Estimating the Effect of Moist Air on Natural Convection Heat Transfer in Electronics Cooling by Ashok Kumar, S

Estimating the Effect of Moist Air on Natural Convection Heat Transfer in Electronics Cooling By Ashok Kumar, S. Robert Bosch Engineering and Business Solutions Limited, Bangalore, India.

Introduction with reference to the dimensionless numbers which are Atmospheric air is a mixture of gases and water vapor as used to characterize the natural convection heat transfer well as a number of pollutants. The amount of water vapor phenomena. A simple relationship for the Nusselt number and pollutants vary from place to place. The concentration in terms of air properties is derived from which the ratio of of water vapor and pollutants decreases with the increase the Nusselt number for moist air to dry air is obtained. This of altitude from the sea level, and above an altitude of article presents the equations for calculating the moist air about 10 km, atmospheric air consists of only dry air [1]. properties. The comparison of the Nusselt number for dry The mixture of dry air and water vapor is known as moist and moist air for different relative humidity and temperature air. Psychrometry is the study of the properties of moist air. conditions is also presented to substantiate the effect of At a given temperature and pressure, the dry air can only humidity on natural convection in thermal performance of hold a certain maximum amount of moisture which is known electronic products. as saturated air. The amount of water vapor present in the air is quantified by the term humidity. The term humidity is Natural Convection and Dimensionless Numbers commonly refers to relative humidity. Relative humidity is When heat is carried by the circulation of fluids, due to defined as the amount of water vapor in a sample ofair buoyancy from density changes induced by heating itself, compared to the maximum amount of water vapor the air then the process is known as free or natural convective heat can hold at any specific temperature in a form of 0 to 100%. transfer. The heat transfer due to free convection is described by Newton’s Law of Cooling, In most of the numerical simulations, the properties of air

(density, specific heat capacity, thermal conductivity and Q = hA(Tw -T∞) (1) viscosity) are assumed as a function of temperature alone. The above assumption is true for dry air but in reality the The rate of heat Q transferred to the surrounding fluid air contains moisture also. Water vapor and dry air have is proportional to the object’s exposed area A, and the different fluid properties; hence with increased relative difference between the object temperature Tw and the fluid humidity, mixture properties such as specific heat, viscosity, free-stream temperature T∞. The constant of proportionality thermal conductivity and density vary accordingly. h is termed the convection heat-transfer coefficient.

A majority of outdoor electronic products work under Over the years; it has been found that average free natural convection with different humidity conditions. As convection coefficient can be represented in the following changes in relative humidity affect the values of thermo- functional form for a variety of circumstances: physical properties, such changes will also affect the natural n convection heat transfer coefficients and hence the cooling Nuf = C(Grf Prf) (2) of electronics. In this study, the air properties are discussed

10 Typically, n = 1/4 and 1/3 for laminar and turbulent flows [2]. values of moist air as it is a mixture of several permanent gases C is a constant. and water vapor. However, the moist air obeys the perfect gas law with accuracy sufficient to engineering calculations up to The subscript f indicates that the properties (air) in 3 bar pressure. For higher accuracy, Goff and Gratch tables the dimensionless numbers are evaluated at the film can be used for estimating moist air properties. These tables temperature. are obtained using mixture models based on fundamental principles of statistical mechanics that take into account the TT∞ + Where, T = W (3) real gas behavior of dry air and water vapor. However, these f 2 tables are valid for a barometric pressure of 1 bar only. Even hL (4) though the calculation procedure is quite complex, using the Nuf = kfluid mixture models it is possible to estimate moist air properties at other pressures also. However, since in most cases the gβ L3 ()TT − ρ2 Gr = W ∞ (5) pressures involved are low, one can apply the perfect gas f µ2 model to estimate psychrometric properties.

µCp Pr = (6) Basic Gas Laws for Moist Air k fluid The total pressure of a mixture of gases is made up by the sum of the partial pressures of the components in the mixture In the above equations, all the physical properties are as known from Gibbs-Dalton’s Law of Partial Pressures. assumed as function of temperature alone. But as discussed, According to this law, the total pressure in a mixture of gases the humidity may play a role. can be expressed as:

Rearranging the Nusselt number equation, Ρ = ΣΡi (10) n ρ 2C  Nu ∝ d pd  (7) Applying this equation to moist air, dry air µ  dk d 

n P = P = P + P (11) 2  tot da wv ρhC ph Nu ∝   (8) humid air µ  hk h  Important Psychrometric Properties The psychrometric properties can be found in any basic air And the ratio of humid air to dry air is, conditioning text book [3].

