text: f 0 nominal drive frequency ( under normal thermal conditions ) as chosen by the manufacturer zero ( f ) term describing the effect frequency has on nominal time delay , δt 0 . equal to δt 0 +( f − f 0 ) α f α { dot over ( m )} proportionality coefficient relating δt to mass flow . similar to fcf in the prior art . α { dot over ( m )} f linear frequency ( temp ) coefficient for α { dot over ( m )} . similar to α in the prior art . fig1 is a perspective view of a first possible exemplary embodiment of a coriolis flow meter embodying the invention . it discloses a flow meter 100 having a flow tube 102 inserted through legs 117 , 118 of base 101 . pick - offs lp 0 and rp 0 and driver d are coupled to flow tube 102 . flow meter 100 receives a process material flow from supply tube 104 and extends the flow through process connection 108 to flow tube 102 . flow tube 102 is vibrated at its resonant frequency with material flow by driver d . the resulting coriolis deflections are detected by pick - offs lp 0 and rp 00 which apply signals over conductors 112 and 114 to coriolis flow meter electronics 121 . coriolis flow meter electronics 121 receives the pick - off signals , determines the phase difference between , determines the frequency of oscillation and applies output information pertaining to the material flow over output path 122 to a utilization circuit not shown . meter electronics 121 is shown in further detail on fig1 . the material flow passes from flow tube 102 and through tube 106 which redirects the material flow through return tube 103 through process connection 107 to exit tube 105 which delivers the material flow to a user application . process connections 107 , 108 , 109 and 110 connect tubes 104 , 105 and 106 to the ends of flow tube 102 and return tube . the process connections have a fixed portion 111 that includes threads 124 . locking holes 130 receive set screws 411 to fixably connect element 111 to base 101 as shown in fig4 . the movable portion of process connections 107 through 110 are threaded onto male threads 124 to connect their respective tubes to the fixed body of the process connection of which the hexagonal nut portion 111 is a part . these process connections function in a manner similar to the well known copper tubing flared process connections to connect tubes 104 , 105 and 106 to ends of flow tube 102 and return tube 103 . details regarding the process connections are further shown in fig4 . in fig2 is a top view of flow meter 100 of fig1 . pick - offs lp 0 and rp 0 and driver d each include a coil c . each of these elements further includes a magnet which is affixed to the bottom portion of flow tube 102 as shown in fig3 . each of these elements further includes a base , such as 143 for driver d , as well as a thin strip of material , such as 133 for driver d . the thin strip of material may comprise a printed wiring board to which coil c and its winding terminals are affixed . pick - offs lp 0 and rp 0 also have a corresponding base element and a thin strip fixed to the top of the base element . this arrangement facilitates the mounting of a driver or a pick - off to be accomplished by the steps of gluing a magnet m to the underside of flow tube , gluing the coil c to a printed wiring board 133 ( for driver d ), positioning the opening in coil c around the magnet m , moving the coil c upwardly so that the magnet m fully enters the opening in coil c , then positioning base element 143 underneath the printed wiring board 133 and gluing these elements together so that the bottom of base 143 is affixed by glue to the surface of the massive base 116 . the male threads 124 of process connections 107 - 110 are shown on fig2 . the inner details of each of these elements are shown on fig4 . opening 132 receives conductors 112 , 113 and 114 . coriolis flow meter electronics 121 of fig1 is not shown on fig2 to minimize drawing complexity . however it is to be understood that the conductors 112 , 113 and 114 extend through opening 132 and further extend over path 123 of fig1 to coriolis flow meter electronics of fig1 . fig3 shows pick - offs lp 0 , rp 0 and driver d as comprising a magnet m affixed to the bottom portion of flow tube 102 and a coil c affixed to the base of each of elements lp 0 , rp 0 and driver d . fig4 is a sectional taken along line 4 - 4 of fig2 . fig4 discloses all the elements of fig3 and further details of process connections 108 and 109 and o - rings 430 . o - rings 430 couple flow tube 102 to base 401 . fig4 further discloses openings 402 , 403 and 404 in base 101 . the top of each of these openings extends to the lower surface of the base of pick - offs lpo , rpo and driver d . the coil c and magnet m associated with each of these elements is also shown on fig . 