R. Balasubramaniam and U. Hegde

National Center for Space Exploration Research, Glenn Research Center, Cleveland, Ohio

S. Gokoglu

Glenn Research Center, Cleveland, Ohio

Analysis of Water Recovery Rate From the

Heat Melt Compactor

NASA/TM—2013-216573

September 2013

AIAA–2013–3393
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R. Balasubramaniam and U. Hegde

National Center for Space Exploration Research, Glenn Research Center, Cleveland, Ohio

S. Gokoglu

Glenn Research Center, Cleveland, Ohio

Analysis of Water Recovery Rate From the

Heat Melt Compactor

NASA/TM—2013-216573

September 2013

AIAA–2013–3393

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

43rd International Conference on Environmental Systems

sponsored by the American Institute for Aeronautics and Astronautics

Vail, Colorado, July 14–18, 2013
Acknowledgments

The authors gratefully acknowledge the support of this work by the Heat Melt Compactor project which is a part of the NASA 

Advanced Exploration Systems program. R. Balasubramaniam and U. Hegde were supported under NASA Contract No. 

NNC08BA08B with the National Center for Space Exploration Research. The authors thank G. Pace and J. Fisher of NASA Ames 

Research Center and E. Golliher, D. Linne, K. Sacksteder and J. Lytle of NASA Glenn Research Center for technical discussions 

and insights during the course of this investigation.

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Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
Human space missions generate trash with a substantial amount of plastic (20% or

greater by mass). The trash also contains water trapped in food residue and paper products

and other trash items. The Heat Melt Compactor (HMC) under development by NASA

Ames Research Center (ARC) compresses the waste, dries it to recover water and melts the

plastic to encapsulate the compressed trash. The resulting waste disk or puck represents an

approximately ten-fold reduction in the volume of the initial trash loaded into the HMC.

In the current design concept being pursued, the trash is compressed by a piston after it

is loaded into the trash chamber. The piston face, the side walls of the waste processing

chamber and the end surface in contact with the waste can be heated to evaporate the

water and to melt the plastic. Water is recovered by the HMC in two phases. The first is

a pre-process compaction without heat or with the heaters initially turned on but before

the waste heats up. Tests have shown that during this step some liquid water may be

expelled from the chamber. This water is believed to be free water (i.e., not bound with

or absorbed in other waste constituents) that is present in the trash. This phase is herein

termed Phase A of the water recovery process. During HMC operations, it is desired that

liquid water recovery in Phase A be eliminated or minimized so that water-vapor processing

equipment (e.g., condensers) downstream of the HMC are not fouled by liquid water and

its constituents (i.e., suspended or dissolved matter) exiting the HMC. The primary water

recovery process takes place next where the trash is further compacted while the heated

surfaces reach their set temperatures for this step. This step will be referred to herein

as Phase B of the water recovery process. During this step the waste chamber may be

exposed to different selected pressures such as ambient, low pressure (e.g., 0.2 atm), or

vacuum. The objective for this step is to remove both bound and any remaining free water

in the trash by evaporation. The temperature settings of the heated surfaces are usually

kept above the saturation temperature of water but below the melting temperature of the

plastic in the waste during this step to avoid any encapsulation of wet trash which would

reduce the amount of recovered water by blocking the vapor escape. In this paper, we

analyze the water recovery rate during Phase B where the trash is heated and water leaves

the waste chamber as vapor, for operation of the HMC in reduced gravity. We pursue

a quasi-one-dimensional model with and without sidewall heating to determine the water

recovery rate and the trash drying time. The influences of the trash thermal properties,

the amount of water loading, and the distribution of the water in the trash on the water

recovery rates are determined.

