## Abstract

The Graetz problem in a transpiration-cooled channel was analytically attacked so as to explore the developing temperature field due to a sudden change in wall temperature of the channel subject to an arbitrary distribution of the local mass flux over the porous wall. Analytical expressions for the developments of the thermal boundary layer thickness, wall temperature, and Nusselt number were obtained for the thermal entrance region, assuming hydrodynamically forced convective flow in a channel with a locally variable blowing mass flux. When the blowing mass flux is kept constant over the wall surface, the cooling by the coolant is less effective near the entrance, thus, exposing to danger of thermal damage. This study reveals that the blowing mass flux graded inversely proportional to one-third power of the axial distance is quite effective to keep the wall temperature uniform. Numerical calculations based on finite volume method were also carried out to verify the analysis. The findings from this study can be applied to possible thermal managements of heat generating stacks such as in EV batteries and PEMFC, in which temperature uniformity is essential for product longevity.

## 1 Introduction

The Graetz problem for the high Prandtl number case refers to the problem of developing temperature field due to a sudden change in wall temperature, in which the flow is already hydrodynamically fully developed, while a thin thermal boundary layer just starts to form there [1]. In fact, the Graetz model serves as a reasonable convective entrance model for all forced convective flows of high Prandtl number fluids in conduits, since the velocity field of such high Prandtl number fluid develops much faster than the temperature field. In these cases, one may assume that the flow in a channel spreads its fully developed velocity profile almost immediately after the entrance, whereas the temperature profile starts to develop right from the entrance. The Graetz problems associated with laminar and turbulent forced convection channel flows are found in many engineering processes and thus have been widely investigated theoretically and experimentally. Furthermore, recent investigations of microchannels have driven an increasing interest in thermal entrance characteristics in laminar flow regimes. A considerable number of reviews on this topic may be found in the literature, e.g., [25].

Due to the advent of advanced permeable wall materials, transpiration cooling is now considered to be one of the most effective ways to thermally protect hot wall surfaces. There exist many cases in which inner wall surfaces are exposed to a high temperature environment, such as in jet engines, rocket engines, and turbines. In such cases, a hot gas enters in a passage whose wall surface can be thermally damaged due to high thermal loads unless some positive cooling is used. In a transpiration cooling system, a coolant fluid from a cold reservoir passes through a permeable wall, effectively absorbing the conductive heat within the permeable wall structure, whose surface is exposed to the hot gas environment. As the coolant exits the permeable wall and then covers the hot inner wall surface, it insulates the wall from the hot gas, reducing the heat flux entering into the wall. The transpiration cooling is considered to be more thermally effective to the film cooling in which the coolant is injected through a limited number of rows of discrete holes on the wall surface.

In this study, the Graetz version of forced convective channel flows subject to locally variable blowing ratio will be treated in consideration of possible engineering applications to controlled transpiration cooling systems. It is well-known that, for the case of uniform blowing over the uniformly porous wall, the fully developed flow field is established in which the velocity profile becomes independent of axial distance, while its bulk velocity increases linearly as the coolant fluid is blown at a constant blowing ratio into the channel [6]. The momentum transfer in channel and duct flows with uniformly porous walls with fluid injection (or suction) has been theoretically attacked by a considerable number of researchers, including Berman [7], Sellars [8], Yuan [9], Donoughe [10], Eckert et al. [11], and White [12]. Zhou and Catton [13] and Zhou et al. [14] introduced a volume averaging theory based modeling and its closure for cooling problems encountered in engineering applications. A volume averaging procedure similar to theirs have been successfully applied for transpiration cooling in a channel to attack the corresponding energy balance equation by Zhang et al. [15] and Bai et al. [16].

Recently, a two-domain analytical approach has been proposed by Nakayama et al. [17] to reveal the fully developed temperature fields in forced convective flows in a channel with uniform porous walls subject to transpiration cooling. Both temperature fields in the porous wall domain and the channel domain with uniform porous walls are treated simultaneously and matched on the interstitial porous wall. However, little has been reported in the literature with thermal entry problem of forced convective flow in a transpiration cooled channel. In many cases of transpiration cooling, the wall is partially cooled by transpiration, where the thermal damage is expected unless some active cooling is taken. Thus, the Graetz version of forced convective channel flows subject to locally graded mass flux must be fully explored if one is to apply such a cooling system to control the wall surface temperature as desired.

