Volume Verses Crank Angle

[MolaKule]

In this Question we will continue to use my 1999 S-10 engine (6-cylinder, 4.3L Vortec) as the question Example.

Data: The total Displacement of the engine is 262.3 cubic inches. Bore is 4.012 inches, Stroke is 3.48 inches, and the Connecting Rod Length is 5.7 inches from center of Wrist-Pin bore to center of Bid-End bore.

Just analyzing one cylinder, determine the volume of that one cylinder, in cubic inches, at a Crank Angle of 225 Degrees.


This question is open to Everyone. Just remember folks, your response goes into a queue so if you don't see it right away, it is awaiting posting and will be posted if it is relevant to the question.

 

[A_Harman]

Is your engine overbored? Stock bore on a GM 4.3 is 4.000".
How is the 225 degree crank angle defined? Is that 225 degrees after TDC Firing or something else?
Do you want the total volume of the cylinder, or just the volume remaining to be swept by the piston based on the crank angle?

 

[MolaKule]

Originally Posted by A_Harman
Is your engine overbored? Stock bore on a GM 4.3 is 4.000".


Dimensions to be used in this problem are taken from the General Motors 4.3 L Powertrain Service manual, TP-6103.

Originally Posted by A_Harman
How is the 225 degree crank angle defined? Is that 225 degrees after TDC Firing or something else?


Quote
At 0 degrees Top Dead Center and at 180 degrees Bottom Dead Center, and with the crankshaft rotating clockwise, the piston motion is virtually zero.


Clockwise rotation of crankshaft is viewed as Clockwise rotation starting from 0.xxx degrees Top Dead Center, to 180 Degrees Bottom Dead Center, then back to 359.xxx degrees Top Dead Center.


Originally Posted by A_Harman
Do you want the total volume of the cylinder, or just the volume remaining to be swept by the piston based on the crank angle?


Hint: The cylinder volume at any point in time is solely a function of the crank angle at any point in time.

 

[SubieRubyRoo]

32.7875 cubic inches. PFM if I get it right, cuz it was an educated guesstimate. Relegate me to the ranks of normal idjit if I guessed wrong.

 

[MolaKule]

Originally Posted by SubieRubyRoo
32.7875 cubic inches. PFM if I get it right, cuz it was an educated guesstimate. Relegate me to the ranks of normal idjit if I guessed wrong.

A good start but not quite, so use your equations and keep cranking.

 

[SubieRubyRoo]

31.66 cubic inches. Final answer. Trig is for smart people

 

[sloinker]

My quick 6th grade math yielded 32.985 but I just woke up and may need to re-think my assumptions.

 

[MolaKule]

A good start but not quite, so use your equations and keep cranking.

The Volume HAS to be a function of (a result of) the crank angle in degrees or radians.

HINT: The Volume of any one cylinder at a crank angle of 180 degrees or BDC is of course 1/6th the volume of 262.3 cubic inches.

At any other crank angle, the volume of any one cylinder is less than 1/6th the volume of 262.3 cubic inches.

 

[A_Harman]

Originally Posted by MolaKule
Originally Posted by A_Harman
Is your engine overbored? Stock bore on a GM 4.3 is 4.000".


Dimensions to be used in this problem are taken from the General Motors 4.3 L Powertrain Service manual, TP-6103.

Originally Posted by A_Harman
How is the 225 degree crank angle defined? Is that 225 degrees after TDC Firing or something else?


Quote
At 0 degrees Top Dead Center and at 180 degrees Bottom Dead Center, and with the crankshaft rotating clockwise, the piston motion is virtually zero.


Clockwise rotation of crankshaft is viewed as Clockwise rotation starting from 0.xxx degrees Top Dead Center, to 180 Degrees Bottom Dead Center, then back to 359.xxx degrees Top Dead Center.


Originally Posted by A_Harman
Do you want the total volume of the cylinder, or just the volume remaining to be swept by the piston based on the crank angle?




Hint: The cylinder volume at any point in time is solely a function of the crank angle at any point in time.


Yes it is, but there is the complicating factor of compression ratio, which requires you to take into account the clearance volume at the top of the stroke, not just the swept volume. Now we're getting into semantics. When you say "cylinder", do you mean a simple geometric cylinder, or an engine cylinder, which includes clearance volume?

