Using a Spreadsheet to Construct a Takeoff Chart |
Harvard Holmes December, 2002 |
Abstract
We take published or experimental data for takeoff performance and combine it with analytical models of performance to generate take off charts. These charts allow easy evaluation of take off distance including the effects of density altitude, gross weight and winds. We follow readily available and published material on aircraft performance; the contribution of this work is to show how to construct these charts using spreadsheets and available performance data.
Introduction
Take off performance is a critical element of aircraft performance. We take existing data and use a spreadsheet and analytic formula to provide a convenient take off chart. It allows estimation of take off runs as functions of density altitude, weight, and wind. Tradeoffs between various factors can be quickly evaluated to provide a safe take off strategy. We follow the method described in reference 1. The examples given here apply to a non-supercharged engine. For a supercharged engine, the reduction in performance with pressure altitude is less than with a normally aspirated engine.
Figure 1. Sample Takeoff Chart
These charts are used by starting at the left hand side with the pressure altitude and temperature to get a starting point which corresponds to density altitude. Using the background grid, this value is carried to the right to the "reference line" for takeoff weight. From the reference line, the curve is followed across and up or down to the actual takeoff weight. This adjusts the takeoff roll for differences from the reference weight. Again using the background grid, carry the value from the weight adjustment across to the wind reference line. Then follow the wind lines across and up or down to account for the wind. This completes the estimate for this chart.
We describe how to set up the spreadsheet using the equations in Reference 1. We also refer to Reference 2. to provide the necessary atmospheric values for density altitude.
International Standard Atmosphere
The International Standard Atmosphere (ISA) is described in terms of 7 layers with different temperature lapse rates and other parameters for each layer. We only need to consider layer 1, from sea level up to 36089 feet. For this layer the reference temperature is 288.15 Kelvin (=518.67 Rankine), and the variation with altitude is theta = (1.0 - h/145442) where h is in feet. The pressure ratio from sea level is delta = (theta)^5.255876. The density ratio from sea level is sigma = (theta)^4.255876. For example, at 10,000 feet, the standard temperature is 288.15 * (1-10000/145442) = 268.34 Kelvin or -4.81 Celsius.
Aircraft performance is commonly characterized using atmospheric density rather than the corresponding atmospheric pressure and temperature. Given the pressure altitude (with Altimeter set to 29.92 in. Hg) and the temperature, we use the relationship PV ~ T (for constant mass) to get that the density ratio = sigma_ISA / (T_ambient/T_ISA). This is equivalent to the pressure ratio (delta) / (T_ambient/T_sea_level). The temperature ratios must be in an absolute scale (Kelvin or Rankine).
Given the density ratio (sigma), the density altitude can be calculated. From sigma, we calculate the ISA temperature ratio theta = sigma ^ (1/4.255876) and then altitude h = (1 - sigma) * 145442.
Aircraft Performance
From Reference 1, the take off roll is commonly characterized as a function of the atmospheric density raised to some negative power: TO_roll = TO_roll_SL * sigma ^ (-k), where k is usually from 1.5 to 2.5. The analytic value of k is 2.0, but observed values of k range from 1.6 up to 2.5. For our 1966 Mooney M20E, k is 2.46 at maximum gross weight (2575), and 1.67 at 2200 lbs. Since the fit is better for the maximum gross weights and a larger value of k is more conservative, we used 2.4 for our charts. To get a value of k, we compared the values in the POH with our calculated values and adjusted k for a good fit. While the POH gives figures for 0, 2500, and 5000 feet altitude, each with ISA temperature and +- 40 deg. F, the overlap in density altitudes is such that only 5 density altitudes are covered. There is no observed scatter in multiple values for similar density altitudes -- something looks "too good" here!
Also per Reference 1, the effect of take off weight is commonly modeled by an exponential function of the weight ratio, eg TO_roll_2200-lbs = TO_roll_reference * (2200/W_reference) ^ (k2). The value of k2 is usually empirically determined, falling in the range of 1.5 to 2.5. For our Mooney again, a value of 2.23 provides the best fit to the POH data.
The effect of wind on take off distances is again modeled as a ratio of groundspeed to airspeed at takeoff raised to a power, e.g., wind_correction = (1 + v_wind/v_to) ^ (-k3). Following Reference 1, we use 1.85 for k3. We also increase the take off velocity with density altitude according to v_to = 1.3 Vs / (sigma ^ 0.5). This is equivalent to the rule of thumb of 1.6% per 1000 feet of altitude increase in true air speed.
