714
Ind. Eng. Chem. Process Des. Dev. 1985, 24, 714-718
Efficiency of Energy Conversion: Rational PreHminary Estimates Isaac H. Lehrer Department of Chemlcal Ewinwdng, Monesh UnlversIW, Clayton, Vlctorla 3 168, Australla
Methods are shown that alkw preliminary estimates of energy converslon efficiency. The methods require knowledge of operating principles, limiting temperatures, and the approximate magnitude of required power input or output. Properly values are not required. The methods are useful for maklng p r e l i m estimates, for e x a m h i i the credibilky of detailed estimates, and for brMging the gap between classroom derhration and achieved values shown in the terature.
Introduction In the selection of process plant, several criteria are used; one of these is efficiency of energy conversion a t various stages in a given process. Useful definitions of such efficiencies are typically 4 = (useful power generated by system)/(rate of energy supply to generating system) (1)
4 = (minimum power calculated for task in idealized conditions) /(actual power required for task) (2) An accurate estimate of 4 requires the manufacturer’s specification of the equipment under consideration. When preparing cost estimates for new ventures, such specifications are not available when particular types of equipment and their manufacturer have not yet been specified. Estimates of production coat are then based on efficiency values quoted for similar plants elsewhere, or on approximate values that are shown in private design data or in the literature, e.g., boiler 0.85, steam turbine 0.45, generator 0.99, electric motor 0.90 (Dixon, 1975); furnaces about 60%, pumps 40-80%, compressors about 60% (Clark, 1976). Such similarity or rule-of-thumb estimates are useful, particularly when employed by an experienced estimator, but they can be unsatisfactory when (a) data are not available for the case under consideration; (b) available data do not define the numerator and the denominator in eq 1and 2 above; e.g., in gas compression, to which does a given efficiency pertain, isothermal or isentropic compression, power supplied to compressor drive shaft or to compressor drive unit? (c) equipment may operate over a range of power requirements and the relation between power and efficiency is not specified; (d) detailed calculations are used to find power in idealized conditions and this precisely evaluated quantity is then multiplied by a coarse rule-of-thumb factor without quantitative theory; such procedure is particularly unsatisfactory in the engineering classroom. It is desirable to have simple rational methods for finding characteristic efficiencies (Lehrer, 1983). Such methods are useful in preliminary process design and also in engineering instruction; they are discussed in the sections that follow. Relevant Quantities and Classification of Energy When discussing efficiency of energy conversion, relevant items are defined by P = useful power obtained from a system, e.g., VAp from a pump for incompressible fluid (3a) W, - P = power lost by fluid action on bounding and submerged surfaces in the system (3b)
w, - w,= power lost by seal and bearing friction in device containing the system, also by auxiliary equipment (3c) W, - W, = transmission loss between system device and ita driver, e.g., belt-, chain-, or gear transmission loss (3d) W4 - W8 = driver system loss due to action of driving medium, e.g., fluid or current (3e) W5 - W, = driver bearing and seal friction loss (3f) W, - w5 = transmission loss in supply of driving medium to driver, e.g., electricity transmission loss between generator and motor (3g)
W, - W6 = electric generator bearing friction loss (3h) w8 - W, = electric generator cooling power (3j) w9- w8
=
transmission loss between generator and its driver (3k)
Wlo - W, = power lost by fluid action on bounding and submerged surfaces of driver (31)
- WlO = power loss due to seal- and bearingfriction in driver (3m)
w11
- Wll = power loss of driving medium in transit from its generator to driver, e.g., pressure drop in steam and heat transfer from steam between boiler and turbine (3111 W13 - W12 = power for auxiliary equipment of driving medium generator (3p) Q14 - W13 = power of driving medium after completing a reversible Carnot cycle = power lost to environment (3q) Q16 - Q14 = power loss in heating medium leaving at temperature above temperature of environment, e.g., flue gas from a furnace (3r) w12
= loss due to thermal leakage from heat source (3s)
816
-
Qi7
- Qie
Q16
power for heat source auxiliaries, e.g., fans and pumps (3t) E - Q17 = power loss in unused part of energy source,e.g., incomplete combustion (3u) In many cases, only some of the above 19 equations require 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
Table I. Typical Relations between Power, Power Loss, and Flow Rate, and Appropriate Efficiency Equation power input = power output = flow rate Ax P (or Q) = Cxm power loss = Bx" = X (loss/output) a x"+" n (a) Resistive Electrical Load Vmud RLcudI* RmurJ2 Z (Rmource/Rioad) = Constant 2 torque N
K@N3
ViI
VO I
torquei N
(b) Centrifugal Pump Impeller, Operating with Negligible Shock Loss KLPN' N KL/Ko = constant
3
(c) Heavy Alternating Current Conductor, Simplified to One Phase, Power Factor = 1 Z 2R I Z2R/(VoI) = ZR/Vo = Kx 2
715
appr effic eq 11 or 15
11 or 15
17
= mg, Friction Coefficient = F, Shaft Diameter = D torqueo/FmgifD 1 17
(d) Roller or Ball Bearing for Horizontal Shaft, Supported Load torqueo N
FmglrDN
N
K~v
K0v3
(e) Turbine Water Supply from Elevated Storage, Loss Coefficient = f CWLf)v3 U C ( K Lf ) / K O= constant approx
Kip0
KopV z TH; Tc > TE Referring to eq 3, 21, and 22, with hot end thermal conductance UsHAsH
At any selected efficiency C # J ~
