[email protected] ENGR-25_Prob_9_15_Solution.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical

[email protected] • ENGR-25_Prob_9_15_Solution.ppt 1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Problem 9.15Problem 9.15SolutionSolution



Simulate ThermoStatic ControlSimulate ThermoStatic Control

By Engineering Analysis the ODE with Highest Order Term ISOLATED

tTTqRCRdt

dTaH

HH

1

Integrate to Isolate T(t) on LHS

dzCR

zTTqRdT

t

HH

aHtT

F

070

dz

CR

zTTqRFtT

t

HH

aH

0

70



Simulate ThermoStatic ControlSimulate ThermoStatic Control

Note for

• T(t) appears on BOTH Side of the Eqn → Use FEEDBACk

• The Integrand is in the form of a SUM

Also the thermostat in this case has a 2F DEADBAND• Implement using SimuLink’s RELAY

function

dz

CR

zTTqRFtT

t

HH

aH

0

70



The Part-a SimuLink ModelThe Part-a SimuLink Model

T-StatScope

simout

Plot Ta & T(t)

1s

IntegratorIC = 70°F

20

FnceR*qm

AmbientTemp, Ta

0.5

1/RC

RELAY → Discontinuities Library

MUX → Commonly Used Blocks Library

SINE WAVE FUNCTION → Math Operations Library

TO WORKSPACE (simout)→ Sinks Library



Test Large & Small FurnacesTest Large & Small Furnaces

The SimuLink Model

T-StatScope

simout

Plot Ta & T(t)

1s


20

FnceR*qm

AmbientTemp, Ta

0.5

1/RC

Fnce RH (ºF-Hr/BTU) qmax (BTU/Hr) R*qmax (ºF)Small 0.0389 514 20Large 0.0389 1028 40



The Output for Small FnceThe Output for Small Fnce

0 5 10 15 20 25 30 35 40 45 5040

45

50

55

60

65

70

75

time (hrs)

Te

mp

era

ture

(F

)

Prob 9.15, part-a Note• Small Fnce canNOT keep

up with heating load when Ta drops below about 55F

• 2 Hour Time-Lag as predicted by

Hr

F

BTU

HrBTU

FCR HH

0.2

1

4.51

1

0389.0

plot(tout, simout(:,1), tout,simout(:,2)), xlabel('time (hrs)'),...ylabel('Temperature (F)'), title('Prob 9.15, part-a')



Low Power FnceLow Power Fnce

0 5 10 15 20 25 30 35 40 45 5040

45

50

55

60

65

70

75

time (hrs)

Te

mp

era

ture

(F

)

Prob 9.15, part-a



Result for Large FnceResult for Large Fnce

0 5 10 15 20 25 30 35 40 45 5040

45

50

55

60

65

70

75

time (hrs)

Tem

pera

ture

(F

)

Prob 9.15, part-a - Lg Fnce

The Large Furnace CAN Keep Up with Heat load at coldest Outside Temps

The RH*qmax Product indicates the MAXIMUM Temp Difference that the Furnace+Insulation combination can accommodate

In This case (T-Ta)min = 70F – (50-10)F = 30F• The Small Fnce is Overwhelmed



High Power FnceHigh Power Fnce

0 5 10 15 20 25 30 35 40 45 5040

45

50

55

60

65

70

75

time (hrs)

Tem

pera

ture

(F

)Prob 9.15, part-a - Lg Fnce



Part b – Daily Energy UsePart b – Daily Energy Use

To Determine Daily Energy Use, E, Need to time-integrate the fnce Power Output over 24hrs

dttqEHrs

24

0

max24

For an Arbitrary time

Refine the Part-a model to• Gain Access to qmax

alone

• Use the integrator Block to find E

dzzqtEt

0

max



The Part-b SimuLink ModelThe Part-b SimuLink Model

1028

qm

1s

TotalEnergy

T-Stat

1/1e5

Scale Outputto Therms

0.0389

Rsimout

Plot Ta & T(t)

1s


DeBugScope

AmbientTemp, Ta

0.5

1/RC

Note the Output scaling to “PG&E” Units• 1 “Therm” = 100 kBTU



Energy UseEnergy Use

Small Fnce Large Fnce

0 4 8 12 16 20 240

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time (hrs)

Cum

ulat

ive

Ene

rgy

Use

(T

herm

s)

Prob 9.15, part-b, Lg Fnce

0 4 8 12 16 20 240

0.02

0.04

0.06

0.08

0.1

0.12

0.14Prob 9.15, part-b, Sml Fnce

time (hrs)

Cum

ulat

ive

Ene

rgy

Use

(T

herm

s)

St-Line → Fnce On 100% of time

Fewer Therms, but Cold Inside

Note Differing Slopes Before & After ~11hr



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

AppendixAppendix

Model Model ConstructiConstructi

onon



Prob_9_15a.mdl (1)Prob_9_15a.mdl (1)

40

FnceR*qm

T-Stat




0.5

1/RC




1s




Prob_9_15a.mdlProb_9_15a.mdl(4)(4)

AmbientTemp, Ta



Prob_9_15a.mdl (5)Prob_9_15a.mdl (5)simout

Plot Ta & T(t)



Part-a Configuration ParametersPart-a Configuration Parameters



Prob_9_15b.mdl (1)Prob_9_15b.mdl (1)

514

qm

1s

TotalEnergy

T-Stat

1/1e5


0.0389

Rsimout

Plot Ta & T(t)

1s


DeBugScope

AmbientTemp, Ta

0.5

1/RC

Non GREEN Blocks are the same as in part-a




514

qm

0.0389

R




1/1e5


1s

TotalEnergy



Part-b Configuration ParametersPart-b Configuration Parameters

Documents

[email protected] ENGR-25_Prob_9_15_Solution.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical