Energy Balance Equations

In total, three different energy balance equations are required for the model of the tubular reactor. One describes the temperature profile in the reactor, one the profile in the wall and one is needed for the coolant, which is operated counter-current wise.

The reactor consists of two nested tubes, the main tube where reactant and product resides, the intermediate tube, where coolant flows co- or counter current wise and the outer tube with the insulation. The heat insulation is made up of two layers, the first one is just a small air gap, the second one consists of insulation material.

First the energy balance for the inner tube, separating coolant from the reaction mixture will be derived, then the distributed model equations for the reactor and the coolant are shown.

Inner Wall. In this section the energy balance for the wall is derived. At first, a two-dimensional model for the temperature in the reactor wall is derived. Then this equation, which is distributed in two parameters (z, r) is transformed into a

semi-dz

Q(z)˙

Q(r)˙

Q(z˙ +dz) Q(r˙ +dr)

r1

r2

dr

(a) 2-dim

dz Q˙_R Q˙_C

Q˙_in Q˙_out sSL

sLG

sair

r2

r3

r4

r5

(b) 1-dim

Figure 2.7: Cut out of the tubular reactor with heat fluxes for the energy balance equation of the wall

lumped equation, distributed only in the axial dimension. Both partial differential model equations are compared and shortcomings of the semi-lumped model are dis-cussed. Then a suitable modification is proposed for the semi-lumped model in order to improve the validity of the model.

At first, the two dimensional model equation is derived. Using Fig. 2.7(a) one starts with

dU_W

dt = ˙Q(z)− ˙Q(z+d z)+ ˙Q(r)− ˙Q(r+dr). (2.28) Herein, Q˙(r+dr)can be expressed using the Fourier law with a Taylor series expan-sion by

−2πλWd z(r+dr)∂Tˆ_W(r+dr)

∂r = −2πλWd z h

r∂Tˆ_W(r)

∂r +dr∂Tˆ_W(r)

∂r +r dr∂²Tˆ_W(r)

∂r² i.

This result and the definition ^dU_dt^W =%WA_Wc_p,W^∂T_∂t^ˆW, which follows from the caloric equation of state, can be used in Eq. (2.28) to derive the general form of the

two-2.2. DETAILEDMODEL OF THETUBULARREACTOR

dimensional temperature distribution in the reactor wall as 2πc_p,W(Tˆ_W)%Wr dr d zdTˆ_W

dt =2πλWr dr d z∂²Tˆ_W

∂z² +2πλWdr d z h∂Tˆ_W

∂r +∂²Tˆ_W

∂r² i,

which finally can be transformed to c_p,W(Tˆ_W)%W

dTˆ_W dt =λW

∂²Tˆ_W

∂z² +λW

∂²Tˆ_W

∂r² +λW

r

∂Tˆ_W

∂r . (2.29) Note that in (2.29) the wall density%W and the heat conductivityλW are constant, in particular the values for steel are%W =7800^kg

m³ andλW =42_{K m}^W . The heat capacity is a linear function of the wall temperature.

Radial boundary conditions for Eq. (2.29) are

−2πλWr₁ ∂Tˆ_W

∂r r=r1

= ˙Q_R(T,Tˆ_W|r=r1) (2.30)

and

−2πλWr₂ ∂Tˆ_W

∂r r=r2

= ˙Q_C(T,Tˆ_W|r=r2) (2.31)

forr =r₁ andr =r₂ respectively. Heat fluxes Q˙_R and Q˙_C transferred at the bound-aries can be calculated by

Q˙_R =k_{i nner}(T− ˆT_W|r=r1) (2.32)

˙

Q_C =k_{out er}(Tˆ_W|r=r2−T_C). (2.33) Herein,T is the reactor temperature,T_C the temperature of the coolant. Moreover, k_{i nner} andk_{out er} are the overall heat transfer coefficients from reactor to the wall and from the coolant to the wall respectively. These coefficients can be calculated by

k_{i nner} = 1

R_i+R_ll,i, (2.34)

k_{out er} = 1

R_sl,i+R_C,i. (2.35)

R_i andR_C,i account for heat transfer by convection, R_ll,i is the conductive heat trans-port through a laminar layer and R_sl,i is the conductive heat transport through a slime layer. They are evaluated by the following expressions (Stephan et al., 2006)