	

	 

Analysis of Water Recovery Rate From the Heat Melt Compactor 

R. Balasubramaniam and U. Hegde 

National Center for Space Exploration Research 

Glenn Research Center 

Cleveland, Ohio 44135 

S. Gokoglu 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 
Nomenclature

C Specific heat, J/(kg-K)

D, D0 Instantaneous and initial diameter of a spherical wet region in the trash, m

H Height of compressed trash, m

k Thermal conductivity, W/(m-K)

Lˆ Latent heat, J/kg

M Mass fraction of water evaporated

Mw Mass of water in the trash, kg

r Radial coordinate, m

R Radius of trash chamber, m

t Time, s

tdry Trash drying time, s

T Temperature, K

Tw Heated wall temperature, K

Ts Saturation temperature, K

x Axial coordinate, m

Subscripts

1 Dry trash region

2 Wet trash region

Symbols

δ Location of liquid/vapor interface, m

 Volume fraction of water

t Volume fraction of trash

ρ Density, kg/m3

ρw Density of liquid water, kg/m3

σ Quantity appearing in Eq. (5), 1/K

φ Vapor quality

I. Introduction

Human space missions generally generate trash with a substantial amount of plastic (20% or greater by

mass). The trash also contains water trapped in food residue and paper products and other trash items.

The Heat Melt Compactor (HMC) under development by NASA Ames Research Center (ARC) compresses

the waste, dries it to recover water, and melts the plastic to encapsulate the compressed trash. The resulting

waste disk or puck represents an approximately ten-fold reduction in the volume of the initial trash loaded

into the HMC.

Figure 1 shows a cut-away view of the initial design concept. The trash is loaded into the waste processing

chamber where it is compressed by the piston. The piston face, the side walls of the waste processing chamber

and also the surface in contact with the waste at the input door can be heated to first evaporate the water

and second to melt the plastic. The major steps in the overall process are:

1. Trash loading into HMC.

2. Primary compaction without heat or with the heaters on but before the waste heats up. This phase

is herein termed Phase A of the water recovery process. Typical compaction time is on the order of 2

minutes.

3. Primary water extraction with secondary compaction. This step will be referred to herein as Phase B

of the water recovery process. Typical time is on the order of 80 minutes in initial tests.

4. Melt encapsulation at elevated temperatures.

5. Waste disk removal and reactor reset for next cycle.

	

	 	
Figure 1. Cut-away view of the HMC design concept.

Phase A needs to be investigated by performing reduced gravity flight tests. In this paper, we develop

analytical models to determine the drying time and water recovery rate from the HMC for Phase B during

reduced gravity operation and compare these results with ground test data.

II. Model of Water Recovery Rate

During Phase B of the HMC process, the heated surfaces are raised to temperatures above the saturation

temperature of water but below the melting temperature of the plastic in the trash. During this phase the

waste chamber may be exposed to different selected pressures such as ambient, low pressure (e.g., 0.2 atm),

or vacuum. The objective is to remove both bound water and any remaining free water by evaporation.

Keeping the temperature below the plastics melting temperature avoids encapsulation of wet trash that may

block vapor escape and reduce the net water recovered.

A. Analysis Objective

The overall objective of the analysis is to determine the drying time and water recovery rate from the HMC

during reduced gravity operation. A second major objective is to determine the trash temperature as the

evaporation process proceeds. It may be expected that higher wall temperatures will hasten the evaporation

process, but wall temperatures must be kept below the melting temperature of the plastic constituents to

avoid trash encapsulation. However, if the bulk of the trash remains much cooler than the wall, it may be

possible to raise the wall temperatures to increase evaporation rates.

B. Model Description

During Phase A, the trash undergoes substantial compaction; only minimal additional compaction takes place

during heating of the trash and water evaporation in Phase B. With the heaters maintained at ∼ 120◦C, safely

below the plastic melt point, plastic constituents may soften somewhat but do not cause any encapsulation

that would hinder vapor escape. Similarly, ground-based experiments suggest that the vapor escape path

	

	 
is not obstructed when the trash is compressed. In some instances, however, it has been noted that the

compacting piston retreats due to build up of some type of resistance in the waste chamber. When this

occurs the measured chamber pressure is still close to the set pressure of evacuation (which may be as low

as 0.2 atm). This suggests that the resistance causing the piston to retreat is not due to excessive build

up of vapor pressure from rapidly evaporating water. It has been suggested that water or liquids trapped

in pouches or containers present in the chamber evaporate and build up local pressure inside these pouches

which resists the compression force of the piston. It is noticed that after some time the piston advances

again presumably as the vapor inside the pouches finally escapes. Events such as these are very dependent

on the trash sample and ultimately do not appear to impact the overall evaporation rate to any significant

degree.

