Radial flow boundary layer development analysis

 

 

D.G. Kröger^I; J.D. Buys^II

^IDepartment of Mechanical Engineering, University of Stellenbosch, Private Bag Xl, Matieland, 7602 South Africa
^IIDepartment of Mathematics, University of Stellenbosch, Stellenbosch

 

 

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ABSTRACT

A boundary layer analysis was performed to determine the pressure differential due to frictional effects, and the heat transfer coefficient during turbulent radial fluid flow between two approximately parallel discs or surfaces. The results of the analysis were applied to flow at the inlet of the collector of a solar chimney power plant and a numerical example is presented to show the effect that various independent variables have on the radial pressure and heat transfer coefficient.

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Nomenclature

b exponent

C_pspecific heat, J/kg-K

d_ehydraulic diameter, m

f friction

ggravitational acceleration, m/s²

Hheight, m

hheat transfer coefficient, W/m²-K

Kloss coefficient.

kthermal conductivity, W/m-K

m mass flow rate, kg/s

Nu Nusselt number, h (r₀ - r)/k

ppressure, N/m²

Pr Prandtl number, μc_p/k

rradius, m

Ra Rayleigh number, gß∆TH³/α-v

Re Reynolds number , ρvd_e/μ

Ttemperature, K

vvelocity, m/s

zco-ordinate

 

Subscripts

c core

clcentreline

ppipe

rradius or rough

ssmooth

0 at outside radius

 

Greek letters

αthermal diffusivity, k/ρ-c_p, m²/s

ß coefficient of volumetric thermal expansion, K^-1 ∆ differential

δboundary layer thickness, m

ϵsurface roughness, m

θangle, o

μ dynamic viscosity, kg/m-s

Vkinematic viscosity, m²/s

ρdensity, kg/m³

τshear stress, N/m²

 

Introduction

When a fluid flows radially between two approximately parallel surfaces or discs a radial pressure gradient exists. If there is a temperature difference between the fluid and the surfaces, heat will be transferred. A practical example where such flow would occur is in the solar collector of a solar chimney power plant as shown schematically in Figure 1. This plant consists of a central chimney which is surrounded by a circular collector having a glass roof. Air is heated in the collector and flows up the chimney due to buoyancy effects. The air stream drives a turbine that is located near the base of the tower.

 

 

To evaluate the performance of such a plant the pressure drop and the heat transfer in the collector must be determined.

 

Analysis

Consider an elementary control volume in the boundary layer on a (smooth or rough) surface in a radial flow field between two discs (see Figure 2). The momentum equation in the radial direction is

 

 

For incompressible flow this reduces to

In the core region outside the boundary layer the flow is essentially frictionless with the result that Bernoulli's equation is applicable;

For fully developed turbulent flow in a pipe at Re_p≈ 10⁵ the velocity distribution can be approximated by a relatively simple empirical equation.¹ For developing flow over a flat surface the analogous velocity distribution (due to the universality of the velocity distribution in a turbulent boundary layer) may be expressed in terms of the core velocity v_c and the boundary layer thickness δas

Substitute (2) and (3) into (1) and integrate, and find

For turbulent flow where the velocity distribution is relatively uniform at any radius (i.e. the boundary layer is relatively thin or the Reynolds number is high) the mass flow rate can be expressed approximately as

To avoid excessively high radial velocities in the collector as the radius decreases, the height of the upper disc or collector roof is assumed to be given by

where b is a parameter (0 < 6 < 1).

Heat transfer coefficient: smooth surfaces

For smooth pipes the Fanning friction factor can be approximated by the following empirical equation¹ in the range 10⁴ < Re_p < 10⁶:

At a pipe Reynolds number of Re_p ≈ 10⁵ the ratio of mean pipe velocity to centreline velocity is

The analogous relation applied to the boundary layer on a flat plate is

Substitute (5), (6), and (7) into (4), and find the differential equation

The solution of this differential equation together with the boundary condition δ(r₀) = 0 (see Appendix A) yields the boundary layer thickness

According to the Colburn analogy for a flat plate¹ the local heat transfer coefficient can be expressed in terms of the local shear stress:

From (5), (7), and (10) it then follows that

The local Nusselt number is defined as Nu = h(r₀ -r)/k. From (6) and (9) it then follows for b ≠0.4122

that

The thermophysical properties in this equation are evaluated at the arithmetic average temperature of the surface and the core.

