In Section 3, a brief validation is described by using both the 3-dimensional discretization and the 2-dimensional discretization. Computations were performed using both methods and were compared to the available experimental and numerical solutions. It can be noted from Fig. 5 that there is a very minimal deviation between the 3D and 2D discretization. Hence, with reduced computational time and reasonably accurate results, the 2D discretization method has been implemented in this current work. The flow is assumed to be two-dimensional and steady. The dimensions of the plate are as follows: the length of the plate is 23 cm in the direction of the flow and 2.5 cm normal to the flow direction. It can be noted from Fig. 6 that a pre-stabilization length of 5 cm has been given to the geometry to obtain a fully developed laminar flow without any reverse flow or vortex generation at the entry length of the plate. Two regimes of heat transfer have been studied in the present investigation. The regimes are forced convection with buoyancy effects and forced convection with negligible buoyancy effects. The heat transfer characteristics have been evaluated over a range of five different ratios of P∞/Pcr starting from a value of 1.1 till 1.5, where the critical pressure (Pcr) of nitrogen is 3.396MPa. Five different wall temperatures were chosen in such a fashion that two of the values were below T∗, one equal to T∗ and two other greater than T∗ with three different values of free stream velocity. These parameters were used in varying combinations to obtain a large data set to predict the heat transfer progression in the supercritical region with utmost accuracy. Specifying appropriate boundary conditions for the flow field is of prime importance to simulate the required case accurately. A velocity inlet boundary condition is applied at the inlet where three different values of free stream velocities were quantified–1.4, 1.6, and 1.8 m/s for the case of forced convection without buoyancy effects and 0.25, 0.3, and 0.35 m/s for the case of forced convection with buoyancy effects. An ambient temperature (T∞) of 110 K was maintained throughout all the case studies. A no-slip boundary condition was specified at the wall which was maintained at an isothermal condition. The region opposite the isothermal wall is specified as a symmetry boundary condition that acts as a far-field. It is of prime importance to stipulate the acceleration due to gravity in the negative y-direction to capture the boundary layer effects with buoyancy forces. Since supercritical fluids have varying properties with temperature, the default ideal gas property values cannot be used in this scenario. Hence, the piece-wise linear option available in the commercial software ANSYS Fluent 19.2 was used to input different values of k,μ,ρ and Cp at different temperatures which were obtained from NIST [21] in such a way that the properties encapsulate the range of the present problem.

An important factor while examining convection heat transfer is the identification of the type of boundary layer, i.e., laminar, or turbulent. Convection transfer rates strongly depend on this factor. The current work completely concentrates on the laminar boundary layer. In a laminar boundary layer, streamlines are highly distinguishable and ordered along the direction of fluid flow. A triggering mechanism is a major cause of the transition from the laminar to the turbulent regime. These triggering mechanisms can be unsteady flow structures, disturbances within the boundary layer, and fluctuations in the free stream. A non-dimensional, definitive way to predict the nature of the boundary layer by using the Reynolds number (Rex<Rex,crfor laminar flows). Where, Rex,cr is the critical Reynolds number. For flows flowing over a flat plate, the critical Reynolds number can vary from 5×105−3×106 depending on the turbulence level of the free stream and the surface roughness [22]. In the present study, an investigation for forced convection heat transfer with laminar flow is examined in detail. To calculate the heat transfer parameters, certain equations have been used by the solver to provide accurate and reliable results. The three most important equations while solving such problems are the continuity, momentum, and energy equations. These three equations constitute and account for the heat transfer using fluids. The continuity equation for the case of a steady laminar incompressible flow without free-surface effects is commonly given by Eq. (2). Most of the solvers make use of the continuity equation derived from the Navier-Stokes equation as shown by Eq. (3) [23]. (2)∇→⋅V→=0 (3)(V→⋅∇→)V→=−1ρ∇→P′+ν∇2V→

Eq. (2) is commonly known as the conservation equation whereas Eq. (3) is known as the transport equation which denotes the transport of the linear momentum throughout the computational domain [23]. The modified form of pressure (P′) is used in these equations due to the absence of free-surface effects. The more advanced form of the momentum and continuity equations used by solvers are represented by the following equations: (4)∂u∂x+∂v∂y+∂w∂z=0

