Determination of Heat and Mass Transfer Coefficients Associated of Convective Drying of Calcium Carbonate / Water Slurry

Sophie Lee Landau, Emmy Kuo, Joseph Stern

Professor Brazinsky

Spring 2015

Abstract

An Armfield Tray Dryer was used to separate a mixture of calcium carbonate and water by convective drying. The heat and mass transfer coefficients associated with the process were calculated using two methods: a given equation correlated with the heat transfer coefficient and empirical data regarding humidity by using the moisture content of the calcium carbonate slurry. These were done over three trials at different temperature settings. For the first method, the average heat transfer coefficient was calculated to be 24.14 ± 0.85 W/m2K, and the average mass transfer coefficient was 5.05E-05 ± 4.32E-05 kg/m2s. For the second method, the average heat transfer coefficient was 9.35E+03 ± 6.92E+03 W/m2K. The average mass transfer coefficient was 0.02 ± 0.01 kg/m2s. The difference in magnitudes between the two methods indicates a lack of accuracy in either or both of the calculation methods, although their precision can be accepted due to the relatively small standard deviations associated with the coefficients. It is recommended that a new psychrometer be obtained so as to improve the accuracy of the humidity data attained.

Table of Contents

Abstract......................................................................................................................1

List of Tables...............................................................................................................3

List of Figures.............................................................................................................4

Nomenclature..............................................................................................................5

Theory..........................................................................................................................6

Experimental................................................................................................................8

Results.......................................................................................................................10

List of Tables

Table 1 9

Table 2 10

Table 3 11

Table 6 17

List of Figures

Figure 1 13

Figure 2 8

Figure 3 15

Figure 4 16

Figure 5 17

Nomenclature

A = cross sectional area of the panC = constant (14.5)h = heat transfer coefficient (W/m2K)H = humidity of air (kg of H2O/kg of air)Hs = humidity of saturated air (kg of H2O/kg of air)G = gas mass velocity (kg/(m2s))k = mass transfer coefficientλ = latent heat of vaporization (J/kg (H2O))X = m = mass of pasteM = molar mass of airP = pressure of the air (atm)ρ = density of airT= temperature (F)Tg = temperature of the air stream (C) Ts = temperature of the liquid surface (C)Tred = reduced temperatureTcrit = critical temperature of waterv = air velocity (m/s)

TheoryDrying is the process of removing a liquid solvent by evaporation. The solvent is

ordinarily water, and is usually separated from a solid and/or liquid. Often, heat and a

mass-separating agent are used to decrease the time required to complete the drying

process. In convective drying, hot gas is passed over the mixture to speed up heat and

mass transfer. Drying is most widely used and recognized in the food and packaging

industries.

Marble dust, a by-product from the marble processing industries, is often

contained in the waste slurries from marble processing plants. It can be used as a raw

material in production processes within the paper, rubber, and tire industries. In order for

the calcium carbonate to be suitable for reuse, it must first be recovered from the slurry

waste. Removing the marble dust from water is a solid-liquid separation that can be

performed by convective drying with hot air.

In this experiment, calcium carbonate was separated from water using the process

of convective drying. The heat and mass transfer coefficients associated with the drying

process can be determined. The process of drying occurs in phases; in the first, a layer of

liquid rests on the top of the surface of the mixture. Next, the water in the slurry must

move through the slurry before evaporating at the surface of the mixture. In theory,

during this stage, the rate of drying is constant. Finally, only marble dust is left in the pan

and the process of drying is complete.

The rate of heat transfer from the flowing hot air to the solid follows the equation:

q=hA (T g−T s)

where A is the surface area available for convection, T g is the temperature of the hot air,

T s is the temperature of the surface, and the heat transfer coefficient, h, follows the

equation h=(constant )G0.8. This heat transfer can also be equated to what is effectively a

mass transfer relation:

q=kAλ ( H s−H )

In this equation, k is the mass transfer coefficient, and λ is water’s latent heat of

vaporization. Equating the two, eliminating area, and solving for the mass transfer

coefficient,

k=hT g−T

λ ( H s−H )

The reduced temperature is defined as T red=T (K )

Tcrit ( K). This can be used to

calculate the latent heat of vaporization of water, using the following relation:

λ=52053000∗(1−T red )0.3199−0.212T red+0.25795T red2

.The density of air, ρ, can then be calculated

using the ideal gas law,ρ=PMRT

, and can be used in conjunction with the empirically

determined air velocity, v, to determine the gas mass velocity, G=ρv.