4 . coriolis flow meter electronics 121 of fig1 is not shown on fig3 and 4 to minimize drawing complexity . element 405 in process connection 108 is the inlet of flow tube 102 ; element 406 in process connection 109 is the outlet of flow tube 102 . the fixed portion 111 of process connection 108 includes male threads 409 which screw into mating threads in receiving hole 420 located in base 401 to attach fixed portion 111 to segment 401 of base 101 . the fixed portion of process connection 109 on the right is similarly equipped and attached by threads 409 into receiving hole 420 located in element 401 of base 101 . fixed element 111 of process connection 108 further includes a threaded portion 124 whose threads receive the movable portion 415 of process connection 108 . process connection 109 is similarly equipped . fixed element 111 of process connection 108 further includes on its left a conical stub 413 which together with movable element 415 acts as a flare fitting to force the right end of input tube 104 over the conical stub 413 of fixed portion 111 . this creates a compression fitting that sealably affixes the flared opening of supply tube 104 onto the conical stub portion 413 of fixed portion 111 of the process connection . the inlet of flow tube 102 is positioned in process connection fixed portion 111 and is flush with face 425 of stub 413 . by this means , the process material delivered by supply tube 104 is received by inlet 405 of flow tube 102 . the process material flows to the right through flow tube 102 to fixed portion 111 of process connection 109 where the outlet 406 of flow tube 102 is flush with face 425 of stub 413 . this sealably affixes the outlet of flow tube 102 to connector 109 . the other process connections 107 and 110 of fig1 are identical to those described for the details of process connections 108 and 109 on fig4 . the invention achieves temperature compensation of the mass flow output of a coriolis flow meter by using drive frequency as an indicator of flow tube temperature change . coriolis flow meters measure mass flow directly by calculating the time delay ( δt ) between the inlet and outlet ends of the active portion of the vibrating flow tube of the coriolis flow meter . an offset time delay at zero flow ( δt 0 ) is measured and subtracted from the calculated time delay during flow to produce a value that is directly proportional to mass flow via a proportionality constant α { dot over ( m )} which is termed calibration factor ( fcf ) in the prior art . fluid density also affects the drive frequency of the flow tube . to understand the sensitivity of frequency to temperature and density , the density effect must be characterized and compared to the temperature effect . the coriolis flow meter of the present invention may be used with slurries whose specific gravities are quite limited . when frequency is to be used for temperature compensation , the affect of changing fluid density must also be evaluated and determined to determine mass flow . the flow tube materials respond to changes in temperature . these changes must be considered to accurately calculate mass flow rate . traditionally , rtd thermal sensors are used to directly measure the temperature . they are positioned on an inactive surface of the coriolis tube . this is generally done outside of the brace bar ( s ). the disclosed coriolis flow meter has a single - straight flow tube . there is no inactive section of the flow tube on which a temperature measurement representative of the flow tube can be made by a thermal sensor positioned on the flow tube without affecting output accuracy . the present invention detects changes in flow tube temperature by monitoring drive frequency . using drive frequency in accordance with the present invention to determine temperature changes and compensate for flow output raises the following issues . the prior art measures temperature using an rtd thermal sensor mounted on a secondary or inactive portion of the flow tube and infers the coriolis flow tube temperature . the method and apparatus of the present invention improves accuracy of the temperature determination and also improves the meter response time in detecting temperature changes . when using frequency as a means to detect and compensate for temperature changes , a taylor series expansion on δt isolates the effects of frequency and mass flow on δt . the expansion is arranged to look like a flow equation and a least - squares pseudo - inverse problem is set up . assume that the pick - off delay , δt , is a function of mass flow and drive frequency . this can be expanded about an operating point with a taylor series : δ ⁢ ⁢ t = f ⁡ ( m . 