To date, there have been limited studies of heat transfer properties of the trash. In particular, questions

on the degree of trash heterogeneity, thermal conductivity and diffusivity variation, and effects of trash

wetness on these properties, as well as vapor path descriptions still exist. There is also uncertainty in the

spatial distribution of the water content in the trash which will likely be very sample dependent. Taking

these uncertainties into account, the modeling activity will proceed from idealized cases such as uniform 1-D

trash configurations to more realistic ones as available information increases and/or more detailed predictions

are needed.

Therefore, the initial model described here in this paper is a 1-D configuration with spatial variations in

the direction of piston compaction alone; radial or transverse variations will be ignored. The methodology

pursued in the model will be similar to the classical Stefan problem where a phase-change front is tracked.

Two situations will be modeled:

Case 1. A pure one-dimensional case where heating is provided only at the two end-faces of the trash as

shown in Fig. 2(a) and

Case 2. A quasi-one-dimensional case where side-wall heating is accounted for as shown in Fig. 2(b). In

both cases the axial extent (denoted as H in Fig. 2) of the trash is assumed constant during the process.

The cross-section of the trash may be a square (or rectangle) or a circle without loss of generality for Case 1.

For Case 2 a circular cross-section has been assumed but the overall results are expected to hold for square

or rectangular cross-sections as well.





	
	

	δ









 



 







(a) (b)

Figure 2. (a) One-dimensional case with heating at the two end-faces (b) Quasi-one-dimensional case with

inclusion of side-wall heating.

1. Case 1: 1-D model

The evaporation front moves from the dry trash (named Region 1) into the wet trash (named Region 2) along

the x-axis. Corresponding properties (such as conductivity and density) will be denoted by the appropriate

subscript. Consider the energy equation for the trash written as:

ρiCi

∂Ti

∂t = ∂∂x 

ki

∂Ti

∂x 

(1)

where i = 1, 2. In this 1-D analysis, appropriately defined average or effective values of the thermophysical

properties are assumed to be known and will be considered as uniform in their respective regions. Statistical

	

	 
distributions of these properties are being obtained through testing at NASA ARC and NASA JSC and may

be incorporated in the model at a later date. No forced or buoyant convective effects are included in the

model. It is assumed that all vapor produced escapes from the trash without blockage, and the chamber

pressure is maintained to be constant.

It may be noted that Eq. (1) ignores heat transfer between the trash and the vapor phase. Experiments

on the ground have shown that the vapor production rate is about 0.03 g/sec from a trash mass of about

500 g. If there is no appreciable build-up of vapor in the waste chamber then the mass of the trash is much

greater than that of the vapor present, justifying the neglect of the trash-vapor heat transfer in the energy

equation for the trash. The same reasoning indicates that the vapor will attain the temperature of the trash

that it passes through prior to leaving the waste chamber. An alternative point of view is that T in Eq.

(1) represents a local temperature in the trash regarded as a porous medium, assuming that the various

constituents of the porous trash (i.e., plastic, water, water-vapor, etc.) are locally in thermal equilibrium

and are therefore at the same temperature.

The initial and boundary conditions are specified as

Initial Condition:

T = T(x, t = 0) for all x

Boundary Conditions:

T = T(x = 0, t) i.e., the wall temperature at x = 0

T = T(x = H, t) i.e., the wall temperature at x = H

Interface conditions are also required at the evaporation front x = δ(t). The first condition is continuity

of temperature between Regions 1 and 2 which is expressed as

Ti = Ts (2)

where Ts is the vaporization temperature (K) at the prevailing chamber pressure. The second condition

balances the heat fluxes from/to Regions 1 and 2 at the interface.

−k1

∂T1

∂x = −k2

∂T2

∂x + ρwLˆ dδdt (3)

Note that in general there will be two evaporation fronts, one emanating from each heated end-wall (i.e.,

x = 0, H).

