Units conversion is often necessary in calculations

Transcription

1 Easy Units Conversion Methodology Igathinathane Cannayen, Departent of Agricultural and Biosystes Engineering, NDSU, Fargo, ND Units conversion is often necessary in culations as any types of units were frequently followed to represent the sae quantity. It is soeties confusing to students to ake this conversion fro one unit to another. A siple ethodology that handles units along with the nubers helps to eliinate the confusion. This ethod can be easily extended logily to ost coon units conversion situations. This paper introduces the principle of the ethod, deonstrates the ethodology using several exaples, outlines the liitations, and infors of the online units conversion tools available to users. This siple ethodology of using the units along with the nubers o ers a way of check and guide the units conversion process. Thus the process of units conversion should be perfored along with the units. The purpose of this paper is to introduce the ethodology, deonstrate the ethodology with nuerous exaples fro siple to advanced conversion, outline the liitations, and point out the availability of other resources of units conversion. 2 Principle of Methodology Introduction Often a physi quantity (represented by a cobination of nuber and unit) needs to be converted fro one unit to other to express it in other fors. For exaple, in US, ass is coonly expressed in pounds () while in kilogras () internationally; siilarly the liquid volue in gallon (gal) as liter (L). Although unit conversions in the for of relationships, charts, and online resources are available, it is useful to understand the logic so that it is possible to convert fro one to any another relevant unit as required. Soeties it causes confusion in the ind as to Whether to ultiply or divide? especially when one knows the conversion forula and works only with nubers, without considering the units in cobination. The drawback in working only with nubers is it will not o er any way of checking the correctness of culation. One siple way of perforing correct conversion is factor-label or unit-factor ethod (Wikipeida, 203). It should also be understood that only a nuber does not ake sense unless accopanied by units (e.g. as a physi quantity 8 akes ore sense than siply 8). Let us consider the following atheati expression: A B () It follows fro the above equation that: A B or B A (2) Which eans that the ratios (A/B or B/A), being unity, when ultiplied or divided with any other quantity will not change the value of that quantity. Extending the analogy to product of two quantities, using Eqn. () and Eqn. (2): A P B Q A P B Q or B Q A P (3) This analogy (Eqn. (3)) can be applied any physi quantity represented by a cobination of a nuber August, 203 Page of 5

2 and an unit, as the nuber and its unit were considered as a ultiplication (product) of these ites. As an exaple, for a coon physi quantity we have: (4) Thus, this unity (Eqn. (4)) representedbytwonubers and their corresponding units, when ultiplied or divided to any other nuber and its unit will not change its value of quantity. This logic fors the ain principle of the units conversion ethodology discussed in the work. As the principle applied is the sae, the ethodology can be extended to other siple or copounds units readily that are not discussed in this work. It is also necessary to know soe of the coon conversion factors to perfor the conversion. These units conversion forulas are available through several standard sources (e.g. textbook appendices). 3 Useful Coon Conversions Soe conversion forulas frequently used are presented in Table. It is not necessary to reeber or have the two-way conversion forulas ( ) ; and ) ), as in the conversion ethodology one conversion forula is su cient. 4 Siple Conversion Exaples Exaple 4. What is the equivalent of 50 pounds () in kilogras ()? The conversion forula known is:. Following the ethod outlined earlier (Eqn. (4)), the cobination ratio (nubers and units) fro the conversion forula can be ultiplied with the physi quantity to be converted, which cancels relevant units, and arrive at the results. Now 50 when ultiplied with the ratio of unity: Upon canceling the relevant units and ultiplying the nubers: (5) In Eqn.(5), correctly using the second ratio of unity Eqn. (4) the final unit was worked out as. Caution: Had the first ratio of unity ( / ) used will lead to: Even though the nuber 0.3 did not provide any way of cross checking, the resulting unit of 2 / is clearly not intended - aking the conversion incorrect. Table : Coon conversions Physi-quantity Basic-unit Conversion Mass: 6ounce tonne g Length: 3.28 ft ile.609k in yard 0.94 Area: ile acre acre foot 2 ha 2.47 acre ha Volue: gal liter ft liter bushel 8 gal 3 000liter Pressure: bar 00 kpa psi 6895Pa kpa 000N/ 2 Energy: Btu kwh 3.6M Power: hp kw kw 000/s Tie: h 60 in in 60s Prefixes: kilo 000; ega illi /000; centi /00 August, 203 Page 2 of 5

3 Exaple 4.2 How any pounds will be 00? Using the conversion forula and appropriate ratio of unity following Eqn. (4) it follows: Advanced Conversion Exaples Exaple 5. What will be the area of 200 acres (a) in square iles and (b) square eters? Using the relevant conversion forulas (Table ): (a) In square iles: 200 acre 640 (b) In square eters: 200 acre ile 2 acre ile ile ha acre , It can be seen in the above culations two forulas were used in succession (acre ) ha ) 2 ) to arrive at the final conversion. It is also possible to obtain direct conversion fro acre to 2. Such chain-conversion can be used in advanced conversions using a set of siple basic unit conversion forulas. This ethodology can be applied to any copound units - even copound hypotheti units involving several basic units. It is also possible to perfor conversion on copound units that are not available fro any online resources. Exaple 5.2 What will be the area of 00 ft 2 in 2? It is possible to use the siple conversion forula between feet and eter ( 3.28 ft) to arrive at the areas in square units. The ratio of nuber and units cobination will be again unity when the ratio was squared as follows: 3.28 ft 3.28 ha ft ft 2 (6) It should be noted that all the ites (nubers and units) should be squared individually in Eqn. (6). Using the Eqn. (6), 00 ft 2 will be converted as: 00 ft Exaple ft Using the siple conversion of in 25.4, what will be the volue of 24,000 cubic in cubic in? Extending the squaring ethod used earlier (Exaple 5.2, Eqn. (6)), volues can be converted by cubing the ratio of unity as follows: Exaple in in in3.465 in 3 What will be the speed of 75 ph in (a) kph and (b) eter per inute? Using siple conversion between ile and kiloeter, tie conversion forulas (Table ), and chainconversion ethod perfors the required conversion. (a) In kph: 75 ile.609 h kph (b) In eter per inute: 75 ile.609 k h ile Exaple 5.5 k ile k h k 60 h in /in in For a wheat variety the seed rate followed is 90 /acre. What will be this seed rate in /ha? Using conversions forulas (Table ) the conversion August, 203 Page 3 of 5