Heat transfer coefficient: rough surfaces

For rough pipe surfaces (ε> 0) the turbulent friction factor can be approximated by

This empirical relation approximately correlates the friction data for rough pipes in the range 0.0001 < ε/d_p < 0.01 and makes possible an analytical solution of the problem. The corresponding pipe shear stress is

With v_p - 0.817v_pcl as before we find the analogous shear stress for turbulent flow over a rough flat plate:

Substituting (5), (6), and (12) into (4), the differential equation for δbecomes

The solution of this equation with δ(r₀) - 0 is (see Appendix A):

Pressure drop: smooth surfaces

To find the radial pressure drop due to friction in the collector, consider a control volume located between two discs as shown in Figure 3. Then

 

 

Substituting (6), (7), and (9) into (15):

and integrating between the outer radius r₀ and any other radius r, we find the pressure drop

The integral in (16) is improper at both the upper limit t = 1and the lower limit t = x when x → 0. For computational purposes (see Appendix B) the integral can be written as

The integrals in (17) both have integrands with finite limits at both endpoints and can be integrated to a high degree of accuracy with a 4-point Gaussian integration rule. Alternatively, the integral can be approximated with an error of at most 5% in the range 0.6 < x < 1, 0 < 6 < 1 by

The loss coefficient based on inlet velocity due to friction on the two smooth surfaces in the collector is

Pressure drop: rough surfaces

Assuming the same roughness on both surfaces, and substituting (6), (12) and (13) into (15), find

The integral is again improper at both limits, and may be written (see Appendix B)

For 0.6 < x < 1 and 0.001 < q <0.1, the expression

approximates the integral with an error of at most 10% in the range 0 < 6 < 1, and less than 2% for 0.4 < 6 < 0.8. The loss coefficient for rough surfaces is

 

Results

The development of the boundary layer during steady radial flow between two surfaces has been analysed. The results of the analysis are employed to predict the change in radial pressure due to frictional effects and the local heat transfer coefficient at the inlet of the collector of a solar chimney power plant.

For smooth surfaces the boundary layer thickness as given by equation (9) is shown graphically in Figure 4 for given values of b, r₀/H₀and μr₀/m. Generally the flow between the surfaces becomes fully developed (δ≈ H₀/2) fairly rapidly, whereafter the present analysis is no longer applicable. The boundary layer growth rate for rough surfaces as given by (13) is even more rapid than that for smooth surfaces. For the conditions specified, δ/H₀is almost independent of the air mass flow rate, as shown in Figure 5.

The corresponding Nusselt numbers as given by (11) and (14) are shown in Figures 6 and 7, respectively. In both cases the Nusselt number decreases with increasing radius but improves with increasing air flow rate. A rough surface results in a higher heat transfer coefficient.

The corresponding loss coefficients are given by (18) and (19), respectively and are presented graphically in Figures 8 and 9.

 

Conclusions

Analytical solutions for predicting the pressure differential due to frictional effects and the heat transfer coefficient during developing radial flow between two essentially parallel discs are obtained. These results can be applied in the performance evaluation of a solar chimney power plant collector.

It should be noted that this analysis does not necessarily entirely model the true nature of the developing flow in the solar collector. Since the Rayleigh number Ra_H » 1708 in this region, secondary flow patterns that affect the pressure drop and tend to augment the heat transfer may be present. When Ra_H > 10⁸ the Nusselt number due to these meandering counterflowing thermals becomes independent of the height of the collector roof above the ground.² Under certain operating conditions in certain areas of the collector the heat transfer due to the secondary flow may be of the same order or greater than that predicted by the present analysis.

 

References

1. Rohsenow WM & Choi HY. Heat, Mass and Momentum Transfer. Prentice Hall, Englewood Cliffs, New Jersey, 1961.         [ Links ]

2. Bejan A. Heat Transfer. John Wiley and Sons, New York, 1993.

 

 

Received April 1999
Final version October 1999

 

 

Appendix A

The solution of the differential equation

may be found as follows. The substitution v - δ^α reduces the equation to the linear equation

The solution of the corresponding homogeneous equation is v_h (r) - Ar^αk;assuming a particular solution of the form v(r) - u (r) r^αk and incorporating the boundary condition δ(r₀) = 0, the solution is found as

 

 

 

 

 

 

 

 

 

 

 

 

Appendix B

Consider the integral

and suppose that γ and Γ are constants such that

Then the integral is improper and integrable at t = 1, and F(x) →∞ when x → 0 if γ > 1. The successive substitutions

reduce the integral to the form

These last integrals have integrands with finite limits at both upper and lower end-points of the integration interval.