In this study, an analytical solution for the temperature field in the thermal entrance region is obtained assuming hydrodynamically fully developed forced convective flow in a channel in which the coolant fluid is supplied through the porous wall at a locally variable blowing mass flux. When the blowing mass flux is kept constant over the wall surface, the cooling by the coolant is less effective near the entrance since the coolant film covering the wall is thinner, and thus the wall temperature is higher there. The task in this study is to find the cooling effectiveness as a function of the level of the blowing mass flux and also a local distribution of the blowing mass flux which makes the wall temperature uniform over the entire transpiration-cooled wall section. Such a local distribution of the blowing mass flux can be experimentally implemented by grading the blowing rate locally by means of variable permeability or local pressure distribution within the coolant reservoir.

Numerical calculations based on finite volume method will also be carried out to verify the analysis. The results in this study can be quite useful for effective cooling systems for heat generating stacks such in EV batteries and PEMFC. In these cases, it is required to control its average temperature to the desired working temperature as close as possible, and at the same time, to keep the temperature distribution as uniform as possible, within a certain allowable deviation from its average [18,19].

## 2 Physical Model for Forced Convective Flow in a Channel Subject to Blowing

Thermal entrance problem known as Graetz problem is considered for the case of transpiration-cooled channel flow. Hydrodynamically fully developed hot fluid of uniform temperature $TFB(0)$ enters at the mass flowrate per unit area $GH (kg/m2s)$ into a channel of height 2H, as shown in Fig. 1. Both upper and lower porous walls are subject to blowing at locally graded mass flux $G(x)($kg/s$m2)$ (though the case of constant G is illustrated in the figure for brevity). The coolant fluid of the thermal properties as the same as those of the hot fluid is supplied from the cold reservoir of constant temperature $Tc$ outside, and enters into the porous wall at the variable mass flux $G(x)$. The temperature of the coolant fluid rises from $Tc$ to $Tw$ as the coolant passes vertically through the porous wall and exits from the inner porous wall surface to the main stream of hot fluid. Thus, the heat transferred from the flowing hot fluid to the porous wall surface, conducts through the porous material, heating up the coolant fluid, as it travels through the porous wall. The unknown temperature distribution over the porous surface $Tw(x)$ will be determined in consideration of the heat balance in both porous wall and channel domains.

Fig. 1
Fig. 1
Close modal

### 2.1 Momentum Balance.

The continuity and momentum equations for the hot fluid in a channel can be given in terms of the velocity components u and v as
$ρf∂u∂x+ρf∂v∂y=0$
(1)
$ρfu∂u∂x+ρfv∂u∂y=−∂p∂x+μf∂2u∂y2$
(2)

White [6] and many others pointed out that similarity velocity profiles exist for hydrodynamically fully developed flows with constant blowing mass flux. Their arguments can be extended to the cases of variable blowing mass flux $G(x)$.

Thus, with the help of the mass conservation equation (1), one may assume that the streamwise velocity distribution is given by
$u(x, y*)=uB(x)f(η*)=uB(0)(1+∫0xF(x)Hdx)f(η*)$
(3)
where $uB(x)$ is the bulk mean velocity, and
$η*=y*H$
(4)
is the dimensionless vertical coordinate measured from the inner porous wall surface, while
$F(x)≡G(x)GH=G(x)ρfuB(0)$
(5)
is the blowing ratio (i.e., injection ratio). The mass conservation equation may be integrated as
$ρfv−G(x)=−∫0y*ρf∂u∂xdy*=−ρfduB(x)dxH∫0η*f(η*)dη*= −G(x)∫0η*f(η*)dη*$
(6)
which satisfies $vη*=1=0$ at the channel center. Equation (6) can be substituted into Eq. (2) to eliminate $v$ as
$Re(f2+(1−∫0η*f(η*)dη*)dfdη*)=Λ+d2fdη*2$
(7)
where
$Re(x)=G(x)Hμf$
(8)
is the Reynolds number based on the blowing mass flux. The dimensionless pressure gradient $Λ$ is defined by
$Λ=−H2μfuB(x)dpdx$
(9)

Equation (7) is exact only for the case of constant $Re$, since the velocity profile function $f$ cannot be a function of $η*$ alone, when $Re$ varies with $x$. However, a fairly accurate integral solution with locally variable $Re$ is possible as long as the blowing ratio is moderate. In fact, it will be shown shortly that the streamwise velocity profile function $f(η*)$ is quite insensitive to the variation of $Re$, and almost solely depends on $η*$.

