Effects of Averaging the Heat Transfer Coefficient on Predicted Material Temperature and Its Gradient

Gas turbines are widely used for power generation and aircraft propulsion. Though tremendous advances have been made since their debut over a half-century ago, there are still incredible opportunities to improve efficiency and service life [1–6]. For example, the thermal efficiency can be increased by further increasing the temperature of the hot gas entering the turbine component from the combustor, which requires advances in high-temperature resistant materials such as nickel- and rhenium-based superalloys and ceramic-matrix composites (CMCs). Also, since a substantial amount of the air entering the compressor is extracted to cool the turbine, cooling designs that can protect the turbine material effectively by using less cooling flow will also improve efficiency.

On cooling, the goal is to ensure that all material temperatures and their gradients to never exceed the maximum allowable for strengths and durability [4–6]. If the temperature exceeds the maximum allowable, then hot spots form, where the material's strength weakens. Temperature gradient is important in affecting thermal stress. Thus, cooling deserves serious attention because advanced turbines are designed to operate near the material's maximum allowable temperature and temperature gradients by using minimum cooling flow. Bottom line is that there is very little room for mistakes in the design of cooling.

Current analysis tools used to design cooling strategies do not always account for the effects induced by individual ribs and pin fins in internal cooling passages. Typically, a bank of ribs, pin fins, and/or their combinations are represented by an effective heat transfer coefficient [4,6], which smears out local variations induced by each rib or pin fin. These variations in the heat transfer coefficient can be as high as a factor of eight or more about a rib or a pin fin because of the stagnation and wake regions about them (see, e.g., Refs. [5,6]). References [7,8] showed that these variations in the heat transfer coefficient could produce hot spots. Thus, it is important to understand the consequences of averaging the heat transfer coefficient.

So far, no studies have been reported on the effects of averaging the heat transfer coefficient on the temperature or its gradient in the turbine material. The objective of this study is to examine the effects of averaging the heat transfer coefficient (HTC) on the temperature and its gradient in a flat plate, made of a nickel-based superalloy that is at steady state and operating under gas-turbine relevant conditions.

The organization of the remainder of this paper is as follows. First, the problem studied is described along with the problem formulation and the numerical method of solution. Afterward, the results generated are presented and discussed.

The problem studied is shown in Fig. 1. It consists of a flat plate with a periodic length L = 20 mm and thickness H, where H/L = 0.05, 0.1, 0.25, and 0.5. This plate is subjected to cooling on one side and heating on the other, where the length L can be viewed as a pitch between successive ribs or pin fins. The plate is made of nickel-based superalloy (Inconel-713LC) with temperature-dependent thermal conductivity [9]. However, the density of the superalloy is assumed to be a constant so that the geometry of the plate does not change due to thermal stresses. On the cooled side of the plate, the averaged HTC is fixed at $h¯$ = 1442.5 W/m² K, and the coolant temperature is fixed at T_c = 400 °C. On the heated side of the plate, either a constant heat flux of q_h″ = 68 W/cm² is imposed or a fixed convective environment with a constant HTC of h = 1167.2 W/m² K and constant hot-gas temperature of T_h = 1482 °C is imposed. Both sets of conditions on the heated side give rise to the same temperature distribution in the plate under steady-state conditions if the average HTC is used on the cooled side (i.e., if variations in the HTC are not accounted for). The heating and cooling conditions were adjusted to ensure that the maximum temperature in the plate, which occurs on the heated side, does not exceed the maximum allowable by the superalloy, if the average HTC is used. The maximum allowable is set at 900 °C.

In this study, $h¯$ = 1442.5 W/m²K, and the following variations were studied: h_H/h_L = 2, 4, and 8; and L_H/L = 0.1 to 0.9 in increments of 0.1. Some of the variations in h_H and h_L as a function of L_H/L are shown in Fig. 2(a) and given in Table 1.

where P = L_H if L_H/L$≤$ 0.5 and P = L − L_H = L_L if L_H/L$>$ 0.5. In this part of the study, $h¯$ was also set at 1442.5 W/m² K, and the following variations were examined: h_H/h_L = 8 and L_H/L = 0.1 to 0.9 in increments of 0.1. The variations in h_H and h_L as a function of L_H/L are shown in Fig. 2(b).

