STU2013-1033

Proposed Criterion for Steady State Parallel Plate Thermal Conductivity Measurement of Liquids and Nanofluids Amin Shams Khorrami1, Atta Sojudi2, Esmaeil Esmaeilzadeh3 1

University of Tabriz; [email protected] 2 Sharif University; [email protected] 3 University of Tabriz; [email protected]

Abstract Steady state parallel plate thermal conductivity apparatus for different liquids and nanofluids has been studied numerically. Single phase model and the most popular theoretical formulas employed to develop the nanofluids heat transfer characteristics. The parallel plate thermal conductivity (PPTC) apparatus consists of a heating element above the test fluid specimen and a closed cooling system below. Because of downward heat transfer (unidirectional heat transfer), convection effects due to gravity can be avoided until the critical Rayleigh number is reached. Correlations derived for critical Rayleigh numbers in order to avoid free convection. Results show that critical Rayleigh numbers behave linearly with respect to Prandtl number for different concentrations of nanofluids and using further additive reduces the Rayleigh number needing to instable the test fluid specimen. It has been shown that the more thermally conductive additive is used, the less the Rayleigh number is needed to instable the suspension. Keywords: Thermal conductivity, nanofluids, measurement, parallel plate, numerical solution Introduction The potential use of nanofluids in thermal systems has been well documented throughout the scientific and industrial communities. The addition of carbon, metal and metal oxide nanoparticles to standard base fluids has been shown to increase the thermal conductivity and heat transfer capabilities of these fluids. Since thermal conductivity is the most important parameter responsible for heat transfer enhancement, many experimental methods have been developed to measure it. In general, experimental methods of obtaining thermal conductivity are divided into two distinct parts: Unsteady methods and Steady methods. Unsteady methods have been used extensively in the literature, like transient hot wire which was used by different authors [1, 2, 3]. TPS method was used by Zhu [4] and Jiang [6]. Temperature oscillation technique was used by Roetzel [7] and Czarnetzki [8] and 3ω method was used by Cahill [9]. Cylindrical cell method is one of the most common steady state methods used for the thermal conductivity measurement of fluids. In this method the nanofluid whose thermal conductivity is to be measured fills the annular space between two concentric cylinders. Kurt has given a detailed description of the equipment [10]. The apparatus for the steady-state parallel-plate method can be constructed on the basis of the Challoner design

[11]. Wang used this method for measuring the thermal conductivity of alumina and copper oxide based nanofluids [12]. The PPTC apparatus mainly governing equation is Furrier heat transfer law for solid continuum. On this basis, measuring temperature gradient on both sides of liquids and nanofluids is possible until it behaves like a solid slab. Because the heat conduction through a stationary fluid is identical to the heat conduction through a solid medium, this concept can be applied to obtain the steady state, parallel-plate method of determining fluid thermal conductivity. So the effects of heating and cooling parts on variation of density must not create instability and Benard cells. It is a fact of history that the necessary care with this instrument has been taken by only a few workers so, only some of the measurements are reliable. The most important limitation of steady state apparatus is the onset of free convection which should be avoided during the experiments. In the present work, the maximum heat flux which should be imposed to the top plate to activate pure conduction within the nanofluid is investigated. Most of researchers have used this technique without any attempt to find the ultimate heat fluxes. Correlations derived for critical Rayleigh numbers in order to avoid free convection and buoyancy forces. So, the derived formulas will help researchers to know the limitation of Rayleigh numbers in which there is no advection within the cavity. Heat transfer characteristics of nanofluids For numerical simulations two approaches have been adapted in the literature to investigate the heat transfer characteristics of nanofluids. The first approach assumes that the continuum assumption is still valid for fluids with suspended nanosize particles. The other approach uses a two phase model for better description of both the fluid and the solid phases, but it isn’t common in the open literature. The use of mixture models to predict effective density, specific heat capacity, and thermal expansion coefficient of binary mixtures has been well verified in the literature. Zhang directly applied mixture models to nanofluids for these properties because they are likely independent of the migration of nanoparticles, e.g., Brownian and thermophoretic motions [14]. However, Heris noticed that in prediction of thermal conductivity of nanofluids the diffusions of nanoparticles should be considered, in addition to the static contribution governed by the mixture models [15]. In the present study, the models proposed by Koo are utilized [16], which have been successfully used by Yu (to investigate numerically the

