Visualization of the Key Air Properties Influencing Derived Functions

Visualization of the Key Air Properties Influencing Derived Functions

1Shweta Singhal (Agrawal) Assistant Professor, Mechanical Engineering Department

MVJ College of Engineering Bangalore, India

2Sayooj N

Module Lead Continental Automotive India

Bangalore, India

3Sandeep J

Thermal Analyst

GKN Aerospace Engine Systems Bangalore, India

Abstract The main objective of this paper is to suggest the best correlation between the temperature and temperature dependent air properties. The main air properties presented in this paper are specific heat, density, dynamic viscosity, kinematic viscosity, thermal conductivity and thermal diffusivity. The reliance of these air properties with temperature is studied at atmospheric pressure. At the end, details of a MATLAB computer program are presented. This computer code would be useful for the students in further detailed studies and hand calculations.

KeywordsTemperature, air properties, specific heat, density, viscosity, thermal conductivity, diffusivity

NOMENCLATURE

Cp= specific heat in kJ/kg-K T = temperature in K

= density in kg/m3

µ = dynamic viscosity in N-s/m2

k = thermal conductivity in W/m-K = kinematic viscosity in m2/s

= thermal diffusivity in m2/s

1. INTRODUCTION

   This section provides a brief survey about the relevant literature and preliminary considerations of various thermodynamic properties of air. The first author Donald W. Mueller, Jr., Hosni I. Abu-Mulaweh presented a study on isentropic compression of a gas with temperature-dependent specic heat capacities [1]. The values of coefficients to solve thermodynamic properties are taken from this paper. The work by Sanford Gordan, Bonnie J. McBride [3] has also given the polynomial equation to solve specific heat, enthalpy and entropy which are dependent on temperature values. Density relation with respect to temperature is given by F J McQuillan, J R Culham, MMYovanovichin their paper [11]. The most satisfactory law of variation of viscosity of a gas with temperature was proposed by W. Sutherland [7]. R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, in their paper [12] have considered an example of molecular momentum transport theory and introduced "Newton's law of viscosity" along with the definition of kinematic viscosity. Cannon John Rozier in his paper [13] had introduced the term heat diffusion which describes the heat distribution in a given region which may be further used with Fouriers law to determine the heat flux. A reliable experimental equation for thermal conductivity covering a wide range of temperature is presented by K Kodaya, N Matsunaga and A Nagashima [6].

2. GOVERNING EQUATIONS (THEORY, EXPERIMENT, EMPERICAL)

   1. Specific heat

      The polynomial governing equation for specific heat as a function of temperature [1] is presented as,

      Cp = R(a1T-2 + a2T-1 + a3+ a4T+ a5T2+ a6T3+ a7T4)

      The coefficients value are given with the attached MATLAB program.

   2. Density

      Density of air can be found using the relation, = P/RT which requires the values of pressure, gas constant along with the temperature. In the relation given below, at atmospheric pressure, density can be found with the help of only temperature. This inverse relationship is given in the paper [11].

      = 351.99/T+ 344.84/T2

   3. Dynamic Viscosity

      The simple expression for a theoretical model is given below for the dynamic viscosity of a gas.

      µ/ µ0 = (T/T0)0.5((1 + C/T0)/ (1 + C/T))

      Where, µ0, T0 and C denote the reference values of the dynamic viscosity, absolute temperature, and the Sutherlands constant respectively. Using Holmans values of µ/ µ0 based on experimental measurements [8], the value of C was calculated to be 113. Based on experimental data from Barus [9], the above equation holds good for a temperature range of 273K to 1200K. Since experimental data is required to specify µ0, the Sutherlands theoretical expression takes a semi- empirical form as shown below:

      µ = a (T1.5/ (C + T))

      Where, a = 1.47 * 10-6 kg/msK0.5, calculated using µ0 at T0. The above equation is further modified by using a multiplication factor, as suggested by James J. Gottlieb, David V. Ritzel [10] such that dynamic viscosity could be calculated for the temperature range of 78K to 2500K.

      µ = a (T1.5/ (C + T))(1+ 1.53*10-4(T/C -1)2)

   4. Kinematic Viscosity

      Referring to Bird et al [12], based on the analogy of Newtons viscosity equation with heat and mass transport, kinematic viscosity, often called as momentum diffusivity, is given by:

      = µ/

   5. Thermal Conductivity

   The keystone of conduction heat transfer is Fouriers law which defines a crucial material/transport property viz.

   thermal conductivity. This property depends on the atomic conservation of energy, it is seen that the thermal diffusivity

   & molecular structure of matter.

   The trend in thermal conductivity among the different types of matter is ksolid>kliquid>kgas

   In gases, the molecular collisions increase with rise in

   is given by:

   = k/ ( Cp)

3. SENSITIVITY ANALYSIS

   temperature which indicates k increasing with temperature.

