Manual of Weighing Applications. Part 1. Density

Transcription

1 Manual of Weighing Application Part 1 Denity

2 Foreword In many common area of application, a weighing intrument or the weight it meaure i jut a mean to an end: The value that i actually ought i calculated from the weight or ma that wa determined by the weighing intrument. That i why thi Manual of Weighing Application decribe the mot widely ued weighing application in a erie of eparate booklet, with each booklet repreenting a complete and independent manual. Each of thee booklet begin with a decription of the general and theoretical bai of the application concerned which in many cae cannot be done without uing formula and equation taken from the field of phyic and mathematic. The equation ued are explained in the text, and the intermediate tep neceary for arriving at the reult given here are alo included. Some reader may get the impreion, at firt glance, that the manual i jut a collection of (many) equation, but we do hope that the mot important point are familiar to all reader, even in thi context. Each eparate booklet alo contain a chapter giving detailed example of application, a well a an index where you can look up keyword to find the information you need. At the back of each booklet, jut before the index, we have included a lit of quetion on the ubject o that reader can check whether they have undertood what they have read and can put it into practice. Thi manual wa written to provide artoriu employee and aociate, a well a any intereted cutomer, with a comprehenive compilation of information on the mot widely ued application, both to introduce the ubject and to increae exiting knowledge in the field, a well a to provide a reference ource. We alo hope that reader who make active ue of thi manual will gain confidence in finding olution when uing thee application and gain a feeling for the poibilitie that thee weighing application open up, o they can begin to create cutom olution where needed. Contribution from uer both in the laboratory and in indutry will help enure that thi manual "live" and grow with ue. We are epecially intereted in receiving your application report, which we would like to include in future edition of the Manual of Weighing Application, to make information about new and intereting application of weighing technology available to all our reader. Marketing, Weighing Technology February 1999 BK VORWOR_E.DOC

3 Symbol Indice a (a) () B b t Air Fluid (owing ytem) Solid Determined in air Determined in a uid Buoyancy Bulk Total Symbol Unit Unit m Ma m kg V Volume V m 3 A Area A m² F Force F N = kg m/ 2 W = m g Weight (force) W = m g N = kg m/ 2 g Gravitational g m/ 2 acceleration p = F/A Preure p = F/A Pa=N/ m² rho T t γ gamma α alpha γ gamma ϕ phi π pi Denity Temperature in Kelvin Temperature in Celiu Specific gravity (old!) Linear expanion coefficient of (of olid) Expanion coefficient (of liquid or gae) Relative humidity Poroity rho T kg/ m 3 ; g/cm 3 K t C γ gamma α alpha γ gamma ϕ phi π pi t=t-273,15 K kp/dm 3 1/K = K -1 1/K = K -1 % Volume-%

4 Content: CONTENTS: THE GENERAL PRINCIPLES OF DENSITY DETERMINATION... 3 EXAMPLES OF APPLICATIONS FOR DENSITY DETERMINATION...3 DENSITY: A DEFINITION...3 UNITS FOR MEASURING DENSITY...4 DEPENDENCE OF DENSITY ON TEMPERATURE...4 THE ARCHIMEDEAN PRINCIPLE...7 GRAVIMETRIC METHODS OF DENSITY DETERMINATION DENSITY DETERMINATION BASED ON THE ARCHIMEDEAN PRINCIPLE...10 Buoyancy Method...10 Diplacement Method...12 Determining the Denity of Air...13 DENSITY DETERMINATION USING PYCNOMETERS...15 Weighing a Defined Volume ("Weight per Liter ")...15 Pycnometer Method...16 OTHER METHODS OF DENSITY DETERMINATION OSCILLATION METHOD...18 SUSPENSION METHOD...18 HYDROMETERS...20 PRACTICAL APPLICATIONS DETERMINING THE DENSITY OF SOLIDS...21 Characteritic Feature of Sample Material...21 Chooing a Denity Determination Method...21 Performing Denity Determination uing the Diplacement Method...21 DETERMINING THE DENSITY OF POROUS SOLIDS...22 Characteritic Feature of Sample Material...22 Chooing a Denity Determination Method...23 Performing Denity Determination uing the Buoyancy Method (in Accordance with European Standard EN 993-1)...23 DETERMINING THE DENSITY OF POWDERS AND GRANULES...25 Characteritic Feature of Sample Material...25 Chooing a Denity Determination Method...25 Performing Denity Determination uing the Pycnometer Method (in Accordance with German and European Standard DIN EN 725-7)...26 DETERMINING THE DENSITY OF HOMOGENOUS LIQUIDS...27 Characteritic of Sample Material...27 Chooing a Denity Determination Method...27 Performing Denity Determination uing the Buoyancy Method...27 DETERMINING THE DENSITY OF DISPERSIONS...28 Characteritic of Sample Material...28 Chooing a Denity Determination Method...29 Performing Denity Determination uing the Diplacement Method...29 ERRORS IN AND PRECISION OF DENSITY DETERMINATION AIR BUOYANCY CORRECTION...31 Diplacement Method...31 Buoyancy Method

5 Pycnometer Method...32 AIR BUOYANCY CORRECTION FOR THE PAN HANGER ASSEMBLY...32 Buoyancy Method...32 Diplacement Method...34 PREVENTION OF SYSTEMATIC ERRORS...35 Hydrotatic Method...35 Pycnometer Method...36 ERROR CALCULATION...36 Buoyancy Method...37 Diplacement Method...45 Pycnometer Method...46 COMPARISON OF DIFFERENT METHODS OF DENSITY DETERMINATION APPENDIX TEMPERATURE DEPENDENCY OF DENSITY...54 HYDROSTATIC DENSITY DETERMINATION ELIMINATION OF THE VOLUMES IN THE EQUATION FOR R...55 AIR DENSITY DETERMINATION...56 AIR BUOYANCY CORRECTION...57 QUESTIONS ABOUT DENSITY TIPS FOR ANSWERING THE QUESTIONS REGISTER

6 The General Principle of Denity Determination Example of Application for Denity Determination Denity i ued in many area of application to deignate certain propertie of material or product. In conjunction with other information, the denity of a material can provide ome indication of poible caue for alteration in product characteritic. Denity determination i among the mot often ued gravimetric procedure in laboratorie. Denity can indicate a change in the compoition of a material, or a defect in a product, uch a a crack or a bubble in cat part (known a void), for intance in anitary ceramic or in foundrie in the iron and teel indutrie. In aluminum foundrie, the melt quality i monitored by taking two ample: one under air preure and one under, for example, 80 mbar preure. Once they have et and cooled, the denity of the ample i determined. The ratio of both denity value provide information on the purity of the melt. Determining the denity of platic ued in engineering can help to monitor the proportion of crytalline phae, becaue the higher geometric order of crytal make them dener than the noncrytalline portion. The denity of gla i determined by both the chemical compoition of the ample and the cooling rate of the melted ma. With porou material, the denity i affected by the quantity of pore, which alo determine certain other qualitie of the material; for intance, the frot reitance of roof tile, or the inulating propertie of wall material uch a clay and lime malm brick or porou concrete. One of the factor meaured to determine the quality of wine i known a the mut weight, which i meaured in degree Öchle thi i alo a denity value, becaue the denity i proportional to the concentration of a ubtance in the olvent (e.g., ugar, alt or alcohol in an aqueou olution). The denity of product alo play an important role in the average weight control of prepackaged product, in thoe cae where a package i filled by weight but mut carry a label indicating the content in volume. Denity: A Definition Denity () i the proportion of ma (m) to volume in an amount of material. The term ued here are defined in the German Indutrial Standard (DIN) = m V Equation 1 Different field of indutry and technology have alo developed variou pecial term relating to denity: Normal denity the denity of gae under normal phyical condition (0 C and 1013 hpa) Tap denity the denity of a powdery material under undefined condition; for example, in hipping (DIN ) or the ma quotient of the uncompreed dry granule in a deignated meauring container divided by the volume of the container (DIN EN ) 3

7 Apparent denity the denity of a powder when filled in accordance with the relevant deignated teting procedure; thi i important for determining the fill quantity of preing mold Bulk denity the ma/volume ratio that include the cavitie in a porou material Solid denity the ma/volume ratio of a porou material; i.e., excluding the pore volume True denity Term till ued for "olid denity" Relative denity the proportion of a denity value to a reference denity 0 taken from a reference material; relative denity i a ratio with the dimenion 1 The value for pecific gravity i till ometime given a well, although it i rarely ued today. W m g γ = = V V Equation 2 In contrat to the denity, thi value indicate the weight force in relation to the volume; i.e., the difference between denity and pecific gravity i that the calculation of pecific gravity include the gravitational acceleration (g). Unit for Meauring denity In the International Sytem of Unit, the unit for denity i kg/m 3 ; the unit ued mot often i g/cm 3 thi correpond to the reult in g/ml. Reult are alo ometime given in kg/dm 3. 1 kg/m 3 = g/cm 3 or: 1 g/cm 3 = 1 kg/dm 3 = 1000 kg/m 3 Dependence of Denity on Temperature The denity of all olid, liquid and gaeou material depend on temperature. Aide from temperature, the denity of gaeou material alo depend on preure. Gae are compreible at "normal" preure; thi mean that air denity change when the air preure change. The normal denity i the denity of a ga (or combination of gae) under normal phyical condition: temperature T = 0 C, preure p = kpa. A general rule i: The higher the temperature, the lower the denity. Material expand when heated; in other word: their volume increae. Therefore the denity of material will decreae a their volume increae. Thi i more noticeable in liquid than in olid, and epecially in gae. The change in denity over a certain temperature interval can be calculated uing the heat expanion coefficient; thi will yield the change in volume of a material in relation to the temperature (ee Appendix, page xx). The following diagram how the denity of variou ubtance calculated in relation to the temperature the x axe how denity in interval of 0.06 g/cm3 (except in the cae of air). A can be een from thee diagram, temperature affect ome ubtance more trongly than other. For denity determination, thi mean that depending on the required accuracy of meaurement, of coure the tet temperature mut be et very preciely and kept contant. In hydrotatic denity determination method, for example, it i uually better to ue water than ethanol a a liquid for buoyancy; when the temperature increae, for example, from 20 C to 4