Spreadsheet Construction
The spreadsheet is constructed on top of a grid, where the vertical values (y) are values of takeoff performance, and the horizontal values (x) are used to specify the values of each parameter that one wishes to compensate for. The first section begins with the construction of the pressure altitude lines. These lines are constructed such that the vertical value at this point will indicate the take off performance at the specified conditions. Subsequent sections will adjust this value for non-standard weights and winds. For this first section, the lines are constructed by calculating the take off performance for selections of pressure altitude and selections of temperatures. In this example, we use altitudes from 0 to 10,000 feet by 2,000, and temperatures from -20 to +40 Celsius by 10. At each point, the density ratio and the take off performance are calculated as described above, and the points of constant pressure altitude are connected. An additional line is calculated to indicate the points of ISA temperature along each pressure altitude line. This line is not needed to use the chart, but gives a nice reference to ISA temperatures.
The next section (from left to right) of the spreadsheet gives lines that allow one to adjust the take off performance for the weight of the aircraft. These lines provide a slope to follow as you move along or between the lines from one weight to another. Since they only provide a relative correction, they only have to have the correct slope and do not have to have any particular absolute value of take off performance. However, it is convenient to locate each line at so that the performance at the reference weight matches a particular density altitude. Here we have chosen density altitudes from 0 up to 14,000 (note there is a 2,000 foot bias in the spreadsheet formula to adjust the column headings of -2,000 to 12,000). For most aircraft, this range should be abbreviated by adjusting the spreadsheet.
In the same way, for the wind correction, we calculate the performance at altitudes of 0 to 12,000 feet at 2,000 foot intervals, and adjust it for winds of -10 kts (tailwind), 0, 10, and 20 kts (headwind). Points at constant altitude are joined by lines.
At this point, the numerical values for the spreadsheet have been plotted. In many cases, the remainder of the work is most easily done by hand, but the example spreadsheet does provide the additional lines needed to help interpret the spreadsheet. Two additional lines are needed to mark the reference points for the weight correction and the wind correction. These are simple vertical lines and could easily be drawn on a printed version of the spreadsheet. One line will be at the reference weight for the aircraft, typically the maximum take off weight, and the other line is at zero wind. It is also conventional to put lines illustrating the use of the chart, a line up from the ambient temperature to the pressure altitude, then a line horizontally across to the weight reference line, then a sloping line following the weight correction down to the take off weight of interest, then another horizontal line to the wind reference line, then a line along the wind correction curves, and the finally a horizontal line from the wind correction to the scale on the right side of the chart for the take off roll. If these lines are to be constructed using the spreadsheet, then it is convenient to overlay the existing data on the chart with a grid that has an x, y coordinate system. These explanatory lines are then added by looking at the chart and deducing the coordinates that will put the lines in the right places.
Finally, values for density altitudes can be added to the left hand side of the chart. The spreadsheet includes these values, but it is also easy to do by hand. For each pressure altitude, select a temperature that corresponds to the standard temperature at that altitude. This will be where the ISA reference line crosses the lines of pressure altitude. From each such point, use a ruler to mark the left edge of the chart at the same vertical position as the standard temperature and pressure. These should be labeled with the value of the pressure altitude curve, since at standard temperature the density altitude is the same as the pressure altitude.
Conclusion
A performance chart can be a nice addition to tabular data. It can give a better feeling for the tradeoffs that one might want to make when take off performance might be marginal. In our case, it provides values for additional weights and it estimates the effects of winds, where no wind correction is provided in the POH.
We believe that the same technique can be applied to the take off distance to reach 50 feet altitude, but we have not done the comparison with multiple POHs to see if the same analytic model is as good a fit to the actual data. For the Mooney illustrated, we find that the take off distance over a 50' obstacle is about 1.72 x the ground roll distance. The Mooney POH does not give any corrections for wind, so you need to check this our before applying it to your airplane.
References:
1. Maj. Russell E. Erb, "A Low Cost Method for Generating Takeoff Ground Roll Charts from Flight Test Data," Society of Flight Test Engineers (SFTE) 27^{th} Annual Symposium, Fort Worth, Texas, November, 1966, http://www.eaa1000.av.org/technicl/takeoff/topaper.htm
2. A Sample Atmosphere Table (US Units) http://www.pdas.com/e2.htm
3. Ilan Kroo, "Standard Atmosphere Computations," http://aero.stanford.edu/StdAtm.html