Analytical models are useful to study theoretically optimum solutions. They give a methodology for design and assessment of attainable systems [2]. Usually, they are developed with the assumption that

- mixing
- boundary layer viscous effects
- trailing edge blockage
- endwall losses

are additive and can therefore be determined independently.

## 1. 1D Mixing of injected flow

Denton [3] gives a method to obtain the mixed flow entropy generation. The method uses the properties resulting from the mixed flow Mach number (*M*_{m}) computed with eq. 1 for two streams. It assumes that:

- the flows initially occupy equal areas
- the flows mix at constant area
- the flows have equal velocity (
*U*) - the flows have equal static pressure (
*p*) - wall friction is negligible
- the process is adiabatic
- the inlet properties are know
- outlet area
*A*_{m}is known

He shows that entropy generated as a result of differences in stagnation pressure is almost independent of the difference in stagnation temperature.

Hartsel [4] provides two mixing models where coolant is injected at a single location into a uniform mainstream flow. These are easier to apply than 1D constant area methods. Moreover, they provide similar results.

Model I (eq. 2) assumes that coolant and mainstream flow mix at constant static pressure.

In the second model (Model II), the coolant is injected at an angle *α*. The model results from Shapiro’s [5]-[6] influence coefficients method. Shapiro’s method is derived by writing nine physical equation and definitions in differential logarithmic form.

This allows to simplify the separation of the resulting thirteen variables.

- state equation
- sound velocity
- M definition
- impulse definition
- continuity equation
- momentum equation
- energy equation
- II law of thermodynamics
- equations for total pressure and total temperature in case of constant
*c*_{p}, and molecular weight

Subsequently, the linear algebraic equations system is solved. Four independent variables (*X’*) are written as a function of the remaining nine dependent ones (*X’’*). The coefficients of the dependent variables are called influence coefficients *δ _{X’,X’’}* and are reported in a table to simplify their manipulation.

Specifically, the influence coefficients *δ*_{p0,A} , *δ*_{p0,T0} and *δ*_{p0,m} (eq. 4) can be found with Shapiro’s method.

The method requires that the change in total pressure (*dp*_{0}) is written in terms of

- change in area
- change in total temperature
- change in mass flow rate (
*dA*,*dT*_{0}and*dm*).

Therefore, that the flow is characterised by constant external and frictional forces.

It is a linear approximation in the independent variables *dT*_{0}/*T*_{0} and *dm*/*m*, and assumes that flows have equal static pressure at injection.

It is more precise than the approach, employed by many workers, of accounting for the mixing losses by considering that they only result from the loss of the coolant kinetic energy. Indeed, this apprach does not account for the static pressure recovery [3].

## 2. Mixed out layer

Hartsel [4] also develops the TOTLOS method (from “TOTal LOSs”) to model more complex geometries where multiple injections and different mainstream conditions are present.

He considers the mainstream displaced by the thickness of a partially mixed coolant layer (Figure 1), which thickness is shown to have little effect on the calculation. Mixing layers then mix with the mainstream at a constant pressure (Model I), which equals the average pressure at the outlet *p*_{out}.

Viscous boundary layer and effects of trailing edge blockage are modelled separately with Stewart’s method [7], and their effect is supposed to be superimposable.

Povey [8] uses the method to model the effect of film cooling on turbine capacity.

Figure 1: TOTLOS mixing layer method (adapted from [4).

## 3. Multiple film cooling rows

The majority of authors employ the Sellers superposition method [9], and its multiple variants, to model the effect of multiple film cooling rows.

Information in the model is transferred in only one direction.

The mean film temperature *T*_{ad} of the first layer is determined from eq. 5. The subscript “ad” derives from reference to the temperature the wall would have if:

- it was adiabatic
- there was a single film

The procedure is repeated for additional injections, by considering that from the second layer *T*_{01 }in eq. 5 is substituted by *T*_{ad} from the later below.

Although Sellers’ method has received conspicuous attention by the research community, its applicability is limited. This is because energy is not conserved in the model. Consequently, it lacks in accuracy.

To overcome the issue, Kirollos et al. [10] developed a new method (Figure 2). An open access MATLAB finite volume script was provided. In their method, enthalpy of the mainstream-coolant system is conserved.

The results where compared with previous models against CFD (CFD represented a converging duct cooled using six rows of staggered cooling holes along the flat wall). They show to have overcome the underprediction of effectiveness.

Figure 2: Schematics of a method for multiple film cooling rows where enthalpy is conserved (from [10]).

## 4. Conjugate models

Kirollos et al. [10] method considers an adiabatic wall. The inclusion of a conductive wall results in a conjugate model that allows studying the simultaneous effect of internal and film cooling. Such models might be used to study:

- the effect that certain unmatched nondimensional parameters have on metal effectiveness [11]
- the effects of a reverse-pass cooling system for improved performance [12]
- the effects on leading-edge deterioration. This could result from cooling holes size reduction due to both mainstream or coolant flow particulates deposition. It causes reduced coolant mass flow for a given pressure ratio across the cooling holes.

Kirollos et al. [12] give an example of a 2D numerical conjugate heat transfer model using finite volume methods. In the model, iteration is used to calculate the steady state temperature field for both fluid and solid.

The model shows that in most situations hot and cold streams flowing in opposite directions outperform the co-current arrangement. This is in agreement with the physics of other heat exchangers.

## 5. Bibliography

[1] Rey L. (2021). Improved assessment and modeling of High Pressure Nozzle Guide Vanes Leading Edge deterioration.

[2] Kirollos B., Povey T. (2016). Cooling optimization theory – part II: optimum internal heat transfer coefficient distribution.

[3] Denton J. (1993). Loss mechanisms in turbomachinery.

[4] Hartsel J. E. (1972). Prediction of effects of mass-transfer cooling on the blade-row efficiency of turbine airfoils.

[5] Shapiro A. H (1954). The Dynamics and Thermodynamics of Compressible Fluid Flow.

[6] Shapiro A. H., Hawthorne W. R., Edelman G. M. (1947). The mechanics and thermodynamics of steady one-dimensional gas flow with tables for numerical solutions.

[7] Stewart Warner L. (1955). Analysis of two-dimensional compressible flow loss characteristics downstream of turbomachine blade rows in terms of basic boundary layer characteristics.

[8] Povey T. (2010). Effect of Film Cooling on Turbine Capacity.

[9] Sellers J. P. (1963). Gaseous film cooling with multiple injection stations.

[10] Kirollos B., Povey T. (2015). An Energy-Based Method for Predicting the Additive Effect of Multiple Film Cooling Rows.

[11] Cartlidge J., Povey T. (2020). Scaling Overall Metal Effectiveness from Highly Matched Test Rig to Engine Conditions.

[12] Kirollos B., Povey T. (2014). Reverse-Pass Cooling Systems for Improved Performance.