ARTICLES

Fully developed laminar convection whit variable thermophysical properties between two heated vertical parallel plates

D.F. Delmastro^†, A.F. Chasseur^‡ and J.C. García^§

^†Centro Atómico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina. delmast@cab.cnea.gov.ar
^‡ Centro Atómico Bariloche, 8400 Bariloche, Argentina. alfredo.chasseur@cab.cnea.gov.ar
^§ Centro Atómico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina. garciajc@cab.cnea.gov.ar

Abstract -In this work, the influence on the flow and heat transfer of the density, viscosity and thermal conductivity temperature dependence was analyzed. A perturbation analysis was applied to a vertical fully developed laminar stationary flow between two heated parallel flat plates. For small values of the dimensionless numbers associated with the buoyancy, viscosity and thermal conductivity changes effects, the influence of the different properties temperature dependence was obtained. Application examples to water and air flows are also presented.

Keywords -Laminar Flows. Heat Transfer. Temperature-Dependent Properties.

I. INTRODUCTION

In many situations of heat transfer applications, like research reactors fuel elements, a coolant fluid flows between heated vertical parallel flat plates. In case of low flow rates and significant heat fluxes, laminar flows with important temperature and properties variations appear. For these temperature-dependent property problems it is useful to know the influence of these variations on the flow and heat transfer behaviors.

One complication of the temperature-dependent property problem is that the properties of different fluids change differently with temperature. For most liquids the viscosity decrease a lot with temperature, the density varies little, while the specific heat and thermal conductivity are relatively independent of temperature. For gases the viscosity and the thermal conductivity increase very much with temperature while density decrease and specific heat varies only slightly. For engineering applications two schemes for correction of the constant-property results are usually used. In the reference temperature method a characteristic temperature is used to evaluate the properties. In the property ratio method all the properties are evaluated at the mean temperature and corrected by multiplying by a function proportional to the ratio of mean to wall temperature or viscosities (Kays, 1966; Kreith, 1973).

The internal laminar heat transfer with temperature dependent properties were numerically investigated using finite difference methods (Swearingen and McEligot, 1971).

Many solutions have been obtained for laminar flow taking into account the effects of temperature-dependent viscosity or density. But generally they do not consider both effects nor the thermal conductivity temperature dependence (Bergles, 1983).

Recently the fully developed laminar free convection with large temperature differences was numerically studied for three fluids taking into account the variation of density, viscosity and thermal conductivity with temperature (Pantokratoras, 2006).

In this work, the influence on the flow and heat transfer of the density, viscosity and thermal conductivity temperature dependence for a mixed convection case is analyzed. A perturbation analysis is applied to a vertical fully developed laminar stationary flow between two heated parallel flat plates. For small values of the dimensionless numbers associated with the buoyancy, viscosity and thermal conductivity changes effects, the linear influence of the temperature-dependent properties is obtained.

II. PERTURBATIVE METHOD

A. Laminar Flow between Two Parallel Heated Flat Plates

Let us consider a fully developed laminar stationary flow between two parallel flat plates a distance 2L apart (Fig. 1). Both plates have a uniform heat flux q". The flow is vertical, upward or downward. The x coordinate is vertical and positive downward. The y coordinate is horizontal and perpendicular to the plates.

Figure 1: Sketch of a heated channel

Let us assume that the density ?, the viscosity µ and the thermal conductivity k are linear functions of the temperature:

where the subscript m means that the properties are evaluated at the mean temperature.

B. Conservation Equations

Considering a Boussinesq approximation (Incropera and DeWitt, 1999), the momentum equation can be written as

(1)

where u is the velocity and p* is define as

The boundary conditions for the momentum equation are:

The energy equation can be written as:

(2)

where c is the specific heat at constant pressure.

The boundary conditions for the energy equation are:

The mean temperature derivative can be obtained integrating Eq.(2) between the center of the channel and one flat plate, considering the symmetry and the uniform heat flux boundary condition:

C. Nondimensional Form

Writing Eq. (1) and Eq. (2) in a nondimensional form, using the references values

we obtain the following expressions

	(3)
	(4)

where y is the dimensionless position, ? is the dimensionless T-Tm, ? is the dimensionless velocity and the friction factor f. The Reynolds number Re, the Richardson number Ri, the viscosity number N_μ and the thermal conductivity number N_k are defined as

D. Perturbation Analysis

If the density, the viscosity and the thermal conductivity are constant and do not depend on the temperature, the dimensionless velocity and temperature should satisfy

	(5)
	(6)

with the following boundary and integral conditions

.

where the subscript 0 means that the properties are constant.