At this point, it is important to note that the design of cooling strategies cannot be decoupled from the convective environment on the heated side of the material being cooled. Without the coupling, the effects of averaging the HTC on the material temperature cannot be shown.

The problem described in the section Problem Description is modeled by the balance of thermal energy in two-dimensions with the Fourier law as the constitutive relation for the conduction heat transfer in the plate, where temperature-dependent thermal conductivity is accounted for (see, e.g., Ref. [10]).

Solutions to the aforementioned governing equations with the boundary conditions on the heated and cooled sides of the plate were obtained by a numerical method. The spatial domain was replaced by uniformly distributed grid points (i.e., x_i = (i − 1) Δx, i = 1, 2,…, IL and y_j = (j − 1) Δy, j = 1, 2,…, JL, where Δx and Δy are the grid spacing in the x and y directions, respectively). The spatial derivatives in the governing equations were approximated by second-order central difference formulas and cast in cell-centered finite-volume form.

Since the thermal conductivity is a function of temperature, the system of algebraic equations that results after replacing derivatives by difference operators is nonlinear. There are several ways to solve this nonlinear system of equations. One is to use a method such as the Newton–Raphson iteration. The other is to treat the thermal conductivity as locally constant at the temperature of the previous iteration so that the system of equations becomes linear. However, iterations must continue until both the temperature and the thermal conductivity in the plate converge. The latter method is much easier to implement and quite stable. Thus, this is the method employed in this study. The solution to the linear system of equations at each iteration was obtained by the Gauss–Seidel method.

In this study, grid independent solutions were obtained by using 2001, 101 uniformly space grid points across the plate (IL = 2001, JL = 101). All solutions given in the results section are iteratively converged on this grid. Convergence is assumed when there is a maximum change in temperature and the thermal conductivity between successive iterations plateaus. In this study, the maximum relative error is less than 10⁻⁸ at convergence.

As noted in the Introduction, the heat transfer coefficient (HTC) in internal-cooling passages can vary appreciably, up to a factor or eight or more about a heat transfer enhancement feature such as a pin fin or a rib because of stagnation and wake regions about a feature. Despite this, the computed or measured HTC is often averaged spatially in the spanwise direction relative to the flow or over some region such as a bank of pin fins or ribs when used in the design of cooling strategies. The objective of this study is to examine the effects of averaging the HTC on the temperature and its gradient in a flat plate at steady-state with relevant gas-turbine operating conditions.

Figure 2 and Table 1 summarize the cases studied to examine the effects of averaging the HTC. The results of this study are presented in Figs. 3–13 and Tables 2–6. In Tables 2 and 3, 1D denotes cases where the average HTC is imposed on the cooled side of the plate so that the conduction in the plate is 1D, and 2D refers to cases where variable HTCs are imposed. In Table 2 and in Fig. 5, the temperature T is normalized by T_max,1D so that wherever $∅$  = T/T_max,1D exceeds unity is where over temperature occurs (i.e., temperature greater than the maximum temperature predicted by using the average HTC). In this study, T_max,1D was set at 900 °C. Table 3 compares the normalized difference between the maximum and minimum temperatures in the plate when variation in the HTC is accounted for and when it is not. In Figs. 7 and 8 and in Table 5, normalized temperature gradients are compared. Figure 9 shows the range of Biot number as a function of L_H/L. In Figs. 10 and 11, normalized temperatures as a function of the plate's thickness are presented. Figures 12 and 13 and Tables 5 and 6 show the effects of abrupt versus gradual change in the HTC on temperature and its gradient. The details of these figures and tables are described below.

