Importable variables and equations¶

This jupyter notebook can be found at: https://github.com/environmentalscience/essm/blob/master/docs/examples/importable_variables_equations.ipynb

Below, we will import some generic python packages that are used in this notebook:

[1]:

# Checking for essm version installed
import pkg_resources
pkg_resources.get_distribution("essm").version

[1]:

'0.4.2.dev5'

[14]:

from IPython.core.display import display, HTML
display(HTML("<style>.container { width:150% !important; }</style>"))

[4]:

from IPython.display import display
from sympy import init_printing, latex
init_printing()
from sympy.printing import StrPrinter
StrPrinter._print_Quantity = lambda self, expr: str(expr.abbrev)    # displays short units (m instead of meter)

[5]:

import scipy as sc
# Import various functions from sympy
from sympy import Derivative, Eq, exp, log, solve, Symbol

[6]:

from essm.variables.utils import generate_metadata_table, ListTable

General physics variables and equations¶

Variables¶

Pre-defined thermodynamics variables can be imported from essm.variables.physics.thermodynamics:

[7]:

import essm.variables.physics.thermodynamics as physics_vars
vars = ['physics_vars.' + name for name in physics_vars.__all__]
generate_metadata_table([eval(name) for name in vars])

[7]:

Symbol	Name	Description	Definition	Default value	Units
$\alpha_a$	alpha_a	Thermal diffusivity of dry air.		-	m$^{2}$ s$^{-1}$
$\lambda_E$	lambda_E	Latent heat of evaporation.		2450000.0	J kg$^{-1}$
$\nu_a$	nu_a	Kinematic viscosity of dry air.		-	m$^{2}$ s$^{-1}$
$\rho_a$	rho_a	Density of dry air.		-	kg m$^{-3}$
$\sigma$	sigm	Stefan-Boltzmann constant.		5.67e-08	J s$^{-1}$ m$^{-2}$ K$^{-4}$
$c_{pa,mol}$	c_pamol	Molar specific heat of dry air. https://en.wikipedia.org/wiki/Heat_capacity#Specific_heat_capacity		29.19	J K$^{-1}$ mol$^{-1}$
$c_{pa}$	c_pa	Specific heat of dry air.		1010.0	J K$^{-1}$ kg$^{-1}$
$c_{pv}$	c_pv	Specific heat of water vapour at 300 K. http://www.engineeringtoolbox.com/water-vapor-d_979.html		1864	J K$^{-1}$ kg$^{-1}$
$C_{wa}$	C_wa	Concentration of water in air.		-	mol m$^{-3}$
$D_{va}$	D_va	Binary diffusion coefficient of water vapour in air.		-	m$^{2}$ s$^{-1}$
$g$	g	Gravitational acceleration.		9.81	m s$^{-2}$
$h_c$	h_c	Average 1-sided convective heat transfer coefficient.		-	J K$^{-1}$ s$^{-1}$ m$^{-2}$
$k_a$	k_a	Thermal conductivity of dry air.		-	J K$^{-1}$ m$^{-1}$ s$^{-1}$
$M_w$	M_w	Molar mass of water.		0.018	kg mol$^{-1}$
$M_{air}$	M_air	Molar mass of air. http://www.engineeringtoolbox.com/molecular-mass-air-d_679.html		0.02897	kg mol$^{-1}$
$M_{N_2}$	M_N2	Molar mass of nitrogen.		0.028	kg mol$^{-1}$
$M_{O_2}$	M_O2	Molar mass of oxygen.		0.032	kg mol$^{-1}$
$N_{Gr_L}$	Gr	Grashof number.		-	1
$N_{Le}$	Le	Lewis number.		-	1
$N_{Nu_L}$	Nu	Average Nusselt number over given length.		-	1
$N_{Pr}$	Pr	Prandtl number (0.71 for air).		-	1
$N_{Re_c}$	Re_c	Critical Reynolds number for the onset of turbulence.		-	1
$N_{Re_L}$	Re	Average Reynolds number over given length.		-	1
$P_a$	P_a	Air pressure.		-	Pa
$P_{N2}$	P_N2	Partial pressure of nitrogen.		-	Pa
$P_{O2}$	P_O2	Partial pressure of oxygen.		-	Pa
$P_{was}$	P_was	Saturation water vapour pressure at air temperature.		-	Pa
$P_{wa}$	P_wa	Water vapour pressure in the atmosphere.		-	Pa
$R_d$	R_d	Downwelling global radiation.		-	W m$^{-2}$
$R_s$	R_s	Solar shortwave flux per area.		-	J s$^{-1}$ m$^{-2}$
$R_u$	R_u	Upwelling global radiation.		-	W m$^{-2}$
$R_{mol}$	R_mol	Molar gas constant.		8.314472	J K$^{-1}$ mol$^{-1}$
$T_0$	T0	Freezing point in Kelvin.		273.15	K
$T_a$	T_a	Air temperature.		-	K
$v_w$	v_w	Wind velocity.		-	m s$^{-1}$
$x_{N2}$	x_N2	Mole fraction of nitrogen in dry air.		0.79	1
$x_{O2}$	x_O2	Mole fraction of oxygen in dry air.		0.21	1

