ANALYTECAL AND GRAPHICAL ENGINEERENG ECONQMLC ANALWIS A3 AWHED TO THE CGMPRESSDN REFRIGERATIQN SYSTEM AND ALMS? CGMFQNERTS Thesls for “10 Degree of pH. D. MiCHEGAN SMTE UNK‘IERSITY George E. Suttan 1957 THFSES This is to certify that the thesis entitled Analytical and Graphical Engineering Economic Analysis as Applied to the Compression Refrigeration System and Allied Components presented by George E. Sutton has been accepted towards fulfillment of the requirements for Ph.D. (169,661,, 14.13. Majéf professor Date Nay 111, 1957 0-169 ANALi'i'ICAL AN D GRAPHICAL ENGINEERING ECONOMIC ANALYSIS AS APPLIED TO THE COMPRESSION REFRIGERATION SYSTEM AND ALLIED COMPONENTS by . f. s‘ George Eé'Sutton A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR or PHILOSOPHY Department of Mechanical Engineering 1957 Copyright by George Edwin Sutton 1959 Analytical and Graphical Engineering Economics as applied to the Compression Refrigeration System and Allied Components By George E3 Sutton ABSTRACT The methods currently available to engineers for selection of mechanical equipment such as air ducts, refrigeration piping, etc. are limited, generally, to published taoles of vague origin. A number of equations are developed relating size to operating and owning costs. These are minimized to produce the optimum size for minimum annual cost. A systematic approach to graphical solution of equations is developed in order to generalize the solutions at cost equations. Linimizing of equations in one, two, and three variables is discussed. Applications of these methods are made to the following: Insulation; Condenser water Rate; Water piping; Discharge, Suction and Liquid Lines; Air Ducts; and Tubular Heat Exchangers. The results indicate that the use of generalized tables for pipe sizing, etc. should be discouraged, and that graphical solutions should be used whenever possible in order to produce the most economic selection. The study of tubular heat exchangers indicates a trend toward a large number of very small tubes, with Reynolds' Numbers in the transition range between laminar and turbulent flow. As a consequence, more study is needed of the character of flow and neat transfer in such small tubes. A I ’1). ‘1 Q i m ’1 j 4 b\‘>...c‘r M” '{J' .- $ WW " _... Major qufessor\ Copyright @ 1957 George E. Sutton Acknowledgements The author wishes to express his gratitude for the capable leadership and ingenuity exercised by Professor Donald J. Renwick, and for the apt guidance of the committee: Dr. James T. Anderson, Professor L. C. Price, Dr. C. P. Wells and Dr. hax Rogers. Gratitude must also be expressed to the Government of the United States for the financial aid through the G. I. Bill of Rights (PL 550) and the Naval Air Reserve, as well as to Michigan State University and the Department of Mechanical Engineering for part time employment, all of which made the pursuit of the degree financially possible. It would be impossible to enumerate all those who, currently, as well as in years past, have, through their direct aid or forbearance, made this work possible. Analytical and Graphical Engineering Economic Analysis as Applied to the Compression Refrigeration System PAGE 1 Chapter I 17 II 23 III 29 IV 50 V and Allied Components by George EEJSutton CONTENTS Introduction Graphical Techniques Minimizing of Equations Equations not Suitable for Graphical Solutions (a) Economic Thickness of Insulation, Walls (b) Economic Thickness of Insulation, Pipe Simple Graphical Solutions (a) Economic Condenser Water Rate (b) Air Duct Sizing Simple Graphical Solutions with Incremental Variable (a) General Pipe Sizing (b) Water Piping (c) Discharge-Lines (d) Suction.Lines (e) Liquid Lines Contents, Page 2. PAGE CONTENTS 76 Chapter VI Equations with Three Variables (a) Simple Heat Exchangers 85 VII’ Summary (a) Effect of Choice Other Than Economic (b) Comparison with Published Tables (0) Future Work 93 Bibliography Page 33 38 47 57 60 63 7O 75 TABLES Title Variation of Refrigeration Effect with Operating Conditions Hc/m for various Operating Conditions 0(t) for Air ¢(t) for water Piping ¢(t) for Flow with Given Pressure Loss ¢(t) for Discharge Lines C(t) for Suction Lines Wt) for Liquid Lines Page 12 13 15 34 39 41 48 50 FIGURES 1-1 1-3 l~4 1-5 1-6 1-7 4-2 4-3 4-4 4-5 Title Solution of the Equation flu) new) Solution of the Equation (1:232 0 Solution of the Equation ab d ‘cd Solution of the Equation =22 cd Solution of the Equation 2 a E 5 : d1.5 cd ' Addition of Functions Solution of the Equation b g? +'e = cd Refrigeration Effect Versus Operation Conditions Economic Condenser Water Temperature-Rise“ Variation of Operating Coat with Condenser water Temperature Rise Variation of Cost with Diameter - Ducts Air Duct Sizing Figures (Cont) Page 55 58 61 64 71 73 76 5-1 5~2 5-3 5-4 5-5 5-6 5-7 Title Economic Water Pipe Size Variation of Cost with Diameter Water Pipe for Specified Pressure Loss Discharge Line Sizing Suction Line Sizing Head Equivalent of Subcooling Liquid Line Sizing Introduction The sizing and selection of mechanical equipment has long been based upon experience and Judgement, neither of which is an easily obtainable commodity. Practice has shown the minimum size consistent with effective Operation. Admittedly, this practice, based upon experience, yields the lowest first cost consistent with Operable equipment. It can be shown, however, that, in many cases, the operating cost is so high that the total owning and operating cost of the equipment over a period of years is far greater than if the economic size were chosen by analytical means. There is another strong reason for examining critically present methods of selection. Many references show tabulated sizes, particularly in the case of pipe, which will yield, according to previous practice, the economic system. These data are often based upon prices of some years past. The cost of electric power has remained essentially constant over many years, while material and installation costs have practically tripled since 1926. These facts would tend to indicate that, in many cases, smaller sizes than previously selected are now Justified economically. The technique of minimizing cost by differentiation of a total cost equation is beset by two disadvantages. First, the equation representing total cost must be derived. Such an equation may, however, be derived approximately, if not exactly, for most applications by use of some ingenuity and 1 ii reasonable approximations. The second disadvantage lies in the usual apprehension of engineers toward higher mathematics. This may be due, largely, to the lack of stress placed upon practical aspects of mathematics in scholastic courses and in the practice of engineering. It is the aim of this treatise to provide a guide to methods of setting up the equations of total cost, and then show how to solve for the economic sizes or quantities. Although the title indicates, correctly, that the apparatus treated relate to the compression refrigeration system, the methods deveIOped may be applied with equal ease to many other engineering systems. In many cases, assumptions are made necessary. These will be clearly indicated as such. The primary value of the literature search in connection with this work has been in providing bases for these necessary assumptions. It is believed that graphical solutions, because of their simplicity and generality, offer a valuable contribution. Graphical techniques will be discussed in detail prior to discussion of Specific equations. Chapter I Graphical Techniques A graph, as used in this work, is defined as a plot of the behavior of some function with variation Of a primary variable. The most common graphs encountered in engineering work are linear scale, semi-logarithmic scale, and log-log scale. Nomographs and alignment charts are not graphs as herein defined, but, more properly, form another class of charts. Linear scales and semi-logarithmic scales have the primary disadvantage that few functions plot as straight lines on them. Log-log scale graphs are advantageous in that any single-termed function, regardless of the number of multiplying and dividing parameters contained and the powers to which the parameters or variables are raised, will plot as a straight line. In