Questions and Answers of Advanced Thermodynamics For Engineers

An inertial observer \(\mathcal{O}\) is midway between two sources of light at rest in her frame with the sources \(2 \mathrm{~km}\) apart. Each source emits a flash of light that reaches
A particle with integer or half-integer spin \(s\) has \(2 s+1\) values of spin with respect to any arbitrary spin quantization axis \(\mathbf{n}\) in the particle's rest frame. If parity is a good
The Poincaré group is a subgroup of a larger Lie group, the conformal group \(C(1,3)\). This fifteen-parameter group consists of the 10-parameter subgroup of Poincaré transformations, the
A mass of \(10 \mathrm{~kg}\) of water at \(0^{\circ} \mathrm{C}\) is brought into contact with a large heat reservoir at \(100{ }^{\circ} \mathrm{C}\).a. When the water has reached \(100^{\circ}
A system contains a fluid at a temperature of \(70^{\circ} \mathrm{C}\) and 1 bar. It undergoes a reversible process during which the temperature of the system remains constant. Given that the heat
Calculate the gain in entropy when \(1 \mathrm{~kg}\) of water at \(30^{\circ} \mathrm{C}\) is converted into steam at \(150{ }^{\circ} \mathrm{C}\) and then superheated to \(300^{\circ}
A mass of a liquid, \(m\), at temperature, \(T_{1}\), is mixed with an equal mass of the same liquid at temperature, \(T_{2}\). The system is thermally insulated. Show that the change of entropy of
A substance has the following physical properties at a certain pressure:Saturation temperature, \(t_{s}=76^{\circ} \mathrm{C} ; h_{\mathrm{fg}}=61.1 \mathrm{~kJ} / \mathrm{kg}\)\(c_{p(\text { liquid
Determine the criteria for equilibrium for a thermally isolated system at (a) constant volume; (b) at constant pressure. Assume that the system isa. constant, and invariant, in composition;b.
Determine the criteria for isothermal equilibrium of a system at (a) constant volume, and (b) constant pressure. Assume that the system is:a. constant, and invariant, in composition;b. variable in
A system at constant pressure consists of \(10 \mathrm{~kg}\) of air at a temperature of \(1000 \mathrm{~K}\). This is connected to a large reservoir which is maintained at a temperature of \(300
A thermally isolated system at constant pressure consists of \(10 \mathrm{~kg}\) of air at a temperature of \(1000 \mathrm{~K}\) and \(10 \mathrm{~kg}\) of water at \(300 \mathrm{~K}\), connected
A thermally isolated system at constant pressure consists of \(10 \mathrm{~kg}\) of air at a temperature of \(1000 \mathrm{~K}\) and \(10 \mathrm{~kg}\) of water at \(300 \mathrm{~K}\), connected
Show that if a liquid is in equilibrium with its own vapour and an inert gas in a closed vessel, then\[\frac{\mathrm{d} p_{v}}{\mathrm{~d} p}=\frac{ho_{v}}{ho_{l}}\]where \(p_{v}\) is the partial
An incompressible liquid of specific volume \(v_{l}\), is in equilibrium with its own vapour and an inert gas in a closed vessel. The vapour obeys the law\[p(v-b)=\Re T\]Show that\[\ln
a. Describe the meaning of the term thermodynamic equilibrium. Explain how entropy can be used as a measure of equilibrium and also how other properties can be developed which can be used to assess
Show that when different phases are in equilibrium the specific Gibbs energy of each phase is equal.Using the following data, show the pressure at which graphite and diamond are in equilibrium at a
Van der Waals equation for water is given by\[p=\frac{0.004619 T}{v-0.0016891}-\frac{0.017034}{v^{2}}\]where \(p=\) pressure (bar), \(v=\) specific volume \(\left(\mathrm{m}^{3} /
A steam turbine operates on a Carnot cycle, with a maximum pressure of 20 bar and a condenser pressure of 0.5 bar. Calculate the salient points of the cycle, the energy addition and work output per
A steam power plant operating on a basic Rankine cycle has the following parameters: maximum (boiler) pressure 20 bar; minimum (condenser) pressure 0.5 bar. Calculate the thermal efficiency of the
