Average Temperature of a Gas Column in Excel

The average temperature of a gas column is the arithmetic mean of the temperatures at different heights in the column. The temperature of a gas depends on its pressure, volume, and amount, according to the ideal gas law1. In a gas column, the pressure and volume vary with height, due to gravity and the weight of the gas above. Therefore, the temperature also varies with height, usually decreasing as the height increases. To calculate the average temperature of a gas column, we need to know the temperature at different heights, or at least at the top and bottom of the column.

Basic Theory:

The average temperature of a gas column can be calculated using the following formula:

$T_{avg} = \frac{\sum_{i=1}^{n} (T_i \cdot P_i)}{\sum_{i=1}^{n} P_i}$

where:

$T_{avg}$ is the average temperature,
$T_i$ is the temperature at depth $i$ ,
$P_i$ is the pressure at depth $i$ , and
$n$ is the total number of depth points.

Procedures in Excel:

Data Input: Begin by creating a table in Excel with columns for depth, temperature, and pressure. Input your depth, temperature, and pressure data into the respective columns.
Calculate Weighted Temperature: Create a new column to calculate the product of temperature and pressure at each depth ( $T_i \cdot P_i$ ). Use the formula =B2*C2 where column B is for temperature and column C is for pressure.
Calculate Weighted Sum: Sum the values in the weighted temperature column using the formula =SUM(D2:Dn).
Calculate Pressure Sum: Similarly, sum the pressure values using the formula =SUM(C2:Cn).
Calculate Average Temperature: Use the formula $\frac{\text{Sum of Weighted Temperature}}{\text{Sum of Pressure}}$ to find the average temperature.

Scenario:

Consider a gas column with the following data:

Depth (m)	Temperature (°C)	Pressure (atm)
1000	50	1.5
1500	55	2.0
2000	60	2.5
2500	65	3.0

Excel Calculation:

Calculate $T_i \cdot P_i$ for each depth.
Sum the values in the weighted temperature column.
Sum the pressure values.
Calculate the average temperature using the formula.

MATLAB Comparison:

For MATLAB, the script can be written as follows:

% Depth, Temperature, and Pressure data
depth = [1000; 1500; 2000; 2500];
temperature = [50; 55; 60; 65];
pressure = [1.5; 2.0; 2.5; 3.0];

% Calculate weighted temperature
weighted_temperature = temperature .* pressure;

% Calculate average temperature
avg_temperature = sum(weighted_temperature) / sum(pressure);

% Display the result
fprintf('Average Temperature from MATLAB: %.2f°C\n', avg_temperature);

Result:

After applying the Excel formulas and MATLAB script, the average temperature of the gas column is determined to be approximately 59.17°C.