Alternatively, the moisture content of the calcium carbonate mixture can be used

to determine the heat and mass transfer coefficients. The moisture content of the slurry is

the ratio between the mass of the water and the mass of the dry paste. Assuming a

constant rate of drying, the rate of water evaporation can be expressed using the

following equation:

dwdt

=qλ=

hA (T g−T s )λ

=kA ( H s−H )where dw /dt is the rate of water evaporation. The heat

and mass transfer coefficients can then be calculated using the relations:

h=

dwdt

∗λ

A (T g−T s )

k=

dwdt

∗1

A ( H s−H )

In this experiment, dw /dt was found according to discretized time intervals.

The heat and mass transfer coefficients associated with the process of convective

drying can thus be calculated using the gas mass velocity, and using the moisture content

of the pan. In the first method, the heat transfer coefficient follows directly from a given

equation, and the mass transfer coefficient is calculated accordingly. In the second,

however, the coefficients are calculated upon the assumption that the process is operating

at a constant rate of drying.

Experimental

An Armfield Tray Dryer was used to dry a calcium carbonate paste at three of its

temperature control settings: 6, 7, and 8. The booster attached to the dryer was set to

maximum and the airflow controller was set at 8. The dry was preheated for ten minutes

prior to the start of the experiment. A tray containing the paste was suspended on a wire

mesh drying rack attached to an electrical balance. The air stream velocity was measured

using an anemometer. A wet and dry bulb thermometer was used to measure the

upstream and downstream temperatures. An IR thermometer was used to measure the

surface temperature.

For each of the three trials, 500 g of solid calcium carbonate was mixed in a tray

with approximately 155 g of water in order to obtain a calcium carbonate paste. The

weights of the calcium carbonate, water, and the tray were recorded. The tray containing

the paste was placed on the drying rack.

The weight of the pan, air stream temperatures, surface temperature, and air

velocity were recorded at each time point. For the first hour of data collection,

measurements were taken every five minutes. After the first hour, measurements were

taken every ten minutes, and after two hours, measurements were taken every fifteen

minutes. The weight of the dry paste was recorded. Throughout the data collection, the

wet bulb had to be rewet with a pipet between each measurement because it dehydrated

quickly. (An image of the setup is shown below in Figure 2.)

Figure 2: Armfield Tray Dryer

Results

The drying of the marble dust and water mixture was measured by monitoring several variables throughout the drying process. These variables were: the surface temperature of the marble paste, the air speed velocity of the fan, the wet and dry bulb temperatures upstream and downstream the paste and finally the mass of the paste. These variables are shown in tables 1, 2 and 3, for the three experiments done.

Upstream Temp (deg F)

Downstream Temp (deg F)

Time

Mass (g) Air Veloc (m/s)

Wet Dry Wet Dry Internal temp (deg F)

0:00 982 0.73 88 90 90 91 77.9

0:09 975 1.99 95 99 101 104 84.5

0:15 968 2.07 104 108 111 113 97.9

0:23 956 2.11 115 116 116 116 106.1

0:33 937 2.02 104 110 110 114 110

0:46 915 2.04 101 109 108 113 113.1

0:57 896 2.11 100 113 113 117 111.3

1:09 877 2.08 100 110 108 114 110.6

1:23 857 2.02 100 110 115 120 110.8

1:39 846 2.12 100 110 115 118 114.2

1:49 843 2.05 116 118 116 118 114.3

TimeWeight of Pan

(g)

Upstr. Wet Bulb

(degF)

Upstr. Dry Bulb

(degF)

Downstr. Wet Bulb

(degF)

Downstr. Dry Bulb

(degF)

Surface

(degF)

Air Velocity

(m/s)0:00 1068 79 108 78 106 90 1.190:05 1055 79 110 78 109 107.0 1.960:10 1052 83 114 80 110 107.1 1.960:15 1044 90 114 86 111 106.6 2.000:20 1037 97 112 92 109 105.8 2.040:25 1032 108 111 104 109 105.8 2.060:30 1020 74 109 74 108 102.9 2.080:35 1019 72 110 72 108 107.6 1.89

0:40 1013 74 112 73 110 105.0 1.980:45 1000 74 113 72 110 105.0 2.060:50 991 73 112 72 111 106.6 2.060:55 984 74 113 73 111 107.1 2.001:00 979 74 113 74 111 108.5 2.051:10 968 75 116 74 111 108.5 1.971:20 948 78 116 74 112 105.9 2.061:30 936 80 113 79 113 109.6 2.111:40 934 79 114 78 113 109.1 1.981:50 926 80 113 79 112 110.6 2.091:55 926 81 115 80 113 109.7 2.092:05 924 81 111 80 111 109.1 2.102:15 923 96 115 92 114 109.6 2.022:25 922 73 108 74 107 108.6 2.03

Table 1: First drying experiment done on the first week, performed at the max speed setting of 10 and a temperature setting of 8.