Each of the above can also be imported one-by-one, using its Name, e.g.:

from essm.variables.physics.thermodynamics import R_mol

Equations¶

General equations based on the above variables can be imported from essm.equations.physics.thermodynamics:

[8]:

import essm.equations.physics.thermodynamics as physics_eqs
modstr = 'physics_eqs.'
eqs = [name for name in physics_eqs.__all__]
table = ListTable()
#table.append(('Name', 'Description', 'Equation'))
for name in eqs:
    table.append((name, eval(modstr+name).__doc__, latex('$'+latex(eval(modstr+name))+'$')))
table

[8]:

eq_Le	Le as function of alpha_a and D_va. (Eq. B3 in :cite:`schymanski_leaf-scale_2017`)	$N_{Le} = \frac{\alpha_a}{D_{va}}$
eq_Cwa	C_wa as a function of P_wa and T_a. (Eq. B9 in :cite:`schymanski_leaf-scale_2017`)	$C_{wa} = \frac{P_{wa}}{R_{mol} T_a}$
eq_Nu_forced_all	Nu as function of Re and Re_c under forced conditions. (Eqs. B13--B15 in :cite:`schymanski_leaf-scale_2017`)	$N_{Nu_L} = - \frac{\sqrt[3]{N_{Pr}} \left(- 37 N_{Re_L}^{\frac{4}{5}} + 37 \left(N_{Re_L} + N_{Re_c} - \frac{\left\|{N_{Re_L} - N_{Re_c}}\right\|}{2}\right)^{\frac{4}{5}} - 664 \sqrt{N_{Re_L} + N_{Re_c} - \frac{\left\|{N_{Re_L} - N_{Re_c}}\right\|}{2}}\right)}{1000}$
eq_Dva	D_va as a function of air temperature. (Table A.3 in :cite:`monteith_principles_2007`)	$D_{va} = T_a p_1 - p_2$
eq_alphaa	alpha_a as a function of air temperature. (Table A.3 in :cite:`monteith_principles_2007`)	$\alpha_a = T_a p_1 - p_2$
eq_ka	k_a as a function of air temperature. (Table A.3 in :cite:`monteith_principles_2007`)	$k_a = T_a p_1 + p_2$
eq_nua	nu_a as a function of air temperature. (Table A.3 in :cite:`monteith_principles_2007`)	$\nu_a = T_a p_1 - p_2$
eq_rhoa_Pwa_Ta	rho_a as a function of P_wa and T_a. (Eq. B20 in :cite:`schymanski_leaf-scale_2017`)	$\rho_a = \frac{M_{N_2} P_{N2} + M_{O_2} P_{O2} + M_w P_{wa}}{R_{mol} T_a}$
eq_Pa	Calculate air pressure. From partial pressures of N2, O2 and H2O, following Dalton's law of partial pressures.	$P_a = P_{N2} + P_{O2} + P_{wa}$
eq_PN2_PO2	Calculate P_N2 as a function of P_O2. It follows Dalton's law of partial pressures.	$P_{N2} = \frac{P_{O2} x_{N2}}{x_{O2}}$
eq_PO2	Calculate P_O2 as a function of P_a, P_N2 and P_wa.	$P_{O2} = \frac{P_a x_{O2} - P_{wa} x_{O2}}{x_{N2} + x_{O2}}$
eq_PN2	Calculate P_N2 as a function of P_a, P_O2 and P_wa.	$P_{N2} = \frac{P_a x_{N2} - P_{wa} x_{N2}}{x_{N2} + x_{O2}}$
eq_rhoa	Calculate rho_a from T_a, P_a and P_wa.	$\rho_a = \frac{x_{N2} \left(M_{N_2} P_a - P_{wa} \left(M_{N_2} - M_w\right)\right) + x_{O2} \left(M_{O_2} P_a - P_{wa} \left(M_{O_2} - M_w\right)\right)}{R_{mol} T_a x_{N2} + R_{mol} T_a x_{O2}}$