0 , f 0 ) + ( m . - m . 0 ) ⁢ ∂ f ∂ m + ( f - f 0 ) ⁢ ∂ f ∂ f + 1 2 ⁢ ( m . - m . 0 ) ⁢ ( f - f 0 ) ⁢ ∂ 2 ⁢ f ∂ m . ⁢ ∂ f + … 1 . 2 equation 1 . 2 is a full expansion and could be taken out to an infinite number of higher order terms . an optimal fit can be achieved by taking terms that effect zero flow and fcf linearly with temperature and terms that effect zero and fcf quadratically with temperature . however , over the operating temperature of the sensor ( 18 - 28 c ) of the present invention , behavior is linear enough to use only frequency terms effecting fcf and δt 0 linearly . doing this and re - labeling the partial derivatives one obtains : δ ⁢ ⁢ t ≅ δ ⁢ ⁢ t 0 + ( m . - m . 0 ) ⁢ α m . + ( f - f 0 ) ︸ zero ⁢ ⁢ term ⁢ ⁢ linear in ⁢ ⁢ frequency ⁢ α f + ( m . + m . 0 ) ⁢ ( f - f 0 ) ︸ flow ⁢ ⁢ term ⁢ ⁢ linear in ⁢ ⁢ frequency ⁢ α m . ⁢ ⁢ f 1 . 3 now , rearrange the terms by grouping those associated with zero flow and those associated with flow : δ t ≅[ δt 0 +( f − f 0 ) α f ]+( { dot over ( m )}−{ dot over ( m )} )[ α { dot over ( m )} +( f − f 0 ) α { dot over ( m )} f ] 1 . 4 for convenience choose nominal { dot over ( m )} 0 = 0 , equation 1 . 4 can then be arranged to look like a flow equation : m . = δ ⁢ ⁢ t - δ ⁢ ⁢ t 0 - ( f - f 0 ) ⁢ α f [ α m . + ( f - f 0 ) ⁢ α m . ⁢ ⁢ f ] 1 . 5 drawing an analogy between equations 1 . 5 and 1 . 6 , it can be seen that the portion of equation 1 . 5 relative to “ zero ” as a function of frequency is : and the portion of equation 1 . 5 relating to “ flow calibration factor ” as a function of frequency is : fcf ⁡ ( f ) = 1 [ α m . + ( f - f 0 ) ⁢ α m . ⁢ ⁢ f ] 1 . 8 rewriting equation 1 . 4 in the form of a vector equation sets up the least - squares pseudo - inverse problem : δ ⁢ ⁢ t = [ 1 ( f - f 0 ) m . m . ⁡ ( f - f 0 ) ] ⁡ [ δ ⁢ ⁢ t 0 α f α m . α m . ⁢ ⁢ f ] 1 . 9 assuming we can record δt and drive frequency from the coriolis flow meter while we record mass flow from a series connected master coriolis flow meter , we can solve for the column vector in equation 1 . 4 by multiplying both sides by the pseudo - inverse of the row vector : [ δ ⁢ ⁢ t 0 α f α m . α m . ⁢ ⁢ f ] = pinv ⁡ ( [ 1 ( f - f 0 ) m . m . ⁡ ( f - f 0 ) ] ) ⁢ δ ⁢ ⁢ t 1 . 10 equation 1 . 10 is the equation we need used by the present invention to optimally characterize the sensor &# 39 ; s temperature dependence . as mentioned earlier , drive frequency has a linear dependence on temperature . because of this , compensating flow for temperature effects using drive frequency should is advantageous . fig5 illustrates the relationship between drive frequency and temperature for two flow tube sensors on lines 501 and 502 . the flow tube sensors were placed in an oven whose temperature was cycled between 15 and 35 c . a linear trend line was fit to each data set of lines 501 and 502 and the specified operating range for the sensor is shown between dotted lines 503 and 504 . using the slope of each trend line 501 and 502 , we can estimate the sensitivity of drive frequency as 14 hz per full scale temperature swing . fig6 graphs drive frequency versus fluid specific gravity for the same two sensors on trend lines 601 and 602 . the sensors are specified to measure flow rate of slurries whose fluid densities are from 1 . 0 to 1 . 3 sg as shown for trend lines 603 and 604 . three fluids were used to span the density operating range of the sensor . three data points were taken for each sensor and a trend line was fit to the data . using the slope of each trend line , we can estimate the sensitivity of drive frequency as 4 hz per fs density swing . this seems significant , but is should be noted that each coriolis flow meter is installed on a particular process using a particular fluid in this range . further , once installed , users typically calibrate their process to the actual flow output , eliminating errors due to new process fluid in the device . fig7 - 12 describe the method by which the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f of flow equation 1 . 