From Table 2, it can be seen that when the average HTC is used on the cooled side of the plate, the maximum and minimum normalized temperature, $∅max$ and $∅min$⁠, are 1 and 0.968, respectively, if H/L = 0.05 and 1 and 0.816 if H/L = 0.25. Also, when the average HTC is used, $∅max$⁠, $∅min$⁠, and the temperature distribution in the plate are identical whether heat flux or convective environment is imposed on the heated side of the plate.

When variations in the HTC on the cooled side is accounted for and constant heat flux is imposed on the hot-gas side, Table 2 shows that $∅max$ and $∅min$ in the plate can be as large as 1.363 and as small as 0.821 if H/L = 0.05 and as large as 1.130 and as small as 0.704 if H/L = 0.25. If a convective environment is imposed, then $∅max$ and $∅min$ can be as large as 1.202 and as small as 0.811 if H/L = 0.05 and as large as 1.079 and as small as 0.684 if H/L = 0.25. With $∅max$  = 1.363 and 1.202, these correspond to 1226.7 °C and 1081.8 °C, which exceed the temperature predicted by the average HTC by as much as 326.7 °C and 181.8 °C, respectively. Since 10–20 °C is considered the limit [2–6], this shows the danger of using the average HTC because it can severely underpredict the maximum temperature in the material.

Tables 2 and 3 and Figs. 3–5 show the normalized temperature and the difference between the maximum and minimum temperatures to be greater if the heat flux is imposed on the heated side of the plate instead of the convective environment when the variation in the HTC on the cooled side is accounted for. This is because when a convective environment is imposed on the heated side, the heat flux on the heated side depends on the HTC as well as the surface temperature. The surface temperature on the heated side will increase (i.e., get hotter) if the cooled side could not remove the thermal energy transferred from the heated side. Similarly, the surface temperature on the heated side will decrease (i.e., get cooler) if the cooled side could remove all of the thermal energy transferred from the heated side and more. Thus, the heat transfer is high where the HTC on the cooled side is high, and the heat transfer is low where the HTC on the cooled side is low. However, if the heat flux on the heated side is fixed, then the temperature on the heated side must increase further where the HTC on the cooled side is low and decrease further where the HTC on the cooled side is high. Thus, the plate is cooler in regions where the HTC on the cooled side is higher and is hotter where the HTC on the cooled side is lower as shown in Figs. 3–5. Basically, when the HTC is variable, the heat transfer from the heated side to the cooled side seeks the path of least resistance.

Figure 6 shows the maximum normalized temperature $∅max$ as a function of L_H/L and h_H/h_L. From this figure, it can be seen that $∅max$ exceeds unity whenever HTC varies and that the magnitude of the over temperature beyond the maximum predicted by using the averaged HTC (i.e., T_max,1D = 900 °C) increases as h_H/h_L increases and is higher if heat flux is imposed on the heated side instead of a convective environment. On the peak $∅max$⁠, it shifts to lower L_H/L as h_H/h_L increases. When the heat flux is imposed on the heated side and h_H/h_L = 8, $∅max$ reaches its peak when L_h/L is between 0.3. When h_H/h_L = 4 and 2, $∅max$ reaches its peak when L_h/L is 0.4 and 0.5, respectively. When a convective environment is imposed on the heated side, $∅max$ reaches its peak at a slightly higher L_H/L value.

where C_o, C₁, C₂, and C₃ are given in Table 4. Equation (3) was derived by nonlinear least-square fit of the simulation data in Table 2. The maximum relative error of the fit is less than 3%.

The model given by Eq. (3) can be used in design tools that employ averaged HTCs if the designer can provide an estimate of h_H/h_L, L_H/L, and the Biot number about a heat transfer enhancement feature such as pin fins or ribs. When the averaged HTC is used, the predicted maximum temperature is denoted as $Tmax.1D$⁠. The actual maximum temperature that account for the variation in HTC induced by a heat transfer enhancement feature is $Tmax=θ Tmax.1D$⁠, where $θ$ is given by Eq. (3).

