05 Transient Energy Balances - Colorado School of Mines

Comments Transient Energy Balances

Updated: September 20, 2017John Jechura ([email protected])

General form of the “stuff” balance equation

2

Rate of Rate Rate Rate of Rate of

Accumulation In Out Generation Consumption

F1

F3F4

F5

F6Q1

Q2

Q3F2


Word form of the energy balance equation

3

Rate of Convection Convection Heat Across Work Across

Accumulation Energy In Energy Out Bounary Boundary

F1

F3F4

F5

F6Q1

Q2

Q3F2


Transient energy balance equation

4

F1

F3F4

F5

F6Q1

Q2

Q3F2

,

: :

,: :

,: :

ˆ ˆ

ˆ ˆˆ ˆ

i j k s mi inlet j outlet k m

i i i j j j k s mi inlet j outlet k m

i i i j k s mi inlet j outlet k m

dEE E Q W

dt

d UFH FH Q W

dt

d HV d UVFH FH Q W

dt dt

K P


Where does heat of reaction come from?Use of heat capacity for enthalpy

The enthalpy form of the energy balance is the same whether or not there is a chemical reaction, heat of mixing, heat of absorption, etc.

These effects are artifacts of describing the enthalpy of a mixture using the heat capacity.

5

ˆ ˆˆ ˆ

ref

ref

T

p ref p p refTP

T

ref p p refT

HC H H C dT C T T

T

H H C dT C T T



No reaction:

With chemical reaction, the reference enthalpies are different:

6

10 1

10 1

10 1

ˆˆ ˆ

ˆ ˆ ˆ

ˆ ˆ

p p ref ref p ref ref

p p

dHV FH FH Q

dtdT

VC F C T T H F C T T H QdtdT

VC FC T T Qdt

10 ,0 1 ,1

10 1 ,0 ,1

ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ

p p ref ref p ref ref

p p ref ref

dTVC F C T T H F C T T H Q

dtdT

VC FC T T F H H Qdt



What if the reactants have different heat capacities then the products?

7

1 10 0 ,0 1 1 ,1

1 10 0 1 1 ,0 ,1

ˆˆ ˆˆ ˆ

ˆˆ ˆ ˆ ˆ

pp ref ref p ref ref

pp ref p ref ref ref

d C TV F C T T H F C T T H Q

dt

d C TV F C T T C T T F H H Q

dt


Where does heat of reaction come from?Enthalpy of formation as reference enthalpy

Total enthalpy is component average of individual enthalpies (if no mixing effect)

For constant heat capacities for each chemical species

8

1 1 1 1

ˆˆN N N N

i i i i p i pi i pii i i i

H m H n H C m C n C

10 1

1,0 ,0 ,1 ,1

1 1

1,0 , 0 ,1 , 1 ,0 ,1

1 1 1

N N

i i i ii i

N N N

i p i ref i p i ref i i F ii i i

dHH H Q

dtdH

n H n H QdtdH

n C T T n C T T n n H Qdt


Where does heat of reaction come from?Heat of formation or combustion as reference

Reference state enthalpies can be the enthalpy of formation or the heat of combustion Take care whether the heat of combustion is reported as a negative or

positive value (positive value consistent with report as heating value)

Can separate the terms for the reacting & non-reacting speciesWritten with heat of combustion as negative value

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Reacting Reacting Reacting

Nonreacting

1,0 , 0 ,1 , 1 ,0 ,1

1 1 1

,0 , 0 11

N N N

i p i ref i p i ref i i c ii i i

N

i p ii

dHn C T T n C T T n n H

dt

n C T T Q


Latent heat effects

There are other latent heat effects that look like “generation” terms – changes in enthalpy not due to temperature changes

Heats associated with phase change, e.g., vaporization (solidification, dissolution, absorption, etc. are similar)

Heat of mixing

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1 1

0 0

, ,

, 0 , 1

vap

vap

TT T

p L p VvapT T T

p L vap p L vapvap

dH C dT H C dT

C T T H C T T

,

1 1

N N

i i i i Rmix mix Ri i

H x H H x H x H


How to calculate the effect of heat of reaction

Heat of reaction effects are determined by the indirect path: Determine enthalpy change to take reactants to reference temperature Determine the change in enthalpy @ reference temperature Determine enthalpy change to take products from the reference temperature

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Figure 5.2

Bioprocess Engineering Principles, 2nd edPauline Doran, Elsevier Science & Technology Copyright © 2012, Elsevier Inc. All rights Reserved.


Calculating heat of reaction from heats of combustion (values in Table C.8)

Use the heat of combustion as the reference

Example: Glucose EthanolC6H12O6 (s) 2 C2H5OH (l) + 2 CO2 (g)

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Compound (state) H0c

C6H12O6 (s) -2805.0 kJ/mol

C2H5OH (l) -1366.8 kJ/mol

CO2 (g) 0 kJ/mol

0 0 0

reactants products

1 2805.0 2 1366.8 2 0

kJ71.4

mol glucose

rxn i c i cH n H n H


Heat of reaction with cell growth

Estimated from available electrons

Heat of reaction with oxygen as electron acceptor (aerobic, combustion))

13

c CH q x

kJ

111 to 115 g.mol

q

2

kJ460

g.mol OrxnH



Heat of combustion for bacteria & yeast

Can be estimated as:

For general biomass(CH1.8O0.5N0.2):

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kJ23.2c bacteria

Hg

kJ21.2c yeast

Hg


kJ21.3c biomass

Hg


Calculating heat of reaction from heats of combustion (values in Table C.8)

Use the heat of combustion as the reference

Example: Glucose EthanolC6H12O6 (s) 2 C2H5OH (l) + 2 CO2 (g)

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Compound (state) H0c

C6H12O6 (s) -2805.0 kJ/mol

C2H5OH (l) -1366.8 kJ/mol

CO2 (g) 0 kJ/mol

0 0 0

reactants products

1 2805.0 2 1366.8 2 0

kJ71.4

mol glucose

rxn i c i cH n H n H


Example energy balance – Continuous Ethanol Fermentation

Saccharomyces cerevisiae is grown anaerobically in continuous culture at 30oC. Glucose is used as the carbon source; ammonia is the nitrogen source. A mixture of glycerol and ethanol is produced.

What is the cooling requirement to keep @ 30oC?

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Compound In [kg/h] Out [kg/h]

Glucose 36.0

NH3 0.40

Cells 2.81

Glycerol 7.94

Ethanol 11.9

CO2 13.6

Produced H2O 0.15Figure 5.9



Example energy balance – Continuous Ethanol Fermentation

We’ll calculate the heat of reaction at the standard temperature (25oC) & assume this is close enough For more accuracy we would need to cool reactants to 25oC, calculate heat

of reaction @ 25oC, & then heat up products to 30oC

Heat of reaction:

Hrxn = -570000 – (-556025)= -13,977 kJ/h

Since negative this heat to beremoved

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Other topics from supplemental text

Use of steam tables How much steam might be needed to heat a reactor or stream How much heat is carried away by evaporated water

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Closing comments

Remember that the heat of reaction is energy released not energy generated Unless we are working with nuclear reactions we do not create nor destroy

energy

Using heat of combustion (i.e., change in enthalpy upon combustion)is a valid reference state, just like enthalpy of formation Take care whether the values are reported as negative (consistent with Hrxn) or positive (reported as heating value) For biological reactions make sure you choose the higher heating value (i.e.,

where any product water is in the liquid state)

Many times you can ignore the actual reactor temperature & just use the heat of reaction at standard conditions

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05 Transient Energy Balances - Colorado School of Mines

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