Thus, an integral solution is sought for the velocity profile function $f(η*)$, in which the pressure drops according to the distribution of the blowing ratio $F(x)$
$p(x)−p(0)=−ΛμfuB(0)εH2(x+∫0x∫0xF(x)Hdxdx)$
(10)
where $Λ$ can be determined for given $Re$. The differential equation (7) can be solved by a standard numerical integration scheme such as Runge–Kutta–Gill method (e.g., Ref. [20]), using the boundary and compatibility conditions as follows:
$dfdηη*=1=0$
(11a)
$fη*=0=0$
(11b)
and
$∫01fdη*=1$
(11c)
Alternatively, Yi [21] demonstrated that sufficiently accurate velocity profiles valid for any finite blowing ratio are obtainable, using the integral version of the momentum equation with the foregoing appropriate conditions. To this end, the following velocity profile function with the unknown velocity gradient at the wall, $f′(0)≡(df/dη*)η*=0,$ is introduced:
$f(η*)=f′(0)η*+12−5f′(0)2η*2−12−4f′(0)3η*3$
(12)
which automatically satisfies the foregoing appropriate conditions given by Eqs. (11a)(11c). The dimensionless pressure gradient $Λ$ may be expressed in terms of $f′(0)$ by writing the momentum equation (2) at the channel wall
$Λ=−d2fdη2η*=0+Redfdηη*=0=(5+Re)f′(0)−12$
(13)
where the conditions (11b)(11c) are used. Thus, the unknown parameter $f′(0)$ can be determined using an integral form of the momentum equation (11) with $Λ given by$ the forgoing equation
$Re∫01(f2+(1−∫0η*f(η*)dη*)dfdη*)dη*=−dfdηη*=0+(5+Re)f′(0)−12$
(14a)
which under the conditions given by Eqs. (11a)(11c) reduces to
$2Re(∫01f2dη)=(4+Re)f′(0)−12$
(14b)
This integral momentum equation, when the velocity profile function (12) is substituted, yields the following quadratic equation in terms of the unknown parameter $f′(0)$:
$2Re(f′(0))2−(131Re+420)f′(0)+12(26Re+105)=0$
(15)
which can readily be solved for given $Re$ as
$f′(0;Re)=420+131Re−4201+1730Re+4195040Re24Re$
(16)
It is well-known (Ref. [6]) that the velocity profiles $f(η*)$ for the cases of infinitely small and large F are given by
$f(η*)=32(1−(1−η*)2 ) : F→0$
(17a)
$f(η*)=π2cos(π2(1−η*)) : F→∞$
(17b)

As presented in Fig. 2, Eq. (16) yields $f′(0;0)=$ 3.0, reducing Eq. (12) to the exact profile, namely, Eq. (17a), for $Re→0$ (i.e., $F→0$), whereas it gives $f′(0;∞)=$ 2.475 for $Re→∞$ (i.e., $F→∞$), which is very close to the exact value $(π/2)2=$ 2.467 based on Eq. (17b).

Fig. 2
Fig. 2
Close modal

As can be observed in Fig. 2, the velocity gradients $f′(0)$ at the wall for these limiting cases, namely, $f′(0;0)=$ 3.0 for $F→0$ and $f′(0;∞)=$ 2.475 for $F→∞$, respectively, are fairly close to each other, despite of being the two limiting cases. In fact, the streamwise velocity profiles are quite alike for all possible cases of moderate blowing ratio. Since the blowing flow ratio F ($∝Re$) in the case of transpiration is typically less than a few percent, the streamwise velocity profiles may be approximated by Eq. (17a) (i.e., $f′(0;0)=$ 3.0) for the all cases of laminar flow in the transpiration-cooled channels.