The general momentum equation in three flow directions, that is x, y and z are as follows: (5)u∂u∂x+v∂v∂y+w∂w∂z=−1ρ∂P′∂x+ν(∂2u∂x2+∂2u∂y2+∂2u∂z2) (6)u∂u∂x+v∂v∂y+w∂w∂z=−1ρ∂P′∂y+ν(∂2v∂x2+∂2v∂y2+∂2v∂z2) (7)u∂u∂x+v∂v∂y+w∂w∂z=−1ρ∂P′∂z+ν(∂2w∂x2+∂2w∂y2+∂2w∂z2)

The following equations are strictly applicable to incompressible flows. These equations must be modified in order for them to be applicable for steady laminar compressible flows. Eqs. (8)–(12) [24] represent the continuity, momentum, and energy equations for compressible flows. (8)∂(ρu)∂x+∂(ρv)∂y+∂(ρw)∂z=0

Similarly, the momentum equations are compiled over three flow directions namely x, y, and z. (9)ρ(u∂u∂x+v∂u∂y+w∂u∂z)=ρgx−∂P∂x+∂∂x(2μ∂u∂x+λ∇→⋅V→)+∂∂y(μ(∂u∂x+∂v∂x))+∂∂z(μ(∂w∂x+∂u∂z)) (10)ρ(u∂v∂x+v∂v∂y+w∂v∂z)=ρgy−∂P∂y+∂∂y(2μ∂v∂y+λ∇→⋅V→)+∂∂x(μ(∂v∂x+∂u∂y))+∂∂z(μ(∂v∂z+∂w∂y)) (11)ρ(u∂w∂x+v∂w∂y+w∂w∂z)=ρgz−∂P∂z+∂∂z(2μ∂w∂z+λ∇→⋅V→)+∂∂x(μ(∂w∂x+∂u∂z))+∂∂y(μ(∂v∂z+∂w∂y))

An important equation while solving any heat transfer-fluid flow problem is the energy transport equation. The energy equation for a compressible laminar flow is given by equation. (12)ρCp(u∂T∂x+v∂T∂y+w∂T∂z)=βT(u∂P∂x+v∂P∂y+w∂P∂z)+∇→⋅(k∇→T)+Φ

The above partial differential equations (Eqs. (9) to (12)) are converted to algebraic equations by the solver’s inbuilt algorithm which computes the required fluid flow and heat transfer parameters. The present investigation incorporates the pressure-based solver which is commonly used to compute characteristics of incompressible flows and slightly compressible flows. Since supercritical fluids portray a trait of varying properties with temperature and pressure, it has been sure that these property variations have been considered by incorporating fifty data points for the independent properties using the piecewise linear option offered by ANSYS Fluent 19.2 [25]. The partial differential equations are solved sequentially by the pressure-based solver which takes the property values at each iteration [25].

Using these models without enabling certain options would not yield the correct results. Hence, the laminar model with viscous effects has been used in this study to actively capture the boundary layer effects. In addition to this, a no-slip boundary condition is specified at the velocity boundary layer and a second-order differential scheme is used to calculate the gradients [26]. Most of the liquid flows can be considered as incompressible. But for gaseous flows, it is important to estimate the Mach number to ascertain it as an incompressible flow [23]. Hence, the Mach numbers have also been calculated to prove the usability of the pressure-based solver as shown in Table 1.

For a flow to be considered incompressible, the Mach number should be less than 0.3 [27]. This assumption is rational at low values of Mach numbers which holds good for the present investigation. It can be observed from Table 1 that for both minimum (0.25 m/s) and maximum (1.8 m/s) values of the velocity, the Mach numbers are significantly less than 0.3 which advocates the use of the pressure-based model for this investigation.

The finite volume method has been used in this study to evaluate the algebraic equations obtained from the partial differential equations computed by the differencing scheme. The coupled algorithm has been used for the computational study. The coupled scheme results in an efficient and robust single-phase implementation of steady flows when compared to the default SIMPLE-type pressure-velocity coupling. This coupled algorithm solves the pressure and momentum-based continuity equations together to reduce the computational time. A second-order upwind scheme was set to discretize the pressure, momentum, and energy equations. A Least Squares Cell-Based approach was used for the gradient spatial discretization. A pseudo-transient analysis was performed to obtain faster and more efficient results. A pseudo-transient approach is a form of implicit under-relaxation for steady-state cases [24].

