Hydrodynamic considerations on the performance of chilled water thermal storage tanks in the discharge cycle

Abstract

The paper analyzes the effect of hydrodynamic viscous boundary layer growth over the performance of a thermal storage tank during the discharge cycle. As well established in Fluid Mechanics textbooks, a viscous boundary layer modifies the velocity profile across the tank section. Consequently, in a discharge cycle the warm water flowing outside the boundary layer (i.e., in the core region) reaches the tank bottom exit faster than that if the down flow water were flowing at the mean velocity based on the discharge flow rate, which is the usual designing and tank selection assumption. Consequently, the storage tank height must be greater than that determined using the simple mean flow velocity. Two controlling parameters appear naturally in the analysis: the Reynolds number based on tank diameter, Re_Ø, which is also associated with the hydrodynamic entry length, and f, which defines the position of a given contact surface from the tank entrance to the hydrodynamic entry length. Results show that the tank loss of capacity due to viscous effects may not be negligible and the selection of a height-to-diameter tank ratio is essential for minimizing those effects.

Appendix A: Correlations for axial velocity along with the hydrodynamic entry length flow inside tubes and flat plates

As a liquid flows inside a cylindrical thermal storage tank, a hydrodynamic boundary layer is formed next to the wall due to viscous effects, giving rise to velocity gradients across the tank section, as depicted in Fig. 10. Outside that boundary layer (fluid core), viscous effects are negligible and the fluid flows with uniform velocity, which increases as the boundary layer grows thicker, due to the mass conservation law. As well known, two flow regimes can be formed and they are reviewed next in the context of the present work.

Fig. 10

Velocity profile along the hydrodynamic entry length

1.1 Laminar flow

Laminar flow inside tubes occurs for a Reynolds number based on tube diameter, Re_Ø, less or equal to 2,300, as well established in Fox et al. [20] and Kays and Crawford [21]. Table 1 shows the values for axial velocity at the fluid core along with the boundary layer thickness given as a function of the distance x _l from the beginning of the boundary layer formation. The data have been extracted from the studies of Hornbeck [24], being valid for Re_Ø higher than 400.

Table 1 Axial velocity at fluid core and boundary layer thickness along the hydrodynamic entry length—laminar flow inside tubes

Hornbeck’s solution [24] requires (x _l /Ø)/Re_Ø going to infinity for reaching the fully developed flow condition. However, according to Langhaar [21], hydrodynamic entry length x _ell can be determined by Eq. (24):

$$\frac{{x_{\text{ell}} /\phi }}{{{\text{Re}}_{\phi } }} = 0.05,\quad {\text{Re}}_{\phi } \le 2,300$$

(24)

Hence, in the first column in Table 1 the value of Eq. (24) was used for (x _l /Ø)/Re_Ø. In the second column, f represents the dimensionless boundary layer distance considered from the beginning of the boundary layer formation, as defined in Eq. (2).

Usually, the hydrodynamic entry length is greater than the usual thermal storage tank height and, therefore, the inside flow never reaches the fully developed flow regime. Using Eq. (24) for Re_Ø equals to 2,300, for example, one has x _ell equal to 115 Ø. If one considers a particular tank with 15 m diameter and 20 m high, the dimensionless length for the height will be equal to 0.01. Therefore, a correlation which represents the values of the axial velocity from the studies by Hornbeck [24] for a f range between 0 and 0.1 as well as possible should be sought. A correlation that meets such conditions is given below:

$$U/U_{0} = \mathop {\left( {373f + 1} \right)}\nolimits^{0 \cdot 1}$$

(25)

The last two columns in Table 1 show the axial velocity calculated from Eq. (25) and the relative deviation, when compared to the values from the studies by Hornbeck [24]. As observed, the values obtained are very reasonable for the considered f range.

1.2 Turbulent flow

The turbulent flow inside tubes is well defined for the Reynolds number based on tube diameter, Re_Ø, equal or higher than 4,000 [19, 20]. For the flow within the hydrodynamic entry length, the equation relating a given length x _t with the axial velocity at the fluid core will be obtained from some analogies with the flow over a flat plate. For the velocity in the fully developed flow, the power law profile was used, with the exponent equal to 1/6, which represents the velocity radial distribution in turbulent flow for Re_Ø equals to 4,000. The other acceptable exponent value is 1/7 that is valid for Re_Ø equals to 110,000 [6], which greatly exceeds the typical Reynolds number values of the thermal storage tank. For flow over a flat plate, the power law profile with the exponent 1/6 was also used. Table 2 shows a summary of relevant Eq. (27) through (30) for flow over flat plates and inside tubes, set side by side, needed for Milaré’s study of analogies [17]. The relative nondimensional  thickness δ* can be defined as the ratio between the boundary layer thickness (within the reference length) and the reference thickness, as it can be seen in Eq. (26):

$$\left( {\left. {\frac{{\mathop \delta \nolimits_{{_{\text{t}}^{*} }} }}{{\mathop \delta \nolimits_{{_{\text{l}}^{*} }} }}} \right)} \right._{{{\text{flat}}\;{\text{plate}}}} = f^{0.3}$$