### 2.2 Energy Balance.

Focusing on the thermal boundary layer developing over the porous wall subject to blowing, the energy conservation equation for the hot fluid flowing in the channel is given by
$ρfcpf∂uTF∂x+ρfcpf∂vTF∂y*=∂∂y*kf∂TF∂y*$
(18)
where the properties of the hot fluid, the density $ρf$, thermal conductivity $kf$, and specific heat capacity at constant pressure $cpf$ of the hot fluid are assumed to be the same as those of the coolant fluid. $u$ and $v$ are the hot fluid velocity components in the x and y directions, while $TF$ is the temperature of the hot fluid. The thermal boundary conditions for the energy equation (18) are given by
$TFx=0=TFB(0)$
(19a)
and
$TFy*=0=Tw(x)$
(19b)
where the wall temperature $Tw(x)$ must be determined such that it satisfies the heat balance between the heat flux on the wall and the enthalpy of the coolant fluid supplied from the reservoir of constant temperature $Tc$. von Wolfersdorf [22] pointed out that conductive heat absorbed into the porous wall, heats up the coolant gas, increasing its temperature from $Tc$ to $Tw$ as it travels through the porous wall. Thus, the following heat balance holds:
$Gcpf(Tw−Tc)=kf∂TF∂y*y*=0$
(20)
As done in the momentum equation, one may apply an integral procedure also to this thermal entry problem, and integrate the energy equation (18) across the developing thermal boundary layer thickness $δ$. Thus, integrating Eq. (18) from $y*=0$ to $δ$, with reference to Fig. 1, along with the continuity equation (1) and boundary conditions (19a) and (19b), one obtains the integral energy equation as follows:
$ddx∫0δρfcpfu(TF−TFB(0))dy*+Gcpf(TFB(0)−Tw)=−kf∂TF∂y*y*=0$
(21a)
which can be combined and integrated with the heat balance equation (20) to give
$∫0δρfu(TF−TFB(0))dy*=(Tc−TFB(0))∫0xGdx$
(21b)
where $u$ in this Graetz problem can be approximated by the linear velocity profile since the thermal boundary layer is much thinner than the channel height
$u(x, y*)=uB(x)f(η*)=uB(0)(1+∫0xF(x)Hdx)f(η*)≅uB(0)(1+∫0xF(x)Hdx)f′(0;0)y*H$
(22)
Hence, by substituting Eq. (22) into Eq. (21b), one obtains
$3(1+∫0xF(x)Hdx)(δH)2∫01ζθ(ζ)dζ=∫0xF(x)Hdx$
(23)

where the approximation $f′(0;Re)≅f′(0;0)=$ 3.0 is used since $f′(0;Re)$ is insensitive to $Re$ ($∝F$) in a moderate range of the blowing ratio, as confirmed in Fig. 2.

The temperature profile may be assumed as
$θ(ζ)≡TF−TFB(0)Tc−TFB(0)=θw((1−ζ)2+c(1−ζ)3−c(1−ζ)4)$
(24)
where
$ζ=y*δ$
(25)
The temperature profile $θ(ς)$ automatically satisfies
$θ(0)=θw$
(26a)
$θ(1)=0$
(26b)
and
$dθdζζ=1=0$
(26c)
Writing the original energy equation (18) at the wall as
$cpfG∂TF∂y*y*=0=kf∂2TF∂y*2y*=0$
(27)
which, when the temperature profile (24) is applied, reduces to
$c=2(Peδ+1)Peδ+6$
(28)
where
$Peδ=cpfGδkf$
(29)

is the Peclet numbers based on $δ$ and the blowing mass flux $G$.

Likewise, the heat balance equation (20) gives
$θw=Tw−TFB(0)Tc−TFB(0)=PeδPeδ+2−c=Peδ(Peδ+6)Peδ2+6Peδ+10:Overall cooling effectiveness$
(30)

which is one of the most important indices for the performance evaluation of transpiration cooling, namely, the overall cooling effectiveness. The foregoing equation in terms of $Peδ∝F$ clearly indicates that $θw$ is zero (i.e., $Tw=TFB(0)$) for $F→0$, and unity (i.e., $Tw=Tc$) for $F→∞$.

Thus, the integral $∫01ζθ(ζ)dζ$ can be evaluated as
$∫01ζθ(ζ)dζ=∫01ζθw((1−ζ)2+c(1−ζ)3−c(1−ζ)4)dζ=θw12(1+c5)=Peδ(7Peδ+32)60(Peδ2+6Peδ+10)$
(31)
which is substituted into Eq. (23) to find
$Peδ3(7Peδ+32)Peδ2+6Peδ+10=20Pe2∫0ξPe dξ1+∫0ξPe dξ$
(32)
where
$Pe(x)=RePr=cpfG(x)Hkf=cpfGHHkfF(x)$
(33)
is the blowing Peclet number based on the blowing mass flux, so that $Peδ=Pe(δ/H)=(cpfGHH/kf)(δ/H)F$, and
$ξ=(xH)(cpfGHHkf)∝x$
(34)

is the Graetz number (note that some define its inverse as Graetz number). According to Bai and Nakayama [23], the linear (Leveque) approximation of the velocity profile provides reasonably accurate results for the entrance region given by $0<ξ<0.1,$ namely, $(x/H)<0.1(cpfGHH/kf)$. This covers a sufficiently large entrance region especially when the channel Peclet number $(cpfGHH/kf)$ is high.