The pseudo transient explicit under-relaxation factors were set to 0.5 for momentum, pressure, and density, 1 for body forces, and 0.75 for energy for attaining solution stabilization. The scaled residuals were to be set to 10−6 from a default value of 10−3 for solving continuity, x, y momentum and the energy equation. Computations performed with individual values of 10−6 and above are known as tightly converged solutions. This in turn indicates that the solution is well normalized and has converged within the required range [24].

Grid independence study plays a vital role in any computational analysis. The purpose of a mesh independence study is to obtain results that do not fluctuate with the element size. This ensures that the solution obtained is uniform throughout with minimal deviation. The geometry in the present study was discretized using the face mesh option along with the inflation layer near the heated surface. Inflation layers are added near the heated surface to capture the slightest variations in the boundary layer. A uniform set of quadrilateral elements has been used to mesh the face of the plate. The quality of the generated mesh was based on the following parameters: skewness, orthogonal quality, and element quality. It is often ascertained as a good practice to keep the average and maximum value of skewness to be less than 0.4 and 0.9, respectively. Similarly, the average and minimum values of both orthogonal and element quality are greater than 0.7 and 0.1, respectively. With drastic property changes and distortions close to the pseudo-critical point with the use of supercritical fluids, it is of prime importance to have a geometry with an appropriate mesh topology as it plays a deterministic role in the heat transfer rates in the flow. Grid independence was achieved by varying the size of each element. The investigation is performed by changing the number of mesh elements from a value of 35,000 to a value of 738,000. Table 2 showcases the various mesh elements used in this study and the corresponding outlet temperatures and heat transfer coefficient values.

The variation of outlet temperature and the heat transfer coefficient is plotted against the mesh density as displayed in Fig. 7. After an element size of 200,000, it can be noted from the above figure that mesh independence is achieved. Therefore, an element size of 345,000 has been used for the rest of the case studies. This has been selected based on the following criteria: grid independence, reduced computational time, and significantly accurate data.

Fig. 8 shows the velocity profile for the case of forced convection without significant buoyancy effects for three values of (Tw−115K(Tw<T∗), Tw−128.24K(Tw=T∗) and Tw−140K(Tw>T∗). It can be seen from these profiles that as Tw increases, the velocity profile becomes flatter thereby increasing the velocity gradient at the wall, which in turn increases the wall-shear stress and the drag coefficient. However, the velocity profiles for the case of forced convection with buoyancy effects, shown in Fig. 9 specify that for the case when Tw−115K(Tw<T∗), i.e., when T∗ lies outside the boundary layer, the velocity profile is smooth, whereas for the other two cases, namely for Tw=T∗ and Tw>T∗, there is distortion in the velocity profile and this distortion occurs at the location where the fluid temperature is in the neighborhood of T∗. This distortion of the velocity profile near T^* can be attributed to the fact that for fluid temperature less than T^*, C_p increases with temperature reaches a maximum at T^* and then decreases with an increase in fluid temperature. A similar trend is observed for the fluid property β which affects the buoyancy force distorting the velocity profiles as well as the temperature at the location where the fluid temperature is close to T^*. This distortion affects the velocity gradient at the wall which affects the wall shear stress. How much the velocity gradient is affected depends on how close the location of T∗ is concerning the heated wall. It can be observed from Fig. 9, that for the case when Tw>T∗, the distortion has considerably reduced the velocity gradient at the wall thereby decreasing the wall shear stress. Thus whenever T∗ lies within the boundary layer, the buoyancy forces play a major role in deciding the wall shear stress.

Temperature profiles for the cases without buoyancy effects are shown in Fig. 10 for three values of wall temperatures, Tw−115 K(Tw<T∗), Tw−128.24 K(Tw=T∗) and Tw−140 K(Tw>T∗). It can be noted from these profiles that as Tw increases the temperature profile becomes flatter thereby increasing the temperature gradient at the wall, which successively increases the heat transfer coefficient and hence the Nusselt number. Also, distortions are not observed in the temperature profile, even when T∗ lies within the boundary layer because the buoyancy forces are negligible to affect the temperature profiles. However, the temperature profiles for the cases of forced convection where buoyancy forces play a vital role, as shown in Fig. 11 indicate that for the case when Tw−115 K(Tw<T∗), i.e., when T∗ lies outside the boundary layer, the temperature profile is smooth without any distortions, but whereas for the other two cases, namely for Tw=T∗ and Tw>T∗, there are distortions in the temperature profiles and this distortion occurs at the location where the fluid temperature is in the neighborhood of T∗. The reason for these distortions of the temperature profiles is the same as those mentioned under the section “velocity profiles”. This distortion will in turn affect the temperature gradient (decreases the temperature gradient) at the wall which sequentially decreases the heat transfer coefficient. The dependency on the location of T∗ concerning the wall accounts for the extent to which the temperature gradient at the wall is affected. Fig. 11 also indicates, that for the case when Tw>T∗, the distortion has considerably reduced the temperature gradient at the wall thereby decreasing the heat transfer coefficient. Thus whenever T∗ lies within the boundary layer, the buoyancy forces play a substantial role in deciding the wall shear stress as well as the heat transfer coefficient. The distortion of both the temperature and velocity profiles due to the location of T∗ within the boundary layer can be attributed to the change in the sign of the slope of dCpdT across T∗.