8 21 C, the denity of the water only decreae by g/cm 3, where the denity of ethanol decreae by g/cm 3 more than 4 time a much. Thi mean that the temperature ha to be controlled more preciely, or a greater error mut be aumed in the reult of the denity determination uing ethanol Denity / g/cm³ Denity /g/cm³ Temperature / C Water (calc. from denity (20 C) and expanion coefficient) Water (PTB-table) Ethanol Temperature / C Figure 1: Temperature dependency of denity for water and ethanol (above) and for teel and Denity /g/cm³ Denity /g/cm³ Steel Temperature / C Aluminum Temperature / C aluminum (below) 5

9 Denity /g/cm³ Denity/g/cm³ Temperature / C Temperature / C Quartz Gla AR-gla (plummet of the Denity Determination Kit) Denity /g/cm³ Denity /g/cm³ Polyethylene Temperature / C Air Temperature / C Figure 2: Temperature dependency of gla, polyethylene and air 6

10 The Archimedean Principle In accordance with the definition of denity a = m/v, in order to determine the denity of matter, the ma and the volume of the ample mut be known. The determination of ma can be performed directly uing a weighing intrument. The determination of volume generally cannot be performed directly. Exception to thi rule include cae where the accuracy i not required to be very high, and meaurement performed on geometric bodie, uch a cube, cuboid or cylinder, the volume of which can eaily be determined from dimenion uch a length, height and diameter. The volume of a liquid can be meaured in a graduated cylinder or in a pipette; the volume of olid can be determined by immering the ample in a cylinder filled with water and then meauring the rie in the water level. Becaue of the difficulty of determining volume with preciion, epecially when the ample ha a highly irregular hape, a "detour" i often taken when determining the denity, by making ue of the Archimedean Principle, which decribe the relation between force (or mae), volume and denitie of olid ample immered in liquid: From everyday experience, everyone i familiar with the effect that an object or body appear to be lighter than in air jut like your own body in a wimming pool. Figure 3: The force exerted by a body on a pring cale in air (left) and in water (right) Both the caue of thi phenomenon and the correlation between the value determined in it meaurement are explained in detail in the following. A body immered in water i ubjected to tre from all ide imultaneouly due to hydrotatic preure. The horizontal tre i in equilibrium, which i to ay that the force cancel each other out. The vertical preure on the immered body increae a the depth of the body under the urface increae. The preure at a certain depth in liquid exerted by the liquid above that point i called weight preure. The weight preure can be calculated from the denity of the liquid, the height of the liquid and the gravitational acceleration: p = g h. The ame preure i exerted on area A at depth h: F = p A = g A h Equation 3 7

11 h 3 h 2 A h 1 with = 1 g/cm 3 and g 10 m/ 2 h p A F 1 1 cm 0.01 N/cm 2 1 cm N 2 10 cm 0.1 N/ cm 2 1 cm N 3 50 cm 0.5 N/ cm 2 1 cm N A = 1g/cm 3 Figure 4: Gradient of preure in liquid F 1 x x + h h F 2 Figure 5: Buoyancy effect The weight preure on the urface of an immered body with area A caue a force of F 1 = A x g to be exerted on the upper urface of the body, and F 2 = A (x+h) g on the lower urface. The reulting force on the body can be calculated from the difference between thee two force: F = F - F re 2 1 [ ] [ ] = A (x+h) g - A x g = A (x+h-x) g = A h g Equation 4 The product of area and height of the body i equal to the volume of the body. Thi volume i in turn equal to the volume of water that i diplaced by the immered body. (A h) = V = V Equation 5 Thu the reulting force i F = V g = F. Equation 6 re B Thi force i known a buoyancy force or imply buoyancy, and it directly include the value for volume to be determined. 8

12 Oberving the ratio of force exerted on the immered body and the water diplaced by the body, it can be een that the force exerted include both the weight W, a downward force, and buoyancy F B, an upward force. The reulting force can be calculated from the difference between thee two force: F re = W - F B. The buoyancy F B exerted on the body i equal to the weight W of the liquid diplaced by the body. W S W = m g = F B F B F B = V g Figure 6: On the Archimedean Principle: The figure on the left repreent the body immered in liquid; on the right, the liquid element. The reult i W = m g= V g Equation 7 If the body and the liquid element are at equilibrium, the buoyancy F B mut, by module, be equal to the weight W ; thu F = W B. Equation 8 The buoyancy i the reult of the level of hydrotatic preure in a liquid. Buoyancy i invere to the weight of a body immered in liquid. Thi explain why a body eem to be lighter in water than in air. Depending on the ratio of body weight to buoyancy, the immered body may ink, oat or be upended. If the buoyancy i le than the weight (F B < W ), the body will ink. In thi cae, the denity of the body i greater than that of the liquid ( > ). The widely ued method of determining denity according to the buoyancy method i uually ued under thee condition. If the buoyancy i equal to the weight (F B = W ), the body remain completely immered and i upended in the liquid. Becaue both the volume and ma of the body are equal to the volume and ma of the diplaced water, it follow that the body and the liquid have the ame denity. There are a number of denity determination procedure that make ue of thi condition (ee Page 18). If the buoyancy i greater than the weight (F B > W ), the body oat; i.e., it rie to the urface of the liquid and remain only partially immered. In fact, it dip o far into the liquid until the weight of that volume of liquid that i diplaced i equal to the weight of the body. In thi cae, the volume of the diplaced liquid i le than the volume of the body (V < V ) and the denity of the liquid i greater than the denity of the body >. Thee are the condition for denity determination uing a hydrometer (ee page 20). 9

13 Gravimetric Method of Denity Determination Denity Determination Baed on the Archimedean Principle The relationhip between the ma, the volume and the denity of olid bodie immered in liquid a decribed by Archimede form a bai for the determination of the denity of ubtance. The difficulty in thi method of denity determination lie in the precie determination of the volume of the ample. When a body i completely immered in liquid, the mode of procedure demand that the volume of the body i equal to the volume of the diplaced liquid. Thu we can derive the following general equation between the denity and ma of a liquid and of a olid, in which the volume are not explicitly named (ee Appendix, page 54 for the derivation): m = Equation 9 m Accordingly, the unknown denity of a olid ubtance can be determined from the known denity of the liquid for buoyancy and two ma value: m = m or m = V Equation 10 Reciprocally, the denity of liquid can be determined from one ma value and the known denity of the immered body. Simple and precie method of ma determination can eliminate the need to meaure volume. Hydrotatic balance and Mohr balance are till ued in ome cae for meauring denity; the Mohr balance, a beam balance ha been widely replaced by the ue of denity et in conjunction with laboratory balance. There are two baically different procedure for hydrotatic weighing method. The actual meaning of the value diplayed on the weighing intrument depend on the mode of procedure ued. The buoyancy method (ee Figure 7 and Figure 8) entail meauring the weight of the body, which i decreaed by buoyancy, while the diplacement method (ee Figure 9) call for the direct meaurement of the weight or ma of the diplaced uid. Other method that are baed on the Archimedean principle include denity determination uing hydrometer (ee page 20) a well a variou upenion method (ee page 18). Buoyancy Method The buoyancy method i often ued to determine the denity of bodie and liquid. The apparent weight of a body in a liquid, i.e., the weight a reduced by the buoyancy force i meaured. Thi value i ued in combination with the weight in air to calculate the denity. 10

14 Weighing pan Figure 7: Baic procedure for the buoyancy method with below-balance weighing Weighing pan Liquid in a beaker on a metal platform; no contact with the weighing pan Figure 8: Baic procedure for the buoyancy method with a frame for hanging the plummet and a bridge for holding the container for liquid In the procedure illutrated in Figure 7 and Figure 8, the value diplayed on the weight readout indicate the ma of the immered body a reduced by buoyancy (ee alo Figure 3). Thi mean that, in light of the equation = (m /m ), the ma of the body weighed in air i known: m = m (a). The ma of the liquid m i not directly known, but i yielded by the difference between the weight of the body in air (m (a) ) and in liquid (m () ): m = m (a) - m (). Thi change Equation 9 for determining the denity of the body into: m(a) = m m (a) (). Equation 11 To determine the denity of a liquid, m i again calculated from the value meaured for ma of the body in air and in liquid m = m (a) - m () and the reult ued in Equation10. The buoyancy method maintain the relationhip for determination of the denity of the liquid: m(a) m = m (a) () m = (a) m V () Equation 12 11

