fluid_mechanics
222 Chapter 5 ■ Finite Control Volume Analysis Section (1) Fixed control volume 30° W 2 T shaft V 1 Section (2) ω D 2 = 2r 2 = 12 in. D 1 = 2r 1 = 10 in. h = 1 in. ω T shaft (a) F I G U R E E5.19 Fixed control volume W 2 W r2 V r2 U 2 V 2 30° V θ2 (b) Then m # The rotor exit blade speed, U 2 , is 16 in.211725 rpm212 rad/rev2 U 2 r 2 112 in./ft2160 s/min2 90.3 ft/s To determine the fluid tangential speed at the fan rotor exit, V 2 , we use Eq. 5.43 to get The vector addition of Eq. 4 is shown in the form of a “velocity triangle” in Fig. E5.19b. From Fig. E5.19b, we can see that To solve Eq. 5 for V 2 we need a value of W 2 , in addition to the value of U 2 already determined (Eq. 3). To get W 2 , we recognize that where V r2 is the radial component of either W 2 or V 2 . Also, using Eq. 5.6, we obtain or since 10.0766 lbm ft 3 21230 ft 3 min2 160 smin2 V 2 W 2 U 2 V 2 U 2 W 2 cos 30° W 2 sin 30° V r 2 m # A 2 V r 2 A 2 2 r 2 h 0.294 lbms (3) (4) (5) (6) (7) (8) where h is the blade height, Eqs. 7 and 8 combine to form m # 2r 2 hV r 2 Taking Eqs. 6 and 9 together we get m # W 2 r2pr 2 h sin 30° rQ r2pr 2 h sin 30° Q 2pr 2 h sin 30° Substituting known values into Eq. 10, we obtain By using this value of W 2 in Eq. 5 we get V 2 U 2 W 2 cos 30° 90.3 ft/s 129.3 ft/s210.8662 64.9 ft/s Equation 1 can now be used to obtain with BG units. With EE units W 2 1230 ft3 min2112 in.ft2112 in.ft2 160 smin22p16 in.211 in.2 sin 30° 29.3 fts 10.294 lbm/s2190.3 ft/s2164.9 ft/s2 W # shaft 332.174 1lbm ft21lbs 2 243550 1ft lb2/1hp s24 (9) (10) 10.00912 slug/s2190.3 ft/s2164.9 ft/s2 W # m # shaft U 2 V 2 31 1slug ft/s 2 2/lb43550 1ft lb2/1hp s24
5.3 First Law of Thermodynamics—The Energy Equation 223 In either case W # shaft 0.097 hp (Ans) COMMENT Note that the “” was used with the U 2 V 2 product because U 2 and V 2 are in the same direction. This result, 0.097 hp, is the power that needs to be delivered through the fan shaft for the given conditions. Ideally, all of this power would go into the flowing air. However, because of fluid friction, only some of this power will produce useful effects (e.g., movement and pressure rise) on the air. How much useful effect depends on the efficiency of the energy transfer between the fan blades and the fluid. 5.3 First Law of Thermodynamics—The Energy Equation The first law of thermodynamics is a statement of conservation of energy. 5.3.1 Derivation of the Energy Equation The first law of thermodynamics for a system is, in words time rate of net time rate of net time rate of increase of the energy addition by energy addition by total stored energy heat transfer into work transfer into of the system the system the system In symbolic form, this statement is D Dt er dVa a Q # in a Q # a a W # in a W # outb sys outbsys sys or D Dt sys er dV1Q # net in W # net2 sys in (5.55) Some of these variables deserve a brief explanation before proceeding further. The total stored energy per unit mass for each particle in the system, e, is related to the internal energy per unit mass, ǔ, the kinetic energy per unit mass, V 2 2, and the potential energy per unit mass, gz, by the equation e ǔ V 2 (5.56) 2 gz The net rate of heat transfer into the system is denoted with and the net rate of work transfer into the system is labeled W # Q # net in, net in. Heat transfer and work