Seven formats of correlations which are available in the literature for supercritical fluids are used to compare the predicted Nusselt numbers with the actual numerical predictions. All seven formats consider the variation of the fluid properties in the correlation in one way or the other. These formats are as follows:

(i) Format F1: This format for forced convection without buoyancy effects [GrxRex2<0.1] considers the variation of fluid properties by introducing four property rations in the correlation and the correlation is given by (13)(Nux)fcF1=a(Rexb)(Pr)c(C¯pCp\infty )d(ρ∞ρw)e(µ∞µw)f(k∞kw)g

Gustavo et al. [26] have shown that the above format gives better agreement with the actual predictions based on their comparison of available data with the various forms of correlations. In the present investigation, based on Nu_x at 540 data points and using multiple linear regression analysis, Eq. (13), takes the following form: (14)(Nux)fcF1=0.398×(Rex0.505)(Pr∞)−0.144(C¯pCp\infty )0.101(ρ∞ρw)−0.015(µ∞µw)−0.0136(k∞kw)−0.1274

The R² value concerning Eq. (14) is found to be 0.9986. Out of 540 data points, 536 data points (99.26%) have deviations less than 5% [% deviation = ((Nu_x) by correlation/(Nu_x) by numerical prediction) – 1) 100] and the remaining 4 data points have deviations between 5% and 9%.

(ii) Format F2: In this format, the correlation for Nu_x has only two property ratios, namely, the specific heat ratio (C¯pCp∞) and the density ratio (ρ∞ρw). This format assumes that the nature of variation of viscosity and thermal conductivity are similar to the variations of density and specific heat for supercritical fluids and hence the specific heat ratio and the density ratio will also account for the variation of viscosity and thermal conductivity. The multiple linear regression analysis for this format yields the following correlation: (15)(Nux)fcF1=0.314×(Rex0.505)(Pr∞)0.277(C¯pCp\infty )0.148(ρ∞ρw)−0.159

The R² value for this correlation is found to be 0.998. The deviation is less than 5% for 538 data points (99.6%) out of 540 points and the remaining two points have deviations between 5 and 8 percent.

(iii) Format F3: This method has been proposed by Ghajar et al. [28] for free convective heat transfer from vertical isothermal flat plates to supercritical fluids. In this method, a characteristic temperature, called ‘reference temperature’, T_ref is chosen at which the properties appearing in the nondimensional groups (Nu, Gr, Pr, etc.) are evaluated. The constant property results at that temperature are then used to evaluate the Nusselt number for supercritical fluids. Typically, this temperature may be the surface temperature T_w, the bulk fluid temperature T_∞, or a temperature in between T_w and T_∞. Based on this concept, T_ref is expressed as follows: (16)Tref=Tw–r[Tw−T∞]

In the investigations (experimental/analytical) reported in the literature for reference temperature, the parameter ‘r’ of Eq. (16) has been assumed to have a constant value. This assumption is valid only when the thermodynamic and transport properties of the fluid under consideration do not vary significantly within the boundary layer. For such cases determination of the reference temperature constant, r is simple and straightforward. But in the case of heat transfer in supercritical fluids, this assumption is not valid as the thermos physical properties of the fluid vary very severely, especially when the fluid pressure is near to the critical pressure and the temperatures T_w and T_∞ are close to the pseudo-critical temperature T^*. The parameter r is no longer a constant and is a complicated function of a dimensionless temperature group expressed as follows: (17)r=f(T∗−T∞Tw−T∞)=f(ΔT*);0≤r≤1

The method involves the following 6 steps:

Step 1: Generate data of local Nusselt number, (Nu_x)_ap for the supercritical fluid, for a given set of conditions either through experimental investigation or by analytical/numerical solution of the governing equations.