2n n  n n Nuhumidair ρ  µ  Cph k  =  h   d    d  (9) (i). Dry bulb temperature (DBT): Nu ρ µ C  k dry air d   h  pd  h  Dry bulb temperature is the temperature of air measured by using a normal thermometer. The dry-bulb temperature is an Equation (9) quantifies the effect of humidity over dry air. indicator of heat content. The following topics describe the procedure to estimate moist (humid) air properties. (ii). Wet bulb temperature (WBT): Wet bulb temperature is associated with the moisture Methods for Estimating Moist Air Properties content of the air. Wet bulb temperature can be measured It is difficult to estimate the exact thermo-physical property with a thermometer that has the bulb covered with a water-

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moistened bandage with air flowing over the thermometer. Wet bulb temperatures are always lower than dry bulb temperatures but they will be identical with 100% relative humidity in the air.

(iii). Dew point temperature (DPT): Dew point is the temperature at which water vapor starts to condense out of the air, the temperature at which air becomes completely saturated. Above this temperature, the moisture will stay in the air.

(iv). Saturated vapor pressure (psat): It is the pressure of a vapor in equilibrium with its non-vapor phases. At any given temperature, for a particular substance, there is a pressure at which the gas of that substance is in dynamic equilibrium with its liquid or solid forms. This is the vapor pressure of that substance at that temperature.

The following equations are from the ASHRAE (American Society of Heating, Refrigeration and Air-Conditioning Engineers) Handbook of Fundamentals [4].

C ln(p )C=1 + +CT + CT2 + sat T 2 3 4 (12) 3 4 5 CT5+CT 5 + CT6+ C 7 ln(T)

The regression coefficients C1 to C7 are given by:

Over ice Over water Constants (-100°C < T < 0°C) (0°C < T < 200°C)

C1 -5.67E+03 -5.80E+03

C2 -5.15E-01 -5.52E+00

C3 -9.68E-03 -4.86E-02

C4 6.22E-07 4.18E-05

C5 2.07E-09 -1.45E-08

C6 -9.48E-13 0

C7 4.163502 6.545967

(v). Relative humidity (Φ): Relative humidity is the ratio of the water vapor pressure

(Pwv), to the vapor pressure of saturated air at the same temperature (Psat), expressed as a percentage. Relative humidity is a relative measure. The moisture-holding capacity of air increases with air temperature. In practice, relative

12 humidity, indicates the moisture level of the air compared Mda=28.97 mole and Mwv=18.015 mole to the airs moisture-holding capacity. Relative humidity is nµ,i T  normally expressed as a percentage. Thus, for saturated air, µ = µ   (19) i 0,i T Φ is 100%. 0  Power law parameters for viscosity [8] Pwv φ = (13) Psat

2 (vi). Humidity ratio (W): µ0 (Ns/m ) T0 (k) nµ (-) It is the ratio of the actual mass of water vapor present in moist air to the mass of the dry air. It can also be expressed -5 µda 1.716 x 10 273 0.666 with the partial pressure of water vapor,

-5 pw v µ 1.12 x 10 350 1.15 W = 0.622 (14) vw ptot -p wv

(vii). Specific heat of humid air (Cph): The mole fraction of dry air and water vapor may be found The specific heat of humid air can be calculated using [5]: using,  1+ 1.858W  = PPtot− ϕ sat Cpm   (15) X =  1+ W  da (20) Ptot

(viii). Specific volume of humid air (v): ϕ Specific volume is defined as the total volume of dry air and Psat Xwv = (21) water vapor mixture per kg of dry air. From the perfect gas Ptot equation

(x). Thermal conductivity of humid air (k ): RT RT 3 h = a = a m /Kg dry air v (16) The thermal conductivity of humid air can be calculated Pa PPtot− w v using [9]: Where R is specific gas constant = 287.05 J/Kg K. a 2 X k k = i i h ∑ 2 (22) i= 1 ϕ The specific heat, thermal conductivity and dynamic viscosity ∑ Xj ij j= 1 of humid air can be calculated as described by J. Zhang et Where, nk,i al. [6]. T  ki= k 0,i   (23) T0 

