Frost Formation on an Airfoil: A Mathematical Model I

82 Pages·2000·4.49 MB·English

https://ntrs.nasa.gov/search.jsp?R=19790014535 2019-03-31T02:38:30+00:00Z
NASA
CR
3129
C.1
NASA Contractor Report 3129
Frost Formation on an Airfoil:
A Mathematical Model I
Mark Dietenberger, Prem Kumar, and James Luers
CONTRACT NASS-3 1294
APRIL 1979
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NASA Contractor Report 3129
Frost Formation on an Airfoil:
A Mathematical Model I
Mark Dietenberger, Prem Kumar, and James Luers
University of Dayton Research hstitute
Dayton, Ohio
Prepared for
Marshall Space Flight Center
under Contract NASS-3 1294
NASA
National Aeronautics
and Space Administration
Scientific and Techoical
Information Office
1979

TABLE OF CONTENTS
SECTION PAGE
1 INTRODUCTION l-1
2 DERIVATION OF THE FROST THERMAL CONDUCTIVITY 2-l
2.1 THE HEAT TRANSFER PROCESSES WITHIN THE FROST LAYER 2-l
2.1.1 Air - Ice Thermal Conductivity, k 2-4
2.1.2 Radiation Effective Conductivity,ek 2-6
2.1.3 Water Vapor Effective Conductivity,rk 2-7
2.1.4 Forced-Air Enthalpy Rate Term, GaCp a?! 2-13
ax
2.2 APPROACHES FOR CALCULATING K 2-15
2.2.1 Brian, et al. Approach 2-15
2.2.2 White's Approach 2-16
2.2.3 Biguria and Wenzel's Approach 2-17
2.2.4 Jones and Parker's Approach 2-18
2.2.5 Summary of the Four Approaches 2-19
2.2.6 The UDRI Approach 2-23
3 HEAT AND MASS TRANSFER COEFFICIENTS 3-l
3.1 LAMINAR NATURAL CONVECTION 3-4
3.2 TURBULENT FORCED CONVECTION 3-5
3.3 SUMMARYO F HEAT AND -MASS TRANSFER COEFFICIENT
EQUATIONS 3-9
4 THE SIMULATION OF FROST FORMATION 4-l
4.1 FROST FORMATION MODEL 4-l
4.2 THE NUMERICAL SCHEME FOR THE FROST FORMATION MODEL 4-3
5 COMPARISON OF THE MODEL WITH THE AVAILABLE EXPERIMENTAL
DATA 5-l
6 SUMMARYA ND RECOMMENDATIONS 6-l
7 REFERENCES 7-l
. . .
111
LIST OF FIGURES
FIGURE NO. PAGE
1 Schematic Diagram of the Heat Transfer
Processes in the Frost Layer 1-3
2 Water Mass Flux Versus Distance in Frost Layer
(Brian, et al. data, Reference 1) 2-9
3 Water Mass Flux Versus Distance in Frost Layer
(Yamakawa, et al. data, Reference 13) 2-10
4 Thermal Conductivity Versus Temperature at
Ice Density 2-21
5 Thermal Conductivity Versus Temperature at
Air Density 2-22
6 Thermal Conductivity Versus Frost Porosity
at 211 Ok 2-24
7 Frost Structure Model of the Present Work 2-25
a Comparison of the Present Frost Thermal
Conductivity Model with Experimental Data
of Brian, et al. (Reference 1) 2-28
9 Relationship Between Nusselt's Number and
Reynolds' Number (Reference 13) 3-7
10 Relationship Between Local Heat and Mass
Transfer Coefficient (Reference 13) 3-a
11 Weight Versus Time for Brian et al. Data
(Reference 1) 5-3
12 Density Versus Time for Brian et al. Data
5-4
(Reference 1)
13 Thickness Versus Time for Brian et al. Data
5-5
(Reference 1)
14 Weight Versus Time for Yamakawa, et al. Data
(Reference 13) 5-a
15 Density Versus Time for Yamakawa, et al Data
(Reference 13) 5-9
16 Thickness Versus Time for Yamakawa, et al.
S-10
Data (Reference 13)
iv
LIST OF FIGURES (CONT.)
FIGURE NO. PAGE
17 Weight Versus Time for Nakamura Data
(Reference 14) 5-12
ia Density Versus Time for Nakamura Data
(Reference 14) 5-13
19 Thickness Versus Time for Nakamura Data
(Reference 14) 5-14
LIST OF TABLES
TABLE NO. PAGE
I Summary of Approaches to Calculating Frost
Thermal Conductivity 2-20
II Data Input to the Frost Formation Model for
Comparison With Brian et al. Data for Forced
Convection (Reference 1) 5-2
III Data Input to the Frost Formation Model'for