addition, multiplication and division may be readily performed. Addition and subtraction of functions may also be performed, but less readily. Log-log graphs are simply plots on special paper, having a logarithmic scale on both abscissa and ordinate. The simplicity encountered with base ten logarithms will be utilized here throughout. A change of one cycle upward or to the right represents a multiplication by ten, while a change of one cycle downward or to the left produces division by ten. This decimal characteristic allows much freedom in using such 1 graphs for calculation. To illustrate the technique of solving an equation of two single-termed functions, consider the following example: ab 2 d2 C where a, b, and c are parameters. That is, they may vary with other conditions, but are not functions of the primary variable, d. The equation may be rewritten: ab cd which is of the form: fl(d) z f2(d) Thus, for given values of a, b, and c, the solution occurs :d. when flzrfe. Figure 1-1 shows a plot of f1(d) and f2(d) against values of d. Since fl is a function of d raised to the -1 power, the slope is -l and, since f2 is a function of d to the1'l power, the slope is+—l. The intersection of the lines representing the two functions will occur at the value of d which produces equality of the two functions, which is the solution of the equation. In this case, the values chosen were: a, 10; b, 10; and c, l; which produces a solution d equal to 10. In order to generalize, or to provide for other values of a, b and c, the graph may be extended to facilitate calculation of the term %9 , for all values of a, b, and c encountered in the particular problem. Let the ranges of the parameters be chosen as follows: IOO IOOO IOO flon LO IO IOO IOOO d FIGURE I-l Solution of the equation: f.(d)=f2(d) IOOOO lb 0.1 S a 10 105. bilOO 1 _<_ c5100 For these ranges of the parameters, the minimum solution for d is 0.1 and the maximum is 31.5. This represents 2% cycles range for d. Thus a solution area required is then 2% by 2% cycles. Prior to demonstration of the method of computing the operating area, that is, the area required to compute the value of fig , it will be necessary to examine the operations of multiplication and division in more detail. As previously mentioned, multiplication and division may be performed by vertical movement, which results in addition or subtraction of logarithms. Figure 1-2 demonstrates the operations involved in computing 2% . Starting with log a, progressing downward, each cycle progressed represents division by 10. Thus division by c may be achieved by prOgressing downward to log 0 and then vertically to any chosen unity. The procedure is simplified if all multiplicands are arranged above a certain unity, and all divisors arranged below the same unity as shown in Figure 2. Further multiplication by b is achieved by progressing upward vertically to log b. The equation representing these Operations is: log a - log c+log b =log 295% Further movement along the lines of slope -1 produces u ooh .domuv 5:33 2.20 cozeow N4 mmaoi ll|||l I0 OOI— | J l n 00:... I'll'l :2: 132.32 division by 10 for each cycle. However, since the decima is arbitrary, due to the uniformity of the cycles, the value at the right ordinate of Figure 2 may be chosen as log ‘15%%553 or log %2 , or any other decimal value. The only requirement is that the other functions in the equation agree in decimals with the first. From observation of the operations, an arbitrary rule may be derived for determining the number of cycles, horizontal and vertical, required for a given operation. Assigning the unity scale as shown, multiplication of "a" by 1 requires na cycles, vertical and horizontal, where na is the range of a in cycles. The division by c requires no cycles, horizontal and vertical. Multiplication by b requires no additional cycles vertically, but requires nb cycles horizontally. It is obvious, then, that the space required will be equal to the sum of the n cycles horizontally, but only the sum of the n's for the maximum range multiplier and maximum range divisor will be required vertically. For the example chosen, the requirements would be Na'+ Nb-+ Nc = 2-+ 1 + 2 = 5 Horizontal Na-+ Nc == 2 + 2 = 4 Vertical The total for the operation and solution becomes 7% by 6% cycles, provided the orientation is as shown. That is, if the solution area is not allowed to overlap the operation area. Experience will show that some overlapping may be tolerated, thus reducing the total space requirements. Figure 1-3 shows the total operation and solution. 0 u ' IOO \ \ ' \ :00 0.: ~ ‘\~ l0 \ \ \ \ /\< I. l0 \ \r / Q2. \\ / \6‘0/0 l \ // \15? \ / / \ \ \ / \ / \ 1' \ a a \ °g\ \ 99° \ x o ¢°o\ K\ \ \ \ 0.! I IO IOO d F l G U R E l- 3 Solution ofthe equation= d=gg The example shown is for a= l, b==10, c==lO. This illustrates, perhaps, the simplist method of locating the scale in the solution area. These values were chosen such that the solution for d would be 1. Thus the two lines representing the functions must intersect at the ordinate representing 1 as the value of d. The minimum area would be utilized if the function equal to d were moved to the heavy dashed line shown, which represents an overlapping of two cycles. The total area could thus be reduced to 5% by 5 cycles. This reduction is difficult to predict in advance of actually laying out the graph, however. A simplification may be made in scale location which will reduce crowding of scales. Since horizontal movement has the same effect as vertical movement, one scale may be placed along the upper abscissa. If the parameter chosen is the primary one, such as flow rate, tonnage, etc., the effect of this parameter upon the solution becomes readily apparent. Figure 1-4 shows the rearrangement of Figure 3 into this pattern. Such methods result in a graph which may be used to solve the equation for all values of the parameters within the chosen ranges. In the simplicity of such solutions lies the major value of the system. Solutions which require the use of more complex parameters may be implemented in equally simple ways. The most generally encountered type involves parameters and variables raised to powers other than unity. IOCPJ I IO b Ido Io A \ \\ \ \ IO \ <50”. \ ’b \ / \ox”? I \ / \ / \ / \\ / Kb \ ,/ 4;, I)\ f. 15%} «7/ \ .O‘ 70’ Voo\\ A \ Q \ \ O.l I d FIGURE l-4 Solution of the equation= @213- 10 With such exponential parameters, it is deemed simplest to treat the parameter with its exponent as a new variable raised to unity power. Thus if a2 were to be a part of the function, a2 would be plotted on the scale, with values of “a”inscribed so that the location of values of "a" would automatically produce a2. Since log a2 is equal to 2 log a, the scale length would be twice the range of "a". If the primary variable appears at any other power than unity, the solution may be handled in two ways. First, the operation may take place at a lepe equal to the power. Second, the variable and its exponent may be treated in the same way as the parameters discussed in the previous para- graph, whereupon the new variable is the variable raised to the appropriate power. The latter system will be used in this treatise. As an example of this technique, consider the equation: 239; d3 c which may be rewritten as: 2 a b - d1‘5 cal-05 Let the parameters have the same ranges as previously. The required operations area will now be, using the upper scale for u '0: Horizontal: (2)+(2)-*l+’2='7 Vertical: 24-1= 3 2 The range of d3 is from (O ll0él01 ; 0.001 to 2 Lloll(1oo).=10,000 which represents values of d1°5 of 11 0.0315 to 100. Thus the solution will require 3% cycles. The total area will be a maximum of 10% by 6% cycles. Figure 1-5 shows the graph for solution of this equation. It has been shown that any equation consisting of two single-termed functions may be readily handled by graphical means. For three or more terms, the equations are somewhat more difficult to solve. Consider the equation: gB-Ie::cd The solution requires that the sum of two functions of d and a parameter "e" be equal to another function of d. Computation of the two functions is readily carried out by previously demonstrated methods. Determining the sum of function of d and the parameter e, which produces another function of d, is not so readily performed on log-log graphs. Note that the sum is a two-termed function, and does not plot as a straight line. In general, the equation to be solved is: fl(d)+ f2(d) = f3(d) or==log(fl+-f2)= log f3 but: log(f1+ r2): log fl(1+ :3.) l f : 10 f + 10 (l+‘ 2 81 a T1) This operation is easily carried out with a pair of dividers. Figure 1-6 shows the method. The dividers are set between f2 and f1, which measures the log £2. If they are reset with the lower leg on any unity, f1 they may be extended by one unit, which gives log(l+-;§ ). Resetting the lower 1 O.