But anyway, I have a spreadsheet that does crank and slider calculations based on rod length, bore, stroke, deck height, piston compression height, etc, and puts out piston position, bore volume above the piston, piston velocity, piston acceleration, and cylinder wall thrust load in G's.

Based on typical 350 Chevy engine dimensions, which I am pretty sure are the same as the 4.3 V6:
DECK HGT= 9.025 STROKE= 3.48
PIST HGT= 1.56 ROD LENGTH= 5.7
BORE DIA= 4.000 R/L= 0.305263158
DECK CLR= 0.025 L/R= 3.275862069

Here are the piston travel and bore volume values:
CRANK ANGLE PISTON TRAVEL VOLUME ABOVE
(DEG) (IN. BELOW DECK) IN^3
0 0.025 0.314159265
1 0.025345903 0.318506008
2 0.026383439 0.331544068
3 0.028112091 0.353266949
4 0.030530997 0.383663825
5 0.033638953 0.42271955
6 0.03743441 0.470414665
7 0.041915477 0.526725415
8 0.047079923 0.591623756
9 0.052925176 0.665077379
10 0.059448328 0.747049726
11 0.066646134 0.837500014
12 0.074515012 0.93638326
13 0.083051053 1.04365031
14 0.092250014 1.159247867
15 0.102107328 1.283118527
16 0.112618103 1.415200814
17 0.123777124 1.555429218
18 0.135578863 1.703734236
19 0.148017473 1.860042419
20 0.161086799 2.024276413
21 0.174780379 2.196355015
22 0.189091448 2.376193219
23 0.204012945 2.563702274
24 0.219537512 2.758789742
25 0.235657506 2.961359555
26 0.252364997 3.171312079
27 0.269651778 3.388544179
28 0.287509369 3.612949288
29 0.305929022 3.844417478
30 0.324901728 4.082835528
31 0.344418221 4.328087007
32 0.364468985 4.580052347
33 0.385044264 4.838608924
34 0.406134062 5.103631147
35 0.427728156 5.374990533
36 0.449816099 5.652555807
37 0.472387228 5.936192985
38 0.495430674 6.225765467
39 0.518935366 6.521134138
40 0.542890041 6.822157458
41 0.56728325 7.128691568
42 0.59210337 7.440590387
43 0.617338606 7.757705721
44 0.642977007 8.079887362
45 0.669006467 8.406983203
46 0.695414739 8.738839341
47 0.722189443 9.075300196
48 0.749318073 9.416208615
49 0.776788008 9.761405994
50 0.804586519 10.11073239
51 0.832700782 10.46402664
52 0.861117884 10.82112648
53 0.889824835 11.18186866
54 0.918808574 11.54608906
55 0.948055983 11.91362284
56 0.977553894 12.28430453
57 1.007289101 12.65796816
58 1.037248366 13.03444738
59 1.067418432 13.41357561
60 1.097786031 13.79518612
61 1.128337897 14.17911219
62 1.15906077 14.5651872
63 1.189941412 14.9532448
64 1.220966612 15.34311896
65 1.252123198 15.73464416
66 1.283398044 16.12765547
67 1.314778085 16.52198869
68 1.346250318 16.91748043
69 1.377801819 17.31396829
70 1.409419747 17.71129089
71 1.441091356 18.10928806
72 1.472804 18.50780091
73 1.504545147 18.90667193
74 1.536302382 19.30574511
75 1.568063417 19.70486605
76 1.599816101 20.10388205
77 1.631548425 20.50264218
78 1.66324853 20.90099745
79 1.694904714 21.29880079
80 1.72650544 21.69590723
81 1.758039343 22.09217394
82 1.789495235 22.48746033
83 1.820862108 22.88162809
84 1.852129148 23.2745413
85 1.883285731 23.66606647
86 1.914321436 24.05607264
87 1.945226044 24.4444314
88 1.975989545 24.83101695
89 2.006602141 25.21570618
90 2.037054252 25.59837869
91 2.067336515 25.97891684
92 2.097439793 26.35720578
93 2.127355172 26.73313352
94 2.157073965 27.10659089
95 2.186587716 27.47747162
96 2.2158882 27.84567236
97 2.244967423 28.21109266
98 2.273817626 28.573635
99 2.302431282 28.9332048
100 2.330801099 29.28971043
101 2.358920018 29.64306319
102 2.386781214 29.99317731
103 2.414378094 30.33996993
104 2.441704298 30.68336114
105 2.468753694 31.02327388
106 2.49552038 31.35963397
107 2.52199868 31.6923701