Comparison With Yamakawa, et al. Data for
Forced Convection in a Duct (Reference 13) 5-7
IV Data Input to the Frost Formation Model for
Comparison With Nakamura Data for Natural
Convection on Vertical Plate (Reference 14) 5-11
V
List of Symbols
b Linear dimension of ice crystals (cm)
C Specific heat of air (J/gOC)
P
C Specific heat of frost (J/gOC)
Pf
D Diffusion coefficient (cm2/s)
D Total.derivative = 3 + U 8
Dt at aax
Hydraulic diameter (cm)
De
D Effective diffusion coefficient of water vapor in
eff
the frost (mm /s)
F Blowing parameter
g Gravitational acceleration (m/s2)
Air mass flow rate per unit area (g/m2s)
Ga
Gr Grashof number
H Height of plate (m)
Heat transfer coefficient (w/m2"C)
hH
Mass transfer coefficient (g/m2s)
hm
Experimental heat transfer coefficient (w/m2"C)
hlt Experimental mass transfer coefficient (g/m2s)
i Enthalpy (per unit mass)
K Thermal conductivity of the frost (w/m"C)
k Thermal conductivity of ice/air structure in the
layer (w/m"C)
Thermal conductivity of air (w/m"(I)
ka
Thermal conductivity of air bubbles (w/m"C)
kb
Thermal conductivity of ice cylinders (w/m"C)
kC
Effective thermal conductivity of air-ice structure (w/mOC
ke
k Effective air thermal conductivity (w/m"C)
eff air
vi
Thermal conductivity of ice (w/m"C)
ki
Lower limit of thermal conductivity for air
kl
bubbles and ice cylinders (w/m"C)
k Thermal conductivity of ice planes (w/m"C)
P
Radiation thermal conductivity (w/m"C)
kr
Thermal conductivity of ice spheres (w/m"C)
kS
Upper limit of thermal conductivity for air
kU bubbles and ice cylinders (w/m"(Z)
Water vapor effective conductivity (w/m"C)
kV
L Latent heat, Le or Ls
Le Lewis number
Latent heat of water evaporation (J/g)
Le
Latent heat of ice sublimation (J/g)
LS
Water vapor mass flux within the frost layer (gh2s)
'd
Water vapor flux at the frost surface (g/m2s)
fidS
Ih Experimental value20f water mass flux through the
ew
frost surface (g/m s)
Total water mass flux in the frost layer (g/m2s)
%
Nu Nusselt number for forced convection
Nusselt number for natural convection based on
N"H
plate height
N”Z Local Nusselt number for natural convection
P Pressure (N/m2)
Pr Prandtl number
Total pressure (N/m21
Pt
Water vapor pressure (N/m2 1
pV
Referenced pressure (N/m2 1
Internal heat generated within the frost layer (w/m31
vii
Constant heat flux at the wall (w/m2)
g0
Radiation heat flux vector .(w/m2)
qr
Re Reynolds number based on hydraulic diameter
Water vapor gas constant = 4.6150 X lo6 erg/C2g
RV
Schmidt number
SC
The Sherwood number for natural convection based
ShH
on height of plate
St Stanton number
T Temperature (OKI
t Time (s)
Ambient air temperature (OKI
Ta
Frost surface temperature ( OK)
TS
Wall temperature CO K)
TW
T* Referenced temperature (OK)
Air velocity (m/s)
'a
X Distance from the wall .(m)
Frost thickness (cm)
X
S
a k
eff aiJki
B Porosity of frost
Proportion of frost volume representing ice
% spheres and ice planes
E Emissivity of frost
0 Fractional volume of ice fragments
Proportion of frost volume representing ice
% cylinders and air bubbles
Air density (g/cc)
pa
Mass density of frost (g/cc)
Pf
Water vapor density (g/cc)
PV
Viii
I
P Density of water vapor at frost surface (g/cc)
vs
-a
Stefan-Boltzmann constant = 0.56697 X 10 w/m2/OC4
CJ
Tortuosity
-r
S
V Kinematic viscosity
Relative humidity in ambient air
@a
X Relative concentration (moles H20/moles air)
w absolute ambient humidity
a
frost surface absolute humidity
Y3
w absolute saturated humidity at wall temperature
W
I