EBK THERMODYNAMICS: AN ENGINEERING APPR
EBK THERMODYNAMICS: AN ENGINEERING APPR
8th Edition
ISBN: 8220100257056
Author: CENGEL
Publisher: YUZU
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Chapter 17.7, Problem 108P
To determine

The rate of heat transfer in the duct.

The pressure drop in the duct.

Expert Solution & Answer
Check Mark

Answer to Problem 108P

The rate of heat transfer in the duct is 834Btu/s.

The pressure drop in the duct is 14.6psia.

Explanation of Solution

Determine the inlet density of air.

ρ1=P1RT1 (I)

Here, the inlet pressure of air is P1, the universal gas constant of air is R, and the inlet temperature of air is T1.

Determine the cross sectional area of duct at inlet.

Ac1=πD24 (II)

Here, the diameter of the duct is D.

Determine the inlet velocity of air.

V1=m˙airρ1Ac1 (III)

Here, the mass flow rate of the air is m˙air and the density of the air is ρ1.

Determine the inlet stagnation temperature of air.

T01=T1+V122cp (IV)

Here, the inlet static temperature of ideal gas is T3, the specific heat of pressure for ideal gas is cp, and the inlet velocity of the ideal gas flow is V1.

Determine the relation of ideal gas speed of sound at the inlet.

c1=kRT1 (V)

Here, the specific heat ratio of air is k, the gas constant of the air is R, and the inlet temperature of the air is T1.

Determine the speed of sound at the inlet.

Ma1=V1c1 (VI)

The inlet velocity of the air flow in the device is V1 and the inlet Mach number is Ma1.

Determine the static temperature in the duct.

T2T1=T2/TT1/T (VII)

Here, the ratio of Rayleigh flow for inlet temperature is T1/T and the ratio of Rayleigh flow for outlet temperature is T2/T.

Determine the static pressure in the duct.

P2P1=P2/PP1/P (VIII)

Here, the ratio of Rayleigh flow for inlet pressure is P1/P and the ratio of Rayleigh flow for outlet pressure is P2/P.

Determine the stagnation temperature in the duct.

T02T01=T02/TT01/T (IX)

Here, the ratio of Rayleigh flow for exit stagnation temperature is T01/T and the ratio of Rayleigh flow for outlet stagnation temperature is T02/T.

Determine the rate of heat transfer of the duct.

Q˙=m˙aircp(T02T01) (X)

Determine the pressure drop of the duct.

ΔP=P1P2 (XI)

Conclusion:

From the Table A-2E, “Ideal-gas specific heats of various common gases” to obtain value of universal gas constant, specific heat of pressure, and the specific heat ratio of air at 80°F temperature as 0.06855Btu/lbmR, 0.2400Btu/lbmR and 1.4.

Substitute 30psia for P1, 0.06855Btu/lbmR for R, and 800R for T1 in Equation (I).

ρ1=30psia(0.06855Btu/lbmR)(800R)=30psia(0.06855Btu/lbmR×(5.40395psiaft31Btu))(800R)=30psia(0.3704psiaft3/lbmR)(800R)=0.1012lbm/ft3

Substitute 4in. for D in Equation (II).

Ac1=π(4in.)24=π(4in.×(1ft12in.))24=0.0872ft2

Substitute 5lbm/s for m˙air, 0.1012lbm/ft3 for ρ1, and 0.0872ft2 for Ac1 in Equation (III).

V1=(5lbm/s)(0.1012lbm/ft3)(0.0872ft2)=(9lbm/s)(0.0.19871lbm/ft)=566.6ft/s

Substitute 800R for T1, 566.6ft/s for V1, and 0.2400Btu/lbmR for cp in Equation (IV).

T01=(800R)+(566.6ft/s)22×(0.2400Btu/lbmR)=(800R)+(321030.3ft2/s2)(0.48Btu/lbmR)=(800R)+(321030.3ft2/s2)×(1Btu/lbm25,037ft2/s2)(0.48Btu/lbmR)=(800R)+(26.71R)

     =826.7R

Substitute 1.4 for k, 0.06855Btu/lbmR for R, and 800R for T1 in Equation (V).

c1=(1.4)(0.06855Btu/lbmR)×(800R)=(1.4)(0.06855Btu/lbmR)×(25,037ft2/s21Btu/lbm)×(800R)=1922241ft2/s2=1386ft/s

Substitute 1386ft/s for c1 and 566.6ft/s for V1 in Equation (I).

Ma1=(566.6ft/s)(1386ft/s)=0.4082

Refer to Table A-34, “Rayleigh flow function for an ideal gas with k=1.4”, to obtain the value ratio of static temperature, pressure, and stagnation temperature at 0.4082 inlet Mach number using interpolation method of two variables.