When solving the above system of equations numerically the interface condition is often replaced by

a temperature dependent specific heat that accounts for latent heat of vaporization in the vicinity of the

evaporation temperature. The description of the specific heat follows Muhieddine et al1 and is given by

C = C2 + (C1 − C2)φ + Lˆ dφ

dT (4)

where φ = U(T − Ts) is the vapor quality and U is the Heaviside unit-step function (whose value is zero for

T <Ts and unity otherwise), and its derivative, the Dirac delta function is approximated by

dφ

dT = σ

√π exp 

−σ2(T − Ts)

2 (5)

where σ = √

1

2ΔT and ΔT is a small temperature interval around Ts.

The numerical solution approach will be most useful when detailed properties of the trash matrix and

spatial distribution of water in the trash matrix are known. For example, if the trash is arranged in layers with

varying water composition the numerical model will then be able to account for these variations without

employing the assumption of homogeneous properties. Similarly, model predictions for a few cases with

different assumed spatial distributions of water in the trash matrix are given below. The numerical approach

is also useful when side-wall temperatures are varied with time throughout the evaporation process (possibly

for improved control). For example, in a practical case, heater cycling may be employed with a PID control

system. The numerical approach can incorporate different time-dependent temperature profiles to simulate

these conditions.

	

	 
It is also useful to obtain analytical solutions of the evaporation process to identify parameters of importance and obtain mathematical expressions of their influence on the evaporation time. This requires some

further simplification but the assumptions made do not take away from the general application of the solution. The primary assumption made will be that the vaporization process is quasi-steady. This assumption

is justified when the motion of the vaporization front is so slow that temperatures in both the wet and

dry trash regions have essentially adjusted to the location of the evaporation front before any appreciable

evaporation has taken place at that location. This assumption is often utilized in the literature and may

be formally justified by comparing the time-scales of the processes. This assumption allows Eq. (1) to be

simplified to a steady diffusion equation given by

d2Ti

dx2 = 0 (6)

A second assumption made is that the wall temperatures at x = 0 and x = H are constant and equal,

i.e.,

T(t, x = 0) = T(t, x = H) = Tw (7)

The constant end-wall temperature assumption means that the initial heating of the end-walls to the

desired process temperature is ignored. This assumption holds formally when the wall heat-up period is

small compared to the overall evaporation time and is consistent with the quasi-steady approximation. The

assumption of equal wall temperatures at the two ends is primarily for algebraic convenience as it allows for

the application of symmetry at x = H/2 so that only one evaporation front needs to be tracked. Thus, the

boundary condition at x = H is replaced by the following symmetry condition.

dT2

dx = 0 at x = H/2 (8)

The numerical and analytical solutions are discussed in Section III.

2. Case 2: Quasi-1-D Model with Side-Wall Heating

Case 1 accounts for heating at the end-walls at x = 0 and x = H. In general, however, the side walls of the

waste chamber will also be heated. This heating is accounted for in Case 2 in a quasi-1-D fashion. This is

done by considering the energy equation for the cross-sectional average of the trash temperature. Formally,

this is done by area integration of the 2-D or 3-D energy equation over the directions orthogonal to the x-axis

and averaging by dividing the result by the cross-sectional area. Under the same assumptions as in Case 1,

the result for an axisymmetric quasi-steady situation is

∂2Ti

∂x2 +

2

R

∂Ti

∂r 

r=R

= 0 (9)

where Ti is now understood to be the cross-section-averaged temperature. The second term in Eq. (9) is

related to the heat flux at the side-wall. Consider the case where the side-wall temperature is held constant

as in the experiments and assume that this temperature is the same as the end-wall temperature Tw. Then

the following approximation may be used

∂Ti

∂r 

r=R

≈ Tw − Ti

R (10)

The same end boundary and interface conditions as in Case 1 apply to Case 2 as well. The solution for Case

2 is also discussed in Section III.

III. Results and Discussion

A. Case 1: Numerical solution

An example solution set for a Case 1 problem is next discussed. The solution presented here was coded

in MATLAB. It may be convenient to code more complicated scenarios with two- and three-dimensional

dependencies in a multi-physics solver such as COMSOL in the future.