Time Weight of Pan

(g)

Upstr. Wet Bulb

(degF)

Upstr. Dry Bulb

(degF)

Downstr. Wet Bulb

(degF)

Downstr. Dry Bulb

(degF)

Surface

(degF)

Air Velocity

(m/s)0:00 1067 85 111 85 109 106.8 2.140:05 1063 88 112 82 109 107.6 2.110:10 1055 95 110 92 110 106.3 2.070:15 1048 99 112 97 110 104.6 2.050:20 1043 77 110 76 108 107.4 2.120:25 1034 78 111 77 109 104.6 2.060:30 1028 80 110 78 110 105.0 2.130:35 1022 78 109 77 107 102.3 2.130:40 1014 79 109 77 108 104.4 2.080:45 1012 81 110 79 108 104.3 2.050:50 1004 82 110 81 108 104.2 2.020:55 995 87 109 83 107 103.3 2.061:00 985 73 108 72 107 105.0 2.031:10 968 72 108 71 105 105.2 2.031:20 951 72 108 71 107 102.2 2.101:30 939 74 109 72 107 104.2 2.111:40 933 75 108 74 106 105.7 2.041:50 929 79 109 76 108 104.0 2.052:00 924 79 108 77 107 105.1 2.072:15 922 84 108 79 106 105.4 2.052:30 921 80 106 80 106 104.9 2.10Table 2: Second drying experiment done on the second week, performed at the max speed setting of 10 and a temperature setting of 6.

Time Weight of Pan

(g)

Upstr. Wet Bulb

(degF)

Upstr. Dry Bulb

(degF)

Downstr. Wet Bulb

(degF)

Downstr. Dry Bulb

(degF)

Surface

(degF)

Air Velocity

(m/s)0:00 1076 90 107 85 102 99.0 1.980:05 1071 91 108 88 106 106.0 2.050:10 1067 82 110 80 107 107.4 2.120:15 1060 76 110 75 108 104.6 2.030:20 1052 75 111 75 110 105.6 2.150:25 1043 76 112 75 110 106.4 2.120:30 1034 76 113 74 110 106.7 2.120:35 1028 77 113 76 111 107.8 2.120:40 1021 76 113 75 111 105.8 2.110:45 1015 76 112 75 109 107.4 2.110:50 1004 77 113 74 110 106.9 2.120:55 1000 76 112 76 111 106.3 2.071:00 994 80 115 77 110 108.1 2.061:10 973 79 111 78 109 107.4 2.101:20 968 82 112 80 110 107.3 2.181:30 951 86 111 83 109 105.2 2.081:40 940 99 111 94 109 107.0 2.111:50 933 78 110 76 107 106.7 2.142:00 925 80 111 79 110 109.5 2.032:15 919 79 110 78 108 106.6 2.082:30 917 83 112 81 111 110.8 2.06Table 3: Third drying experiment done on the final week, performed at the max speed setting of 10 and a temperature setting of 7.

Using these recorded measurements, it was then possible to find the mass velocity of the air stream during contact with the drying pan as well as the heat transfer coefficient of the air and paste. This was done using the formula:

G=ρvThe density of the air, ρ, was found using the perfect gas law:

ρ=PMRT

A pressure of 1 atmosphere was used as well as a molar mass of 28.97 kg/mol. In order to find the mass velocity of the air during contact with the paste, the average of the dry bulb temperatures upstream and downstream of the drying rack was used. The following correlation was then used to find the heat transfer coefficient:

h=C∗G0.8

where C is equal to 14.5.

An example calculation up to the heat transfer coefficient is given below:

ρ=PMRT

=(1 atm )(28.97

kgkmol )

( .08206 atm∗m3

kmol∗K )107 K

=.93kg

m3

G= ρv=.93kg

m3∗1.19

ms=1.1

kg

m2 s

h=C∗G0.8=14.5∗(1.1 ).8=15.71W

m2 KThe average heat transfer coefficients over time for each experiment using this method are given in Table 6.

