(*^ ::[ frontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.1"; macintoshStandardFontEncoding; keywords = "Title, Subtitle, Subsubtitle, Section, Subsection, Subsubsection, Text, Small Text, Input, Output, Message, Print"; paletteColors = 128; showRuler; automaticGrouping; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, italic, e6, 18, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 14, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 12, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, italic, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 9, "Geneva"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, L-5, 12, "Times"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 9, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 10, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; ; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = leftheader, inactive, L2, 12, "Times"; ; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; ; fontset = leftfooter, inactive, L2, 12, "Times"; ; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; ; ] :[font = title; inactive; preserveAspect; startGroup; ] Calculation of Particle Removal Velocities in the Mecklenburg Bight Based on the Thorium Deficit :[font = smalltext; inactive; preserveAspect; center; ] by ;[s] 2:0,1;2,0;4,-1; 2:1,13,10,Geneva,0,9,0,0,0;1,16,12,Times,1,14,0,0,0; :[font = subtitle; inactive; preserveAspect; startGroup; ] Joachim Gruber Arbeitsbereich Umweltschutztechnik Technische Universität Hamburg Harburg Eißendorfer Straße 40 D 21073 Hamburg Tel: (040) 7718 2701 E-Mail: Gruber@TU-Harburg.d400.de ;[s] 2:0,1;181,0;182,-1; 2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0; :[font = input; preserveAspect; center; endGroup; ] Zwischenbericht Teilprojekt C: Oktober 1994 - Dezember 1994 :[font = subtitle; inactive; preserveAspect; startGroup; ] Abstract :[font = smalltext; inactive; preserveAspect; ] The temporal development of the thorium activity formed by decay of the natural uranium in sea water was modeled with a dynamical model derived from Kershaw and Young ["Scavenging of Th 234 in the Eastern Irish Sea" J. Environ. Radioactivity 6, 1 - 23, 1988]. It is a system of 6 ordinary linear first order differential equations. The independent variable is the time t, the dependent variables are the soluble and adsorbed thorium and uranium activities in the water column and the sediment. Consistently with the experimental errors, the system of equations can be simplified without loss of accuracy. This way all system parameters become experimetally accessible. In the end, the sedimentation rate has to be estimated from a single equation in which we have the following three unknowns: (1) the time t after the perturbation of the settling velocity v of the suspended matter, (2) the ratio of the thorium and the uranium activities immediately befor the perturbation, (3) the settling velocity v. Since the system is undetermined (three unknowns in a single equation) we have to enter probable values for some of the unknowns. The simplification and the use of probable values exceeds the mentioned model of Kershaw and Young. In addition to the undeterminedness of the system the following errors contribute to the error of the here calculated sedimentation rates: (1) the systematic and statistical experimental errors and (2) the uncertainties resulting from not having experimentally resolved the spatial and temporal variability of the thorium and uranium activities. The sedimentation velocities v thus calculated are up to two orders of magnitude smaller than the settling velocities calculated after Aldredge and Gotschalk ["In-situ settling behaviour of marine snow" Limnol. Ocea- nog., 33, 339-351, 1988] in stagnant water without resuspension based on the density and size of the particulate matter. :[font = smalltext; inactive; preserveAspect; ] Die zeitliche Entwicklung der aus natürlichem Uran im Seewasser gebildeten Thorium- aktivität wurde mit einem an Kershaw und Young ["Scavenging of Th 234 in the Eastern Irish Sea" J. Environ. Radioactivity 6, 1 - 23, 1988] angelehnten neuen dynamischen Modell dargestellt. Es ist ein System von 6 gewöhnlichen linearen Differentialgleichungen erster Ordnung. Die unabhängige Variable ist die Zeit t, die abhängigen Variablen sind die gelösten und adsorbierten Thorium- und Uranaktivitäten in der Wassersäule und im Sedi- ment. Das Gleichungssytem kann im Einklang mit den Meßfehlern ohne Verlust von Genauigkeit vereinfacht werden, sodaß alle Systemparameter experimentell zugänglich werden. Die Sedimentationsrate muß dann aus einer Gleichung abgeschätzt werden, in der folgende drei Unbekannte stehen: (1) die Zeit t nach Störung der Absinkgeschwindigkeit v des Schwebstoffs, (2) das Verhältnis der Thorium- und der Uran-Aktivität unmittelbar vor dieser Störung, (3) die Absinkgeschwindigkeit v. Die Unterbestimmtheit des Systems (drei Unbekannte in einer Gleichung) muß durch statistische Wahrscheinlichkeitsbetrachtungen beseitigt werden. Die Vereinfachung des Differentialgleichungssystems und die Beseitigung der Unterbe- stimmtheit führt über das Modell von Kershaw und Young hinaus und ist wissenschaftliches Neuland. Zusätzlich zu dieser Unterbestimmtheit des Systems tragen zum Fehler der berechneten Sedimetationsraten bei: (1) die systematischen und statistischen Meßfehler und (2) die Unsicherheiten, resultierend aus der experimentell nicht bestimmten, also unbe- kannten räumlichen Verteilung der Aktivitäten. Die so berechneten Absinkgeschwindigkeiten v liegen bis zu zwei Größenordnungen unter den nach Aldredge und Gotschalk ["In-situ settling behaviour of marine snow" Limnol. Ocea- nog., 33, 339-351, 1988] aus der Dichte und Größe berechneten Sinkgeschwindigkeiten in ruhendem Wasser ohne Resuspension. :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] Table of Contents :[font = text; inactive; preserveAspect; ] The report has the following sections: ;[s] 1:0,1;39,-1; 2:0,13,9,Times,0,12,0,0,0;1,11,8,Times,0,9,0,0,0; :[font = text; inactive; preserveAspect; ] 1. Notation 1.1 The Interactive Mathematica Report 1.1.1 Mathematica's Text Structure 1.2 Abbreviations 1.3 Variables 1.3.1 Explicit Explanation of Some Frequently Used Variables 2. Definitions 2.1 Definitions Including Units (define) 2.2 Definitions Excluding Units (defineU) 2.4 Defintions Concerning Step One Model 2.4.1 Thorium Activities as a Function of Time and Removal Rate 2.4.2 Time Asymptotic Thorium Activities 2.5 Two-Compartment Model 2.5.1 MOST Data Processing 2.5.2 MOST Data Plots 2.6 Removal of Definitions from Their Algebraic Symbols (definex) 2.7 Printing of Data (pdata) 2.8 Conversion between Concentration and Activity 3. Data 3.1 Reference Case 3.1.1 Reference Case Data (data) 3.1.2 Removal of Data Values from Their Symbols (datax) 3.2 Kershaw and Young Data 3.3 MOST Data 4. Compartment Model (Exact System) 4.1 Basic Definitions and Assumptions 4.2 Set of Model Equations 4.3 Graphical Form of Model Equations 4.4 Mathematica Form of System and Typical Solution 4.5 Transition Rates Between Compartments 4.6 Initial Value Problem: Reference Case: Solution of Complete Set of Differential Equations 4.6.1 Water Column 4.6.2 Sediment 4.7 Solution and Transition Rates: Graphical Representation 5. Simplification of the Model: Step One - Omission of the Sediment 5.1 Graphical Form of Model System 5.2 Mathematical Form of Model System 5.3 Analytical Mathematica Solution 5.3.1 Defintions of Solutions of Step One Model 5.4 Inital Value Problem, Reference Case: Solution of Sediment Free Set of Equations 5.4.1 Results 5.4.1.1 Activities 5.4.1.2 Time Development of Thorium Distribution Coefficient 5.4.1.3 Comparison of Solutions of Full and Half System 5.5 Time Asymptotic Activities 5.5.1 Defintions 5.5.2 Young Kershaw 5.5.3 Again: Definition of Thime Asymptotic Activities 5.5.3.1 Soluble and Adsorbed Activities 5.5.3.2 Total Activity 5.5.4 Theorem 5.5.6 Proof 6. Simplification of the Model: Final Step - Adsorption Equilibrium 6.1 Defintion 6.2 Theorem 6.3 Model Equations 6.4 Proof of Theorem (of Section 6.2): Evaluation of Adsorption Equilibrium Solutions 6.4.1 Variation of Sedimentation Rate in Removal Rate Constant S/hx 6.4.2 Variation of Thorium Distribution Coefficient kd 6.4.3 Summary of Sedimentation and Distribution Coefficient Cases ;[s] 9:0,1;48,2;59,1;75,2;89,1;1107,2;1118,1;1593,2;1604,1;2607,-1; 3:0,13,9,Times,0,12,0,0,0;5,11,8,Times,0,9,0,0,0;4,11,8,Times,2,9,0,0,0; :[font = input; preserveAspect; endGroup; endGroup; ] :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] 1. Notation :[font = subtitle; inactive; preserveAspect; ] 1.1 The Interactive Mathematica Report ;[s] 3:0,0;20,1;31,0;39,-1; 2:2,19,14,Times,1,18,0,0,0;1,19,14,Times,3,18,0,0,0; :[font = smalltext; inactive; preserveAspect; ] This is an interactive report. It has been written for the Mathematica high level language [available for Macintoshs, PC's and Unix machines from Wolfram Research Inc., 100 Trade Center Drive, P.O.Box 6059, Champaign, IL 61821, USA, E-Mail: info@wri.com (in the USA), info-euro@wri.com (in Europe)], which uses a notation very similar to the conven- tional notation in mathematics. Whenever the Mathematica notation seemed obscure for a newcomer in this field, it has been supplemented by the original mathematical notation. The advantage of such an interactive report over a conventional one is that the statements are mathematically complete and consistent. The reader can ask every question concerning notation, definition and results. SHe can have each variable plotted, thus being able to quickly study the mathematical background. In this way, the report is a simulator of the system presented. ;[s] 7:0,0;59,1;71,0;91,1;92,0;398,1;409,0;909,-1; 2:4,13,10,Geneva,0,9,0,0,0;3,13,10,Geneva,2,9,0,0,0; :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 1.1.1 Mathematica's Text Structure :[font = smalltext; inactive; preserveAspect; endGroup; endGroup; ] The Mathematica documents used here are called notebooks. Each line in a notebook is processed by the Mathematica interpreter. Text in this notebook is either active or inactive. The latter is ignored by the interpreter. So, inactive text is comparable to comment in e.g. Fortran programs. Active text is executed by the interpreter. It is comparable to executable text in a Fortran program. 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solution", DE .............. set of differential equations describing compartment system in Section 4 (exact model) f ............... subscript meaning "per gsusp or per gsed", gsed ............ gramm sediment, gsusp ........... gramm suspended matter, i ............... superscript, i can be d or p, L ............... liter = 1000 cubic centimeter, Lliq ............ Liter of liquid, in water column Lliq = Lsyst, in sediment Lliq = e Lsed, Lsed ............ Liter of sediment volume, Lsyst ........... Liter of system volume, l ............... subscript meaning "per Lliq", m ............... meter, mBq ............. 0.001 Bequerel, N ............... number of sampled locations or depths, p ............... superscript meaning "on particulate matter or solid phase", part ............ particles, r or ^........... suffix or subscript meaning normalized, thtr = tht/uto, s ............... superscript meaning "in sediment", in Figures represented as line above letter, sol1 ............ solution of the Step One Approximation of DE, syst ............ system, in water column: water plus suspended particles, in sediment: sediment body plus pore water, t ............... subscript "total", ct = cd + cp, c being any activity, Th .............. index or word meaning "thorium", total act. ...... sum of soluble and particulate or sediment particle bound act. U ............... superscript or word meaning "uranium", yr .............. year, x ............... subscript meaning experimental value, o ............... subscript, abbreviation for t = 0, 1 ............... suffix meaning Simplification Step One, Dc ............. experimental error of c, - ............... dimensionless quantity, {a,b,c} ......... 3-component vector, abcissa in plot unless otherwise noted: time t axis, unit: day, ordinate in plot unless otherwise noted: activity axis, unit: mBq/L ;[s] 21:0,0;37,2;38,0;668,2;669,0;1120,1;1121,3;1122,0;1127,1;1128,0;1130,1;1131,3;1132,0;1491,1;1492,0;1496,1;1497,0;1501,1;1502,0;1925,2;1926,0;2184,-1; 4:10,11,8,Courier,0,9,0,0,0;6,17,11,Courier,32,9,0,0,0;3,14,10,Symbol,0,9,0,0,0;2,17,11,Courier,64,9,0,0,0; :[font = subtitle; inactive; preserveAspect; startGroup; ] 1.3 Variables :[font = smalltext; inactive; preserveAspect; leftNameWrapOffset = 3; rightWrapOffset = 390; fontName = "Courier"; ] conv .............:= lambda/Lambda = 1/86.4 (mBq day), dthrthr[i, k] ... experimental error of thtras at location and time {i, k}, hker, hsus....... N-component vector of water depth at sampling locations, for Young Kershaw's and MOST's samples (m), respectively, Kdp, Kdp ........ rate of transition of thorium from soluble to particulate phase (day-1), Kpd, Kpd ........ rate of transition of thorium from particulate to soluble phase (day-1), Kd .. ........... partitioning coefficient of Th in seawater (-), Kds ............. partitioning coefficient of Th in sediment (-), KdU, KdU ........ partitioning coefficient of U in seawater (-), kdU, kdU ........ partitioning coefficient of U in seawater (Lliq/gsusp), KdsU, KdsU ...... partitioning coefficient of U in sediment (-), kd, kdth ........ partitioning coefficient of Th in seawater (Lliq/gsusp), kdsU, kdsU ...... partitioning coefficient of U in sediment (Lliq/gsed), kds... .......... partitioning coefficient of Th in sediment (Lliq/gsed), Lambda .......... decay constant (particles/day), lambda .......... decay constant (mBq), point ........... point in thtras vs. kdth plot, experimental values, rect ............ point enlarged to show experimental errors of thtras and kdth, sal[i] .......... salinity at set i of locations (%o), Shx, shx ........ particulate matter removal rate constant (1/day), = v/x, t ............... time after last disturbance of settling behavior, Th .............. concentrations of thorium (part/Lsyst), th .. ........... activities of thorium (mBq/Lsyst), thlthp .......... vector composed of N two-component vectors, each component of which has again two experimentally determined components: {{{thd, Dthd}, {thp, Dthp}}, {{thd, Dthd}, {thp, Dthp}}, ... }, thrv ............ {{tht/ut}, {tht/ut}, ...}, tlocs ........... N-component vector of sampling locations, ulup ............ exp. determined vector composed of N two-component vectors, each component of which has again two experimentally determined components: {{{ud, Dud}, {up, Dup}}, {{ud, Dud}, {up, Dup}}, ... }, thtv ............ exp. determined vector composed of N two-component vectors: {{tht, Dtht}, {tht, Dtht}, ...}, U ............... concentrations and uranium (part/Lsyst), u ............... activities of uranium (mBq/Lsyst), u[U] ............ activity (mBq/L) of U (mol/L), uthlist ......... exp. determined vector composed of N two-component vectors: {{ut, tht},{ut, tht}, ...}, utv ............. exp. determined vector composed of N two-component vectors: {{ut, Dut}, {ut, Dut}, ...}, v ............... effective particulate matter settling velocity (m/day), = shx h, x ............... suspended load (gsusp/Lsyst), xs .............. exp. determined vector composed of N two-component vectors of suspended load: {{x, Dx}, {x, Dx}, ...}, e, eps ......... exp. determined volume of liquid in 1 L of sediment (Lliq/Lsyst), 1 - e, onemeps volume of solid in 1 L of sediment (Lsed/Lsyst), r, rho ........ specific weight of sediment (gsed/Lsed). ;[s] 65:0,0;284,2;286,0;387,1;389,0;393,2;395,0;497,1;499,0;504,1;505,0;570,1;571,0;636,2;637,0;701,2;702,0;775,1;776,2;777,0;915,1;916,2;917,0;988,1;989,0;1829,3;1830,0;1842,3;1843,0;1857,3;1858,0;1870,3;1871,0;2224,3;2225,0;2235,3;2236,0;2248,3;2249,0;2259,3;2260,0;2395,3;2396,0;2408,3;2409,0;2847,3;2848,0;2858,3;2859,0;3157,3;3158,0;3166,3;3167,0;3177,3;3178,0;3183,1;3184,0;3282,3;3283,0;3292,1;3293,0;3345,3;3346,0;3352,1;3353,0;3406,-1; 4:32,11,8,Courier,0,9,0,0,0;10,17,11,Courier,32,9,0,0,0;6,17,11,Courier,64,9,0,0,0;17,14,10,Symbol,0,9,0,0,0; :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 1.3.1 Explicit Explanations of Some Frequently Used Variables :[font = smalltext; inactive; preserveAspect; endGroup; endGroup; endGroup; ] error[a] ........ relative error of a, = Da/a, Thst0, Thst0..... total Th concentration in sediment at time t = 0 (particles/Lsyst), Thsd, Thsd0...... Th concentration in sediment water (part/Lsyst), Thsp, Thsp ...... Th concentration on sediment particles (part/Lsyst), Tht0, Tht0 ...... total Th concentration in water column at time t = 0 (particles/Lsyst), thp0f, thp0f .... Th activity on suspended matter (mBq/ gsusp), thsp0f, thsp0f... Th activity on sediment particles (mBq/gsed), thd, thd......... Th activity in water column (mBq/Lsyst), thdas ........... Th activity for t -> €, thd1, thd1s[t,shx]soluble Th activity in sol1, thp, thp ........ Th activity on suspended matter (mBq/Lsyst), thpas ........... Th activity on suspended particulate matter for t-> €, thp1, thp1s[t,shx]particulate matter bound Th activity in sol1, tht[t] .......... total Th activity in Step One Model, thtas ........... total Th activity for t->€, thtr, thtrc ..... ratio of total Th and U activity, thtras[i] ....... ratio of experimentally determined total Th and U activity at location i, thtrv[points] ... vector of thtr-values evaluated at points, thtv[i] ......... vector of total Th activity experimentally determined at suite i of locations, tht1s ........... total Th activity in water column, tht1r ........... total Th activity/total U activity in water column, Ud0, Ud0......... U concentration in water column (part/Lsyst), Up0, Up0 ........ U concentration on suspended matter (part/Lsyst), UMost ........... soluble U concentration (mol/L) in samples taken at locations in Mecklenburger Bucht, uMost ........... soluble U concentration UMOst, expressed in mg/L, Usp0, Usp0 ...... U concentration on sediment particles (part/Lsyst), Ust, Ust ........ total U concentration in sediment (part/Lsyst), Usd0, Usd0....... U concentration in sediment water (part/Lsyst), usd0l, usd0l..... U activity in sediment water (mBq/Lliq), Ut, Ut .......... total U concentration in water column (part/Lsyst), utv[i] .......... vector of total U activity experimentally determined at suite i of locations, ;[s] 50:0,1;41,0;42,1;49,2;50,3;52,1;153,2;155,1;220,2;222,1;291,3;293,1;399,2;400,3;402,1;463,2;465,3;467,1;527,2;528,1;623,0;624,1;675,2;676,1;806,0;807,1;971,0;972,1;1438,2;1439,3;1440,1;1502,2;1503,3;1504,1;1754,0;1755,1;1761,2;1763,3;1764,1;1831,2;1832,3;1833,1;1897,2;1899,3;1900,1;1963,2;1965,3;1967,1;2022,3;2023,1;2208,-1; 4:5,14,10,Symbol,0,9,0,0,0;21,11,8,Courier,0,9,0,0,0;13,17,11,Courier,32,9,0,0,0;11,17,11,Courier,64,9,0,0,0; :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] 2. Definitions :[font = smalltext; inactive; preserveAspect; ] Two type of definitions are used: defineU includes the units, define does not include the units. :[font = input; initialization; preserveAspect; ] *) sf := ScientificForm; (* :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.2 defineU :[font = input; preserveAspect; endGroup; ] Clear[defineU]; defineU := ( Kpd := Kdp/Kd; Kspd := Ksdp/Kds; Kd := kd x; Kds := onemeps rho / eps kds; KdU := kdU x; KdsU := onemeps rho / eps kdsU; lambdaTh := LambdaTh conv ; lambdaU := LambdaU conv; onemeps := (1 - eps (Lsyst/Lliq)) (Lsed/Lsyst); Thp0 := thp0 /lambdaTh ; Thsp0 := thsp0 / lambdaTh ; thp0 := thp0f (mBq/gsusp) x; thsp0 := thsp0f (mBq/gsed) onemeps rho; Thd0 := Thp0/Kd; Thsd0 := Thsp0/Kds; Ud0 := ud0 (mBq/Lliq) 1 (Lliq/Lsyst) / lambdaU; Usd0 := usd0 / lambdaU; usd0 := eps usd0l; Up0 := KdU Ud0; up0 := KdU ud0; Usp0 := KdsU Usd0; usp0 := KdsU usd0; Ut := Ud0 + Up0; ut0 := ud0 + up0; ); definex; defineU; :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.3 define :[font = input; preserveAspect; ] Clear[define]; define := ( Kpd := Kdp/Kd; Kspd := Ksdp/Kds; Kd := kd x; Kds := onemeps rho / eps kds; KdU := kdU x; KdsU := onemeps rho / eps kdsU; lambdaTh := LambdaTh conv ; lambdaU := LambdaU conv; onemeps := (1 - eps); Thp0 := thp0 /lambdaTh ; Thsp0 := thsp0 / lambdaTh ; thp0 := thp0f x; thsp0 := thsp0f onemeps rho; Thd0 := Thp0/Kd; Thsd0 := Thsp0/Kds; Ud0 := ud0 / lambdaU; Usd0 := usd0 / lambdaU; usd0 := eps usd0l; Up0 := KdU Ud0; up0 := KdU ud0; Usp0 := KdsU Usd0; usp0 := KdsU usd0; Ut0 := Ud0 + Up0; ut0 := ud0 + up0; ); definex; define; :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.4 Step One Model :[font = smalltext; inactive; preserveAspect; ] These definitions will be introduced in Section 5 :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.4.1 Thorium Activities as a Function of Time and Removal Rate :[font = input; inactive; preserveAspect; height = 32; ] datax; definex; Clear[sol1]; sol1 = {{thda[t] -> -((lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*Shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* ud0)/(2*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (lambdaTh*ud0)/(conv*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*Shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* ud0)/(2*conv*Kdp*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Kpd* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Shx* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 )/(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/2 + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Kpd* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 )/(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Shx* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 )/(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 )/(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp), thpa[t] -> -((KdU*lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*ud0)/ (conv*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (E^ (((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))*t)/2)*((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0)/ (conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)))}}; :[font = input; inactive; preserveAspect; height = 32; ] Clear[thd1]; thd1[t_] := -((lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*Shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)*ud0)/ (2*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (lambdaTh*ud0)/(conv*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (KdU*Kpd*lambdaTh*Shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (KdU*lambdaTh*Shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)*ud0)/ (2*conv*Kdp*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Kpd* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Shx* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/2 + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Kpd* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)*Shx* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp) - (E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))))/(2*Kdp) :[font = input; inactive; preserveAspect; height = 27; ] Clear[thd1s]; thd1s[{t_, shx_}] := -((lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (KdU*lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*lambdaTh*shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)*ud0)/ (2*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))) + (lambdaTh*ud0)/(conv*(Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))) + (Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (Kdp*KdU*lambdaTh*ud0)/ (2*conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (KdU*Kpd^2*lambdaTh*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (KdU*lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (KdU*Kpd*lambdaTh*shx*ud0)/ (conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (KdU*lambdaTh*shx^2*ud0)/ (2*conv*Kdp*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (KdU*lambdaTh*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)*ud0)/ (2*conv*Kdp*(Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))) + (E^(((-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2) - (E^(((-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0)))/ (64*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)*Kpd* (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) + (E^(((-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)*shx* (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0)))/ (64*conv^2*Kdp*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) - (E^(((-Kdp - Kpd - 2*LambdaTh - shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)) - (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))))/2 + (E^(((-Kdp - Kpd - 2*LambdaTh - shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)*Kpd* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)) - (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))))/(2*Kdp) + (E^(((-Kdp - Kpd - 2*LambdaTh - shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)*shx* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)) - (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))))/(2*Kdp) - (E^(((-Kdp - Kpd - 2*LambdaTh - shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* ((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) /(conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)) - (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))))/(2*Kdp) :[font = input; inactive; preserveAspect; height = 32; ] Clear[thp1]; thp1[t_] := -((KdU*lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))) + (KdU*lambdaTh*ud0)/ (conv*(Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (KdU*lambdaTh*Shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) ) + (E^(((-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0)))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2)) + E^(((-Kdp - Kpd - 2*LambdaTh - Shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^ (1/2))*t)/2)*((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0)/ (conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)) - (-4*conv*(Kdp - Kpd - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*Shx*thp0 + conv*LambdaTh*Shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + Shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*Shx*thd0 - conv*LambdaTh*Shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*Shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*Shx + LambdaTh*Shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*Shx + 2*Kpd*Shx + Shx^2)^(1/2))) :[font = input; inactive; preserveAspect; height = 32; ] Clear[thp1s]; thp1s[{t_, shx_}] := -((KdU*lambdaTh*ud0)/ (conv*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))) + (KdU*lambdaTh*ud0)/ (conv*(Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))) - (2*Kdp*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) - (Kdp*KdU*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (KdU*Kpd*lambdaTh*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (KdU*lambdaTh*shx*ud0)/ (conv*(Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) ) + (E^(((-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))*t)/2)* (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0)))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2)) + E^(((-Kdp - Kpd - 2*LambdaTh - shx - (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^ (1/2))*t)/2)*((conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0)/ (conv*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)) - (-4*conv*(Kdp - Kpd - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (conv*Kdp*LambdaTh*thp0 + conv*Kpd*LambdaTh*thp0 + conv*LambdaTh^2*thp0 + conv*Kdp*shx*thp0 + conv*LambdaTh*shx*thp0 - Kdp*lambdaTh*ud0 - Kdp*KdU*lambdaTh*ud0 - KdU*lambdaTh*LambdaTh*ud0 ) + 8*conv*Kdp*(-Kdp - Kpd - 2*LambdaTh - shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (Kdp + Kpd + 2*LambdaTh + shx + (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))* (-(conv*Kdp*LambdaTh*thd0) - conv*Kpd*LambdaTh*thd0 - conv*LambdaTh^2*thd0 - conv*Kdp*shx*thd0 - conv*LambdaTh*shx*thd0 + Kpd*lambdaTh*ud0 + KdU*Kpd*lambdaTh*ud0 + lambdaTh*LambdaTh*ud0 + lambdaTh*shx*ud0))/ (32*conv^2*(Kdp*LambdaTh + Kpd*LambdaTh + LambdaTh^2 + Kdp*shx + LambdaTh*shx)^2* (Kdp^2 + 2*Kdp*Kpd + Kpd^2 - 2*Kdp*shx + 2*Kpd*shx + shx^2)^(1/2))) :[font = input; inactive; preserveAspect; endGroup; ] Clear[tht1s, tht1r]; tht1s[{t, shx}] := thd1s[{t, shx}] + thp1s[{t, shx}] ; tht1r[{t, shx}] := tht1s[{t, shx}] / (ud0 + up0); :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.4.2 Time Asymptotic Thorium Activities :[font = input; preserveAspect; ] Clear[thdas, thpas, thtas]; thdas := (LambdaTh*ud0)/(Kdp + LambdaTh) - (Kpd*(-(Kdp*LambdaTh*ud0) - LambdaTh*(Kdp + LambdaTh)*up0))/ ((Kdp + LambdaTh)*(-(Kdp*Kpd) + (Kdp + LambdaTh)*(Kpd + LambdaTh + Shx))); thpas := -((-(Kdp*LambdaTh*ud0) - LambdaTh*(Kdp + LambdaTh)*up0)/ (-(Kdp*Kpd) + (Kdp + LambdaTh)*(Kpd + LambdaTh + Shx))); thtas := (LambdaTh ((Kdp + Kpd + LambdaTh) (ud0 + up0) + Shx ud0)/ ((Kdp + Kpd + LambdaTh) (LambdaTh + Shx) - Shx Kpd)) :[font = input; preserveAspect; endGroup; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.5 Two-Compartment Model :[font = smalltext; inactive; preserveAspect; ] These definitions will be introduced in Section 6 :[font = input; preserveAspect; ] Clear[tht, thtr, thtrc]; tht[t_] := ( (LambdaTh*ut0)/(LambdaTh + Shx/(1 + 1/Kd)) + (E^((-LambdaTh - Shx/(1 + 1/Kd))*t)*(LambdaTh*tht0 + Shx/(1 + 1/Kd)*tht0 - LambdaTh*ut0))/(LambdaTh + Shx/(1 + 1/Kd)) ); thtr[t_] := (tht0 / ut0 Exp[-(LambdaTh + Shx/(1 + 1/Kd))t] + LambdaTh / (LambdaTh + Shx/(1 + 1/Kd)) (1 - Exp[-(LambdaTh + Shx/(1 + 1/Kd))t] ) ); thtrc[{t_, shx_}] := ( ((1 - E^(-((LambdaTh + shx/(1 + 1/Kd))*t)))*LambdaTh)/(LambdaTh + shx/(1 + 1/Kd)) + tht0/(E^((LambdaTh + shx/(1 + 1/Kd))*t)*ut0) ); :[font = input; noFill; preserveAspect; ] Clear[thtrv, error]; thtrv[points_] := Map[thtrc, points]; error[a_] := Table[(a[[n+1]] - a[[n]])/a[[n]], {n, Count[a, _._] - 1}] :[font = input; preserveAspect; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.5.1 MOST Data Processing :[font = input; preserveAspect; ] Clear[kdth, thtv, utv]; thtv[i_] := ctv[clcp[thlthp[i], xs[i]]]; utv[i_] := ctv[clcp[ulup[i], xs[i]]]; kdth[i_, j_] := kdth[i, j] = thlthp[i][[j,2,1]]/thlthp[i][[j,1,1]] :[font = input; preserveAspect; ] Clear[add, divide]; add[{x1_, x2_}] := x1 + x2; divide[{x1_, x2_}] := x2/x1; xmax[i_] := Count[xs[i], _._]; Clear[clcp, ctv]; (*gives vector of soluble and susp particle concs (mBq/L) at the depth for which vec is the ulup or thlthp vector and csusp is the xs vector*) clcp[vec_, csusp_] := ( u = Table[{vec[[ii, 1]], csusp[[ii, 1]] vec[[ii, 2]]}, {ii, Count[csusp, _._]}]; For[j = 1, j <= Count[csusp, _._], j++, dup = u[[j, 2, 1]] (csusp[[j, 2]] / csusp[[j,1]] + vec[[j, 2, 2]] / vec[[j, 2, 1]] ); u[[j, 2, 2]] = dup ]; u ); ctv[clcpv_] := Map[add, clcpv] (*mBq/L*); Clear[uthlist, thtras, dthrthr]; uthlist[i_] := Table[{utv[i][[jj, 1]], thtv[i][[jj, 1]]}, {jj, xmax[i]}] thtras[k_] := thtras[k] = Map[divide, uthlist[k]] (*reduced Th activity = Th activity / U activity*); dthrthr[i_, k_] := thtv[i][[k, 2]]/thtv[i][[k, 1]] + utv[i][[k, 2]]/utv[i][[k, 1]] (*relative error of thtras*) :[font = input; preserveAspect; endGroup; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.5.2 Most Data Plots :[font = smalltext; inactive; preserveAspect; ] Plots of the thorium/uranium activity ratio vs. kd use the following graphics definitions :[font = input; preserveAspect; ] point[i_, j_] := Point[{kdth[i,j] , thtras[i][[j]]}]; rect[i_, j_] := Rectangle[0.7{kdth[i,j] , thtras[i][[j]]}, 1.3{kdth[i,j] , thtras[i][[j]]}] :[font = input; preserveAspect; endGroup; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.6 definex :[font = text; inactive; preserveAspect; ] The procedure definex cancels all definitions in define and defineU. :[font = input; preserveAspect; endGroup; ] Clear[definex]; definex := (Clear[onemeps, Kpd, Kd, Kds, KdU, KdsU, Kspd, lambdaTh , lambdaU, Tht0, Thst0, Thp0 , Thsp0, thp0, thsp0 , Thd0, Thsd0, Ud0, Usd0, usd0, Up0, up0, Usp0, usp0, Ut0 , ut0, Ust0, ust0 ]; ); definex; :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.7 pdata :[font = input; preserveAspect; ] dataU; defineU; Clear[pdata]; pdata := ( Print[" LambdaU LambdaTh Shx Kdp Kpd Ksdp Kspd "]; Print["_____________________________________________________________________ "]; Print[ LambdaU, ", ", sf[N[LambdaTh]], ", ", sf[Shx], ", ", sf[Kdp], ", ", sf[Kpd], ", ", sf[Ksdp], ", ", sf[Kspd]]; Print[" "]; Print["=================================================================== "]; Print[" "]; Print[" x rho kd kds kdU kdsU"]; Print["_________________________________________________________________"]; Print[x, ", ", rho, ", ", kd, ", ", kds, ", ", kdU, ", ", kdsU]); :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 2.8 Conversion Between Concentration and Activity :[font = input; preserveAspect; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.1.1 Definition :[font = smalltext; inactive; preserveAspect; ] Let C be the number of particles l the radioactive decay constant, measured in units 1/day and c the activity measured in mBq. The conversion factor conv is defined as ;[s] 3:0,0;36,1;37,0;172,-1; 2:2,13,10,Geneva,0,9,0,0,0;1,14,10,Symbol,0,9,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 96; pictureHeight = 22; ] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.229167 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth 0.000000 setlinewidth 0.000000 0.229167 moveto 0.000000 0.229167 lineto stroke 0.010417 setlinewidth 0.005208 0.078125 moveto 0.171875 0.078125 lineto stroke /Symbol findfont 12 scalefont setfont [(l)] 0.000000 0.052083 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(C)] 0.072917 0.062500 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(c)] 0.052083 -0.031250 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.166667 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.197917 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(=)] 0.229167 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.302083 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(c)] 0.333333 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(o)] 0.385417 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(n)] 0.447917 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(v)] 0.510417 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.572917 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.604167 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.635417 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.666667 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.697917 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.729167 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.760417 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.791667 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.822917 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.854167 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(\()] 0.885417 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [( )] 0.927083 0.020833 -1 -1 Mshowa /Times findfont 12 scalefont setfont [(\))] 0.958333 0.020833 -1 -1 Mshowa %% End of Graphics MathPictureEnd %% End of picture :[font = input; preserveAspect; endGroup; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 2.1.2 Application: Uranium Concentration and Activity :[font = input; preserveAspect; ] data; define; :[font = input; preserveAspect; startGroup; ] Solve[lambdaU U mol/L 6 10^23 part/mol == u mBq/L, U] :[font = output; output; inactive; preserveAspect; endGroup; ] {{U -> (3.412262310710576*10^-10*mBq*u)/part}} ;[o] -10 3.41226 10 mBq u {{U -> -------------------}} part :[font = input; preserveAspect; startGroup; ] dataU; defineU; lambdaU :[font = output; output; inactive; preserveAspect; endGroup; ] (4.88434509104195*10^-15*mBq)/part ;[o] -15 4.88435 10 mBq ----------------- part :[font = smalltext; inactive; preserveAspect; ] From the uranium concentration (U, mol/L) the uranium activity (u, mBq/L) can be calculated as :[font = input; preserveAspect; ] Clear[u]; u[U_] := lambdaU U mol/L 6 10^23 part/mol :[font = input; preserveAspect; startGroup; ] u[U] :[font = output; output; inactive; preserveAspect; endGroup; ] (2.930607054625169*10^9*mBq*U)/L ;[o] 9 2.93061 10 mBq U ----------------- L :[font = input; preserveAspect; endGroup; endGroup; endGroup; ] :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] 3. Data :[font = smalltext; inactive; preserveAspect; rightWrapOffset = 388; ] As with the definitions, there are also two data types used: 3.1 dataU include the units, 3.2 data do not. datax removes the numerical values from the data, leaving only the algebraic symbols. The values for the ad- and desorption rate constants are good first guesses based on laboratory experiments. 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 pop grestore 1.00000 setgray 0.811813 0.179945 moveto 0.995879 0.179945 lineto 0.995879 0.254121 lineto 0.811813 0.254121 lineto closepath fill 0.633242 0.199176 moveto 0.820055 0.199176 lineto 0.820055 0.273352 lineto 0.633242 0.273352 lineto closepath fill /Helvetica findfont 12 scalefont setfont [(Nov. 93 )] 0.634615 0.239011 -1 -1 Mshowa /Helvetica findfont 12 scalefont setfont [(Kd = 50 L/g)] 0.634615 0.200550 -1 -1 Mshowa /Helvetica findfont 12 scalefont setfont [(Feb. 94 )] 0.189560 0.612637 -1 -1 Mshowa /Helvetica findfont 12 scalefont setfont [(Kd = 1600 L/g)] 0.189560 0.574176 -1 -1 Mshowa 0.361264 0.457418 moveto 0.570055 0.457418 lineto 0.570055 0.534341 lineto 0.361264 0.534341 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MathPictureEnd %% End of picture :[font = smalltext; inactive; preserveAspect; leftWrapOffset = 55; rightWrapOffset = 413; ] Fig. 1: Graphical representation of the soluble and adsorbed thorium activities, determined in November 1993, February and April 1994. :[font = smalltext; inactive; preserveAspect; ] The uranium partitioning coefficient (kdU = 1 L/g) has been taken from Kershaw and Young [Kershaw, P., A. Young, Scavenging of 234Th in the eastern irish sea, J. Environ. Radioactivity 6, 1 - 23, 1988]. It will be shown that kdU has only little influence on the results. It is arbitrarily assumed that the partitioning coefficients in the sediment are the same as in the water column. It will be shown that the sediment has little influence on the thorium activities in the water column. ;[s] 5:0,0;40,1;42,0;227,1;228,0;490,-1; 2:3,13,10,Geneva,0,9,0,0,0;2,19,13,Geneva,64,9,0,0,0; :[font = input; preserveAspect; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 3.1 Reference Case :[font = input; preserveAspect; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 3.1.1 data :[font = input; preserveAspect; ] Clear[dataU]; dataU := (conv = 1/86.4 (mBq/(part/day)) ; LambdaTh = Log[2.]/24 (day^-1) ; LambdaU = Log[2.]/(4.5 10^9 365) (day^-1); x = 0.003 (gsusp / (Lsyst)) ; eps = 0.3 (Lliq/(Lsyst)) ; rho = 1200 (gsed/(Lsed)) ; Shx = 500 (gsusp/(m^2 yr)) (yr/(365 day)) / (21 m 10^3 (Lsyst/m^3) x ); Kdp = 24 day^-1; Ksdp = 24 day^-1; kd = 1000 Lliq 1 (Lsyst/Lliq) / gsusp; kds = 1000 Lliq / gsed; kdU = 1. Lliq 1 (Lsyst/Lliq) / gsusp; kdsU = 1. Lliq / gsed; ); datax; dataU; :[font = input; preserveAspect; ] Clear[data]; data := (conv = 1/86.4 (*mBq/(part/day)*) ; LambdaTh = Log[2.]/24 (*day^-1*) ; LambdaU = Log[2.]