(*^ ::[ 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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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 = .