Recalculate P3.1 assuming that the pump efficiency, \(\eta_{\mathrm{P}}=0.8\), and the turbine efficiency, \(\eta_{\mathrm{T}}=\) 0.9. Comment on the effect on the thermal efficiency of the plant,
Recalculate P3.2 assuming that the pump efficiency \(\eta_{\mathrm{P}}=0.8\), and the turbine efficiency \(\eta_{\mathrm{T}}=\) 0.9. Comment on the effect on the thermal efficiency of the plant, and
The engine designed by Lenoir was essentially an atmospheric engine based on the early steam engines. In this, a combustible mixture was contained in a cylinder: it was ignited and the pressure
A Lenoir engine (described in P3.5) operates with inlet conditions of \(p_{1}=1 \mathrm{bar}\) and \(T_{1}=27^{\circ} \mathrm{C}\). The energy added to the charge is \(1000 \mathrm{~kJ} /
A cycle is proposed as a development of the Lenoir cycle, in which the working fluid is expanded isentropically from its peak pressure down to a point where its temperature is equal to \(T_{1}\), the
The condenser pressure of the turbine in P3.2 is reduced to 0.15 bar. Calculate the same parameters for this cycle as in the previous example. Why have the parameters improved so much? [28.97\%;
Both cycles in P3.2 and P3.9 resulted in extremely 'wet' steam (low quality) at the exit to the turbine. This would cause erosion of the blades, and should be avoided. One way of achieving this is to
Problems P3.2, P3.9 and P3.10 have shown how the efficiency of a basic Rankine cycle can be improved, but even after superheating the steam leaving the turbine is still wet. This situation could be
Recalculate P3.11 with the pressure at which steam is reheated and reduced to 5 bar. What have been the benefits of using this lower pressure?[32.36\%; 51.40\%; \(1101 \mathrm{~kW} /(\mathrm{kg} /
Problem P3.12 seems to demonstrate that the efficiency of the reheated Rankine cycle gets better as the work distribution between the high pressure (HP) and low pressure (LP) turbines becomes more
What has the development of the basic Rankine cycle carried out in Problems P3.9-P3.14 shown you about the effect of the salient parameters on the efficiency of the cycle? Evaluate the mean
An air-standard Diesel cycle operates with a compression ratio, \(r=10: 1\). If the initial conditions at bdc are \(1 \mathrm{bar}\) and \(27^{\circ} \mathrm{C}\), and the energy addition is \(2000
A piston-cylinder assembly contains \(3 \mathrm{~kg}\) of air at 15 bar and \(620 \mathrm{~K}\). The environment is at a pressure of 1 bar and \(300 \mathrm{~K}\). The air is expanded in a fully
Air passes slowly through a rigid control volume A, as shown in Fig. P4.2(a), in a hypothetical, fully reversible, steady-flow process between specified stable end states 1 and 2 in the presence of
Confirm that item (iv) in P4.2 is equal to \(\left(b_{1}-b_{2}\right)\), where \(b=h-T_{0} s\), the specific steady-flow availability function. (Note that, since \(T_{2}=T_{0}\) and \(p_{2}=p_{0}\),
A mass of \(0.008 \mathrm{~kg}\) of helium is contained in a piston-cylinder unit at \(4 \mathrm{bar}\) and \(235^{\circ} \mathrm{C}\). The piston is pushed in by a force, \(F\), until the cylinder
A system at constant pressure consists of \(10 \mathrm{~kg}\) of air at a temperature of \(1000 \mathrm{~K}\). Calculate the maximum amount of work which can be obtained from the system if the
A thermally isolated system at constant pressure consists of \(10 \mathrm{~kg}\) of air at a temperature of \(1000 \mathrm{~K}\) and \(10 \mathrm{~kg}\) of water at \(300 \mathrm{~K}\), connected
An amount of pure substance equal to \(1 \mathrm{kmol}\) undergoes an irreversible cycle. Neglecting the effects of electricity, magnetism and gravity, state whether each of the following
A gas turbine operates between an inlet pressure of 15 bar and an exhaust pressure of 1.2 bar. The inlet temperature to the turbine is \(1500 \mathrm{~K}\) and the turbine has an isentropic
