301

3. Heat and mass transfer 3.1 Convection 3.1.1 Model based on hydraulic diameter

One approach to estimate the heat transfer coefficient in a packed bed is given by Jeschar (1964), in which a packed bed can be described as a bundle of parallel pipes. The heat transfer coefficient is based on the established Nusselt correlation

Re005.01

PrRe12.12Nu2

1

3

1

2

1

(3-1)

where ψ is the void fraction of the packed bed. The Nusselt number is defined as:

G

dNu

(3-2)

where d is the size of the particle and λG is the gas thermal conductivity. The Reynolds number Re in Eq. (3-1) is defined as:

d.wRe (3-3)

where w is the gas velocity if no packing was present (superficial velocity), which is described in chapter 2, and υ is the gas kinematic viscosity. The Prandtl number is defined as:

G

pGG cPr

(3-4)

where ρG is the density and cpG is the specific heat capacity of the gas. 3.1.2 Model based on the flow over a single particle

Another common model to determine the heat transfer coefficient in a packed bed is given by Gnielinski (1978), where the Nusselt number is based on the cross flow over a simple sphere. The laminar and turbulent Nusselt functions for a cross-flow are given as

3

1

2

1

lam PrRe664.0Nu (3-5)

and

)1(PrRe443.21

PrRe037.0Nu

3/21.0

8.0

turb

(3-6)

The Nusselt and Reynolds number are defined as before. From the above equations,

302

the Nusselt function for a single sphere in a packed bed can be estimated as:

)1(5.11)NuNu(2Nu 2/12turb

2lam (3-7)

Figure 3-1 compares both Nusselt functions for void fractions of 0.4 and 0.6 and a Prandtl number for gas of 0.7. Both models give a similar result. Therefore, the Nusselt function described in Eq.(3-1) is preferred, because it is a slightly simpler equation.

Figure 3-1: Comparing two heat transfer models 3.1.3 Influence of shape and size distribution

The packed bed of kilns consists of particles with a size distribution. If only a mean heat transfer coefficient is required for the bed, the Nusselt and Reynolds numbers have to be formed with the Sauter-diameter, which was already explained with Eq.(2-10) for the pressure drop.

1

1

1

i

in

i dV

Vd (3-8)

If the temperature of every individual particle is required, the individual heat transfer has to be calculated, for which the Nusselt and Reynolds numbers have to be determined with the specific diameter. 3.1.4 Mass transfer coefficient

The convective mass transfer in a packed bed can be calculated from the Sherwood function, which is analogized to the Nusselt function as:

303

Re005.01

ScRe12.12Sh2

1

3

1

2

1

(3-9)

The Sherwood number is defined as:

D

dSh

(3-10)

where D is the binary diffusivity of the reacting gas component in the kiln flow.

1

2

Dn

ooAirCO T

TDD (3-11)

The Schmidt number is defined as:

DSc

(3-12)

Becaue PrSc it follows

pc

(3-13)

3.1.5 Gas mixture properties

The material property values for the dimensionless numbers have to be calculated at the gas temperature T with the following equations given by Müller (1968). These values can be approximated with an error less than ±3 %.

n

oo T

T,

n

oo T

T (3-14)

1

oo T

T

,

cn

opop T

Tcc

(3-15)

1n

oo T

T

,

cn1n

oo T

Taa

,

1n

oo

D

T

TDD

(3-16)

where T0=273 K. The material properties of the main gases at the temperature T0 = 273 K and the exponents are summarized in Table 3-1. The Prandtl number is temperature independent.