Step 2: Select a value of r (one can start with r = 1), determine T_ref using Eq. (16), and find the required fluid properties at T_ref.

Step 3: Using these fluid properties, Calculate the dimensionless numbers, namely (Nu_x)_Ref, (Rex)Ref and Prref for all the data points: (18)(Nux)Ref=hxkref,(Rex)Ref=U∞xνref,Prref=µref(Cp)refkref

Step 4: Assume the correlation for forced convection to be of the form (19)(Nux)Ref=a(Rex)Refb(Prref)c(T∞ΔT)d

This form is similar to the one proposed by Gajhar et al. [28] for laminar-free convection from a vertical isothermal flat plate to supercritical fluids. The dimensionless numbers namely, (Nu_x)_Ref, (Rex)Ref and Prrefare determined for each data point and using this data the constants a, b, c and d are determined using multiple linear regression analysis. For the present problem, these constants are found to be

a = 0.275; b = 0.505; c = 0.371; d = 0.033

Then using Eq. (19) with the values of a, b, c, and d, the local Nusselt number (Nu_x)_cor is calculated. This value is next compared to the actual predictions (Nu_x)_ap. If the two do not match within the permissible deviation [say maximum % deviation shall be≤10%], change the value of r and repeat the steps 2 to 4 till the agreement is satisfactory. If r^* is the value of r which gives satisfactory agreement between (Nu_x)_cor and (Nu_x)_ap, then obtain r^* for each data point and plot r^* vs. ΔT^* for each pressure of the supercritical fluid. This plot can be used to find r^* for any value of ΔT^*. For the given data, the plots of r^* vs. ΔT^* for one pressure of supercritical nitrogen is shown in Fig. 19 for 37.35 bar. For the remaining 4 pressures the plots are not shown as, for all these pressures r^* is found to be 1 for all values of ΔT^*. These plots are used to find r^* for any given value of ΔT^* and hence Tref∗ at which the fluid properties to be evaluated can be determined. Then the final form of the correlation will be

(20)(Nux)fcF3=0.275×(Rex)Ref0.505(Prref)0;371(T∞ΔT)0.033

The R² value with respect to Eq. (20) is found to be 0.998. The deviations are <5% for 525 data points (97.5%) out of 540 points and the remaining 15 points have deviations between 5 and 7 percent. In Eq. (20), T_∞ appears explicitly and this is not desirable. Hence the dimensionless T_∞ which appears in Eq. (20) is replaced by ∆T^*. In that case the regression analysis yields. (20a)(Nux)fcF3=0.277×(Rex)Ref0.505(Prref)0.386(ΔT∗)0.033

The R² value with respect to Eq. (20a) is found to be 0.9984. Comparison of Nusselt numbers obtained using Eq. (20a) with the actual predictions indicates that only 4 data points out of 540 have deviations between 5 and 8 percent and the rest have deviations within ±5 percent.

(iv) Format with correlation based on enthalpy ratios (Format F4): This format, which uses enthalpy ratios in the correlation to account for the variation of fluid properties, has been used by Rousselet et al. [29] for predicting the Nusselt number based on their experimental results for free convection from horizontal cylinders to supercritical carbon dioxides. Further, they have proposed two correlations one for the cases for which T_w does not lie within the boundary layer (i.e., T_∞ < T_w < T^*) and the second correlation for cases for which T_w ≥ T^*. The format of these two correlations is as follows: (21)(Nux)fc4a=a1(Rexb1)(Pr∞)c1(iwic)d1(ici∞)e1

valid for cases for which T_∞ < T_w < T^*, and for cases for which T_w ≥ T^* > T_∞ the proposed correlation is of the form (22)(Nux)fc4b=a2(Rexb2)(Pr∞)c2(iwic)d2(ici∗−i∞)e2

In the present investigation, 342 data points are available for which the condition T_w < T^* is satisfied. Hence using the format given in Eq. (21), regression analysis yields the following: (23)(Nux)fcF4a=0.32×(Rex0.50)(Pr∞)0.049(iw∗ic∗)−0.033(ic∗i∞∗)0.013

The R² value with respect to Eq. (23) is found to be 0.9998. All the data points have deviations that are less than ±2%.

