Heat and mass transfer during the warming of a bottle of beer

The warming of a bottle of beer during a Friday evening happy hour directly involves transport phenomena, such as mass transfer due to condensation of air humidity on the bottle surface and heat transfer from the ambient to the bottle, which occurs by free convection and water condensation. Both processes happen simultaneously and are directly associated with the heat and mass transfer coefficients involved, which are affected by the ambient humidity and temperature. Several runs were made in several ambient conditions by exposing a cold bottle of beer to varied temperature and humidity and measuring the temperature of beer and the mass of water condensed on the bottle surface over time. From these measures, a theoretical and experimental methodology was developed and applied for the evaluation of the heat and mass transfer coefficients that govern this process. Both the relative humidity and ambient temperature exert a significant influence on the convective heat transfer coefficient. However, the mass transfer coefficient is affected only by the temperature.


Introduction
Brazil is one of the largest consumers of beer in the world.Therefore, pleasing the final consumer is a task that requires knowledge of the market and technical knowledge of how the product behaves in different ambient conditions.
The warming of beer is a known undesirable effect for the consumer.Knowing its mechanisms and identifying its main causes are key to appropriately tackling this problem.
The warming of a bottle of beer is thought to be due mainly to two heat transfer processes, natural convection and condensation, which are influenced mainly by the air temperature and relative humidity.
Natural convection heat transfer always occurs from a body immersed in a stagnant fluid whose temperature is different from that of the body, such as a bottle of beer in the ambient.Due to the different temperatures of the air and the bottle surface, the density of the air near the bottle changes, resulting in a descending movement of the colder and heavier air near the bottle surface.
The determination of the heat exchange rate requires knowing the natural convection heat transfer coefficient.For practical applications, we can use Newton's Law of Cooling (KREITH, 2003) The heat and mass transfer phenomena are frequently associated and may occur in common daily situations, such as the warming of a bottle of beer, in which both phenomena are simultaneous until the system reaches thermodynamic equilibrium.
Condensation occurs when air with a certain amount of humidity makes contact with the cold surface of the bottle of beer.From this point on, a liquid film forms and flows continuously on the surface (INCROPERA;DAVID, 2003).
The liquid film is formed due to the mass transfer from the water in the air to the surface of the bottle.This process is called convective mass transfer and is described by an equation similar to Newton's Law of Cooling, (CREMASCO, 2002), Equation 2.
Mass transfer by natural convection occurs due to the difference in water concentration between the bottle surface and the surrounding gas.In turn, the concentration of water on the bottle surface can be evaluated using saturation pressures obtained by Antoine's equation (SMITH et al., 2000) and the appropriate coefficients.
Parallel to the mass transfer, heat is transferred by condensation, as is described by Equation 3. The heat transfer rate on the surface can also be evaluated using Equation 4, if the water condensation rate ( cond m ) is known.
According to Fortes et al. (2006), in most processes the condensation heat transfer has a greater magnitude than the convective heat transfer.
A theoretical-experimental methodology has been developed to evaluate the heat and mass transfer coefficients in the initial instants of the warming of a bottle of beer, in which the condensation heat transfer prevails, and to verify the influence of ambient temperature and humidity on the heat and mass transfer coefficients.

Experimental apparatus
In this work, the experimental module illustrated in Figure 1 was used, which is constituted of a bottle of beer and a plastic tray placed under the bottle of beer, which served as a condensate collector.A Gehaka analytical balance (1  0.01 g), a graduated mercury thermometer (-10 to 60ºC) set on a rubber stopper, and a stopwatch were also used.

Experimental procedure
A full bottle of beer with the thermometerstopper was placed in a refrigerator and kept there until it reached the temperature of -2°C.
The full bottle with the thermometer was taken to the balance room and the bottle surface was wiped dry.The bottle was placed in the condensatecollecting tray in the balance plate and the balance and the stopwatch were reset.
From this point on, the temperature and accumulated condensate mass were measured in regular intervals of time corresponding to 0.5°C until the system reached room temperature.
The ambient temperature, relative humidity, and pressure were measured simultaneously.
Several assays were performed to estimate the influence of the ambient temperature and humidity on the process in a wide range of ambient temperature and relative humidity conditions (Table 2).