| I to ICC b IOO IO IO l 5| 1 l 141 ID d FIGURE l-S Solution of the equation: _ 02b . 3875 = “"5 IOO ’b\ ’I ’9 ” I a?’ I’- -" ” e— \ ‘ \\ \\ \\ \ I _ FIGURE I- 6 Addition of functions Step It Set dividers between fl and f2 2: Set lower leg on unity 3* Open upper leg by one unit 4= Reset lowerleg on emalle: function Upper leg indicates sum l4 leg on f1, the upper point will rest on the point at which the ordinate is log (fl+ f2), as shown in the derivation. As a numerical example, assume f2: 12 and fl: 6. Setting the dividers between them, and then placing the leg on any unity, the span will show an upper reading of 2, which indicates the span to be log 2. Extending the upper leg to 3 will cause the span to be log 3. Adding this to 6 will give log (6)(3) or log,18. Nets that the sum of 12 and 6 is l8, so the solution is correct. For subtraction of two functions, the method is essentially the same, since: log (f2 - f1) =log fl[-§.—2- - l] 1 fr : + 3- _ The calculation is made as before, except that the divider span is reduced by one unit, rather than expanded as before. The example chosen for graphical solution, ilug-re: cd is shown on Figure 1-7. The ranges of the parameters are: 10 9 a 5100 0.1 $ b 5:10 1 5 e 6 100 10 6 c 3 100 In this case the additive term is a constant for a given solution. Let the values of the parameters be: a :10, b=1, 0 =10, e =100 for simplicity. The solution for these values will be slightly greater than ten, since: (%fl+10310* In order to insure correct results, the scales of loom I 10 T ' :00 b '0 ‘ l0 / l I l ‘\ fiirf f2 2 ' IOO . l I e lo g l , I | l I l l l 0.: I I0 IOO IOOO d FIGURE l-7 Solution of the equation: ab T+G=Cd 16 f1, f2, and f3 must be known. These may be arbitrarily chosen, but must be identical. If they are arbitrarily chosen, the location of the scale for the primary variable, d, may be found by taking the case where e=.O, whence: AggiOd; d= 1 This automatically locates the scale for c. The previous discussion provides ample techniques for the solutianof practically any equation of four or less terms. Since no equations have thus far been encountered in this work which necessitated use of more than three terms, no further development is deemed necessary. Chapter II Minimizing of Equations . In general, differentiation of a function with respect to a variable, and setting this derivative equal to zero will give the value of the variable which will produce either a maximum or a minimum of the function. Equations for total cost of a mechanical component may usually be written as the sum of two functions of the primary variable, one being the annual owning cost and the other being the annual operating cost. These are usually a direct function and a reciprocal function, both of which are smooth con- tinuous functions. The maximums will occur in such cases at infinite and zero values of the primary variable, and there will be only one minimum. As a simple example of such an equation, consider the following: — 13. Ct‘ Ad+ d It is desired to compute the value of d which will produce a minimum total cost. Differentiating, and setting the derivative equal to zero: dot B .__.: A - : dd 52 O This is the equation which must be solved for the value of d which will produce a minimum cost. The solution is as ./E d' A In the case of two independent variables, the partial follows: derivatives of the function with respect to each of the two 17 18 variables may be set to zero, and the equations produced thus solved simultaneously to yield the solution for the values of the variables to produce either a maximum or a minimum. The same reasoning as applied to a single variable indicates that the solution will yield a minimum. As an example, consider an equation: ct: AL3d - B(Ld+ L3) The solution is as follows: act.) . 2 .. 2 (3L d Bid. (1 B( 9L ) act)- 3.. g (5.5L..AL BLO AL3=BL ii and: 3A(IT32<-B(d+-3—) 3 Ed: ade-S.2 2 Bd=3 K2,A (182 If the two variables are not independent, the previous method will generally not lead to a solution. In this case the total differential of the function must be set to zero and the differential equation solved. The total differential of a function may be written as follows: do, = (9%Md1” (%E§)L dd Consider the general equation: f(L,d) = r1 (L,d)-r r2 (L,d) l9 Differentiating, and setting the total differential equal to Zero 3 ar.[(afi)d. (Lf2)d]dL+[(§_a.f3-)L+( (af2)L ]dd- 0 This may be written, symbolically: M d Ld-N d d The test for exactness of a differential equation is: $4-939. ad" oL Applying this test: 32f1 62f2 5% 321‘s m * amaj= am + 37:. In practically all the cases encountered in Mathematics the order of differentiation has no effect, so that, generally the equality will exist in the previous expression. The method of solution for an exact differential equation is as follows:* F(L. d)- A] MdL+/d[1¢- H/LMdLJdd Const. Applying this to the general equation: F(L, A)” Afl L39] dL-I-[d{[3:l 6:2- iii/11:33:“; 3%) <11) dd ; Const. * Kells, L. M., Elementary Differential Equations, McGraw- Hill: New York, 1947, pp 44-46. 20 d F(d.L)= 1‘ +f2+ éfi bfa- a f dd 1 [{m—tfi‘] '35 1* 2]} = 1‘14. f2+fdiafl +af2 -bfl _ are dd 3’ d a d ‘S‘c'i 35" = fl-o-f2 = Constant This has the same form as the original equation with the function set equal to some undetermined constant. With no additional conditions, the value of the constant cannot be determined by elegant methods, but may be determined by successive approximations if necessary. If any constraint on the system is known, the method of Lagrange* offers a solution. For two or more variables this method is applicable. As in the previous case, assume that the function Ct (d,L) is given, and that the constraint is: G(d,L) = constant Lagrange's method consists of minimizing the combined function: F :Ct(d,L)+ Y G(d,L) where Y is an arbitrary constant, called Lagrange's multiplier. Differentiation of the function yields two equations, with the constraint furnishing the additional equation necessary to complete the system of simultaneous equations. The system is: F - (“at ° F - (7)6 o G(d,L) :1 Const. g—Aglggggg;§§lggl3§, A.E. Taylor, Ginn and Co., Boston, 1955. pp 198—201 21 Solution of this system may yield multiple results, which may be maxima, minima, or neither. Classification may be achieved by the following:* Let: 2 A. 32%(61. b) B ,gectgapz ’ c,act(a.b) D: B2 _ AC ad2 ’ adaL a1? ’ where (a,b) are the values of (d,L) found as a solution of the system. A If: D4 u (b) Pipe Insulation. Pipe insulation is usually carried out by one of two means: First, pre-cast insulation, available in incremental thicknesses; and second, by paste insulation which may be applied in the desired thickness, and allowed to harden in place. The equation for cost may be simplified if the cost may be expressed in terms of the volume of the insulation, say in cents per year per cubic foot. It may be further simplified if the thermal resistance of the pipe or duct is negligible, and only the effect of film resistances and of the insulation need be considered. In most cases, the thermal resistance of the interior film is also negligible with respect to that of the insulation and the exterior film. Utilizing the above simplifications, the cost of insulation may be expressed as: 2 2 01:1: L7T(do - <11) (4) 141; where: d0 =the unknown outside diameter of the insulation, inches. d1..the inside diameter of the insulation, inches. A = cost of insulation, cents per year per cubic foot L =.length of pipe, feet. The heat transfer may be written, using the above simplifications: 28 fl'IUAtm 1' 12 1 do 3' + 1n __] d h Zk [“ O 0 d1 where: tm::the logarithmic mean temperature difference between the interior fluid and the outside air. ho: exterior film conductance, BTU/hr-ft2-OF. The cost of heating or cooling may be written as: Bh,c 11" L Atm (8760) F Ch,c 12 1000 -—- -ln d0 [.dhole< H; where: Eh: use factor and the tot- l cost2 is: C Afi(d02 «112) L 8.76TPBFu LAtm 1': T4) 144 4' page 1“ 2-3] Differentiating with respect to do' and setting to zero: as 27TAd L 1 .... . o _ 8.767rB Fu LAt mfg-2011+ __ ddo " W ° “0 0 l2 ‘* 1 in do 2 h k [C1002 d1 12 [don *ailc' 1“ 3-] 0 d1 (4)(J44)(8.76) BFu‘Atm d1 12 z — 2 Ad [1:3, - ozfio 0 Since the equation contains the logarithm, it can be solved only by some form of successive ap roximation, and does not lend itself to graphical solution. 