Abstract:
One aspect of the invention provides a flow sensing apparatus including: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second thermoelectric sensors, the heating element being substantially equally spaced from the first and second thermoelectric sensors; a heating element temperature sensor for sensing a temperature of the heating element; and a control device configured to maintain the heating element at a substantially uniform temperature.

Description:
RELATED APPLICATIONS 
       [0001]    This application claims priority to U.S. provisional patent application Ser. Nos. 61/287,949, filed Dec. 18, 2009; 61/288,024, filed Dec. 18, 2009; and 61/288,044, filed Dec. 18, 2009, the entire disclosures of which are incorporated herein by this reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    All flow sensors that employ the differential thermal anemometry technique suffer from a limited linear range. 
         [0003]    Typically, these are thermal flow sensors that operate as shown in  FIG. 1 . A discrete thermal plug introduced into a liquid filled tube/channel will disperse in both the upstream and downstream directions due to thermal conduction or diffusion, respectively. In the case where a discrete section of the fluid in the tube/channel is continuously heated, a temperature profile similar to C a  in  FIG. 1  will develop (i.e., a zero flow condition). The shape of this temperature profile will depend upon the amount of heat added to the fluid and the upstream and downstream temperatures of the liquid. Assuming identical upstream and downstream fluid temperatures, under this zero-flow condition, liquid temperatures measured at P 1  and P 2  will be equal as thermal diffusion will be equal in both directions. If the liquid in the tube/channel is permitted to flow, the fluid temperatures at P 1  and P 2  will now also depend upon the rate of liquid flux and the resulting heat convection. As liquid begins to flow past the heated zone, a temperature profile similar to C B  in  FIG. 1  will develop since, in addition to the symmetrical diffusion of the heat, asymmetrical convection of the heated fluid will occur in the direction of the fluid flow. Therefore, under flowing conditions, fluid temperatures measured at P 1  and P 2  will be different. Temperature measurements made at P 1  and P 2  can be sampled, subtracted and amplified electronically in situ to allow a high degree of common-mode noise rejection which will allow discrimination of extremely small upstream and downstream temperature differences. 
         [0004]    Although the differential temperature is proportional to flow at low flow rates, at elevated flow rates, this relationship becomes nonlinear. At low flow rates, the temperature of the heater is essentially constant as the convection of heat carried away from the heater due to the flowing liquid is relatively small. A thermal flow sensor operated in this low flow range has essentially a linear response of upstream/downstream ΔT with flow. At higher flow rates, this convective heat removal from the heater becomes non-trivial. If constant power is applied to the heater, its temperature will decrease. As the temperature of the heater decreases, the upstream/downstream temperatures will necessarily converge and the ΔT response to flow rate changes will decrease. At extremely high flow rates, this convection will become so large that an increase in flow rate will actually cause a decrease in ΔT. 
         [0005]    It is this phenomenon that limits the effective range of any sensor design based on thermal anemometry. It is not uncommon for flow sensors to be used outside their linear range by using a polynomial function to linearize the raw sensor response. While this is effective, it requires processing of the analog signal and can only extend the flow sensor range to the point where the AT response to flow is still measurable and positive. Also, while a flow sensor can be designed appropriately to handle large fluid flows without decreasing the heater temperature, this typically requires a more massive heater with larger surface area to be used. Such a heater would make the sensor unsuitable for measuring low fluid flow rates as its response to flow changes would be very slow. 
         [0006]    Accordingly, there is a need for devices and methods that extend the linear range of flow sensing apparatus for both low and high flow applications. 
       SUMMARY OF THE INVENTION 
       [0007]    One aspect of the invention provides a method for determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The method includes: actuating the heating element; receiving temperature information from the first temperature sensor and the second temperature sensor at a zero flow rate; flowing a first known flow rate through the flow sensor to calculate a flow scalar value; flowing a second known flow rate through the flow sensor, the second known flow rate exceeding a maximum linear range flow rate of the flow sensor; receiving temperature information from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor; computing a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor; flowing a fluid sample through the flow sensor; and calculating a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor, temperature information from the heating element temperature sensor, the flow scalar value, and the heating element compensation factor, thereby determining a flow rate through the flow sensor. 
         [0008]    This aspect of the invention can have a variety of embodiments. The method can further include the steps of: calculating a difference ΔT 0  between the second temperature sensor and the first temperature sensor at a zero flow state; and receiving a temperature H 0  of the heating element at the zero flow state from the heating element temperature sensor. 
         [0009]    The method can further include the step of: calculating a difference ΔT between the second temperature sensor and the first temperature sensor while the flow sensor is exposed to the first known flow rate. The first known flow rate can be substantially equal to a maximum linear range flow rate of the fluid sensor. 
         [0010]    The flow scalar value can be calculated by dividing the known flow rate by the difference between ΔT and ΔT 0 . 
         [0011]    The method can further include the step of: receiving a temperature H of the heating element from the heating element temperature sensor while the flow sensor is exposed to the second known flow rate. The heating element compensation factor can be defined by the expression (1+α(H 0 −H)), wherein α is a constant according to an expression 
         [0000]    
       
         
           
             
               α 
               = 
               
                 
                   
                     F 
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         T 
                       
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           T 
                           0 
                         
                       
                     
                   
                   - 
                   1 
                 
                 
                   
                     H 
                     0 
                   
                   - 
                   H 
                 
               
             
             , 
           
         
       