Write the formula of interpolation method of two variables.

y2=(x2x1)(y3y1)(x3x1)+y1 (XII)

Here, the variables denote by x and y is ratio of stagnation temperature and Mach number.

Show the Mach number at 0.3 and 0.4 as in Table (1).

S. No

Mach number

(x)

ratio of stagnation temperature

(y)

10.40.529
20.4082y2=?
30.50.6914

Calculate ratio of static temperature, pressure, and stagnation temperature at 0.3268 inlet Mach number using interpolation method.

Substitute 0.4 for x1, 0.4082 for x2, 0.5 for x3, 0.529 for y1, and 0.6914 for y3 in Equation (XII).

y2=(0.40820.4)(0.69140.5290)(0.50.4)+0.5290=0.5434

From above calculation the ratio of stagnation temperature at 0.4082 is inlet Mach number 0.5434.

Repeat the Equation (XII), to obtain the value of inlet ratio of static temperature and pressure at 0.4082 inlet Mach number as:

T1/T=0.6310P1/P=1.946

From the Table A-34, “Rayleigh flow function for an ideal gas with k=1.4”, to obtain the value of the outlet ratio of temperature, pressure, and velocity at 1 outlet Mach number as:

T2/T=1P2/P=1V2/V=1

Substitute 800R for T1, 1 for T2/T, and 0.6310 for T1/T in Equation (VII).

T2(800R)=10.6310T2=(800R)0.6310T2=1268R

Substitute 30 psia for P1, 1 for P2/P, and 1.946 for P1/P in Equation (VIII).

P2(30psia)=11.946P2=(30psia)1.946P2=15.421psiaP215.4psia

Substitute 826.7R for T02, 1 for T02/T, and 0.5434 for T01/T in Equation (IX).

T01(826.7R)=10.5434T01=(826.7R)0.5434T01=1521.34RT011521R

Substitute 5lbm/s for m˙air, 0.2400Btu/lbmR for cp, 1521R for T02 and 826.7R for T01 in Equation (X).

Q˙=(5lbm/s)(0.2400Btu/lbmR)(1521826.7)R=(5lbm/s)(0.2400Btu/lbmR)(694.3R)=834Btu/s

Thus, the rate of heat transfer in the duct is 834Btu/s.

Substitute 30psia for P1 and 15.4psia for P2 in Equation (XI).

ΔP=(3015.4)psia=14.6psia

Thus, the pressure drop in the duct is 14.6psia.