/(4.5 10^9 365) (*day^-1*); x = 0.003 (*gsusp / Lsyst*) ; eps = 0.3 (*Lliq/Lsyst*) ; rho = 1200 (*gsed/Lsed*) ; Shx = 500 (*gsusp/(m^2 yr) yr*) 1/(365 (*day*)) / (21 (*m *)10^3 (*Lsyst/m^3*) x ); Kdp = 24 (*day^-1*); Ksdp = 24 (*day^-1*); kd = 1000 (*Lliq 1 (Lsyst/Lliq) / gsusp*); kds = 1000 (* Lliq / gsed*); kdU = 1 (*Lliq 1 Lsyst/Lliq) / gsusp*); kdsU = 1 (*Lliq / gsed*); ); datax; data; :[font = input; preserveAspect; endGroup; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 3.1.2 datax :[font = input; preserveAspect; ] Clear[datax]; datax := (conv = .; LambdaTh = . ; LambdaU = .; x = . ; eps = .; rho = . ; Shx = .; thp0f =.; thsp0f =.; Kdp = .; Ksdp =.; kd = .; kds =.; kdU =.; kdsU =.; ); datax :[font = input; preserveAspect; endGroup; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 3.2 Kershaw and Young Data :[font = smalltext; inactive; preserveAspect; fontName = "Courier"; ] ;[s] 1:0,1;3,-1; 2:0,11,8,Courier,0,9,0,0,0;1,12,9,Courier,0,10,0,0,0; :[font = input; preserveAspect; ] datak := ( thlthp[1] = {{{0.8, 0.1}, {1318, 41}}, {{0.6, 0.1}, {1217, 19}}, {{1.1, 0.1}, {1737, 24}}, {{4.1, 0.2}, {6438, 86}}, {{5.4, 0.2}, {24274, 329}}}; thlthp[2] = {{{1.1, 0.1}, {1228, 32}}, {{0.6, 0.1}, {1556, 44}}, {{1.5, 0.1}, {1915, 29}}, {{4.0, 0.2}, {5773, 73}}, {{6.8, 0.3}, {16575, 251}}}; ulup[1] = {{{40.0, 1.3}, {48.9, 2.1}}, {{39.8, 1.2}, {64.1, 2.5}}, {{38.1, 0.5}, {29.8, 1.6}}, {{39.2, 1.4}, {29.3, 2.8}}, {{40.9, 1.5}, {115, 10.0}}}; ulup[2] = {{{38.6, 1.4}, {38.5, 1.9}}, {{42.0, 1.6}, {128, 3.4}}, {{39.8, 1.5}, {23.3, 1.8}}, {{40.9, 1.9}, {33.1, 2.9}}, {{37.7, 1.3}, {85.2, 7.2}}}; xs[1] = 10^-3 {{6.8, 0.21}, {6.78, 0.20}, {3.58, 0.20}, {0.99, 0.11}, {0.64, 0.15}}; (*g/L*) xs[2] = 10^-3 {{6.51, 0.25}, {6.90, 0.25}, {2.85, 0.14}, {1.78, 0.2}, {0.36, 0.12}}; hker = {21, 21, 26, 36, 61}; (*m*) ); datak; :[font = input; preserveAspect; ] datakx := (Clear[thlthp, ulup, xs]; hker =.); datakx; :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 3.3 MOST Data :[font = input; preserveAspect; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 3.3.1 Conversion of MOST Uranium Concentrations into Activities :[font = input; inactive; preserveAspect; ] Needs["Miscellaneous`ChemicalElements`"] :[font = input; inactive; preserveAspect; startGroup; ] AtomicWeight[Uranium] :[font = output; output; inactive; preserveAspect; endGroup; ] 238.0289 ;[o] 238.0289 :[font = smalltext; inactive; preserveAspect; ] The following uranium concentrations (uMost, mg/L) were measured with ICP/MS coupling. ;[s] 3:0,0;45,1;46,0;88,-1; 2:2,13,10,Geneva,0,9,0,0,0;1,14,10,Symbol,0,9,0,0,0; :[font = input; preserveAspect; ] uMost = {2.47, 2.45, 1.78, 1.68, 1.77, 1.68, 1.61, 1.67, 1.68, 1.62, 1.37, 2.13, 2.17, 2.08, 1.72, 1.66, 1.12, 1.50, 1.58, 2.16, 2.23, 2.19}; :[font = smalltext; inactive; preserveAspect; ] Conversion of the units mg into mol is achieved with the function UMost ;[s] 3:0,0;24,1;25,0;72,-1; 2:2,13,10,Geneva,0,9,0,0,0;1,14,10,Symbol,0,9,0,0,0; :[font = input; preserveAspect; startGroup; ] UMost = 10^-6/238 uMost :[font = output; output; inactive; preserveAspect; endGroup; ] {1.03781512605042*10^-8, 1.029411764705883*10^-8, 7.478991596638654*10^-9, 7.058823529411765*10^-9, 7.436974789915967*10^-9, 7.058823529411765*10^-9, 6.764705882352941*10^-9, 7.016806722689076*10^-9, 7.058823529411765*10^-9, 6.806722689075631*10^-9, 5.756302521008405*10^-9, 8.94957983193277*10^-9, 9.11764705882353*10^-9, 8.73949579831933*10^-9, 7.226890756302521*10^-9, 6.974789915966387*10^-9, 4.705882352941177*10^-9, 6.302521008403362*10^-9, 6.638655462184876*10^-9, 9.07563025210084*10^-9, 9.36974789915966*10^-9, 9.20168067226891*10^-9} ;[o] -8 -8 -9 -9 -9 -9 {1.03782 10 , 1.02941 10 , 7.47899 10 , 7.05882 10 , 7.43697 10 , 7.05882 10 , -9 -9 -9 -9 -9 -9 6.76471 10 , 7.01681 10 , 7.05882 10 , 6.80672 10 , 5.7563 10 , 8.94958 10 , -9 -9 -9 -9 -9 -9 9.11765 10 , 8.7395 10 , 7.22689 10 , 6.97479 10 , 4.70588 10 , 6.30252 10 , -9 -9 -9 -9 6.63866 10 , 9.07563 10 , 9.36975 10 , 9.20168 10 } :[font = smalltext; inactive; preserveAspect; ] The function u converts the units mol into mBq :[font = input; preserveAspect; startGroup; ] Map[u, UMost] :[font = output; output; inactive; preserveAspect; endGroup; ] {(30.41428329800072*mBq)/L, (30.16801379761205*mBq)/L, (21.91798553459161*mBq)/L, (20.68663803264826*mBq)/L, (21.79485078439727*mBq)/L, (20.68663803264826*mBq)/L, (19.82469478128791*mBq)/L, (20.56350328245392*mBq)/L, (20.68663803264826*mBq)/L, (19.94782953148225*mBq)/L, (16.86946077662388*mBq)/L, (26.22770179139333*mBq)/L, (26.72024079217066*mBq)/L, (25.61202804042165*mBq)/L, (21.1791770334256*mBq)/L, (20.44036853225959*mBq)/L, (13.79109202176551*mBq)/L, (18.47021252915023*mBq)/L, (19.45529053070491*mBq)/L, (26.59710604197633*mBq)/L, (27.45904929333668*mBq)/L, (26.96651029255933*mBq)/L} ;[o] 30.4143 mBq 30.168 mBq 21.918 mBq 20.6866 mBq 21.7949 mBq 20.6866 mBq 19.8247 mBq {-----------, ----------, ----------, -----------, -----------, -----------, -----------, L L L L L L L 20.5635 mBq 20.6866 mBq 19.9478 mBq 16.8695 mBq 26.2277 mBq 26.7202 mBq 25.612 mBq -----------, -----------, -----------, -----------, -----------, -----------, ----------, L L L L L L L 21.1792 mBq 20.4404 mBq 13.7911 mBq 18.4702 mBq 19.4553 mBq 26.5971 mBq 27.459 mBq -----------, -----------, -----------, -----------, -----------, -----------, ----------, L L L L L L L 26.9665 mBq -----------} L :[font = section; inactive; preserveAspect; startGroup; ] 3.4.1 Summary of Conversions: :[font = input; preserveAspect; startGroup; ] MatrixForm[Table[{uMost[[i]] ug/L, Map[u, UMost][[i]]}, {i, Count[UMost, _._]}]] :[font = output; output; inactive; preserveAspect; endGroup; ] MatrixForm[{{(2.47*ug)/L, (30.41428329800072*mBq)/L}, {(2.45*ug)/L, (30.16801379761205*mBq)/L}, {(1.78*ug)/L, (21.91798553459161*mBq)/L}, {(1.68*ug)/L, (20.68663803264826*mBq)/L}, {(1.77*ug)/L, (21.79485078439727*mBq)/L}, {(1.68*ug)/L, (20.68663803264826*mBq)/L}, {(1.61*ug)/L, (19.82469478128791*mBq)/L}, {(1.67*ug)/L, (20.56350328245392*mBq)/L}, {(1.68*ug)/L, (20.68663803264826*mBq)/L}, {(1.62*ug)/L, (19.94782953148225*mBq)/L}, {(1.37*ug)/L, (16.86946077662388*mBq)/L}, {(2.13*ug)/L, (26.22770179139333*mBq)/L}, {(2.17*ug)/L, (26.72024079217066*mBq)/L}, {(2.08*ug)/L, (25.61202804042165*mBq)/L}, {(1.72*ug)/L, (21.1791770334256*mBq)/L}, {(1.66*ug)/L, (20.44036853225959*mBq)/L}, {(1.12*ug)/L, (13.79109202176551*mBq)/L}, {(1.5*ug)/L, (18.47021252915023*mBq)/L}, {(1.58*ug)/L, (19.45529053070491*mBq)/L}, {(2.16*ug)/L, (26.59710604197633*mBq)/L}, {(2.23*ug)/L, (27.45904929333668*mBq)/L}, {(2.19*ug)/L, (26.96651029255933*mBq)/L}}] ;[o] 2.47 ug 30.4143 mBq ------- ----------- L L 2.45 ug 30.168 mBq ------- ---------- L L 1.78 ug 21.918 mBq ------- ---------- L L 1.68 ug 20.6866 mBq ------- ----------- L L 1.77 ug 21.7949 mBq ------- ----------- L L 1.68 ug 20.6866 mBq ------- ----------- L L 1.61 ug 19.8247 mBq ------- ----------- L L 1.67 ug 20.5635 mBq ------- ----------- L L 1.68 ug 20.6866 mBq ------- ----------- L L 1.62 ug 19.9478 mBq ------- ----------- L L 1.37 ug 16.8695 mBq ------- ----------- L L 2.13 ug 26.2277 mBq ------- ----------- L L 2.17 ug 26.7202 mBq ------- ----------- L L 2.08 ug 25.612 mBq ------- ---------- L L 1.72 ug 21.1792 mBq ------- ----------- L L 1.66 ug 20.4404 mBq ------- ----------- L L 1.12 ug 13.7911 mBq ------- ----------- L L 1.5 ug 18.4702 mBq ------ ----------- L L 1.58 ug 19.4553 mBq ------- ----------- L L 2.16 ug 26.5971 mBq ------- ----------- L L 2.23 ug 27.459 mBq ------- ---------- L L 2.19 ug 26.9665 mBq ------- ----------- L L :[font = input; preserveAspect; endGroup; endGroup; ] :[font = subsubtitle; inactive; preserveAspect; startGroup; ] 3.4 DataVectors :[font = section; inactive; preserveAspect; startGroup; ] 3.5.1 Definition :[font = smalltext; inactive; preserveAspect; ] Let the dates and locations of the samples be given by the vector {{date, location}, {date, location}, ...