Integrating Eq. (5) and Eq. (6) ?₀, ?₀, f₀ ·Re and Nu₀ are obtained

Linearizing Eq. (3) and Eq. (4) about the constant properties velocity and temperature leads to the linear differential equation system (Aziz and Na, 1984)

	(7)
	(8)

where d are the perturbation produced by the density, viscosity and thermal conductivity temperature dependence.

Integrating Eq. (8) considering the boundary conditions

the velocity perturbation is obtained

(9)

Integrating Eq. (7) considering the boundary conditions

the temperature perturbation is obtained

(9)

The friction factor perturbation can be obtained considering that the integral of the velocity perturbation is zero

The Nusselt number can be obtained as

III. RESULTS

A. Ri·Re, N_μ and N_k influence

In Figs. 2, 3, 4 and 5, the influence of the dimensionless numbers Ri·Re; N_μ and N_k, in the velocity and temperature profiles is shown. In each figure two dimensionless numbers remain with the value of zero while the other takes -1, 0 and 1 value. In Fig. 2 the variation of the dimensionless velocity profile for different values of Ri·Re is shown. The value of Ri·Re is zero when the density of the fluid does not change with temperature. For a heated channel, positive values of Ri·Re usually mean that the flow is downward, while negative values mean that the flow is upward. The influence of Ri·Re is not important, and this influence is even smaller in the temperature profile.

Figure 2: Density changes influence in the velocity profile.

Figure 3: Viscosity changes influence in the velocity profile.

Figure 4: Viscosity changes influence in the temperature profile.

Figure 5: Thermal conductivity changes influence in the temperature profile.

Figure 3 shows the variation of the dimensionless velocity profile for different values of N_μ. The value of N_μ is zero when the viscosity of the fluid does not change with temperature. It can be seen that this number has a more significant influence. Positive numbers of N_μ produce more flat profiles due to the viscosity decrease near the wall. The opposite happens with negative numbers of N_μ. According to Eq. (9), the velocity profile is not affected by N_k (i.e. the thermal conductivity variations). From Eq. (9) the perturbations of the velocity in the center of the channel and the velocity derivative on the wall can be obtained.

Positive values of N_μ reduce the central velocity and increase the velocity near the wall, giving more flat profiles. Negative values produce the opposite effect. The Ri·Re influence is smaller and opposite.

In Fig. 4 the influence of N_μ in the temperature profile is shown. Positive values of N_μ reduce the difference between the mean temperature and the wall temperature. The opposite effect occurs for negative values of N_μ.

Figure 5 shows the influence the N_k in the temperature profile. For positive values of N_k the thermal conductivity increases near the wall and therefore the difference between the wall temperature and the central temperature decreases.

From Eq.(10) the perturbations of the temperature in the center and the wall of the channel can be obtained.

Positive values of N_μ reduce the temperature. Negative values produce the opposite effect.

Positive values of N_k increase a little the central temperature and reduce the temperature near the wall. Negative values produce the opposite effect. The change near the wall is much bigger than in the center.

Positive values of Ri·Re increase the temperature but its influence is very small.

B. Water flow application

Considering that water is a typical fluid in heat transfer applications, in Figs. 6, 7, 8 and 9, examples of influence of properties temperature dependence in the friction factor, Nusselt number, velocity and temperature profiles are shown. In Fig. 6 the variation of the f·Re factor with the Reynolds number and the heat flux for water at atmospheric pressure with a mean temperature of 305K is shown. The module of the dimensionless numbers was kept below the unity. In this range, f·Re does not depend on the Reynolds number but only on the heat flux (i.e. N_μ).

Figure 6: Friction factor dependence for water flow at Tm = 305K.

Figure 7: Nusselt number dependence for water flow at Tm = 305K.

Figure 8: Dimensionless velocity profile for water (Tm = 305K; Re = 2100; q" = 20kW/m²).

Figure 9: Dimensionless temperature profile for water (Tm= 305K; Re = 2100; q" = 20kW/m²).

Figure 7 shows the Nusselt number dependence with the Reynolds number and the heat flux for water with a mean temperature of 305K. The value range for Ri·Re is between -1 and 1, and for N_μ and N_k is between 0 and 1. It can be seen that the influence of the Reynolds number is small, and the Nusselt number depends mainly on the heat flux. The Nusselt number increase with the heat flux except for low Reynolds number where a different trend is observed. The origin of this behavior is that for low Reynolds numbers the buoyancy forces are more important than the change in the friction forces.