It is proposed to improve the energy utilisation of a steel works by transferring the heat from the gases leaving the blast furnace at \(600{ }^{\circ} \mathrm{C}\) to those entering the furnace at
Find the maximum and maximum useful, specific work \((\mathrm{kJ} / \mathrm{kg})\) that could be derived from combustion products that are (a) stationary and (b) flowing in an environment under the
In a test of a steam power plant (Fig. P5.1), the measured rate of steam supply was \(7.1 \mathrm{~kg} / \mathrm{s}\) when the net rate of work output was \(5000 \mathrm{~kW}\). The condensate left
Again neglecting the change in state of the feed water in passing through the feed pump, calculate the thermal efficiency of an ideal Rankine cycle in which the conditions of the fluid at inlet to
Neglecting the temperature rise of the feed water in passing through the feed pump, calculate the mean temperature of heat reception in the ideal Rankine cycle of P5.1. Thence make a second,
Air enters the compressor of a simple gas turbine at a pressure of \(1 \mathrm{bar}\) and a temperature of \(25{ }^{\circ} \mathrm{C}\). The compressor has a pressure ratio of 15 , and an isentropic
A steam turbine operates on a superheated Rankine cycle. The pressure and temperature of the steam leaving the boiler are 10 bar and \(350^{\circ} \mathrm{C}\) respectively. The specific steam
depicts a closed cycle gas turbine operating on the Joule cycle (i.e. constant pressure heat addition and rejection, and isentropic compression and expansion). Energy is added to the working fluid
Explain why the Carnot cycle efficiency is unrealistically high for a real engine. Introducing the concept of external irreversibility, evaluate the efficiency of an endoreversible engine at maximum
A heat engine operates between two finite reservoirs, initially at 800 and \(200 \mathrm{~K}\), respectively. The temperature of the hot reservoir falls by \(1 \mathrm{~K}\) for each \(1
Closed cycle gas turbines operate on the internally reversible Joule cycle with an efficiency of\[\eta_{\text {Joule }}=1-\frac{1}{r_{p}^{(\kappa-1) / \kappa}}\]where \(r_{p}=\) pressure ratio of the
Explain why the Carnot cycle overestimates the thermal efficiency achievable from an engine producing power output. Discuss why external irreversibility reduces the effective temperature ratio of an
The operating processes of a spark-ignition engine can be represented by the Otto cycle, which is internally reversible and gives a thermal efficiency of\[\eta_{\text {Otto
It is required to specify an ideal closed cycle gas turbine to produce electricity for a process plant. The first specification requires that the turbine produces the maximum work output possible
For a van der Waals gas that obeys the state equation\[p=\frac{R T}{v-\mathrm{b}}-\frac{\mathrm{a}}{v^{2}}\]shows that the coefficient of thermal expansion, \(\beta\), is given by \[\beta=\frac{R
Prove the general thermodynamic relationship\[\left(\frac{\partial c_{p}}{\partial p}\right)_{T}=-T\left(\frac{\partial^{2} v}{\partial T^{2}}\right)_{p}\]and evaluate an expression for the variation
Show that if the ratio of the specific heats is 1.4 , then\[\left(\frac{\partial p}{\partial T}\right)_{s}=\frac{7}{2}\left(\frac{\partial p}{\partial T}\right)_{v}\]
Show that (a) h-u-T2 a(f/T) ---7 ((^/7). - ("x/7)) ((g/T)
Show\[\begin{aligned}\text { (a) } T \mathrm{~d} s & =c_{p} \mathrm{~d} T-T\left(\frac{\partial v}{\partial T}\right)_{p} \mathrm{~d} p \\\text { (b) } T \mathrm{~d} s & =c_{v}\left(\frac{\partial
Show that, for a pure substance,\[\left(\frac{\partial s}{\partial T}\right)_{p}=\left(\frac{\partial s}{\partial T}\right)_{v}+\left(\frac{\partial s}{\partial v}\right)_{T}\left(\frac{\partial
Assuming that entropy is a continuous function, \(s=s(T, v)\), derive the expression for entropy change\[\mathrm{d} s=c_{v} \frac{\mathrm{d} T}{T}+\left(\frac{\partial p}{\partial T}\right)_{v}