There are 198 data points for which the condition is T_w ≥ T^*. Using Eq. (22) for these data points gives correlation as (24)(Nux)fcF4=0.21(Rex0.504)(Pr∞)1.049(iwic)−0.180(ici∗−i∞)−0.0837

The R² value with respect to Eq. (24) is 0.996. Only one data point has deviation >10% (=10.3%), 27 data points have deviations between 5% and 9% and the remaining points have deviations <5%.

(v) Format using constant property correlation with four property ratios (Format F5): In this format, the correlation for the Nusselt number for supercritical Nitrogen is expressed in terms of the Nusselt number for constant property fluids. If the four property ratios are used in the correlation, then this format will be of the form (25)(Nux)fcF5=0.942×((Nuxfccp)1.009×(Cp0.098)×(ρ∗)−0.022×(k∗))−0.121

In Eq. (25), (Nux)xfccp represents the correlation for Nusselt number for laminar forced convection for constant property fluids and is given by [22,29] (26)(Nux)xfccp=0.332×Rex0.5×Pr∞0.3333)

The R² value with respect to Eq. (25) is found to be 0.9887. Out of 540 data points, only 6 points have deviations between 5 and 7 percent, and the rest have deviations less than 5%.

(vi) Format using constant property correlation with two property ratios (Format F6): In this format, the correlation for Nu_x is similar to Eq. (25), but has only two property ratios, namely, the specific heat ratio (C¯pCp∞) and the density ratio (ρ∞ρw). The regression analysis yields the correlation as (27)(Nux)xWBF6=0.926×((Nu)xfccp)1.009)×Cp∗0.1479×ρ∗−0.1589

The R² value with respect to Eq. (27) is found to be 0.8345. Only 4 data points have deviations between 5 and 8 percent and the rest of the points have deviations that are less than 5 percent.

(vi) Format using reference temperature concept and the dimensionless temperature ΔT^* (Format F7T^*). In the format F3 (Eq. (19)), the variation of the fluid properties is considered in the correlation by selecting a proper reference temperature to evaluate the fluid properties, in addition to the dimensionless temperature T_∞/∆T. This dimensionless temperature has T_∞ explicitly appearing in the correlation and this is not desirable. Instead, the dimensionless temperature, ∆T^* (∆T^* = [(T^* − T_∞)/∆T]) can be used. In that case, regression analysis yields the correlation as (28)(Nux)xfcF7=0.297×Reref0.505×Prref0.327×ΔT∗0.031

The R² value with respect to Eq. (28) is found to be 0.998. Only 4 data points out of 542 have deviations between 5% and 10% and the rest have deviations within ±5%.

The local heat transfer coefficients obtained using the correlations of various formats except the correlation based on enthalpy ratios, are compared with the actual numerical predictions. The reason for comparing the heat transfer coefficients and not the Nusselt numbers is because except for the reference temperature method, the fluid properties are evaluated at T∞ for other formats, whereas in the reference temperature format, the fluid properties required in the correlation are evaluated at the reference temperature which may or may not be the same as T_∞. Table 3 shows the number of data points which have, (i) deviations that are >10%, (ii) the number of points with deviations between 5 and 10 percent, and (iii) the number of points with deviations ≤5%. It can be seen from this table that all formats of correlations predict local Nusselt numbers which agree with the actual predictions within ± ±10%. The correlations as per format fcF2, format fcF3, format fcF6 and format fcF7 predict Nusselt numbers that agree with the actual predictions within the ±5% for a maximum number of data points (99.26%) and hence recommended.

The procedure followed to find the correlations for laminar forced convection without buoyancy effects is also followed for forced convection with buoyancy effects except that an additional parameter Gr_x is introduced in the correlation in all the formats to account for the buoyancy effects.

(i) Format 8: This format is similar to format 1 (Eq. (13)) except that one more parameter Gr_x is introduced to account for the buoyancy effects. Thus, the format of the correlation is (29)(Nux)wbF1=a1(Rexb1)(Grx)c1(Pr)d1(C¯pCp\infty )e1(ρ∞ρw)f1(µ∞µw)g1(k∞kw)h1

The constants a₁ to h₁ in Eq. (29) are obtained using multiple linear regression analysis for 540 data points and the correlation obtained is (30)(Nux)wbF1=0.481×(Rex0.528)(Grx)−0.008(Pr)−0.659(C¯pCp\infty )0.397(ρ∞ρw)−0.108(µ∞µw)−0.178(k∞kw)0.025

The R² value with respect to Eq. (30) is found to be 0.999. The deviations of the data points with respect to Eq. (30) are ≤2.5% for 520 data points (96.3%) out of 540 points and for the remaining 20 points the deviations lie between 2.5% and 6%.

