CALCULATING CAPACITY TRENDS IN ROTARY DRYERS

CALCULATING CAPACITY TRENDS IN ROTARY DRYERS

C.R.F. PACHECO¹ and S.S. STELLA²

¹ Escola Politécnica da USP - Departamento de Engenharia Química,

Caixa Postal 61548 - CEP 05424-970

Phone [phone] - Fax [phone] - S.Paulo - SP- Brazil

² Hercules Inc., Av. Roberto Simonsen, 500 CEP 15140-000 -Paulínea - SP - Brazil

(Received: August 15, 1997; Accepted: April 19, 1998)

Abstract - This paper provides a methodology developed for the calculation of the feed rate and of the exit air conditions in an adiabatic rotary dryer, which operates with granular, non-porous solids having only unbound surface moisture. Some aspects related to the algorithm are also discussed in greater detail, such as the behavior of the wet-bulb temperatures along the dryer and the selection of initial values for the iterative loops. The results have been compared with published data from commercial rotary dryers, and predictions compare within 10% of the available data. The methodology can be used to evaluate trends in the behavior of a rotary dryer where the operating parameters vary, and it is useful for the practical engineer, who has to manage several problems commonly encountered in the operation of a rotary dryer installed in a chemical plant.

Keywords: Rotary dryers, drying systems, drying analysis.

INTRODUCTION

Ongoing market globalization has been pushing companies for a reduction in prices, along with an increase in quality. For any industrial operation this scenario implies a reduction of production costs and a tightening of the specification ranges.

The rotary dryer is a piece of equipment which is of relatively common use in the chemical process industries, due to its simplicity and versatility in handling different types of solids. The ability to estimate its operating characteristics is of major importance either in the production planning of an existing plant or in the design of a new one.

The purpose of this paper is to develop an algorithm for estimating the production capacity of existing rotary dryers.

For a practical engineer involved with either operation or design of chemical plants, this algorithm may be helpful in several situations:

1. If one or more process conditions of an existing rotary dryer change, what is the new capacity for the same product specifications?

2. During the process of purchasing a new rotary dryer several proposals are normally received. Can the offered systems do the specific job?

3. There is a possibility of buying a second hand rotary dryer from a given supplier. In this case, how much product in specific conditions could be processed?

Literature on rotary dryers is primarily focused on design methodology and on fundamental parameters needed to understand the physical phenomena involved in this equipment. In this paper we consider an alternative point of view which represents a contribution to the analysis of the performance of existing rotary dryers.

We developed a new algorithm using an integral analysis methodology and assuming an adiabatic dryer operating with granular, non-porous solids. A comparison between the results obtained using our procedure and the available data from commercial rotary dryers was shown to be quite satisfactory for the purposes above.

This paper is organized as follows: firstly, we present the equations which describe the behavior of a rotary dryer; secondly, a careful discussion of the temperature profiles in the drying region is performed. Then, we develop the structure of the algorithm and an analysis of the physical conditions which establish the restrictions needed to assure convergence. A comparison to commercial rotary dryers is performed, and finally, an example of utilization is given. A copy of the executable program is available to the reader upon request.

THEORETICAL BACKGROUND

Tsao and Wheelock (1967) presented a set of general equations which describe the behavior of a rotary dryer.

Here, we use those equations to calculate the production capacity of an existing rotary dryer of known diameter and length in different scenarios.

A rotary dryer operating with granular, non-porous solids with unbound surface moisture may be divided, in a simple scheme, in three zones:

I - a first one where the solids are heated to the wet-bulb temperature of the drying air without losing any moisture,

II - a second one where the solids lose all the desired moisture while remaining at the wet-bulb temperature of the air and

III - a third one where again the temperature of the solids rises without any further moisture loss.

Figure 1 sketches the temperature profiles in the three before mentioned zones of such simplified model, for both counter flow and parallel flow. In the equations below, the distinction between the two stream arrangements is made labeling the variable S_g which assumes the value +1 for the counter flow arrangement and -1 for the parallel flow.

Figure1: Temperature profiles in the three zones of the rotary dryer as proposed by the model.