The Dieterici equation for a pure substance is given by\[p=\frac{\Re T}{v-b} \mathrm{e}^{-\frac{a}{\Re T v}}\]Determine(a) the constants \(a\) and \(b\) in terms of the critical pressure and
Derive expressions for \(\left(\frac{\partial c_{v}}{\partial v}\right)_{T}\) for substances obeying the following laws:(a) \(p=\frac{\Re T}{v-b} \mathrm{e}^{-\frac{a}{\Re T v}}\)(b) \(p=\frac{\Re
The difference of specific heats for an ideal gas, \(c_{p, m}-c_{v, m}=\Re\). Evaluate the difference in specific heats for gases obeying (1) the van der Waals and (2) the Dieterici equations of
Derive an expression for the law of corresponding states for a gas represented by the following expression:\[p=\frac{\Re T}{v-b}-\frac{a}{T v^{2}}\]\[\left[p_{\mathrm{R}}=\frac{8 T_{\mathrm{R}}}{3
Show, for a gas obeying the state equation\[p v=(1+\alpha) \Re T\]where \(\alpha\) is a function of temperature alone, that the specific heat at constant pressure is given by\[c_{p}=-\Re T
The virial equation of state is\[p v=\Re T\left(\mathrm{~b}_{1}+\frac{\mathrm{b}_{2}}{v}+\frac{\mathrm{b}_{3}}{v^{2}}+\ldots . .\right)\]Compare this equation with van der Waals equation of state and
The equation of state for a certain gas is\[\frac{p v}{\Re T}=1+p \mathrm{e}^{-\mathrm{A} T}\]where A is constant.Show that if the specific heat at constant pressure at some datum pressure \(p_{0}\)
How can the equation of state in the form of a relationship between pressure, volume and temperature be used to extend limited data on the entropy of a substance.A certain gas, A, has the equation of
A gas has the equation of state\[\frac{p v}{\Re T}=\mathrm{a}-\mathrm{b} T\]where \(\mathrm{a}\) and \(\mathrm{b}\) are constants. If the gas is compressed reversibly and isothermally at the
A closed vessel of \(0.15 \mathrm{~m}^{3}\) capacity contains a mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and air, the air being \(20 \%\) in excess of that required for chemically correct
An engine runs on a rich mixture of methyl and ethyl alcohol and air. At a pressure of 1 bar and \(10^{\circ} \mathrm{C}\) the fuel is completely vapourised. Calculate the air-fuel ratio by volume
An engine working on the constant volume (Otto) cycle has a compression ratio of 6.5 to 1 , and the compression follows the law \(p V^{1.3}=\mathrm{C}\), the initial pressure and temperature being 1
A compression-ignition engine runs on a fuel of the following analysis by weight: carbon \(84 \%\), hydrogen \(16 \%\). If the pressure at the end of combustion is 55 bar, the volume ratio of
The exhaust gases of a compression-ignition engine are to be used to drive an exhaust gas turbo-supercharger. Estimate the mean pressure ratio of expansion and the isentropic enthalpy drop per kmol
(a) An amount of substance equal to 2 kmols of an ideal gas at temperature \(T\) and pressure \(p\) is contained in a compartment. In an adjacent compartment is an amount of substance equal to \(1
The exhaust gas from a two-stroke cycle compression-ignition engine is exhausted at an elevated pressure into a large chamber. The gas from the chamber is subsequently expanded in a turbine. If the
The following data refer to an analysis of a dual combustion cycle with a gas having specific heats varying linearly with temperature:The pressure and temperature of the gas at the end of compression
Distinguish between an ideal and a perfect gas and show that in both cases the specific entropy, \(s\), is given by\[s=s_{0}+\int_{T_{0}}^{T} \frac{\mathrm{d} h}{T}-\Re \ln
A jet engine burns a weak mixture of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and air, with an equivalence ratio, \(\phi=2\). The products of combustion, in which dissociation may be
The products of combustion of a jet engine have a molecular weight, \(m_{w}\), of 30 and a molar specific heat at constant pressure given by \(c_{p, m}=3.3 \times 10^{4}+15 T \mathrm{~J} /