5 are derived using the calibration procedure of the present invention . these calibration constants are derived using equation 1 . 10 . once derived , these are used in flow equation 1 . 5 together with the measured values of drive frequency f and δt ( flow induced time delay ) to determine the measured flow rate { dot over ( m )}. this is described in detail in the following paragraphs . the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f are derived using a four level procedure as next described . a master coriolis flow meter and the coriolis flow meter to be calibrated are connected in series and a test flow is applied . the flow data corresponding to runs 1 - 4 of table 1 below is used for the calibration . the test matrix for identifying mass flow of the master test meter and temperature / frequency contributions to δt is shown in table 1 . the measured mass flow from the master coriolis flow meter and the series connected tested coriolis flow meter are shown in fig7 . the measured mass flow from the master coriolis flow meter and the series connected test coriolis flow meter using the four test runs of table 1 are shown in fig7 . the four runs are shown of fig7 together with the corresponding values of temperature , drive frequency and { dot over ( m )} for the master coriolis flow meter are used . line 703 represents the flow rate of the master coriolis flow meter . fig8 illustrates the data obtained for the four tests runs of fig7 . this is the data profile that is used , as subsequently described for equation 1 . 10 , to derive the values of the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f . line 803 represents the flow rate of the master coriolis flow meter . fig9 shows the results of using both frequency and temperature ( rtd ) compensation for steady flow states . both methods give a good data fit for calibration of the coriolis flow meter during steady state flow conditions . fig9 shows drive frequency on line 901 , temperature on line 902 , flow rate { dot over ( m )} for each run on line 903 and the error rate on line 905 . the results of both methods of compensation are represented by line 905 since they are both congruent for steady state flows . this accuracy is indicated by zero error for both methods across line 905 all four test runs . meter electronics 121 , shown on fig1 and 15 , executes the data processing functions required to implement the operations shown on fig7 - 9 . fig1 is a block diagram 1000 describing how the method and apparatus of the present invention derives the calibration constants . fig1 discloses a plurality of processing or program steps each representing one or more program instructions stored in memory 1502 of coriolis flow meter electronics 121 . the instructions are executed by processor 1502 of the coriolis flow meter electronics 121 with the results either being stored in memory 1501 of the flow meter electronics or being outputted to a user over path 122 . the process of fig1 derives the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f . the derived calibration constants are inputted to fig1 where they are used to derive a temperature compensated mass flow rate { dot over ( m )}. on fig1 , element 1002 receives the input pick - off signals and from lpo and rpo of fig1 and applies them to element 1004 which uses the received information to derive the detected δt and the drive frequency f of the flow meter under test . element 1006 receives from element 1003 the signals representing the measured flow rate { dot over ( m )} of the series connected reference flow meter . element 1006 receives the above indicated information from elements 1004 and 1003 and uses them to derive the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f by solving equation 1 . 10 . the derived calibration constants are extended from element 1006 to element 1106 of fig1 . element 1102 receives the input pick - off signals from pick - offs lpo and rpo of fig1 for the flow meter under tests . this information is extended to element 1104 which derives the δt and the drive frequency f for the flow meter under test and extends them to element 1106 . element 1106 receives the outputs of elements 1104 and 1006 to derive the temperature compensated mass flow rate of the flow meter under test by using the equation 1 . 5 indicated in element 1106 . the terms δt and δt 0 are derived by element 1006 . the expression f − f 0 in both the numerator and dominator of equation 1 . 