4 is obtained as follows: 90 Exaple 5.6 acre acre ha /ha ha A vehicle o ers 3 iles per gallon (pg) ileage. What will be the ileage in k per liter (kpl)? Using conversions forulas (Table ) the conversion is obtained as follows: 3 ile gal k ile k 3.8 kpl liter gal liter (b) In k/h: 20 kw 000. kw s s 000 k k h (c) In hp: Exaple kw hp , 250 k/h hp kw hp Express the orific value of a bioass of 6.93 egajoules/kilogra (M/) in (a) and British theral unit/pound (Btu/) and (b) kiloorie/kilogra (k/). s h Exaple 5.7 What will be the equivalent of 800 Btu in ories ()? Fro conversions forulas (Table ), we have Btu and connected by joules (). Thus applying chainconversion ethod the result will be: 800 Btu 055 Btu , Also: Exaple , k 20.8 k What will be 20 kilowatt (kw) in (a) joule/second (/s), (b) kiloorie/hour (k/h), and (c) horsepower (hp)? Fro conversions forulas (Table) the conversions are obtained as follows: (a) In /s: 20 kw , 000 /s (/s) kw (/s) Fro conversions forulas (Table ) we obtain the following conversions: (a) In BTU/: 6.93 M , (b) In k/: 6.93 M , Liitations M 055 Btu Btu 7, 278 Btu/ M 4.84 k 4, 046 k/ 000 k Unit conversion ethodology deonstrated thus far will work only when the units converted are proportional, and both units vary linearly starting fro zero. Most of the units follow this principle aking this ethod valid for wide variety of applications. One failiar exception is the teperature units, naely, Fahrenheit ( F), Celsius ( C), and Kelvin. A teperature of 0 C refers to 32 F or 273 K, as opposed to proportional units when one is zero the other will be zero as well (e.g. 0 ft 0 ile 0 ). Thus, the teperature units have o sets - August, 203 Page 4 of 5

5 eaning with inter-conversions both units do not start fro zero. Therefore, a single constant cannot ake the conversion, as seen in the following teperature conversion forulas: F.8 C + 32 (7) C( F 32)/.8 K C (8) One can see fro the above conversion forulas that an addition or subtraction operation is involved and these conversions have two constants. For interconversion of teperature units, these forulas involving o sets should be used. For copound units involving teperatures, the o sets ( 32 in Eqn. (7) and in Eqn. (8)) can be ignored as shown subsequently. Let F ini be the initial and F fin be the final teperature in F, and C ini and C fin are the corresponding teperatures in C. Substituting these in Eqn. (7) we get: F ini.8 C ini + 32 F fin.8 C fin + 32 Subtracting the first relation fro the second. (F fin F ini ).8 (C fin C ini )+( 32 32) F di.8 C di (9) It can be seen that the di erence in teperatures in F and C (Eqn. (9)) are connected with a single constant (.8 ) without an o set ( 32 ). A teperature di erence in F, however is.8 ties greater than the sae di erence in C. Thus, theeqn. (9) can be considered as a basic conversion forula analogous to others (Table ). Siilarly, observing Eqn. (8) the teperature di erences connecting C and K units have no constants (C dif K dif ) involved, hence are sae (e.g. / C /K). Exaple 5.0 What will be the heating rate of 5 F/in in C/hour? A5F/in rate of heating eans the di erence in teperature is 5 units of F/in (regardless of initial and final teperatures). Using Eqn. (9) and regular units conversion ethodology we have: 5 F di in.8 C di F di 60 in h 540 C/h 7 Other Resources Several online resources and useful tools, such as conversion charts, tables, forulas, and web-based culators or converters are available now-a-days to the users. For generality, units conversion tables, forulas, and charts usually provide conversion for an unit quantity and requires sing for other or interediate quantities; however, the web-based converters do autoatic sing. A basic search of the words unit conversion in Google will also brings in a web-based online culator. Online culators are convenient, quick, o er several optional units, and they do sing culations. 8 Suary General users and especially students can be benefited fro the understanding of the logic of the ethodology of units conversion (factor-label or unit-factor) deonstrated. This ethodology of carrying the units with nubers, as a ratio of unity, derived fro the conversion forula, works well with ost units. The ethod involves only basic ultiplication, division, and canceling the units to finally arrive at the required results. All units with single constant conversion forulas of the style: Unit Constant Unit2; can be utilized in this ethod. The ethod is suited both for siple direct or advanced units conversions even that not available fro other sources. Chainconversion strategy deonstrated can be applied for advanced conversion involving copound units coposed of several basic units. Square and cubic units can be derived fro the regular units by squaring and cubing the quantities. One failiar liitation of the ethod is the inter-conversion of teperature units, naely, Fahrenheit, Celsius, and Kelvin. These units conversion forula are characterized by an o set and requires two constants; however, the teperature di erences have no o sets. For quick units conversion, several online resources such as web-based converters or culators and conversion charts or tables can be used. References Wikipedia. Units conversion by factor-label; conversion_by_factor-label. Accessed on August 20, 203. August, 203 Page 5 of 5