304

Gas

kmol

kg

M~

3

0

m

kg

Kkg

kJ

c 0p

cn

Km

W0

n

sm

mg0

n

Pr

N2 28 1.26 1.00 0.11 0.024 0.76 16.8 0.67 0.70 CO 28 1.26 1.00 0.12 0.024 0.78 16.8 0.67 0.70 Air 29 1.29 1.00 0.10 0.025 0.76 17.4 0.67 0.70 O2 32 1.44 0.90 0.15 0.025 0.80 19.7 0.67 0.70 CO2 44 1.98 0.84 0.30 0.017 1.04 14.4 0.77 0.73 H2O 18 0.81 1.75 0.20 0.016 1.42 8.7 1.13 0.95 Table 3-1: Material properties of gases at T0 = 273 K according to Müller

(1968) The properties of gas mixtures can be calculated with the following formulas:

iiM x~ (3-17)

iiM x~ (3-18)

iiM x~ (3-19)

iipiM

ipipM x~cxcc1

(3-20)

where xi and xi are the mass and volume concentration, respectively, of the gas component i. 3.1.6 Influencing parameter

From the equations of the hydraulic model and the gas properties, the convective heat transfer coefficient can be approximated as:

1.0

0o

0STP

43.0

0

0

5.0

o

STP

76.0

0

0

T

Tw005.0

T

T

d

w)1(

T

T

d2

(3-21)

Figure 3-2 shows the influence of the superficial velocity on the heat transfer coefficient. Calculations were done for air at a temperature of 600 oC, particle sizes varying from 30 mm to 120 mm and a void fraction of 0.4. It can be seen that the heat transfer coefficient increases significantly with increasing the superficial velocity. It can also be seen that a smaller particle size results in a higher heat transfer coefficient.

305

Figure 3-2: Heat transfer coefficient as a function of superficial gas velocity Figure 3-3 shows the influence of the gas temperature on the heat transfer coefficient. Calculations were done for a velocity of the gas at STP of 0.6 m/s, and other inputs were kept the same as before. It can be seen that the increase of the gas temperature leads to a significant increase in the heat transfer coefficient.

Figure 3-3: Influence of gas temperature on heat transfer

306

3.2 Radiation 3.2.1 Radiative heat transfer coefficient

The convective heat transfer is slightly superposed by radiation. This is emitted mainly by the CO2 from the decomposition and the fuel combustion, the H2O from the fuel combustion and the solid fuel particles. For comparison with the convective heat transfer and for simulation of processes, it is more convenient to use a heat transfer coefficient by radiation, which is defined as

)TT()TT( sgeffsg44 . (3-22)

where eff is the effective emissivity. This emissivity can be approximated by

1111

sg

eff (3-23)

where s and g are the emissivity of the solid and gas, respectively. Therewith follows

3

3

2

23 1

g

s

g

s

g

sgeff

T

T

T

T

T

TT . (3-24)

Because the absolute temperatures of the gas and the solid surface don’t differ much, the radiative heat transfer coefficient can be approximated by

111

4 3

sg

gT. (3-25)

3.2.2 Mean beam length

The beam length of gas radiation depends on the size and shape of the gas space. This space is depicted in Figure 3-4 for spherical particles. Figure 3-4: Mean beam length in a packed bed

s

A

307

The mean beam length of the gas space s can be determined with sufficient accuracy from the formula:

particle

gas

A

V49.0s

(3-26)

where V stands for the volume of the gas space and A for the surface of the gas volume. For the arrangement of figure 3-4 follows (d as particle diameter)

d15.0

d

d619.0s

2

3

. (3-27)

3.2.3 Gas emissivity

The emissivity of CO2 and H2O can be approximated for high temperatures with the following equation:

TBexpA (3-28)

Gas A B ps in barm T in K

CO2 20.0sp36.0 19.04 sp104.3 > 0.002 < 0.1 > 1300

084.0sp28.0 11.04 sp101.4 < 0.1 < 10

H2O 46.0sp69.0 22.04 sp107.3 > 0.002 < 0.1 > 700

23.0sp41.0 46.04 sp101.2 < 0.1 < 2

Table 3-2: Values for approximation of the emissivity of CO2 and H2O with

p s in bar m where p is the partial pressure and s the equivalent beam length. For a gas mixture is valid:

OHCOOHCOg 2222 . (3-29)