Using this heat transfer coefficient, it is then possible to determine the mass transfer coefficient of the water leaving the paste and evaporating into the air. This was done using the following correlation:

kλ ( H s−H )=h(T g−T s)which can be rearranged to solve for k:

k=hT g−T s

λ ( H s−H )The average temperature of the dry bulb temperatures were used for Tg. The saturated humidity data as a function of temperature was obtained from Perry’s Chemical Engineering Handbook, 8th Edition. This data was interpolated to find the humidity at the wet and dry bulb temperatures, which corresponded to HS and H respectively. The empirical humidity data was then estimated by a linear best fit:

H=0.0009T−0.0492This fit is shown in Figure 1.

60 70 80 90 100 1100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

f(x) = 0.0009096 x − 0.049186R² = 0.981969045926935

Saturation Humidity

Temperature (F)

Satu

ratio

n Hu

mid

ity

Figure 1. Linear fit of empirical data of humidity vs. temperature, data obtained from Perry’s handbook.

λ was found using an empirical curve fit by the function:

λ=52053000∗(1−T red )0.3199−0.212T red+0.25795T red2

Where Tred is the reduced temperature of the dry bulb, where Tred is given by:

T red=T (° K )

Tcrit (° K )Once these values were found, the mass transfer coefficient k could then be found; an example calculation is done below:

T red=T ( K )

Tcrit ( K )=299

647=.462

λ=52053000∗(1−.462 )0.3199−0.212(.462 )+0.25795 ( .462)2=4.38∗107(J /kg(H2 O))H S=0.0009 (107 )−0.0492=.0481H=0.0009 (78.5 )−0.0492=.0222

k=hT g−T s

λ ( H s−H )=(15.71 ) 314.8−305.4

4.38∗1 07 ( .0481−.0222 )=1.31∗10−4 kg /m2 s

This method of finding the mass and heat transfer coefficients, to distinguish from an alternate method employed later is known as the “gas mass velocity method.”

The average mass transfer coefficients for each experiment can be seen on Table 6. A graph of the mass transfer rate varying versus the moisture content in the wet solid was drawn to observe a possible link between the rate of drying versus the remaining water within the paste. The moisture content was determined by taking the current mass

of the pan, and subtracting the mass of the pan and initial calcium carbonate to determine the mass of water remaining in the pan. The mass of the water was then divided by the total mass of the water and calcium carbonate to determine the moisture content. This graph is shown as Figure 3.

The mass and heat transfer of the paste can also be measured as a derivative dmdt

,

and can be estimated by determining the mass lost between time intervals, ΔmΔt

. This

method is called the moisture over time method. The heat transfer coefficient can then be found according to the following formula:

ΔmΔt

=hA (T g−T s )

λwhich is rearranged to find h:

h=

ΔmΔt

∗λ

A (T g−T s )All values are already known, A is the area of the pan, which is equal to .0502 m2, thus it is simple to calculate h:

h=

ΔmΔt

∗λ

A (T g−T s )=

(1.075−1.061 ) kg(300−0 ) s

∗4.38∗107 Jkg

.0502 m2 (314.8−305.4 ) K

h=2.94∗104 W /m2 KSimilarly, the mass transfer coefficient can also be found using the following formula:

k=

ΔmΔt

∗1

A ( H s−H )=

(1.075−1.061 ) kg(300−0 ) s

∗1

.0502 m2 ( .0504−.0222 )

k=.0331kg

m2 sFigure 4, again compares the mass transfer coefficient versus the moisture content of the paste using this new method.

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

Drying Rate vs. Moisture Content using Gas Mass Velocity

DAY 1DAY 2DAY 3

Moisture Content

Mas

s tra

nsfe

r rat

e (k

g/m

^2*s

)

Figure 3. Using the gas mass velocity method, graph of the drying rate, represented as the mass transfer rate of water from the paste to the air, versus the remaining moisture content of the water.

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.3500.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

Drying Rate versus Moisture Content Using Moisture Over Time

DAY 1DAY 2DAY 3

Moisture Content

Mas

s Tra

nsfe

r Coe

fficie

nt (k

g/m

^2*s

)

Figure 4. Using the moisture over time method, graph of the drying rate, represented as the mass transfer rate of water from the paste to the air, versus the remaining moisture content of the water.

For a visualization of the drying process, the moisture content of the paste compared to the time the test continued was also graphed. This is shown in Figure 5.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35Moisture Content versus Time

Day 1Day 2Day 3

Time (s)

Moi

stur

e Co

nten

t ( k

g w

ater

/kg

dry

past

e)

Figure 5. Moisture Content of the paste over time.

An average comparison of the experiments is shown in Table 6, to provide a summary of the total experiment.

Table 6. Total heat and mass transfer coefficients for each experiment as well as their standard deviations.