In both cases the following parameters are assumed to be known:

• dryer geometry: length (Z) and internal diameter (D);

• for the solids: specific heat (C

  _S) , inlet and outlet moisture content (X

  ₁ and X

  ₂ for counter flow or X

  ₂ and X

  ₁ for parallel flow) and temperature (T

  _S1 and T

  _S2 for counter flow or T

  _S2 and T

  _S1 for parallel flow);

• for the drying air: mass flow rate (G), inlet temperature (T

  ₂), ambient temperature (T

  ₃) and relative humidity (W

  _R3).

The algorithm calculates the mass flow rate of solids (L) that can be processed in the dryer and the air temperature (T₁) and humidity (W₁) at the outlet.

For the situations sketched in Figure 1 the following equations apply:

Overall water balance:

(1)

Overall enthalpy balance:

(2)

where: H_S = enthalpy of the solids

H = enthalpy of the air, calculated from its temperature and humidity

Enthalpy balance between points 2 and 4:

(3)

and between 4 and 5:

(4)

The kinetics of the process is expressed in terms of the Number of Heat Transfer Units (N_T) and the Length of a Transfer Unit (L_T).

The Number of Heat Transfer Units (N_T) is related to the fraction of the initial heat transfer driving force existing at the air outlet. This is defined as:

(5)

In order to integrate equation (5), Tsao and Wheelock (1967) have assumed that the heat capacities of both streams have little variation along the dryer. With this assumption, the temperature profiles become linear and integration for each zone of the dryer gives, respectively:

(6)

(7)

(8)

The Length of a Transfer Unit (L_T) is defined by:

(9)

where: C_G = specific heat of the air, kJ kg^-1oC^-1

G_S = air mass velocity, kg m^-2 s^-1,

U_a = overall heat transfer coefficient, W m^-3oC^-1.

Several methods for estimating the overall heat transfer coefficients have been described and were summarized by Baker (1983). According to him, none of the correlations reviewed in his article can be recommended with reasonable degree of confidence. However, the correlation proposed by Friedmann and Marshall (1949) is considered the most reliable, since it is based on extensive and careful experimental data. Their correlation has the following form:

(10)

where K=244.7 for U_a in W m^-3oC^-1, G_S in kg m^-2 s^-1 and D in m.

The above correlation is valid for peripheral shell speeds between 0.2 and 0.5 m s^-1 and holdups between 2 and 8 %.

The constant (K) takes into account factors that could influence the available heat transfer area, namely: particle size distribution, shell rotation speed, material holdup in the dryer and shape and number of flights.

The dryer length is related to N_T and L_T by:

, (11)

where

(12)

With this set of equations, it is possible to fully describe the behavior of a rotary dryer.

Analysis Of The Temperature Profiles

One point that needs to be discussed more profoundly is the behavior of the wet-bulb temperature along zone II of the dryer.

The temperature profiles shown in Figure 1 suggest that the wet-bulb temperature decreases in the direction of the air flow along zone II. In fact, the wet-bulb temperature can rise or fall depending on the operating conditions. This behavior may be understood in the following way:

The effect of drying on a material for a counter flow arrangement implies that:

X₅ > X₄ (13)

Developing this inequality using the enthalpy balance between points 4 and 5, one obtains:

(14)

where: C_w = water specific heat, kJ kg ^oC^-1

T_W = air wet-bulb temperature, ^oC

Rearranging the above inequality and taking into account that, for a given pressure, H can be calculated as a function of T_W only, the results are:

(15)

where:

(16)

The left-hand side of inequality (15) is a function of T_W4 and T_W5, where r is a parameter. Therefore, we can write:

f(T_W4 ,T_W5) >0 (17)

As stated above, drying can only occur if condition (17) is met.

Figure 2 shows a plot of the above inequality for three different values of r, in the full lines represents the region where f(T_w4, T_w5) >0 and in the dotted lines the negative one. The behavior of f(T_W4,T_W5) and the resulting wet-bulb temperature profile are summarized as below.

The same analysis would show a similar behavior for a dryer operating in parallel flow arrangement.

value of r

f(T_W4,T_W5)

wet-bulb temperature along zone II

0.04

>0 for any T_W4 > T_W5

increases

0.4

>0 for T_W4 > T_W5 in some regions and for

T_W5 > T_W4 in other regions

may increase or decrease

4.0

>0 for any T_W5 > T_W4

decreases

Figure 2a: Study of the variation of the function f(T_W4 , T_W5 ) for a selected value of the parameter r= 0.04.