R_i = 1

αiU_i αi = N u_iλi

2(r₁−s_ll) U_i =2π(r₁−s_ll) R_ll,i = s_ll,i

λll,iU_m,ll,i U_m,ll,i = 2π(r₁−(r₁−s_ll)) ln_r ^r1

1−sll

R_sl,i = s_sl,i

λsl,iU_m,sl,i U_m,sl,i = 2πs_sl ln^r2+_r^ssl

2

R_C,i = 1

αC,iU_C,i αC,i = λC,iN u_C 2(r₃−r₂−2s_sl) U_C,i =2π(r₂+s_sl)

Axial boundary conditions for Eq. (2.29) are

−πλW(r₂²−r₁²)∂Tˆ_W

∂z = ˙Q|_z⁻

k (2.36)

and

−πλW(r₂²−r₁²)∂Tˆ_W

∂z = ˙Q|_z⁺

k. (2.37)

For the internal boundaries,Q˙|_z⁺

k = ˙Q|_z⁻

k+1 with the module indexk=2,...,15. For k=1 andk=16, the outermost heat fluxes (Q˙_z−

1 andQ˙_z+

16 respectively) are calculated from

Q˙|z^•_k =k_wallπ(r₂²−r₁²)(Tˆ_W|z^•_k−T_amb). (2.38) The heat transfer coefficient k_wall accounts for heat transfer by convection, by con-duction and by radiation, hence

k_wall = 1

R_{b,i so}+R_b,amb, (2.39)

2.2. DETAILEDMODEL OF THETUBULARREACTOR

where

R_{b,i so}= s_air λair

R_b,amb= 1

λairN uair

2(r2−r1) +εσ^T_T^{i so}_{i so}^{4 −}₋^T_T^amb_amb⁴ .

Using a mathematical model for the energy balance of the reactor wall which is distributed both in axial and radial direction, would drastically increase the size of the overall model. Of course,DIVAis capable of solving such large systems, depending on the available hardware. However, due to memory limitations, at that time it has not been possible to solve such a detailed system on a standard PC. So, in order to receive a reasonable size of the dynamic mathematical model, a semi-lumped version of Eq. (2.29) is needed. Therefore, a radial average operator is introduced and defined by

T_W = 2π A_W

Z r2

r1

rTˆ_Wdr= 2 r₂²−r₁²

Z r2

r1

rTˆ_Wdr. (2.40) Application of Eq. (2.40) to Eq. (2.29) results in

c_p,W%W

∂T_W

∂t =λW

∂²T_W

∂z² +2πλW

A_W Z r2

r1

r∂²Tˆ_W

∂r² dr+2πλW

A_W Z r2

r1

∂Tˆ_W

∂r dr

=λW

∂²T_W

∂z² +2πλW

A_W h

r∂Tˆ_W

∂r ir2

r1

− Z r2

r1

∂Tˆ_W

∂r dr+ Z r2

r1

∂Tˆ_W

∂r dr

=λW

∂²T_W

∂z² + 1 A_W

n

− ˙Q_C+ ˙Q_R

o. (2.41)

The lumped version of the boundary conditions (Eqns. (2.32) and (2.33)) is Q˙_R =k_{i nner}(T −T_W),

Q˙_C =k_{out er}(T_W−T_C),

with the same heat transfer coefficients defined in Eqns. (2.34) and (2.35).

Hence, the lumped version of the energy balance reads

%Wc_p,W∂T_W

∂t −λW

∂²T_W(z)

∂z² = k_{i nner}

π(r₂²−r₁²)(T−T_W)− k_{out er}

π(r₂²−r₁²)(T_W−T_C) (2.42) which can also be derived directly using an energy balance equation for a one dimen-sional model as depicted in Fig. 2.7(b).

By comparison of (2.29) and (2.42) withk_{i nner} and k_{out er} defined in (2.34) and (2.35), one realizes that in the equation distributed in only one parameter, the radial heat transfer is not properly accounted for. Since this heat transfer coefficient and the thickness of the wall are quite large, it is not acceptable to drop that term. In order to include it, the following procedure is proposed. The idea is to make a very simple and rough discretization in radial direction. Therefore, the wall is separated artificially into two layers. Doing this, implies the assumption that the temperature in between the two layers remains constant over the whole diameter. Next, the coefficients for the radial heat transfer have to be included into (2.42). So the suggestion is to change in Eq. (2.42) only the global heat transfer from the inner side to the wall (ki nner) and from the outer side to the wall (kout er). The modified heat transfer coefficients then read

k_{i nner} = 1

R_i+R_ll,i+R_{r ad,i} (2.43)

R_{r ad,i} =

1

2(r₂−r₁)