Determination of K m and h cond
The mass of condensed water was correlated by fitting a polynomial of appropriate order as a function of time, M(t).
The condensate flow rate (m cond ) can thus be evaluated by deriving M(t) in relation to time, as shown in Equation 5. )) (  (5) The condensation heat transfer coefficient (h cond ) can be determined from the condensate mass flow rate (m cond ) by equaling Equations 3 and 4, and Equation 6.
)) ( ( In turn, the molar rate of condensation of water on the bottle surface (m cond /PM H2O ) is equal to the mass transfer rate from the air to the surface, (N A A 0 ), which gives Equation 7.
The substitution of Equation 7 in Equation 2gives Equation 8 and the value of K m .

 
Both C A∞ and C AP (t) were obtained with Equations 9 and 10, while the water saturation pressures were estimated with Antoine's equation (Equation 11).
Contribution of q cond to q total The heat transferred to the bottle by convection was calculated using the empiric Equation 12, as proposed by Fortes et al. (2006).
Considering that the total heat received by the bottle of beer (q total ) is the sum of q conv and q cond , the total percentage of heat received by the bottle by convection (q cond ) in relation to the total heat (q total ) can be estimated using Equation 13. 100 q q q (%) q cond conv cond cond   (13)

Results and discussion
As previously mentioned, the condensation heat (q cond ) may contribute to the heat received by the bottle of beer to a larger extent than the heat received by convection (q conv ).
To determine these parameters, Table 1 was constructed using Equations 4, 12, and 13.The percent contribution of the condensation heat in relation to the total heat involved in the bottle warming process over time was thus estimated.
Table 1.Heat rates involved in the warming of the bottle of beer.Time (min.) q cond (J s -1 ) q total (J s -1 ) q cond (%) (J s Table 1 shows that the behavior of q cond and q total = q cond + q conv during the initial 15 min. of warming of the bottle of beer.These data correspond to assay 2, Table 2.The interval of time of study was established so that the contribution of the condensation heat transfer rate was greater than 99.66% of the rate of total heat received by the bottle of beer in any given instant.In this way, the complete analysis was carried out in the initial 15 min., as shown in Table 1. In this period of time, both the mass of condensed water and the temperature of the beer increased and could be correlated by fitting a 2 nd Acta Scientiarum.Technology Maringá, v. 32, n. 2 p. 153-157, 2010 order polynomial, as shown in Figure 2, and with Equations 14 and 15, whose coefficients of correlation were over 0.98.M = -3.95x10 - t²+1.58x10 - t -3.89x10 -4 ( 14) T s = -1.66x10 -6 t² + 4.68 x10  The behavior of the heat and mass transfer coefficients as a function of the temperature and the relative humidity is shown in Figures 3 and 4. Both coefficients decreased as the temperature of the surface of the bottle of beer increased.
Figure 3a shows the influence of relative humidity on K m .The values of K m obtained in assays 2 and 3 were compared, Table 2, and it was concluded that the mass transfer coefficient is hardly affected by the relative humidity of the air.
Figure 3b compares assays 2 and 4. Both the relative humidity and the temperature of the ambient air varied, as shown in Table 2.It is known that the air humidity does not affect the mass transfer significantly; therefore, the different behaviors shown in Figure 3b result from the influence of the ambient temperature.
Similarly, Figure 4a, which shows the influence of the relative humidity on the heat transfer coefficient, compares the behavior of h cond obtained in assays 2 and 3 for the same ambient temperature.It can be observed that the higher the relative humidity is, the larger the value of K m is.It is also observed that h cond is influenced by the relative humidity of the air.Figure 4b shows the influence of temperature on the heat transfer coefficients in assays 2 and 4 in different T amb and RH conditions.As RH affects h cond , it can be inferred that the effect of RH opposes that of the ambient temperature.Thus, while RH increase leads to an increase in h cond , the increase in the ambient temperature results in a decrease of the heat transfer coefficient.

Conclusion
Heat transfer by condensation determines the rate of warming of the bottle in the initial 15 min.
In this period of time, the temperature and the accumulated condensed mass vary quadratically with time.
In contrast to the relative humidity, ambient temperature affects K m significantly in the range of experimental conditions that was investigated.
Both the relative humidity and the ambient temperature influence h cond significantly; however, the results suggest that the increase in the relative humidity of the air is associated with an increase in h cond , while an increase in the ambient temperature results in a decrease of h cond .

Figure 1 .
Figure 1.Scheme of the experimental module.

Figure 2 .
Figure 2. accumulated mass (a) and temperature (b) as a function of time.

Figure 3 .
Figure 3. Influence of relative humidity (a) and the ambient temperature (b) on the mass transfer coefficient.

Figure 4 .
Figure 4. Influence of relative humidity (a) and ambient temperature (b) on the heat transfer coefficient.