15 V i the known volume of the plummet ued to determine the liquid denity. Thu the denity of a ubtance can be determined in two weighing operation. Diplacement Method The diplacement method i another way that the Archimedean Principle i ued in determining the denity of bodie and liquid. The procedure for the diplacement method entail determining the ma of the liquid diplaced by the body. A container of liquid i placed directly on the weighing pan while the body i immered. In mot cae, the body i upended from a hanger aembly. When the body i immered in the liquid, it diplace a volume of liquid V with denity and ma m. The buoyancy force exerted on the body i F B = V g = m g. Becaue the weight of the body W = m g i carried by the hanger aembly and the balance i not loaded, the balance readout directly indicate the ma of the liquid m auming the weight of the container wa tared beforehand. Figure 9: Baic mode of procedure for the diplacement method Thi mean that, in the cae of the diplacement method, Equation 9 and 10 (ee page 10) can be directly applied for denity determination. For the denity of a olid: m = m Equation 13 while for the denity of a liquid: m m = = m V Equation 14 If you ue a plummet with a known volume V, the unknown denity of a liquid can be calculated from jut one meaurement. 12

16 Determining the Denity of Air To convert a weight value to the true ma value, you mut know the value for the air denity. Thi value can vary over the coure of a day by an average of ± 0.05 mg/cm 3 in relation to the normal denity of 1.2 mg/cm 3. Thi i why the air denity mut be determined at the time the value i required, with a relative uncertainty factor of < The air denity a depend on the temperature T, the preure p and the relative humidity of the air ϕ. There are variou approximation formula ued to determine the denity of air a dependent on air preure, temperature and humidity, and even ome which conider of the CO 2 -content of the air (ee alo "The Fundamental of Weighing Technology," pp ) It i alo poible to determine the denity of air (with a 1 % margin of error) with high-reolution weighing intrument. Thi i done uing 2 calibrated weight that are each made of different material, with different denitie (for example, aluminum and teel). Thi determination method i alo baed on the Archimedean Principle. Becaue air i made up of matter, a body in air i ubject to buoyancy jut a it i in liquid. The ame regularitie apply a thoe decribed in the chapter entitled "The Archimedean Principle", on page 7. Oberving firt in a vacuum an aluminum cylinder with a denity Al 2.7 g/cm 3 (Figure 10, left), we ee that it i in equilibrium with a tandardized weight of the ame ma ( N = g/cm 3 ). F BN = a. V N. g F BAl = a. V Al. g G N = m N. g G Al = m Al. g G N = m N. g G Al = m Al. g Figure 10: The effect of buoyancy on weighing in a vacuum (left) and in air (right) Oberving the ame circumtance in air (Figure 10, right), rather than in a vacuum, we ee that the aluminum cylinder and the tandardized weight are no longer in equilibrium. Thi i due to the difference in buoyancy force caued by the different material denitie and volume. To determine what ma m N hold the aluminum cylinder (m AL ) in equilibrium in air that ha a denity a, all effective force are oberved at equilibrium: W F = W F N BN Al BAl mn g a VN g = mal g a VAl g weight buoancy weight buoancy Equation 15 m N N m Converion with V N = and V Al Al = then yield Al 13

17 m m 1 Al = N 1 a N a Al Equation 16 m N i the weight value W. The weight value i in general equal to the readout on the weighing intrument. The weight value W m 1 a Al Al = Al i not contant; rather, it i dependent on the air a 1 N denity on the weather, o to peak. Thi relationhip applie in a imilar manner for the teel cylinder in the weight et for the determination of air denity: W m 1 a St St = St. From thee 2 a 1 equation, a relation to the determination of the air denity can be derived (ee Appendix, page 57): N a = m W m W Al St St Al ma l WSt mst W Equation 17 Al Al St W St and W Al are the weight value currently meaured. m St and m Al are calculated according to the following formula, uing the conventional ma and the denitie of the certified weight: m M 1 St = St kg/m kg/m 3 1.2kg/m St or m M 1 Al = Al kg/m kg/m 3 1.2kg/m Al Equation 18 The conventional ma M of a weight i not the ma of the weight itelf, but rather i equal to the ma of the reference weight (tandard ma) which under certain defined condition 1 i in equilibrium with the weight being meaured. The air denity a can be calculated from the conventional ma value given in the weight et for the determination of air denity for the weight (deignated the characteritic value of the weight), the material denitie of the weight and the current weight value. A number of Sartoriu weighing intrument have the formula for calculating the air denity, including the value ST = g/cm 3 and Al = g/cm 3, integrated in their oftware. The current air denity value can be aved and i then ued to convert weight value to the actual mae of the ample weighed, uing the formula derived at the beginning of thi chapter: m W 1 a N = v. a 1 x 1 Temperature T = 20 C Denity of the tandard ma at 20 C: Ν = 8000 kg/m 3 Air denity a = 1.2 kg/m 3 14

18 Denity Determination uing Pycnometer A pycnometer i a gla or metal container with a preciely determined volume, ued for determining both the denity of liquid and diperion by imply weighing the defined volume (ee alo next chapter), but epecially for determination of the denity of powder and granule. Pycnometer can alo be ued in determining the denity of the olid phae in a porou olid, but the ample mut firt be cruhed or ground to the point where all pore are opened. The pycnometer ued in different area of application have different hape and tandard. During meaurement, it i important to make ure that all weighing operation are performed at a contant temperature and that there i no air trapped either in the liquid or between the ample particle. Fill level Fill level Figure 11: Different gla pycnometer for denity determination in laboratorie: The pycnometer made according to Gay-Luac, DIN (c) and to Hubbard, DIN (f) are ued for determining the denity of olid; the volume indicated applie to complete filling after the topper i inerted. The pycnometer made according to Bingham, DIN (b) and to Sprengel, DIN (d) have a line marking the fill level for the defined volume; the Reichauer, DIN (a) and Lipkin, DIN (e) pycnometer are marked with cale for checking the fill level. Weighing a Defined Volume ("Weight per Liter ") An epecially imple gravimetric method for determining the denity of owing ubtance (liquid, powder, dipere ytem) i to weigh a ample with a defined volume. In thi cae, the ample i placed in a container that ha a defined volume, and the ma of the ample (after taring) i determined by weighing. The denity can eaily determined according to = m/v. Different tandardized container are available for thi purpoe in different branche of indutry; for example, a pherical 1 l-container for determining the denity of cat material (lip) in the ceramic indutry. In the lime indutry, the tap denity of unhydrated lime granule i determined uing a tandardized procedure, in which both the container for the ample and the procedure for filling the container are preciely defined. US and Britih tandard call for the ue of cylindrical tainle teel container, called pecific gravity cup, with variou volume and error margin. 15

19 Pycnometer Method The pycnometer method i a very precie procedure for determining the denity of powder, granule and diperion that have poor owability characteritic. The pycnometer method i more labor-intenive and far more time-conuming than the buoyancy and diplacement method. Thi method alo entail the difficulty of precie volume determination of a powder ample V for denity determination of the olid. The need for explicit determination of the volume of powder or granule can generally be avoided by performing 3 weighing operation and uing an auxiliary liquid with a known denity. m V = Equation 19 V ge m 1 V, m 2, V, m, Figure 12: Pycnometer with content The volume of the olid V can only be determined indirectly: V = Vge V Equation 20 The procedure i a follow: V Firt the pycnometer i completely filled with liquid, and the ma of the liquid in the pycnometer determined. Once thi value ha been determined, the volume of the pycnometer V ge i known. ge m1 = Equation 21 Next (after the pycnometer i emptied, cleaned, dried and brought to the required temperature) the pycnometer i filled to about 2/3 with ample material; thi yield the ma of the powder m. The next tep i to fill the pycnometer the ret of the way with liquid and weigh it again, which give the combined ma of the ample with the liquid m (+). The ma of the liquid m 2 can be calculated from thi data m2 = m(+ ) m Equation 22 V which alo yield the volume V of the liquid in the pycnometer filled with water and liquid m m -m 2 (+) = =. Equation 23 16

20 The volume of the powdered ample V, the value actually ought, i yielded by the difference between the total volume V ge and the volume of the liquid V. V = Vge V V m m -m 1 (+) = Equation 24 Uing the volume V in the original equation = m /V reult in the following converion 2 m = m -m + m 1 (+ ) or = m2 m + m m Equation 25 with the mae m 1, m 2 and m 3 in the order of the procedural tep: m 1 m 2 m 3 ma of the liquid in the pycnometer filled completely with liquid ma of the ample material ma of the ample and liquid together in the pycnometer Thu thi procedure repreent another method for determining denity "via detour," i.e. uing a erie of ma determination meaurement. 2 m = = V m m m = m -(m m ) = (+ ) 1 (+ ) m -(m m ) m m m 1 1 (+ ) 17

