Analysis of pumped heat energy storage (PHES) process

using explicit exponential matrix solutions (EEMS)

Fan Ni and Hugo S. Caram*

Department of Chemical Engineering, Lehigh University, 18015

Contents

• Background

• Introduction

• Mathematical Model

• Results and Discussion

• Conclusions

Background

• Electric Energy Storage Incentives: • 1. provide a way of grid management to compensate the peak

demand;

• 2. promote the utilization of intermittent renewable energy like

wind, solar and wave energy ;

Pumped heat energy storage (PHES)

T0

W’

TH

Traditional

thermal

energy

storage

W

TH

W

W’’

T0 TH

T0

T0TH

PHES

0( )p HW mc T T H pQ mc dT

0'

H

T TW

Q T

Carnot efficiency for

fixed cold sink temperature

If the process is reversible

: W’’ =W

Basic difference compared with traditional thermal energy storage:

Comparison with other energy storage

methodsAdvantages:

– No special geological formation requirement

– Less environmental concerns

Heater

Cooler

Electric Energy

Turbomachines

T2nom = 500 oC

T0nom = 20 oC

Loading step:

High pressure tank

(HP tank)Low pressure tank

(LP tank)

-72 ~ -5oC

433 ~ 500oC870 ~ 980oC

20 ~ 136oC

A. White, et al., Applied Thermal Engineering (2012).

T. Desrues, et al., Applied Thermal Engineering (2010).

Electric energy is converted into

sensible heat stored in the solid

material

Temperature is regulated at heater

and cooler to maintain a constant

operating condition and eliminate

turbomachine irreversiblities

Delivery step:

Heater

Cooler

Electric Energy

Turbomachines

T2nom = 500 oC

T0nom = 20 oC

Low pressure tank

(LP tank)

High pressure tank

(HP tank)

-72 ~ -5oC

433 ~ 500oC870 ~ 980oC

20 ~ 136oC

Electric energy is retrieved from

sensible heat stored in the solid

material

T. Desrues, et al., Applied Thermal Engineering (2010).

Irreversibilities of Turbomachines

1 1 1

/ ( )Hout in

L

Pw T T

P

1

( ) H

L

P

P

= /p vC C

1

/ )Hin out

L

PT T

P

（

Compressors:

Turbines:

2

1

( ) ( )H Hdelivery Loading

L L

P P

P P

Polytropic efficiency: ζ=0.9

PL Tin PH Tout

PH Tin PL Tout

Specific heat ratio:

Different pressure ratios for Loading and Delivery

T. Desrues, et al., Applied Thermal Engineering (2010).

Thermodynamic limit

Work done during loading:1

2 0[ ( 1) (1 )]p nom nomW c T T

Work received during delivery:1

2 0'' [ ( ' 1) ( ' )]p nom nomW c T T

Efficiency:

When T0nom = 293K T2nom = 773K ψ = 1.55 ψ’ = 1.72 ζ=0.9

1

2 0

1

2 0

( ' 1) ( ' )''

( 1) (1 )

nom nom

nom nom

T TWEff

WT T

T. Desrues, et al., Applied Thermal Engineering (2010).

Eff = 0.83

Model details

• Assumptions:– Ideal gas law

– Adiabatic system

– No axial dispersion term

– No accumulation of heat in the gas phase of each

compartment

– Constant pressure and mass flow rate

– The properties are independent on temperature or pressure

– The pressure change between steps is neglected for the

current analysis

Discretized heat transfer model

n=1 n=N-1 n=NT0nom/α

Loading

Delivery

n=N n=N-1 n=2 n=1

n=N+1 n=2N-1 n=2N

wT2nom

T0nomn=2N-1n=2N

T2nom

( )ns ps n n

dTC ha T

dt

1( ) ( )g g pg n n n nU C hal T

θn is the temperature of gas, Tn is the temperature of solid

Dimensionless

transform 1n n nT

nn n

dTT

d

/

1 /

N

N

1

g g pg

haL

U c

s ps

hat

c

Dimensionless length: Dimensionless time:

Loading step:

1 1 0

2 2

1

1 1

1

2 2

/1 0 0 0

1 0 0 0 0

0 0 0

1 0 0 0

0 0 0 1 0 0 0

0

0 0 0 1

nom

n

n n

n n

n

n n

A

T T T

T T

dT T

dT T

T T

2

0

0

2

2

/

/

nom

n

nom

nom

n

nom

F

T

T

T w

T w

d ( )( )

d

TAT F 1 1

0( ) ( )e AT T A F A F

DAEs:

Explicit exponential matrix solution:

Delivery step:

1 1 0

2

2 2

1

1 1

1

2 2

1 0 0 0

1 0 0 0 0

0 0 0

1 0 0 0

0 0 0 1 0 0 0

0

0 0 0 1

nom

n

n n

n n

n

n n

A

T T T

T T

dT T

dT T

T T

0

0

2

2

'

nom

n

nom

nom

n

nom

F

T

T

T

T

d ( )( ) '

d

TAT F

Explicit exponential matrix solution:

DAEs:

1 1

0( ) ( ') 'e AT T A F A F

Cyclic steady state solution• Tssc : The solid temperature distribution after the delivery step

• Tssh : The solid temperature distribution after the loading step

1 1( )ssh ssce AT T A F A F

1 1( ') 'ssc sshe A

T M MT A F A F

0 1

1

1 0

M

11 1 1 1( ) ' 'ssc e e e e

A A A AT I M M M M A F A F A F A F

1

1 1 1 1' 'ssh e e e e

A A A A

T I M M M A F A F A F A F

where

Global efficiency

2 1 2

1 1 2 1 2

1 2

1 2

1 2

1 2

1

1( )

( )1

( )

net cool cool

net hot hot tot cool hot

g g pg cool g g pg cool

s ps HP LP g g pg cool g g pg hot

cool cool

HP LP cool hot

W Q QEff

W Q Q E Q Q

U c T t U c T t

c L T T U c T t U c T t

T T

T T T T

Qhot1Wnet1 Qhot2

Wnet2

Qcool2Qcool1

Loading Delivery

1 2 2 2

1 1 1

1 =cool cool net hot

tot cool net hot

Q Q W QEff

E Q W Q

Defined by Desrues et al.*:

The energy stored in the process during the loading

step is:

1 1 1 2 2 2tot hot net cool net hot coolE Q W Q W Q Q

Neglecting the energy change of the gas we have:

( )tot HP LP s ps HP LPE E E c LS T T

Modified definition:

where Qcool1, Qcool2, Qhot1, Qhot2, Wnet1 , Wnet2

and Ttot are defined as , , ,

, , ,1coolT 2coolT 1hotT

2hotT ( )Comp TubT T ( )Tub CompT T ( )HP LPT T

,

,

,

,

,

Ttot= is used as the total temperature difference to quantify the storage capacity( )HP LPT T

*T. Desrues, et al., "A thermal energy storage process for large scale electric applications," Applied Thermal Engineering, vol. 30, pp. 425-432, 2010.

Change of internal energy

Results and Discussion

Cyclic steady state temperature distribution of the solid in the LP tank and the HP tank

0 50 100 150 200 250 300 350 400

200

300

400

500

600

700

800

900

1000

1100

1200

1300

from the bottom of the LP tank to the bottom of the HP tank

Tem

per

atu

re/K

Gas flow direction

=3/4

=0=/4

=/2

=

=0

=/4

=/2

=3/4

=

Figure 1 Temperature profile during the loading and the delivery step after the cyclic steady state is reached

when PR=3, PR’=3.88, T0nom =298.15K, T2nom= 773.15K, π=100 and Λ=200

0 50 100 150 200 250 300 350 400

200

300

400

500

600

700

800

900

1000

1100

1200

1300

from the bottom of the LP tank to the bottom of the HP tank

Tem

per

atu

re/K

=

=5/4=3/2

=7/4=2

=2

=7/4=3/2

=5/4

=

Gas flow direction

Comparison with literature reported Data

*T. Desrues, et al., "A thermal energy storage process for large scale electric applications,"

Applied Thermal Engineering, vol. 30, pp. 425-432, 2010.

Λ=100 π=40

Eff=0.678

Eff*=0.667

The solid temperature distribution at cyclic steady state

0 50 100 150 200

200

300

400

500

600

700

800

900

1000

1100

1200

1300

from the bottom of the LP tank to the bottom of the HP tank

Tem

per

atu

re/K

After loading step

After delivery step

After loading step *

After delivery step *

Process gas selection

Ttot* Eff Qcool1

* Qcool2* Qhot1

* Qhot2* Wnet1

* Wnet2*

Argon 37.8 0.731 2.03 9.03 1.21 1.25 38.6 30.0

Air 24.4 0.698 1.69 6.56 1.16 1.19 24.9 19.0

Comparison of Air and Argon as the process gas

Table 1 Comparison of Argon and air when PR=3, PR’=3.88, T0nom =298.15K, T2nom= 773.15K, π=100 and Λ=200*×103K

γair =1.40 γargon =1.67

Gas density : 1.2 kg m-3 for Air

1.7 kg m-3 for Argon (20 oC 1 atm)

For the same ΔP the peripheral velocity of the rotors is lower

Bernoulli’s principle 2P v

Effects of changing the loading step pressure ratio:

PR

2 4 6 8 10 12

0.74

0.76

0.78

0.8

0.82

0.84

PR

Glo

bal

eff

icie

ncy

2 4 6 8 10 120

0.5

1

1.5

2

2.5

x 105

To

tal

tem

per

atu

re d

iffe

ren

ce/

oC

3 4 5 6 7 8 9 10 11 121200

1400

1600

1800

2000

2200

2400

PRT

he

max

imu

m s

oli

d t

emp

erat

ure

/ K

Figure 2 Effects of PR on the global efficiency, the total temperature difference and the maximum solid temperature

when T0nom=298.15K, T2nom= 773.15K, π=150 and Λ=300

Effects of changing the dimensionless numbers: π

and Λ on the capacity and efficiency

The dimensionless step duration:

The dimensionless tank length:

step

s ps

hat

c

g g pg

haL

U c

0 100 200 300 400 500 6000

2

4

6

8

10

12

14

16x 10

4

To

tal

tem

per

atu

re d

iffe

ren

ce T

tot/

K

=50

=100

=200

=300

=400

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Glo

bal

eff

icie

ncy

=50

=100

=200

=300

=400

Figure 3 Effects of π and Λ on the total temperature difference and global efficiency when T0nom=298.15K, T2nom= 773.15K, PR1=3 and PR2=3.88

Ideal thermal wave

Figure 4 The relationship between the average temperature difference and π/Λ

when T0nom=298.15K, T2nom= 773.15K, PR1=3 and PR2=3.88

Neglecting the thermal gradient when the tank is fully used:

0

g pg g ps sU t c Lc

0 ps s

g pg g

Lct

U c

0/ /step g pg g

step

ps s

t U ct t

Lc

tot HP LPT T T = 386 K

Average temperature difference:

0 0.5 1 1.50

50

100

150

200

250

300

350

400

tstep

/t0

=50

=100

=200

=300

=400

Ideal thermal wave

Effects of the heat transfer coefficient

and solid surface area

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0.68

0.69

0.7

0.71

0.72

0.73

0.74

/o or /

o

Glo

bal

eff

icie

ncy

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0.6

0.8

1

1.2

1.4

1.6x 10

5

To

tal

tem

per

atu

re d

iffe

ren

ce/

oC

Figure 5 Effects of ha on the efficiency and the total temperature difference when T0nom=298.15K, T2nom= 773.15K, PR1=3, PR2=3.88, πo=150 and Λo=200

Case study

159.7HP LPT T K

( ) / 2=22.2 /2 2

tot HP LPenergy ps HP LP

E E Ec T T kWh t

V V

Data* for long heat regenerators packed with uniformly sized spherical basaltic beach stones

For the solid:

dp=0.08m ε=0.4 a=3(1- ε )/R=45 m-1 ks=0.5 W m-1 K-1 ρs=912 kg m-3 Cps=1000 J kg-1 K-1 Ug=0.4 m s-1 at 20 oC

For Argon at 20 oC and 1 atm:

µ= 2.2×10-5 kg m-1 s-1 kg=0.017 W m-1 K-1 ρg=1.7 kg m-3 Cpg=521 J kg-1 K-1

For the process parameters:

PR=3, PR’=3.88, T0nom =298.15K, T2nom= 773.15K

Using the correlation from Levenspiel:

The simulation results show that the global efficiency is 0.64 and the average temperature difference is

The stored energy density:

Suppose we choose π=30

W m-2 K-1

*O. Levenspiel, "Design of long heat regenerators by use of the dispersion model," Chemical Engineering Science, vol. 38, pp. 2035-2045, 1983.

Lead acid battery : 30 kWh/ t

max max( ) / 2 53.6 /ps HP LPc T T kWh t

Conclusions

• 1. Cyclic steady state solutions of the PHES process are

obtained using the explicit exponential matrix form

• 2. A thorough dimensionless analysis procedure is established

• 3. Increasing PR and ha will increase the capacity and

efficiency, gas with higher heat capacity ratio and density is

prefered.

11 1 1 1( ) ' 'ssc e e e e

A A A AT I M M M M A F A F A F A F

1

1 1 1 1' 'ssh e e e e

A A A A

T I M M M A F A F A F A F

g g pg

haL

U c

s ps

hat

c

Dimensionless length: Dimensionless time:

Air Reversible Energy Storage:Loading

23.1 bar, 537 C

23 bar, 50 C C

1.1 bar, 30 C

22.1 bar, 44 C 1.13 bar, -115 C

Air Reversible Energy Storage:Delivering

23.1 bar, 532 C

Small expander, lower power of compression

• Questions?