(ii) Format 9: If it is assumed that the variation of viscosity and conductivity can be accounted for by using the density ratio and specific heat ratio in the correlation, then multiple linear regression analysis gives the following correlation: (31)(Nux)wbF2=0.298×(Rex0.554)(Grx)−0.017(Pr)0.011(C¯pCp\infty )0.171(ρ∞ρw)−0.153

The R² value with respect to Eq. (31), is found to be 0.998. For all the 540 data points the deviations are ≤5%.

(iii) Format 9.1 (Correlation using reference temperature method): In this format, for forced convection with buoyancy effects, the correlation is assumed to be of the form (32)(Nux)wbF3∗=a2(Grx)Refb2(Rex)Refc2(Prref)d2(T∞ΔT)e2

In Eq. (32), all the dimensionless numbers, namely, (Grx)Ref, (Rex)Ref, Prref and (Nu_x)_WBF9 are first calculated for all the data points using some reference temperature, say T_ref = T_∞ and using multiple linear regression analysis, the constants a₂ to e₂ are found. The constants are found to be as follows:

a₂ = 0.383; b₂ = 0.00042; c₂ = 0.504; d₂ = −0.399; e₂ = 0.062.

Then r^* which is the optimum value of r is found for each Δ^* and for each pressure and the plot of r^* vs. ΔT^* is obtained. It has been found that for all pressure except p = 44.145 bar, the value of r^* = 1 for all values of ΔT^* and hence the plot of r^* vs. ΔT^* is shown in Fig. 20, only for p = 44.145 bar. From this plot r^* can be obtained for any ΔT^* and using Eq. (16) the reference temperature is determined to get the fluid properties at that temperature. Then the final form of correlation obtained is

(33)(Nux)wbF3=0.383(Grx)Ref0.00042(Rex)Ref0.504(Prref)−0.399(T∞ΔT)0.062

The R² value with respect to Eq. (33), is found to be 0.996. It is found that 523 data points (96.9%) out of 540 points have deviations <5% and the remaining 17 data points have deviations between 5% and 10%. T_∞ explicitly appears in Eq. (33) and this is not desirable. Instead, if dimensionless temperature ∆T^* is used in the correlation, then regression analysis yields the correlation as (33a)(Nux)wbF3∗=0.383(Grx)Ref0.00042(Rex)Ref0.504

The R² value with respect to Eq. (33a) is found to be 0.995412. The Nusselt number predicted using Eq. (33a) agrees with the actual numerical predictions within ±5 percent for 497 points out of 537 points, and for the rest of the data points, the deviations are between 5 and 10 percent.

(iv) Format 10 (Correlation using enthalpy ratios): In this format, two correlations are proposed, one for the cases for which the data points have T_w < T^* (cases for which T^* lies outside the boundary layer) and the other correlation for which T_w ≥ T^* (cases for which T^* lies inside the boundary layer).

In the present investigation, 342 data points have T_w < T^* and the format of the correlation is (34)(Nux)wbF4a=a3(Grxb3)(Rexc3)(Pr∞)d3(iw∗ic∗)e3(ic∗i∞∗)f3

Regression analysis, using 342 data points, Eq. (34) reduces to (35)(Nux)wbF4=0.245×Grx−0.0068×Rex0.5242×Pr∞0.246×((iw∗ic∗)−0.028)×((ic∗i∞∗)−0.028)

with R² = 0.9992. Nusselt numbers predicted using Eq. (33) agrees with the actual numerical predictions within ±2 percent. For the case of remaining 198 data points for which T_w ≥ T^*, the correlation is assumed to be of the form of (36)(Nux)wbF4=a3(Grxb3)(Rexc3)(Pr∞)d3(iw∗ic∗)e3(ic∗i∗∗−i∞∗)f3

Regression analysis using Eq. (36) for these 198 points yields the correlation as (37)(Nux)wbF4=0.696×(Grx−0.0006)×(Rex0.521)×(Pr∞−1.231)×(iw∗ic∗)−0.156×(ic∗i∗∗−i∞∗)0.405

with R² = 0.994. Comparison of Nusselt numbers obtained using Eq. (37) with the actual numerical predictions indicate that 10 data points out of 197 points have deviations between 6% and 7% and the rest of data points have deviations less than 6%.

