{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Climbing Flight\n", "\n", "In climbing flight, $T>D$, so the aircraft cannot maintain equilibrium in straight and level flight.\n", "\n", "The aircraft *ascends* with climb angle $\\bar{\\gamma}$, with the horizontal component of the aircraft weight opposing the thrust.\n", "\n", "\n", "{figure} ../Images/ClimbForces.png\n", "---\n", "height: 300px\n", "name: ClimbForces\n", "---\n", "Forces on aircraft in a climb\n", "\n", "\n", "{admonition} What's actually happening in a climb?\n", ":class: dropdown\n", "\n", "Using the same reasoning as before, you can see that the aircraft is converting vertical kinetic energy into gravitational potential energy.\n", "\n", "\n", "\n", "## Climb Angle\n", "\n", "As for the glide angle, the climb angle can be determined by resolving forces perpendicular to the flight path\n", "\n", "$$L=W\\cos\\bar{\\gamma}$$\n", "\n", "and parallel to the flight path\n", "\n", "$$T-D=W\\sin\\bar{\\gamma}$$\n", "\n", "and from trigonometry, the climb angle is simply\n", "\n", "$$\\sin\\bar{\\gamma}=\\frac{T-D}{W}$$\n", "\n", "the rate of climb is $V_{climb}$ and is\n", "\n", "$$V_{climb}=V\\sin\\bar{\\gamma}=\\frac{V\\left(T-D\\right)}{W}$$\n", "\n", "which gives the rate of increase of GPE\n", "\n", "$$\\underbrace{W\\cdot V_{climb}}_{\\substack{\\text{Rate of increase}\\\\\\text{of potential energy}}} = \\underbrace{T\\,V}_{\\substack{\\text{Thrust}\\\\\\text{Power}}} - \\underbrace{D\\,V}_{\\substack{\\text{Drag}\\\\\\text{Power}}}$$\n", "\n", "## Climb Performance\n", "\n", "The **maximum climb angle** requires the maximum **excess thrust**\n", "\n", "The **maximum rate of climb** requires the maximum **excess power** \n", "\n", "This should feel intuitively correct to you, based upon what we know about glide angle/rate and $D_{min}/P_{min}$ and - obviously - these do not occur at the same speed. These depend on the powerplant type, and individual engine characteristics.\n", "\n", "For propeller and turbojet engines, there are some simplifications that can be made about the propulsor that allows easy determination of *one* of the parameters - *excess thrust* for a turbojet, and *excess power* for a turboprop. \n", "\n", "### Powerplant assumptions\n", "\n", "For **turbojet aircraft** and **low bypass ratio turbofan aircraft** is is assumed that **thrust remains constant with speed**, and accordingly power *increases* with speed.\n", "\n", "For **turboprop aircraft** and **high bypass ratio turbofan aircraft** is is assumed that **power remains constant with speed**, and accordingly thrust *falls* with speed." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": false, "tags": [ "hide-input", "remove-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "150.0" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "W" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "55" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "S" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "0.008" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "CD0" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "0.05500000000000001" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "K" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "1.55" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "Clmax" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "17.0" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "TA" } }, "output_type": "display_data" }, { "data": { "application/papermill.record/text/plain": "1.7" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "PA_constant" } }, "output_type": "display_data" } ], "source": [ "## Initialise aircraft parameters\n", "from myst_nb import glue\n", "from random import seed\n", "from random import randint\n", "import numpy as np\n", "\n", "# Make aircraft parameters\n", "W = np.arange(120e3, 190e3, 10e3)\n", "S = np.arange(45, 85, 5)\n", "CD0 = np.arange(0.008, 0.022, 0.002)\n", "K = np.arange(0.025, 0.065, 0.005)\n", "Clmax = np.arange(1.45, 1.95, 0.05)\n", "TA = np.arange(15e3, 25e3, 2e3)\n", "\n", "\n", "\n", "def getrandomvalue(vectin):\n", " val_out = vectin[randint(0, len(vectin)-1)]\n", " return val_out\n", "\n", "\n", "# Get a random parameter\n", "W = getrandomvalue(W)\n", "S = getrandomvalue(S)\n", "CD0 = getrandomvalue(CD0)\n", "K = getrandomvalue(K)\n", "Clmax = getrandomvalue(Clmax)\n", "TA = getrandomvalue(TA)\n", "PA_constant = TA*100\n", "\n", "glue(\"W\", W/1e3, display=False)\n", "glue(\"S\", S, display=False)\n", "glue(\"CD0\", CD0, display=False)\n", "glue(\"K\", K, display=False)\n", "glue(\"Clmax\", Clmax, display=False)\n", "glue(\"TA\", TA/1e3, display=False)\n", "glue(\"PA_constant\", PA_constant/1e6, display=False)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Climb Curves - Turbojet\n", "\n", "For a turbojet aircraft with a drag equation described by:\n", "\n", "$$C_D = C_{D0} + K\\cdot C_L^2$$\n", "\n", "with $C_{D0}$={glue:text}CD0:1.3f, and $K$={glue:text}K:1.3f, a wing area of {glue:text}S:1.0f$\\text{m}^2$, a weight of {glue:text}W:1.3fkN, and a maximum lift coefficient of {glue:text}Clmax:1.3f, capable of producing a thrust of {glue:text}W:1.3fkN at a given altitude, the climb rate and angles can be taken from the *difference between* the thrust available/required curves.\n", "\n", "These plots are sensible to produce in EAS, for hopefully obvious reasons (if you're unsure why, ask on Slack).\n", "\n", "{admonition} Try and reproduce the plot below (ON YOUR OWN)\n", ":class: dropdown\n", "\n", "The source for this plot isn't included on the website (but it is included on GitHub). You should try and reproduce the plot using *some* computational method - Excel, MATLAB, Python using matplotlib, plotly, or whatever.\n", "\n", "If you just download the source, you might trick yourself that you know what's going on - but you'll not actually learn anything.\n", "\n", "It would be better to do some work to learn some coding and equation manipulation *now*, than lose your first graduate job when they find out that you bluffed your way through your undergraduate degree. That is, I don't care whether you've done your own homework when it comes in - but I promise that cheating on any of my homeworks, graded or not, will not be worth the trade-off you'll make for time now vs. being a shit engineer as a graduate.\n", "\n", "Rant aside, I've also chosen not to make the source easily-available for some of these plots because I chose to use *plotly* to enable you guys to pan/zoom/interact with them online. I am far less confident with plotly than I am with matplotlib, so my code probably isn't anything to learn from technique-wise.\n", "\n", "\n", "You can hover over the plots and check the values to see if you get the same answers as the ones I've produced.\n", "\n", "Do your best to reproduce these plots as they may help you with a future homework - discuss on Slack and help each other if in doubt how to complete." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "remove-input" ] }, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "name": "$T_R$", "type": "scatter", "x": [ 0.1, 0.4002002002002002, 0.7004004004004003, 1.0006006006006005, 1.3008008008008007, 1.601001001001001, 1.901201201201201, 2.201401401401401, 2.5016016016016014, 2.8018018018018016, 3.102002002002002, 3.402202202202202, 3.702402402402402, 4.002602602602602, 4.302802802802802, 4.6030030030030025, 4.903203203203202, 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import plotly.graph_objects as go\n", "\n", "\n", "rho_sl = 1.2255\n", "\n", "# Get drag curve parameters\n", "A = CD0 * 0.5 * rho_sl * S\n", "B = K * W**2 / 0.5 / rho_sl / S\n", "\n", "# Get minimum drag and power speeds\n", "Vmd = (B/A)**.25\n", "Vmp = (B/3/A)**.25\n", "\n", "md = A * Vmd**2 + B * Vmd**-2\n", "dmp = A * Vmp**2 + B * Vmp**-2 \n", "\n", "# Make a velocity vector\n", "VE = np.linspace(0.1, 300, 1000)\n", "\n", "D = A*VE**2 + B*VE**-2\n", "P = A*VE**3 + B*VE**-1\n", "\n", "# Power available\n", "PA = TA * VE\n", "\n", "# Thrust and power deltas\n", "dT = TA - D\n", "dP = PA - D*VE\n", "\n", "# Get the parameters to label - first best angle\n", "v_bestclimb_angle = Vmd\n", "d_bestclimb_angle = md\n", "dd_bestclimb_angle = max(dT)\n", "dp_bestclimb_angle = PA[VE==Vmd] - md*Vmd\n", "p_bestclimb_angle = md*Vmd\n", "\n", "# Then best climb rate - this has to be determined from the vectors\n", "v_bestclimb_rate = VE[dP == max(dP)][0]\n", "d_bestclimb_rate = D[dP == max(dP)][0]\n", "p_bestclimb_rate = P[dP == max(dP)][0]\n", "dp_bestclimb_rate = max(dP)\n", "dd_bestclimb_rate = TA - max(dP)/v_bestclimb_rate\n", "\n", "\n", "########################################################\n", "\n", "# Make a plot - first do thrust\n", "fig = go.Figure()\n", "fig.add_trace(go.Scatter(x=VE, y=D, name=\"$T_R$\"))\n", "fig.add_trace(go.Scatter(x=[min(VE), max(VE)], y=[TA, TA], mode='lines', name=\"$T_A$\"))\n", "fig.add_trace(go.Scatter(x=VE, y=dT, name=\"$T_A - T_R$\"))\n", "\n", "# Annotations - best angle\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_angle], y=[d_bestclimb_angle],\\\n", " mode=\"markers+text\", text=\"$\\\\bar{\\\\gamma}_{max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\", marker=dict(\n", " color='LightSkyBlue'\n", " )))\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_angle], y=[dd_bestclimb_angle], mode=\"markers\", text=\"$\\\\theta_{max}$\", textposition=\"top center\", name=\"Annotation\", marker=dict(\n", " color='LightSkyBlue'\n", " )))\n", "\n", "# Annotations - best rate\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_rate], y=[d_bestclimb_rate],\\\n", " mode=\"markers+text\", text=\"$V_{climb,max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\",\\\n", " marker=dict(color='DarkBlue')))\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_rate], y=[dd_bestclimb_rate],\\\n", " mode=\"markers\", text=\"$V_{climb,max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\",\\\n", " marker=dict(color='DarkBlue')))\n", "\n", "# Dress the plot\n", "fig.update_layout(\n", " title=\"Thrust Available/Required for Turbojet\",\n", " xaxis_title=\"Airspeed m/s EAS\",\n", " yaxis_title=\"Thrust Available/Required \"\n", ")\n", "\n", "\n", "fig.update_yaxes(range=[0, 3 * TA])\n", "\n", "\n", "for trace in fig['data']: \n", " if(trace['name'] == \"Annotation\"): trace['showlegend'] = False\n", "\n", "fig.show()\n", "\n", "########################################################\n", "# Make a plot - second do power\n", "fig = go.Figure()\n", "fig.add_trace(go.Scatter(x=VE, y=P, name=\"$P_R$\"))\n", "fig.add_trace(go.Scatter(x=VE, y=PA, mode='lines', name=\"$P_A$\"))\n", "fig.add_trace(go.Scatter(x=VE, y=dP, name=\"$P_A - P_R$\"))\n", "\n", "\n", "\n", "# Annotations - best angle\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_angle], y=[p_bestclimb_angle],\\\n", " mode=\"markers+text\", text=\"$\\\\bar{\\\\gamma}_{max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\", marker=dict(\n", " color='LightSkyBlue'\n", " )))\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_angle], y=[dp_bestclimb_angle], mode=\"markers\", text=\"$\\\\theta_{max}$\", textposition=\"top center\", name=\"Annotation\", marker=dict(\n", " color='LightSkyBlue'\n", " )))\n", "\n", "# Annotations - best rate\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_rate], y=[p_bestclimb_rate],\\\n", " mode=\"markers+text\", text=\"$V_{climb,max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\",\\\n", " marker=dict(color='DarkBlue')))\n", "fig.add_trace(go.Scatter(x=[v_bestclimb_rate], y=[dp_bestclimb_rate],\\\n", " mode=\"markers\", text=\"$V_{climb,max}$\",\\\n", " textposition=\"top center\", name=\"Annotation\",\\\n", " marker=dict(color='DarkBlue')))\n", "\n", "fig.update_yaxes(range=[0, 1 * max(PA)])\n", "\n", "fig.update_layout(\n", " title=\"Power Available/Required for Turbojet\",\n", " xaxis_title=\"Airspeed m/s EAS\",\n", " yaxis_title=\"Power Available/Required \"\n", ")\n", "for trace in fig['data']: \n", " if(trace['name'] == \"Annotation\"): trace['showlegend'] = False\n", "\n", "fig.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Climb Curves - Turboprop\n", "\n", "For the same aircraft, with a turboprop capable of producing a constant *power* of {glue:text}PA_constantMW, the thrust and power curves can be shown similarly: \n" ] }, { "cell_type": 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