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` transient features of natural convection heat transfer of nanofluids in a bottom heated triangular enclosure [17]. Properties of nanoparticles are listed below: Table 1: Thermo physical properties of nanoparticles [23]

Material Al2O3 CuO TiO2 SiC

ρ (kg/m3) 3890 6320 4157 3160

Cp (j /kg.K) K (W/m.K) 880 531.8 710 675

35 76.5 8.4 490

Governing equations Energy and momentum transport equations have been solved numerically, in which all equations are related to free convection basis. The continuity (eq. 1), NavierStokes (eqs. 2 and 3) and energy equations have been solved for the apparatus for both liquid (eq. 4) and solid (eq. 5) parts.    u     v   0 x y

(1)

  P  2u  2u  uv   ( uv)     ( 2  2 ) x y x x y

(2)

  P  2v  2v ( uv)  (  uv)    ( 2  2 )   g x y y x y

(3)

   2T  2T ( uh)  (  vh)  K ( 2  2 ) x y x y 2 2 T T  0 x 2 y 2 Ra 

2

g  H 4 q 2CP K 2

(4) (5) (6)

Boundary Conditions The parallel plate thermal conductivity (PPTC) apparatus consists of a heating element above the test fluid specimen and a closed cooling system (mixture of ice and water) below. This creates a temperature difference across the test fluid which is measured by thermocouples embedded in the parallel plates. This temperature difference is then used along with the heating power to calculate the thermal conductivity of the test fluid. Because heat is transferred downward (unidirectional heat transfer), convection effects due to gravity can be avoided until the critical Rayleigh number is reached. The PPTC apparatus is also insulated, restricting all heat conduction from the heater to the ice-water mixture in axial direction. Figure 1 shows the schematic of the PPTC apparatus. For the parallel plates, we have selected silver in order to reduce radial heat conduction through plates because of its high thermal conductivity and low specific heat capacity. Diameter of the apparatus is 100mm and the gap height is 4mm. The parallel plates are 1mm thick to avoid multidirectional heat transfer (1mm<<100mm). The bottom and top of domain are taken as constant temperature and constant heat flux, respectively. The uniform wall temperature (UWT) of the cooled wall was kept to 273 K (ice and water mixture) and the uniform heat flux (UHF) condition was imposed to the heated wall. All boundary conditions except for the heated and cold walls are regarded as adiabatic walls.

Figure 1: The PPTC apparatus

Computational details A numerical study was carried out using a commercial CFD program FLUENT 6.2. The GAMBIT software was used to generate the 2-D mesh of a vertical cylindrical cavity based on the size of the test facility which is shown in fig. 2. The steady segregated solver was used with first order upwind algorithm for momentum and energy in laminar model. For pressure discretization the PRESTO algorithm was adopted while for pressure–velocity coupling discretization the SIMPLEC algorithm is used. Laminar model has been selected among viscous models and boussinesq approximation is applied to consider convection effects. Beyond the critical Rayleigh, no convergence occurred and this shows the disturbance in the specimen and the viscous model should be changed to a turbulent model to converge the solution.