(26)

where indexes “l” and “t” have been added to δ* to identify laminar and turbulent flow.

Table 2 Equations for flows over flat plate and inside tubes

1.3 Analogies between flow over a flat plate and inside tube

Equations for flows over a flat plate and inside a tube are similar and two possible analogies are usually used for describing them, as discussed next.

1.3.1 First analogy

For the laminar regime, let one adopt the velocity profile of the flat plate to represent the velocity radial distribution inside the boundary layer, along with the hydrodynamic entry length inside tubes. It is first necessary to verify if the velocity profile of the fully developed flow is met. Making δ in Eq. (28) equal to r ₀ (when the flow becomes fully developed, the boundary layer thickness grows to the tube ratio) and y equal to (r ₀ − r), one obtains Eq. (32). The second step is to verify if applying Eq. (28) inside the boundary layer, at a given cross section within the entry length, along with the equation of continuity, one obtains the axial core velocity value equal to the values obtained from the studies by Hornbeck [24]. In fact, the values obtained for axial velocities are quite close to the ones obtained from Hornbeck’ studies as shown in the last column in Table 1 (less than 3 % deviation for the range of interest and less than 6 % for the entire range). Therefore, for laminar flow inside tubes and within the hydrodynamic entry length, Eq. (28) can represent the velocity inside the laminar boundary layer.

By comparing equations for turbulent flow over a flat plate and inside a tube, one can notice that they are very similar to each other in a similar fashion to the ones for the laminar flow case. Also, if one assumes in Eq. (30) the same simplifications assumed in Eq. (28) for the laminar case, one obtains Eq. (34). Therefore, as the first analogy, Eq. (30) will be assumed to represent the velocity radial distribution inside the boundary layer for the hydrodynamic entry length for turbulent flow inside tubes.

1.3.2 Second analogy

Analyzing Table 2, the particularity between Eqs. (27) and (29) (flow over a flat plate) is verified to be very similar to the particularity between Eqs. (31) and (33) (flow inside tubes). And so is the relationship between Eqs. (28) and (30) to the particularity between Eqs. (32) and (34). From these comparisons, as the second analogy, the relation between the relative thickness and the dimensionless length for flows over flat plate, Eq. (26), will be assumed to be also valid for flows inside tubes. Therefore:

$$\left. {\left( {\frac{{\mathop \delta \nolimits_{{_{\text{t}}^{*} }} }}{{\mathop \delta \nolimits_{{_{\text{l}}^{*} }} }}} \right.} \right)_{{{\text{inside}}\;{\text{tubes}}}} = f^{0.3}$$

(35)

1.4 Correlations for axial velocity at fluid core inside the tube

From the above analogies and the values from Table 3, one can obtain, along with the equation of continuity, the values for axial velocity and relative thickness, as a function of the dimensionless position f from the beginning of the boundary layers, for turbulent flow.

Table 3 Axial velocity at fluid core and boundary layer thickness along the hydrodynamic entry length—turbulent flow inside tubes

The hydrodynamic entry length in turbulent flow is far smaller than that of the laminar one. Using Eq. (16) Re_Ø equal to 5,000, for example, one has x _elt equal to 5.23 Ø. If one considers the same tank from the laminar example, the dimensionless length for the height will be equal to 0.255. However, larger values of f are possible.

Therefore, a correlation that represents the values of the axial velocity for a largest f range as well as possible should be sought. A correlation that meets such a condition is given below:

$$U/U_{0} = \mathop {\left( {17.8f + 1} \right)}\nolimits^{0.08}$$

(36)

The last two columns in Table 3 show the axial velocity calculated from Eq. (36) and the relative deviation, when comparing those results with the values from the analogies. As observed, the values obtained from that equation are very reasonable for the entire f range. However, Eq. (36) does not obey one boundary condition, that is: its derivative relative to f, for f equals 1, should be zero, but it does not vanish. Its application should thus be limited, say, for f in the range between 0 and 0.9.