There are four main power draws during operation of the WAVAR device: the microwave heating unit, the actuators to seal the regeneration chamber and rotate the bed, the computer control system and sensors, and the fan. The desorption process was outlined in Section 4, though no efficiency factors (such as microwave efficiency and extraction efficiency) were included in that analysis. The mechanisms to rotate the adsorption bed and seal the regeneration chamber require much less power than the microwave itself, and since the two are never running concurrently, the microwave heating requirements dominate. The power requirements for the computer and sensors can be considered constant and minimal (~5 Watts). The largest power draw for the system is that of the fan, and determining its power requires examination of all the pressure drops throughout the system.
Identification of the sources of pressure loss through the system is simple. A quick look at Figure 20 shows the two main sources of pressure drop, D P: the filter and the adsorption bed. A third source, the ductwork, is comparatively negligible.12 All sources of D P are functions of the flow velocity.
Filter D
P
Filter pressure drop
is proportional to the flow velocity and dependent on the type of filter
medium. For the pressure drop calculations, a Filtrete Type G filter from
3M was chosen.34 Filtrete
is an electrostatically enhanced non-woven fiber and is available in numerous
grades, each having different filtration efficiencies and associated pressure
drop. For WAVAR applications on Mars, a Filtrete G-200 will provide 95%
efficiency, or greater, over the applicable velocity range. Based on pressure
drop data listed in Ref. , which was then dimensionalized by density, a
linear pressure drop correlation was determined as:
(3)
This relation will give pressure drop in Pascals provided fluid density r and fluid velocity V are in SI units. Filtrete has been reported to have a longer life and greater temperature stability compared to similar media34 and should be acceptable for the ambient conditions that WAVAR will see on Mars.
Bed D
P
There are a number
of ways to estimate pressure drop across a packed bed. If the flow through
the bed can be considered purely laminar, the Poiseuille flow equation
can be used:35
(4)
where L is the bed depth, m is the viscosity, V is the flow velocity, and D is the pellet diameter. However, a laminar flow assumption is not always valid, especially for tightly packed beds with small pellet diameters. Ergun added a term to the Poiseuille equation to account for turbulent flow.36
(5)
where e is the void fraction, r is the fluid density, f is the Ergun friction factor given below, L is the bed depth, m is the viscosity, V is the flow velocity, and D is the pellet diameter. The Ergun friction factor is
(6)
where ReP is the Reynolds number based on bed particle diameter and superficial fluid velocity (flow speed without bed).
The first term of Ergun’s friction factor represents laminar flow while the second represents turbulent flow. The coefficients 150 and 1.75 were determined experimentally by Ergun.36,37 The Ergun equation is very sensitive to small changes in void fraction e , requiring that a very accurate value for bed voidage be available. If no such measurement exists, then another correlation should be used. For LTA zeolites of nominal pellet size 3.25 mm. the typical void fraction is between 0.34 and 0.32.37 However, within this range, the Ergun equation appears to overestimate the pressure drop when compared to other correlations.
A more recent correlation attributed to Gupta and Thodos makes use of the Chilton-Colburn J Factor Analogy.38,39 They recommended a correlation for gas flow in a packed bed of spheres in the form
(7)
where J is the Colburn J Factor, e is the void fraction, and ReD is the Reynolds number based on pellet diameter and upstream velocity. To develop a pressure drop relation from this, the Chilton-Colburn J Factor Analogy for mass transfer is used38,39
(8)
where Ff is the Fanning friction factor (often simply referred to as the friction coefficient). Then the pressure drop equation for laminar flow is used35
(9)
with the final correlation becoming
(10)
with r , V, L, e , D, and ReD defined as above. This relationship is effective for Reynolds numbers between 90 and 4000.
A correlation that is valid for Reynolds numbers less than 40 begins with the same pressure drop relation as Ergun with a simpler friction factor attributed to Chilton and Colburn.37
(11)
This expression, while attractive in its simplicity, would underestimate pressure drop drastically for higher ReD. The operating conditions on Mars generally result in values for ReD of between 20 and 120. For higher ReD, Chilton and Colburn derived the following model for pressure drop37
(12)
Figure 21 shows a comparison of several of the pressure drop models for a void fraction of 0.33. The two Chilton-Colburn models above (for ReD<40 and ReD>40) are labeled as CC<40 in CC>40, while the Colburn-J model is shown labeled as ColJ. The Chilton-Colburn models seem to agree well with the Colburn-J model for the assumed void fraction and the plotted range of ReD, while the Ergun correlation is much higher than the other models. For higher void fractions (>0.4), the Ergun correlation more closely matches the other models.
The two correlations with close agreement (Chilton-Colburn and Colburn-J) both have ReD raised to a power. However, a simpler linear approximation is desired to enable the combination of the filter and bed pressure drop correlations into a single expression for power as a function of flow velocity. Such an expression can be derived by increasing the constant term in the low ReD Chilton-Colburn correlation. If the model is taken as
(13)
the simplicity of a linear model can be retained for use with the assumed void fraction (e = 0.33) as an acceptable (and conservative) correlation within the desired flow regime. This final model is plotted in Figure 21 and will be considered valid only for ReD less than 120.
All the models detailed here have their roots in experimental measurements, but most of the research with packed bed flow is with atmospheric pressures or higher, and at ambient temperatures. While it is risky to apply any correlation outside of the range in which it was developed, there is little choice until low-pressure, low-temperature packed bed pressure drop is experimentally investigated and a more accurate empirical correlation derived.
Power Calculation
With the expressions
for the D
P introduced by the filter and the bed, a calculation of the power requirement
for the fan is possible. For the performance calculations presented here,
a four-bladed propeller (NACA 5868-9, Clark-Y section) was used to model
the fan blades. A fan of this type with a 35°
blade will yield an efficiency of 85% with an advance ratio Jadv
of 1.5.40
The dimensionless advance ratio Jadv is defined as
(14)
where V is the flow velocity, n is the number of revolutions per unit time period, and D is the diameter of the propeller. To maintain this advance ratio and thereby the maximum propeller efficiency, the ratio of V to nD must remain constant. The speed of the fan motor would therefore be continuously adjusted to maximize efficiency for the required velocity.
The power requirement for the fan can be found using classical momentum theory in propeller analysis. The power required by the propeller equals the product of the thrust produced (flow area times pressure differential) and the velocity through the propeller (V3 = V2) divided by efficiency.40 Thus
(15)
The power required for the fan is then dependent on the total D P, and the pressure drop for both the filter and the adsorption bed are functions of velocity, as discussed above. Equation 15 can now be rewritten as
(16)
With an expression for the fan power incorporating the two D P sources, the total system power requirements can be modeled and performance calculations made.
Temperature Effects
Temperature across
the filter is considered constant (T2 = T1
= Tamb). The temperature across the fan increases from
the basic adiabatic energy balance:41
(17)
where Cp is the specific heat of the atmosphere and D W is the specific energy of the flow (kJ/kg). As mentioned above, the velocity across the fan is considered constant40 (V3 = V2), so Eqn. 17 becomes:
(18)
The exothermic adsorption
process, and the cooling of bed sectors after they leave the regeneration
chamber, heat the temperature at the outlet (T4).
Because of the low mass flow rates of H2O,
the adsorption process does not heat the flow to any appreciable extent
(less than 0.5 K for frost point temperatures of 200 K). Cooling of the
just-regenerated sectors will result in heating of the flow immediately
aft of the regeneration chamber, but that flow quickly leaves the system
and will not adversely affect the performance of the WAVAR.
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