29 Chapter IV: Simple Graphical Solutions Many cost equations may be written as equations involving one independent variable only, or reduced to such equations by approximations. Such were the equations developed in Chapter III. These may or may not lend themselves to graphical solution. Two such situations, peculiar to refrigeration and air conditioning, are treated in this chapter. (a) Economic Condenser Water Rate Temperatures may be measured more easily and less expensively than flow rates. For a given fluid, rate of flow, and rate of heat transfer the temperature rise in the condenser will be constant. Thus, the solution for the economic temperature rise suggests itself. Assuming a well designed condenser,(see Chap.VI) the only effects produced by varying the condenser water flow rate would be to change the cost of water, and to change the condensing pressure and temperature in the refrigeration cycle. The latter would create a change in compressor work, thus directly affecting the cost of operation of the compressor motor. Water costs may be simply described as: w c”: :050/ where: wf, rate of flow, lb/hr A =.cost of water, cents/1000 gallons 30 «Azdensity of water, lb/gallon For any refrigeration cycle, the rate of water flow may be expressed as a function of the Operating conditions of the system. The heat dissipated in the condenser, per pound of refrigerant is: He: (In/7g) (tc-ts)+Qr where: m =work per degree temperature difference between condensing temperature and suction temperature, °F. 77: overall compressor efficiency for the Open system or combined motor and compressor efficiency for a hermetic system. t =condensing temperature, 0F. c tscsuction temperature, °F. Qr=refrigeration effect, the heat absorbed in the evaporator per pound of refrigerant. The rate of refrigerant flow will be: - 12000 T where T = tons Since the heat absorbed by water must equal the heat given up by the refrigerant in the condenser; using the average specific heat of water as l BTU/lb.-°F: wfw (tc-tw’ ) wfr HO: wa A tw W [(%)(tc-ts)+ Qr] =weight of flow of water, lb/hr where: wa tw’:;inlet water temperature plus the terminal temperature difference. 31 Solving for the water flow rate: “1‘“ %§%%%_%; > U727”) (to-ts“ Qr] Thus, the cost of water is: _ A 12000 ‘I' (m) J W ‘ (10094) Qrztc" 't'w/ ) (tC'ts)+ Qr The work required is: Wk=Wfr m(tc-t8) __12000 T m (to-ta) Qr and the cost of work is: ka , B(l2000)T m(tc-t3) 7° (Buffer where: .B= cost of electricity, cents/KW-hr 3413==conversion factor, BTU to Kw-hr ?Q=overall motor-compressor efficiency The total operating cost becomes: CT8(T0%07) }%%2—2—_%w/) [(%) )(tc‘ts)+ Q.\]4-7 +4W'EL‘B" 1203224 c't ) = 128201. {1000/a (13‘?th [(m (t H't8)+Q1’-‘]7W7 3 it } Using methods outlined in Chapter II, the minimum cost condition may be determined as follows: d Ct- 12000 ‘I' tc-tw )(7? EL”??? ('33 -’° 5“er “a tc 1000/ (to 4“; )2 W .__s___ (mytc’tw ) - (1: CM... > - (tw’ -t. > J Qr + 1000/{6? (ta-tw’ )2 70 B3 15 32 A I!) (twl-ts) 4» Q]? - B m 1000/ [(1 (‘4th j - 37.0.7.3 13 (A tw)2 : A(3413) 70 [(2) (tw’ "ts)+ Qr] 1000 B/m 7 0 Using an average density of water as 8.33 lb/gallon, the equation becomes: 2 _ 41 A (1’!) ’ (Atw) - (8-3—38) 77‘ 1:7? (13w “'03) + bur B “m with A tw in 0F. This equation is completely general, regardless of the refrigerant, for a single stage system. Table 4-1 shows the values of Qr and‘N7for Dichlorodifluoromethane for various condensing and suction temperatures, assuming 90 superheat at the suction inlet and 9° subcooling at the condenser outlet. Figure 4-1 shows a plot of Qr with to and ts, Table 4-1 VARIATION OF REFRIGERQTIOI‘I EFFECT WITH OPERATION CONDITIONS (0F) (0F) (BTU/lb) (BTU/lb-OF) (BTU/lb) —40 80 18.47 0.15592 50.59 90 19.80 .15251 48.15 100 20.92 .1494} 45.76 110 22.11 .14740 43.31 120 23.21 .14506 40.82 130 24.30 .14294 38.31 -20 80 14.61 0.14610 52.96 90 15.83 .14391 50.52 100 17.05 .14192 48.13 110 18.16 .13969 45.66 120 19.24 .1374} 43.19 130 20.52 .1554? 40.68 0 80 11.30 0.14125 55.23 90 12.49 .13878 52.79 100 15.69 .13690 50.40 110 14.79 .15445 47.93 120 15.85 .13208 45.56 130 16.90 .15000 42.95 + 20 80 8.09 0.13485 57.65 90 9.29 .13271 55.21 100 10.46 .15075 52.82 110 11.56 .12844 50.35 120 12.60 .12600 47.88 150 15.65 .12409 45.37 -r40 80 5.15 0.12785 59.90 90 6.34 .12680 57.46 100 7.49 .12485 55.07 110 8.59 .12271 52.60 120 9.65 .12038 50.13 130 10.66 .11844 47.52 33 Refrigeration effect (BTU/ Lb.) 62 60 Suction temperature (°F) , , FIGURE 4-l , . , Refrigeration effect versus operating conditions 35 The terminal temperature difference will be essentially constant for a given condenser design. Thus, the economic temperature rise of the water appears to be primarily a function of inlet water temperature and suction temperature. It would appear, then, that the water flow rate should be controlled by a valve sensitive to this temperature difference, rather than to condensing pressure. Unfortunately, the form of the equation renders it difficult to adapt to graphical solution. However, by certain appr ximations, another form of equation may be derived which may be readily solved by graphical methods. If all other parts of the cycle are assumed to be unaffected by slight changes in condensing pressure, the part of the work affected is: m (Zitw) The water flow rate may be further simplified by using Hc to indicate the amount of heat to be dissipated in the condenser per pound of refrigerant, and treating it as a parameter, rather than a variable. The water cost would then be: - 12000 T H A . C“'(W) “1800 48—1233) ———-—-‘f T H° / . “irAt‘w The cost of the variable part of the work is: Wk 9,. 3413 7,, 36 Thus : 0 T , + 8—530 Q'I‘Atw 3413 Qr 776 : 12000 T_ + B m 413w BBUZEW 52313 7?, Minimizing: I my 12000 T - A He B m ] ._._... a 0: ———-———- + d Atw Qr 8530 (41mg 32:13 7]. (At )2:34137?‘A HQ 3 A e H w 8330 B m (0.411) _§Z_ fig 37 In order to use a plot of this equation effectively, values of Hc/m must be known. This ratio is a function of the refrigerant used, the suction and condensing temperatures, and the isentropic compression efficiency. Table 4-2 shows values of this function for various isentropic efficiencies, suction and condensing temperatures for Dichlorodifluoromethane. It should be noted that the variation within a given isentropic efficiency is slight. Considering the data for 7? of 0.9, the maximum variation about a mean value of 481.08 is 5.93%. Taking the square root reduces this variation with respect to Atw to 2.96%. This is of the order of accuracy of plotting numbers on logarithmic graph paper with 2é-inch cycles. Thus, the final equation to be plotted is: A1 (Atw)2= c B with the values of 0 being: Figure 4-2 shows the solution of this equation for a typical 10 ton system. Table 4-2 Hc/m t t ’71 5 c 0.7 0.8 0.9 -40 30 500.13 478.69 461.99 90 501.87 478.63 460.57 100 506.26 481.23 461.75 110 508.14 481.34 460.52 120 510.00 481.39 459.19 130 510.84 481.95 456.90 -20 80 505.34 487.47 473.37 90 508.16 488.57 473.28 100 510.57 489.15 472.45 110 512.56 489.87 471.33 120 514.30 489.27 469.84 130 514.58 487.78 466.97 0 80 505.27 491.04 479.93 90 508.94 492.87 480.40 100 511.03 493.13 479.25 110 513.65 494.01 478.69 120 516.35 494.92 478.27 130 516.08 492.92 474.84 4.20 80 513.31 502.56 494.25 90 516.01 503.50 493.78 100 518.24 504.02 492.85 110 520.55 504.52 491.98 120 522.86 505.00 491.11 130 522.77 503.10 487.87 1-40 80 522.41 515.26 509.67 90 524.61 515.69 508.68 100 526.88 516.14 507.81 110 528.64 516.18 506.40 120 530.74 516.45 505.32 130 530.65 514.61 503.02 515.06 496.37 481.08 38 ()1 O at) N 0 got 0 to 09 4—\ 0.”