   For gases, variation of k with temperature can be presented both theoretically and experimentally.

   Theoretical: Thermal conductivity of gases depicting the effect of temperature, pressure and chemical species may be explained in terms of kinetic theory of gases [5],

   k = (cvmfp)

   where, v is mean molecular speed &mfp, the mean free path which denotes the average distance travelled by a molecule prior to a collision given by,

   mfp = kb T/(1.414d2 p)

   where, kb is Boltzmanns constant, d is the diameter of the gas molecule & p is the pressure.

   Experimental: The uncertainty over thermal conductivity of air is large compared to other thermal properties. This is quite evident from the trace of extensive amounts of experiments carried out during the mid-19thcentury [4]. In fact, the precariousness in experimental determination is larger than the theoretical way of arriving at a value of k as a function of temperature due to the difficulty exhibited in measuring it. In order to cover a wider temperature range & reliability of results, the below equation derived from the extensive study done by K Kodaya, N Matsunaga and A Nagashima [6] is used.

   k (Tr,r) = [k0 (Tr) + k (r)]

   1. Graphical representation

      Cp-DHB

      Cp-DHB

      1.200

      1.200

      Poly. (% error)

      Poly. (% error)

      0.80

      0.60

      0.80

      0.60

      1.150

      1.150

      0.40

      0.40

      1.100

      1.100

      0.20

      0.20

      Specific Heat (kJ/kg K)

      Specific Heat (kJ/kg K)

      % Error

      % Error

      The graphs showing the variation of air properties with respect to the temperatures are shown below and a comparison with the values available from the heat and mass transfer data hand book by C P Kothandaraman and S Subramanyan is done for the validation of all the correlations presented on this paper. The trend of errors is also plotted in the graphs to clearly understand the range of temperatures over which the correlations are reliable.

      1.250

      CP

      1.00

      1.250

      CP

      1.00

      1.000

      -0.40

      1.000

      -0.40

      0.950 -0.60

      0 200 400 600 800 1000 1200 1400

      Temperature (K)

      0.950 -0.60

      0 200 400 600 800 100 1200 1400

      Temperature (K)

      1.050

      1.050

      0.00

      -0.20

      0.00

      -0.20

      Fig.1. Specific heat vs Temperature

      1.40

      0.00

      1.40

      0.00

      Where,

      1.20

      -0.04

      1.20

      -0.04

      Density

      Density-DHB

      Poly. (% error)

      Density

      Density-DHB

      Poly. (% error)

      1.00

      1.00

      -0.08

      -0.08

      Density (kg/m^3)

      Density (kg/m^3)

      k0 (Tr) = C1Tr + C0.5 Tr0.5 + 4 C Ti

      5

      0.80

      0.80

      -0.12

      -0.12

      0.60

      0.60

      -0.16

      -0.16

      % Error

      % Error

      k (r) =

      D i

      i=0 i r

      Tr = T//T*

      and r= /*

      i=1 i r

      The list of constants to be used to arrive at final k values is as given in the following table 1.

      TABLE II. CONSTANTS CATERING EXPERIMENTAL EQUATION

      Constants catering experimental equation
      T* 132.5 K	D1 0.4022
      * 314.3 kg/m³	D2 0.3566
      25.9778×10-3 W/m.K	D3 -0.1631
      C1 0.2395	D4 0.1380
      C0.5 0.0064	D5 -0.0201
      C0 1.0000
      C-1 -1.9261
      C-2 2.0038
      C-3 -1.0755
      C-4 0.2294

      F. Thermal Diffusivity

      In heat transfer, the ability of any material to conduct thermal energy relative to its ability to store it is measured as thermal diffusivity. In the one dimensional heat equation derivation by Cannon [13] from Fourier's law and

      0.40

      -0.20

      0.40

      -0.20

      0.20

      -0.24

      0.20

      -0.24

      0.00 -0.28

      0 200 400 600 800 1000 1200 1400

      Temperature(K)

      0.00 -0.28

      0 200 400 600 800 1000 1200 1400

      Temperature(K)

      Fig. 2. Density vs Temperature

      55 Dynamic Viscosity

      0.00

      30

      Thermal Diffusivity (m^2/s) x10-6

      Thermal Diffusivity (m^2/s) x10-6

      Thermal Diffusivity

      2.30

      50 Dynamic Viscosity- DHB

      Dynamic Viscosity (N s/m^2) x10-6

      Dynamic Viscosity (N s/m^2) x10-6

      Poly. (% error)

      45

      -0.30

      -0.60

      25

      Thermal Diffusivity-DHB Poly. (% error)

      1.60

      0.90

      % Error

      % Error

      40 -0.90 20 0.20

      35 -1.20

      30 -1.50

      25 -1.80

      20 -2.10

      15 -2.40

      10 -2.70

      0 200 400 600 800 1000 1200 1400

      Temperature (K)