For given $ξ and Pe(x) ($i.e., graded distribution of the blowing mass flux $G(x)$ or $F(x)$), the value of the right-hand side in Eq. (32) can readily be calculated. Then, the local value $Peδ(x)$ on the left-hand side in Eq. (32) can be determined iteratively. Once $Peδ(x)$ is found, the overall cooling effectiveness $θw(x)$ is given by Eq. (30), while Nusselt number $NuH$ of interest can be evaluated from
$NuH=−kf∂TF∂y*y*=0Hkf(Tw−TFB(0))=Hδ(2−c)=10PePeδ(Peδ+6)$
(35)

## 3 Results and Discussion

### 3.1 Uniform Blowing.

When the blowing mass flux $G(x)$ is constant over the porous wall (i.e., $Pe(x)=cpfG(x)H/kf=const.$) Eq. (32) reduces to an algebraic equation
$Peδ3(7Peδ+32)Peδ2+6Peδ+10=20Pe3ξ1+Peξ$
(36)
For the limiting case of no blowing, $G∝F→0$, both $Peδ$ and $Pe(>Peδ)$ become much less than unity. Thus, Eq. (36) reduces to
$ξ≅425(PeδPe)3$
(37)
Substituting this into Eq. (35), one finds
$NuH=10PePeδ(Peδ+6)≅5Pe3Peδ=53×(425ξ)13=0.905ξ1/3:G∝F→0$
(38)

which is fairly close to the exact solution for the Graetz channel flow without blowing subject to constant heat flux [24], namely, $NuH=0.939/ξ1/3$, indicating the soundness of the present approximate analysis. Equation (38) based on the Leveque approximation holds over the entrance region from $0<ξ<0.1$.

For the general cases of the dimensionless mass flux, $Pe$ = 0.1, 0.5, and 1.0, the dimensionless thermal boundary layer thickness $Peδ(ξ)$, the overall cooling effectiveness $θw(ξ)$, and the local Nusselt number $NuH(ξ)$ are plotted against the Graetz coordinate $ξ$ in Figs. 3(a)3(c), respectively, using the corresponding algebraic equations (36a), (30), and (35). As Fig. 3(a) shows, the thermal boundary layer thickness $δ∝ Peδ=cpfGδ/kf$ grows downstream in a manner similar to the case of no blowing, namely, $δ∝ξ1/3$. Naturally, $Peδ=Pe(δ/H)$ is higher for higher $Pe$.

Fig. 3
Fig. 3
Close modal

As illustrated in Fig. 3(b), the overall cooling effectiveness $θw(ξ)$ can be translated into the dimensionless wall temperature by $Tw(ξ)−Tc/TFB(0)−Tc =1−θw(ξ)$. $θw(ξ)$ approaches unity when $Tw(ξ)$ reaches the coolant temperature $Tc$, while it vanishes when there is no cooling effectiveness with $Tw(ξ)$ being the same as the hot fluid temperature $TFB(0)$. Naturally, when the blowing mass flux $G∝Pe$ is higher, the level of the overall cooling effectiveness $θw(ξ)$ is higher. These figures clearly indicate that the thermal resistance increases downstream as the thermal boundary layer thickens, enhancing the overall cooling effectiveness $θw(ξ)$ downstream. As a result, the wall temperature $Tw(ξ)$ is higher near the entrance and lower downstream. Thus, possible thermal damage is expected at the thermal entrance region around $ξ=0$.

The local Nusselt number variation is presented in Fig. 3(c), which shows that $NuH$ depends almost solely on the Graetz coordinate $ξ$ and is very much insensitive to the blowing mass flux $G∝Pe$. Since the Nusselt number is high at the entrance, the convective heat transfer rate from the hot fluid to the porous wall is high, heating the wall surface up. On the contrary, it becomes low downstream as the coolant fluid film covers over the wall thick, protecting the wall surface thermally.

### 3.2 Nonuniform Blowing.

For thermal management of heat generating stacks such in EV batteries and PEMFC, it is essential to control not only its average temperature (i.e., the average $θw$ over the wall) to a desired working temperature as close as possible but also the temperature distribution as uniform as possible, within a certain allowable deviation from its average. Thus, the task in this study is to find the relationship between the overall cooling effectiveness and the intensity of the blowing mass flux, and also its corresponding blowing mass flux distribution over the wall, which eventually accomplish these requirements for proper thermal managements.

To this end, one may revisit the energy balance relationship as given by Eq. (20), which may be translated in terms of the local thermal boundary layer thickness $δ(x)$ as follows:
$(TFB(0)−Tw(x))δ(x)kf≅(Tw(x)−Tc)1G(x)cpf$
(39)
This relationship is illustrated in Fig. 4, in which the highest potential of the hot fluid $TFB(0)$ and the lowest potential of the coolant $Tc$ are jointed at the potential $Tw(x)$ at the wall by the thermal resistances $δ(x)/kf$ and $1/G(x)cpf$ in series. Obviously, the ratio of these two resistances must be kept constant in order to make $Tw(x)$ unchanged over the wall surface such that
$Peδ(x)=cpfG(x)δ(x)kf=const$
(40)
Fig. 4
Fig. 4
Close modal
This preliminary consideration prompts one to grade the blowing mass flux to follow a 1/3 power function of the distance as follows:
$G(ξ)=G0/ξ1/3$
which provides a higher blowing mass flux near the entrance, thickening the coolant film to reduce the wall temperature more there. For this mass flux distribution, the blowing Peclet number varies over the wall as
$Pe(ξ)=cpfG(ξ)Hkf=(cpfG0Hkf)/ξ1/3$
(41)
which reduces Eq. (32) to
$Peδ3(7Peδ+32)Peδ2+6Peδ+10=30(cpfG0Hkf)31+32(cpfG0Hkf)ξ2/3$
(42)
Note the denominator in the right-hand side term, $uB(ξ)/uB(0)=1+(3/2)(cpfG0H/kf)ξ2/3$, stays almost unity, since the blowing ratio is usually less than a few percent. Thus, as expected, $Peδ(ξ)=cpfG(ξ)δ(ξ)/kf$ in the left-hand side term remains nearly constant over the wall $(i.e., δ(x)∝ξ13)$ such that, for the first approximation, one obtains
$Peδ(ξ)=cpfG(ξ)δ(ξ)kf≅Peδ(0)$
(43)
$θw(ξ)≅Peδ(0)(Peδ(0)+6)Peδ(0)2+6Peδ(0)+10$
(44)
from Eq. (30) and
$NuH(ξ)=10(cpfG0Hkf)Peδ(0)(Peδ(0)+6)ξ1/3$
(45)
from Eq. (35) where
$Peδ(0)3(7Peδ(0)+32)Peδ(0)2+6Peδ(0)+10=30(cpfG0Hkf)3$
(46)

Once $Peδ(0)$ is determined from the implicit algebraic equation (46) for the given intensity of the blowing mass flux $(cpfG0H/kf)$, one can readily evaluate the dimensionless thermal boundary layer thickness $δ(ξ)/H=Peδ(0)/(cpfG0H/kf)ξ1/3.$ Then, the overall cooling effectiveness $θw(ξ)≅const.$ is given by Eq. (44), while the local Nusselt number $NuH(ξ)∝1/ξ13$ is determined from Eq. (45).

Alternatively, when the desired level of the overall cooling effectiveness $θw$ (i.e., the wall temperature $Tw=Tc+(TFB(0)−Tc)(1−θw)$) is specified, $Peδ(0)$ is readily found by solving the quadratic equation (30) as
$Peδ(0)=3(1+θw91−θw−1)$
(47)

Then, the required level of the blowing mass flux $(cpfG0H/kf)$ can be evaluated from Eq. (46). With $Peδ(0)$ and $(cpfG0H/kf)$ thus evaluated, the local Nusselt number $NuH(ξ)$ can finally be estimated from Eq. (45).

