Differential Equations Preview

\frac{d (T1)}{d (t)} = \frac{W \cdot Cp \cdot (T0 - T1) + UA \cdot (Tsteam - T1)}{M \cdot Cp}

\frac{d (T2)}{d (t)} = \frac{W \cdot Cp \cdot (T1 - T2) + UA \cdot (Tsteam - T2)}{M \cdot Cp}

\frac{d (T3)}{d (t)} = \frac{W \cdot Cp \cdot (T2 - T3) + UA \cdot (Tsteam - T3)}{M \cdot Cp}

W = 100

Cp = 2

T0 = 20

UA = 10

Tsteam = 250

M = 1000

Boundaries

t (0) = 0

T1 (0) = 20

T2 (0) = 20

T3 (0) = 20

t (f) = 200

Program validated. Ready to Solve.

# Example 11 - ODE System. Heat Exchange
# Verified Final Values: T1 = 30.9524, T2 = 41.3832 , T3 = 51.3174
# Ref.: Comput. Appl. Eng. Educ. 6: 175, 1998

d(T1)/d(t)=(W*Cp*(T0-T1)+UA*(Tsteam-T1))/(M*Cp)
d(T2)/d(t)=(W*Cp*(T1-T2)+UA*(Tsteam-T2))/(M*Cp)
d(T3)/d(t)=(W*Cp*(T2-T3)+UA*(Tsteam-T3))/(M*Cp)

W=100
Cp=2.0
T0=20
UA=10.
Tsteam=250
M=1000
t(0)=0
T1(0)=20
T2(0)=20
T3(0)=20
t(f)=200

Calculated values

	Variable ▲	Initial	Minimal	Maximal	Final Value
1	t	0	0	200	200
2	T1	20	20	30.95	30.952381
3	T2	20	20	41.38	41.38322
4	T3	20	20	51.32	51.31735

Chart 1

Data Table (39 records)

#	t	T1	T2	T3
1	0	20	20	20
2	2.375716	22.417967	22.693256	22.714602
3	4.2931666	23.974298	24.76353	24.872118
4	6.350934	25.330263	26.836124	27.13647
5	8.557984	26.493187	28.860992	29.483126
6	10.934045	27.477774	30.800299	31.88762
7	13.503506	28.29939	32.621086	34.318214
8	16.298563	28.974144	34.296684	36.738266
9	19.363625	29.518517	35.807262	39.108425
10	22.763958	29.94904	37.14028	41.389442
11	26.432835	30.269817	38.24635	43.458527
12	30.374357	30.501146	39.131596	45.2695
13	34.6301	30.663757	39.820164	46.809853
14	39.248165	30.77466	40.338676	48.07838
15	44.289352	30.847706	40.71521	49.08613
16	49.83104	30.893887	40.97749	49.85457
17	55.86551	30.921343	41.149185	50.40552
18	62.21251	30.936443	41.252922	50.7696
19	68.87918	30.944468	41.313244	51.000454
20	75.545845	30.948452	41.34585	51.13595
21	82.21251	30.95043	41.363365	51.21447
22	88.87918	30.951412	41.372715	51.259476
23	95.545845	30.9519	41.37768	51.285027
24	102.21251	30.952143	41.38031	51.299416
25	108.87918	30.952263	41.3817	51.307457
26	115.54584	30.952322	41.382423	51.31192
27	122.21251	30.952353	41.382805	51.314384
28	128.87918	30.952366	41.383003	51.31574
29	135.54584	30.952374	41.38311	51.31648
30	142.21251	30.952377	41.383163	51.31688
31	148.87918	30.95238	41.38319	51.317097
32	155.54584	30.95238	41.383205	51.317215
33	162.21251	30.952381	41.383213	51.31728
34	168.87918	30.952381	41.383217	51.317314
35	175.54584	30.952381	41.383217	51.317333
36	182.21251	30.952381	41.38322	51.31734
37	188.87918	30.952381	41.38322	51.317345
38	195.54584	30.952381	41.38322	51.31735
39	200	30.952381	41.38322	51.31735

Raw data presented with 39 records.

Constant Values

Constant	W	Cp	T0	UA	Tsteam	M
Value	100	2	20	10	250	1000

Settings and Hints

	Name/Source	Value	Type
1	Total number of equations	9	Setting
2	Number of differential equations	3	Setting
3	Number of explicit equations	6	Setting
4	Reporting digits	10	Setting
5	Minimal License	Free - 2	Setting
6	Elapsed time	0.084 ms	Setting
7	Solution method	RKF_45	Setting
8	Step size guess. h	6.6666667	Setting
9	Truncation error tolerance. eps	0.000001	Setting
10	Good steps	47	Setting
11	Bad steps	0	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) ; T1, T2, T3
2	Info	-Infor- Explicit variables (6) ; W, Cp, T0, UA, Tsteam, M
3	Info	-Infor- Independent variable = t

Calculated values

Data Table (39 records)

Constant Values

Settings and Hints

Messages

Source