108 2.548183141 32.02141374
109 2.574068533 32.34669917
110 2.599649846 32.66816343
111 2.624922283 32.98574624
112 2.649881263 33.29939003
113 2.674522412 33.60903984
114 2.698841562 33.9146433
115 2.722834749 34.21615057
116 2.746498203 34.51351431
117 2.769828351 34.8066896
118 2.792821809 35.09563391
119 2.815475375 35.38030702
120 2.837786031 35.66067099
121 2.859750932 35.93669008
122 2.881367405 36.20833069
123 2.902632943 36.47556132
124 2.923545198 36.73835247
125 2.944101981 36.99667662
126 2.964301252 37.25050814
127 2.984141115 37.49982322
128 3.003619819 37.74459982
129 3.022735743 37.98481762
130 3.041487401 38.2204579
131 3.059873429 38.45150354
132 3.077892583 38.67793891
133 3.095543736 38.89974984
134 3.112825868 39.11692352
135 3.129738065 39.32944845
136 3.146279512 39.5373144
137 3.162449488 39.74051231
138 3.178247363 39.93903426
139 3.19367259 40.13287338
140 3.208724703 40.32202382
141 3.223403312 40.50648066
142 3.237708097 40.68623989
143 3.251638803 40.86129831
144 3.265195239 41.03165351
145 3.27837727 41.19730379
146 3.291184815 41.35824814
147 3.30361784 41.51448615
148 3.31567636 41.66601798
149 3.327360427 41.81284429
150 3.338670133 41.95496625
151 3.349605603 42.09238542
152 3.360166992 42.22510375
153 3.370354482 42.35312352
154 3.380168278 42.47644732
155 3.389608605 42.59507796
156 3.398675705 42.7090185
157 3.407369835 42.81827216
158 3.415691262 42.9228423
159 3.423640263 43.02273239
160 3.431217119 43.11794598
161 3.438422116 43.20848663
162 3.445255539 43.29435797
163 3.451717675 43.37556356
164 3.457808804 43.45210695
165 3.463529204 43.52399161
166 3.468879142 43.59122091
167 3.473858878 43.65379813
168 3.478468663 43.71172639
169 3.482708732 43.76500867
170 3.486579309 43.81364777
171 3.490080602 43.85764631
172 3.493212802 43.8970067
173 3.495976084 43.93173114
174 3.498370606 43.96182158
175 3.500396502 43.98727974
176 3.502053892 44.00810712
177 3.503342872 44.02430491
178 3.504263517 44.03587408
179 3.504815882 44.04281531
180 3.505 44.045129
181 3.504815882 44.04281531
182 3.504263517 44.03587408
183 3.503342872 44.02430491
184 3.502053892 44.00810712
185 3.500396502 43.98727974
186 3.498370606 43.96182158
187 3.495976084 43.93173114
188 3.493212802 43.8970067
189 3.490080602 43.85764631
190 3.486579309 43.81364777
191 3.482708732 43.76500867
192 3.478468663 43.71172639
193 3.473858878 43.65379813
194 3.468879142 43.59122091
195 3.463529204 43.52399161
196 3.457808804 43.45210695
197 3.451717675 43.37556356
198 3.445255539 43.29435797
199 3.438422116 43.20848663
200 3.431217119 43.11794598
201 3.423640263 43.02273239
202 3.415691262 42.9228423
203 3.407369835 42.81827216
204 3.398675705 42.7090185
205 3.389608605 42.59507796
206 3.380168278 42.47644732
207 3.370354482 42.35312352
208 3.360166992 42.22510375
209 3.349605603 42.09238542
210 3.338670133 41.95496625
211 3.327360427 41.81284429
212 3.31567636 41.66601798
213 3.30361784 41.51448615
214 3.291184815 41.35824814
215 3.27837727 41.19730379
216 3.265195239 41.03165351
217 3.251638803 40.86129831
218 3.237708097 40.68623989
219 3.223403312 40.50648066
220 3.208724703 40.32202382
221 3.19367259 40.13287338
222 3.178247363 39.93903426
223 3.162449488 39.74051231
224 3.146279512 39.5373144
225 3.129738065 39.32944845
226 3.112825868 39.11692352
227 3.095543736 38.89974984
228 3.077892583 38.67793891
229 3.059873429 38.45150354
230 3.041487401 38.2204579
231 3.022735743 37.98481762
232 3.003619819 37.74459982
233 2.984141115 37.49982322