Figure 2: (2-D) mesh of a vertical cylindrical cavity

Independence test of grid density Dependence of numerical accuracy on the spatial resolution is tested for water, ethylene glycol and engine oil at Ra=100 which is not nearby the critical Rayleigh number, even if the heater would exists at the bottom. All computed thermal conductivities are compared to Incropera [13] . It is shown that the greatest relative deviation is less than 0.16%, which is negligible for engineering applications. The spatial 108×180 is adopted for all the cases studied. Using the finest resolution obtained the critical Rayleigh number for test fluids including Water, Ethylene glycol and Engine Oil is are 569588, 147673 and 37216 respectively. Also, linear behavior of temperature distribution along the vertical mid line for Ra below the critical Rayleigh number is illustrated in Fig 2 for ethylene glycol. It can be seen that beyond the RaC the test is not suitable and deviates from linear behavior.

Table 2: Critical Rayeigh numbers’ correlations for different Nanofluids

Figure 3: Vertical mid line temperature distribution from bottom to top of the enclosure for pure ethylene glycol at Ra=147000 (below RaC) and Ra=150000 (beyond RaC)

Results and Discussion In this section, we have used 3 base fluids for 4 different nanoparticles, creating 5 different volume fractions for each nanofluid. Ethylene glycol, liquid water and engine oil are the base fluids and SiC, Cuo, Al2O3 and TiO2 are four different additive particles. All 12 nanofluids’ thermo physical properties have been calculated for 0.01 to 0.05 volume fractions. We have derived correlations for critical Rayleigh numbers of ethylene glycol, liquid water and engine oil based suspensions. It can be seen that the thermal conductivity of all five concentrations for all nanofluids are distributed around straight lines. The more the volume fraction is, the lower Rayleigh number is needed to instable the test fluid. Reducing the Prandtl number through adding more particles, causes the momentum diffusivity to be reduced with respect to thermal diffusivity and then low values of Rayleigh numbers will disturb the test fluid specimen. It can be seen that for all nanofluids of low concentration, critical Rayleigh number is independent of nanomaterial, so for very low concentration of every nanofluid, we reach to the base fluid’s properties. Table 2 correlates the computed Critical Rayleigh numbers with Pr of each nanofluid. It should be noted that these critical values for Ra might be treated as conservative values, Because Brownian and thermophoretic motions may reduce this critical value. So the two phase modeling and experimental works are being done to investigate this statement, in which the results will be represented in near future.

Nanofluid Correlation SiC + Water RaC = (1.7523Pr-6.575)×105 SiC + Ethylene glycol Rac = (0.0278Pr-2.4698) ×105 SiC + Engine Oil Rac = 2.6Pr+2252.5 CuO + Water RaC = (0.6417Pr+1.2724)×105 CuO + Ethylene glycol RaC = (0.0938Pr+2.8264)×104 CuO + Engine Oil RaC = (0.0001Pr+2.2905)×104 TiO2 + Water RaC = (1.2343Pr-2.9185)×105 TiO2 + Ethylene RaC = (0.166Pr-8.2685)×104 glycol TiO2 + Engine Oil RaC = 3.01Pr-864.08 Al2O3 + Water RaC = (1.421Pr-4.2353)×105 Al2O3 + Ethylene RaC = (0.0198Pr-1.299)×105 glycol Al2O3 + Engine Oil RaC = (0.004Pr-1.3218)×104 Conclusions Steady state parallel plate thermal conductivity apparatus for 12 different nanofluids in 5 different volume fractions has been simulated numerically. Theoretical formulas are used to calculate thermal characteristics of nanofluids for 0.01 to 0.05 volume fractions. Up to the critical Rayleigh number, we have a unidirectional conduction heat transfer. Critical Rayleigh numbers were detected for these nanofluids and new correlations derived for them to estimate when the free convection heat transfer starts. Results show that distribution of all nanofluids is around some specified lines and the more additive is used, the low critical Rayleigh number is needed to activate Bénard cells and the more thermal conductive additive is used, the less Rayleigh number is needed to instable the suspension. List of Symbols Cp g h K

Specific heat Gravity Enthalpy Thermal conductivity

P

Pressure

Pr Ra T

Prandtl number Rayleigh number Temperature

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