Calculate the lower heat of reaction at constant volume for benzene \(\mathrm{C}_{6} \mathrm{H}_{6}\) at \(25^{\circ} \mathrm{C}\). The heats of formation at \(25^{\circ} \mathrm{C}\) are: benzene,
The heat of reaction of methane \(\left(\mathrm{CH}_{4}\right)\) is determined in a constant pressure calorimeter by burning the gas as a very weak mixture. The gas flow rate is \(70 \mathrm{~L} /
In an experiment to determine the calorific value of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) with a bomb calorimeter the mass of octane was \(5.42195 \times 10^{-4} \mathrm{~kg}\), the
A vessel contains a mixture of ethylene \(\left(\mathrm{C}_{2} \mathrm{H}_{4}\right)\) and twice as much air as that required for complete combustion. If the initial pressure and temperature are 5
A gas engine is operated on a stoichiometric mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and air. At the end of the compression stroke the pressure and temperature are \(10 \mathrm{bar}\) and
A gas injection system supplies a mixture of propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) and air to a spark-ignition engine, in the ratio of volumes of 1:30. The mixture is trapped at 1
A turbocharged, intercooled compression ignition engine is operated on octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and achieves constant pressure combustion. The volumetric compression
One method of reducing the maximum temperature in an engine is to run with a rich mixture. A spark-ignition engine with a compression ratio of 10:1, operating on the Otto cycle, runs on a rich
A gas engine with a volumetric compression ratio of 10:1 is run on a weak mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and air, with an equivalence ratio, \(\phi=0.9\). If the initial
A jet engine burns a weak mixture \((\phi=0.32)\) of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) and air. The air enters the combustion chamber from the compressor at \(10 \mathrm{bar}\)
A gas engine is run on a chemically correct mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and air. The compression ratio of the engine is 10:1, and the trapped temperature and pressure at inlet
A cylinder contains \(1 \mathrm{~kg}\) carbon dioxide, and this is compressed adiabatically. Show the pressure, temperature and specific volume are related by the equation\[\frac{1-\alpha}{\alpha}
(a) Calculate the equilibrium constant, \(K_{p}\), at \(2000 \mathrm{~K}\) at the reference pressure of \(1 \mathrm{~atm}\) for the reaction\[\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \Leftrightarrow
If it is assumed that the enthalpy of reaction, \(Q_{p}\), is a constant, show that the value of \(K_{p}\) is given by\[K_{p}=\mathrm{e}^{-Q_{p} / \Re T+k}\]where \(k\) is a constant.For a particular
A stoichiometric mixture of propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) and air is burned in a constant volume bomb. The conditions just prior to combustion are 10 bar and \(600
Methane \(\left(\mathrm{CH}_{4}\right)\) is burned with \(50 \%\) excess air. The equilibrium products at a pressure of 10 bar and a temperature of \(1600 \mathrm{~K}\) contain \(\mathrm{CO}_{2},
The exhaust gas of a furnace burning a hydrocarbon fuel in air is sampled and found to be \(13.45 \% \mathrm{CO}_{2} ; 1.04 \% \mathrm{CO} ; 2.58 \% \mathrm{O}_{2} ; 7.25 \% \mathrm{H}_{2} \mathrm{O}
A weak mixture of propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) and \(50 \%\) excess air is ignited in a constant volume combustion chamber. The initial conditions were \(1 \mathrm{bar}\)
A mixture of propane and air with an equivalence ratio 0.9 (i.e. a weak mixture) is contained in a rigid vessel with a volume of \(0.5 \mathrm{~m}^{3}\) at a pressure of 1 bar and \(300

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