Variables for leaf chamber model¶

These refer to the model by Schymanski & Or (2017) and ongoing work.

Leaf chamber insulation¶

Variables related to the model by can be imported from essm.variables.chamber.insulation:

[12]:

import essm.variables.chamber.insulation as chamber_ins
vars = ['chamber_ins.' + name for name in chamber_ins.__all__]
generate_metadata_table([eval(name) for name in vars])

[12]:

Symbol	Name	Description	Definition	Default value	Units
$A_i$	A_i	Conducting area of insulation material.		-	m$^{2}$
$c_{pi}$	c_pi	Heat capacity of insulation material.		-	J K$^{-1}$ kg$^{-1}$
$dT_i$	dT_i	Temperature increment of insulation material.		-	K
$L_i$	L_i	Thickness of insulation material.		-	m
$lambda_i$	lambda_i	Heat conductivity of insulation material.		-	J K$^{-1}$ m$^{-1}$ s$^{-1}$
$Q_i$	Q_i	Heat conduction through insulation material.		-	J s$^{-1}$
$rho_i$	rho_i	Density of insulation material.		-	kg m$^{-3}$

Leaf chamber mass balance¶

Variables related to the model by can be imported from essm.variables.chamber.mass:

[13]:

import essm.variables.chamber.mass as chamber_mass
vars = ['chamber_mass.' + name for name in chamber_mass.__all__]
generate_metadata_table([eval(name) for name in vars])

/home/stan/Programs/essm/essm/variables/_core.py:89: UserWarning: "essm.variables.leaf.energy_water:L_A" will be overridden by "essm.variables.chamber.mass:<class 'essm.variables.chamber.mass.L_A'>"
  instance[expr] = instance

[13]:

Symbol	Name	Description	Definition	Default value	Units
$F_{in,mol,a}$	F_in_mola	Molar flow rate of dry air into chamber.		-	mol s$^{-1}$
$F_{in,mol,w}$	F_in_molw	Molar flow rate of water vapour into chamber.		-	mol s$^{-1}$
$F_{in,v}$	F_in_v	Volumetric flow rate into chamber.		-	m$^{3}$ s$^{-1}$
$F_{out,mol,a}$	F_out_mola	Molar flow rate of dry air out of chamber.		-	mol s$^{-1}$
$F_{out,mol,w}$	F_out_molw	Molar flow rate of water vapour out of chamber.		-	mol s$^{-1}$
$F_{out,v}$	F_out_v	Volumetric flow rate out of chamber.		-	m$^{3}$ s$^{-1}$
$H_c$	H_c	Chamber height.		-	m
$L_A$	L_A	Leaf area.		-	m$^{2}$
$L_c$	L_c	Chamber length.		-	m
$n_c$	n_c	molar mass of gas in chamber.		-	mol
$P_{w,in}$	P_w_in	Vapour pressure of incoming air.		-	Pa
$P_{w,out}$	P_w_out	Vapour pressure of outgoing air.		-	Pa
$R_{H,in}$	R_H_in	Relative humidity of incoming air.		-	1
$T_d$	T_d	Dew point temperature of incoming air.		-	K
$T_{in}$	T_in	Temperature of incoming air.		-	K
$T_{out}$	T_out	Temperature of outgoing air (= chamber T_a).		-	K
$T_{room}$	T_room	Lab air temperature.		-	K
$V_c$	V_c	Chamber volume.		-	m$^{3}$
$W_c$	W_c	Chamber width.		-	m

Bibliography¶

See https://essm.readthedocs.io/en/latest/api.html#bibliography

[ ]:

Symbol	Name	Description	Definition	Default value	Units
$\alpha_l$	alpha_l	Leaf albedo, fraction of shortwave radiation reflected by the leaf.		-	1
$\epsilon_l$	epsilon_l	Longwave emmissivity of the leaf surface.		1.0	1
$\rho_{al}$	rho_al	Density of air at the leaf surface.		-	kg m$^{-3}$
$a_s$	a_s	Fraction of one-sided leaf area covered by stomata. (1 if stomata are on one side only, 2 if they are on both sides).		-	1
$a_{sh}$	a_sh	Fraction of projected area exchanging sensible heat with the air.		2.0	1
$C_{wl}$	C_wl	Concentration of water in the leaf air space.		-	mol m$^{-3}$
$E_l$	E_l	Latent heat flux from leaf.		-	J s$^{-1}$ m$^{-2}$
$E_{l,mol}$	E_lmol	Transpiration rate in molar units.		-	mol s$^{-1}$ m$^{-2}$
$g_{bw,mol}$	g_bwmol	Boundary layer conductance to water vapour.		-	mol s$^{-1}$ m$^{-2}$
$g_{bw}$	g_bw	Boundary layer conductance to water vapour.		-	m s$^{-1}$
$g_{sw,mol}$	g_swmol	Stomatal conductance to water vapour.		-	mol s$^{-1}$ m$^{-2}$
$g_{sw}$	g_sw	Stomatal conductance to water vapour.		-	m s$^{-1}$
$g_{tw,mol}$	g_twmol	Total leaf layer conductance to water vapour.		-	mol s$^{-1}$ m$^{-2}$
$g_{tw}$	g_tw	Total leaf conductance to water vapour.		-	m s$^{-1}$
$h_c$	h_c	Average 1-sided convective heat transfer coefficient.		-	J K$^{-1}$ s$^{-1}$ m$^{-2}$
$H_l$	H_l	Sensible heat flux from leaf.		-	J s$^{-1}$ m$^{-2}$
$L_A$	L_A	Leaf area.		-	m$^{2}$
$L_l$	L_l	Leaf width as characteristic length scale for convection.		-	m
$N_{Gr_L}$	Gr	Grashof number.		-	1
$P_{wl}$	P_wl	Water vapour pressure inside the leaf.		-	Pa
$r_{bw}$	r_bw	Boundary layer resistance to water vapour, inverse of $g_{bw}$.		-	s m$^{-1}$
$R_{la}$	R_la	Longwave radiation absorbed by leaf.		-	W m$^{-2}$
$R_{ld}$	R_ld	Downwards emitted/reflected global radiation from leaf.		-	W m$^{-2}$
$R_{ll}$	R_ll	Longwave radiation away from leaf.		-	W m$^{-2}$
$R_{lu}$	R_lu	Upwards emitted/reflected global radiation from leaf.		-	W m$^{-2}$
$r_{sw}$	r_sw	Stomatal resistance to water vapour, inverse of $g_{sw}$.		-	s m$^{-1}$
$r_{tw}$	r_tw	Total leaf resistance to water vapour, $r_{bv} + r_{sv}$.		-	s m$^{-1}$
$T_l$	T_l	Leaf temperature.		-	K
$T_w$	T_w	Radiative temperature of objects surrounding the leaf.		-	K