In order to confirm the validity of the foregoing arguments based on $G(ξ)=G0/ξ1/3$, Eq. (42) is iteratively solved without approximations (with retaining the denominator in the right-hand side term as it is) over the range of $ξ$ = 0.005–0.10 for the cases of the blowing mass flux intensity $(cpfG0H/kf)=$ 0.1, 0.5, and 1. The resulting streamwise variations of $Peδ(ξ)$, $θw(ξ)$, and $NuH(ξ)$ are presented in Figs. 5(a)5(c), respectively. In these figures, the approximate results based on $Peδ(0)$ are indicated by dashed lines, while those based on the local value $Peδ(ξ)$ iteratively obtained are indicated by solid lines. As expected, $Peδ(ξ) ∝G(ξ)δ(ξ)$ in Fig. 5(a) remains almost constant over the wall, despite of slight decrease in $Peδ(ξ)$ downstream. As a result, the wall temperature (i.e., $1−θw(ξ))$ is kept nearly constant, as can be confirmed in Fig. 5(b), while $NuH(ξ)$ decreases as increasing the blowing mass flux intensity, as indicated in Fig. 5(c), which is consistent with the well-known fact reported by Kays and Crawford [25] for transpiration cooling.

Fig. 5
Fig. 5
Close modal

As demonstrated in Fig. 5(b), the blowing mass flux graded according to $G(ξ)=G0/ξ1/3$ is found quite satisfactory to keep the wall temperature uniform. Furthermore, the figure indicates that one can control the wall temperature freely to a desired level by adjusting the blowing mass flux intensity $(cpfG0H/kf)$. Naturally, the higher $(cpfG0H/kf)$ brings out the lower wall temperature, as illustrated in Fig. 6, in which $(cpfG0H/kf)$ is plotted against the wall temperature (i.e., $1−θw(ξ)),$ using Eqs. (47) and (46). The figure shows that the blowing mass flux intensity $(cpfG0H/kf)$ increases steeply from unity, as one tries to reduce the wall temperature to $Tw(ξ)−Tc/TFB(0)−Tc<0.3$. Thus, in engineering applications, $(cpfG0H/kf)$ should be set below $(cpfG0H/kf)<1$, since 70% (or less) overall thermal effectiveness (i.e., $(Tw(ξ)−Tc)$ > 0.3$(TFB(0)−Tc)$) is usually quite sufficient for most cases of thermal managements. The figure also indicates that, in this range, the sensitivity $−dθw/d(cpfG0H/kf)$ is high so that the wall temperature can easily be set to a desired level by adjusting the blowing mass flux intensity $(cpfG0H/kf)$.

Fig. 6
Fig. 6
Close modal

### 3.3 Comparison of Numerical and Analytical Results.

Finally, numerical simulations based on the finite volume method [20] were carried out for a limited number of cases to verify the validity of the present analysis. The in-house computer code SAINTS [20] based on staggered grids was used for numerical calculations. Appropriate velocity boundary treatments such as no-slip streamwise velocity component and variable vertical velocity component are applied on the wall boundaries, while the thermal boundary condition was set according to the heat balance relationship on the permeable wall, as given by Eq. (20). In actual numerical computations, the variables normalized by the references, such as $ρf, uB(0)$, H and $(TFB(0)−Tc),$ were used so that the convergence was measured in terms of the maximum change in each normalized variable during an iteration. The maximum change allowed for the convergence check was set to an arbitrarily small value such as 10−5.

The wall temperature variations $(1−θw)$ for the case of $GHH/μf=100$ and $cpfμf/kf=1$ (i.e., $cpfGHH/kf=100$) were obtained for the variable mass flux distribution $G(ξ)=G0/ξ1/3$ with $(cpfG0H/kf)$  = 0.1, 0.5, and 1.0, and compared with those based on the present analysis in Fig. 7. The channel length is set to 10H such that the whole channel stays within the entrance region, namely, $0≤ξ≤0.1.$ As will be shown shortly, the blowing parameter $(cpfG0H/kf)$  = 0.1, 0.5, and 1.0 correspond with the cases of uniform blowing ratio of F = 0.32%, 1.6%, and 3.2%, respectively. As predicted in the analysis, the wall temperature in fact stays nearly constant over the channel wall. The levels predicted by both numerical and analytical methods agree well with each other. The temperature rise toward the entrance indicated by the numerical method is due to the fact that the limiting condition $G0/ξ1/3ξ→0→∞$ can never faithfully be described in the finite volume method.