Figure 2b: Study of the variation of the function f(T_W4 , T_W5 ) for a selected value of the parameter r= 0.4.

Figure 2c: Study of the variation of the function f(T_W4 , T_W5 ) for a selected value of the parameter r= 4.0.

ALGORITHM STRUCTURE

The algorithm developed in this work calculates the product flow rate and the exit air conditions for an adiabatic rotary dryer operating with granular, non-porous solids having only unbound surface moisture.

The methodology assumes the knowledge of the following data:

• dryer geometry:

- length (Z)

- internal shell diameter (D)

• solids conditions:

- specific heat (C_S)

- dry-basis moisture content at the inlet (X_i) and at the outlet (X_o)

- temperature at the inlet (T_Si) and at the outlet (T_so )

• drying air conditions:

- pressure (P)

- surrounding air temperature (T₃) and relative humidity (W_R3)

- temperature (T₂) after the air heater

- mass flow rate (G)

• stream arrangement: counter flow or parallel flow

Then the following assignments are done, based on the stream arrangement:

variable

counter flow

parallel flow

S_G

+1

-1

T_Si

T_S1

T_S2

X_i

X₁

X₂

T_So

T_S2

T_S1

X_o

X₂

X₁

The algorithm follows the steps:

1.Calculate:

- the air mass velocity G_S by the expression G / (p D² / 4).

- the volumetric heat transfer coefficient U_a using equation (10).

- the length of the transfer unit L_T using equation (9).

- the number of heat transfer unit N_T using equation (11).

2.With P, T₃ and W_R3, calculate all psychrometric properties for the air at point 3 (ambient air) (psychrometric chart).

3.With P, T₂ and W₂ = W₃, calculate all psychrometric properties for the air at point 2 (dryer inlet) (psychrometric chart).

4.Assume initial values of T_W4 and the solids mass flow rate L.

5.Calculate:

- W₁ using equation (1), the overall water balance.

- the solids enthalpy at point 2 H_S2 by the expression ( C_S + X₂) T_S2.

6.Assume T_S4 = T_w4 (model hypothesis).

7.Calculate:

- H_S4 the solids enthalpy at point 4 by the expression ( C_S + X₂) T_S4.

- H₄ using equation (3), enthalpy balance between points 2 and 4.

8.Assume W₄ = W₂ (model hypothesis).

9.With P, H₄ and W₄, calculate all psychrometric properties for the air at point 4, and in particular the recalculated value T_w4c (psychrometric chart).

10.Use the values of T_w4 and T_w4c to calculate a new value of T_w4 and return to step 6 until convergence is attained.

11.Assume the initial value of T_w5.

12.Assume T_S5 = T_W5 (model hypothesis).

13.Calculate:

- H_S5 the solids enthalpy at point 5 by the expression ( C_S + X₁) T_S5.

- H₅ using equation (4) enthalpy balance between points 4 and 5.

14.Assume W₅ = W₁ (model hypothesis).

15.With P, H₅ and W₅, calculate all psychrometric properties for the air at point 5 and in particular the recalculated value T_w5c (psychrometric chart).

16.Use the values of T_w5 and T_w5c to calculate a new value of T_w5 and return to step 12 until convergence is attained.

17.Calculate the Number of Heat Transfer Units for zones I, II and III.

- N_TIII using equation (8).

- N_TII using equation (7)

- N_TI using equation (12)

18.Assume the initial value of T₁.

19.Calculate T_1C using equation (6) in the form below:

20.Use the values of T₁ and T_1c to calculate a new value of T₁ and return to step 19 until convergence is attained.

21.With P, T₁ and W₁, calculate all psychrometric properties for the air at point 1, and in particular H₁ (psychrometric chart).

22.Calculate:

- H_S1 the solids enthalpy at point 1 by the expression ( C_S + X₁ ) T_S1.

- The reiterative value of the solids mass flow rate L_c using equation (2) overall enthalpy balance.