	

	 
In the example simulation, the two end heating surfaces are raised to their final temperature (393 K)

over a period of approximately 500 seconds as shown in Fig. 3. These surfaces and the trash complex are

initially at 293 K. Calculated temperature profiles in the trash complex at various times are shown in Fig.

4. The profiles are symmetric about x = H/2 because of the assumed symmetry of end-wall conditions. For

the results shown in Fig 4., we have used σ = 1/

√2 K−1 and assumed the thermophysical properties of wet

and dry trash to be the same. The grid spacing in the x direction in the numerical solution is 0.1 mm and

is sufficient to resolve the liquid-vapor interface. The following suggested values for the various parameters

corresponding to tests conducted under 1-g conditions are used (from Pace et al2, Pace3 and Hong4):

C 1200 J/kg

k1 0.2 W/(m-K)

ρw 1000 kg/m3

 0.182 (which corresponds to a chamber water mass of 150 g)

t 0.425 (which corresponds to a chamber trash mass of 350 g)

Lˆ 2357620 J/kg

H 0.0254 m

R 0.1016 m

Tw 393 K

Ts 333 K (which corresponds to a chamber pressure of 0.2 atm)

Figure 3. Assumed temperature-time profile of the heated walls.

	

	 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 280

300

320

340

360

380

400

x/H

Temperature, K

0 s

250 s

500 s

750 s

1000 s

Figure 4. Calculated transient temperature profiles in the trash complex at various times.

0 10 20 30 40 50 60 70 80 90 0

20

40

60

80

100

Time, Minutes

Percent Water Recovered, %

Numerical Model

Analytical Model

Figure 5. Comparison of numerical and analytical model predictions of water recovered with time.

	

	 
Figure 6. Experimental water recovered with time for horizontal and vertical configurations (from Pace et

al1.)

During the transient phase of heating, evaporation commences as soon as the temperature of the endwalls exceeds the vaporization temperature (assumed to be 333 K, i.e., 60 C) at the operational chamber

pressure (assumed to be 0.2 atm). The progress of the vaporization front may be tracked by following the

kink in the temperature profiles. Note that the temperature profiles become flat in the wet trash once the

transient heating of the end-walls is complete. Thus, there is a linear temperature variation from the wall

temperature to the vaporization temperature at the vaporization front in the dry trash region and a constant

temperature equal to the vaporization temperature in the wet trash region. It will be shown below that this

is consistent with the analytical model.

B. Case 1: Analytical solution

The analytical solutions are as follows:

Region 1 (Dry trash: 0 <x<δ)

T1 = Tw − (Tw − Ts)

δ x (11)

Thus, in the dry trash region, the temperature decreases linearly from the hot-wall temperature Tw to the

temperature Ts at the evaporation front.

Region 2 (Wet trash, δ < x < H/2)

T2 = Ts (12)

Thus, in the wet trash region, the temperature is constant at the temperature Ts pending the arrival of the

evaporation front. Note that in the quasi-steady approximation, the sensible heating of the wet trash from

its initial temperature to the saturation temperature is accomplished much earlier than the time of arrival

of the evaporation front.

The solutions for Regions 1 and 2 as shown above may be compared to the numerically computed

temperature profile trends in Fig. 4 which shows the same behavior once the transient heating of the walls

to their final temperatures is complete.

	

	 
The interface evaporation condition (Eq. (3)) yields the following differential equation for the evaporation

front location δ:

ρwLˆ dδ

dt = k1

(Tw − Ts)

δ (13)

Integration of the above equation provides δ as a function of time

δ =



2k1(Tw − Ts)t

ρwLˆ (14)

Therefore, the location of the evaporation front is proportional to the square root of time.

Under the assumptions of the model, the cumulative mass fraction of water, M, evaporated from the

trash (relative to the total initial mass of water in the trash) is twice that evaporated from the half region

0 < x < H/2 and is given by the expression

M = 4

H



k1(Tw − Ts)t

2ρwLˆ (15)

The calculated time variation of the cumulative water out from the numerical model is compared with the

analytical model result in Fig. 5. Except for a small shift in time which accounts for the heat-up of the

end-walls in the numerical model both cases are similar and exhibit the square-root dependence on time.