234 2.964301252 37.25050814
235 2.944101981 36.99667662
236 2.923545198 36.73835247
237 2.902632943 36.47556132
238 2.881367405 36.20833069
239 2.859750932 35.93669008
240 2.837786031 35.66067099
241 2.815475375 35.38030702
242 2.792821809 35.09563391
243 2.769828351 34.8066896
244 2.746498203 34.51351431
245 2.722834749 34.21615057
246 2.698841562 33.9146433
247 2.674522412 33.60903984
248 2.649881263 33.29939003
249 2.624922283 32.98574624
250 2.599649846 32.66816343
251 2.574068533 32.34669917
252 2.548183141 32.02141374
253 2.52199868 31.6923701
254 2.49552038 31.35963397
255 2.468753694 31.02327388
256 2.441704298 30.68336114
257 2.414378094 30.33996993
258 2.386781214 29.99317731
259 2.358920018 29.64306319
260 2.330801099 29.28971043
261 2.302431282 28.9332048
262 2.273817626 28.573635
263 2.244967423 28.21109266
264 2.2158882 27.84567236
265 2.186587716 27.47747162
266 2.157073965 27.10659089
267 2.127355172 26.73313352
268 2.097439793 26.35720578
269 2.067336515 25.97891684
270 2.037054252 25.59837869
271 2.006602141 25.21570618
272 1.975989545 24.83101695
273 1.945226044 24.4444314
274 1.914321436 24.05607264
275 1.883285731 23.66606647
276 1.852129148 23.2745413
277 1.820862108 22.88162809
278 1.789495235 22.48746033
279 1.758039343 22.09217394
280 1.72650544 21.69590723
281 1.694904714 21.29880079
282 1.66324853 20.90099745
283 1.631548425 20.50264218
284 1.599816101 20.10388205
285 1.568063417 19.70486605
286 1.536302382 19.30574511
287 1.504545147 18.90667193
288 1.472804 18.50780091
289 1.441091356 18.10928806
290 1.409419747 17.71129089
291 1.377801819 17.31396829
292 1.346250318 16.91748043
293 1.314778085 16.52198869
294 1.283398044 16.12765547
295 1.252123198 15.73464416
296 1.220966612 15.34311896
297 1.189941412 14.9532448
298 1.15906077 14.5651872
299 1.128337897 14.17911219
300 1.097786031 13.79518612
301 1.067418432 13.41357561
302 1.037248366 13.03444738
303 1.007289101 12.65796816
304 0.977553894 12.28430453
305 0.948055983 11.91362284
306 0.918808574 11.54608906
307 0.889824835 11.18186866
308 0.861117884 10.82112648
309 0.832700782 10.46402664
310 0.804586519 10.11073239
311 0.776788008 9.761405994
312 0.749318073 9.416208615
313 0.722189443 9.075300196
314 0.695414739 8.738839341
315 0.669006467 8.406983203
316 0.642977007 8.079887362
317 0.617338606 7.757705721
318 0.59210337 7.440590387
319 0.56728325 7.128691568
320 0.542890041 6.822157458
321 0.518935366 6.521134138
322 0.495430674 6.225765467
323 0.472387228 5.936192985
324 0.449816099 5.652555807
325 0.427728156 5.374990533
326 0.406134062 5.103631147
327 0.385044264 4.838608924
328 0.364468985 4.580052347
329 0.344418221 4.328087007
330 0.324901728 4.082835528
331 0.305929022 3.844417478
332 0.287509369 3.612949288
333 0.269651778 3.388544179
334 0.252364997 3.171312079
335 0.235657506 2.961359555
336 0.219537512 2.758789742
337 0.204012945 2.563702274
338 0.189091448 2.376193219
339 0.174780379 2.196355015
340 0.161086799 2.024276413
341 0.148017473 1.860042419
342 0.135578863 1.703734236
343 0.123777124 1.555429218
344 0.112618103 1.415200814
345 0.102107328 1.283118527
346 0.092250014 1.159247867
347 0.083051053 1.04365031
348 0.074515012 0.93638326
349 0.066646134 0.837500014
350 0.059448328 0.747049726
351 0.052925176 0.665077379
352 0.047079923 0.591623756
353 0.041915477 0.526725415
354 0.03743441 0.470414665
355 0.033638953 0.42271955
356 0.030530997 0.383663825
357 0.028112091 0.353266949
358 0.026383439 0.331544068
359 0.025345903 0.318506008