23.Use the values of the solids mass flow rate L and L_c to calculate a new value of L and return to step 5 until convergence is attained.

SELECTION OF INITIAL VALUES FOR THE ITERATIVE LOOPS

The algorithm has four iterative loops (T_W4, T_W5, L and T₁) and some precautions had to be taken in the selection of the initial values in order to assure the convergence of the loops. The use of constant values would not assure convergence for any arbitrary set of operating conditions. To overcome this difficulty, the initial values are calculated based on the analysis of physical processes occurring in the dryer.

In that sense, the first estimate for T_W4 is made under the following assumptions (refer to Figure 3):

1. At any point in the dryer the temperature T

   _S of the solids must be above the dew point temperature T

   _D of the air in order to avoid condensation and in particular for point 4, (T

   _S4 > T

   _D4).

2. T

   _S4 = T

   _W4 as assumed by Tsao and Wheelock (1967).

3. T

   _D4 = T

   _D2 since there is no change on moisture content between points 2 and 4 (the dew point for the air inlet is represented by point 6 in

   Figure 3).

4. Consequently, the air is cooled at constant moisture from point 2 to point 4, T

   _W4 < T

   _W2.

The first three conditions imply that T_W4 > T_D2, and condition 4 implies that T_W4 < T_W2. Therefore, the first estimate for T_W4 is taken as the average between T_W2 and T_D2.

The first estimate for L is calculated from the global mass balance, choosing a value for W₁, based on the following assumptions:

1. W

   ₁ > W

   ₂ because moisture is added to the air during the drying process.

2. The value for T

   _W4 estimated as described above, determines the maximum outlet moisture attainable in an ideal drying process (shown in

   Figure 3 as point 8). Then, it follows that W

   ₁ < W

   ₈.

Therefore, W₁ is taken as the average between W₂ and W₈, and the value of L is then calculated.

Since T_W5 is close to T_W4, the initial value for T_W5 is taken as equal to T_W4 (as in fact the temperatures would be the same if the change in the total enthalpy of the solids stream were negligible in an adiabatic dryer).

The initial value of T₁ is obtained from equation (6). But since it cannot be put in a form where T₁ is isolated on one side, an iterative loop is used. Among the several forms of rearranging equation (6) in order to have T₁ on the right-hand side, we have chosen this particular one with a unique root:

(17)

Figure 3: First estimates for moist air wet-bulb temperature T_W4 and moist air absolute humidity W₁.

Since T₁ must be greater than T_S1, its initial value is taken slightly above T_S1.

The use of these criteria on the selection of the initial values for the iterative loops assures that the algorithm works properly.

COMPARING THE RESULTS OF THE METHOD WITH PUBLISHED DATA

As we discussed above, this algorithm was developed by assuming an adiabatic dryer. In reality, commercial rotary-dryers have heat losses to the environment, and therefore, they do not present rigorous adiabatic behavior.

Nevertheless, a comparison between predictions from our algorithm and real world was made to evaluate the difference between the ideal and real situations.

In order to perform such a comparison, a data set published by Perry (1984) (table 20-13, page 20-33) for seven rotary dryers operating in parallel flow was used. This table furnishes:

• dryer length and diameter

• for the solids:

- inlet and outlet temperatures

- inlet and outlet wet-basis moisture content

- outlet solids mass flow rate (wet-basis)

• for humid air:

- inlet and outlet temperatures

• evaporation rate

Table 1 shows for the seven dryers the data needed to use the algorithm and the results obtained.

The solids moisture content and its mass flow rate are presented in dry-basis as requested by the algorithm.

The specific heat of the solids was not given, and we assumed: 1 kJ kg^-1oC^-1, which is a representative value for several materials (Perry (1984) table 20-14).

The air mass flow rate was estimated using the evaporation rate and the following approximation:

(18)

where: E = evaporation rate in kg s^-1

l = water latent heat of evaporation at (T_S1+T_S2)/2 in kJ kg^-1

As can be seen from Table 1, the feed rates agree to within 10% and the air outlet temperatures agree to within 5%. The results obtained are lower consistently than the real ones, but these do not affect the validity of an evaluation of trends on dryer operation. In addition, the agreement remains along a broad range of dryer sizes.