transfer are considered “” going into the system and “” coming out. Equation 5.55 is valid for inertial and noninertial reference systems. We proceed to develop the control volume statement of the first law of thermodynamics. For the control volume that is coincident with the system at an instant of time (5.57) Furthermore, for the system and the contents of the coincident control volume that is fixed and nondeforming, the Reynolds transport theorem 1Eq. 4.19 with the parameter b set equal to e2 allows us to conclude that or in words, 1Q # net in W # net2 sys 1Q # net in in D Dt sys er dV 0 0t cv er dV cs erV nˆ dA the time rate the time rate of increase of the total stored of increase of the total energy of the contents stored energy of the control volume of the system W # net2 coincident in control volume the net rate of flow of the total stored energy out of the control volume through the control surface (5.58)
- Page 196 and 197: 172 Chapter 4 ■ Fluid Kinematics
- Page 198 and 199: 174 Chapter 4 ■ Fluid Kinematics
- Page 200 and 201: 176 Chapter 4 ■ Fluid Kinematics
- Page 202 and 203: 178 Chapter 4 ■ Fluid Kinematics
- Page 204 and 205: 180 Chapter 4 ■ Fluid Kinematics
- Page 206 and 207: 182 Chapter 4 ■ Fluid Kinematics
- Page 208 and 209: 184 Chapter 4 ■ Fluid Kinematics
- Page 210 and 211: 186 Chapter 4 ■ Fluid Kinematics
- Page 212 and 213: 188 Chapter 5 ■ Finite Control Vo
- Page 214 and 215: 190 Chapter 5 ■ Finite Control Vo
- Page 216 and 217: 192 Chapter 5 ■ Finite Control Vo
- Page 218 and 219: 194 Chapter 5 ■ Finite Control Vo
- Page 220 and 221: 196 Chapter 5 ■ Finite Control Vo
- Page 222 and 223: 198 Chapter 5 ■ Finite Control Vo
- Page 224 and 225: 200 Chapter 5 ■ Finite Control Vo
- Page 226 and 227: 202 Chapter 5 ■ Finite Control Vo
- Page 228 and 229: 204 Chapter 5 ■ Finite Control Vo
- Page 230 and 231: 206 Chapter 5 ■ Finite Control Vo
- Page 232 and 233: 208 Chapter 5 ■ Finite Control Vo
- Page 234 and 235: 210 Chapter 5 ■ Finite Control Vo
- Page 236 and 237: 212 Chapter 5 ■ Finite Control Vo
- Page 238 and 239: 214 Chapter 5 ■ Finite Control Vo
- Page 240 and 241: 216 Chapter 5 ■ Finite Control Vo
- Page 242 and 243: 218 Chapter 5 ■ Finite Control Vo
- Page 244 and 245: 220 Chapter 5 ■ Finite Control Vo
- Page 248 and 249: 224 Chapter 5 ■ Finite Control Vo
- Page 250 and 251: 226 Chapter 5 ■ Finite Control Vo
- Page 252 and 253: 228 Chapter 5 ■ Finite Control Vo
- Page 254 and 255: 230 Chapter 5 ■ Finite Control Vo
- Page 256 and 257: 232 Chapter 5 ■ Finite Control Vo
- Page 258 and 259: 234 Chapter 5 ■ Finite Control Vo
- Page 260 and 261: 236 Chapter 5 ■ Finite Control Vo
- Page 262 and 263: 238 Chapter 5 ■ Finite Control Vo
- Page 264 and 265: 240 Chapter 5 ■ Finite Control Vo
- Page 266 and 267: 242 Chapter 5 ■ Finite Control Vo
- Page 268 and 269: 244 Chapter 5 ■ Finite Control Vo
- Page 270 and 271: 246 Chapter 5 ■ Finite Control Vo
- Page 272 and 273: 248 Chapter 5 ■ Finite Control Vo
- Page 274 and 275: 250 Chapter 5 ■ Finite Control Vo
- Page 276 and 277: 252 Chapter 5 ■ Finite Control Vo
- Page 278 and 279: 254 Chapter 5 ■ Finite Control Vo
- Page 280 and 281: 256 Chapter 5 ■ Finite Control Vo
- Page 282 and 283: 258 Chapter 5 ■ Finite Control Vo
- Page 284 and 285: 260 Chapter 5 ■ Finite Control Vo
- Page 286 and 287: 262 Chapter 5 ■ Finite Control Vo