Fig. 7
Fig. 7
Close modal

In Fig. 8, the temperature field numerically obtained for the case of graded mass flux distribution $(cpfG0H/kf)$  = 1.0 is presented along with the case of constant blowing ratio $F$ in which the mass flowrate at $ξ=0.1$ coincides with that of graded mass flux, such that

Fig. 8
Fig. 8
Close modal
$F=10.1∫00.1GGHdξ=10.1∫00.1(cpfG0Hkf)(cpfGHHkf)1ξ1/3dξ=10.1∫00.111001ξ1/3dξ=0.032$

The velocity and temperature fields for the case of constant blowing mass flux and that of graded blowing mass flux are presented in Figs. 8(a) and 8(b), respectively. The higher blowing mass flux near the entrance for the case of variable blowing mass flux suppresses the wall temperature rise there, resulting the wall temperature much lower there than the corresponding temperature for the case of constant blowing mass flux. On the other hand, the wall temperature downstream for the case of variable blowing mass flux in Fig. 8(b) stays comparatively higher than that for the case of constant blowing mass flux in Fig. 8(a). It can be confirmed from the temperature field presented in Fig. 8(b) that the temperature can be kept nearly constant over the wall by controlling the blowing mass flux locally.

## 4 Conclusions

Thermal entry problems associated with transpiration cooling were analytically treated assuming hydrodynamically fully developed forced convective flow in a channel in which the coolant fluid is supplied through the porous wall at a locally variable blowing mass flux. First, the hydrodynamically fully developed velocity fields in a channel were investigated allowing an arbitrary distribution of local blowing mass flux. The velocity profiles were found fairly insensitive to the blowing mass flux for a practical range of the blowing intensity. Second, the energy balance equation in the entrance region of the channel was considered along with the heat balance equation between the enthalpy increase of the coolant and the convective heat flux over the porous wall, in order to elucidate the effects of the blowing mass flux distribution on the wall temperature distribution. The analysis clearly shows that the uniform blowing results in a thin coolant film near the inlet and a thick one downstream, yielding a high wall temperature at the inlet, whereas the variable blowing rate inversely proportional to one-third power of the distance remedies the situation keeping the wall temperature uniform everywhere.

## Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Nomenclature

$c$ =

shape parameter for the temperature profile

$cpf$ =

specific heat of fluid at constant pressure (J/kg K)

$f$ =

$u/uB$ velocity profile function

$F$ =

$G/GH$ blowing ratio

$f′(0)$ =

$df/dη*η*=0$ velocity gradient at the wall

$G$ =

blowing mass flux (kg/m2s)

$GH$ =

hot fluid mass flow rate per unit area at the inlet (kg/m2s)

$G0$ =

blowing mass flux intensity $G(ξ)=G0/ξ1/3$

H =

half of the channel height

$kf$ =

thermal conductivity of the fluid (W/m K)

$NuH$ =

$−kf(∂TF/∂y*)y*=0H/kf(Tw−TFB(0))$ Nusselt number

$p$ =

pressure (Pa)

$Pe$ =

$cpfGH/kFe$Peclet number based on $G$

$Peδ$ =

$cpfGδ/kf$ Peclet number based on $G$ and $δ$

$Pr$ =

$cpfμf/kf$ Prandtl number

$Re$ =

$GH/μf$ blowing Reynolds number

$Tc$ =

coolant reservoir temperature (K)

$TF$ =

hot fluid temperature (K)

$TFB$ =

hot fluid bulk mean temperature (K)

$Tw$ =

channel wall temperature (K)

$u$, v =

x and y velocity components (m/s)

$uB$ =

bulk mean velocity of hot fluid (m/s)

$x$ =

streamwise coordinate (m)

$y$ =

vertical coordinate (m) from the channel center

$y*$ =

vertical coordinate (m) from the inner wall

### Greek Symbols

Greek Symbols
$δ$ =

thermal boundary layer thickness $ζ=y*/δ$

$η*$ =

$y*/H$ dimensionless vertical coordinate

$θ$ =

$θ =TF−TFB(0)/Tc−TFB(0)$ dimensionless temperature profile of the hot fluid

$θw$ =

$Tw−TFB(0)/Tc−TFB(0)$ overall cooling effectiveness

$Λ$ =

$Λ =(−H2/μfuB(x))dp/dx$ dimensionless pressure gradient

$μf$ =

fluid viscosity (Pa·s)

$ξ$ =

$(x/H)/(cpfGHH/kf)$ Graetz number

$ρf$ =

fluid density (kg/m3)

### Subscripts

Subscripts
B =

bulk mean

c =

cold reservoir side

f =

fluid

F =

hot fluid

$w$ =

wall

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