Experimental Data Used For Running the Program Xi : 0.3333 Xo:: 0.005 CS: 1 kJ kg-1oC TSi: 27 oC TSo: 65 oC T2 : 165 oC P: 101234 Pa Dryer #1 #2 #3 #4 #5 #6 #7 D m 1.219 1.372 1.524 1.839 2.134 2.438 3.048 Z m 7.62 7.621 9.144 10.698 12.192 13.716 16.767 G kg s-1 0.957 1.276 1.595 2.233 2.871 3.829 6.062 Feed Rate Lexperim. kg s-1 0.113 0.144 0.176 0.263 0.351 0.451 0.715 Lcalculated kg s-1 0.108 0.136 0.174 0.244 0.316 0.416 0.650 Difference % 4.42 5.55 1.13 7.22 9.97 7.76 9.09 Air Outlet Temperature T1 experim. oC 71 71 71 71 71 71 71 T1 calculated oC 67.7 72.6 70.4 70.3 69.7 70.7 71.9 Difference % 4.67 -2.25 0.84 0.99 1.83 0.42 -1.27

Experimental Data Used For Running the Program
Xi : 0.3333
Xo:: 0.005
CS: 1 kJ kg-1oC
TSi: 27 oC
TSo: 65 oC
T2 : 165 oC
P: 101234 Pa
Dryer		#1	#2	#3	#4	#5	#6	#7
D	m	1.219	1.372	1.524	1.839	2.134	2.438	3.048
Z	m	7.62	7.621	9.144	10.698	12.192	13.716	16.767
G	kg s-1	0.957	1.276	1.595	2.233	2.871	3.829	6.062
Feed Rate
Lexperim.	kg s-1	0.113	0.144	0.176	0.263	0.351	0.451	0.715
Lcalculated	kg s-1	0.108	0.136	0.174	0.244	0.316	0.416	0.650
Difference	%	4.42	5.55	1.13	7.22	9.97	7.76	9.09
Air Outlet Temperature
T1 experim.	o C	71	71	71	71	71	71	71
T1 calculated	o C	67.7	72.6	70.4	70.3	69.7	70.7	71.9
Difference	%	4.67	-2.25	0.84	0.99	1.83	0.42	-1.27

Table 1: Comparative results between published and calculated values for seven commercial.

Rotary dryers

THE EXECUTABLE PROGRAM

An executable program, which runs in a user-friendly environment, where all the operations can be followed in a single screen page, is available upon request to the reader.

This program uses a previous algorithm developed by Pacheco (1995) which calculates the moist air properties and plays the role of the psychrometric chart. For the convergence of the iterative loops, the Wegstein method as exposed by Franks (1972) was used.

The executable program is composed of two files:

• ROTDRYER.EXE - the executable file

• DATA.DAT - a file containing the last set of data entered in the program

The program is started by typing ROTDRYER

<enter> at the DOS prompt. The program screen, shown in Figure 4, is composed of three sets of lines:

• 13 lines which contain input data that can be changed by the user

• 7 lines containing the results of the calculations

• one line for modification of any of the inputs

The lines containing the results display the product flow rate, the exit air conditions, and the accordance with the following guidelines given by practice:

• 0.1 £ D/Z £ 0.25

• 0.28 £ G

  _S £ 13.9 kg s

  ^-1 m

  ^-2

• 1.5 £ N

  _T £ 2.0

To change one of the inputs, just type the number of the item and press <enter>. A new line appears, showing the parameter to be changed.

Type the new value, and press <enter>. The program then calculates the new outputs. Repeat the process to modify other parameters.

Figure 4:
Shows program single screen page, with the values of dryer #7.

The results can be printed out by pressing the" Print Screen" key. To exit the program, type "0" (zero) and <enter>.

EXAMPLE

Consider a rotary dryer operating with the conditions displayed in Figure 4. Suppose that, for some reason, the inlet moisture of the solids increases from 33.3% to 40%. What could be done to obtain the same product flow rate (0.65 kg s^-1) with the same outlet moisture (5%)? It is also assumed that the system has some flexibility for changing the air flow rate and inlet temperature.