This finding gives confidence that the evaporation rate calculated by the analytical model provides the right

dependencies on the trash parameters. The overall time for evaporation, tdry, is obtained by setting δ = H/2.

This yields

tdry = ρwLHˆ 2

8k1(Tw − Ts) (16)

Noting that the mass of water in the trash is Mw = πρwR2H, we may write

tdry = MwLHˆ

8πR2k1(Tw − Ts) (17)

Equations (16) and (17) for the drying time are the primary result for the 1-D analysis described in this

paper. With the values for the trash complex properties given earlier, the time for evaporation is calculated

from Eq. (16) as tdry = 2852 s. While this time is very comparable to experiments conducted in both

horizontal and vertical orientations of the HMC (see Fig. 6), the behavior of the cumulative water output

curve corresponds better with the horizontal orientation case. The horizontal test result does appear to

exhibit a behavior similar to a square-root time dependence; however, the vertical test result appears to be

more piece-wise linear in time. It has been surmised that in the vertical case liquid water may have been

discharged into the annulus between the waste chamber wall and the outside wall and evaporated there via

boiling. However, further investigation of this problem would likely require some repeat testing.

It is instructive to consider the factors that influence the drying time according to Eq. (16) or (17).

When the volume fraction  is specified, the drying time is proportional to  and to the square of the axial

height, H, of the compressed trash. Reducing H to the extent possible, i.e., without impeding the flow of

vapor from the trash complex, will be beneficial to reducing the drying time. When the mass of water in

the trash is specified, the drying time is linearly proportional to Mw and H. The drying time is inversely

proportional to the thermal conductivity of the dry trash, k1, and the difference between the wall and

vaporization temperatures. While the wall and vaporization temperatures are controllable, the value of the

thermal conductivity needs to be experimentally determined. Knowledge of the statistical distribution of k

will be useful for making further predictions. It is clear from Eq. (16) that a given uncertainty in k will

translate to a corresponding uncertainty in the drying time. Recent work at JSC to measure the thermal

conductivity of trash after melting and resolidification indicates values for conductivity for the dry trash in

the range 0.11 to 0.15 W/m-K at T = 20◦C and 0.14 to 0.165 W/m-K at T = 50◦C for various samples

(Hong4). We have used k1 = 0.2 W/m-K in our calculations. For example, a 20 percent lower conductivity

(i.e., 0.16 W/m-K vs 0.2 W/m-K as utilized above) yields a 25 percent higher drying time (i.e., 3565 s vs

2852 s) with all other parameters the same.

It appears, given the agreement between model and experimental results, that the neglect of convective

effects in the model is acceptable for predicting the experimental results. It will be useful to confirm this with

	

	 

additional model comparisons with experimental results as they become available. If the model is validated,

an important conclusion may be made that if the trash complex properties are not significantly different in

microgravity then the drying time in microgravity will be of the same order as in 1-g on the ground.

C. Case 1: Effect of non-uniform water distribution

The numerical and analytical results presented above assume that , the volume fraction of water in the

trash is initially spatially uniform. In reality, this assumption may not be valid. Each batch of trash loaded

into the HMC is likely to have variations in both the amount of water it contains and also how the water

is distributed within the trash, depending on how the individual water-containing components of the trash

accumulated.

Consider, for example, a sinusoidally varying initial water volume fraction given by the expression (x) =

1

2 π ̄ sin πxH . The water distribution is zero at the two ends of the trash (x = 0, π) and peaks in the middle.

 ̄ is the mean water volume fraction in the trash. The time-evolution of the location of the liquid/vapor

interface may be determined by using (x) given above in Eq (13). Defining  ̄δ = δ

H and t

 ̄ = t

τ where

τ = ρ ̄ wLHˆ 2

2k1(Tw−Ts) , the equation for  ̄δ may be written as

 ̄δ

d ̄δ

dt

 ̄ = 1

π sin(π ̄δ) (18)

with the initial condition  ̄δ = 0 at t

 ̄= 0. The drying time is determined as the time when  ̄δ = 1

2 . Numerical

calculations show that the scaled drying time is t

 ̄dry = 0.318, and the physical drying time is tdry = 3631

s for parameter values as in the case of uniform water distribution (for which the drying time is 2852 s).

