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.
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 . 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., [2–5].
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 . 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 , Sellars , Yuan , Donoughe , Eckert et al. , and White . Zhou and Catton  and Zhou et al.  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.  and Bai et al. .
Recently, a two-domain analytical approach has been proposed by Nakayama et al.  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 enters at the mass flowrate per unit area 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 kg/s (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 outside, and enters into the porous wall at the variable mass flux . The temperature of the coolant fluid rises from to 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 will be determined in consideration of the heat balance in both porous wall and channel domains.
2.1 Momentum Balance.
White  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 .
Equation (7) is exact only for the case of constant , since the velocity profile function cannot be a function of alone, when varies with . However, a fairly accurate integral solution with locally variable is possible as long as the blowing ratio is moderate. In fact, it will be shown shortly that the streamwise velocity profile function is quite insensitive to the variation of , and almost solely depends on .
As presented in Fig. 2, Eq. (16) yields 3.0, reducing Eq. (12) to the exact profile, namely, Eq. (17a), for (i.e., ), whereas it gives 2.475 for (i.e., ), which is very close to the exact value 2.467 based on Eq. (17b).
As can be observed in Fig. 2, the velocity gradients at the wall for these limiting cases, namely, 3.0 for and 2.475 for , 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 () in the case of transpiration is typically less than a few percent, the streamwise velocity profiles may be approximated by Eq. (17a) (i.e., 3.0) for the all cases of laminar flow in the transpiration-cooled channels.
2.2 Energy Balance.
where the approximation 3.0 is used since is insensitive to () in a moderate range of the blowing ratio, as confirmed in Fig. 2.
is the Peclet numbers based on and the blowing mass flux .
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 clearly indicates that is zero (i.e., ) for , and unity (i.e., ) for .
is the Graetz number (note that some define its inverse as Graetz number). According to Bai and Nakayama , the linear (Leveque) approximation of the velocity profile provides reasonably accurate results for the entrance region given by namely, . This covers a sufficiently large entrance region especially when the channel Peclet number is high.
3 Results and Discussion
3.1 Uniform Blowing.
which is fairly close to the exact solution for the Graetz channel flow without blowing subject to constant heat flux , namely, , indicating the soundness of the present approximate analysis. Equation (38) based on the Leveque approximation holds over the entrance region from .
For the general cases of the dimensionless mass flux, = 0.1, 0.5, and 1.0, the dimensionless thermal boundary layer thickness , the overall cooling effectiveness , and the local Nusselt number are plotted against the Graetz coordinate 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 grows downstream in a manner similar to the case of no blowing, namely, . Naturally, is higher for higher .
As illustrated in Fig. 3(b), the overall cooling effectiveness can be translated into the dimensionless wall temperature by . approaches unity when reaches the coolant temperature , while it vanishes when there is no cooling effectiveness with being the same as the hot fluid temperature . Naturally, when the blowing mass flux is higher, the level of the overall cooling effectiveness is higher. These figures clearly indicate that the thermal resistance increases downstream as the thermal boundary layer thickens, enhancing the overall cooling effectiveness downstream. As a result, the wall temperature is higher near the entrance and lower downstream. Thus, possible thermal damage is expected at the thermal entrance region around .
The local Nusselt number variation is presented in Fig. 3(c), which shows that depends almost solely on the Graetz coordinate and is very much insensitive to the blowing mass flux . 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 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.
Once is determined from the implicit algebraic equation (46) for the given intensity of the blowing mass flux , one can readily evaluate the dimensionless thermal boundary layer thickness Then, the overall cooling effectiveness is given by Eq. (44), while the local Nusselt number is determined from Eq. (45).
Then, the required level of the blowing mass flux can be evaluated from Eq. (46). With and thus evaluated, the local Nusselt number can finally be estimated from Eq. (45).
In order to confirm the validity of the foregoing arguments based on , Eq. (42) is iteratively solved without approximations (with retaining the denominator in the right-hand side term as it is) over the range of = 0.005–0.10 for the cases of the blowing mass flux intensity 0.1, 0.5, and 1. The resulting streamwise variations of , , and are presented in Figs. 5(a)–5(c), respectively. In these figures, the approximate results based on are indicated by dashed lines, while those based on the local value iteratively obtained are indicated by solid lines. As expected, in Fig. 5(a) remains almost constant over the wall, despite of slight decrease in downstream. As a result, the wall temperature (i.e., is kept nearly constant, as can be confirmed in Fig. 5(b), while 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  for transpiration cooling.
As demonstrated in Fig. 5(b), the blowing mass flux graded according to 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 . Naturally, the higher brings out the lower wall temperature, as illustrated in Fig. 6, in which is plotted against the wall temperature (i.e., using Eqs. (47) and (46). The figure shows that the blowing mass flux intensity increases steeply from unity, as one tries to reduce the wall temperature to . Thus, in engineering applications, should be set below , since 70% (or less) overall thermal effectiveness (i.e., > 0.3) is usually quite sufficient for most cases of thermal managements. The figure also indicates that, in this range, the sensitivity is high so that the wall temperature can easily be set to a desired level by adjusting the blowing mass flux intensity .
3.3 Comparison of Numerical and Analytical Results.
Finally, numerical simulations based on the finite volume method  were carried out for a limited number of cases to verify the validity of the present analysis. The in-house computer code SAINTS  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 , H and 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 for the case of and (i.e., ) were obtained for the variable mass flux distribution with = 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, As will be shown shortly, the blowing parameter = 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 can never faithfully be described in the finite volume method.
In Fig. 8, the temperature field numerically obtained for the case of graded mass flux distribution = 1.0 is presented along with the case of constant blowing ratio in which the mass flowrate at coincides with that of graded mass flux, such that
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.
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.
shape parameter for the temperature profile
specific heat of fluid at constant pressure (J/kg K)
velocity profile function
velocity gradient at the wall
blowing mass flux (kg/m2s)
hot fluid mass flow rate per unit area at the inlet (kg/m2s)
blowing mass flux intensity
- H =
half of the channel height
thermal conductivity of the fluid (W/m K)
Peclet number based on
Peclet number based on and
blowing Reynolds number
coolant reservoir temperature (K)
hot fluid temperature (K)
hot fluid bulk mean temperature (K)
channel wall temperature (K)
- , v =
x and y velocity components (m/s)
bulk mean velocity of hot fluid (m/s)
streamwise coordinate (m)
vertical coordinate (m) from the channel center
vertical coordinate (m) from the inner wall
thermal boundary layer thickness
dimensionless vertical coordinate
dimensionless temperature profile of the hot fluid
overall cooling effectiveness
dimensionless pressure gradient
fluid viscosity (Pa·s)
fluid density (kg/m3)
- B =
- c =
cold reservoir side
- f =
- F =