 

[MolaKule]

Originally Posted by A_Harman


...Yes it is, but there is the complicating factor of compression ratio, which requires you to take into account the clearance volume at the top of the stroke, not just the swept volume. Now we're getting into semantics. When you say "cylinder", do you mean a simple geometric cylinder, or an engine cylinder, which includes clearance volume?...



As of now, we're basing the problem on a "simple geometric cylinder" with a total displacement volume of 262.3 inches^3 as given in the first post.

 

[A_Harman]

Originally Posted by MolaKule
Originally Posted by A_Harman


...Yes it is, but there is the complicating factor of compression ratio, which requires you to take into account the clearance volume at the top of the stroke, not just the swept volume. Now we're getting into semantics. When you say "cylinder", do you mean a simple geometric cylinder, or an engine cylinder, which includes clearance volume?...



As of now, we're basing the problem on a "simple geometric cylinder" with a total displacement volume of 262.3 inches^3 as given in the first post.


So it would be 39.015 cubic inches.

 

[Leo99]

I'll throw in my guess of 27.32 cu in.

 

[FowVay]

my calc. is 32.99 cu in of volume remaining in the cylinder with the piston on the up stroke (either compression or exhaust).

 

[Ded Mazai]

39,22

 

[sloinker]

So if the total volume of any one cylinder is 1/6th of the total volume and that volume is at maximum at BDC and diminishes as you head to top dead center the 225 degrees I believe is halfway between zero and maximum volume. I am probably missing something with the connecting rod angle or something silly that I am not taking in account. I will let brighter minds prevail on this I suppose. 21.858 is my final grasp at this straw....

 

[CR94]

I get a preliminary result of 40.73 in³, but didn't double-check my calculations yet. That assumes 4.000" bore, which is necessary to get the claimed displacement with 3.48" stroke. This does not include clearance volume.

 

[CR94]

Correction to my previous estimate, in which I found an error: 38.99 in³, about the same as A Harmon's table shows.

 

[MolaKule]

Thanks to all who participated in the Question of The Day and it was good to see the many number of responses.

Those who gave an answer of 39.xx in.^3 or cubic inches deserve a big

Notice - I didn't ask about the Clearance Volume Vc or give the compression Ratio as that is for another QOTD and to keep this problem constrained within terms of the simpler Slider-Crank mechanical model.

We know the maximum volume of each cylinder is 262.3 in.^3/6 = 43.72 in.^3 or cubic inches and this has to occur at 180 Degrees of crank angle or at Bottom Dead Center (BDC). At any other Crank Angle the Volume has to be less than 43.72 in.^3.

We also know that maximum piston travel will be from Top Dead Center (TDC) to BDC or from BDC to TDC.

What we need to know is: What is the piston position (x) for any given Crank Angle or x(Theta)?

Once we know the piston position x we can multiply that by the Bore area BA, since Area times Length = Volume.

Data Given ("*" means Multiplication):

B = 4.012; Total Diameter of Bore in inches
S = 3.48; Stroke in inches
VD = 262.3; Total Cubic inches of Displacement
N = 6; Number of cylinders
L = 5.7; Length of connecting in inches from bore to bore
225 Degrees Crank Rotation.

Determine Total Displacement per cylinder
Vp = VD/N = 43.72 in.^3;

Determine Bore Area;

BA = pi/4*(B^2) = 12.6419 in.^2 or square Inches;

Determine Crank Radius from Stroke:
a = S/2 = 1.74 inches; Crank Radius;

Convert Degrees to Radians:
ThetaDeg = 225;
ThetaRad = ThetaDeg/57.3 = 3.9267 Radians;

Determine Position of Piston verses Crank Angle from Geometry of Problem :
x = a + L - [SQRT(L^2 - a^2*sin(ThetaRad)^2] - a*cos(ThetaRad) = 3.11 inches;

Since V is a function of x as in V(x), determine Volume for x position at 225 degrees of Crank Angle.
V = x*BA = 39.2533 or 39.3 cubic inches or in.^3 is the cylinder Volume at 225 degrees of Crank Angle which means the piston is on its up stroke travelling towards TDC.

The volume would be the same at 135 degrees of crank angle but the piston would be on its down stroke towards BDC.