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Chapter 17 Solutions

EBK THERMODYNAMICS: AN ENGINEERING APPR

Ch. 17.7 - Prob. 11PCh. 17.7 - Prob. 12PCh. 17.7 - Prob. 13PCh. 17.7 - Prob. 14PCh. 17.7 - Prob. 15PCh. 17.7 - Prob. 16PCh. 17.7 - Prob. 17PCh. 17.7 - Prob. 18PCh. 17.7 - Prob. 19PCh. 17.7 - Prob. 20PCh. 17.7 - Prob. 21PCh. 17.7 - Prob. 22PCh. 17.7 - Prob. 23PCh. 17.7 - Prob. 24PCh. 17.7 - Prob. 25PCh. 17.7 - Prob. 26PCh. 17.7 - Prob. 27PCh. 17.7 - The isentropic process for an ideal gas is...Ch. 17.7 - Is it possible to accelerate a gas to a supersonic...Ch. 17.7 - Prob. 30PCh. 17.7 - Prob. 31PCh. 17.7 - A gas initially at a supersonic velocity enters an...Ch. 17.7 - Prob. 33PCh. 17.7 - Prob. 34PCh. 17.7 - Prob. 35PCh. 17.7 - Prob. 36PCh. 17.7 - Prob. 37PCh. 17.7 - Prob. 38PCh. 17.7 - Air at 25 psia, 320F, and Mach number Ma = 0.7...Ch. 17.7 - Prob. 40PCh. 17.7 - Prob. 41PCh. 17.7 - Prob. 42PCh. 17.7 - Prob. 43PCh. 17.7 - Prob. 44PCh. 17.7 - Prob. 45PCh. 17.7 - Prob. 46PCh. 17.7 - Is it possible to accelerate a fluid to supersonic...Ch. 17.7 - Prob. 48PCh. 17.7 - Prob. 49PCh. 17.7 - Consider subsonic flow in a converging nozzle with...Ch. 17.7 - Consider a converging nozzle and a...Ch. 17.7 - Prob. 52PCh. 17.7 - Prob. 53PCh. 17.7 - Prob. 54PCh. 17.7 - Prob. 55PCh. 17.7 - Prob. 56PCh. 17.7 - Prob. 57PCh. 17.7 - Prob. 58PCh. 17.7 - Prob. 59PCh. 17.7 - Prob. 62PCh. 17.7 - Prob. 63PCh. 17.7 - Prob. 64PCh. 17.7 - Prob. 65PCh. 17.7 - Air enters a nozzle at 0.5 MPa, 420 K, and a...Ch. 17.7 - Prob. 67PCh. 17.7 - Are the isentropic relations of ideal gases...Ch. 17.7 - What do the states on the Fanno line and the...Ch. 17.7 - It is claimed that an oblique shock can be...Ch. 17.7 - Prob. 73PCh. 17.7 - Prob. 74PCh. 17.7 - For an oblique shock to occur, does the upstream...Ch. 17.7 - Prob. 76PCh. 17.7 - Prob. 77PCh. 17.7 - Prob. 78PCh. 17.7 - Prob. 79PCh. 17.7 - Prob. 80PCh. 17.7 - Prob. 81PCh. 17.7 - Prob. 82PCh. 17.7 - Prob. 83PCh. 17.7 - Prob. 84PCh. 17.7 - Air flowing steadily in a nozzle experiences a...Ch. 17.7 - Air enters a convergingdiverging nozzle of a...Ch. 17.7 - Prob. 89PCh. 17.7 - Prob. 90PCh. 17.7 - Consider the supersonic flow of air at upstream...Ch. 17.7 - Prob. 92PCh. 17.7 - Prob. 93PCh. 17.7 - Prob. 96PCh. 17.7 - Prob. 97PCh. 17.7 - Prob. 98PCh. 17.7 - Prob. 99PCh. 17.7 - What is the effect of heat gain and heat loss on...Ch. 17.7 - Consider subsonic Rayleigh flow of air with a Mach...Ch. 17.7 - What is the characteristic aspect of Rayleigh...Ch. 17.7 - Prob. 103PCh. 17.7 - Prob. 104PCh. 17.7 - Air is heated as it flows subsonically through a...Ch. 17.7 - Prob. 106PCh. 17.7 - Prob. 107PCh. 17.7 - Prob. 108PCh. 17.7 - Air is heated as it flows through a 6 in 6 in...Ch. 17.7 - Air enters a rectangular duct at T1 = 300 K, P1 =...Ch. 17.7 - Prob. 112PCh. 17.7 - Prob. 113PCh. 17.7 - Prob. 114PCh. 17.7 - What is supersaturation? Under what conditions...Ch. 17.7 - Prob. 116PCh. 17.7 - Prob. 117PCh. 17.7 - Steam enters a convergingdiverging nozzle at 1 MPa...Ch. 17.7 - Prob. 119PCh. 17.7 - Prob. 120RPCh. 17.7 - Prob. 121RPCh. 17.7 - Prob. 122RPCh. 17.7 - Prob. 124RPCh. 17.7 - Prob. 125RPCh. 17.7 - Using Eqs. 174, 1713, and 1714, verify that for...Ch. 17.7 - Prob. 127RPCh. 17.7 - Prob. 128RPCh. 17.7 - 17–129 Helium enters a nozzle at 0.6 MPa, 560...Ch. 17.7 - Prob. 130RPCh. 17.7 - Prob. 132RPCh. 17.7 - Prob. 133RPCh. 17.7 - Nitrogen enters a convergingdiverging nozzle at...Ch. 17.7 - An aircraft flies with a Mach number Ma1 = 0.9 at...Ch. 17.7 - Prob. 136RPCh. 17.7 - Helium expands in a nozzle from 220 psia, 740 R,...Ch. 17.7 - 17–140 Helium expands in a nozzle from 1 MPa,...Ch. 17.7 - Air is heated as it flows subsonically through a...Ch. 17.7 - Air is heated as it flows subsonically through a...Ch. 17.7 - Prob. 145RPCh. 17.7 - Prob. 146RPCh. 17.7 - Air is cooled as it flows through a 30-cm-diameter...Ch. 17.7 - Saturated steam enters a convergingdiverging...Ch. 17.7 - Prob. 151RPCh. 17.7 - Prob. 154FEPCh. 17.7 - Prob. 155FEPCh. 17.7 - Prob. 156FEPCh. 17.7 - Prob. 157FEPCh. 17.7 - Prob. 158FEPCh. 17.7 - Prob. 159FEPCh. 17.7 - Prob. 160FEPCh. 17.7 - Prob. 161FEPCh. 17.7 - Consider gas flow through a convergingdiverging...Ch. 17.7 - Combustion gases with k = 1.33 enter a converging...
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