5 are obtained by element 1106 from element 1104 . the term α f is obtained by the element 1004 . the terms α { dot over ( m )} and α { dot over ( m )} f are obtained from element 1104 . element 1106 employs all of the terms on the right side of equation 1 . 5 to provide a temperature compensated mass flow rate output to a user over output 122 of fig1 from meter electronics 121 . fig1 shows the method the present invention employs to derive the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f using equation 1 . 10 . the derivation of the calibration constants δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f of the flow meter under test ( fmut ) requires : the actual flow measurement from a series connected reference flow meter connected in series with the coriolis flow meter under test . temperature measurement from the fmut . drive frequency measurement from the fmut . these measurements are made while the fmut is subjected to process conditions shown in table 1 as shown below . the actual reference mass flow and fluid temperature may differ from the ideal reference mass flow and the fluid temperature due to limitations in the controllability of the flow stand on which the fmut is being calibrated . for each run of table 1 , multiple measurements of actual mass flow , the fmut temperature and the fmut drive frequency are made . let : m =# of measurements of the three aforementioned quantities during run 1 . n =# of measurements of the three aforementioned quantities during run 2 . o =# of measurements of the three aforementioned quantities during run 3 . p =# of measurements of the three aforementioned quantities during run 4 . during the calibration runs of fig8 , the reference mass flow { dot over ( m )}, the fmut temperature and the fmut drive frequency are each be measured m + n + o + p times . it is required that m + n + o + p be at least equal to the number of parameters being solved using equation 1 . 10 . however , it is recommended that the sum m + n + o + p be much greater than the number of parameters being solved for . this is the definition of a least - squares problem , which is what we are trying to solve . to eliminate confusion , let &# 39 ; s assume m = n = o = p , so that the total number of measurements of reference mass flow , the fmut temperature and the fmut frequency is 4 times m . dimensionally , the calibration problem shown in equation 1 . 10 , dictated by table 1 and depicted in fig8 is also shown in fig1 . once the data of fig8 is taken and the equation 1 . 10 is solved , four values are obtained , namely δt 0 , α f , α { dot over ( m )} , α { dot over ( m )} f . these are the desired calibration constants . the term δt 0 is the zero offset of the fmut . this value comes from zeroing the meter in the traditional sense ( i . e . pushing the zero button ). the term α f is the linear frequency ( hence temperature ) effect on zero offset of the fmut . the term α { dot over ( m )} is the flow cal factor ( fcf ) of the fmut . the term α { dot over ( m )} f is the linear frequency ( hence temperature ) effect on fcf of fmut . these calibration constants can now be used in equation 1 . 5 , to calculate fmut mass flow compensated for frequency ( hence temperature ) effects . meter electronics 121 , shown on fig1 and 15 , executes the data processing required to implement the operations shown on fig1 . fig1 shows the results of using frequency determination to achieve thermal compensation as compared to the use of rtd thermal sensors to monitor flow tube frequency and provide thermal compensation . fig1 clearly shows the merit of frequency based temperature compensation as compared to the use of the rtd . line 1303 represent the mass flow provided by the master coriolis flow meter . solid line 1308 portrays the results obtained using rtd sensors . dotted line 1309 illustrates the use of frequency monitoring as provided in accordance with the present invention . line 1303 portrays the drive frequency . line 1302 portrays the temperature . line 1301 portrays the resultant mass flow rate . in the straight portions of line 1303 , the results of the master , the rtd and the use of frequency overlay one another the results diverging only in the portions represented by portions 1305 , 1306 , and 1307 which represent sudden changes in mass flow rate . element 1305 contains lines 1303 , 1308 , and 1309 . line 1303 represents the mass flow rate of the master . line 1308 represents the response of the rtd . line 1309 represents the response using frequency monitoring . it can be seen that the line 1303 and 1309 responses are essentially the same in the period of transition represented by element 1305 . in can also be seen that the response 1308 of element 1305 indicates that the response of the rtd diverges considerably from that compared to that of the master . the same observation may be made with respect to elements 1306 and 1307 where the results obtained using frequency compensation in portrayed by line 1309 follow more closely the response of the master as portrayed by line 1303 then does the results portrayed by the line 1308 representing the results obtained using rtd thermal compensation . the mass flow rate { dot over ( m )} 1 ,{ dot over ( m )} 2 ,{ dot over ( m )} 3 , and { dot over ( m )} 4 on fig . 13 correspond to the correspondingly designated elements on fig7 , 8 , and 9 . the conclusion to be reached from an analysis of fig1 is that the use of frequency monitoring to provide thermal compensation is far superior to that of the use of rtd when accuracies is desired during transient conditions of the mass flow rate . meter electronics 121 , shown on fig1 and 15 , executes the data processing required to implement the operations shown on fig1 . fig1 discloses the comparative accuracies of the response using frequency monitoring as compared to the use of rtd . line 1401 portrays the drive frequency , line 1402 portrays the temperature , and line 1403 portrays the mass flow rate . line 1414 portrays the compensation error produced by the use of frequency monitoring as compared to the use of rtd . it can be seen , use of rtd versus frequency monitoring produces comparable results when the mass flow rate is essentially constant . however , the elements 1405 , 1406 , and 1407 indicate conditions where the mass flow rate is undergoing transients . as shown for the element 1405 , the response of using frequency monitoring 1415 corresponds to the mass flow rate 1414 of the master more closely than does the response 1405 of the rtd . meter electronics 121 , shown on fig1 and 15 , executes the data processing required to implement the operations shown on fig1 . as shown on fig1 , meter electronics 121 includes a processor 1501 and a memory 1502 . processor 1501 may comprise a conventional cpu . alternatively it may comprise a special or a general - purpose processor or a dsp . memory 1502 may comprise any type of memory system , such as flash memory or a conventional rom and ram , for the storing information on both a long - term and short - term basis . meter electronics 121 receives input information over path 123 and applies its output information over path 122 to a user . meter electronics 121 receives input signals over path 123 from pickoffs lpo and rpo . these pickoff signals are shown on fig1 in element 1002 . the pickoffs signals are applied from element 1002 to element 1004 which performs the indicated functions and outputs its information to element 1006 . elements 1002 and 1004 are duplicated on fig1 as elements 1102 and 1104 . element 1006 on fig1 receives the outputs of elements 1004 and 1003 and derives the indicated calibration constants . memory 1502 and processor 1501 of fig1 are used for this function . element 1106 on fig1 receives the outputs of element 11006 and element 1104 and derives the indicated the mass flow rate by solving equation 1 . 5 . these functions are performed by the memory 1502 and the processor 1501 of fig . 15 . memory 1502 and processor 1501 of meter electronics 121 are also used to perform the various computations and functions shown in on fig5 , 6 , 7 , 8 , 9 , 12 , 13 and 14 . it is to be expressly understood that the claimed invention is not to be limited to the description of the preferred embodiment but encompasses other modifications and alterations within the scope and spirit of the inventive concept . while specific relationships and equations have been described in connection with the invention , it is to be understood that the invention includes and may be practiced using modifications of the disclosed equations and relationships . also although the method and apparatus is shown in conjunction with a coriolis flow meter , it should be under stood that the method and apparatus of the invention can be used with a coriolis flow meter of any type including a coriolis flow meter having metal , plastics or glass flow tube ( s ).