We see that the drying time is longer for the sinusoidal water distribution than that for a uniform water

distribution. The reason may be explained as follows. In the sinusoidal case, there is more water that is

initially present away from the heaters than near the heater surfaces. Since the heat flux from the heater is

equal to k1(Tw−Ts)

δ , we see that the heat flux is reduced as δ (and consequently the distance from the heater)

increases. Thus, for a sinusoidal water distribution, more heat is required to vaporize the water precisely

when the heat flux is low. As a result, the time taken to evaporate the water increases.

The explanation described above enables us to determine the minimum and maximum drying time for a

given amount of water in the trash, when the water volume fraction distribution is unknown, but restricted to

be one-dimensional. The minimum drying time would occur when all the water is contiguous and is located

closest to the heater. Since our analysis is based on two heaters at either ends of the trash, the minimum

drying time would occur when equal amounts of water ( 1

2 πR2H) occupy a layer of thickness H2(1−t) next

to the heaters, where t is the trash volume fraction (assumed to be spatially constant). The time evolution

of the location of the liquid/vapor interface is as described in Eq (13). The interface is at x = 0 at t = 0

and moves to x = H

2(1−t) at the time the trash is totally dry. The drying time can be calculated to be

tdry,min = 2H2ρwLˆ

8k1(1 − t)(Tw − Ts) = MwLHˆ8πR2k1(1 − t)(Tw − Ts) = 1 − t

tdry (19)

Thus the minimum drying time is smaller than the drying time when the water distribution is uniform (Eq

16) by a factor of 

1−t .

The maximum drying time can be similarly determined. It occurs when water is present as a contiguous

layer of thickness H midway between the heaters. The maximum drying time can be obtained to be

tdry,max = (2 −  − 2t)H2ρwLˆ

8k1(1 − t)(Tw − Ts) = (2 −  − 2t)MwLHˆ8πR2k1(1 − t)(Tw − Ts) =

2 −  − 2t





tdry,min (20)

For the values of the parameters used to calculate the drying time for a uniform water distribution (for

which tdry = 2852 s), the minimum and maximum drying times are obtained to be 914 s and 4857 s,

respectively. The minimum drying time is somewhat larger than the initial heat-up time to establish quasisteady temperature profiles in the trash. The ratio of maximum to minimum drying time is 2−−2t

 = 5.3

for  = 0.182 and t = 0.425. For low water content, this ratio can be quite large. Note that for the above

estimates (especially for the maximum drying time) we are restricted to a strictly 1-D distribution of water,

because the theory we have developed is only applicable in this situation. In reality, the water distribution in

	

	 
the trash will be three-dimensional, and the maximum drying time will likely be different than that predicted

above.

Since a sphere encloses a maximum volume of water to be evaporated with minimum surface area available

for heating, we anticipate that a spherical wet region in the trash of an appropriate size might yield the

maximum drying time. The drying time for a spherical wet region with an initial diameter D0 that is

embedded in the trash may be determined as follows. Assume that the trash is heated uniformly to a

temperature Tw by the heaters. To easily calculate the rate of heat transfer to the sphere, we assume that

the trash is of infinite extent. Under the quasi-steady assumption, the spherical wet region will be at a

uniform temperature Ts, and the temperature profile in the vicinity of the wet region in conductive heat

transfer is spherically symmetric and is given by

T(r) = Tw − (Tw − Ts)

D

2r (21)

where r is the radial coordinate from the center of the sphere and D is the instantaneous diameter of the

wet region. Noting that the volume fraction of water in the wet region is 1 − t, the instantaneous diameter

can be calculated by an overall energy balance for the sphere

−πD2k1

∂T

∂r 

r= D

2

= d

dt 

πD3

6 (1 − t)ρwLˆ



(22)

D2 = D2

0 − 8k1(Tw − Ts)t

(1 − t)ρwLˆ (23)

where D0 is the initial diameter of the wet region. The drying time is given by

tdry,sphere = (1 − t)ρwLDˆ 2

0

8k1(Tw − Ts) (24)

Since the diameter of the largest spherical wet region must equal the trash height H, the maximum drying

time is

tdry,max,sphere = (1 − t)ρwLHˆ 2

8k1(Tw − Ts) (25)

and is 9110.4 s for t = 0.425, which is nearly twice as large as the maximum drying time form the 1-D

theory.

