Differential Equations Preview

\frac{d (T)}{d (W)} = \frac{0.8 \cdot (Ta - T) + rA \cdot delH}{CPA \cdot FA0}

\frac{d (x)}{d (W)} = \frac{- rA}{FA0}

\frac{d (y)}{d (W)} = \frac{- 0.015 \cdot (1 - 0.5 \cdot x) \cdot (\frac{T}{450})}{2 \cdot y}

FA0 = 5

Ta = 500

delH = - 40000

CPA = 40

k = 0.5 \cdot \exp (5032 \cdot (\frac{1}{450} - \frac{1}{T}))

CA = \frac{0.271 \cdot (1 - x) \cdot (\frac{450}{T})}{1 - 0.5 \cdot x} \cdot y

CC = \frac{0.271 \cdot 0.5 \cdot x \cdot (\frac{450}{T})}{1 - 0.5 \cdot x} \cdot y

Kc = 25000 \cdot \exp (\frac{delH}{8.314} \cdot (\frac{1}{450} - \frac{1}{T}))

rA = - k \cdot ({CA}^{2} - \frac{CC}{Kc})

Boundaries

W (0) = 0

T (0) = 450

x (0) = 0

y (0) = 1

W (f) = 20

Program validated. Ready to Solve.

# Example 22 - ODE System
# Reversible Reaction in a PBR
# Verified Final Values: T = 1149.6, x = 0.725004, y = 0.766736
# Ref.: Comput. Appl. Eng. Educ. 6: 176-178, 1998

d(T)/d(W)=(.8*(Ta-T)+rA*delH)/(CPA*FA0)
d(x)/d(W)=-rA/FA0
d(y)/d(W)=-0.015*(1-.5*x)*(T/450)/(2*y)
FA0=5
Ta=500
delH=-40000
CPA=40
k=.5*exp(5032*(1/450-1/T))
CA=.271*(1-x)*(450/T)/(1-.5*x)*y
CC=.271*.5*x*(450/T)/(1-.5*x)*y
Kc=25000*exp(delH/8.314*(1/450-1/T))
rA=-k*(CA^2-CC/Kc)
W(0)=0
T(0)=450
x(0)=0
y(0)=1
W(f)=20

Calculated values

	Variable ▲	Initial	Minimal	Maximal	Final Value
1	CA	0.271	0.03509	0.271	0.035085615
2	CC	0	0	0.05107	0.046250127
3	k	0.5	0.5	479	451.19418
4	Kc	25000	35.27	25000	37.34643
5	rA	-0.03672	-0.9715	0.003342	0.0033423484
6	T	450	450	1165	1149.5994
7	W	0	0	20	20
8	x	0	0	0.7272	0.7250039
9	y	1	0.7667	1	0.7667365

Chart 1

Chart 2

Data Table (42 records)