Among the several possibilities, here we will only discuss two solutions:

a) Increasing the air flow rate by 10% (from 6.062 to 6.668 kg s^-1) and increasing the inlet air tempertaure from 160 to 181 ^oC. The resulting product flow rate in this case would still be 0.65 kg s^-1.

b) Increasing the air inlet temperature from 160 to 190 ^oC, keeping the air flow rate constant. The resulting product flow rate is also 0.65 kg s^-1.

Although both alternatives satisfy the condition of maintaining the production rate and the product specifications, the first one does the job with an N_T of 1.41, while the second one does it with an N_T of 1.53. This means that the second alternative uses energy in a more efficient way, and therefore, should be chosen. In fact, the first option operates with an air flow rate of 6.67 kg s^-1 and gives an air exit temperature of 79.4 ^oC, while the second operates with a lower air flow rate (6.06 kg s^-1) and a lower air exit temperature of 78.4 ^oC.

This was an example of an application of the presented algorithm, but several others can be devised.

CONCLUSIONS

This algorithm, in spite of being rather simple since it consists basically of a set of algebraic equations, permits a quick look at the performance of a rotary dryer.

However, several details had to be examined in order to assure that the algorithm would work properly. The examination of these details actually uncovered rather interesting aspects of the behavior of a rotary dryer.

The agreement between the algorithm and real data shows that our approach can help the practical engineer to obtain a rapid diagnosis of the performance of a rotary dryer.

NOMENCLATURE

C_G Air specific heat, kJ kg^-1oC^-1

C_S Solids specific heat, kJ kg^-1oC^-1

C_W Water specific heat, kJ kg^-1oC^-1

D Internal rotary dryer diameter, m

E Evaporation rate, kg s^-1

G Air mass flow rate, dry-basis, kg s^-1

G_S Air mass velocity, dry-basis, kg s^-1 m^-2

H Moist air enthalpy, kJ kg^-1

H_S Solids enthalpy, kJ kg^-1

K Constant for the heat transfer coefficient

L Solids mass flow rate, dry-basis, kg s^-1

L_T Length of a Transfer Unit

N_T Number of Heat Transfer Units

P System total pressure, Pa

P_p Water vapor partial pressure, Pa

P_ss: Water vapor pressure at dry-bulb temperature, Pa

P_su Water vapor pressure at wet-bulb temperature, Pa

r (G/L)/(C_S+C_WX₅)

S_G Counter flow; parallel flow identification variable

T Moist air dry-bulb temperature, ^oC

T_D Dew point temperature, ^oC

T_S Solids temperature, ^oC (Tsi- inlet; Tso- outlet)

T_W Moist air wet-bulb temperature, ^oC

U_a Volumetric heat transfer coefficient, W m^-3oC^-1

W Moist air absolute humidity, kg/kg

W_R Moist air relative humidity, %/100

X Solids moisture content, dry-basis, kg/kg (X_i- inlet; X_o- outlet)

Z Dryer shell length, m

Greek letters

l Water latent heat of evaporation at (T_S1+T_S2)/2 (kJ kg^-1)

• Baker,C.G.J., Cascading Rotary Dryers, in Advances in Drying, McGraw-Hill, Vol 2, ch 1 (1983).
• Franks, R.G.E., Modeling and Simulation in Chemical Engineering. Wiley-Interscience, (1972).
• Friedmann, S.J. and Marshall, W.R., Studies in Rotary Drying Part II - Heat and Mass Transfer, Chem.Eng.Prog. Vol.45, pp.573-588 (1949).
• Pacheco, C.R.F., Carta Psicrométrica para Computadores Pessoais - PC for PC, Revista Brasileira de Eng. Química, Vol.XV, No.2, pp.21-26 (Nov.,1995).
• Perry, R.H.; Green, D.W. and Maloney, J.O., Perry's Chemical Engineers'Handbook, McGraw-Hill Book Co, 6^th ed., New York (1984).
• Tsao, G.T. and Wheelock, T.D., Drying Theory and Calculations. Chem.Eng. , pp. 201-214 (June 19,1967).

Publication Dates

• Publication in this collection
  27 Oct 1998
• Date of issue
  Sept 1998

History

• Accepted
  19 Apr 1998
• Received
  15 Aug 1997