Table 3 shows the difference between the maximum and minimum temperatures in the plate normalized by the maximum difference predicted by using the average HTC, denoted as $Δ∅max$⁠, as a function of L_H/L and h_H/h_L. If heat flux is imposed on the heated side, this table shows that the normalized difference can be as high as 16.3 if H/L = 0.5 and 2.16 if H/L = 0.25. If a convective environment is imposed on the heated side, then the normalized difference can be as high as 11.7 if H/L = 0.5 and 2.04 if H/L = 0.25. Such differences between the maximum and minimum temperatures can produce significant thermal stresses, and so need to be accounted for.

To further examine the effects of variations in the HTC, the normalized temperature gradient in the plate, denoted as λ, is given in Table 5 and in Figs. 7 and 8. In this table, the maximum temperature gradient in the plate when the average HTC is used is |ΔT|_max,1D = 28,142 °C/mm if H/L = 0.05 and (|ΔT|_max,1D = 27,074 °C/mm if H/L = 0.25. From Table 5, the maximum λ, λ_max, can be as low as 0.16 to as high as 6.42 if H/L = 0.05 and as low as 0.16 to as high as 6.70 if H/L = 0.25. Thus, using average HTCs can significantly underpredict temperature gradients and hence thermal stresses in the material. In addition, when the average HTC is used, the orientation of the principle stress acts only in the y direction (i.e., across the plate), whereas when variable HTCs are accounted for, the thermal stress acts not only in the y direction but also in the x direction (i.e., across and along the plate).

Figure 7 shows the normalized temperature gradient, $λ$⁠, for h_H/h_L = 8 with L_H/L ranging from 0.1 to 0.9. By comparing Fig. 7 with Fig. 2, it can be seen that the higher $λ′s$ occur in regions where the HTC changes abruptly. From Fig. 5, it can be seen that this region has $∅$ less than unity. Thus, the highest temperature gradients and hence the highest thermal stresses occur in regions where the HTC changes the most and where the temperatures are typically lower.

Figure 8 and Table 5 show the peak normalized temperature gradient, λ_max, as a function of L_H/L and h_H/h_L. From there, it can be seen that λ_max is highest when h_H/h_L = 8 and L_H/L = 0.1. As L_H/L increases from 0.1 to 0.9 and as h_H/h_L decreases from 8 to 2, λ_max decreases from 6.42 to 1.39 if H/L = 0.5 and from 6.70 to 1.32 if H/L = 0.25. This is because for a given h_H/h_L, (h_H − h_L)/$h¯$ decreases as L_H/L increases, indicating that the temperature gradient in the material is a strong function of the variation in the HTC. Also, the peak normalized temperature gradient is higher if heat flux is imposed on the heated side instead of a convective environment.

On temperature gradients, Table 5 and Figs. 7 and 8 clearly show the deficiencies of using average HTCs. In particular, when the average HTC is used, the temperature gradients across the plate cannot be discerned. Also, the magnitudes of the gradients are severely underpredicted.

This section examines the effects of H/L as well as h_H/h_L, which strongly affect the Biot number, a parameter that links the hot-gas side, the cooled side, and the plate. In this subsection, the effects of the Biot number are examined by varying H − H/L = 0.05, 0.1, 0.25, and 0.5—where L = 20 mm, h_H/h_L = 8, and L_H/L = 0.1. With these H values and using averaged HTCs (1442.5 W/m² K for the cooled side and 1167.2 W/m² K for the heated side), the Biot number varies from 0.026 to 0.26. As shown in Fig. 9, the actual Biot number based on h_H and h_L can be as low as 0.027 and as high as 0.40.

