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This section is devoted to mathematical analysis of the effects of EMFs exposure. The electromagnetic (EM) and heat transfer (HT) phenomena rule thermal BEs due to EMF exposure. The temperature increase ΔT in the tissue will be determined by the second phenomenon. The corresponding heat source will be the power dissipated in the material concerned, which can be considered by the first phenomenon. The volume density of this power Pd for dielectric materials (biological tissues) and the corresponding specific absorption rate (SAR) are given by:
$\rm P_{d} = \omega \cdot \varepsilon''\cdot E^{2}/2 $ (1) $\rm SAR = P_{d} /\rho = \omega \cdot \varepsilon''\cdot E^{2}/ (2\rho) $ (2) Equations (1, 2), the parameters: ε'' is the imaginary part of the complex permittivity of the absorbing material and ρ (in kg/m3) is the material density. The variable ω is the angular frequency = 2πf, f is the frequency (Hz) of the exciting EMF, E the absolute peak value of the electric field strength (V/m) and SAR (in W/kg). The power dissipation (in W/m3) given by Eqn (1) relates to foremost dielectric heating of EMF energy loss. Note that the imaginary part ε'' of the (frequency-dependent) permittivity ε (in F/m) is a measure for the ability of a dielectric material to convert EMF energy into heat, also named dielectric loss. The real part ε' of the permittivity is the normal effect of capacitance and results in non-dissipative reactive power.
The power dissipations given by Eqn (1) as well as the SAR given by Eqn (2) will be used in the coupling of EM and HT equations.
The governing equations are the Maxwell EM equations[22] and the HT adapted plant bio-heat tissue equation. The junction between EMF and ΔT is Pd or SAR, which can be obtained from EM equations and used as input to the HT equation.
EM equations
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The system of EM equations can be formulated mathematically in different functional forms of the problem considered. One of the most common is the basic full-wave electromagnetic formulation for harmonic fields, which seems simpler to explain. The EM equations can be given by:
$ \bf\bigtriangledown \times \bf H = J $ (3) $ {\bf J} = \sigma {\bf E} + j \omega {\bf D} + {\bf J}_{\bf e} $ (4) $ {\bf E} = -\bigtriangledown V - j \omega {\bf A} $ (5) ${\bf B} = \bigtriangledown \times {\bf A} $ (6) In EM Eqns (3–6), H and E are the magnetic and electric fields in A/m and V/m, B and D are the magnetic and electric inductions in T and C/m2, A and V are the magnetic vector and electric scalar potentials in W/m and volt. J and Je are the total and source current densities in A/m2, σ is the electric conductivity in S/m, and ω is the angular frequency. The symbol
is a vector of partial derivative operators, and its three possible implications are gradient (product with a scalar field), divergence and curl (dot and cross products respectively with a vector field). The magnetic and electric comportment laws respectively between B/H and D/E are represented by the permeability μ and the permittivity ε (in H/m and F/m).$\bigtriangledown $ The input source term in EM Eqns (3−6) is the excitation current density Je = σ Ee = j ω De = j ω ε Ee. The choice of the form of the source term depends on the nature of the exposure, near or far field, see the section - EMF-PBH model solution strategy - relative to the solution technique.
The dissipated power Pd and the SAR given by Eqns (1 and 2) can be determined via the E value obtained from the solution of Eqns (3−6).
HT equations
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Considering the HT problem, in general the amount of heat quantity absorbed by an element of tissue or more generally of a lossy dielectric can be given as:
$\rm \Delta Q = c m_{s} \Delta T $ (7) In Eqn (7), Q is the heat energy absorbed or dissipated in joule (J), ms is the mass of the substance in (kg), ΔT is the change in substance temperature in (°C), c is the specific heat of the substance (the heat required to change a substance unit mass by one degree) in J/(kg °C). The corresponding power will be ΔP = ΔQ/ Δt. The volume specific power will be:
$\rm \Delta P_{v} = \Delta P/ v = c (m_{s} /v) (\Delta T/\Delta t) = c \rho (\Delta T/\Delta t) $ (8) In Eqn (8), P is the power in watt (W), t is the time in (s), Pv is the power per unit volume in (W/m3), v is the volume in m3, and ρ is the density in kg/m3.
The HT equation giving the volume specific power of Eqn (8) in its differential form is given by:
$\rm c \rho \partial T/\partial t = \bigtriangledown \cdot (k \bigtriangledown T) $ (9) In Eqn (9), k is thermal conductivity in W/ (m∙°C)
Considering the case of tissues in plants, we have to consider in Eqn (9) the internal heat source Pi and the involved convective heat transfer via sap fluid corresponding to the considered part of the plant (leaf, flower petal, stem, branch, trunk, etc.). The convective heat transfer coefficient in the sap hs in W/(m2∙°C) can be defined through ΔP = hs S ΔT and from Eqn (8):
$\rm \Delta P_{v} = h_{s} (S/v) \Delta T = (h_{s} / \chi) \Delta T = c_{s} \rho_{s} (\Delta T/\Delta t) $ (10) Then:
$\rm h_{s} = c_{s} \rho_{s} p_{s}\chi $ (11) In Eqns (10, 11), S is the surface of plant part concerned in m2, χ is the thickness of the plant part in m. Also cs, ρs, ps are respectively sap, specific heat in in J/(kg °C), density in kg/m3, perfusion rate in 1/s. Note that in the present work the vulnerable exposed plant parts are mostly leaves and flower petals.