! C 30 8 Electric cost (cents/kwhr) .0 at FIGURE 4-2 Economic condenser water temperature rise (Dichlorodifluoromethane) l5 Temperature rise (°F) 4 I 40 The latter simplification leads to slightly higher values of (it. However, since the total cost equation contains the primary variable to only the first power, the minimum is not sensitive. Figure 4—3 shows a plot of total cost versus temperature rise of the condenser water. It can be seen that there is little difference in total cost for a value ten degrees different from the optimum. However, this difference, of the order of 0.26, on an hourly basis will amount to a considerable difference during the course of a year. A variation of 15 - 20° inlfit would add approximately 10% to the total yearly cost. 0 perat i ng cost (Ce nts/hr.) 3| 30 29 t j _\ f ; ,// i 26 \ r 25 e i i : /§ g 24 J ' i J/ | t 23 l , 20 30 40 50 so 70 so 90 Condenser water temperature rise (°F) FIGURE 4-3 Variation of operatinq cost with condenser water temperature rise 42 (b) Air Ducts The equation for cost of owning and operating air ducts may be reduced to one in which diameter is the independent variable. Rectangular ducts may be reduced to an equivalent diameter for equal capacity and pressure loss? so that solution for the economic diameter will lead, through proper conversion, to the economic rectangular section. The cost of material and fabrication may be expressed as a function of the area of the sheet metal involved, say in cents per square foot. The owning cost would be, then: Co -'- W cent s/year 12 where: A = ¢/sq.ft.-year L = length, feet d : diameter, inches The value of cost, A, is a discontinuous function, since it increases abruptly with a change in allowable minimum gauge. This variation can be expressed, awkwardly, as a function of diameter, but it is so insensitive when so expressed, that it adds little to the accuracy of the calculations. Thus, it will be treated here as a parameter. The pressure loss in a duct may be expressed as: r 1...,/v2 2 AP; 2 8c 0 1b/ft * A.S.H.A.E. Guide, American Society of Heating and Air Conditioning Engineers, New York, 1956, pp 737-9. #3 where: f: friction factor Le= equivalent length of straight duct, feet V: velocity, feet per second J”=density, pounds per cubic foot D: diameter, feet The velocity may be expressed in terms of diameter, rate of flow, and density, as follows: V: -3%X- feet/second where: Q==rate of flow, cubic feet per minute. A: area of flow, square feet. Substituting this in the pressure loss equation: f La!” 9 f L,,/Q2 ZXP‘W 2 D 0A '-§—-——-2 50 7 00 chA The friction factor may be accurately approximated, over reasonable ranges, by a function of Reynolds' Number. For Reynolds' Numbers between 104 and 106, a good approx- imation is:* - 0.2 g 0.2/10-2 (R°)O.2 D6.2v0.2/002 where: .A(:viscosity, lb./foot-hour V = velocity, feet per hour *Thermodynamics of Fluid Flow, Newman A. Hall, Prentice Hall, New York, 1951, pp 30-1. 44 D = diameter, feet /0= density, pounds per cubic foot Substituting the value of velocity in terms of area: v=;§%9— feet/hour f _ (021/r 0°2A°°2 - Do.2(6o)o.2Qo.a/oo.2 Substituting this value of f in the pressure loss equation. A1: =(O.2)Le flQl.8/O.8/9 0,2 (2)sc (60)2-2 91.2 A1.8 ‘3 2108 28c (60)2'2 Dl.a[H2D ] (o 2)(4)1.8Le/’o.§/90.2 Q1.8 ’ 2 . 2 2 1 8 4 8 lb/ft 23c (60) ° (77) ' D ' Since the pressure drop is usually quite small in comparison to the total pressure in ducts of reasonable length, the process of flow may be approximated as a constant volume process, so that the work will be: wk zé—Ii— ft-lb/lb. In The cost of producing flow is then: Cw 1‘ BAP wf _ cents/hour 7o A778)(3413) where: B -cost of electricity, cents per kilowatt-hour \I Wf flow rate, pounds per hour overall fan efficiency 0 bllt: Wf :60 Q/ 45 Thus the annual work cost is: 8760 BF “Wk ‘ 313 77 7"“ 6% P (0.2)(4)1-8 1.6/0.8/1 0.2 Q2.8 (8760) ch (60)1.2(77)l.8 (34.3)(778)170 94-8 cents/year where: Fuzzuse factor, fraction of time system is in use. Expressing diameter in inches, and flow rate as Q/lOOO, a number of more convenient magnitude: : 4.422 x 106 Bpufl (9.9/4, 0.2 Le (Q/lOOO)2’9 cents/year Wk 70 (14.8 C The total cost of owning and Operating the duct is: 4.422x106 BFu /0.8/,o.2 Le 77 O d&.8 Minimizing cost with respect to diameter, in accordance with _ AL7Td (Q/lOOO)2'8 cents/year the methods discussed in Chapter II: dCT ALTT 2.123 x 107 BFuPO'B/‘ma L. w 2 8 15‘0: 12 770 (153 (41/1000) d5-8= 8.106x107 BFuPO‘BfiO'e Le (Q/1000)2'8 A720 ('17) The group of factors: 8.106 x 107 /0'8/”°°2 is a function of operating conditions only, and primarily of temperature only at low pressures. These may be combined as ¢(t)' so that the equation becomes: d5'8 = 4%) B F11 (Le) (c/1000)2°8 p. ’70 46 The factor ¢(t) is shown in Table 4-3 for various temperatures at atmospheric pressure. The cost factor A should include cost of insulation, since the relative cost of heat loss for various size ducts has not been previously considered. As an approximation, a large duct may be treated as a flat surface for purposes of estimating the economic insulation thickness. For example, consider a duct which is to carry 3000 cubic feet per minute of air at 60°F. in a room at 80°F. Using mineral wool, the economic thickness is approximately: x=o,2 (8760 13 0.2 20 - 0.2 7 J (120! 0' 0" ;:(0L.0"'2‘)‘§'_125 “311, 5 r l .19“ The same costs were used as in the example of Chapter 111(3). Thus the thickness of insulation used would probably be 1 inch. The cost per square foot of duct surface would thus be increased by approximately l¢ per year over metal and fabricating costs. The approximate cost of metal and fabrication is 2¢ per year per square foot, so the total cost factor A will be approximately 3¢ per year per square foot. Using B==2¢ per kilowatt-hour, Fh==0.7, and.'?fl-O.3, the solution for straight duct (£3 = 1) is: L d- 26.78 inches. Figure 4-4 shows the effect of diameter upon total cost for the example.ch0sen. It should be noted that the curve is quite flat for about three inches of diameter, but Table 4—3 VARIATION OF ¢(t) WITH TEMPERATURE (3F) (lb/ff—hr) (lb/id) ¢ (t) 40 0.043 0.0793 5.688 x 106 60 0.044 0.0765 5.540 x " 80 0.045 0.0734 5.395 x " 100 0.046 0.0708 5.265 x " 120 0.047 0.0684 5.144 x " 140 0.044 0.0661 5.047 x " 160 0.050 0.0640 4.948 x " 180 0.051 0.0620 4.853 x " 200 0.052 0.0601 4.735 x " 47 I; l .l l I III. ll " iv ll! [lilt- ! I i 1 iii- 2 113-125; lll,ltl.tII|.T.ll-l|i -lll iii.--liL -3 It‘ll 11"ij llli Ill iii 3 L i i 1" I‘ll".itltlll"l tlr 'l‘llu.‘l '11."! Ii} I 40 30 (m) Diameter 20 \ |.| liillt' IIIIII 4. i.tl I - 3: ti-..- 1 I I l i i i i l 1w!ill-..| ill! It'll \\I‘ o 411 ii lili it: w illli i I it IiiiiililT il Iii 0 O 0 0 0 O m 0 0 0 0 0 0 0 0 5 0 5 O 5 0 5 0 6 6 5 5 4 4 3 3 Curiae": «moo .93... 250% FIGURE 4-4 Variation of cost with duct diameter 49 increases above and below these values rather rapidly. Thus, the choice of diameter is rather critical. A choice two inches on either side of the economic diameter will result in less than four per cent increase in total cost, while a choice four inches on either side results in approximately ten per cent increase in total cost. Figure 4-5 shows the solution of the previous example. This offers the general solution for pressures not differing appreciably from atmospheric. 4 There are many other mechanical systems which may be treated in the manner shown in this chapter, but their presentation would be a repetition of similar procedures. If the equation for total cost can be reduced to a pair of functions of one variable, the methods illustrated should serve to lead to a general graphical solution. Duct diameter (inches) FIGURE 4‘5 Air duct sizing 5000 6000 (cu.ft./min.) 2000 Air flow I5 3 2 04 o 2 CL 3-5.3.53 2-5.3333 .aEmCE Boo 038E .moo Ego _ _ _ . _ . . . . 2 8 6 4 3 2 m w 0. m 0 0. 0. 0. 0 5.63 ea: 656:3 co... Chapter V. Simple Graphical Solutions with Incremental Variable - Pipe Sizing Selection of pipe for minimum cost operation does not differ essentially from selection of air duct size, except that pipes are available only in standard incremental sizes. Some difficulty is encountered in expressing the cost of pipe as a function of diameter. However, a good approximation may be made by assuming that the cost may be expressed as "A" cents per foot per inch in diameter per year. As before, the cost is the sum of the owning costs and Operating costs. The owning cost is: Op: ALd cents/year The operating cost is the cost of forcing the fluid through the pipe. The pressure drop due to friction may be expressed as: A 1P= fL. / V2 #/ft2 2613 where: f = friction factor L = equivalent length, ft. P.