21 Other Method of Denity Determination There are alo other method of denity determination that are baed on the Archimedean Principle. The denity of air can be determined uing two olid bodie of different denitie (e.g., 2 weight made of different metal). Denity can alo be determined by radioactive aborption by the material being teted. The aborption of the radiation will depend on the ma aborption coefficient, thickne of layer and denity of the material. Once the ma aborption coefficient and thickne a well a the phyical interrelationhip are known, the denity of the ubtance can be calculated. On magnetic ample, the magnetic force can alo be utilized in denity determination of olid or liquid. Ocillation Method The ocillation method i widely ued to determine the denity of homogenou liquid. Thi procedure i not uitable for ue with upenion or emulion which, becaue they are made up of different phae, could eparate. The ample being teted i placed in a meauring chamber (uually a U-haped gla tube) and mechanically vibrated. Calculation of the denity ue the phyical interrelationhip between the frequency of the ocillation and the ma of the ocillation channel (the U-haped tube with the ample). The equipment mut be calibrated uing liquid that have a known denity and a vicoity imilar to that of the ample material. Supenion Method The upenion method make ue of the Archimedean Principle in the pecial cae of upenion in which the denitie of the liquid and of the upended olid are equal. The denity of the olid body can be determined by etting the denity of the tet liquid o that the ample body reache a tate of upenion. The denity etting of the tet liquid can alo be achieved by mixing two liquid of different denitie; the denity of the ample i then determined from the proportion of the liquid mixed, or with the ocillation method (ee page 18) or the diplacement method. Denity Gradient Column With a denity gradient column, two liquid of different denitie are layered in a gla tube o that over time, diffuion reult in a vertical denity gradient (a continuou change of the denity throughout the height of the column). Small olid of variou denitie are then upended at variou height, with each height indicating a particular denity. Colored gla bead of known denitie are available for calibration. In addition to the denity of mall bodie (uch a fiber, powder particle, mall piece of metal or platic foil) thi method can alo be ued to determine the denity of drop of liquid of coure the liquid teted hould be inoluble in the tet liquid. 18

22 Schlieren Method If you fill a capillary tube with liquid and hold it horizontally immered in another liquid, the liquid will only ow horizontally from the tube if the denitie of the two liquid are equal. If the denity of the liquid owing from the tube i lower or higher than that of the liquid in which the tube i immered, chlieren (treak) will form, owing upward in the former cae and downward in the latter. 19

23 Hydrometer Hydrometer, alo known a pindle, are imple meauring intrument for determining the denity of liquid or diperion. They are form of plummet that oat on the urface and then ink to a certain level, depending on the denity of the liquid. The denity of the liquid can be determined from the depth the plummet ink (from the volume of the diplaced liquid) by comparing the height of the liquid in the container to the cale marked on the hydrometer. For certain application, there are alo hydrometer that how the concentration of a given ubtance in an aqueou olution; for intance; ugar (in a accharimeter), alcohol (in an alcoholometer), battery acid or anti-freeze. Thermohydrometer for ue with milk and kimmed milk Thermometer cale Thermometer cale Hydrometer cale 6.25 Hydrometer cale Thermometer capillary Weighting device Thermometer filling Figure 13: Special hydrometer with integrated thermometer in accordance with DIN for determining the denity of milk and kimmed milk the denity i dependent on the fat content of the milk. 20

24 Practical Application Determining the Denity of Solid Characteritic Feature of Sample Material Solid bodie retain their volume and hape under atmopheric preure. Example of olid for which it can be ueful to know the denity include metal, gla and platic. Thee olid may be made up of only one or many phae; one phae may alo be embedded in another (for intance, in fibergla-reinforced platic) or the different phae may be interlocking, uch a the many mall crytal in a homogenou metallic material. An important factor in chooing a uitable ample for denity determination i the quetion of whether the denity i required a a characteritic of a material or whether denity determination i performed to check for defect in a material. The choice of procedure for denity determination will depend on thi factor a well. Chooing a Denity Determination Method The bet procedure for denity determination on olid are the buoyancy and diplacement method, both of which are baed on the Archimedean Principle. Prerequiite for thee method i the ue of a liquid for buoyancy that doe not react with the ample material, but wet it thoroughly. The upenion method, for example, i widely ued in the gla proceing indutry. Gla ample are placed in an organic liquid in which they oat at room temperature. Becaue the denity of the liquid i 100 time more temperature-dependent than that of gla, the gla can be brought to the point where it i upended within the liquid by lowly increaing the temperature in the tet ytem; in thi way, the denity of the gla can be determined. Performing Denity Determination uing the Diplacement Method Equipment Required Weighing intrument Thermometer Stand with holding device for ample Beaker with liquid for buoyancy that ha a known denity ditilled water for all material that do not react with water Preparation of the Sample, Teting Procedure and Evaluation The beaker i placed on the pan of the balance and the ample-holding device i immered in the liquid, to the ame depth that it will later be immered with the ample on it. The weighing intrument i tared. The ample i placed next to the beaker on the weighing pan. The ma of the ample in air m(a) i determined. The ample i placed in the holding device on the tand and immered in the liquid. The weight readout how the ma of the diplaced liquid m. 21

25 m(a) The denity of the ample i calculated according to = m. Determining the Denity of Porou Solid Characteritic Feature of Sample Material There are a number of term ued in connection with the denity of porou material, uch a olid denity, true denity, bulk denity, apparent poroity, open poroity and cloed poroity. Porou olid conit of one or more olid phae and pore. Pore are cavitie filled with air (or other ga). Thee opening are found either between individual crytal in the olid material, or a ga bubble in gla phae; i.e., olidified in a non-crytalline form. Thu there are baically two form of pore: open and cloed. Among open pore, in turn, there are alo different type; for intance, there are pore through which liquid may ow, and aturable pore. With thee deignation, the type of oaking medium and other condition mut be given (e.g., water at a temperature of 22 C and preure of 2500 Pa). The term "pore" i ued for opening or gap from 1 nm to 1 mm. Opening larger than 1 mm are referred to a crack or void; thoe under 1 nm are defect in the crytal lattice. Pore are an important element in the micro tructure of many material; the quantity, type, hape, orientation, ize and ize ditribution of pore ignificantly affect many important characteritic of a material; for example, the frot-reitance of roof tile, or inulating propertie of lime malm brick or porou concrete. Such characteritic a mechanical olidity and corroion reitance are alo affected by pore in a material. Figure 14: Microtructure of a porcelain plate. Magnification: approximately 80x. Left: porcelain with irregular pore between phae; right: glaze layer melted during firing, with cloed pherical pore (bubble); extreme right: ynthetic rein a embedding medium for the polihing preparation The denity determination procedure will depend on whether the value ought i the denity of the olid matter only or the denity of the material including pore; after all, it may be important to know the poroity of the material. The denity of the olid material (not the olid body), which ued to be termed the "true denity" i now imply referred to a "denity": t = m / V olid. The pore are not included in thi meaurement. "Bulk denity" i the quotient of ma and total volume of a ample: b = m / V b. The bulk denity i an average of the denity of the olid and the ga found in the pore. 22

26 "Open poroity" i the ratio of the volume of open pore to the total volume of the porou body, in percent: π a = V a / V b. "Cloed poroity" i the ratio of the volume of cloed pore to the total volume of the porou body, in percent: π f = V f / V b. The apparent poroity i the ratio of the volume of all pore to the total volume of the material, in percent: π t = V t / V b. The apparent poroity i the um of open and cloed poroity: π t = π a + π f. Chooing a Denity Determination Method Both the buoyancy and the diplacement method are uitable for denity determination on porou olid. The olid denity can alo be determined uing the pycnometer method, by firt grinding the ample until the grain ize i roughly equal to the pore ize. To determine the bulk denity of porou material, the ample can alo be covered in a wax or latex coating or layer to prevent liquid from entering open pore (ee for example the German Indutry Standard (DIN) 2738). Denity determination can then be performed uing the buoyancy method. Performing Denity Determination uing the Buoyancy Method (in Accordance with European Standard EN 993-1) Figure 15: Mode of procedure for denity determination uing the buoyancy method and the Denity Determination Set from artoriu Equipment Required Drying oven; temperature: 110 ± 5 C Weighing intrument: margin of error: 0.01 g Frame to be placed over the weighing pan: included with the Denity Determination Set Vacuum generator with adjutable preure and preure gauge Thermometer with an error margin of 1 C Liquid for aturation ditilled water for all material that do not react with water 23

27 Deicator. Preparation of the Sample Shape and ize (total volume between 50 cm 3 and 200 cm 3 ) of the ample, a well a the number of ample to be teted, are defined in the Standard. Teting Procedure and Evaluation The ample i firt dried in the drying oven until it reache a contant ma and then cooled to room temperate in the deicator. The ma of the ample i then determined in air uing the weighing intrument. m 1 The ample i then evacuated under preciely defined condition and aturated (in the vacuum) until the open pore a tated in the tet pecification are filled with the aturation liquid. The apparent ma of the aturated ample i then determined uing a hydrotatic balance (or uing the Denity Determination Set). The ample mut be completely immered in a beaker filled with the aturation uid for buoyancy. m 2 The temperature of the aturation liquid mut be determined. Then the ma of the aturated ample i determined by weighing in air. Liquid that remain on the urface of the ample mut be removed with a damp ponge before weighing. The weighing operation mut be performed quickly, to avoid lo of ma due to evaporation. m 3 The denity of the aturation liquid mut be meaured or taken from a table of denity value at defined temperature. The bulk denity b in g/cm 3 i calculated a follow: b = m m1 m 3 2 Equation 26 The open poroity π a in volume percent i calculated a follow: = m m m m 100 Equation 27 π a The apparent poroity π t i calculated a follow: π t = t t b 100 Equation 28 The apparent poroity i the um of open and cloed poroity (π t = π a + π f ); thu it follow that for the cloed poroity π f : π = π π Equation 29 f t a m 1 m 2 m 3 Ma of the dried ample Apparent ma of the aturated ample weighed in liquid Ma of the aturated ample weighed in air 24