}. Then the following samples exist :[font = smalltext; inactive; preserveAspect; ] data[1] = {{11/93, 12o}, {11/93, 12c}, {2/94, 12a}, {2/94, 23o}, {2/94, 23b}, {2/94, 12o}, {4/94, 12o}, {4/94, 12a}, {4/94, 23b}, {4/94, 12c}}. data[2] = 12c {{27/6/94, 16 m}, {27/6/94, 22 m}, {28/6/94, 20 m}, {28/6/94, 24 m}, {29/6/94, 20 m}, {29/6/94, 24 m}, {30/6/94, 20 m}, {30/6/94, 24 m}, {1/7/94, 20 m}} data[3] = {{11/94, 12o}, {11/94, 23a}, {11/94, 23c}} :[font = input; preserveAspect; ] Clear[tlocs]; tlocs[1] = {{XI/93, 12o}, {XI/93, 12c}, {II/94, 12a}, {II/94, 23o}, {II/94, 23b}, {II/94, 12o}, {IV/94, 12o}, {IV/94, 12a}, (*{IV/94, 23b},*) {IV/94, 12c}}; tlocs[2] = {{27 VI, 94, 16 m}, {27 VI, 94, 22 m}, {28 VI, 94, 20 m}, {28 VI, 94, 24 m}, {29 VI, 94, 20 m}, {29 VI, 94, 24 m}, {30 VI, 94, 20 m}, {30 VI, 94, 24 m}, {first VII, 94, 20 m}}; tlocs[3] = {(*{XI/94, 12o}, *){XI/94, 23a}, {XI/94, 23c}}; :[font = input; preserveAspect; startGroup; ] MatrixForm[tlocs[1]] :[font = output; output; inactive; preserveAspect; endGroup; ] MatrixForm[{{XI/93, 12*o}, {XI/93, 12*c}, {II/94, 12*a}, {II/94, 23*o}, {II/94, 23*b}, {II/94, 12*o}, {IV/94, 12*o}, {IV/94, 12*a}, {IV/94, 12*c}}] ;[o] XI -- 93 12 o XI -- 93 12 c II -- 94 12 a II -- 94 23 o II -- 94 23 b II -- 94 12 o IV -- 94 12 o IV -- 94 12 a IV -- 94 12 c :[font = input; preserveAspect; startGroup; ] MatrixForm[tlocs[2]] :[font = output; output; inactive; preserveAspect; endGroup; ] MatrixForm[{{27*VI, 94, 16*m}, {27*VI, 94, 22*m}, {28*VI, 94, 20*m}, {28*VI, 94, 24*m}, {29*VI, 94, 20*m}, {29*VI, 94, 24*m}, {30*VI, 94, 20*m}, {30*VI, 94, 24*m}, {first*VII, 94, 20*m}}] ;[o] 27 VI 94 16 m 27 VI 94 22 m 28 VI 94 20 m 28 VI 94 24 m 29 VI 94 20 m 29 VI 94 24 m 30 VI 94 20 m 30 VI 94 24 m first VII 94 20 m :[font = input; preserveAspect; startGroup; ] MatrixForm[tlocs[3]] :[font = output; output; inactive; preserveAspect; endGroup; ] MatrixForm[{{XI/94, 23*a}, {XI/94, 23*c}}] ;[o] XI -- 94 23 a XI -- 94 23 c :[font = smalltext; inactive; preserveAspect; ] Uranium has not been determined in {4/94, 23b}, therefore the data {4/94, 23b} will not be included in the evaluations. ;[s] 2:0,0;118,1;120,-1; 2:1,13,10,Geneva,0,9,0,0,0;1,13,10,Geneva,0,10,0,0,0; :[font = input; preserveAspect; ] Clear[datasuse, thlthp, ulup, sal, xs, hsus]; datasuse := ( thlthp[1] = { {{21.7, 4.0}, {922, 480}}, {{17.04, 1}, {374, 220}}, {{4.75, 3.0}, {3952, 2200}}, {{1.44, 0.9}, {6125, 3420}}, {{1.38, 1}, {6288, 3500}}, {{1.84, 1.32}, {2082, 1160}}, {{5.46, 3.0}, {1737, 1000}}, {{5.01, 2.8}, {3206, 1790}}, (*{{7.48, 2.50}, {5478, 1830}},*) {{3.81, 2.0}, {2333, 1300}} }; thlthp[2] = { {{3.32, 0.3 3.32}, {2912, 0.3 2912}}, {{4.85, 0.3 4.85}, {2367, 0.3 2367}}, {{4.67, 0.3 4.67}, {3541, 0.3 3541}}, {{3.27, 0.3 3.27}, {3585, 0.3 3585}}, {{4.57, 0.3 4.57}, {4966, 0.3 4966}}, {{4.16, 0.3 4.16}, {2326, 0.3 2326}}, {{4.05, 0.3 4.05}, {8388, 0.3 8388}},{{4.16, 0.3 4.16}, {2323, 0.3 2323}}, {{6.72, 0.3 6.72}, {2044, 0.3 2044}} }; thlthp[3] = { (*{{4.97, 0.3 4.97}, {4607, 0.3 4607}}, *) {{3.52, 0.3 3.52}, {4969, 0.3 4969}}, {{4.59, 0.3 4.59}, {4367, 0.3 4367}} }; ulup[1] = { {{16.9, 1.7}, {0.05, 0}}, {{20.0, 1.2}, {0.05, 0}}, {{27.78, 1.7}, {0.05, 0}}, {{27.0, 1.7}, {0.05, 0}}, {{23.63, 1.5}, {0.05, 0}}, {{25.6, 1.7}, {0.05, 0}}, {{20.7, 1.7}, {0.05, 0}}, {{21.3, 1.7}, {0.05, 0}}, (*{{0, 0}, {0, 0}},*) {{20.6, 1.7}, {0.05, 0}} }; ulup[2] = 1.05 { {{16.61, 0.3 16.61}, {0.001, 0.0001}}, {{23.04, 0.3 23.04}, {0.001, 0.0001}}, {{19.68, 0.3 19.68}, {0.001, 0.0001}}, {{23.82, 0.3 23.82}, {0.001, 0.0001}}, {{19.75, 0.3 19.75}, {0.001, 0.0001}}, {{24.04, 0.3 24.04}, {0.001, 0.0001}}, {{20.85, 0.3 20.85}, {0.001, 0.0001}}, {{24.53, 0.3 24.53}, {0.001, 0.0001}}, {{21.32, 0.3 21.32}, {0.001, 0.0001}} }; ulup[3] = { {{20.7, 2}, {0.001, 0.0001}}, {{19.8, 2}, {0.001, 0.0001}} }; sal[1] = {20.61, 20.59, 20.88, 23.90, 23.77, 21.30, 20.00, 19.80, (*17.44, *) 20.40}; sal[2] = {16.61, 23.04, 19.68, 23.82, 19.75, 24.04, 20.85, 24.53, 21.32}; sal[3] = {18.06, 17.60}; (*g/L*) xs[1] = (10^-3 {{3.35, 0.3}, {8.94, 0.20}, {1.01, 0.20}, {0.92, 0.2}, {0.78, 0.15}, {1.59, 0.2}, {3.05, 0.2}, {1.57, 0.2}, (*{1.10, 0.2},*) {4.00, 0.2}} ); xs[2] = (10^-3 {{1.10, 0.1}, {1.34, 0.1}, {0.82, 0.08}, {0.8, 0.08}, {0.82, 0.08}, {1.52, 0.15}, {0.49, 0.05}, {1.51, 0.15}, {2.2, 0.22}} ); xs[3] = (10^-3 {{0.54, 0.05}, {0.2, 0.02}} );(*g/L*) hsus[1] = {25, 25, 24, 23, 26, 25, 25, 25, (*20, *) 25}; hsus[2] = {16, 22, 20, 24, 20, 24, 20, 24, 20}; hsus[3] = {22, 14}(*m*) ); datasuse; :[font = input; preserveAspect; endGroup; endGroup; endGroup; endGroup; ] :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] 4. Compartment Model :[font = smalltext; inactive; preserveAspect; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 4.1 Basic Definitions and Assumptions :[font = smalltext; inactive; preserveAspect; rightWrapOffset = 387; endGroup; ] The dynamical model describing the evolution of the thorium activity is based on the following definitions and assumptions: (1) The water column extends by definition in the direction of the movement of the suspended matter (sedimentation path). Therefore, suspended particles do not leave or enter the water column through its side walls. Its length h is therefore at least of the size of the water depth. (2) Suspended particles leave the (bottom of the) water column through a process called sedimentation. They can re-enter the water column through the bottom plane by the process called resuspension. The concentration of suspended particles in the water column has reached a value x that is (a) constant in time (during a season) and (b) spatially homogenous, i.e. independent from the position within the water column. This assumption is a consequence of the technical limitations of the sampling procedure: At any one location one water sample is taken only once during a season at a maximum of two water depths, i.e. close to the water surface and close to the sea bed. Due to currents, (*) the two water samples are probably not be from the same water column as defined in (1). Thus, it is assumed that at sampling time one water sample is representative of the entire water column of length h. (*) the particulate matter in the water sample might not have been embedded in the water of the sample, because the trajectories of the currents might cross the walls of the above defined water column. Because the suspended particles are parachute like and the size of the parachutes depend on the chemistry of the sea water, this effect varies with the chemical composition of the water. In any case, the thorium activity being associated mostly with the particulate matter might not have been generated from the uranium activity measured in the water sample. (3) The particle speed along the sedimentation path is v. The downward flux of suspended matter, S, is therefore v x. According to (2a) the upward particle flux is also v x (otherwise x would not be constant over time). (4) Both uranium and thorium exist as soluble and adsorbed species. (a) The ratio of adsorbed and soluble uranium concentrations is constant in space and time and given by a concentration independent partitioning coefficient kdU. (b) a-decay of an uranium atom adsorbed on particulate matter and the associated recoil of the resulting thorium atom is assumed to not remove the thorium atom from the particulate matter surface, although the recoil energy (order of magnitude millions of electron volts) probably surpasses the thorium binding energy (order of magnitude fractions of an electron volt). (c) The solid phase thorium activity is disturbed by sedimentation and resuspension. (d) Thorium is exchanged from liquid to solid phase and vice versa with concentration independent exchange rates Kdp and Kpd, respectively. The ratio of the rates is the thorium partitioning (or distribution) coefficient, Kd = Kdp/Kpd. (5) The thorium concentrations result from the balance between (a) generation by a-decay of uranium, (b) loss by b-decay, (c) loss and gain resulting from ad- and desorption and (d) loss from the water column by sedimentation and gain by resuspension of particles loaded with thorium. The thorium loss through sedimentation is the product of the downward particle flux, S = v x, and the thorium concentration per unit mass of suspended matter, Thp/x, i.e. (S/x) Thp (mol/gsusp) (see (2), and (4)) . Because the latter originates largely from the total uranium concentration this amounts to assuming that the thorium loss is largely independent from x, i.e. v Thp (mol/Lsyst). The thorium gain through resuspension is the product of the upward particle flux, S = v x, and the thorium concentration per unit mass of sediment particles, Thsp/(r(1 - e)), i.e. (S/x) (x/(r(1 - e) Thsp) (see (2), and (4)). ;[s] 69:0,0;133,4;145,0;497,4;510,0;594,4;606,0;706,4;722,0;749,4;769,0;1896,4;1901,0;2139,4;2159,0;2160,4;2167,0;2303,4;2315,0;2330,2;2331,0;2339,1;2340,0;2416,4;2422,0;2891,4;2905,0;2907,3;2909,0;2915,3;2917,0;3022,3;3024,0;3026,3;3028,0;3078,4;3085,0;3101,5;3111,0;3115,1;3116,0;3147,5;3148,6;3149,5;3155,0;3191,5;3209,0;3250,5;3263,0;3276,5;3288,0;3486,2;3487,0;3504,2;3506,0;3701,2;3703,0;3877,2;3879,0;3881,1;3882,0;3887,1;3888,0;3907,1;3908,0;3913,1;3914,0;3918,2;3920,0;3945,-1; 7:34,13,10,Geneva,0,9,0,0,0;6,14,10,Symbol,0,9,0,0,0;6,19,13,Geneva,32,9,0,0,0;4,19,13,Geneva,64,9,0,0,0;12,13,10,Geneva,1,9,0,0,0;6,13,10,Geneva,2,9,0,0,0;1,14,10,Symbol,2,9,0,0,0; :[font = subtitle; inactive; preserveAspect; startGroup; ] 4.2 Set of Model Equations :[font = smalltext; 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The upper part describes the water column, the lower part the sediment. The part emphasized by heavy lines is model of Kershaw and Young [loc. cit.]