#	W	T	x	y	k	CA	CC	Kc	rA
1	0	450	0	1	0.5	0.271	0	25000	-0.0367205
2	0.488281	453.7995	0.0037055223	0.9963192	0.5490739	0.26724488	0.0004969825	22859.342	-0.03921475
3	1.1549476	459.39706	0.009187227	0.9912309	0.6285042	0.26191455	0.0012142902	20089.145	-0.043114867
4	1.8216144	465.54404	0.015233981	0.9860655	0.72630554	0.25631887	0.0019825809	17494.715	-0.047717724
5	2.488281	472.3446	0.02195159	0.9808171	0.8486029	0.2504173	0.0028102174	15076.059	-0.053214725
6	3.1549478	479.93268	0.029475845	0.97547865	1.0042806	0.24415961	0.003707693	12833.574	-0.059868805
7	3.8216143	488.4834	0.037984207	0.970042	1.2065967	0.23748283	0.004688383	10768.088	-0.06804923
8	4.488281	498.23062	0.047713406	0.9644971	1.4759978	0.23030663	0.0057696463	8880.883	-0.07828765
9	5.1549478	509.49405	0.05898677	0.958832	1.845242	0.22252686	0.006974472	7173.715	-0.091371275
10	5.8216143	522.72284	0.072258	0.95303154	2.3692145	0.21400599	0.008334022	5648.811	-0.10850312
11	6.488281	538.5675	0.08818375	0.9470766	3.1448662	0.20455869	0.00989166	4308.8086	-0.13158737
12	7.1549478	558.00446	0.10774853	0.9409423	4.3545723	0.19393024	0.011709534	3156.5813	-0.16375467
13	7.8216143	582.55707	0.13248716	0.9345954	6.368058	0.1817646	0.013879607	2194.8228	-0.21034999
14	8.488281	614.6989	0.16488993	0.92799056	10.003681	0.1675616	0.016542263	1425.1322	-0.28075618
15	9.00488	647.3705	0.1978309	0.9226555	15.121112	0.15472798	0.019079503	960.076	-0.36171025
16	9.466287	684.8777	0.23564322	0.91769594	23.14497	0.14158176	0.021824092	639.0667	-0.46315968
17	9.929125	733.4338	0.2845846	0.91250086	37.644867	0.12655321	0.025170758	401.3914	-0.60054874
18	10.346994	789.4181	0.34100413	0.9075898	61.237484	0.11138628	0.028818984	252.0753	-0.75276655
19	10.732073	852.41187	0.40449083	0.902858	98.084114	0.09642084	0.032746222	160.66724	-0.89189506
20	11.108027	922.4572	0.47511804	0.8980432	153.55815	0.08173127	0.036991175	104.66386	-0.9714967
21	11.523968	1001.4424	0.5548726	0.8925139	236.11597	0.06695438	0.04173091	69.36568	-0.9164325
22	11.984996	1074.9148	0.62934077	0.8861987	332.8772	0.05437687	0.04616313	49.949604	-0.67662275
23	12.199408	1100.5029	0.6554334	0.8832162	371.15518	0.050162416	0.047709394	45.012707	-0.54053503
24	12.464947	1124.3297	0.67991143	0.8794948	408.9201	0.046261366	0.049132705	41.02977	-0.3854582
25	12.70839	1139.5588	0.6957563	0.8760619	434.12436	0.043739382	0.05001247	38.749256	-0.270227
26	12.946203	1149.6754	0.7064864	0.87269217	451.32486	0.042010207	0.05055923	37.336094	-0.18535578
27	13.188331	1156.4338	0.71387756	0.86924666	463.0184	0.040785298	0.050879814	36.434048	-0.12360264
28	13.440883	1160.8279	0.71893716	0.8656383	470.70782	0.03990373	0.051035333	35.86478	-0.07969766
29	13.709024	1163.5162	0.7223359	0.8617917	475.446	0.0392595	0.051066272	35.522972	-0.049328554
30	13.997875	1164.9617	0.72454894	0.85763043	478.00433	0.0387776	0.051000483	35.34117	-0.028973157
31	14.313058	1165.4967	0.7259227	0.85306925	478.95303	0.038402744	0.050856873	35.27424	-0.015812155
32	14.661204	1165.3582	0.726711	0.8480059	478.70734	0.038093112	0.05064726	35.29155	-0.0076474003
33	15.050553	1164.709	0.7270976	0.84231186	477.5566	0.037816346	0.050377294	35.372852	-0.002815904
34	15.491769	1163.651	0.7272118	0.83581847	475.68442	0.037546586	0.05004674	35.50595	-0.00010230494
35	15.99913	1162.2329	0.72713906	0.8282969	473.1811	0.037261903	0.04964908	35.685524	0.0013462708
36	16.59228	1160.4535	0.7269288	0.81942725	470.0501	0.03694177	0.04917039	35.91276	0.0021000577
37	17.258945	1158.3855	0.7266192	0.80935705	466.42526	0.036585465	0.048620276	36.17956	0.0025019008
38	17.925613	1156.2817	0.7262684	0.7991757	462.75354	0.03622742	0.04805954	36.453983	0.0027468763
39	18.59228	1154.1548	0.7258887	0.788879	459.0572	0.035865564	0.047488756	36.734585	0.0029447887
40	19.258945	1152.0082	0.7254836	0.7784623	455.34274	0.035499047	0.0469079	37.02104	0.0031319452
41	19.925613	1149.8423	0.7250535	0.7679207	451.61157	0.035127416	0.04631674	37.31343	0.003320914
42	20	1149.5994	0.7250039	0.7667365	451.19418	0.035085615	0.046250127	37.34643	0.0033423484

Raw data presented with 42 records.

Constant Values

Constant	FA0	Ta	delH	CPA
Value	5	500	-40000	40

Settings and Hints

	Name/Source	Value	Type
1	Total number of equations	12	Setting
2	Number of differential equations	3	Setting
3	Number of explicit equations	9	Setting
4	Reporting digits	10	Setting
5	Minimal License	Free - 2	Setting
6	Elapsed time	0.143 ms	Setting
7	Solution method	RKF_45	Setting
8	Step size guess. h	0.37193559	Setting
9	Truncation error tolerance. eps	0.000001	Setting
10	Good steps	49	Setting
11	Bad steps	3	Setting
12	Calculated Intermediate data points	50	Setting
13	Version	7.0.73	Setting
14	#@chart_size = 428,300		Default (array2)
15	#@chart_interpolate = 2		Default (integer)
16	#@chart_curve_width = 2		Default (integer)
17	#@report_fix_digits = 8		Default (integer)
18	#@report_show_charts = true		Default (boolean)
19	#@report_show_data_points = true		Default (boolean)
20	#@report_show_intermediate_data_points = true		Default (boolean)
21	#@report_show_source = true		Default (boolean)
22	#@report_show_header = false		Default (boolean)
23	#@report_show_settings = true		Default (boolean)
24	#@chart_all_curves = false		Default (boolean)
25	#@chart_curves_stiff_grouping_percent = 50		Default (integer)
26	#@chart_y_curves =		Default (array2)
27	#@chart_log_scales = 000		Default (string)

Messages

	Severity	Description
1	Info	-Infor- Differential variables (3) ; T, x, y
2	Info	-Infor- Explicit variables (9) ; FA0, Ta, delH, CPA, k, CA, CC, Kc, rA
3	Info	-Infor- Independent variable = W

Calculated values

Data Table (42 records)

Constant Values

Settings and Hints

Messages

Source