      Fig.3. Dynamic viscosity vs Temperature

      -0.50

      % Error

      % Error

      15

      -1.20

      10 -1.90

      -2.60

      5

      -3.30

      0 -4.00

      0 200 400 600 800 1000 1200 1400

      Temperature (K)

      Fig.6. Thermal Diffusivity vs Temperature

      200

      Kinematic Viscosity(N S/m^2) x10-6

      Kinematic Viscosity(N S/m^2) x10-6

      180

      160

      140

      120

      100

      80

      60

      40

      20

      0

      Kinematic Viscosity Kinematic Viscosity-DHB Poly. (% error)

      0.00

      -0.30

      -0.60

      -0.90

      % Error

      % Error

      -1.20

      -1.50

      -1.80

      -2.10

      -2.40

      -2.70

      Property	Temperature (K)	Error (%)
      Specific Heat	273 1100 Above 1100	±0.2 Up to 1.2
      Density	273 1300	-0.2 to 0.03
      Dynamic Viscosity	273 1300	-0.5 to -2.5
      Kinematic Viscosity	273 1300	-0.5 to -2.8
      Thermal Conductivity	273 700 700 1300	-0.6 to -4.1 -4 to +1.1
      Thermal Diffusivity	273 1300	-4.1 to 0.8

      Property	Temperature (K)	Error (%)
      Specific Heat	273 1100 Above 1100	±0.2 Up to 1.2
      Density	273 1300	-0.2 to 0.03
      Dynamic Viscosity	273 1300	-0.5 to -2.5
      Kinematic Viscosity	273 1300	-0.5 to -2.8
      Thermal Conductivity	273 700 700 1300	-0.6 to -4.1 -4 to +1.1
      Thermal Diffusivity	273 1300	-4.1 to 0.8

      -3.00

4. OBSERVATIONS

   The comparative studies shown above have red curve corresponding to the values obtained from heat transfer data hand book, the blue curve corresponds to the values obtained from correlation suggested in the paper and the black curve shows the trend of error with respect to change in temperature values.

   From Figure 1 it is observed that the errors are within ±0.2% for the temperature range of 273 K to 1100 K and go higher up to 1.2%beyond elevated temperatures of 1100 K. Similarly, for other properties the error limits are summarized in the table2 below.

   0 200 400 600 800 1000 1200 1400

   Temperature(K)

   Fig.4. Kinematic viscosity vs Temperature

   0.09 2.30

   Thermal Conductivity

   TABLE II. PROPERTY VS TEMPERATURE AND ERROR

   Thermal Conductivity (W/mK)

   Thermal Conductivity (W/mK)

   0.08 Thermal Conductivity- DHB Poly. (% error)

   0.07

   0.06

   0.05

   0.04

   0.03

   0.02

   0.01

   0.00

   1.60

   0.90

   0.20

   % Error

   % Error

   -0.50

   -1.20

   -1.90

   -2.60

   -3.30

   -4.00

   It is clear that the difference between values of air properties obtained using the correlations and the data hand book is negligible since the error, as shown above, is less than 5% in

   any case. The maximum error of about 4% is observed in

   0 200 400 600 800 1000 1200 1400

   Temperature (K)

   Fig.5. Thermal Conductivity vs Temperature

   the case of thermal conductivity and thermal diffusivity within a range of 650K to 750 K only.

5. CONCLUDING REMARKS

We have presented the correlations for specific heat, density, dynamic viscosity, kinematic viscosity, thermal conductivity & thermal diffusivity varying with temperature over the range of 273-1300K . Graphs have been plotted showing the variation of these air properties

with respect to temperature. For validation of results the [3] Sanford Gordan, Bonnie J. McBride, computer program for

curves have been compared with the values given in heat and mass transfer data hand book by C P Kothandaraman and S Subramanayan. The correlations suggested in this

paper are found satisfactory when compared with the data

calculation of complex chemical equilibrium composition and application, NASA reference publication 1311, 1994

1. W G Kannuluik and E H Carman, The Temperature Dependence of the Thermal Conductivity of Air, 1951

2. Theodore L.Bergman, Adrienne S. Lavine, Fundamentals of Heat

   hand book air properties. Finally,a MATLAB code for students is given to strengthen their understanding of these

and Mass Transfer.Wiley, 62-70, 2017.

K Kodaya, M Matsunaga and A Nagashima, Viscosity and Thermal

Conductivity of Dry Air in the Gaseous Phase, 1985

thermodynamic properties.

Further study is needed to accommodate the impact of varying the pressure and chemical composition of air on the correlations presented.

ACKNOWLEDGMENT

Authors of this paper would want to record their indebtedness toDepartment of Mechanical engineering, MVJ College of Engineering Bangalore, Continental India and GKN Aerospace India for providing the facilities to carry out this research.