. :[font = smalltext; inactive; preserveAspect; ] The following modifications of the appearance of (1) - (12) will be preformed, which transforms the set of equations into an equivalent one (1') - (12'): (1) multiplication of (1) - (4) with the thorium decay constant lTh, (2) replacement: lTh Th = th, (3) replacement: LambdaU U = u/conv, (4) in water column and in sediment: (a) for soluble (i = d) and adsorbed (i = p) uranium Ui(t) ª Ui0 due to the long half life of uranium, (b) Up0 = KdU Ud0 ;[s] 21:0,0;220,1;221,2;223,0;242,1;243,2;245,0;387,3;388,0;392,1;394,0;396,3;397,2;398,0;442,3;443,2;445,0;449,3;450,0;452,3;453,2;455,-1; 4:8,13,10,Geneva,0,9,0,0,0;3,14,10,Symbol,0,9,0,0,0;5,19,13,Geneva,64,9,0,0,0;5,19,13,Geneva,32,9,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 255; pictureHeight = 27; startGroup; ] %! %%Creator: 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Graphics MathPictureEnd %% End of picture :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 4.4 Mathematica Form of System and Typical Solution ;[s] 3:0,0;4,1;15,0;52,-1; 2:2,19,14,Times,1,18,0,0,0;1,19,14,Times,3,18,0,0,0; :[font = output; output; inactive; preserveAspect; ] eqf := { thd'[t] == lambdaTh/conv ud0 - (Kdp + LambdaTh) thd[t] + Kpd thp[t], thp'[t] == lambdaTh/conv KdU ud0 + Kdp thd[t] - (Kpd + LambdaTh) thp[t] + Shx (x/(rho onemeps)thsp[t] - thp[t]), thsd'[t] == lambdaTh/conv usd0 + (Kspd thsp[t]) - (Ksdp + LambdaTh) thsd[t], thsp'[t] == lambdaTh/conv KdsU usd0 - (Kspd + LambdaTh) thsp[t] - Shx (x/(rho onemeps)thsp[t] - thp[t]) + Ksdp*thsd[t], thd[0] == thd0, thp[0] == thp0, thsd[0] == thsd0, thsp[0] == thsp0} ;[o] lambdaTh ud0 lambdaTh KdU ud0 x thsp[t] lambdaTh usd0 lambdaTh KdsU usd0 x thsp[t] eqf := {thd'[t] == ------------ - (Kdp + LambdaTh) thd[t] + Kpd thp[t], thp'[t] == ---------------- + Kdp thd[t] - (Kpd + LambdaTh) thp[t] + Shx (----------- - thp[t]), thsd'[t] == ------------- + Kspd thsp[t] - (Ksdp + LambdaTh) thsd[t], thsp'[t] == ------------------ - (Kspd + LambdaTh) thsp[t] - Shx (----------- - thp[t]) + Ksdp thsd[t], thd[0] == thd0, thp[0] == thp0, thsd[0] == thsd0, thsp[0] == thsp0} conv conv rho onemeps conv conv rho onemeps :[font = input; preserveAspect; ] datax; definex; ud0=.; usd0l =.; Clear[eqf]; eqf := { thd'[t] == lambdaTh/conv ud0 - (Kdp + LambdaTh) thd[t] + Kpd thp[t], thp'[t] == lambdaTh/conv KdU ud0 + Kdp thd[t] - (Kpd + LambdaTh) thp[t] + Shx (x/(rho onemeps)thsp[t] - thp[t]), thsd'[t] == lambdaTh/conv usd0 + (Kspd thsp[t]) - (Ksdp + LambdaTh) thsd[t], thsp'[t] == lambdaTh/conv KdsU usd0 - (Kspd + LambdaTh) thsp[t] - Shx (x/(rho onemeps)thsp[t] - thp[t]) + Ksdp*thsd[t], thd[0] == thd0, thp[0] == thp0, thsd[0] == thsd0, thsp[0] == thsp0} :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 4.5 Transition Rates Between Compartments :[font = input; preserveAspect; startGroup; ] dataU; defineU; pdata; (*initial values = 50 % of time asymptotic values*) ud0 = udo mBq/Lsyst; usd0l = usdol (mBq/Lliq); thd0 = 1.6 mBq/Lsyst; thp0 = 4.8 mBq/Lsyst; thsd0 = 0.0066 mBq/Lsyst; thsp0 = 8400 mBq/Lsyst; :[font = print; inactive; preserveAspect; endGroup; ] LambdaU LambdaTh Shx Kdp Kpd Ksdp Kspd _____________________________________________________________________ -13 -2 -2 -6 4.22007 10 2.88811 10 2.17439 10 24 8. 24 8.57143 10 -------------, ------------, ------------, ---, ---, ---, ------------ day day day day day day day =================================================================== x rho kd kds kdU kdsU _________________________________________________________________ 0.003 gsusp 1200 gsed 1000 Lsyst 1000 Lliq 1. Lsyst 1. Lliq -----------, ---------, ----------, ---------, --------, ------- Lsyst Lsed gsusp gsed gsusp gsed :[font = input; preserveAspect; startGroup; ] sf[eqf] :[font = output; output; inactive; preserveAspect; endGroup; ] ScientificForm[{Derivative[1][thd][t] == (0.02888113252333105*mBq*udo)/(day*Lsyst) - (24.02888113252333*thd[t])/day + (8.*thp[t])/day, Derivative[1][thp][t] == (0.0000866433975699932*mBq*udo)/(day*Lsyst) + (24*thd[t])/day - (8.02888113252333*thp[t])/day + (0.02174385736029571*(-thp[t] + (3.571428571428572*10^-6*gsusp*thsp[t])/gsed))/day, Derivative[1][thsd][t] == (0.00866433975699932*mBq*usdol)/(day*Lsyst) - (24.02888113252333*thsd[t])/day + (8.57142857142857*10^-6*thsp[t])/day, Derivative[1][thsp][t] == (24.26015131959808*mBq*usdol)/(day*Lsyst) + (24*thsd[t])/day - (0.02888970395190248*thsp[t])/day - (0.02174385736029571*(-thp[t] + (3.571428571428572*10^-6*gsusp*thsp[t])/gsed))/day, thd[0] == (1.6*mBq)/Lsyst, thp[0] == (4.8*mBq)/Lsyst, thsd[0] == (0.006600000000000001*mBq)/Lsyst, thsp[0] == (8400*mBq)/Lsyst}] ;[o] -2 1 2.88811 10 mBq udo 2.40289 10 thd[t] 8. thp[t] {thd'[t] == -------------------- - ------------------ + ---------, day Lsyst day day -5 8.66434 10 mBq udo 24 thd[t] 8.02888 thp[t] thp'[t] == -------------------- + --------- - -------------- + day Lsyst day day -6 -2 3.57143 10 gsusp thsp[t] 2.17439 10 (-thp[t] + --------------------------) gsed ---------------------------------------------------, day -3 1 8.66434 10 mBq usdol 2.40289 10 thsd[t] thsd'[t] == ---------------------- - ------------------- + day Lsyst day -6 8.57143 10 thsp[t] --------------------, thsp'[t] == day 1 -2 2.42602 10 mBq usdol 24 thsd[t] 2.88897 10 thsp[t] --------------------- + ---------- - -------------------- - day Lsyst day day -6 -2 3.57143 10 gsusp thsp[t] 2.17439 10 (-thp[t] + --------------------------) gsed 1.6 mBq ---------------------------------------------------, thd[0] == -------, day Lsyst -3 4.8 mBq 6.6 10 mBq 8400 mBq thp[0] == -------, thsd[0] == ------------, thsp[0] == --------} Lsyst Lsyst Lsyst :[font = input; preserveAspect; endGroup; ] :[font = subtitle; inactive; preserveAspect; startGroup; ] 4.6 Initial Value Problem, Reference Case: Solution of Complete Set of Differential Equations :[font = smalltext; inactive; preserveAspect; ] The dynamical behavior of the system will be calculated for a typical set of parameters (see data) and initial condition. 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_____________________________________________________________________ -13 -2 -2 -6 4.22007 10 , 2.88811 10 , 2.17439 10 , 24, 8., 24, 8.57143 10 =================================================================== x rho kd kds kdU kdsU _________________________________________________________________ 0.003, 1200, 1000, 1000, 1, 1 :[font = input; preserveAspect; startGroup; ] Clear[thd, thp, thsd, thsp]; Clear[soln]; soln = NDSolve[eqf, {thd, thp, thsd, thsp}, {t, 0, 5/LambdaTh}] :[font = output; output; inactive; preserveAspect; endGroup; ] {{thd -> InterpolatingFunction[{0., 173.1234049066756}, {{0., 0., {1.6, 0}, {0}}, {3.377225438504785*10^-6, 0., {1.600001794533378, 0.531363218258484}, {0}}, {6.754450877009571*10^-6, 3.377225438504785*10^-6, {1.600003588899195, 0.5313136034535618}, {0}}, {0.00001013167631551435, 6.754450877009571*10^-6, {1.600005383097471, 0.5312639940145516}, {0}}, {0.00001350890175401914, 0.00001013167631551435, {1.600007177128222, 0.5312143899408568}, {0}}, 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The numbers in the boxes above or next to the arrows are the trans- ition rates (unit: 1/day) between the compartments. 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Thus, the entire sediment part of the model is without influence on the water column part. :[font = input; preserveAspect; endGroup; endGroup; ] :[font = title; inactive; Cclosed; preserveAspect; startGroup; ] 5. 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