), enthalpy transport (i.e., the heat content that moves with the molecules)

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1 Steady-state conseration statements for a composite of ces and airspace In steady state, conseration of moecues requires that the tota fux into a representatie oume of mesophy is equa to the fux out storage is zero In a oume of mesophy howeer, composed of both iquid gas phases, ony the tota fux, not the indiidua apor or iquid fuxes, is consered (ie, moecues can jump from one phase to another) Thus, a oca change in the iquid fux (due to eaporation or condensation) must be baanced by an equa and opposite change in the apor fux, 0 = J tota z J z = A J z Conseration of therma energy baances conduction in the iquid and apor phases ( q,q ), enthapy transport (ie, the heat content that moes with the moecues) (1) in the iquid and apor fuxes ( H J, H J ), and the absorbed short wae radiation Q In steady state, q 0 = z A q z A H J z A H J z + Q! (2) Combining energy conseration (2) and moecuar conseration (1) eads to the introduction of the atent heat of aporization as the difference in heat content between the two phases, λ = H H, q 0 = z A q z + Aλ J z + Q! (3) This equation says that in the mesophy, absorbed short wae radiation can be dissipated by an increase in heat conduction in either phase, or by the change of a iquid fux into the apor phase Here we hae negected the conection of enthapy in the absence of a phase change that accompanies a fux of moecues through a temperature fied, as the effect is sma reatie to conduction and atent transport (Suppementa Text S2) Non-dimensiona parameters and forms of the goerning equations To gain insight into which process, heat conduction or atent heat transport, is more important, we rescae temperature and water potentia in reference to their aues at the ascuar pane (subscript o) and normaize by their characteristic gradients We do not yet know what the atter are; their particuar form wi be defined ater to satisfy the

2 constraint that the deriaties are order one For a domain of thickness L in z, the transformed ariabes are, Z = z L, Θ = T T o ΔT c, Ψ = ψ ψ o Δψ c (4) Normaizing the energy oad by either the conductie or atent heat fux in (3), and using moecuar conseration (1), eads to separate (but inked) goerning equations for the potentia and temperature fieds (Suppementa Text S2) With the transformed ariabes, the water potentia fied is goerned by, 0 = 1+ A + A k A λ + A + A λ For conenience we abe the term in brackets, 2 Ψ Z +!QL 2 (5) 2 λ Δψ c 1+ A + A A λ + A + A λ (6) Equation (5) says that for a radiation oad QL! in mesophy tissue to be consumed by oca eaporation λ Δψ c / L, must be equa to one, as the deriatie term is order one by construction The ratios that appear in represent heat conduction oer particuar forms of atent heat transport, and are anaogous to Bowen ratios in ecophysioogica (ie, zero thickness) eaf energy baances (Lambers et a, 1998) can be thought of as describing the tendency of interna eaf tissue toward dissipating a therma oad by interna heat conduction (dominant when is arge) ersus interna atent heat transport (dominant when it is cose to one) is not itsef sufficient to determine where eaporation occurs within a eaf, boundary conditions at the eaf surfaces matter, but it pays an important roe The equation goerning the temperature fied foows as, A 0 = 1+ λ A χ ψ 2 Θ Z ( )ΔT c (7) We again define, 1+ λ T T A (8)

3 To the extent that the energy oad is baanced by therma conduction, must be one Howeer, there is no new information in regarding the baance of heat conduction ersus atent heat transport, as it can be shown that Π 1 T + Π 1 ψ = 1 The definitions for the characteristic gradients that make the deriatie terms order one can now be found as, ΔT c = 2 2 ( ), Δψ = c λ (9) Combining (7) through (9) reduces the goerning equations to, for the potentia fied, and simiary 0 = 2 Ψ Z 2 +1 (10) for the temperature fied Boundary conditions and soutions for a non-transpiring epidermis For a non-transpiring surface, moecuar conseration says that the sum of the apor and iquid fuxes arriing at the epidermis is equa to zero (modeing eaf temperatures beow the ambient dew point, as for exampe occur during foiar uptake of water from fog, woud require modifying this boundary condition) Formay, this condition is stated, (11) ( J J ) z=l = 0 (12) Therma energy conseration requires that the sum of net ong wae radiatie transfer and conduction across the boundary ayer to the enironment baance interna conduction and the transport of enthapy H with the moecuar fux from the mesophy, ( q q + J H J H ) z=l = ( q c + q r ) z=l (13) Insertion of the expressions for the moecuar fuxes (12) and (15) and re-arrangement of (12) and (13) yied the required two conditions on potentia and temperature at the non-transpiring surface At the ascuar pane boundary, we abe the water potentia and temperature that exist there (z=0) as, 0 = 2 Θ Z 2 +1 ψ(z = 0) = ψ o, T (z = 0) = T o, (14) with the understanding that these are among the unknowns to be soed for For water potentia, the non-dimensionaized boundary conditions are then,