REFERENCES

1. Donald W. Mueller, Jr., Hosni I. Abu-Mulaweh , Compression of an Ideal Gas with Temperature-Dependent Specic Heat Capacities, American Society for Engineering Education Annual Conference & exposition, 2005.

2. Frederick 0. Smetana, Howard N. Fai rchi Id I1, Glenn L. Martin, Equilibrium Concentrations of N H and its Decomposition Products Elevated Temperatures and Pressures, NASA N-73-32031, 1960

1. W. Sutherland, The viscosity of gases and molecular force,

   Philosophical Magazine Series 5, 36:223, 507-531, 1893

2. Silas W. Holman, The effect of temperature on viscosity of air and carbon dioxide, Philosophical Magazine Series 5, Vol. 21, 1886

3. C. Barus, Viscosity of gases at high temperature on viscosity and pyrometric use of the principle of viscosity, American Journal of Science 3rd Series, Vol. 35, 1888

4. James J. Gottlieb and David V. Ritzel. Barus, A semi- empirical equation for the viscosity of air, Suffield Technical Note no. 454, 1979

5. F J McQuillan, J R Culham, M MYovanovich, Properties of dry air at one atmosphere, Microelectronics Heat Transfer Lab, University of Waterloo, Ontario, June 1984

6. R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot, Transport Phenomena, 2nd Edition, ISBN: 978-0-470-11539-8, 2002

7. Cannon, John Rozier, The onedimensional heat equation, Encyclopedia of Mathematics and its Applications, 23, Reading, MA: Addison-Wesley Publishing Company, Advanced Book Program, ISBN 0-201-13522-1, 1984

Appendix A

% Input Parameters

T=[100,200,300,400,500,600,700,800,900,1000]; % Kelvin

% Specific heat Cp

% For T value less than 1000K a1 = 1.009950160e+04;

a2 = -1.968275610e+02; a3 = 5.009155110e+00; a4 = -5.761013730e-03; a5 = 1.066859930e-05; a6 = -7.940297970e-09; a7 = 2.185231910e-12;

R = 0.287; % KJ/KgK

% Enthalpy

b1 = 6.462263190E+03; b2 = -8.147411905E+00;

% Dynamic Viscosity a = 1.47*10^(-6);

C = 113;

% Thermal Conductivity C1= 0.2395;

C0 = 0.0064;

for i=1:10

Tr = T(i)/132.5; j=1;

Cp(i) = R*(a1*T(i)^(-2) + a2*T(i)^(-1) + a3 + a4*T(i) + a5*T(i)^2 + a6*T(i)^3 + a7*T(i)^4); % Specifi Heat KJ/KgK

h(i) = R*T(i)*(-a1*T(i)^(-2) + a2*T(i)^(-1)*log(T(i)) + a3 + a4*T(i)/2 + a5*T(i)^2/3 + a6*T(i)^3/4 + a7*T(i)^4/5 + b1/T(i)); % Enthalpy KJ/Kg

s(i) = R*(-a1*T(i)^-2/2 – a2*T(i)^(-1) + a3*log(T(i)) + a4*T(i) + a5*T(i)^2/2 + a6*T(i)^3/3 + a7*T(i)^4/4 + b2); %

Entropy KJ/KgK

rho(i) = 351.99/T(i)+ 344.84/T(i)^2; % Density Kg/m^3

Mu(i) = 1.47*10^(-6)*(T(i)^(1.5))/(113+T(i));

Nu(i) = Mu(i)/rho(i); % Kinematic Viscosity

C2(i) = 1*Tr^(0);

C3(i) = -1.9261*Tr^(-1);

C4(i) = 2.0038*Tr^(-2);

C5(i) = -1.0755*Tr^(-3);

C6(i) = 0.2294*Tr^(-4); D1(i) = 0.4022*rho(i)/314.3;

D2(i) = 0.3566*rho(i)/314.3^2; D3(i) = -0.1631*rho(i)/314.3^3; D4(i) = 0.138*rho(i)/314.3^4; D5(i) = -0.0201*rho(i)/314.3^5;

K0(i) = (C1*Tr) + C0*(Tr)^0.5 + C2(i)+C3(i)+C4(i)+C5(i)+C6(i);

K1(i) = D1(i)+D2(i)+D3(i)+D4(i)+D5(i); C = 0.025978;

K(i) = C*(K0(i)+K1(i)); % Thermal Conductivity

alpha(i) = K(i)/(rho(i)*Cp(i)*1000); % Thermal diffusivity

Property(i,1) = T(i); Property(i,2) = Cp(i); Property(i,3) = h(i);

Property(i,4) = s(i); Property(i,5) = rho(i); Property(i,6) = Mu(i); Property(i,7) = Nu(i); Property(i,8) = K(i); Property(i,9) = alpha(i);

end