Dynamic Energy Balance

Dynamic Energy Balance

• Last time:

si

iRi

T

T

iiooii WQVr

HdTCpCdt

dTCpn

o

well-mixed CSTR w/flow & reaction

• for multiple reactions:

Vr

HVr

HtermRxni

iRi

i

iRi

2

2

2

1

1

1

rxn #

Simplifications

• ACC Term:

dt

dTCpV

dt

dTCpn

so

CpV

or

CpVCCpn

mii

m

iii

:

For moderate T changes use average Cpi

assume Cpi are constant

Simplifications

• Also:

oiioo

omo

T

T

iioo

TTCpC

or

TTCpdTCpCo

• Simplified Form:

si

iRiomom WQV

rHTTCp

dt

dTCpV

well-mixed CSTR w/flow, reaction & avg. Cp

In-Class ExerciseLiquid flows continuously into an initially empty tank, which contains a full-depth

heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow into the discharge pipe, but heating is maintained.

 

How long does it take the system to reach steady state, what is the final exit temperature, and how long does it take before the tank “overflows”?

vo To

vT

Steam inTs

Condensate

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

Energy Balance on Vessel Jacket

• If Constant Tj: TTUAQ j

• Jacket energy balance:

assume j1 = j2

Tj2 = constant

TTUATTCpdt

dTCpV jjjmjjj

jmjjj 1

Assumptions:

•jacket well insulated

•liquid in jacket is well mixed

•Vj (Volume of jacket fluid) is constant

j is constant

•no work nor reaction in jacket

si

iRiomom WQV

rHTTCp

dt

dTCpV

Energy Balance on Vessel Jacket(especially if Tj varies significantly)

(for small systems, low flow, or not well mixed)

• If Tj not constant:

TTUAQ j

221 jj

j

TTT

assume j1 = j2

i.e. constant

[1]

Energy Balance on Vessel Jacket(Tj not constant)

• Energy balance on heat transferred:

Rate of energyloss by jacket fluid

21 jjmjjjj TTCpTTUAQ

221 jj

j

TTT

Rate of energytransfer from jacket to the reactor

=

[1]

[2]

from [1] 12 2 jjj TTT then into [2]

Energy Balance on Vessel Jacket(Tj not constant)

UACp

UATTCpT

mjjj

jmjjjj

2

2 1

TTUACp

CpUAQ j

mjjj

mjjj

12

2

TTUAQ j

Kj

TTUAKQ jj 1

jj fK

Effect of Flow on Kj

UACp

CpK

mjjj

mjjjj

2

2so as j Kj ??

j

Energy Balance on Vessel Jacket(Tj not constant)

• Energy balance on jacket:

Assumptions:

•jacket well insulated

•liquid in jacket is more like plug flow (i.e cooling coil)

•Vj (Volume of jacket fluid) is constant

j is constant

•no work nor reaction in jacket

TTUATTCpdt

TdCpV jjjmjjj

jmjjj 21

12 2 jjj TTT and:

Small Vessel Jackets

• For small vessels or high pressure systems the thickness of the vessel wall can be significant…thus one needs to consider the thermal capacity of the wall.

jwoowiiw

www TTAhTTAhdt

dTCpV

• Energy balance for jacket:

• Energy balance on wall:

jwoojjmjjjj

mjjj TTAhTTCpdt

dTCpV 1 Tj constant

In-Class Exercise Liquid flows continuously into continuous stirred tank reactor, which is fully-jacketed and

well-mixed. At a certain time, reactant A is introduced into the feed liquid, such that the volumetric flowrate remains constant.

1) Show that the steady state solution to the problem gives a reactor T of ~331 K.

2) With no control (Kc=0), use the dynamic model to find the SS solution.

3) Add proportional control to the dynamic model. Examine the effect of varying Tset from 300 - 375 K, and Kc from -1 to 5.

4) Add integral control to the controller equation. What effect does this have?

o, To, Cao T

TETC

jo, Tjo

V, TCa, Cb

j, Tj

In-Class Exercise

TTUAKQ joj

UACp

CpK

mjjj

mjjjj

2

2

• From before:

• Let:

UUAK j

• So: TTUQ jo

In-Class Exercise

• P only controller: with: Kc = controller gain setco TTKUU

• PI controller:

t

setI

csetco dtTT

KTTKUU

0with: I = integral time

• Let:

t

set dtTTerrsum0

dtTTerrsumd setor:

• Then: setTTdt

errsumd initial condition?

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise : SS Solution

In-Class Exercise: SS Solution

In-Class Exercise: dynamic - SS

In-Class Exercise : dynamic - SS

In-Class Exercise

In-Class Exercise