4 Ψ Z=1 Z = (q + q ) c r, Ψ(Z = 0) = 0, (15) for temperature, Θ Z=1 Z = (q + q ) c r, Θ(Z = 0) = 0 (16) The soutions to (10) and (11) for a domain bounded by the ascuar pane and a nontranspiring surface foows from integration and the appication of the boundary conditions (15) and (16) The potentia profie is gien by, and the temperature profie foows as, T (z) = T o Boundary conditions and soutions for a transpiring epidermis For a domain bounded by a transpiring surface, moecuar conseration baances interna transport and transpiration E at the surface, (17) (18) ( J J ) z=l = E z=l (19) Conseration of therma energy simiary has the addition, compared to (13), of the fux of enthapy accompanying transpiration, QL ψ(z) = ψ o + 2 z 2 λ 2L z 2 L + L(q + q ) c r z λ L, ( q q + J H J H ) z=l = ( EH + q c + q r ) z=l (20) Unike a standard ecophysioogica eaf energy baance anaysis (Lambers et a, 1998), a atent heat term does not appear directy as a term in the energy baance at the transpiring surface; rather, it enters as a difference in the enthapy between the apor and iquid phases The non-dimensionaized boundary conditions are found as, ( ) Ψ Z=1 Z = q + q c r λ E for the potentia profie, and for temperature, ( ) Θ Z=1 Z = q + q c r 2 z 2 ( ) 2L z 2 L L(q c + q r ) ( ) ( ), Ψ(Z = 0) = 0, (21) λa λ E, Θ(Z = 0) = 0 (22) ( ) The soutions to (10) and (11) can then be found in dimensiona form as, z L

5 QL ψ(z) = ψ o + 2 z 2 λ 2L z 2 L + L(q + q ) c r LE ( ) z λ λ A χ T L, (23) for the potentia profie, and for the temperature profie, T (z) = T o 2 z 2 ( ) 2L z 2 L L(q c + q r ) ( ) + Lλ E ( ) z ( ) L For both transpiring and non-transpiring surfaces it wi be noted that the soutions contain the surface energy fuxes as measured ariabes These fuxes, as we as E, can aso be expressed in terms of their dependence on surface and ambient temperatures, and combined with the soution forms aboe, to be soed as a system of simutaneous equations Description of surface fuxes With the subscript L denoting a quantity eauated at the surface, the net radiatie ong wae fux q r from a ower eaf surface to its surroundings (assumed to be at uniform temperature T sur ), depends on their respectie emissiities and the temperature difference in a highy non-inear fashion; (24) q r = IR σt 4 L a IR sur σt 4 sur (25) Here the absorptance of the eaf is assumed to be equa to its emissiity as we as the emissiity of the surroundings, a IR = IR = sur, and σ is Stephan-Botzmann constant For a eaf surface facing the sky, the reeant ambient temperature is that of the we-mixed air near the eaf, q r = IR σt 4 L a IR atm σt 4 air (26) The effectie emissiity atm of the sky is defined by an empirica function that depends on the apor pressure and temperature near the earth s surface, and the fraction of coud coer fc (0 to 1) (Campbe & Norman 1998), atm = 084 f c + ( f c )172 p χ atm air 10 3 T air 1/7 (27)

6 Conduction across the eaf s boundary ayer to the we-mixed air, at T=T air, is described by, q c = k T δ (T T ), L air (28) with the boundary ayer thickness δ gien by an empirica dependence on eaf size and wind speed according to (Nobe 2005), δ = c u w (29) Finay, transpiration can be expressed in terms of the temperature and potentia at the surface, and the ambient air temperature, as E = g s g b 1 (χ L χ air ), (30) with the boundary ayer conductance g b in series with stomata conductance g s gien by, g b = c(t )D (T ) air air δ Distribution of short wae radiation absorption in the body Whie the exchange of ong wae therma radiation with the enironment occurs on the upper and ower epiderma surfaces, we regard the absorption of soar radiation as a process distributed through the body of a eaf (Pieruschka et a 2010) Our reasoning is as foows The high absorbance of eaes across the isibe range of waeengths (04 to 07 µm) is due to photosynthetic and protectie pigments distributed through the thickness In the range of 07 to 13 µm, characterized by ery ow absorbance, the sma amount of absorption is ikey distributed through the thickness due to scattering Beyond 13 µm, the absorption spectrum of eaes is simiar to an equiaent thickness of water, again suggesting a process distributed through the eaf thickness (Kniping 1970) Neertheess, the distribution of ight absorption within eaes is compex, first increasing with depth in the paisade (as chorophy concentrations increase) before decaying exponentiay through the rest of the thickness, such that about 80% of the energy absorption occurs in the paisade, and 20% in the spongy mesophy in spinach eaes (Vogemann 2002) For the sake of simpicity, we approximate energy absorption (31)

7 as constant through the thickness within either the paisade or spongy, with the distribution of the tota soar radiation SR absorbed by a eaf foowing that of spinach, Paisade Spongy Q d L d = 08SR, Q b L b = 02SR (32) (33) Here the subscripts b,d refer to the domains beow (abaxia) and aboe (adaxia) the ascuar pane of the eaf With a SR the absorptance of a eaf for short wae soar radiation (ie, PAR and NIR), and r the abedo or refectance of the surroundings, net absorbed soar radiation SR can be estimated from ight meter readings of photosynthetic fux density PPFD by, PPFD SR = a SR (34) J mo -1 (1+ r) A F, e e where 045 represents the fraction of incident soar energy in PAR, and is the energy per moe of PAR (Nobe 2005, Campbe 1998) As we are interested in the energy oad to be dissipated as heat, we define SR to be net the energy stored in chemica bonds by photosynthesis A e or re-emitted by fuorescence F e The former can be estimated from the assimiation rate [mo m -2 s -1 ] times the energy stored in chemica bonds per moe of assimiated CO 2, 479 kj mo -1 (Nobe 2005), whie the atter is typicay ony ~1% of absorbed soar energy Goba conseration statements and cosure Haing defined absorption in the body in terms of PPFD, and the surface fuxes in terms of T and ψ L, for a eaf composed of an abaxia (b) and an adaxia (d) domain we hae L four equations for the soutions to the temperature and potentia profies, and six unknowns: T Ld,ψ Ld,T o,ψ o,t Lb,ψ Lb To cose the system we turn to two goba conseration statements for therma energy and number of moecues First, the tota amount of absorbed short soar radiation must equa the sum of the therma fuxes form both surfaces, or SR = q cb + q cd + q rb + q rd +λ E b +λ E d (35) Second, the tota number of water moecues eaing the eaf as apor must baance the tota number entering as iquid, or

8 E b + E d = J x ςh A (ψ o ψ r ) (36) The transfer coefficient h A is the hydrauic conductance of the eaf ascuature, normaized to the eaf area, as estimated from ein cutting experiments, and ς is a correction factor accounting for the effect of ein spacing on the aerage potentia at the ascuar pane (Rockwe et a 2014b) The reseroir potentia represents a known source, such as the parent stem potentia, as estimated by a bagged (non-transpiring) eaf With the enironmenta ariabes SR,T air,t sur,ψ r,χ air known, the temperature, potentia, and water apor concentration profies through the eaf thickness can be soed for as a system of six simutaneous equations with six unknowns ψ r

ANALYSIS OF FLOW INSIDE THE FOCUSING TUBE OF THE ABRASIVE WATER JET CUTTING HEAD

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