^{1}

^{*}

^{1}

The Chao and Fagbenle’s modification of Merk series has been employed for the analysis of forced convection laminar thermal boundary layer transfer for non-isothermal surfaces. In addition to the Prandtl number (Pr) and the pressure gradient (∧), a third parameter (temperature parameter, γ ) was introduced in the analysis. Solutions of the resulting universal functions for the thermal boundary layer have been obtained for Pr of 0.70, 1.0 and 10.0 and for a range of ∧ . The results obtained for the similarity equations agreed with published results within very close limits for all the ∧’s investigated.

Heat transfer in Newtonian fluids from external surfaces of bodies of various geometries has been the subject of numerous investigations during the past decades, since the pioneer work of Prandtl [

A procedure which belongs to the category of “wedge” methods and which provides a rigorous refinement of the local similarity concept is that of Merk [

An advance in the accuracy of boundary-layer series solutions was therefore made possible by Merk in 1959. He refined the “wedge method” proposed by Meksyn by choosing to treat the wedge parameter,

Some of the latest applications of the Merk-Chao-Fagbenle series solution technique have been universal boundary layer analyses of the mixed convection to Newtonian fluids by Cameron, M.R. et al. [

The purpose of the present investigation is to employ the Merk-Chao-Fagbenle procedure for the analysis of thermal boundary layer on non-isothermal surfaces by providing firstly the sequence of the differential equations governing the universal functions associated with the method and, secondly, to provide a tabulation of these and other related functions.

With the availability of such tabulation, the determination of the local wall shear and the surface heat transfer rates over the non-isothermal simple geometrical surfaces become a simple matter, once the outer stream velocity distribution is known. The development of the boundary layer and details of the velocity and temperature fields can be obtained with equal ease.

Here, consideration is given to the conservation equations for steady, laminar, non-dissipative, constant property boundary layer flow over two-dimensional or rotationally symmetrical (or axisymmetric) bodies of non-uniform surface temperature,

The boundary conditions considered by Merk-Chao-Fagbenle are,

where

The continuity equation is identically satisfied by introducing a stream function

In (6) and other equations which follow, one needs only to set

Following Chao and Fagbenle, the

A dimensionless stream function f is introduced, such that

From (6), (7) and (8), we obtain

where

is the wedge variable.

It was named by Go rtler as the principal function.

Using (9) in (1), the momentum equation with associated boundary conditions may be reduced to the following system:

and the boundary conditions are

The primes denote differentiation with respect to

The quantity

The above momentum Equation (11) is unchanged for the uniform surface temperature problem.

In order to transform Equation (3), we write

The boundary conditions are:

For

For

The dimensionless temperature is defined as:

Substituting (13) in (16), gives

Making use of (9) and (17) the transformation of (3) is now easily obtained as

while the boundary conditions (4) and (5) become:

The temperature parameter,

It can now be seen that with the variable wall temperature proposed in (13), Equation (18) contains an additional term:

With

Herewith, the transformations are complete and we have now to solve (11) with boundary conditions (12) and (18) with boundary conditions (19).

The appropriate series solution for (11) and (12) is that used by Chao and Fagbenle. It is given by

The appropriate series solution for (18) and (19) according to Chao and Fagbenle is stated as follows:

Upon substituting (21a) into (11) and (12) and collecting terms, free of

An inspection of the above set of equations shows that they can be integrated as if they were ordinary differential equations since, for any given stream wise location,

For energy Equations (18) and (19), series (21b) leads to hierarchy of differential equations which depends on 3 parameters Pr,

We seek a better series solution for (18) and (19). A better series solution which leads to only a 2-parameter universal functions dependence is:

Substituting (26) in (18) and (19) leads to the following hierarchy of differential equations:

With boundary conditions:

for

Thus all the temperature functions in the sequence dependent on 2 parameters,

The dependence of the temperature functions on the temperature parameter

With the form of the modified Mark-Chao-Fagbenle series proposed in Equations (21), (26), the higher-order solutions depend on the lower-order solutions, but not vice-versa. An inspection of the resulting equations also shows that there are no terms containing the derivatives of the parameter

When the solutions for the various

The local Nusselt number may be defined by

and one finds, after transformation

Chao and Fagbenle [

The

We first convert the equations stated above into the following simultaneous linear equations of first order as follows.

Then the Newtons method is applied to transform the problem into initial value one, where the initial conditions are:

Hence,

A computer program was written for the calculation of

The computation began with

Very high accuracy is required of the basic function

The iteration was to continue until the following conditions was met:

where

The results obtained (for the similarity equations) as a result agreed with the published data of Chao and Fagbenle within very close limits for all

The integration of the

A method of parameter differentiation was employed for the evaluation of these derivatives.

The integration of the

equation, the evaluation of the derivatives

of parameter differentiation.

The integration step size

The above choice of

The iterative routine for the computation of

. Wall derivatives of universal velocity functions

. Wall derivatives of universal velocity functions | Chao and Fagbenle [1974] | Present Results |
---|---|---|

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.85 0.90 0.95 1.00 | 0.216361405970 0.319269759903 0.400322595443 0.469599988372 0.531129630467 0.587035219193 0.686708181014 0.731940848489 0.774754580283 0.815491778633 0.854412131156 0.891758591602 0.927680039803 0.995836440590 1.05980777319 1.12026765739 1.14934554399 1.17772781922 1.20546125464 1.23258765687 | 0.216362100000 0.319270000000 0.400323600000 0.469600600000 0.531129700000 0.587034100000 0.686707100000 0.731939900000 0.774753200000 0.815490500000 0.854420200000 0.891757500000 0.927680300000 0.995835900000 1.059807000000 1.120268000000 1.149346000000 1.177728000000 1.205462000000 1.232588000000 |

. Wall derivatives of temperature function

. Wall derivatives of temperature function | Chao and Fagbenle [1974] | Present Results |
---|---|---|

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.85 0.90 0.95 1.00 | 0.4093363128 0.4367975318 0.4553719553 0.4695999884 0.4811774669 0.4909492790 0.5068537821 0.5135083784 0.5195184185 0.5249948751 0.5300221727 0.534660954 0.5389789351 0.5467728601 0.5536608195 0.5598232273 0.5626748955 0.5653917079 0.5679851340 0.5704652525 | 0.409336900000 0.436797900000 0.455372800000 0.469600600000 0.481177600000 0.490949300000 0.506853800000 0.513508600000 0.519518400000 0.524995000000 0.530022100000 0.534666100000 0.538979300000 0.546773100000 0.553661300000 0.559823600000 0.562675400000 0.565392200000 0.567986000000 0.570466000000 |

We can actually assess the accuracy of the present numerical program for the two basic functions

It is seen that for the entire range of

Computed values of

Computer printouts of the velocity functions

In design and other technological applications, it is, in most cases, the surface characteristics, such as the local frictional drag and the heat and mass transfer coefficients, which are of interest. The prediction of these quantities requires only the information on the wall derivatives:

It should be emphasized that the Merk-Chao-Fagbenle’s method as employed in this work is strictly applicable to incompressible, uniform property laminar boundary layer flows.

Since all tabulated data are available for various values of

At the forward stagnation point,

sional or axisymmetrical boundary layers.

With the aid of the universal functions the local heat (or mass) transfer coefficient can be obtained as follows:

where

and

for various values of the Prandt number.

The universal functions

The Chao and Fagbenle’s modification of Merk series was employed for the analysis of forced convection thermal boundary layer transfer for non-isothermal surfaces. The sequence of the differential equations governing the universal functions associated with the method has been presented together with the tabulation of these functions.