28 r t r r b Denity of the olid, determined according to EN (or calculated from the compoition) Denity of the uid for buoyancy Bulk denity of the ample One of the numeric value often given for ceramic for the open poroity, in addition to the value lited above, i the water aborption. The water aborption in percent i yielded by the difference in ma between the aturated ample and the dried ample, relative to the ma of the dried ample. The reulting figure can be ued in dividing ceramic into "dene" and "porou" grade. Additional information about the type and ize ditribution of the pore can be gained uing a mercury poroimeter: The porou ample are put under preure with mercury, whereby the preure i increaed at certain tage o that, depending on the pore diameter, a certain number of the pore are filled with mercury. Thi can yield information on the proportion and diameter of open pore that are acceible from the outide. Another method for determining number, hape and ize of pore i image analyi, the quantitative tatitical evaluation of polihed ection under a microcope, imilar to the image hown in Figure 14. Determining the Denity of Powder and Granule Characteritic Feature of Sample Material The term powder refer to "a heap of particle, uually with dimenion maller than 1 mm." Granule are larger particle than thoe that make up a powder. The term "granule" ha different meaning in different area of application: Material made up of "econdary" particle, which in turn are made up of agglomerated particle of a fine powder; or Material that wa heated to the melting point and then cooled very quickly, cauing it to take on a teardrop hape; for example, intermediate product in the platic or porcelain enamel indutrie. Chooing a Denity Determination Method The pycnometer method i the only method that can be ued with powder or granule. Performing Denity Determination uing the Pycnometer Method (in Accordance with German and European Standard DIN EN 725-7) Figure 16: Pycnometer with integrated thermometer 25

29 Equipment Required Ditilled water and another liquid, uch a ethanol Pycnometer with thermometer and idearm with polihed gla topper Water bath Vacuum pump Weighing intrument; error margin: g Teting Procedure and Evaluation The pycnometer mut be carefully cleaned and dried; it i then filled with ditilled water, evacuated under preciely defined condition, and brought to within ± 0.1 K of the required temperature in a water bath. Then the pycnometer i filled. The volume of the pycnometer i calculated from the ma of the water at the tet temperature: V pycnometer = m water / water. The pycnometer i then dried, filled with ethanol and weighed, following the ame procedure a that decribed above. The denity of the ethanol at the tet temperature can be calculated from the ma of the ethanol and the volume of the pycnometer: ethanol = m ethanol / V pycnometer. m ethanol = m 1. Once the pycnometer ha been cleaned and dried again 3, it i loaded with about 10 g (at a denity between 2.5 and 4 g/cm 3 ) of the powder, which ha been dried at a temperature 10 K below the decompoition point of the powder. m 2 Enough ethanol i now added to the pycnometer to wet the powder; the pycnometer i then evacuated and hook to releae a many air bubble a poible. More ethanol i added to the pycnometer; after thi ha been heated to the tet temperature, the pycnometer i completely filled. The total ma of powder and ethanol i determined. m 3 m2 The data i evaluated uing the formula = ethanol m + m m material i given with a preciion of g/cm , the denity of the ample Determining the Denity of Homogenou Liquid Characteritic of Sample Material Homogenou liquid are relatively imple ytem; unlike diperion, they are alway ingle-phae ytem. When ubtance are mixed, uch a water with alcohol or ugar with water, thi i referred to a the olution of one ubtance in the other. A genuine olution i clear, the particle of the diolved ubtance are preent a molecule or ion in the olution. 3 The latet weighing intrument from artoriu come equipped with denity determination oftware that eliminate with time-conuming tep for drying in the drying oven and cooling in the deicator; the program ha a econd tare memory which can be ued to tare the weight of water remaining in the pycnometer. Thi coniderably implifie work in the laboratory. 26

30 The denity of a olution depend on the concentration of the diolved ubtance; in other word, when the interrelationhip are known, the denity value can be ued to derive the concentration of the olution. At 20 C mot uid have a denity between 600 kg/m 3 and 2000 kg/m 3 or 0.6 g/cm 3 to 2.0 g/cm 3. The denity of uid i far more temperature-dependent than that of olid. Thi mean that the temperature mut alway be monitored carefully and, if neceary, the ample heated or cooled to the required temperature. Chooing a Denity Determination Method There are everal method that can be ued to determine the denity of liquid, including the hydrometer, pycnometer, ocillator, buoyancy and diplacement method. The choice of method depend, among other thing, on the degree of preciion required and the amount of ample material available. Performing Denity Determination uing the Buoyancy Method Figure 17: Determining the denity of a liquid uing the buoyancy method Equipment Required Weighing intrument Denity Determination Set Plummet with known volume (10 cm 3 in the artoriu Denity Determination Set; ee Figure 17) Thermometer In ome cae: water bath for adjuting the temperature of the ample Preparation of the Sample, Tet Procedure and Evaluation Poition an empty beaker on the bridge and hang the plummet from the frame provided in the Denity Determination Set. Tare the weighing intrument with the plummet. Fill the beaker with the liquid to be teted up to a level 10 mm higher than the plummet. 27

31 The negative value hown on the weight readout correpond to the buoyancy of the plummet in the liquid. The denity of the liquid i calculated by dividing the meaured value by the volume of the plummet = m V TK. Determining the Denity of Diperion Characteritic of Sample Material Dipere ytem or diperion are combination of two or more phae, each of which i inoluble in the other(). One phae, called the diperion medium, i alway contiguou, while the other phae or phae are preent in the medium in the form of finely ditributed iolated particle. In a colloid diperion, the particle are generally between 1 µm and 1 nm in ize. If the particle are larger than > 1 µm, thi i referred to a a coare diperion; if particle are maller than < 1 nm, it i a molecular diperion. There are many example of diperion, becaue "diperion" i the generic term for all ytem, independent of the tate of the phae. Different type of diperion include: Supenion Emulion Foam Mit Smoke Mixture of olid particle in a liquid Example: "Diperion" paint, ceramic lip, abraive liquid cleaner, toothpate, ink, etc. Mixture of two liquid that are mutually inoluble, where one i preent in the form of finely ditributed minucule drop in the other Example: Creme, lotion, mayonnaie, milk, the claic oil-and-vinegar alad dreing, etc. Mixture of ga bubble in a liquid (or a olid) Mixture of mall drop of liquid in a ga phae Mixture of olid particle in a ga phae The term "tability" in reference to diperion i omewhat problematic, becaue thee are actually untable ytem. Thi can be een in their tendency to eparate. The term "table upenion" and "table emulion" are often ued to refer to ytem that remain contant over a certain period of time. Chooing a Denity Determination Method Many of the ame method ued on liquid or olid can alo be ued for determining the denity of diperion. The bet choice for a given ample material will depend on the conitency of the ample. The ocillation method i not well-uited for ue here, for a number of reaon. The many phae boundarie are a diadvantage; the vicoity ha an effect on the meaurement, and the vibration during meaurement can promote eparation of the phae, which mean the value obtained will not be repreentative of the overall average. Hydrometer can be ued, but it mut be enured that the upenion or emulion doe not how ign of eparating. 28

32 The pycnometer can alo be ued on diperion, jut a with denity determination on liquid or powder. Different container are ued in different branche of indutry; the container may be filled to the rim or up to a marking with the ample material and weighed. The potential eparation of the ample phae during meaurement i not a problem with thi procedure. The buoyancy or diplacement method can be ued for many upenion and emulion. In thi cae, too, care mut be taken to avoid eparation of the ample phae; the ow behavior of the upenion mut alo allow the plummet to ink quickly. Performing Denity Determination uing the Diplacement Method Figure 18: Denity determination uing the diplacement method, with a gamma phere a a plummet, affixed to a holder mounted next to the weighing intrument Equipment Required Weighing Intrument Plummet with known volume Holder Thermometer In ome cae: water bath for adjuting the temperature of the ample Preparation of the Sample, Tet Procedure and Evaluation Bring the ample to the required temperature and place it in a beaker. Place the beaker on the weighing pan and tare the weighing intrument. Immere the plummet in the tet ubtance up to the marking. The weight readout how the ma of the diplaced liquid directly (ee Page 12). The denity of the ample i determined by dividing the meaured value by the volume of the plummet = m V TK. 29

33 Error in and Preciion of Denity Determination In the two previou chapter (Fehler! Verweiquelle konnte nicht gefunden werden. and Fehler! Verweiquelle konnte nicht gefunden werden.) the fundamental of thee two hydrotatic denity determination method were explained and the formula for calculation of the denity were derived. If a high degree of preciion i required, the exiting condition mut be allowed for. Technically peaking, the weighing intrument doe not how the ma of the ample the value given in the equation but rather the weight value for the ample in air. For more precie reult, ue the weight value obtained after air buoyancy correction. When uing the buoyancy method, the immerion level of the pan hanger aembly i affected when the ample i immered, which produce additional buoyancy. Thi mut alo be conidered in more precie calculation. In general, denity determination procedure require very careful work; it i epecially important to make ure the temperature remain contant during teting. Bubble entering the liquid when the ample i immered can alo affect reult; bubble adhering to the tet piece will ditort the meaurement reult. Air Buoyancy Correction For high-preciion denity determination, it i important to note that the weighing intrument doe not directly determine the ma of the ample, but it weight value. Thi value i dependent on the air denity, which in turn depend on the preure and temperature and mut be corrected by the value for air buoyancy. Between the ma of a olid body m and it weight value in air W; that i, under conideration of the air buoyancy on the ample, the following relationhip i generally valid ( G = denity of the tandard): m W 1 = v 1 a G a. Equation 30 Diplacement Method If you include thi equation in the calculation of the denity when the diplacement method i ued, the equation for determining the denity of the olid body follow with allowance for the air buoyancy (ee Appendix, page 58, for derivation). m = = a a m ( - ) W + Equation 31 W Thi i the manner in which denity i calculated by the oftware that come with artoriu weighing intrument. 30