λWU_{m,r ad,i} U_{m,r ad,i} =

π

2(r₂−r₁) ln^r2_2r^+r1

1

(2.44)

and in analogy

k_{out er} = 1

R_sl,i+R_C,i+R_{r ad,o} (2.45)

R_{r ad,o}=

1

2(r₂−r₁)

λWU_{m,r ad,o} U_{m,r ad,o}=

π

2(r₂−r₁) ln_r^2r2

2+r1

. (2.46)

The other heat transfer resistors (R_i,R_ll,i,R_sl,i andR_C,i) can still be calculated by the

2.2. DETAILEDMODEL OF THETUBULARREACTOR

x dz x

Q˙R

H˙(z) H(z˙ +dz)

s_LG

r1

Q˙_amb

(a) Reactor

x dz x

Q˙C

Q˙_amb

H˙C(z)

H˙C(z+dz)

s_SL sSL

sair

r3

r4

r5

(b) Coolant

Figure 2.8: Cut out of the tubular reactor with heat fluxes for the energy balance equations (if a separate energy balance for the reactor wall is included) previously introduced expressions. So, now it is possible to use the simple one dimen-sional energy balance (2.42) without neglecting the heat transport in radial direction through the wall.

Reactor. It is assumed that the temperature in one cross section is constant, so that there is no additional radial temperature distribution. The energy balance for the reactor and the coolant are similar to derive, so that here only the reactor is shown to some extent of detail.

If the reactor wall is modelled separately, one can write for the energy balance equation of the reactor using the dashed box in Fig. 2.8(a) as boundary

dU_R

dt = ˙H(z)− ˙H(z+d z)−pd V_R

dt − ˙Q_R, (2.47)

whereH˙(•)is the enthalpy of the flux entering and leaving the small element

respec-tively. Q˙_R takes the heat exchange to the thick inner reactor wall into account. U_R denotes the inner energy of the boxed element. In this equation, the kinetic energy

1

2m˙v² and the potential energym g h are neglected, it is assumed that gravity has no significant influence on the chemical process in the tubular reactor.

Hence, using a Taylor series expansion for H˙(z+d z)and withU_R =H_R−pV_R this equation can be written as

d H_R

dt = −∂H˙(z)

∂z d z+V_Rd p

dt − ˙Q_R.

Introducing H_R =m_Rh=%A d z h, H˙ = ˙m h=%Avh and (2.15) one ends up with

%Ad z dh

dt +v∂h

∂z

=V_Rd p

dt − ˙Q_R.

The molar enthalpyh is a function of the intensive variables p and T as well as the composition of the mixture. So the total derivative ofh

dh= ∂h

∂T _p,w

i

| {z }

cp

d T+ ∂h

∂p T,wi

| {z }

≈0

d p+

I

X

i=1

∂h

∂wi

_p,T

| {z }

¯ h_i

dwi.

Here, the dependency with respect to the pressure pcan be neglected, since it is required for gases and steams only, if pressure or temperature are close to a phase boundary or pressure is close to the critical pressure or exceeds it. Since neither of this holds for the LDPE production, the balance equations transforms to

%A d z



c_p ∂T

∂t +v∂T

∂z

+

J

X

j=1

h¯_j ∂wj

∂t +v∂wj

∂z



=A d zd p

dt − ˙Q_R.

The sum can be rewritten using (2.24), so for the left-hand side, one ends up with

%A d z



c_p ∂T

∂t +v∂T

∂z

+1

%

J

X

j=1

(1h_{r eac})jr_j



= A d zd p

dt − ˙Q_R.

For the heat of reaction, only the following reactions are assumed to have a significant

2.2. DETAILEDMODEL OF THETUBULARREACTOR

influence,

• initiation (see reaction scheme (2.1) and (2.4)),

• propagation (see (2.5)) and

• termination (see (2.6) and (2.7))

Moreover, the heat of reaction is assumed to be the same for these reactions, in other words(1h^{i ni t}_{r eac})≡(1h_{r eac}^{pr op})≡(1h_{r eac}^{t er m}). Additionally, the pressure dynamics is very fast, so one can safely suppose that a quasi-stationary approach for the pressure dy-namic holds.

˙

Q_R comprises the heat transfer caused by convection and the thermal transfer through a thin laminar layer. Both effects are combined in the inner heat transfer resistork_{i nner}. Hence

Q˙_R=k_{i nner}(T−T^∗)d z.