308

Figure 3-5: Emissivity CO2 of carbon dioxide at p = 1 bar as a function of the

temperature T and the particle diameter as a parameter

Figure 3-6: Emissivity H2O of water at p = 1 bar as a function of the

temperature T and the particle diameter as a parameter

309

3.2.4 Dust emissivity

Dust is able to increase the radiation from hot gases significantly. Radiation of the solid particles distributed in the gas is a function of the wave length, optical characteristics and the size of the particles. The dust concentration is the substantial parameter for the total emitted radiation. In the following, two computation models are described for the determination of the emissivity of dust. Simplified model for small dust concentrations For applications in which the amount of the exponent in the following equations does not exceed a value of 0.5, the emissivity st of the dust according to Biermann and Vortmeyer at negligible backscattering can be calculated with the equations:

sBd2

3kexp1 d

32d

dd (3-30)

where

dd d

1

3

2A

(3-31)

and

31 pabs dQk

absQ - relative active cross-section for absorption of the dust

Bd kg/m3 dust concentration Asp m2/kg specific surface of the dust s m equivalent layer thickness dd m mean dust particle diameter k m-1/3 material constant dependent on absQ

d kg/m3 dust density These values were experimentally determined by Biermann for limestone and coal dust:

Coal Limestonest [kg/m3] 2200 2700 A [m2/kg] 56.0 38.7 dp [10-6 m] 12.2 14.4·

AQabs [m2/kg] 14.4 5.84

absQ 0.257 0.150

Table 3-3: Data for limestone and coal dust

dgdgdg (3-32)

if 1

310

dgdg (3-33)

The approximate value for lime dust concentration in a lime shaft kiln: 2%mass of CaO is 0.0084 kglime dust/m

3 at STP, = 1.1 for weak gas

Figure 3-7: Coal dust emissivity as a function of its concentration and

limestone particle diameter

3.2.5 Influence of radiation

radconv (3-34)

3

1111

4 g

elimgas

T

(3-35)

311

Figure 3-8: Gas radiation coefficient as a function of gas temperature T and

particle diameter d

Figure 3-9: Heat transfer coefficients as a function of temperature and particle

diameter

312

3.3 Heating and Cooling of particles

3.3.1 Transient heat transfer coefficient

In packed beds the solid particles are usually warmed up or cooled down through the heat exchange with gas passing counter-current. Often only the mean temperature of the particle can be considered and not the temperature field within the particle because along with the calculation of the axial temperature profile, the Fourier differential equation would have to be solved, and this requires a lot of effort. For these purposes the overall heat transfer coefficient is introduced. Figure 3-10: Temperatures in particle in counter-current heat exchanger

The overall heat transfer has to describe the convective heat transfer and the heat conduction within the particle. Figure 3-11 shows the temperature profile within the particle when it is cooling down. The core temperature TC is always higher than the surface (wall) temperature Tw.

Figure 3-11: Temperature profiles within the solid particle during the cooling

process. The convective heat transfer is defined with the particle surface and the gas temperature as:

gw TTq . (3-36)

The differential equations previously given are valid only for the mean temperature of the particle Ts. The heat transfer between the gas and the mean solid temperature is described as follows:

gs TTuq (3-37)

where u is the overall heat transfer coefficient, which includes heat conduction from

T

Tg Tg Ts Tw

Tc

d/2 r 0 d/2

z

T

0 L

Tg

Tc Ts

Tm

Tg0

Tp0

313

the particle’s core to its surface and convective heat transfer from the particle’s surface to the gas phase. This overall heat transfer coefficient can be described with the formula below:

s

du

21

1 (3-38)

in which is the transient factor, d is the particle diameter and s is the solid conductivity. For a counter current flow with a capacity flow ratio of 1, the transient factor equals:

spherefor5

cylinderfor4

platefor3

.