D. Case 2: Effect of side-wall heating

For ease of comparison with Case 1 results as well as for algebraic simplicity, the properties in the wet and

dry trash regions will be assumed to be the same. Then, the analytical solutions are as follows:

Region 1 (Dry trash,0 <x<δ)

T1 = Tw − (Tw − Ts)

sinh(ax)

sinh(aδ) (26)

where a = √2/R.

Region 2 (Wet trash, δ < x < H/2)

T2 = Tw − (Tw − Ts)

cosh a

 H

2 − x



cosh a

 H

2 − δ

 (27)

The equation for δ is given by

	 δ

0

sinh(aδ) cosh (a (H/2 − δ)) dδ = k1a(Tw − Ts) cosh 

a H

2



ρwLˆ t (28)

Noting that the aspect ratio H

2R is small (i.e. of order 0.1 for cases of interest) compared to unity allows much

simplification. Carrying terms up to order  H

2R

2 yields the following result for δ which may be compared

to the result for Case 1 (see Eq. 14)

δ =




1 +

H2

4R2



2k1(Tw − Ts)t

ρwLˆ (29)

	

	 	
In fact, dividing Eq. (29) by Eq. (14) yields the ratio of evaporation front location with and without side-wall

heating to be 

1 + H2

4R2 . Thus, side-wall heating only provides a small benefit when the aspect ratio of the

compacted trash is small. In general, for all aspect ratios, side-wall heating increases the energy input to the

trash and leads to faster propagation of the evaporation front and decreased drying time compared to results

without side-wall heating. If the aspect ratio is not small, Eq. (29) will not be valid and the calculations

will need to be modified. Note that for any aspect ratio, if the side wall is well insulated, the 1-D analysis

of Case 1 holds, and the drying time can be calculated from Eq. (16). Note also that the side-wall heating

does not affect the square root dependence on time for small aspect ratio geometries as shown by Eq. (29).

IV. Conclusion

We have developed a model for the drying of wet trash in a heat melt compactor neglecting the effects

of convective heat transfer. The model predictions and trends are in reasonable agreement with results from

tests conducted in normal gravity. The model is intended, however, to predict the drying time in reduced

gravity. In the absence of convective effects, the model results do not explicitly depend upon the gravity level.

Gravity effects can appear implicitly due to differences in trash properties, chiefly controlled by variations in

the water spatial distribution within the trash. When the movement of the vaporization front in the trash

is slow compared with thermal conduction time scales, explicit expressions have been derived that relate

the water recovery rate and the drying time to the geometric and process operating parameters. The onedimensional model predictions show that the drying time is chiefly controlled by the thermal conductivity

of the dry trash, which would be independent of the gravity level, and the distribution of the water volume

fraction, which may be highly g-level dependent. Variations in the water distribution, which likely depend

on how the trash is mixed and loaded into the HMC, can result in large differences in the drying time,

especially for low water content. The model demonstrates that loading the wetter trash closer to the heater

ends of HMC and having relatively drier trash in the central portion would help reducing the drying time.

Sidewall-heating of the trash is shown to have only a small benefit when the length to diameter ratio of the

compacted trash is a small quantity. These results are expected to be generalized to 2-D and 3-D situations

as well.

References

1Muhieddine, M., Canot, E., and March, R., Various Approaches for Solving Problems in Heat Conduction with Phase

Change, International Journal of Finite Volume Method, Vol. 6, No. 1, 2008. 2Pace, G. S., Fisher, J., Delzeit, L., Alba, R., and Polonsky, A., Development of a Plastic Melt Waste Compactor for

Human Exploration Missions A Progress Report. AIAA 2010-6010, American Institute of Aeronautics and Astronautics, 2010. 3Pace, G., Private Communication: HMC Waste Model Thermal Properties Test 8-27-2008 new cotton and air.xls. 4Hong, A., Private Communication: Heat Melt Compactor Thermal Conductivity Update, April 2, 2012.