(v) Format 11 (Correlation using constant property correlation and four property ratios):

In this format, the correlation for the Nusselt number is expressed in terms of the correlation for constant property fluids with additional parameters to account for the variation of the fluid properties. Thus, the correlation takes the form (38)(Nux)wbF5=a4((Nux)mccp)b4(C¯pCp\infty )c4(ρ∞ρw)d4(µ∞µw)e4(k∞kw)f4

In Eq. (38), (Nux)mccp represents the local Nusselt number for constant property fluids for mixed convection from a horizontal isothermal plate with heated surface facing downwards and is given by [29] (39)(Nux)WBcp=((Nux)fccp3+(Nux)nccp3)1/3

In Eq. (39), (Nux)fccp represents local Nusselt number for forced convection for laminar flow over a flat plate for constant property fluids and (Nux)nccp represents the local Nusselt number for free convection for laminar flow over a horizontal isothermal flat plate with the heated surface facing downwards. The Nusselt numbers (Nux)fccp and (Nux)nccp are given by [29] (40)(Nux)WBF5=0.2025×(Grx∗Pr∞)0.25

(Nux)fccp and (Nux)nccp are calculated for each data point and substituted in Eq. (39) to get (Nux)mccp for each data point and using these values along with Eq. (38) regression analysis is carried out to get the correlations as (41)(Nux)WBF5=1.16×((Nux)WBcp)0.961(C¯pCp\infty )0.339(ρ∞ρw)−0.048(µ∞µw)−0.204(k∞kw)−0.033

The R² value concerning Eq. (41) is found to be 0.996. Comparison of Nusselt number obtained using.

Eq. (41) with the actual numerical predictions indicates that 3 data points (0.56%) have deviations >10%, 12 data points (2.22%) have deviations between 5% and 10% and the rest (97.22%) have deviations <5%.

(vi) Format 12 (Correlation using constant property correlation with only two property ratios):

This format assumes that the nature of the variation of viscosity and thermal conductivity are similar to the variations of density and specific heat for supercritical fluids and hence the specific heat ratio and the density ratio will also account for the variation of viscosity and thermal conductivity. The multiple linear regression analysis for this format yields the following correlation: (42)(Nux)WBF6=0.926×((Nux)WBcp0.990187)×(Cp∗0.123848)×(ρ∗−0.1635)

The R² value concerning Eq. (42) is found to be 0.999969. Eq. (42) predicts the Nusselt numbers which are within ±5% for 515 out of 537 data points and the rest of the data points the deviations are between 5 and 15%.

(vii) Format 12 (Correlation using constant property correlation and ∆T^*): For this format, regression analysis yields the correlation as (43)(Nux)WBF7∗=((Nuref)WBcp)0.973139×(ΔT∗)0.075684

The R² value concerning Eq. (43) is found to be 0.999948. The Nusselt numbers predicted by Eq. (43) agree with the actual numerical predictions within ±5% for 497 out of 537 data points and for the rest of the data points the deviations are between 5% and 15%.

All the correlations except the correlation based on enthalpy ratios are compared with the actual numerical predictions. The comparison shown in Fig. 21 is based on heat transfer coefficients and not on the Nusselt numbers as explained earlier and is due to the fact that except for the reference temperature method, the fluid properties are evaluated at T∞ for other formats, whereas in the reference temperature format, the fluid properties required in the correlation are evaluated at the reference temperature which may or may not be the same as T_∞. The correlation based on enthalpy ratios is not included in this comparison as, under this format, two separate correlations based on enthalpy ratios are proposed in this comparison, one for the cases for which T_w < T^* and the other correlation for cases for which T_w ≥ T^*. Fig. 21 indicates that except for 6 data points (1.11%), the ratio of hx based on the correlation to hx obtained in actual numerical predictions varies between 0.9 and 1.1. In other words, the local heat transfer coefficients obtained by various correlations agree with the actual numerical predictions within ±10%. Table 4 shows the number of data points that have (i) deviations that are >10%, (ii) the number of points with deviations between 5% and 10%, and (iii) the number of points with deviations ≤5%. It can be seen from this table that all formats of correlations predict local Nusselt numbers which agree with the actual predictions within ±10%. The correlations as per format 7 and format 8 predict Nusselt numbers which agree with the actual predictions within ±5% for a maximum number of data points (99.44% of data points) and hence recommended.