On the other hand, we have to consider in Eqn (9) the external heat source related to the EMF exposure, Pd or SAR given by Eqns (1 and 2).
Under these conditions, Eqn (9) will be extended to a plant bio-heat (PBH) equation, which could be presented as follows:
$\rm c \rho \partial T/\partial t = \bigtriangledown \cdot (k\; \bigtriangledown T) + P_{d} + P_{i} + c_{s} \rho_{s} p_{s} (T_{s} - T) $ (12) In Eqn (12), Pi is the tissue internal (latent heat flux) heat source in W/m3, Ts and T are respectively the sap fluid temperature and the local temperature of tissue in °C. Equation (12) corresponds to a bio-heat plant tissues accounting for EMF exposure. This equation has a form similar to the Penne’s bio-heat equation[9, 10, 23] related to human tissues and convective heat transfer in blood. As mentioned earlier, plant sap plays the role of blood in animals. In addition, phloems and xylems for sap play the role of arteries and veins for blood. Note that in Penne’s bio-heat equation, the term related to tissues metabolic heat corresponds to plant tissues internal heat in Eqn (12). As well, the terms representing convection heat transfer correspond respectively in the two equations to animal blood and plant sap. The orders of magnitude of the relative values of the parameters related to animal and plant tissues as well as blood and sap are obviously different due to the different nature of animal and plant tissues. The different values of parameters involved in Eqn (12) relative to Pi, cs, ρs, and ps depend on the plant type and the specific part of this plant. These values could be found in textbooks or could be measured.
EMF-PBH model solution strategy
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Equations (3–6 and 12) can be solved in a coupled way. Given the geometric complexity and inhomogeneity of tissue, the solution must be local in the tissue using discretized 3D techniques as finite elements[24−30] in the appropriate element of the tissue. The discretized 3D elements are volume parts enclosed in surface elements, each encircled by edge elements, each ended by two nodes. For example, a tetrahedral element involves four triangular faces, six straight edges, and four nodes. The fields could be defined on nodes, edges, faces or volume depending on the nature of the field as requirements of continuity, etc.[31]. The coupling of the EMF and PBH equations is weak in nature due to the distant values of their time constants[9,10,23]. Thus, an iterative solution provides in the tissue the local distributions of the induced values of E, B, and J, and hence Pd, SAR, and ΔT. The parameters concerned are those of the tissue properties, ε, Pi, cs, ρs, ps, etc. The geometry involved is that related to the shape of the portion of tissue concerned. Exposure conditions are taken into account via the nature of the EMF source (strength and frequency) and the exposure interval. Note that consideration of the exposure source is different for near or far field cases. For near-field radiation, the source is generally involved in the solution domain as a focused field. In the case of the far field, which is generally of homogeneous value, the source is imposed uniformly over the entire exposed surface of the object. Figure 2 summarizes the EMF-PBH models, weak coupled solution strategy accounting for the nature of the exposure source and the tissue parameters. The induced fields are controlled vs thresholds in the whole tissue.
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All data generated or analyzed during this study are included in this published article.
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About this article
Cite this article
Razek A. 2024. Analysis and control of ornamental plant responses to exposure to electromagnetic fields. Ornamental Plant Research 4: e009 doi: 10.48130/opr-0024-0007
Analysis and control of ornamental plant responses to exposure to electromagnetic fields
- Received: 05 January 2024
- Accepted: 18 February 2024
- Published online: 02 April 2024
Abstract: All plants, including ornamentals, are subject to different stressors related to their environment. These can disrupt their progress, including their ability to generate pollen and reproduce successfully. These stressors may involve inert natural impacts such as temperature, deficiency and salinity levels of water. They may also involve natural active effects such as herbivores and pathogens. Other types of accidental natural effects, such as storms and lightning, could also occur. There are also artificial effects that shape the environment of human activity. Nowadays, the environment of electromagnetic fields, and in particular those at high frequency, are an important element. This contribution aims to propose an analysis and a strategy for controlling the biological effects of plant exposure to such fields. The analysis takes into account the nature of the exposure source, the properties of the exposed materials and the exposure conditions. This is accomplished through the governing equations related to electromagnetic and heat transfer phenomena permitting the prediction of biological thermal effects of the exposure. The involved coupled phenomena solution approach makes it possible to predict, verify, and provide means of protection against exposure. The proposed survey is supported by a review of the literature on the various subjects concerned.
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Key words:
- EMF exposure /
- Daily RF devices /
- Plant tissues /
- Biological effects /
- Thermal behavior /
- EMF model /
- Bio-heat transfer model /
- Coupled strategy