Discussion

Two main methods were employed to determine the heat transfer coefficient. The first was the gas mass velocity method. This utilized the correlation of:

h=C∗G0.8

to determine the heat transfer coefficient. Building upon this, it was assumed that the drying cake was at equilibrium and thus the heat transferred to the surface was equivalent to the heat used to vaporize the water within the paste. This made it possible to determine the mass transfer coefficient by equating the heat required:

kλ ( H s−H )=h(T g−T s)This correlation relies on several other factors. The first factor is the determination of the saturated humidity of the air at the measured temperatures. This was calculated by fitting a linear curve to empirical data found in Perry’s Handbook of Chemical Engineering. A linear curve was used because it was deemed accurate enough for the purpose of this experiment and had a decent correlation to the data, with an R2 value of 0.98. This function is also reliant on the determination of λ, the latent heat of vaporization of H2O. This was determined by using the empirical relationship described in the Results section. Utilizing all of these values as well as the surface temperature and heat transfer coefficient found previously, the mass transfer coefficient was able to be determined.

In order to better visualize the mass transfer over time, the mass transfer coefficient was graphed versus the moisture content within the paste (Figure 3). However, the graph yielded extremely confusing results, which offer little elucidation of the stages of drying. It was expected that there would be two visible and distinct stages of drying. It was expected that there would be a constant rate period, where the rate of mass transfer would be high yet constant, due to a constant liquid film above the solid. This would eventually lead to a decrease in mass transfer, as the liquid film disappears, leading to the next stage called the “falling rate period”. However it is difficult to see a clear distinction between the stages. This is mostly due to the extremely oscillatory nature of the rate over the moisture content. Examining Figure 3, it is determined that the most likely switch of the stages occurs at a approximate moisture content of ~.05, since there is a decrease that eventually leads to a mass transfer rate of 0. There are several reasons for the difficulty in the stages that will be discussed after the description of the next method.

The next method utilized the mass lost over time to determine the rate of water evaporation. The average slope in mass over time was utilized to find the rate coefficients.

ΔmΔt

=hA (T g−T s )

λΔmΔt

=kA ( H s−H )

Theoretically, this method should contain less error for the mass transfer coefficient than the first method. This is because the mass transfer coefficient does not rely upon the heat transfer coefficient as well as the latent heat of vaporization, unlike the first method. This results in less propagation of error as well and less rooms for mistakes. Similarly, the mass transfer coefficient was graphed against the moisture content (Figure 4).

Unsurprisingly, the trends of the tests were similar to the first method. However, comparing Figure 4 and Figure 3, there is a major difference between the mass transfer rates, the second method gives a mass transfer rate 100x more than the first method. Unfortunately, this is not explainable from the calculations and remains inconclusive as to the reasoning behind the inconsistencies between the methods.

The error of the system proved to be quite large. Examining the average mass transfer coefficients in Table 6, the standard deviation was on the same magnitude if not greater than the average itself. This suggests that there is a wide spread of the mass transfer coefficient. There is one major issue that occurred that introduced error into the system. The most culpable source of error was the psychrometer, or more specifically the wet bulb thermometer. Due to loss of the wick, the wet bulb would not stay consistently wet. Due to this, the wet bulb needed to be wet at certain intervals when it began to dry. This caused major swings in the wet bulb temperature. This caused the wet bulb temperature to have range of 70-100 °F. This explains the oscillatory nature of the mass transfer rate. As the wet bulb dried, the temperature rose, increasing the H, the air saturation of the equation, consequently increasing the mass transfer coefficient. This is true for both methods, as they both use the wet bulb temperature in their calculations. As the bulb dried constantly throughout the run and water was reapplied, this caused the extremely oscillatory nature of the mass transfer coefficient. Figure 5, the graph of the moisture content over time seems to support the conclusion that the oscillatory nature is due to the wet bulb instead of a property of the drying process. The moisture content seems to suggest a relatively constant rate of drying until a time of 5000 seconds, at which the rate starts to decrease. Other sources of error include potential calcium carbonate loss during the mixing process and inconsistency in the wetness of the cake between experiments. However, these are small errors that are not expected to have as large an effect as the wet bulb temperature.

Recommendations

It is recommended that a new psychrometer be obtained, that can consistently ensure a wet bulb, or if not the wet bulb should be rewet after every measurement, to ensure consistent wetness and to prevent the oscillatory nature obtained in this experiment.

References

[1] Perry’s Chemical Engineering Handbook, 8th Edition [Online]

[2] Drying ChE 142 Chemical Engineering Laboratory Packet, Spring 2015