eq_Rs_enbal	Calculate R_s from energy balance. (Eq. 1 in :cite:`schymanski_leaf-scale_2017`)	$R_s = E_l + H_l + R_{ll}$
eq_Rll	R_ll as function of T_l and T_w. (Eq. 2 in :cite:`schymanski_leaf-scale_2017`)	$R_{ll} = a_{sh} \epsilon_l \sigma \left(T_l^{4} - T_w^{4}\right)$
eq_Hl	H_l as function of T_l. (Eq. 3 in :cite:`schymanski_leaf-scale_2017`)	$H_l = a_{sh} h_c \left(- T_a + T_l\right)$
eq_El	E_l as function of E_lmol. (Eq. 4 in :cite:`schymanski_leaf-scale_2017`)	$E_l = E_{l,mol} M_w \lambda_E$
eq_Elmol	E_lmol as functino of g_tw and C_wl. (Eq. 5 in :cite:`schymanski_leaf-scale_2017`)	$E_{l,mol} = g_{tw} \left(- C_{wa} + C_{wl}\right)$
eq_gtw	g_tw from g_sw and g_bw. (Eq. 6 in :cite:`schymanski_leaf-scale_2017`)	$g_{tw} = \frac{1}{\frac{1}{g_{sw}} + \frac{1}{g_{bw}}}$
eq_gbw_hc	g_bw as function of h_c. (Eq. B2 in :cite:`schymanski_leaf-scale_2017`)	$g_{bw} = \frac{a_s h_c}{N_{Le}^{\frac{2}{3}} c_{pa} \rho_a}$
eq_Cwl	C_wl as function of P_wl and T_l. (Eq. B4 in :cite:`schymanski_leaf-scale_2017`)	$C_{wl} = \frac{P_{wl}}{R_{mol} T_l}$
eq_Pwl	Clausius-Clapeyron P_wl as function of T_l. (Eq. B3 in :cite:`hartmann_global_1994`)	$P_{wl} = p_1 e^{- \frac{M_w \lambda_E \left(- \frac{1}{p_2} + \frac{1}{T_l}\right)}{R_{mol}}}$
eq_Elmol_conv	E_lmol as function of g_twmol and P_wl. (Eq. B6 in :cite:`schymanski_leaf-scale_2017`)	$E_{l,mol} = \frac{g_{tw,mol} \left(- P_{wa} + P_{wl}\right)}{P_a}$
eq_gtwmol_gtw	g_twmol as a function of g_tw. It uses eq_Elmol, eq_Cwl and eq_Elmol_conv.	$g_{tw,mol} = \frac{g_{tw} \left(P_a P_{wa} T_l - P_a P_{wl} T_a\right)}{R_{mol} T_a T_l \left(P_{wa} - P_{wl}\right)}$
eq_gtwmol_gtw_iso	g_twmol as a function of g_tw at isothermal conditions.	$g_{tw,mol} = \frac{P_a g_{tw}}{R_{mol} T_a}$
eq_hc	h_c as a function of Nu and L_l. (Eq. B10 in :cite:`schymanski_leaf-scale_2017`)	$h_c = \frac{N_{Nu_L} k_a}{L_l}$
eq_Re	Re as a function of v_w and L_l. (Eq. B11 in :cite:`schymanski_leaf-scale_2017`)	$N_{Re_L} = \frac{L_l v_w}{\nu_a}$
eq_Gr	Gr as function of air density within and outside of leaf. (Eq. B12 in :cite:`schymanski_leaf-scale_2017`)	$N_{Gr_L} = \frac{L_l^{3} g \left(\rho_a - \rho_{al}\right)}{\nu_a^{2} \rho_{al}}$

Symbol	Name	Description	Definition	Default value	Units
$alpha_l$	alpha_l	Leaf albedo, fraction of shortwave radiation reflected by the leaf.		-	1
$R_d$	R_d	Downwelling global radiation.		-	J s$^{-1}$ m$^{-2}$
$R_u$	R_u	Upwelling global radiation.		-	J s$^{-1}$ m$^{-2}$
$R_{la}$	R_la	Longwave radiation absorbed by leaf.		-	J s$^{-1}$ m$^{-2}$
$R_{ld}$	R_ld	Downwards emitted/reflected global radiation from leaf.		-	J s$^{-1}$ m$^{-2}$
$R_{lu}$	R_lu	Upwards emitted/reflected global radiation from leaf.		-	J s$^{-1}$ m$^{-2}$
$S_a$	S_a	Radiation sensor above leaf reading.		-	J s$^{-1}$ m$^{-2}$
$S_b$	S_b	Radiation sensor below leaf reading.		-	J s$^{-1}$ m$^{-2}$
$S_s$	S_s	Radiation sensor beside leaf reading.		-	J s$^{-1}$ m$^{-2}$

Importable variables and equations¶

General physics variables and equations¶

Variables¶

Equations¶

Variables for leaf chamber model¶

Leaf chamber insulation¶

Leaf chamber mass balance¶

Bibliography¶

Navigation

Related Topics

Importable variables and equations¶

General physics variables and equations¶

Variables¶

Equations¶

Variables and equations related to plant leaves¶

Variables for leaf energy and water balance¶

Equations for leaf energy and water balance¶

Variables for leaf radiative balance¶

Variables for leaf chamber model¶

Leaf chamber insulation¶

Leaf chamber mass balance¶

Bibliography¶