storage

H, W m^-2 enthalpy flux density

K_H, m² s^-1 molecular or eddy diffusivity for heat

K_v, m² s^-1 molecular or eddy diffusivity for vapor

R_n, W m^-2 net radiation

T, °C temperature

c_p, J mol^-1 K^-1 specific heat of air

e, kPa water vapor pressure

e_s, kPa saturation vapor pressure at air temperature

h, relative humidity

p, Pa atmospheric pressure

r_H, m² s mol^-1 resistance to heat transport between canopy layers

r_v, m² s mol^-1 resistance to vapor transport between canopy layers

r'_H, m² s mol^-1 leaf or residue boundary layer resistance for heat transport

r'_v, m² s mol^-1 leaf or residue boundary layer and surface resistance for vapor transport

r_HT, m² s mol^-1 Thevenin equivalent resistance for heat transport

r_vT, m² s mol^-1 Thevenin equivalent resistance for vapor transport

s, °C^-1 slope of the saturation mole fraction function, /p

z, m distance

, kPa °C^-1 de_s/dT, slope of the saturation vapor pressure function

, product operator (successive terms multiplied together)

, °C^-1 psychrometer constant, c_p/

*, °C^-1 apparent psychrometer constant, K_H/K_v

, J mol^-1 latent heat of vaporization for water

, mol m^-3 molar density of air

*, °C equivalent temperature

_a, °C equivalent temperature of air above canopy

_o, °C equivalent temperature of soil surface

_T, °C Thevenin equivalent temperature

	REFERENCES

TOP ABSTRACT INTRODUCTION EQUATIONS FOR SENSIBLE AND... DECOUPLING THE HEAT AND... APPLICATION TO TRANSPORT IN... CALCULATION OF THE THEVENIN... THEVENIN EQUIVALENT CIRCUITS AS... IMPLEMENTATION GAPS AND UNRESOLVED ISSUES APPENDIX: REFERENCES

 

• Bristow, K.L., G.S. Campbell, R.I. Papendick, and L.F. Elliott. 1986. Simulation of heat and moisture transfer through a surface residue–soil system. Agric. For. Meteorol. 36:193–214.
• Campbell, G.S. 1985. Soil physics with BASIC: Transport models for soil–plant systems. Elsevier, New York.
• Campbell, G.S., and J.M. Norman. 1998. An introduction to environmental biophysics. 2nd ed. Springer Verlag, New York.
• Chen, J. 1984. Uncoupled multi-layer model for the transfer of sensible and latent heat flux densities from vegetation. Boundary-Layer Meteorol. 28:213–225.
• Lueg, R.E. 1963. Basic electronics for engineers. Int. Textbook Co., Scranton, Pa.
• Monteith, J.L. 1973. Principles of environmental physics. Elsevier, New York.
• Monteith, J.L. 1980. The development and extension of Penman's evaporation formula. p. 247–253. In D. Hillel (ed.) Applications of soil physics. Academic Press, New York.
• Norman, J.M., and G.S. Campbell. 1983. Application of a plant–environment model to problems in irrigation. p. 155–188. In D. Hillel (ed.) Advances in irrigation. Vol. 2. Academic Press, New York.
• Raupach, M.R. 1987. A Lagrangian analysis of scalar transfer in vegetation canopies. Q. J. R. Meteorol. Soc. 113:107–120.
• Raupach, M.R. 1989. A practical Lagrangian method for relating scalar concentrations to source distributions in vegetation canopies. Q. J. Roy. Meteorol. Soc. 115:609–632.
• Warland, J.S., and G.W. Thurtell. 2000. A Lagrangian solution to the relationship between a distributed source and concentration profile. Boundary-Layer Meteorol. 96:453–471.

This article has been cited by other articles:

				V. S. Manoranjan, A. R. Kemanian, R. L. Orozco, and G. S. Campbell Comment on "Modeling Sensible and Latent Heat Transport in Crop and Residue Canopies" by G.S. Campbell. Agron. J. 95:1388-1392 (2003). Agron. J., October 31, 2006; 98(6): 1664 - 1664. [Full Text] [PDF]

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