34 Buoyancy Method If you include the equation m W 1 = v 1 determination uing the buoyancy method a G a for m (a) and m () in the equation for denity m(a) = m m (a) (), mathematical converion reult in W(a) = ( a) + a. Equation 32 W W (a) () the formula for calculating the denity of olid bodie with allowance for air buoyancy. Thi i the manner in which the air buoyancy i accounted for in the oftware that come with artoriu weighing intrument (for detail, ee the operating intruction for the intrument in quetion). Moreover, another correction factor i included in the calculation which take into account the additional buoyancy caued by the immerion of the wire on the pan hanger aembly (ee below). Pycnometer Method The formula for calculating denity with the air buoyancy correction uing the pycnometer method i: W2 = ( a) + a. Equation 33 W + W W Thi i the formula ued by the oftware that come with artoriu weighing intrument. Air Buoyancy Correction for the Pan Hanger Aembly Buoyancy Method For high-preciion meaurement, another conideration beide the air buoyancy i the additional buoyancy of the pan hanger wire, caued by the fact that the height of the liquid i increaed when the ample i immered which mean the wire are deeper under the urface than without the ample. (The buoyancy of the pan hanger aembly i not included in the denity calculation if the weighing intrument i tared with the pan hanger aembly.) d h D Figure 19: Diagram illutrating the calculation of the buoyancy caued by increaed height of liquid when ample i immered With the procedure for uing the buoyancy method with the Denity Determination Set, the wire of the pan hanger aembly are deeper under the urface of the water when the ample i immered 31

35 becaue the volume of the ample caue more liquid to be diplaced. Becaue more of the wire i immered, it caue more buoyancy; the additional buoyancy can be calculated and corrected for in the reult. The amount of volume by which the liquid increae in a container with diameter D correpond to the volume V in Figure 1 V = π D² h 4. Equation 34 The volume V of the ample i V m = (a) m () Equation 35 Thee two volume are identical, V = V. Uing the above equation and olving for h, the increae in height of the liquid, yield h (m m ) 4 (a) () = π D² Equation 36 The buoyancy force F BD exerted on two wire of diameter d at the uid level h i π d² F = V g = (2 h) g Equation 37 BD D 4 and with the value for h included yield F 2 d² (m(a) m ())4 BD = π g 4 π D² F BD = 2 π d 2 (m(a) m ())4g 4 π D2 2 = 2 d D (m m ) g 2 (a) () Equation 38 Thi mean that the buoyancy caued by the wire i proportional to the relation between the diameter of wire and beaker. In addition to the buoyancy of the ample which i to be determined the meaured value alo include part of the "wire buoyancy;" thu the wire buoyancy mut be ubtracted from the meaured value to yield the buoyancy of the ample alone F BS (corr): [ ] F (corr) (m m ) (2 D² d² m m ) g BS = (a) () (a) () F (corr) (1 2 D² d² ) (m m ) g BS = (a) () correction factor Equation 39 When the diameter of the wire and the beaker are known, the factor by which the value meaured for buoyancy mut be multiplied can be calculated. In the Denity Determination Set 32

36 from artoriu, the wire diameter i d = 0.7 mm, the beaker diameter D = 76 mm, and the pan hanger aembly ha two wire. Thu the correction factor i: Corr = 1 2 D² d² = ² = ² The maller the diameter d of the wire, the larger the diameter D of the beaker and the fewer wire on the pan hanger aembly, the nearer the correction factor will be to 1; i.e., the correction i negligible. Thee condition are eay to create when performing denity determination uing the below-cale or below-balance weighing method. Returning to the formula for denity determination uing the buoyancy method: For the buoyancy m(a) correction for the wire, the equation = yield m(a) m() m(a) = [m m ] Corr (a) () or, when the air buoyancy i alo accounted for, W(a) = ( a) + a Equation 40 (W W ) Corr (a) () Thi i the formula ued for denity determination in the oftware that come with artoriu weighing intrument. The weighing intrument can work with the preet value for a tandard air denity of a = g/cm 3 and a correction factor of for the artoriu Denity Determination Set. Alternatively, uer-defined correction factor can be entered. Diplacement Method Error caued by the additional buoyancy when the pan hanger aembly for the ample or line or wire of the pan hanger can be eliminated at the outet by immering the pan hanger aembly jut a deep in the liquid when it i weighed empty or tared a it will be later with the ample on it. In addition, the tet condition can be arranged to make the correction factor 1; i.e., by uing large container diameter, mall pan hanger aembly diameter, only one pan aembly if poible. The equation for calculation of the olid body denity uing the diplacement method with allowance for air buoyancy and the correction factor for the ample hanger i: W(a) = ( a) + a Equation 41 W Corr () Thi i the formula ued by the oftware that come with artoriu weighing intrument. The default etting in thi oftware include a numerical value of 1.0 for the correction factor and ra = g/cm3 for the tandard air denity. Uer-defined value can alo be entered. When the diplacement method for denity determination on liquid i ued, including on paint and varnihe, the plummet i uually made of metal (gamma phere) and i tapered in one ection (ee Figure 20); the nominal volume of the plummet i calculated to the middle of the tapered portion. There are different ize of plummet available for ue with different degree of urface 33

37 tenion in the liquid being teted. The inuence of the bulge of liquid around the tem of the plummet i alo accounted for. Area for labeling Label according to ection to 12 mm d = 3mm with a 100-ml plummet d = 1mm with a 10-ml plummet 100 ml or 10 ml up to the middle of the tapering ection. Figure 20: Plummet (in accordance with DIN Part 3) for determining the denity of paint, varnihe, and imilar coating material uing the diplacement method. Prevention of Sytematic Error Hydrotatic Method To limit the error in denity determination with different hydrotatic procedure, the following hould be oberved: The temperature mut be kept contant throughout the entire experimental procedure. With water a buoyancy medium, for example, a temperature variation of 0.1 C change the denity by to g/cm 3 ; with alcohol, by g/cm 3. The meauring intrument hould be loaded exactly in the center, to maximally limit off-center loading error. When weighing in air uing the artoriu Denity Determination Set with the ample on top of the frame, eccentric poitioning of the ample due to the hape of the frame caue force to be conducted to two external point on the bae of the frame reulting in a greater torque than with the ue of "normal" weighing pan with the ame deviation from the center. After ubmerion in the liquid there mut be no air bubble on the ample or on the pan hanger aembly. Thee caue an additional buoyancy and falify the meaured weight. To prevent thi, one can wet the ample in a eparate beaker or in an ultraound bath. Error due to adheion of liquid to the wire of the pan hanger aembly can be prevented by taring the weighing intrument with ubmered pan hanger aembly before meauring. 34

38 Air buoyancy caue an error of denity of g/cm3 (correponding with air denity under normal condition); therefore in calculating the denity, the equation hould take air buoyancy into account (ee p. 31, "Air Buoyancy Correction"). After ubmering the ample in the container, the level of the liquid rie o that the wire of the pan hanger aembly caue additional buoyancy. Depending on the diameter of the beaker ued and the number of wire of the aembly, thi additional buoyancy can be corrected (ee p. 32 Air Buoyancy Correction for the Pan Hanger Aembly.) Pycnometer Method To limit the error in determining denity with the pycnometer method, the following hould be oberved: The temperature mut be kept contant throughout the entire experimental procedure; the temperature of the ample mut be carefully controlled. With water a the liquid medium for example, a temperature change of 0.1 C change the denity by to g/cm 3 ; with alcohol, by g/cm 3. There hould be no air bubble in the liquid medium or on the ample. The air buoyancy caue an denity error of g/cm 3 (correponding to air denity under normal condition); therefore in calculating the denity, the equation hould take air buoyancy into account (ee p. 33, "Air Buoyancy Correction"). When proper care i exercied uing the pycnometer, thi method can be ued for highly accurate determination of the denity of material. Error Calculation With careful work and with the prevention of the above-mentioned ytematic error, the error of denity determination can be calculated according to the rule of error reproduction. The denity error i baed mainly on meauring reult in ma determination. Generally, for the determination of the total error F of a dimenion that i calculated from everal meauring value: With the um (and the difference) the abolute ingle error add up quadratically: 2 2 F = F + F +... Equation With product (and quotient) the relative ingle error add up quadratically. (The relative error i the abolute error in relation to the meaured value.): F F1 F = F F1 + F2 Buoyancy Method Equation 43 For the denity determination of olid bodie uing the buoyancy method the following relationhip m(a) W(a) i applied = or = ( a) + a m m (W W ) Korr (a) () (a) () 35

