Wine bottles are never completely filled: a small volume of air is left in the glass bottle's cylindrically shaped neck (inner diameter d = 18.5mm) to allow for wine's fairly large coefficient of thermal expansion. The distance H between the surface of the liquid contents and the bottom of the cork is called the "headspace height" , and is typically H = 1.5cm for a 750mL bottle filled at 20degC. Due to its alcoholic content, wine's coefficient of volume expansion is about double that of water; in comparison, the thermal expansion of glass can be neglected.
Estimate H if the bottle is kept at 11degC.
Estimate H if the bottle is kept at 32degC.
Begin by looking up the thermal expansion coefficient of water.
You can find it at
http://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion
and many other places.
I get 2.07*10^-4 C^-1 at 20 °C
For wine, it would then be
beta = 4.14*10^-4 C^-1 at 20 °C
That coefficient applies to the volume, not the linear length of the wine in the neck of the bottle. At 11 K (a 9 C reduction from 20), the volume will shrink from 750 ml to
750 * (1 - 4.14*10^-4*9) = 747.2 ml
Now assume the area A of the neck of the bottle stays the same. The volume occupied by wine shrinks 2.8 cm^3.
Divide that by the cross sectional area of the neck of the bottle to see how much the liquid level drops.
A = (pi/4)(1.85)^2 = 2.69 cm^2
The liquid level drops and the "head space" lengthens by 2.8/2.69 = 1.04 cm
Perform a similar calculation for the higher temperature.
To estimate the headspace height (H) at different temperatures, we can use the principle of thermal expansion. The change in volume for both wine and air will be considered, assuming the glass volume remains constant.
Step 1: Calculate the change in volume of the wine due to temperature change.
Given:
Initial temperature (T1) = 20°C
Final temperature (T2) = 11°C or 32°C
Coefficient of volume expansion for wine (β_wine) = 2 x coefficient of volume expansion for water (β_water)
The change in volume of the wine (ΔV_wine) can be calculated using the formula:
ΔV_wine = V_wine * β_wine * (T2 - T1)
Step 2: Calculate the change in volume of the air due to temperature change.
The change in volume of the air (ΔV_air) can be calculated using the ideal gas law:
ΔV_air = V_air * α_air * (T2 - T1)
Note: The coefficient of volume expansion for air (α_air) is approximately equal to 1/T, where T is the temperature in Kelvin.
Step 3: Calculate the total change in volume of both the wine and air.
ΔV_total = ΔV_wine + ΔV_air
Step 4: Calculate the new headspace height (H) at the final temperature.
H2 = H1 * (V_total - ΔV_total) / V_total
where,
H1 = Initial headspace height (1.5 cm)
V_total = Total volume of the bottle (750 mL)
Let's calculate the estimated headspace height H when the bottle is kept at 11°C and 32°C.
1. Estimate H when the bottle is kept at 11°C:
Step 1: Calculate the change in volume of the wine:
ΔV_wine = V_wine * β_wine * (T2 - T1)
= 750 mL * (2 * β_water) * (11 - 20) / 100
= -135 mL
Step 2: Calculate the change in volume of the air:
ΔV_air = V_air * α_air * (T2 - T1)
= V_air * (1/T2 - 1/T1) * (T2 - T1)
= V_air * (1/284 - 1/293)
= 750 mL * (1/284 - 1/293)
≈ -3.16 mL
Step 3: Calculate the total change in volume of both the wine and air:
ΔV_total = ΔV_wine + ΔV_air
= (-135 mL) + (-3.16 mL)
≈ -138.16 mL
Step 4: Calculate the new headspace height (H) at 11°C:
H2 = H1 * (V_total - ΔV_total) / V_total
= 1.5 cm * (750 mL - (-138.16 mL)) / 750 mL
≈ 1.67 cm
Therefore, the estimated headspace height (H) when the bottle is kept at 11°C is approximately 1.67 cm.
2. Estimate H when the bottle is kept at 32°C:
Repeat steps 1-4 using T2 = 32°C.
Step 1: Calculate the change in volume of the wine (ΔV_wine)
...
You can follow the same process to estimate the headspace height (H) when the bottle is kept at 32°C.
To estimate the headspace height (H) in the wine bottle when it is kept at different temperatures, we need to consider the thermal expansion of wine and the change in volume due to temperature changes.
First, let's calculate the volume change of the wine when the temperature changes from 20°C to 11°C and from 20°C to 32°C. We'll assume the bottle is initially filled to the brim.
Given:
Initial temperature (T1) = 20°C
Final temperature (T2):
- T2 = 11°C (case 1)
- T2 = 32°C (case 2)
Initial headspace height (H1) = 1.5 cm
Inner diameter of the bottle neck (d) = 18.5 mm
To estimate H, we can use the ideal gas law, which states that the change in volume of a gas is proportional to the change in temperature:
ΔV = V * α * ΔT
Where:
ΔV = Change in volume
V = Initial volume
α = Coefficient of volume expansion
ΔT = Change in temperature
Since wine has a higher coefficient of volume expansion than water (roughly double), we can assume α for wine is approximately 2 times α for water. The coefficient of thermal expansion for water is approximately 0.00021 per degree Celsius.
First, we need to convert the headspace height (H1) to volume (V1). Assuming a cylindrical shape:
Initial volume (V1) = π * (d/2)^2 * H1
Now, let's calculate the change in volume (ΔV) for each case:
Case 1: T2 = 11°C
ΔT1 = T2 - T1 = 11°C - 20°C = -9°C
ΔV1 = V1 * α * ΔT1
Case 2: T2 = 32°C
ΔT2 = T2 - T1 = 32°C - 20°C = 12°C
ΔV2 = V1 * α * ΔT2
Finally, let's calculate the new headspace height (H2) for each case. Assuming the bottle maintains its cylindrical shape and the overall volume remains constant:
New volume (V2) = V1 + ΔV
New headspace height (H2) = V2 / (π * (d/2)^2)
Now, we can substitute the values and calculate the estimates for H in each case.