Figure 10 shows the temperature profiles on the heated and cooled sides of the plate as a function of H/L, which in turn is a function of the Biot number. From this figure, it can be seen that the temperature variation along the surface of the plate—both the heated and the cooled sides—reduces as H/L or Biot number increases except on the cooled side, where the HTC changes abruptly. Also, the temperature difference between the heated and cooled sides of the plate at any x/L increases with H/L, which is expected since the Biot number increases with H/L.

Figure 11 compares the maximum and minimum normalized temperatures, $∅max$ and $∅min$⁠, in the plate as a function of H/L with L_H/L = 0.1 and h_H/h_L = 8. From this figure, it can be seen that $∅max$ reduces from 19.4% to 4.1% as H/L increases if constant heat flux is imposed on the heated side and from 11.3% to 1.9% if convective environment is imposed. For $∅min$⁠, it is not a monotonic function of H/L. It has a peak near H/L = 0.1, which corresponds to a Biot number of 0.052 based on the averaged HTCs.

The discussion presented so far is for an abrupt or step change in the HTC from high to low. Since abrupt changes will never occur in practice, the question is will having a more gradual change in the HTC invalidate all of the discussions and conclusions made so far based on a study involving only abrupt changes in the HTC. This section addresses this question by examining effects of abrupt versus gradual changes in the HTC on the temperature distribution. This study examines the gradual change based on sine waves with a period of 2L_H, if L_H/L is less than or equal to 0.5 and a period of 2(L − L_H) = 2L_L if L_H/L is greater than 0.5, where L_H/L = 0.1 to 0.9 so that a comprehensive set of periods is studied from as small as 0.1 L to as large as L. On the variation in the HTC, only the worst-case scenario is examined, namely when the variation in HTC is h_H/h_L = 8. The abrupt and gradual changes in the HTC studied are shown in Figs. 2(a) and 2(b). Results for abrupt changes in the HTC were already presented. Results for the gradual changes in the HTC are presented in Figs. 12 and 13 and in Tables 5 and 6. Figure 12 shows the heat flux on the cooled side of the plate; Fig. 13 shows the temperature on the cooled and heated sides of the plate; Table 6 gives $∅max$ and $∅min$⁠, and Table 7 gives $λmax$ and $λmin$⁠.

From Fig. 12, it can be seen that with a gradual change in the HTC on the cooled side, the heat flux on the cooled side becomes smooth. From Fig. 13 and Table 6, it can be seen that the minimum temperature in the plate, which occurs on the cooled side, is essentially unaffected by abrupt or gradual changes in the HTC. The maximum temperature, however, is affected. With gradual changes in the HTC, the maximum temperature in the plate, which occurs on the heated side, is always lower than when there is an abrupt change. For the range of parameters studied, the maximum reduction in the maximum temperature is 7.85% when switching from abrupt to gradual change in the HTC. Table 6 gives the relative difference in the maximum and minimum temperatures when switching from abrupt to gradual. From this table, it can be seen that maximum change in the minimum temperature is 1.14%, and the maximum change in the maximum temperature is 7.85% if constant heat flux is imposed on the heated side. Thus, the maximum normalized temperature in the plate reduces from 1.363 to 1.320, which translates to an over temperature of 288.0 °C instead of 326.7 °C (i.e., temperature exceeding the maximum predicted by using an average HTC), indicating that over temperature is still a major issue. If the heated side of the plate has an imposed constant convective environment, then the over temperature is 163.8 °C instead of 181.8 °C when switching from abrupt to gradual change in the HTC. Table 7 gives the relative difference in the maximum and minimum temperature gradients when switching from abrupt to gradual. The maximum normalized temperature gradient can be as high as 6.42 if the change in HTC is abrupt and 3.66 if gradual. This reduction from 6.42 to 3.66 is because with a gradual change in the HTC, the heat flux at the interface where the HTC changes is broadened from one point to a region of thickness L_H (see Fig. 12).