(ix). Dynamic viscosity of humid air (µh): Power law parameters for thermal conductivity [8] The dynamic viscosity of humid air can be calculated using [7]: 2 X µ k (W/mK) T (k) n (-) µ = i i 0 0 k h ∑ 2 i= 1 (17) ∑ Xjϕ ij j= 1 k 0.0241 273 0.81 Where, da 2 − 1  1 1    2   2 4 1 M µ Mj  k 0.0181 300 1.35 ϕ = 1+i   1 + i   (18) vw ij         8 Mj µj Mi       

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Steps to calculate the moist air properties 0.032 Φ=0% Φ=10% 1. Calculate P using Equation 12 for the given dry bulb 0.03 Φ=20% sat Φ=30% temperature, Φ=40% 0.028 Φ=50% 2. Calculate Pwv using Equation 13 for the given relative Φ=60% humidity, Φ=70% 0.026 Φ=80% Φ=90% 3. Calculate W using Equation 14 using the above Pwv value, Φ=100% 4. Calculate Cp of humid air using Equation 15, 0.024

5. Calculate v of humid air using Equation 16. Inverse of v will Thermal conductivity of moist air (W/mK) 0.022 give the humid air density.

6. Calculate µ of humid air using Equation 17-21. 0.02 -40 -20 0 20 40 60 80 100 7. Calculate k of humid air using Equation 18, 20-23. Temperature (°C)

Figure 3.Thermal conductivity of moist air for different relative Results and discussion humidity and temperature

1.7 2.20E-05 Φ=0% Φ=10% Φ=20% 1.5 2.00E-05 Φ=30% ) 2

m Φ=40% ) Φ=50% da

3 3 1.3 m Φ=0% Φ=60% 1.80E-05 Φ=70% Φ=10% Φ=80% Φ=20% 1.1 Φ=90% Φ=30% Φ=100% Φ=40% 1.60E-05

0.9 Φ=50%

Density of moist air (kg/ Φ=60% Dynamic viscosity of moist air (Ns/ Φ=70% 1.40E-05 0.7 Φ=80% Φ=90% Φ=100% 0.5 1.20E-05 -40 -20 0 20 40 60 80 100 -40 -20 0 20 40 60 80 100 Temperature (°C) Temperature (°C)

Figure 1. Density of moist air for different relative humidity and Figure 4. Dynamic viscosity of moist air for different relative humidity temperature and temperature

2.1 0.4

0.35 1.9 Φ=0% Φ=10% 0.3 Φ=0% Φ=20% 1.7 Φ=10% 0.25 Φ=30% Φ=20% Φ=40% Φ=30% 1.5 Φ=50% 0.2 Φ=40% Φ=60% Φ=50% Φ=70% 0.15 Φ=60% 1.3 Φ=80% Φ=70% % change in Prandtl number Φ=80% Φ=90% 0.1 Specific heat of moist air (KJ/kg da K) Φ=90% Φ=100% 1.1 Φ=100% 0.05

0.9 0 -40 -20 0 20 40 60 80 100 -40 -20 0 20 40 60 80 100 Temperature (°C) Temperature (°C)

Figure 2. Specific heat of moist air for different relative humidity and Figure 5. Percent change in Prandtl number for different relative temperature humidity and temperature

14 0.14 increases, the water content in the mixture increases which in turn increases the specific heat of the mixture. The thermal 0.12

Φ=0% conductivity of the humid air decreases with an increase in Φ=10% r 0.1 moisture content as the thermal conductivity of water vapor is Φ=20% Φ=30% less than dry air and with an increase in temperature the mole 0.08 Φ=40% fraction of water vapor and the thermal conductivity of dry air Φ=50% 0.06 Φ=60% increases. The dynamic viscosity of the humid air decreases Φ=70%

% change in Nusselt numbe Φ=80% with increase in relative humidity, as the water content in 0.04 Φ=90% the mixture increases with increase in humidity. Recall that Φ=100% 0.02 the above fluid properties constitute the prandtl and Nusselt

0 number. So the prandtl and Nusselt number increases with -40 -20 0 20 40 60 80 100 Temperature (°C) an increase in relative humidity and temperature.