As in conversion of high-grade energy to work, prima facie characteristic efficiencies are: at maximum power +wia,mx
Therefore, substituting the values of x in eq 9, taking the ratio of the resulting values of P
at maximum value of (power)(efficiency) @( Wia+)MAX
p+i
power at efficiency $i/maximum power = -= PMAX
(24)
When the heat source is the latent heat of a vapor condensing at constant pressure, a well-stirred fluid, or Us, and Ts, are simple functions of temperature, eq 23, and efficiencies defined by eq 24 can be evaluated quickly,
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 717
provided that a reasonable value of TCis available. When USHASH = constant
d(W134) --0
dTH and W134 has maximum value at 1 T H = -[(8TsTc + Tc2)1/2- Tc] 2
(28)
The value of T H must be lower than the maximum economic service temperature of the material of construction of the system under consideration. When both Ts and Us, are not even approximately constant, their value to be used in eq 23 and others is more difficult to specify. Representative, yet still simple approximations are Ts = ( T s ~ f + i TH)/~ (29) where Ts= maximum temperature of the energy source, e.g., the adiabatic flame temperature in combustion USHASH= constant (Ts - TH) = CsH(Ts - TH) (30)
Example 3. Estimate the efficiency (power at generator terminals/calorific value of fuel used per unit time) of a 375-MW generator driven by a steam turbine. Steam is generated by heat transfer from a gas-fired furnace where the adiabatic flame temperature is 2200 K. Materials of construction dictate a maximum steam temperature of 550 “C = 823 K. At the intended location, design dry-bulb temperature is 308 K, mean annual temperature TB= 290 K, design wet-bulb temperature is 295 K. For the fluidmechanical losses in the turbine system during conversion of pressure energy to shaft energy via kinetic energy, n = 3. From eq 15 and eq 3, #(p+)- = (Wg/ Wlo) = 4/5 = 0.80. For conversion of low-grade energy to work in the given combination of systems, neither Ts nor Us, is constant, and like TH, must be used at a characteristic, representative value. Using eq 33 and 34, T H = (TswTc)1/2. From the given data, Tsw = 2200 K, and Tc = 308 K is a reasonable approximation. Then T H = 823 K, which is just within the permissible range. Using eq 21, the estimated effi= (W13/Q14) = 1 - (308/823) = 0.626. ciency, With regard to energy loss in the furnace exit gas, a practical exit temperature is the boiling point of water at atmospheric pressure. Allowing for cooling in the chimney, exit temperature of combustion products = TOUT = 393 K. Then (&14/&15) = (TSMAX - TOUT)/(TSMAX - TE) = (2200 - 393)/(2200 - 290) = 0.946. The remaining losses are dictated mainly by economic considerations and therefore the concomitant efficiencies are given by eq 17, eq 9b or 9c being applicable.
= 0.98 375000 + 0.084 In an installation of the given size and function, it can be assumed that there is complete combustion most of the time, so Q17/E = 1 and also from eq 3 (power at generator terminals) / (calorific value of fuel/time) = W6/E = (0.8)(0.626)(0.98)s(0.946)(1) = 0.40 Note that the heat engine efficiency, usually estimated from enthalpy values, is (0.8)(0.626) = 0.50; Le., two terms, but no property values, are required. Example 4. Estimate the fraction of power lost in a large high-voltage supply system. Considering step-up transformer, line, and step-down transformer, there are three losses in series. All can be described by eq 9c be= VlinJ - PR, cause, simplified, & VJ, = Vlin$, Vline,outl c $ Vline,outl ~ = V212.For eq 9c, i.e., system 111, efficiency = @DES. Because power P is large, = 0.98 (eq 17). 4 = (0.98)3 = 0.94. Hence the estimated fractional loss between step-up transformer inlet and step-down transformer outlet is 0.06. Example 5. Estimate the efficiency of a pump/motor unit similar to that in Example 1. Pumping the same fluids, but with fluid power output P = 1.0 kW. From eq 17,4DEs = 0.98(1- (0.084/1.084) = 0.904. The other efficiencies are the same as in Example 1. Then P/ W5 = (4/5)(1)(0.904)3 = 0.59 for the low-viscosity fluid and (2/3)(1)(0.904)3= 0.49 for the high-viscosity fluid. Conclusion It is desirable to have rational and simple methods of estimating efficiencies of energy conversion, even at a preliminary stage. In many cases, power can be described by (specific energy)(flow rate), where specific energy itself may be a function of flow rate. Methods are shown for estimating efficiency of converting high-grade energy, low-grade energy, and combinations for these, yielding
(
= 0.98 1 - -
Via differentiation of eq 31 with respect to TH, at maximum value of W13 TH
= (1/4)(Tc
+ (Tc2 + 8 T s ~ f i T c ) ” ~ )
(32)
Via differentiation of eq 33 with respect to TH, at maximum value of (W134)
TH = (TsMAxTc)~’~
(34)
Application of the foregoing argument is shown in Examples 2 and 3. Example 2. Eatimate the mechanical power recoverable from an exothermic process in a backmix fluidized bed operating at 690 K. Heat removal rate is to be loo00 kW and process fluids can be cooled to say 310 K without special measures. Referring to eq 3, the appropriate efficiency is (Wg/Q17). In the fluidized bed, both Us, and T H are constant; therefore eq 28 can be used, from which T H at maximum value of W134 is 517 K. Inherent system losses are (Q14- W13) and (Wlo- Wg), the concomitant efficiencies are: from eq 21, c$Carnot = 1 - (310/517) = 0.40; from eq 15, q5(p+)w = 0.80, with n = 3 for the power recovery device and P = W,. From the problem statement, ( 8 1 5 - QI4) = 0. For the remaining losses, use eq 17 0.084 0.084 lo4
+
EZ
0.98 =
WlO w11
--
wll
w12
--=-
Q15
= -Q16
W12 w13 Ql6 Q17 Estimated power recovery = (0.4)(0.8)(0.98)5(lo4) = 2893 kW.