- density, lb/ft3 V =velocity, ft/sec. D =diameter, ft. The work required, since the change in fluid volume is small, is very nearly: wk g A? — §%§. ft-lb/lb 52 The friction factor, f, may be approximated, as in the case of ducts, by: a ' (Rab so that the work is: b 2-b . - a1! LV __ "k - agnpbwsooyb/b ft #/# In most cases, the work is done directly or indirectly by electric motors, so that the work cost may be related to electric costs. On a yearly basis, the work cost is: (8760) B Pu wfflb Le v2"b CWK‘ . b 17b b (341J)(778)(2s)(36oo) 7°12 /’ where : B selectric costs, ¢/kw-hr. Fa: use factor, fraction of time system is in use. Wf - weight of flow, pounds/hr. o : overall pumping efficiency The velocity is related to the weight of flow, flow area and density by: 2 Wf 1/AV: ff DV 4W 4w V: ____2f ft/hr - 1' fVD ' /fi62(3600) ft/sec ._ -b Wk‘ (8760) a s Fu wffl'b L,(4)2 ‘3 wf2 (3600)2(3413)(778)(28)7o DMD/27,24: D4-2b :. (8760)(4)2‘b(12)5‘b amen/(b L wf3'b (3600)2(3413)(778)(zgme-b 70/2 <15“D 53 In order to determine the value of a and b, the charts on pages 30 and 31 of Thermodynamics of Fluid Flow, by Newman Hall, Prentice Hall, New York, 1951, were used, as well as other references. The values will be discussed for each case. The total cost, in cents per year, is as follows: m + (8760)(4)2'b(12) 5"basrugb L,wf3"b 2 2-b ' 2 S—b (3600) (3413)(778)(2s)7T 70/ a Minimizing, with respect to diameter: ct. (L°t}AL_ (5-b)(8760)(4)2-b aBFuflbLe wa-b(12,5;b (3600)2(3413)(77g)77-2-b 70/2 d 6—}: M Q; = (5-b)(8750)(4)2'b(12)5'baBFu’qu°wf3-b (3600)2(3413)(778)(25)(77)2-b/2 L (12.522 This may be written: 3-b ¢(EE) BFu wf Equation 5-1 L —' ‘73 d T h 3 w m (amazeomf'b (12)5"°/(‘° (8600)2 (3413) (778) (25) 7r 2"? 2 Note thati¢ is a constant times a factor which is a function of the operating conditions only, and may be evaluated with respect to temperature and pressure. This equation is valid for all systems involving only direct work which may be chargeable to electric power, or fuel cost, and in which the work may be accurately approximated ,4? as 73—3 Special cases applying this general equation, as 54 well amz<3ther situations will be considered in the remainder of this chapter. (b). Water Piping Since steel pipe is used extensively for water service, the ikurtors a and b may be approximated as 0.185 and 0.23 respectively. The general equation may be applied, since 'this 1J3 a simple problem of balancing pumping costs against pipe costs. Thus: . 4, _lawnsvso)(4)1'7711s24~77<77s)(64.4) 1-77 Jig; (7.526 x 10"?) The general equation (S-l) becomes: 2.77 A(12.885=¢ BFu wf (.112) L 7? 2 .8 85 d Figure 5-1 shows the solution plot for this equation. Approximate values of "A" are shown as a curve on the pipe cost scale. It should be treated as an approximation only, since the values of cost for a given diameter vary from locality to locality and year to year. In addition, variation of the number of fittings for a given length will cause a variation in the unit cost. One method of approximation which yields reasonable results is to assume that the cost of fittings varies directly as the number of equivalent feet for pressure loss calculations. This assumption can be shown to be fairly accurate for an average number of fittings. The result of this assumption is to cause the ratio Le/L to be equal to unity. Wotertemp. L/Le :3 (°F) 300 3500 2000 3000 4000 Weight of flow (lb/ht) 6000 8000 i 4 FIGURE 5-l Economic wafer pipe size l5.000 20, 000 40.000 60.000 IO0,000 | 4 IE 2 Nominal pipe diameter (inches) 2 I. 2 Use factor Pump eff. -5 *5 8g 4: 0| 03 ...x r..\ *‘2 O c _20 “pg 56 The value of "A" can, thus, be estimated with good accuracy, by calculating the total cost of fittings and pipe for an average run; adding an installation cost; and dividing by diameter in inches and by the expected life in years. Table 5-1 shows the calculation of 47with temperature, at atmospheric pressure. Since there is only a negligible change in viscosity and density for liquid water under the influence of higher pressures, a liberal extension of the use of this table for higher pressures is Justified. The example shown is for a flow of‘3000 pounds per hour of water at 40°F., with an electric cost of 2¢ per kilowatt hour, a pump efficiency of 0.7 and a use factor of 0.9. Figure 5-2 shows the effect upon total cost of choosing oiiher diameters for this example with an equivalent length of? 100 ft. Note the effect of choices smaller than the economic size. If the water is supplied at an initial pressure, so that Iluuping is not required, the economic size of pipe is the Shhallest which will pass the required weight of flow 111Lillizing the available pressure. Using the pressure drOp e"quation: 2 AP .- fie-‘e. a": ”“2 fL v2 ‘ ‘31s?— ”“2 TABLE 5-1 @(t) for Water Piping 57 “75.?” 8:32:73, (has 4"“ 32 4.33 62.42 2.706 x 1610 40 3.75 62.42 2.617 50 3.17 62.38 2.522 60 2.71 62.34 2.436 70 2.37 62.27 2.368 80 2.08 62.17 2.305 90 1.85 62.11 2.247 100 1.65 61.99 2.197 110 1.49 61.94 2.157 120 1.36 61.73 2.119 130 1.24 61.54 2.089 140 1.14 61.39 2.059 150 1.04 61.20 2.027 160 0.97 61.01 2.008 170 0.91 60.79 1.994 180 0.84 60.57 1.971 190 0.79 60.35 1.957 200 0.74 60.13 1.942 Total cost (cents/yr.) isoo I ISO l 3 i400 _.___. ..... IZOO 1 l IOOO l 800 T i i, / 600 // \PL/ 200 .1L .3. l. l. J. ‘3 2 4 ' 4 '2 2 22 Nominal diameter (in.) FIGURE 5-2 Variation of cost with diameter ‘where: f :friction factor Le=:length, equivalent ‘20: density, #/ft3 V: velocity, ft/sec D =diameter, feet d =diameter, inches Using the previous approximation for the friction factor: AP: o.185//°-23 lag/V2 . (0.185)(12)0-23 fl0.23 Le/0.77 v1.77 245 (3500)0.23 al.23 fa flfldat3600) : (0.185) (576)1-77(12)0-23/9 0.23Lewf1.77 248 (3600)2 77‘ 1077/ alt-77 ft/sec ASP Thus: c14.77= (0.185)(576)1-77(12)°-23/( 0-23L,wf1-77 24s (3600)2 rr 1'77 /4 1> : -7 flo.23 1.. £1.77 [(3.317 x 10 ) 7—] AP L. (W137 ¢ A4,— Table\512 shows the values of ¢>for various temperatures. Figure 5—3 shows the solution of this equation. The example shown is for 2000 pounds per hour of water at #OoF, with an available pressure loss of 20 pounds per square inch. TABLE 5-2 @(t) for Water Flow with Given Pressure Loss WategF‘gemp . 4’ (t) 32 8.38 x 10-9 40 8.12 50 7.82 60 7.56 70 7.34 80 7.11 90 6.93 100 6.78 110 6.62 120 6.51 130 6.40 140 6.29 150 ' 6.18 160 6.10 170 6.03 180 5.95 190 5.88 200 5.80 60 for lithe. I500 2000 Water temp. (°F) 6 ( |b./sq.in.) w w 0 o) L 3 w w d) L & Weight of flow (Iii/hr.) 3000 4000 6000 8000 20.000 60.000 I00 gth In.) Equivalent Ien Nominal diameter (inches) FIGURE 5-3 Woler pipe for specified pressure loss V" “'5‘: ”.st I”. 1" '_1 . ~v|E 1.7": u. .3 afar, «49‘ (c) Compressor Discharge Lines-Dichlorodifluoromethane. Assuming that the effect of pressure loss in the compressor discharge line is only to increase compressor work, with no appreciable change in efficiency, the general equation may be used. I In this case, copper tubing is in fairly general use, so that the values of a and b used in determining the average value of the friction factor should be 0.0653 and 0.228 respectively“. For these values, the solution becomes: . 2 , (3600)2(3413)(778)(297157727L/202-885 However, the weight of flow may be related to the tonnage of the system by the relationship: 2O wf=1130rT The equation, thus becomes: 8 255.} PT .1772 Ad2.886 _, (255.3)2°772(4.772)(8760) (4)1~772r12)4-77fl.06531551ev 7m. ' (3600)2(3418)(778)(25) 7’1.772,L/2d2.886 01": 2.886 Le BF T2'772 M = 4’ -- J "682. 6 L 77 a 4); (6.2863 x 104) where: I2.772 A 0.228 ,4 2 Values of4are shown in Table 5-3. The solution of the ”nation is shown in Figure 5-4. The example shown is for “Refrigeration and Air Conditioning, R.C. Jordan and G.B. Priester, Prentice-Hall, New York, 1949, p. 151. 62 Condensing Tem . (02’? 