In Figure 8 the velocity profiles for an upward and downward externally heated water flow are shown. The mean temperature of the water is 305K, the Reynolds number is 2100 and the heat flux is 20 kW/m². The velocity profile is more flat than for a constant properties one, independently of the flow sense, but the influence is more notorious for an upward flow. The origin of the main behavior is the change in the viscous forces, while the smaller difference related to the flow sense is due to the buoyancy force.

Figure 9 show examples for an upward and a downward water flow. The mean temperature of the water is 305K, the Reynolds is 2100 and the heat flux is 20kW/m². The shapes are very similar but the wall temperature is lower when the properties temperature dependence is considered.

These results are in good agreement with the numerical results that are obtained using a finite difference method that considers the properties nonlinear temperature dependence (Delmastro et al., 2007). In Figs. 10 and 11 the velocity and temperature profiles obtained using both methods are compared for a downward externally heated water flow. In Table 1 the Nusselt number and f·Re obtained using both methods are also compared. For the perturbation analysis, properties temperature dependence were linearized considering the differences between the mean temperature and the constant properties wall temperature.

Figure 10: Dimensionless temperature profile for a downward water flow (Tm = 305K; Re = 2100; q" = 20kW/m²). Comparison of results obtained with perturbative and finite differences methods.

Figure 11: Dimensionless velocity profile for a downward water flow (Tm = 305K; Re = 2100; q" = 20kW/m²). Comparison of results obtained with perturbative and finite differences methods.

Table 1: Nusselt number and f·Re for a downward water flow (Tm = 305K; Re = 2100; q" = 20kW/m²). Results obtained for finite differences and perturbative methods.

C. Air flow application

In Figs. 12 and 13, the influence of the temperature-dependent properties on the velocity and temperature profiles is shown for an air flow application.

Figure 12: Dimensionless velocity profile for air (Tm = 300K; Re = 94.4; q" = 2.17kW/m²).

Figure 13: Dimensionless temperature profile for air (Tm=300K; Re = 94.4; q" = 2.17kW/m²).

In Fig. 12 the velocity profiles for an upward and a downward externally heated air flow at atmospheric pressure are shown. The mean temperature of the air is 300K, the Reynolds number is 94.4 and the heat flux is 2.17kW/m². The velocity profile is sharper than for the constant properties one, independently of the flow sense, but the influence is more notorious for a downward flow. The main origin of the behavior is the change in the viscous forces, while the smaller difference related to the flow sense is due to the buoyancy force.

Figure 13 shows the temperature profiles for the same examples. The changes in the temperature profile are mainly due to the thermal conductivity increase near the wall.

For these examples (Re = 94.4; q" = 2.17kW/m²), f·Re and the Nusselt number have a different trend and increase, having the opposite trend than for the water case. The value of f·Re obtained is 102.5, and the value of Nusselt number obtained for the upward flow is 8.666 while for the downward flow is 8.748.

IV. CONCLUSIONS

The influence of the density, viscosity and thermal conductivity temperature dependence in the flow and heat transfer in a fully developed laminar stationary flow between two heated parallel flat plates was analyzed.

For small values of the dimensionless numbers associated with the buoyancy, viscosity and thermal conductivity changes effects, Ri·Re, N_μ and N_k, the influence of the different properties temperature dependence was obtained.

In this range the buoyancy effect is generally not significant, but it can change the sign of the variation of the Nusselt number for very low Reynolds numbers.

Viscosity changes have important effects in the velocity profile, friction factor and Nusselt number.

Thermal conductivity changes have important effects in the temperature profile and Nusselt number.

REFERENCES

1. Aziz, A., and T.Y. Na, Perturbation methods in heat transfer, Hemisphere Publishing Corporation (1984).         [ Links ]

2. Bergles, A.E., "Prediction of the effects of temperature-dependent fluid properties on laminar heat transfer", Low Reynolds number flow heat exchangers, Hemisphere Publishing Corporation (1983).         [ Links ]

3. Delmastro, D.F., A.F. Chasseur, J.C. García, N. Silin and V.P. Masson, "Estudio de la influencia de la dependencia de las propiedades del agua con la temperatura en un flujo laminar descendente entre placas paralelas," XXXIV Reunión Anual de la AATN, Buenos Aires, Argentina (2007).         [ Links ]

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8. Swearingen, T.B., and D.M. McEligot, "Internal Laminar Heat Transfer With Gas-property Variation," J. of Heat Transfer ASME, 128, 405-408 (2006).         [ Links ]

Received: March 12, 2008.
Accepted: June 20, 2008.
Recommended by Subject Editor Alberto Cuitiño.