Study on heat transfer coefficients during cooling of PET bottles for food beverages

Abstract

The heat transfer properties of different cooling systems dealing with Poly-Ethylene-Terephthalate (PET) bottles were investigated. The heat transfer coefficient (U_g) was measured in various fluid dynamic conditions. Cooling media were either air or water. It was shown that heat transfer coefficients are strongly affected by fluid dynamics conditions, and range from 10 W/m² K to nearly 400 W/m² K. PET bottle thickness effect on U_g was shown to become relevant under faster fluid dynamics regimes.

Appendix

1.1 Internal heat transfer coefficient

The theoretical case of pure diffusive heat transfer inside the bottle was considered.

Although this condition is not expected to be realistic for our case studies, its analysis is still very useful for data interpretation. Indeed, it allows one to define a limit heat transfer coefficient: any experimentally found U_g, if higher than the one calculated for pure diffusive heat transfer, would imply natural convection is occurring within the bottle. External heat transfer was considered extremely fast and PET wall was lumped with internal distilled water. This latter approximation is legit as thermal diffusivity of PET and water are very close (α_w = 1.43 × 10⁻⁷ and α_PET = 1.5 × 10⁻⁷) [18]. However, if those thermal diffusivities were not similar, by means of dimensional analysis the PET layer could be substituted with a fictitious water layer [19], thus solving a single diffusion equation instead than two distinct differential equations. In the case of pure conduction, the heat equation written in cylindrical coordinates is [13]:

$$\frac{\partial T}{\partial t} = \alpha *\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial T}{\partial r}} \right)$$

(9)

which must be solved with the following boundary conditions: at t = 0, T = T₀; at r = 0, ∂T/∂r = 0; at r = r₁ ≈ r₂, the conductive heat flux must be equal to the external convective heat flux. The characteristic time for this system, defining the shape of the T versus t curve, can be estimated as τ = R²/α [13]. When using the aforementioned values, calculated τ is around 7600 s. According to theory, after a time close to τ, equilibrium is achieved [20].

A simulation with COMSOL Multiphysics package, assuming pure heat conduction, gave nearly the same results (Fig. 9). The external heat transfer coefficient was set to 1000 W/m² K, to highlight the internal resistance.

In this way, the maximum cooling time was estimated, related to the worst heat transfer conditions inside the bottle. A lumped system with the same τ, would be characterized by a U_g in the order of 10⁻² W/m² K, therefore any bigger U_g experimentally measured, would reasonably imply the presence of convection phenomena inside the bottle. Typical values for heat transfer coefficient of liquids in natural convection conditions are in the range 10–100 W/m² K [13]. Collected data always led to U_g in this order of magnitude, thus confirming the presence of natural convection phenomena inside the bottle.

1.2 External heat transfer coefficient

To get an estimate of external heat transfer coefficients during static cooling in air and water, different empirical correlations were employed. Nusselt number (${\text{Nu}} = \frac{\text{hD}}{\text{k}}$, where D is the cylinder diameter) around horizontal cylinders can be calculated by means of empirical correlations Eq. 10 and 11 [21, 22].

$$Nu = \left\{ {0.6 + \frac{{0.387Ra_{{}}^{1/6} }}{{\left[ {1 + \left( {0.559/Pr} \right)^{9/16} } \right]^{8/27} }}} \right\}^{2}$$

(10)

$$Nu = C\left( { \Pr } \right)*Ra^{0.25}$$

(11)

In Eq. 11, C is a constant depending on Prandtl number value ($\Pr = \frac{{\mu {\text{c}}_{\text{p}} }}{\text{k}}$, where μ is the fluid viscosity), which is 0.436 for air and 0.52 for water at ambient conditions. Rayleigh number is defined as Ra = Gr*Pr, with Gr being the Grashof number ($Gr = \frac{{g\beta \left( {T_{wall} - T_{\infty } } \right)D^{3} }}{{\nu^{2} }}$, where g is the gravitational acceleration, β the volumetric thermal expansion coefficient, ν the kinematic viscosity, T_wall the temperature at bottle surface and T_∞ the bulk temperature of cooling fluid).

In order to estimate Nu value in the case of vertically positioned bottle, Eq. 12 was used, which is suitable for vertical surfaces [21] and Eq. 13, for vertical cylinders [22].

$$Nu = \left\{ {0.825 + \frac{{0.387Ra_{{}}^{1/6} }}{{\left[ {1 + \left( {0.492/Pr} \right)^{9/16} } \right]^{8/27} }}} \right\}^{2}$$

(12)

$$Nu = \frac{4}{3}\left[ {7Gr\frac{{Pr^{2} }}{{5\left( {20 + 21Pr} \right)}}} \right]^{0.25} + \frac{{4H\left( {272 + 315Pr} \right)}}{{35D\left( {64 + 63Pr} \right)}}$$

(13)

Results are summarized in Table 1, Theory and Methodology. Calculations for water led to h₂ two orders of magnitude higher than in the case of air.

Calculations worked out in order to estimate heat transfer coefficient in air dynamic cooling were based on the work by [23] on heat transfer from a heated rotating cylinder in still air. During rotation at 200 rpm, Reynolds number around bottle is about 2500, while Gr number is nearly 100,000. In this condition, the system cannot be considered as ruled by forced convection neither by natural convection only, and both Re and Gr numbers are needed in order to estimate Nusselt number. For the considered case, Nu is about 40, thus resulting in a heat transfer coefficient h₂ of about 17 W/m² K.

1.3 Heat transfer across PET layer

The scenario herein presented highlights how much PET layer thickness affects global heat transfer coefficient. Writing Eq. 3 as:

$$\theta = e^{{\left( { - \frac{{U_{g} A}}{{mc_{p} }}t} \right)}}$$

(14)

where θ is (T−T_c)/(T₀−T_c), the time required to reach a certain dimensionless temperature θ can be calculated as:

$$t = - \frac{{mc_{p} }}{{U_{g} A}}\ln \theta$$

(15)

By substituting Eq. 1, Eq. 6 was derived:

$$t = - \frac{{mc_{p} }}{A}\left( {\frac{1}{{h_{1} }} + \frac{1}{{h_{2} }} + \frac{\delta }{k}} \right)\ln \theta$$

(16)

it can be noticed how cooling time is dependent on PET thickness δ. At high fluid dynamics regimes, where 1/h₁ and 1/h₂ are much smaller than δ/k, this effect becomes more important, and t is significantly influenced by δ [18].