Figure 6. Percent change in Nusselt number for different relative humidity and temperature Conclusions The effect of relative humidity and surface temperature The percent change in Prandtl number is calculated as on the Nusselt number was studied using simple Nusselt and the same, way the % change in Nusselt number also number correlation and moist air properties. The relative calculated. humidity was varied from 0 to 100% and the temperature

Figures 1 to 6 show the effect of relative humidity on fluid properties. As expected, it is found that relative humidity has a significant effect on Nusselt and Prandtl numbers, especially at high temperature. In general, it is observed that as the relative humidity increases, the Nusselt and prandtl numbers increase. The density of humid air decreases with the increase in relative humidity, as the mass fraction of water vapor is less than that of air. Also, the decrease in density of the mixture increases the convection currents producing a larger Nusselt number. The specific heat of humid air increases with an increase in humidity. As humidity

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November 2008 |Qpedia 15 Thermal Analysis

was changed from -40°C to 100°C. It is observed that, C.P Arora, Tata McGraw-Hill publications. variation of relative humidity and temperature resulted in a 4. Psychrometric equations http://www.ageng.ndsu.nodak. change of thermo-physical properties and hence increased edu/envr/PsycEquations.htm the Nusselt number in the order of 13%, compared to dry air 5. Gater, Roger A, Engineering Thermodynamics 1996, pg in laminar flow conditions for 100°C, 100% humid air. The 13-17 effect of humidity is significant at higher temperatures, as 6. Jiehei Zhang., Ajaykumar Guptha., John Baker., Effect of in the case of automotive electronics which experiences an Relative Humidity on the Prediction of natural Convection ambient close to 100°C. Heat Transfer Coefficients, Journal of Heat Transfer Engineering., Vol.28, Issue 4, pp335-341,2007. Nomenclature: 7. Buddenberg, J. W., and Wilke, C. R., Calculation of Gas Mixture Viscosities, Ind. Eng. Chem., vol. 41, no. 7, pp. Q Heat transfer rate 1345–1347, 1949.

Tw Wall temperature 8. White, F. M., Viscous Fluid Flow, 2nd ed., McGraw-Hill,

T∞ Free stream temperature New York, pp. 15–45, 1991.

Nuf Nusselt number at film temperature 9. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Pr Prandtl number Phenomena,Wiley & Sons, New York, p. 258, 1960. Gr Grashof number L Characteristic length About the Author

Kfluid Thermal conductivity of fluid Ashok Kumar serves as a thermal g Acceleration due to gravity specialist in the automotive electronics

β Volumetric thermal expansion coefficient = 1/Tf group at BOSCH. He is responsible for ρ Density thermal analysis of Automotive Electronics μ Dynamic viscosity components both at package as well as

Cp Specific heat capacity system level. Prior to joining BOSCH, P Pressure of the mixture he worked as a research scientist in

Pi Partial pressure of gas i Electronics Packaging division at SAMEER-

Ptot Total barometric pressure CEM, Chennai. His interests are in the area of phase change heat

Pda Partial pressure of dry air transfer and package thermal-mechanical design. Ashokkumar

Pwv Partial pressure of water vapor received his B.E in Mechanical Engineering from Barathiyar p Saturated vapor pressure of water [Kpa] University, Coimbatore, India in 1998 and M.E in Refrigeration and sat K Temperature [K] Air conditioning from Anna University, Chennai, India in 2004. M Molar mass, mole Contact information:

References: Ashok Kumar, S. 1. Online Course Material on Psychrometry http:// Robert Bosch Engineering and Business Solutions Ltd. www.nptel.iitm.ac.in/courses/Webcourse-contents/ (RBEI/END2), IIT%20Kharagpur/Ref%20and%20Air%20Cond/pdf/ Gold Hill Square, 690, Hosur Road, Bommanahali, R&AC%20Lecture%2027.pdf Bangalore-560068, India. 2. Incropere, F.R., and Dewitt, D.P., Fundamentals of Heat Tel: +91 80 4191 6442 and Mass transfer, 5th ed., John Wiley & Sons, New Fax: +91 80 4191 6666 York. Mobile: +91 9880309656, 3. Refrigeration and Air Conditioning Second Edition by E-mail: [email protected]