39 Becaue the correction factor for the pan hanger aembly and the denity of air have no noticeable effect on the error of denity, they do not need to be regarded in error calculation. If one ue the baic rule of error calculation, the abolute error of the denominator [m(a)-m()] i calculated next: [ ] 2 2 m m = m + m Equation 44 (a) () (a) followed by the total relative error of denity / () = 2 + m + (a) (m(a) m ()) m(a) m() 2 2 Equation 45 The total error of weighing in air ( m (a) ) i the um of reproducibility and a linearity error of one digit regardle of the type of weighing intrument, becaue differential weighing i performed. The maximum error of weighing in liquid ( m (a) - m () ) i on average aumed to be 10 time a great a with weighing in air thi aumption i baed on many meaurement in denity determination uing the buoyancy method. For the error of liquid denity determination, the value of g/cm 3 for water and g/cm 3 for ethanol are applied; that i the value of a thermometer reading error of ± 0.1 C which correpond to a temperature variation during meaurement of ± 0.1 C. The following figure (ee Figure 21: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity ) how the relative error of olid denity determination with the buoyancy method in water or ethanol, depending on the ample ize and the denity of the ample example, uing artoriu weighing intrument with different readabilitie and weighing capacitie. It i clear that the error in denity determination i dependent on the denity of the ample to a ignificant degree: the lower the denity of the ample, the greater the error of the end reult 4. A further figure (ee figure 27) how the error in denity determination of liquid uing the buoyancy method for liquid denitie between 0.5 and 2.2 g/cm 3 with the ue of the gla plummet included with the artoriu Denity Determination Set. The plummet ha a volume of ( ) cm 3 and a denity of 2.48 g/cm 3 with a tolerance of 0.5 mg in relation to the buoyancy of water. Here too the error of denity i dependent on the denity value of the invetigated ample. 4 For the calculation of m() = m(a) 1 i plugged in for m(); for the denity of water 1.0 g/cm 3 i ued, for the denity of ethanol g/cm 3 36

40 0.8% Readability: 1 mg Sample Denity = 2 g/cm3 0.7% Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 0.6% Relative Error of Denity / 0.5% 0.4% 0.3% Water = 1 g/cm 3 0.2% 0.1% 0.0% Sample Size / g Figure 21: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 37

41 0.3% Readability: 0,1 mg Sample Denity = 2 g/cm3 Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 Relative Error of Denity / 0.2% 0.1% Water = 1 g/cm 3 0.0% Sample Size / g Figure 22: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 38

42 0.05% Readability: 0,01 mg Sample Denity = 2 g/cm3 Sample Denity = 5 g/cm3 0.04% Sample Denity = 10 g/cm3 Relative Error of Denity / 0.03% 0.02% Water = 1 g/cm % 0.00% Sample Size / g Figure 23: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 39

43 0.8% 0.7% Readability: 1 mg Sample Denity = 2 g/cm3 Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 0.6% Relative Error of Denity / 0.5% 0.4% 0.3% Ethanol = 0,79 g/cm 3 0.2% 0.1% 0.0% Sample Size / g Figure 24: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 40

44 0.3% Readability: 0,1 mg Sample Denity = 2 g/cm3 Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 RelativeError of Denity / 0.2% 0.1% Ethanol = 0,79 g/cm 3 0.0% Sample Size / g Figure 25: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 41

45 0.05% 0.04% Readability: 0,01 mg Sample Denity = 2 g/cm3 Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 Relative Error of Denity / 0.03% 0.02% Ethanol = 0,79 g/cm % 0.00% Sample Size / g Figure 26: Solid denity determination uing the buoyancy method the relative error of denity i dependent on the ample ize and the ample denity 42

46 0.20% 0.19% 0.18% Readability: 0,01 mg Readability: 0,1 mg Readability: 1 mg Relative Error of Denity / 0.17% 0.16% 0.15% 0.14% 0.13% 0.12% 0.11% 0.10% Liquid Denity / g/cm³ Figure 27: Liquid denity determination uing the buoyancy method the relative error of denity i dependent on the ample denity and the readability of the weighing intrument for denity determination with the artoriu Denity Determination Set 43

47 Diplacement Method For olid body denity determination uing the diplacement method the following relationhip i applied, in which air denity a i taken a contant and not ued in the error calculation. m = = a m ( - ) W + W = 2 + m + (a) m m(a) m 2 a () () 2 5 In comparion with the buoyancy method, it i triking (ee Figure 21 through 26) that olid body denity determination uing the buoyancy method reult in a maller error than the diplacement method (ee Figure 28) Aide from thi it i clear that uing the buoyancy method the mitake decreae with increaing ample denity, wherea uing the diplacement method it increae with increaing ample denity. 0.8% Readability: 1 mg Sample Denity = 2 g/cm3 0.7% Sample Denity = 5 g/cm3 Sample Denity = 10 g/cm3 0.6% Relative Error of Denity / 0.5% 0.4% 0.3% Water = 1 g/cm 3 0.2% 0.1% 0.0% Sample Size / g Figure 28: Solid denity determination uing the diplacement method the relative error of denity i dependent on the ample ize and the ample denity 5 With m() = m (a) 44

48 Pycnometer Method For denity determination uing the pycnometer method, apply the relationhip m W 2 2 = m1+ m2 m and = ( a) + a. 3 W + W W The error of air denity i neglected again. To get the total error of the weight value, a linear error of one digit wa added to the reproducibility of the weighing intrument type. For the error of liquid denity a value of g/cm 3 i ued, which correpond to the value of a fale readout of the thermometer of ± 0.1 C or a temperature change of ± 0.1 C. The liquid denity with 1.0 g/cm 3 for water a medium i ued in the equation below.. Next the error of the denominator [m 1 +m 2 -m 3 ] i calculated [ ] 2 2 m + m m = m + m + m and then the total relative error of denity / 2 = + m 2 + (m1 + m2 m 3) + m2 (m1 m2 m 3) 2 2. The following diagram (ee Figure 29 to 31) for weighing intrument with different readabilitie and weighing capacitie how the relative error of denity determination dependent on the ample ize and the ample denity. 45

49 0.05% 0.04% Readability: 1 mg Sample Denity = 4 g/cm³ Sample Denity = 3 g/cm³ Sample Denity = 2 g/cm³ Sample Denity = 1 g/cm³ Relative Error of Denity / 0.03% 0.02% Water = 1 g/cm % 0.00% Sample Qantity / g Figure 29: Denity determination uing the pycnometer method the relative error of denity i dependent on the ample ize and the ample denity. 46

50 0.010% Readability: 0,1 mg Sample Denity = 4 g/cm³ Sample Denity = 3 g/cm³ Sample Denity = 2 g/cm³ Sample Denity = 1 g/cm³ Relative Error of Denity / 0.005% Water = 1 g/cm % Sample Quantity / g Figure 30: Denity determination uing the pycnometer method the relative error of denity i dependent on the ample ize and the ample denity. 47

51 0.005% Readability: 0,01 mg Sample Denity = 4 g/cm³ Sample Denity = 3 g/cm³ Sample Denity = 2 g/cm³ Sample Denity = 1 g/cm³ Relative Error of Denity / 0.004% 0.003% Water = 1 g/cm % Sample Quantity / g Figure 31: Denity determination uing the pycnometer method the relative error of denity i dependent on the ample ize and the ample denity. 48

52 Comparion of Different Method of Denity Determination On the following page different method of denity determination are contrated, o that the mot important advantage and diadvantage may be recognized at a glance. 49

53 Buoyancy method Diplacement method Pycnometer method Weighing a defined volume Suitable for: Solid Solid Solid Powder, Granule Powder (tap denity) Liquid Liquid Liquid Liquid Diperion Diperion Diperion Diperion Gae Advantage Diadvantage Suitable for almot all Suitable for almot all ample type ample type Suitable for all ample type Flexible with regard to ample ize Flexible with regard to ample ize Weighing intrument already available Weighing intrument already available Weighing intrument already available Weighing intrument already available Quick proce Quick proce Accurate method Quick proce All method epecially eay to perform uing weighing intrument with integratedoftware with uer guidance prompt and evaluation of reult Solid and liquid mut be brought to a defined temperature Large volume ample required for uid denity determination Care mut be taken to avoid evaporation Sample mut be wet very Sample mut be wet very carefully carefully Solid and liquid mut be brought to a defined temperature Labor-intenive time-conuming Bubble mut not be trapped Bubble mut not be trapped Bubble mut not be trapped 50

54 Buoyancy method Diplacement method Pycnometer method Weighing a defined volume Uncertainty of Dependent on the weighing intrument ued and the ample Dependent on the weighing meaurement * amount and ample denity intrument ued Readability 1 mg Solid: Solid: Liquid, Diperion <0,2% for m>50g (water), <0,4% for m>10g <0,02% for m>20g =5g/cm³ Liquid: ** <0,20% for =1,3g/cm³ Readability 0,1 mg Solid: Solid: <0,005% for m>10g <0,1% for m>5g <0,02% for m>50g (water), =5g/cm³ Liquid: ** <0,11% for <1,8g/cm³ Readability 0,01 mg Solid: <0,003% for m>10g <0,10% for m > 5g (water) <0,15% for m > 5g (ethanol) Liquid: ** ~0,1% for <1,5g/cm³ Dependent on the weighing intrument ued <1% for V=100ml, <0,1% for V=1000ml and <2g/cm 3 *For more exact reult ee the chapter entitled "Error in and Preciion of Denity Determination" **With the plummet from the Sartoriu Denity Determination Set 51