Thus, abrupt changes in the HTC can overpredict the severity of the over temperature by up to 7.85%. It also overpredicts the maximum temperature gradient, where the relative difference can be as high as 79.6%. Since the gradual changes in the HTC studied are fairly comprehensive, the aforementioned represents worst-case corrections for results based on abrupt changes.

A computational study was performed to examine the effects of averaging the HTC on the predicted temperature and its gradient in a flat plate that is heated on one side by a constant heat flux or a fixed convective environment and cooled on the other side by a convective environment that has variable HTC. Results obtained showed that if the averaged HTC is used on the cooled side instead of the actual variable HTC, then the cooling deemed adequate will produce considerable over temperature. For the cases where the HTC changes abruptly from high to low, the over temperature can be as much as 36.3% for H/L = 0.05 and 13.0% for H/L = 0.25 if constant heat flux is imposed on the heated side and as much as 20.2% for H/L = 0.05 and 7.9% for H/L = 0.25 if constant convective environment is imposed on the heated side. The over temperature is less if the change in the HTC is gradual instead of abrupt, but not significant. As the plate's Biot number based on the average HTC increases from 0.026 to 0.26, the over temperature was found to reduce from 19.4% to 4.1% if a constant heat flux is imposed and from 11.3% to 1.9% if convective environment is imposed. When the HTC changes abruptly from high to low, the temperature gradient was found to increase by a factor of 6.42 for H/L = 0.05 and 6.70 for H/L = 0.25 if constant heat flux is imposed on the heated side and by 6.11 for H/L = 0.05 and 5.93 for H/L = 0.25 if constant convective environment is imposed. The temperature gradient is less if the change in the HTC from high to low is gradual instead of abrupt by as much as 79.6%.

Since averaging the HTC could produce significant errors in the predicted temperature and its gradients in the turbine material, it is important to account for these effects when averaged HTCs are used to design cooling strategies. A reduced-order model was developed in this study to account for the effects of averaging the HTC on the predicted maximum temperature.

This work was supported by the Ames Laboratory and the National Energy Technology Laboratory with funding from the Department of Energy under Contract No. DE-AC02-07CH11358/Agreement No. 26110-AMES-CMI. The authors are grateful for this support. The authors are also grateful to Robin Ames and Rich Dennis for the discussions on the research and the results.

Bi_avg =

Biot number based on $h¯$ and k = 25 W/m K: Bi = (L/k)/(1/h_h + 1/ $h¯$⁠)

Bi_H =

Biot number based on h_H and k = 25 W/m K: Bi = (L/k)/(1/h_h + 1/h_H)

Bi_L =

Biot number based on h_L and k = 25 W/m K: Bi = (L/k)/(1/h_h + 1/h_L)

$h¯$ =

average HTC on the cooled side

h_h =

heat transfer coefficient (HTC) on the heated side with imposed convection

h_H, h_L =

high HTC and low HTC on the cooled side

k =

thermal conductivity of plate's material

L =

periodic length of a flat plate

P =

period of sinusoidal change in HTC from h_H to h_L (Eq. (2))

q″ =

wall heat flux

T =

temperature

T_h =

hot-gas temperature on the heated side with imposed convection

T_max =

maximum temperature in the plate

T_min =

minimum temperature in the plate

x, y =

coordinate along and across plate (Fig. 1)

Greek Symbols

$Δ∅max$ =

normalized difference between maximum and minimum temperature: $Δ∅max=(Tmax−Tmin)/(Tmax−Tmin)1D$

$θ$ =

correction factor to account for variation in the HTC (Eq. (3))

$λ$ =

normalized temperature gradient: $λ=|∇T|/|∇T|max,1D$

λ_max/min =

normalized maximum/minimum temperature gradient, $λmax/min=|∇T|max/min/|∇T|max,1D$

ρ =

density, kg/m³

$∅$ =

normalized temperature based on  °C: $∅=T/Tmax,1D$

∅_max,min =

normalized maximum or minimum temperature based on  °C: $∅max or min=Tmax or min/Tmax,1D$