80 90 100 110 120 130 TABLE 5-3 Tem (0F ~40 ~20 0 20 40 ~40 -20 0 20 40 ~40 ~20 0 20 40 -40 ~20 20 ~40 ~20 20 40 ~40 ~20 20 40 $(t) for Discharge Lines Suction go 63 ¢(t) 1.423 1.208 1.024 8.511 7.394 1.284 1.068 8.970 7.53 6.491 1.157 9.389 7.772 6.447 5.457 1.056 8.522 7.086 5.962 5.023 9.823 7.992 6.390 5.272 4.451 8.836 6.830 5.900 4.779 4.007 X>< N>¢N N>(1 -77 ) m (t -t8) X :11 [73757374 P ] AC12.886 = $513er .2) T2. 772 d2. 886 where: ¢:[l§7602(3.1221(2§§./3)(4. 7.721/0 2281'2'772] 1 4.772 (1- ) (t -t ) X[77W+ an1 m c 8/ 2762/(400228 r. 20772 __ +(4. 772)(1-7m )m(tc c-ts)-/ : —WW 1/4 77 79/' P1 Note that ¢ is a function only of the operating conditions, and is, thus, a parameter rather than a variable. Table 5-4 shows this parameter for various suction and condensing temperatures. Figure 5-5 shows the graphical solution of the equation. The example is for a 5 ton system Operating at 100°F condensing and.40°F suction, with an electric cost of 2¢/KW-hr., a use factor of 0.8, a compression efficiency of 0.8 and a ratio of length to equivalent length of 0.8. The choice would be 1% inch nominal tubing. Note the increased effect of Operating conditions on the economic line size, compared to that upon the discharge line selection. TABLE 5-4 4} (t) for Suction Lines Suction Condensing (p ) Tem . Tem . (t (°F7 (°F§ -40 80 3.288 90 3.950 100 4.721 110 5.776 120 6.970 130 8.704 -20 80 0.938 90 1.123 100 1.345 110 1.623 120 1.967 130 2.390 0 80 0.303 90 0.362 100 0.434 110 0.523 120 0.628 130 0.779 20 80 0.0986 90 0.119 100 0.146 110 0.172 120 0.207 130 0.252 40 80 0.0372 90 0.0449 100 0.0542 110 0.0655 120 0.0791 130 0.0961 70 6.63 mm: e._\._ 0 8 6 4. I. O 5 i | 253354 IOO 2 80 i l-L 4 2 Nominal tube diameter (inches) 50 so i l 40 3O 20 IS 6789|O (tons) SYSfem capacity 5 4 0.5 A..__._-3x\mEoov 356:3 .moooEoflm cEmmwano FIGURE 5—5 Suction line sizing 72 (e) Liquid Line SiZing-Dichlorodifluoromethane Here, the problem is similar to that of service water piping, where a given pressure may be dissipated. In this case, the amount of subcooling obtained in the condenser or receiver, represents the maximum pressure differential which may be dissipated without obtaining flashing in the expansion valve. The pressure equivalent of subcooling depends upon the condensing temperature and the number of degrees of subcooling. Figure 5-6 shows this relationship in terms of feet of head. If a vertical rise exists between the condenser and the expansion valve, this must be subtracted from the equivalent rise to give the not head available for pipe friction loss. The relationship between subcooling and head is sufficiently near linear that only small error is obtained by assuming linearity, that is: AH = A Atsub at any given condensing temperature. Also: 7 ' /2 4.772 d so that: Ali/2 1.772 : ¢ Lo '1' AH 20 ,/ / / c f/ r m //// 4 47:. ill/Jlllj .1 in] Q +7 w {W Wt . 8 1 1 .. w W i, 1 4 H M . _ / 4 2 _ t b 0 a m a m w m. m. m o 522v 32. 23:24 9 Subcoofin (’F) FIGURE 5-6 Head equivalent of subcooling 74 The values of (Dfor various operating conditions are shown in Table 5-5. Figure 5-7 shows the solution of the equation. It should be noted that undersizing of the line will produce flashing at the expansion valve. Thus it is advisable to choose the size above the solution if the solution falls between two nominal sizes. TABLE 5-5 4m.) for Liquid Lines Suction Condensing Tem . Tem . @(t) (0F (0F -40 80 4.045 x 10-4 90 4.523 100 4.692 110 4.895 120 5.125 -20 80 3.731 x 10'4 90 4.154 100 4.291 110 4.458 120 4.638 130 6.044 0 80 3.463 x 10-4 90 3.842 100 3.945 110 4.091 120 4.219 130 5.620 20 80 3.210 x 10-4 90 3.550 100 3.639 110 3.749 120 3.863 130 4.972 40 80 2.999 x 10'4 90 3.307 100 3.380 110 3.469 120 3.562 130 4.643 75 TABLE 5-5 $(t) for Liquid Lines Suction Condensing Tem . Tem . (Mt) (or? (OF —40 80 4.046 x 10-4 90 4.523 100 4.692 110 4.895 120 5.125 130 6.826 —20 80 3.731 x 10'4 90 4.154 100 4.291 110 4.458 120 4.638 130 6.044 0 80 3.463 x 10-4 90 3.842 100 3.945 110 4.091 120 4.219 130 5.620 20 80 3.210 x 10-4 90 3.550 100 3.639 110 3.749 120 3.863 130 4.972 40 80 2.999 x 10-4 90 3.307 100 3.380 110 3.469 120 3.562 130 4.643 75 Allowable head loss (feet) 9,25 System capacity (tons) 4O 50 60 80 FIGURE 5-7 Liquid line sizmg 600 800 I000 hes Nominal tube (inc diameter IOOO (feet) IOO Equivalent length Chapter VI: Tubular Heat Exchanger The determination of the length, diameter, and number of tubes for a tubular heat exchanger is somewhat more complex than problems previously discussed, particularly from the difficulty in assessing owning costs. Assume that the cost of the shell and tubing can be expressed as: 01 :AlNLd and that the cost of making the end connections can be expressed as: 02==A2N The cost of friction is the same as in previous piping problems, assuming cOpper tubing, except that the number of tubes is not necessarily one. As before (see Pages 51-2): "k =%‘-%3 ft-lb/lb : I‘wav2 2gD (0.0653) wav2 (33,0.228 26D (0.0653)/’r0'228 L wt. vii-77?- 72% (3600)0.2287f1.772/o0.228 131.2be but v = W: __ Wf 4 ‘ 3500/“ " 3600 /N n02 ft-lb/hr Substituting this value of v ; (0,0553y/40.228 L Wf2'772 (4)1.772 (28)(3600)27/ 1-772/2 “1.772 D4.772 "k 77 78 Thus: =(0.0653)(4)1'772(8760)(l2)4.772/40.228wa2.772 7] (28)(3600)2(3413)(778)771-775/2 1715772 oft-772 For the sake of generality, let: L cwk ‘91 d4.772N1.772 Thus, the total cost may be expressed as: c e A NLd1iA Nail» L t 1 2 1 d4.772N1.772 (6-1) Two constraints apply to the system. First, all of the variables must have values greater than zero. Second, the heat balance must be satisfied. writing the heat balance: - : UNIVdL chAt - UAAtm T- Atm (6-2) where: Ni. 3 weight of flow, #/hr. 0 = specific heat, BTU/lb-o OF A t = temperature change of fluid, U: overall coefficient of heat transfer, BTU/hr-ft2-0F zitmz-logarithmic mean temperature difference, OF. It is assumed that the ratio U/hi, where 111 is the heat transfer coefficient at the interior surface of the tubes, can be approximated from knowledge of similar systems, and U is the overall coefficient, defined as: U:= 1 resistances U can then be expressed in terms of N, L, and d as follows: 0. h1 = 1%; 0.023% [Re]0°8 [P1707 fl“ where: %%¥?, dimensionless —§Z , dimensionless 0.8 UO8/O.8 004'fioe4 U ; LL 0.023) [E f3 V j C .6 0.8 0.4 0.8 t UHF] H no.2 where: Re: Reynolds Number 0 Pr"Prandtl Number Thus: K; conductivity of fluid, BTU/hr-ftB-OF/ft c = specific heat, BTU/lb-OF fl: density, lb/ft3 v = velocity, ft/hr ,4/:viscosity, lb/ft-hr D : diameter, ft. but: My: wf 4 _ wf(576) fA :/1\117D"’5 .fofdQ and: u.- 3%.. (0.03)(12,0.2(4)0.800.4Ko.6wfo.8 1 (3600)0-8(25)°-8770-8 N0-8d1.§/90-4 Substituting for U in Equation 6-2 and simplifying: N0.2 L (10.8 : ¢2 This is the form of the second constraint. Using Lagrange's method, as given in Chapter II: a1?) _ A m + _ 1.772 ¢1L 0.2 XL - 1 A2 4’ :: (3N le N2.772d4.772 No.8do.8 O 80 (1) A1N2 772L d5-776 +112 112'772d4 772 1. 772¢1L o. 33311909721313 972.0 :A Nd no. 2 )g(-%)N1d 15+W2+W = 0 (3d,) N 1L:A1NL' 4 772611}: 0.8 mm"-2 (3) A1212'772d5'772 .. 4.772 451 .. 0,8 XN1-972d3-972 : 0 These three equations, and the second constraint: NO'2L é d0.8 : form a system of four simultaneous equations in N, L, d, and b’ . Combining (2) and (3): ¢1* {N1.972d3.972 = 4.772% _ 0.8N1.972d3.972 1.8 X 111-972d3-772 = -5.772 4’1 X- -3.267¢1 "N1.972:3}972 Substituting this value in (2): 2.772 5.772 - AlN d + 431 - 3.267 ¢l - N2'772d5'772 : 2.267 92 A1 Substituting this expression in (1): 2.267 ¢ L +2.267¢1A 1 2 .. 1.772 L + 0.2) -3.267 )L: 0 -0.158 ¢1L+2.267 4’1 A2 d Al '7 0 Ld - — 0.15? A1 ' 14348 3'1“ 81 Combining this equation, and the constraint: (10.8 L : ¢2 N002 Ld= 14.348 32. ¢ 91.2.5.3. A1 ' 2 110-2 110-2: ¢2A1 (11.8 (14.3483112 N: ¢2Al 5 d9 11.348 A2 But: N2.772d5.772 : 2.267 £21 A1 (15.772: 2.267 $1 A1N2'772 .- 2-2574>1 14.348 A2 ”'86 1 A1 492% d24.948 <130.72 : 2.267% 14.348 A2 13-86 A1 ¢2A1 0.0 26 0. d _, (2-257) 3 451 0326 19.348A2 0-4512 A10.0326 2A1 0.0326 0.4512 - 3.416 $11 A2 0.4838 0.4512 A1 4’2 L _, 14.348 12 A16. and N : (¢2Al ) 5 (19 14.348 A2 5 5 1.650 x 10-5 ¢2 A1 d9 A25 82 These are left in this form by intent, since the solution for d will not necessarily yield a nominal diameter. Thus, the nominal diameter nearest the ideal should be used in the equations for L and N. In order to examine these results, consider as an example the following: Water at an average temperature of 60°F. Wt =3600 lb/hr. At. _ 1 7: 0.8 Atm . F11: 0.8 A1. 