55 Hydrometer Ocillation-method Denity gradient column Suitable for: Solid (mall ample ize) Liquid (Homogeneou) liquid (Liquid) (Diperion) Advantage Eay meaurement Quick proce Inexpenive plummet Small ample ize: approx. 1ml Quick proce Several ample can be checked imultaneouly Diadvantage In diperion: meauring error can be caued by eparation of the component in the ample Denity i lightly inuenced by the vicoity of the ample Expenive equipment mut be purchaed Time-conuming experiment preparation Uncertainty of meaurement 0.1 to 10 kg/ m³ or to 0.01 g/cm³ for r=0.6 to 2.0 g/cm³ Hydrometer are uited to certain area of ample urface tenion There are emauring device with uncertainy of meaurement rated to 0.001g/cm 3, g/cm 3 or g/cm 3 Calibrated gla reference with denitie of 0.8 to 2.0g/cm 3 ± g/cm 3 52

56 Appendix Temperature Dependency of Denity The dependence of denity on temperature may be calculated with the aid of the volume expanion coefficient γ. In general, the expanion coefficient i only given for a et temperature range (e.g. 20 C to 100 C) for which a linear approximation i permiible. Numerical value for γ of different gae and liquid may be found in phyical chemitry table. Uually, with olid bodie the expanion coefficient α i given. To convert the linear calculation into the volume expanion coefficient, ue the relationhip γ 3 α. The denity of a ubtance at temperature T 2 can be calculated with the aid of the temperature T 1 and the volume expanion coefficient: (T 1) (T 2) =. 1 + γ (T T )

57 Hydrotatic Denity Determination Elimination of the Volume in the Equation for With complete ubmerion of the body in the liquid, the experimental method reult in the volume of the olid and the volume of the liquid being equal. One can derive a relationhip between the mae and denitie of the two ubtance, in which volume i no longer explicitly included. m m = V = V otherwie: V m = V from V = V m m = m = For determination of the olid denity, it follow: m = m Or for determination of the liquid denity: m m = =. m V 54

58 Air Denity Determination The relationhip for determining air denity i derived from the equation for the relationhip between the ma and the weight value for aluminum and for teel: W Al 1 = mal 1 W m 1 St = St 1 a Al a N a St a N 1 m 1 Al = W 1 Al 1 m 1 St = W 1 St a Al a N a St a N Standardizing the equation to 1 and joining them give: m W St St 1 1 a St a N m 1 Al = W 1 Al a Al a N a Al a ( ) ( ) mst W 1 m = W 1 St St Al a a ( ) St Al ( ) St Al W m 1 = W m 1 Al St a a WAl mst WAl mst = WSt mal WSt mal l St a a WSt mal WAl mst = W m W m Al Al St A St Al Al St WSt mal WAl m WSt mal WAl mst = a Al St St a = W m l W m WSt mal WAl mst ( ) Al St St A Al St 55

59 Air Buoyancy Correction With the example of denity determination by the diplacement method, the formula for calculating the denity with regard to air buoyancy i derived: Denity i calculated by m =, m for the ma the following relationhip that decribe the dependency of the ma on air denity i ubtituted: m W 1 a G = v a 1 a a W (1 ) (1 ) G = a a (1 ) W (1 ) a W (1 ) = a (1 ) W W = a W 1 W = ( a) W a 1 W W ( ) 1 = G a a ( a) = W a W ( ) a a W = a W ( ) = W a a W ( ) = W W ( ) + a a 56

60 Quetion About Denity 1. How i denity defined and what i the unit of denity? 2. How doe denity change when temperature increae? 3. Why do bodie appear to be lighter in water than in air when they are weighed? Decribe the phenomenon. 4. When do bodie "oat" in a liquid? Which tatement then i valid for the denity of liquid and olid? 5. What i the difference between the tructure of the experiment for denity determination uing the buoyancy and uing the diplacement method, and what do the meaured value include with each method? 6. With which method can the denity of uid be determined? Why are uid denitie determined; which concluion can one reach from the meaured value? 7. What i a diperion and how can one meaure it denity? 8. What i the difference between denity and bulk denity how can one determine the denity of porou material? 9. What i the denity of air; when mut one know the air denity and how can one determine it? 10. How do you determine the denity of powder? 11. Which product or material propertie can be controlled by meauring denity? 12. What relevance ha denity determination in the realm of the prepackage indutry for average weight control? 13. On what i the accuracy of denity determination dependent and how doe one meaure the error of denity value? 14. Explain the meaning of the individual ymbol and factor in the following formula; what i included in Corr? Rho = (Wa * (Rho - LA)) / ((Wa - W) * Corr) + LA Rho = (Wa * (Rho - LA)) / (W * Corr) + LA Rho = (Wa * (Rho - LA)) / (W + Wa - Wr) + LA 15. Which advantage are offered by the application tare memory in the LA/FC weighing intrument in denity determination with the pycnometer method? 16. On product uch a mutard, the amount of the full package mut be given in ml. During filling and checkweighing, however, the weight (or ma) i checked. The denity i neceary a a converion factor between ma and volume. Which denity determination method do you recommend to your cutomer when high accuracy i deired? 57

61 when a quick reult i important? 17. A client want to know the denity of melted gla (in a laboratory oven with the opening on top) at 1200 C what do you recommend? 58

62 Tip for anwering the quetion 1. Denity = Ma / Volume 1000 kg/m³ = 1 kg/dm³ = 1 g/cm³ = 1 g/ml 2. The denity decreae (ee p. 3). 3. Reultant force = Weight force minu buoyancy force. The buoyancy i dependent on the hydrotatic preure in the liquid p = g h; F = p A = g h A = g V (ee p. 7 8) 4. The denity of the olid body and the liquid are the ame (ee p. 9). 5. The beaker with liquid for buoyancy tand on the weighing pan uing the diplacement method, it ha no contact with the weighing pan uing the buoyancy method. Meaured value correpond with the ma of the diplaced uid uing the diplacement method. Meaured value correpond with the ma reduced by the buoyancy in the buoyancy method (ee p. 10). 6. Buoyancy method, diplacement method, weighing a defined volume, hydrometer, ocillation method, upenion method (not with mixable uid). Concluion from the concentration relationhip, AWC: gravimetric rather than volumetric filling; ee p. 27 and table, p Multi-phae ytem coniting of continuou phae (matrix) and one or everal finely divided phae (dipered phae) (ee p. 28) Choice of the method i dependent on the accuracy required, the conitency and owing property of the diperion: either a for liquid or the pycnometer method. Be careful of the inuence of phae eparation uing the different method, with the pycnometer method the eparation ha no effect on the reult. 8. With porou material: bulk denity relate to the total volume including open and cloed pore true denity relate to the olid volume. Denity determination method with porou ample: Buoyancy method (ee p. 22) or pycnometer method for determination of true denity after grinding the ample to a powder. Advantage of the pycnometer method: a higher meaured accuracy i poible, exiting cloed pore are not attributed to the olid. 9. Luft = 0,0012 g/cm³ With analytical and microbalance the weighed value mut be corrected to yield the ma accurately. With two weight of different denitie and thereby different volume, and of more or le the ame ma, air denity can be determined with a microbalance. The conventional ma value of the weight and the denity of the material mut be known before air 59

63 denity can be calculated from the meaured value (ee p. 13 included in the oftware of Sartoriu micro- and ultra-microbalance). 10. Uing the pycnometer method. For a decription, ee p. 16, Poroity, void, crytal content (crytalline phae have a higher denity than non-crytalline gla phae), cooling rate with gla, concentration of an ingredient in a olution, olid component in upenion... ee p Volumetry i replaced by gravimetry, a more accurate and impler meauring method. The proportionality factor of ma and volume i the denity (verified device mut be ued). 13. Careful maintenance of the experimental condition, for example temperature... ee p. 35. The error of the meaured value i dependent on readability, repeatability... of the weighing intrument. The total error of the reult mut be calculated uing the rule of error reproduction ee p See intruction for the LA and FC weighing intrument model. 15. Save time becaue the procedure for drying the pycnometer can be kipped; ee p ??? 17.??? 60

64 Regiter A Adheion 33 Air buoyancy 29; 34 Air Buoyancy Correction 29 Air denity 12; 29 Apparent denity 2 Apparent poroity 22 Archimedean Principle 5 B Bulk denity 2; 22 Buoyancy 6; 7 Buoyancy method 9 Buoyancy of the pan hanger aembly 30 C Cloed pore 21 Cloed poroity 22 Coare dipere ytem 27 Colloid dipere ytem 27 Conventional ma 13 Correction factor 32 D Denity gradient column 17 Denity of a olution 26 Denity of liquid 26 Diperion 27 Diplacement method 9; 11 E Emulion 27 Error calculation 34 Error reproduction 34 F Foam 27 G Gamma phere 32 Granule 24 H Heat expanion coefficient 2 Homogenou liquid 17 Hydrometer 19 Hydrotatic balance 9 Hydrotatic weighing method 9 L Liquid 25 M Mit 27 Mohr balance 9 Molecular dipere ytem 27 N Normal denity 1; 2; 12 O Off-center load error 33 Open pore 21 Open poroity 22 Ocillation method 17 P Plummet 28 Pore 21 Poroity 21 Porou material 21 Powder 24 Pycnometer method 15 Pycnometer 14 R Relative denity 2 S Sartoriu Denity Determination Set 32 Schlieren method 18 Smoke 27 Solid denity 2 Specific gravity 2 Spindle 19 Surface tenion 33 Supenion 27 Supenion method 17 T Tap denity 1 Temperature dependence on 2 Temperature variation 33; 35 Total error 34 True denity 2; 21 61

65 W Water aborption 24 Weight preure 5 Weight value 13; 29 62