These values can be taken with a sufficient accuracy for a capacity flow ratio up to 1.5. The influence of the transient heat conduction within the particle will be illustrated in the following two graphs. Figure 3-12 shows the values of the transient heat transfer coefficient into the particle. The mean conductivity of lime is assumed to be 0.6 W/m/K. For small particle diameters, the heat conduction within the particle is much higher than the convective heat transfer coefficient; it decreases dramatically with the increasing particle diameter. For big particles, the values are similar to the convective heat transfer coefficient, and they decrease slightly with the increasing particle diameter. In principle, the preheating zone can be described with the same set of equations. Therefore, Figure 3-12 includes the values of the transient heat transfer coefficient for limestone, whose mean conductivity is assumed to be 1.5 W/m/K. The trend is the same as the values calculated for the cooling zone, but the values are much higher.

314

Figure 3-12: Values of transient heat transfer coefficient As it is not possible to describe real lime particles as spheres, plates or cylinders, a mean value of = 4 was taken for the following calculations of the cooling zone length. The influence of the heat conduction within a particle is explained in Figure 3-13, where the ratio of the overall heat transfer coefficient to the convective heat transfer coefficient is shown. The mean gas temperature in the cooling zone was taken as a parameter. For fine particles the overall heat transfer coefficient is only 10 – 20 % smaller than the convective heat transfer coefficient, while for the big particles the overall heat transfer coefficient is reduced by 30 – 40 %.

Figure 3-13: Influence of the heat conduction within a particle on the overall

heat transfer

315

3.3.2 Temperature difference within particles

The previous equations are valid only for the mean solid temperature. The surface is always colder than the core. In the following, the temperature difference between the core and the surface will be estimated. For this purpose the linear temperature change will be considered, which is the case for = 1. Assuming that the particles are plates, the temperature difference can be calculated with the following equation:

2

22

1

sv

cTT g

pcs (3-39)

where

STPL

g wL

Tv

. (3-40)

Here vs is the heating rate in K/s, TL the lime temperature change, L the cooling zone length, s the thickness of the plate and ws the velocity of the solid.

Figure 3-14: Temperature difference within particle for different particle sizes and cooling zone lengths

The temperature differences within lime particles calculated with this equation are shown in Figure 3-14. Although the lime particles are not plates, these values can be helpful to create a feeling for how large the differences are. It is clear from Figure 3-14 that the longer the cooling zone is the smaller the temperature differences are within the particle. For example, the temperature difference for particles with a diameter of 120 mm is ~60 K and for particles with a diameter of 60 mm is ~20 K given a cooling zone length of 6 m. The mean output temperature for =1, a cooling zone length of 6 m and an air input temperature of 20°C for these particles are 105°C and 50°C, respectively. The temperature differences at a discharge can be higher than the temperature difference within a particle.

316

Figure 3-15: Temperature difference within a particle as a function of particle diameter

3.3.3 Basic equations for temperature profiles

The principal mean temperature profiles of the lime (solid s) and the air (gas g) in the cooling zone and the symbols used are shown in Figure 3-16 The inlet and the outlet temperatures are influenced by the length of the zone and the amount of air.

Figure 3-16: Schematic temperature profiles of gas and particle for heating and

cooling

The profiles of the mean solid and gas temperature are obtained from the energy balance. The change of the enthalpy flow is equal to the transferred heat flow:

0

gsg

pgg TTL

Au

dz

dTcM (3-42)

T T

Tg0

0 L z 0 z L

Tg Ts0 TgL

Ts0

TsL Tg0

Ts

TgL

Heating Cooling

317

0

sgs

pss TTL

Au

dz

dTcM (3-43)

where M is the mass flow, cp is the specific heat capacity, and u is the overall heat transfer coefficient, which includes convective heat transfer from the gas phase to the particle surface and heat conduction within the particle. These equations are presented in the literature on heat transfer for heat exchangers. The heat transfer area of the lime particles A is given by the following equation:

OLAA f 1 (3-44) in which Af is the kiln cross-section area, L is the zone length, is the void fraction, and O is the specific surface area of lime. The specific surface area of the particle depends on the shape and on the size of the particle. The value of the surface area is unknown because of the irregular shape of the individual particles. Moreover, this area depends on the particle size, whose value is distributed between the upper and lower mesh size. Therefore, the surface area is a parameter with a probability function that is unknown. Measurements have shown that particles classified with mesh sizes between 60 mm and 80 mm have deviations in the surface area of a factor of 2.5. Only for particles with a defined shape can the surface area be calculated. For spheres it is obtained by:

d

6O (3-45)

where d is the particle diameter. In the following the lime particles are considered as spheres with a diameter of the mean mesh size. Consequently, the resulting temperatures can only describe an average temperature. Therefore, the temperature distribution between the smallest and the largest particle cannot be calculated. In order to be independent of the size of the kiln, the mass flows in the following are related to the kiln cross-section area Af:

fA

Mm

. (3-46)

The temperature changes in both phases are now given by:

01 gsg

pgg TTOudz

dTcm (3-47)

01 sgs

pss TTOudz

dTcm (3-48)

The boundary conditions for the cooling of particles are in many cases the solid input temperature:

00 ss TzT (3-49)

and the gas input temperature:

318

gLg TLzT (3-50)

In order to calculate the mean solid temperature, the overall heat transfer coefficient has to be taken into account. It describes the convective heat transfer and the heat conduction within the particle:

s

2d11

u

(3-51)

in which is the transient factor, d is the particle diameter and s is the solid conductivity. For a counter current flow with a capacity flow ratio of 1, the transient factor is as follows:

sphereafor5

cylinderafor4

plateafor3

.

3.3.4 Analytical solution

The temperature profiles in the cooling zone could be calculated with the analytical solution for a counter current heat exchanger if the heat transfer coefficient and the specific heat capacity were constant. These dimensionless equations are:

St1exp

St1exp

, (3-52)

in which

pgg

pss

cM

cM

(3-53)

which is the ratio of the heat capacity flows,

pgg cm

L1OuSt

(3-54)

which is the Stanton number and is the dimensionless temperature. The dimensionless temperature can be calculated with the following formula for cooling

0sgL

0ssL

TT

TT

(3-55)

and preheating

319

sL0g

sL0s

TT

TT

(3-56)

For 1 , equation (3-49) has to be replaced by:

St1St

. (3-57)

In the following, the results of the analytical and numerical calculations will be compared. Therefore, the length of the cooling zone will be considered. From equations (3-43) to (3-46) the cooling zone length is obtained:

for 1

ln

1Ou

cm

1

1L pgg

(3-58)

for 1

1Ou

cm

1L pg

. (3-59)

Because the heat transfer coefficient depends on the temperature, as explained before, the mean value mean is used:

gL0gmean TT21

(3-60)

Here, the heat transfer coefficient at the inlet and the outlet of the cooling zone, (Tg0) and (TgL), have to be determined using the Nusselt function with the material properties at Tg0 and TgL, respectively.

Figure 3-19: Deviations between the cooling zone lengths calculated with the

numerical and the analytical solution.

0.9

0.925

0.95

0.975

1

1.025

1.05

30 45 60 75 90 105 120d in mm

Lan

alyt

ical /

Lnu

mer

ical

= 1

= 1.05

= 1.1

= 1.8

= 1.4

TsL = 50°CTs0 = 1200°C

320

Figure 3-19 shows the ratios of the cooling zone lengths calculated with the analytical solution to the lengths calculated using the numerical method. The maximum deviation between the analytical and numerical solution resulted with the ratio of the heat capacity flows = 1.4. It can be seen that the deviations are in the range +2.5 % to -7.5 %. The values of TsL and Ts0, given as an example in Figure 7, correspond to a very high solid temperature change during the cooling process. Smaller temperature changes result in lower deviations from the numerical solution. Therefore, the analytical solution with respect to the mean heat transfer coefficient can be used as a good approximation for describing the real process. Analytical calculations using a heat transfer coefficient for the mean gas temperature gave much higher deviations.