718
Ind. Eng. Chem. Process Des.
Dev., Vol. 24,
No. 3, 1985
values within the range of pertinent data. In conversion of high-grade energy, the relation between input, loss, and output is given by eq 9. Systems described by eq 9 can be classified into three types, depending on loss/output ratios and the relition between flow rate and loss as given by index n. For each type of system, there is an appropriate efficiency. For system I, described by eq 9a, efficiency is given by eq 11 if maximum power is envisaged, or, as is more likely, eq 15, which applies when the product (power)(efficiency) is to have maximum value. For systems I, the trade-off between power and efficiency is limited in practice, as is evident in Table II. For system 11, described by eq 9b and system 111, described by eq 9c, efficiency is given by eq 17. For these, there are no inherent limitations to efficiency other than economic considerations. In conversion of low-grade to high-grade energy, two basic efficiencies are: (i) $bt, eq 21, when power = QH(1 - (Tc/TH)) is at maximum, with TH evaluated by eq 25 or eq 32; (ii) $-, eq 21, when (power)(efficiency)= QH(1 - Tc/TH))2has maximum value, with TH evaluated by eq 28 or eq 34. When improved efficiency can be achieved by choice of arrangements, materials, and auxiliaries, such as heat recovery, thermal insulation, balanced draft, eq 17 can be usbd provided that there are no constraints, e.g., minimum exit temperature specified in Example 3. The methods shown in this paper require knowledge of principles of operation, limiting temperatures, and for eq 17, magnitude of power input or output; such information can be expected to be available even at a preliminary stage. The methods do not require property values; thus estimates can be made and/or checked using only few data. The arguments used, though elementary, introduce a logical prediction of practical efficiency when details of process and equipment are not yet available. They may also help to bridge the gap between classroom derivations and data shown in the literature. Nomenclature A = specific energy, ML2Px-I ASH = heat transfer area between heat source and working fluid, L2
B = power loss factor, ML2t%-” C = power output factor, ML2t-3x-m CsH = constant, M L 2 r 3 T 2 E = power source, ML2t-3 f = friction factor, loss coefficient F = mechanical fridtion coefficient g = gravity acceleration, I = electric current, A K = constant in Table I, dimensions to fit context n = flow rate index N = revolutions/time, t-l p = pressure, ML-W2 P = power output = useful high-grade power, given value in eq 17, ML2t-3 Q = low-grade power, ML2t-3 R = electrical resisthce, ohms T = temperature, T US, = heat transfer coefficient between heat source and working fluid, M P T 1 u = velocity, Lt-’ V = electric potential, V V = volume flow rate, L3t-’ W = high-grade power = rate of work, ML2t-3 x = flow rate, power/specific energy, quantity of ... t-l Greek Letters A = difference p = density, ML-3 $ = efficiency Subscripts C = at low temperature in Carnot cycle E = environment H = at high temperature in Carnot cycle S = energy source
Literature Cited Clark, J. P. M M E C H 1976, 6, 23. Dixm, J. R. “Thermodynamics. I. An Introduction to Energy”: Rentlce Hall: Englewood Cllffs, NJ, 1975. Lehrer, I. H. “The Eleventh Australian Conference on Chemical Engineering, CHEMCA 83”; EA Books: St. Leonard, N.S.W., 1983 p 433. mum. H. T.; Plnkerton, R. C. Am. Scl. 1955, 43,331. Tribus. M. “Thwmostatlcsand Thermodynamics”; D. Van Nostrand Co. Inc.: Princeton, NJ, 1961; Chapter 16.11.
Received for review March 12,1984 Revised manuscript received August 13,1984 Accepted August 29, 1984