2(tube cost) :2 13.72¢/in-ft-year A2: 1 cent/tube-year hi - D -1 U -2 ft B = 2¢/Kw-hr From the previous chapter: (pl;- .0653 (8760)(4)1-772(12)4-772 BFu fi0.228wf2.772 (3600)2 (3413)(778)(2g)77‘1o77277/02 fi0.228 Wf2.772 3.652 x 10-8 B Eu We and (0.023)(12)°-2 (4)0-8 x0-6 {1— Atm 0.6 0.4 ‘ 1553 0 {3‘6 “1"” .131 4.12.. K ’ U At m 83 For water at 60°F: j“ 2.71 lb/ft-hr /= 64.34 1b/rt3 K = 0.344 BTU/hr-ftZ-oF/ft c = 1.000 BTU/lb-oF For these values: 451. 0.160 $2 = 45.43 d g13.416)(0.16O)O'032611)O'4512 (13.72)°-4838 (45.43)°-‘*512 = 0.1804 inches This is smaller than commercially available 1/8 inch nominal copper tubing, which has, for type K, an inside diameter of 0.185 inches. If 1/8 inch were used: _ 14. 48 1 - L - ET3T7§7%éT%867_ - 5.65} ft. m = (1.65 x 10’5)(g5.43)5(13.7215 (0.1gg)9 (1) and 42 tubes H Assuming the same conditions, with W: = 360,000 lb/hr t1. 5.589 x 104 ¢ 2 -‘ 114.1 d.: l3.416)(8.559 x 10510'0326 £120.4512 (13.72)0-4838 (114,1)0.4512 ; 0.1805" This is almost identical to the previous result. Thus L would have the same value, and: 5 114.1 1 . 2 9 (109.1)5 (0.185)9 4,120 tubes A check of Reynold's numbers indicates, for the first case: Re==2644, and for the second case: Re: 2693. For both cases, the flow should be laminar, rendering the use of the McAdams equation for heat transfer invalid. In addition, there has been little study of heat transfer in the diameter range encountered here, so that the validity may also be challenged on that count. The assumption was made that'Ei may be approximated in liquid-liquid heat exchangers, using larger tubes. In the case of liquid-gas the ratio would be of the order of 100. The effect of a change to this ratio would be to reduce d to approximately one seventh of the values found for liquid- liquid for the same rate of flow. The number of tubes would 'be increased by a factor of approximately (1)9 (50)5, or '78. The length would be increased by about77 to l. Doubt as to the validity of the assumption of Bi :notwithstanding, the trend of solutions is obviouslg toward a large number of very small tubes. Since the ZflPoblem of support of such tubes is a difficult one, the (Knnpact heat exchanger composed of expanded and/or corrugated plates appears to offer a step in the right direction. Keys and London* have performed studies of compact exchangers, but the tubes used were fairly large compared to the solutions found in the examples previously offered in this chapter. It is obvious that, until a general equation for corrective heat transfer is evolved, there can be no general solution for the economic configuration of heat exchangers. In each category, the empirical equation which most nearly fits the situation should be used to evaluate the expression: 0 = %-1- hi(N,L,d) The solution for a minimum cost can then be carried out by the methods of this chapter. * Compact Heat Exchangers, W.M. Keys and A.L. London, National Press, Palo Alto, Calif., 1955. Chapter VIIi Summary The preceeding work has opened many opportunities for further study and re-examination of proper choices in such components as ducts, piping, etc. The methods developed are not limited to the problems discussed, but have widespread application in the field of engineering economics. It is hoped that the methods are so presented as to enable engineers to make use of them in their own applications. A summary of the results is included in this chapter. (a) Effect of choice other than economic size. The general behavior of the total cost with the primary variable is easy to deduce from the cost equation in the case of the single variable. In the case of condenser water, the variable appears as (st and (£%—), so that the variation of cost for choices above and belo: the optimum will be lines with equal slopes on logarithmic paper. Thus the minimum would not be expected to be sensitive to changes in the variable. Figure 4-3 shows a typical cost curve for this type equation. The form of the cost equation for planar insulation is of the same form in the variable,thickness, so similar results would be expected. However, for cylindrical insulation, the first term involves the variable to the second power and the second term involves the first power and the logarithm. The generalized equation for the variable part 86 87 of the cost is: CTV:¢1 (102 +¢2 do 0 d In d 0 o The latter term varies approximately as the reciprocal logarithm, so that it would not be sensitive. In this case, oversizing would be more expensive than undersizing, since the first term is in the square of the variable. Flow problems may be reduced to an equation of the type: CT =¢ld 1'43...— d4: Oversizing will approach a first power increase but undersizing will approach approximately a 4.8 power, or very steep, increase. Figure 5-2 shows a typical variation with an equation of this type. With a four-dimensional equation such as that for a heat exchanger, it is virtually impossible to determine by inspection the effect of varying any one of the variables, since, in this case, they are not independent. It would be necessary to investigate each by successive trials. (b) Comparison with published tables. This is not meant to be a reflection on any individual or individuals, since the tables to which referred are of unknown origin. Many tables have been reproduced so often as to lose all reference to their origin. 88 Each of the situations discussed previously will be compared to available data. (1) Insulation. No tables have been encountered covering economic thickness of insulation. A nomograph for the solution of economic pipe insulation was found *. The form of equation used is unknown, and comparisons have not been made. (2) Air duet sizing. There are two possible bases of comparison: pressure loss and velocity. For the parameters used in the example on Figure 4-4, the following holds: Diameter Velocit Friction loss CFM (inches) (ft/min. 4 (in.water/lOO ft) 300 9.5 ' 608 0.13 3000 27 754 0.028 10000 53 553 0.008 These velocities are within the ranges recommended for residences.** (5) Condenser Water. No tables or charts comparable to that deve10ped have been encountered. However, the equation used is similar to that of Jordan and Priester,*** and gives comparable results. * Fundamentals of Power Plant Engineering, George E. Hemp, National Press, Millbrae, Calif., 1949, p. 197 ** ASHAE Guide, 1956, p. 747 and 735. *** Refrigeration and Air ConditiOning, R. C. Jordan and G. B. Priester, Prentice-Hall, New York, 1948, P 244-5. 89 (4) Water Piping. Perry* gives a nomograph for generalized flow, derived from a more simplified equation. The results are near those obtained from Figure 5-1, but not so accurate. (5) Discharge Lines. Using Figure 5-4, and the parameters used in the example shown, the following is a comparison, for a condensing temperature of JOSOF, with Table A.26, Jordan and Priester**: Line Size Tons Capacity Tons Capacity (Nominal (Fig. 5-4) (Jordan & Priester) type K) 3/8 103 "‘ 1/2 2.2 1.43 3/4 5.1 2.97 l 8.0 5.05 1 1/4 14.9 7.72 1 1/2 22 10.92 2 39 19.2 2 1/2 61 32.2 3 93 51.5 3 1/2 135 72.0 4 180 95.8 The higher capacity shown from Figure 5-4 may be partially explained by the fact that the additional work imposed upon the compressor is such a small fraction 3 Chemical—Engineers Handbogg, John H. Perry, McGraw~Hill, fiEfi"Y5fE,.195o, pp 384-6. **,§g§riggration and gig Conditioning, R. C. Jordan and G.B. Priester, Prentice—Hall, New Ybrk, 1943, p. 491. 90 of the total work as to be of minor importance compared to the cost of the tubing. The increased cost of pipe compared to the relatively stable cost of electricity may also be one contributing factor to the increase in apparent capacity. It should be noted that the cost of electricity used was 2¢/Kw;hr. If electricity is more costly, these capacity values will be reduced, and vice versa. (6) Suction Lines. Comparing the results of Figure 5-5 with the same table: Tube Size Tons Capacity Pressure drop per (nominal) (105° Condensing 100 ft.(psi) for 40° Suction) comparable capacity, . Jordan a Priester 3/8 0.27 2 1/2 0.45 2 3/4 1.0 2 l 1.65 1 1 1/4 3.0 0.6 1 1/2 4.4 0.8 2 7.7 0.7 2 1/2 13 0.6 3 18 <0.5 3 1/2 26