Here, depending on the inclusion of the energy balance for the reactor wall, T^∗ de-notes either the average temperature of the inner reactor wallT_W or the coolant tem-peratureT_C. Moreover, the overall heat transfer coefficientk_{i nner} is

• either to be taken from Eq. (2.43) (if the additional energy balance equation for the wall is included into the dynamic model), or

• defined by the following equation, if the large capacitance of the inner reactor wall is neglected for the transient behavior,

k_{i nner} = 1

R_i+R_ll,i+R_st,i+R_sl,i+R_C,i (2.48) with

R_st,i =r₂−r₁

λst,i . (2.49)

Finally, one ends up with

%c_p ∂T

∂t +v∂T

∂z

= 1

%

J

X

j=1

(1h_{r eac})jr_j−k_{i nner}

A (T−T^∗) (2.50)

I C :T(z,0)=T₀(z) (2.51)

BC :T(0,t)=T_{i n}(t) (2.52)

Note again, that depending on the application, k_{i nner} is defined by either (2.43) with T^∗=T_W or by (2.48) withT^∗=T_C. Moreover, all parameters depend on temperature and pressure, some are also dependent on the composition of the reactor mixture.

Coolant. For the coolant some simplifying assumptions hold as well. Again, it is assumed to have plug flow without any radial temperature distribution. The kinetic and potential energy can be neglected. Additionally, pressure and mass flux are kept constant and no reaction takes place. It is important to note that the flux direction is counter-current wise. Hence the energy balance equation for the dashed box in Fig. 2.8(b) reads

%Cc_pC ∂T_C

∂t −vC∂T_C

∂z

= k_{out er}

A_C (T_∗−T_C)−k_amb

A_C (T_C−T_amb) (2.53)

I C :T_C(z,0)=T_C,0(z) (2.54)

BC:T_C(L,t)=T_{C,i n}(t) (2.55)

Here,k_{out er} andk_ambare the overall heat transfer coefficients accounting for the heat transfer to reactor or wall and to the ambience. Depending on the inclusion of the energy balance equation for the inner reactor wall,k_{out er} is either defined by Eq. (2.45) with T_∗ = T_W or by k_{out er} = k_{i nner} with k_{i nner} from Eq. (2.48) and with T_∗ = T. The heat transfer coefficient to the ambience k_ambcan be evaluated by the following expression

k_amb= 1

R_C,o+R_sl,o+R_st,o+R_air+R_{i so}+R_amb. (2.56) The overall heat transfer coefficientk_ambhence includes the following effects

2.2. DETAILEDMODEL OF THETUBULARREACTOR

• heat transfer by convection (R_C,ofrom coolant to outside and R_ambfrom insu-lation to outside)

• heat transfer by conduction (R_sl,othrough the outer slime layer, R_st,o through the outer wall, R_air through the air gap and R_{i so}through the insulation)

• heat transfer by radiation (in particular, E_s, which is one summand in of R_amb) The resistances are calculated by

R_C,o= 1

αC,oU_o αC,o= λCN u_C 2(r₃−r₂−2s_sl) U_C,o=2π(r₃−s_sl) R_sl,o= s_sl

λslU_m,sl U_m,sl = 2πs_sl ln_r ^r3

3−ssl

R_st,o= s_st,o

λst,oU_m,st,o U_m,st,o=2π(r₄−r₃) ln^r_r⁴

3

R_air = s_air

λairU_m,air U_m,air =2π(r₄+s_air−r₄) ln^r4+_r^sair

4

R_{i so}= s_{i so}

λi soU_{m,i so} U_{m,i so}=2π(r₅−(r₄+s_air)) ln_r ^r5

4+sair

R_amb= 1

αairU_amb+E_s αair =λairN u_air 2r₅ U_amb=2πr₅

E_s =2πr₅ε σ T_{i so}⁴ −T_amb⁴ T_{i so}−T_amb.

The overall heat transfer coefficient R_amb includes both free convection and radia-tion. For the free convection, N u_air = N u(Gr_air,Pr_air). The Grashof number is calculated using Gr_air =8 ^%_η^air

air

2βair

g r₅³(T_{i so}−T_amb). In all equations T_{i so} =50^◦C.

The Prandtl number is evaluated by Pr_air = ^ηair_λ^c_air^P,air. The Nusselt number is com-puted using Eq. (25) in Sec. Gb7 of VDI-Wärmeatlas (1991). The heat transfer by radiation is calculated using the Stefan-Boltzman constantσ =5.6704 10^{−8 W}

m²K⁴ and the emissivity ε=0.93, i.e. it is assumed, that emissivity does not depend on the wavelength.

Im Dokument Modelling and analysis of a production plant for low density polyethylene (Seite 51-62)