- Page 288 and 289: 264 Chapter 6 ■ Differential Anal
- Page 290 and 291: ) 266 Chapter 6 ■ Differential An
- Page 292 and 293: 268 Chapter 6 ■ Differential Anal
- Page 294 and 295: 270 Chapter 6 ■ Differential Anal
5.3 First Law of Thermodynamics—The Energy Equation 223<br />
In either case<br />
W # shaft 0.097 hp<br />
(Ans)<br />
COMMENT Note that the “” was used with the U 2 V 2<br />
product because U 2 and V 2 are in the same direction. This result,<br />
0.097 hp, is the power that needs to be delivered through the fan<br />
shaft for the given conditions. Ideally, all of this power would go<br />
into the flowing air. However, because of <strong>fluid</strong> friction, only some<br />
of this power will produce useful effects (e.g., movement and pressure<br />
rise) on the air. How much useful effect depends on the efficiency<br />
of the energy transfer between the fan blades and the <strong>fluid</strong>.<br />
5.3 First Law of Thermodynamics—The Energy Equation<br />
The first law of<br />
thermodynamics is<br />
a statement of conservation<br />
of energy.<br />
5.3.1 Derivation of the Energy Equation<br />
The first law of thermodynamics for a system is, in words<br />
time rate of net time rate of net time rate of<br />
increase of the energy addition by energy addition by<br />
total stored energy heat transfer into work transfer into<br />
of the system the system the system<br />
In symbolic form, this statement is<br />
D<br />
Dt er dVa a Q # in a Q # a a W # in a W # outb<br />
sys<br />
outbsys<br />
sys<br />
or<br />
D<br />
Dt sys<br />
er dV1Q # net<br />
in<br />
W # net2 sys<br />
in<br />
(5.55)<br />
Some of these variables deserve a brief explanation before proceeding further. The total stored<br />
energy per unit mass for each particle in the system, e, is related to the internal energy per unit<br />
mass, ǔ, the kinetic energy per unit mass, V 2 2, and the potential energy per unit mass, gz, by the<br />
equation<br />
e ǔ V 2<br />
(5.56)<br />
2 gz<br />
The net rate of heat transfer into the system is denoted with and the net rate of work transfer<br />
into the system is labeled W # Q # net in,<br />
net in. Heat transfer and work transfer are considered “” going into<br />
the system and “” coming out.<br />
Equation 5.55 is valid for inertial and noninertial reference systems. We proceed to develop<br />
the control volume statement of the first law of thermodynamics. For the control volume that is<br />
coincident with the system at an instant of time<br />
(5.57)<br />
Furthermore, for the system and the contents of the coincident control volume that is fixed and<br />
nondeforming, the Reynolds transport theorem 1Eq. 4.19 with the parameter b set equal to e2 allows<br />
us to conclude that<br />
or in words,<br />
1Q # net<br />
in<br />
W # net2 sys 1Q # net<br />
in<br />
in<br />
D<br />
Dt sys<br />
er dV 0 0t cv<br />
er dV cs<br />
erV nˆ dA<br />
the time rate<br />
the time rate of increase<br />
of the total stored<br />
of increase<br />
of the total <br />
energy of the contents<br />
stored energy<br />
of the control volume<br />
of the system<br />
W # net2 coincident<br />
in control volume<br />
<br />
the net rate of flow<br />
of the total stored energy<br />
out of the control<br />
volume through the<br />
control surface<br />
(5.58)