     
         [0000]    and wherein F is the second known flow rate. The second known flow rate F can be about three times the maximum linear range flow rate of the fluid sensor. 
         [0012]    The flow rate can be defined by (ΔT −ΔT 0 )(1+α(H 0 −H)) times the flow scalar value. 
         [0013]    Another aspect of the invention provides a flow sensing apparatus including: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second temperature sensors, the heating element being substantially equally spaced from the first and second temperature sensors; a heating element temperature sensor arranged to measure the temperature of the heating element; and a computing device. The computing device can be configured to: calculate a flow scalar value based on temperature information received from the first temperature sensor and the second temperature while the flow sensor is exposed to a first known flow rate; calculate a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to a second known flow rate; and calculate a flow rate based on outputs from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor, the flow scalar value, and the heating element compensation factor. 
         [0014]    This aspect of the invention can have a variety of embodiments. The computing device can be further configured to: calculate a difference ΔT 0  between the second temperature sensor and the first temperature sensor at a zero flow state; and obtain a temperature H 0  of the heating element at the zero flow state from the heating element temperature sensor. 
         [0015]    The computing device can be further configured to calculate a difference ΔT between the second temperature sensor and the first temperature sensor at a non-zero flow state 
         [0016]    The computing device can be further configured to calculate the flow scalar value by dividing a first known flow rate by the difference between ΔT and ΔT 0 . The first known flow rate can be substantially equal to a maximum linear range flow rate of the fluid sensor. 
         [0017]    The computing device can be further configured to obtain a temperature H of the heating element at the non-zero flow state from the heating element temperature sensor. The heating element compensation factor can be defined by the expression (1+α(H 0 −H)), wherein α is a constant according to an expression 
         [0000]    
       
         
           
             
               α 
               = 
               
                 
                   
                     F 
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         T 
                       
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           T 
                           0 
                         
                       
                     
                   
                   - 
                   1 
                 
                 
                   
                     H 
                     0 
                   
                   - 
                   H 
                 
               
             
             , 
           
         
       
     
         [0000]    and wherein F is the second known flow rate. The second known flow rate F can be about three times the maximum linear range flow rate of the fluid sensor. 
         [0018]    The flow rate can be defined by (ΔT−ΔT 0 )(1+α(H 0 −H)) times the flow scalar value. 
         [0019]    Another aspect of the invention provides a flow sensing apparatus including: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second thermoelectric sensors, the heating element being substantially equally spaced from the first and second thermoelectric sensors; a heating element temperature sensor for sensing a temperature of the heating element; and a control device configured to maintain the heating element at a substantially uniform temperature. 
         [0020]    This aspect of the invention can have a variety of embodiments. The control device can be an operational amplifier. The control device can be a proportional-integral-derivative (PID) device. 
         [0021]    Another aspect of the invention provides a method for determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The method includes: flowing a fluid sample through the flow sensor; maintaining the heating element at a substantially uniform temperature; receiving temperature information from the first temperature sensor and the second temperature sensor; and calculating a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor, thereby determining a flow rate through the flow sensor. 
         [0022]    Another aspect of the invention provides a kit including: (a) a flow sensing apparatus comprising: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second temperature sensors, the heating element being substantially equally spaced from the first and second temperature sensors; a heating element temperature sensor arranged to measure the temperature of the heating element; and a computing device; and (b) instructions for installation and/or use. The computing device can be configured to: calculate a flow scalar value based on temperature information received from the first temperature sensor and the second temperature sensor while the flow sensor is exposed to a first known flow rate; calculate a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to a second known flow rate; and calculate a flow rate based on outputs from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor, the flow scalar value, and the heating element compensation factor. 
         [0023]    Another aspect of the invention provides a High Performance Liquid Chromatography (HPLC) device including a flow sensing apparatus comprising: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second temperature sensors, the heating element being substantially equally spaced from the first and second temperature sensors; a heating element temperature sensor arranged to measure the temperature of the heating element; and a computing device. The computing device can be configured to: calculate a flow scalar value based on temperature information received from the first temperature sensor and the second temperature sensor while the flow sensor is exposed to a first known flow rate; calculate a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to a second known flow rate; and calculate a flow rate based on outputs from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor, the flow scalar value, and the heating element compensation factor. 
         [0024]    Another aspect of the invention provides a kit including: (a) a flow sensing apparatus comprising: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second thermoelectric sensors, the heating element being substantially equally spaced from the first and second thermoelectric sensors; a heating element temperature sensor for sensing a temperature of the heating element; and a control device configured to maintain the heating element at a substantially uniform temperature; and (b) instructions for installation and/or use. 
         [0025]    Another aspect of the invention provides a High Performance Liquid Chromatography (HPLC) device including a flow sensing apparatus comprising: a fluid channel that allows a fluid to flow in a first direction; a first temperature sensor arranged at a first position along the fluid channel; a second temperature sensor arranged at a second position along the fluid channel and separated from the first sensor by a predetermined distance along the fluid channel; a heating element arranged between the first and second thermoelectric sensors, the heating element being substantially equally spaced from the first and second thermoelectric sensors; a heating element temperature sensor for sensing a temperature of the heating element; and a control device configured to maintain the heating element at a substantially uniform temperature. 
         [0026]    Another aspect of the invention provides a method for determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The method includes: actuating the heating element; receiving temperature information from the first temperature sensor and the second temperature sensor at a zero flow state; calculating a difference ΔT 0  between the second temperature sensor and the first temperature sensor at a zero flow state; receiving a temperature H 0  of the heating element at the zero flow state from the heating element temperature sensor; flowing a first known flow rate through the flow sensor to calculate a flow scalar value, the first known flow rate within a maximum linear range flow rate of the flow sensor; calculating a difference ΔT between the second temperature sensor and the first temperature sensor while the flow sensor is exposed to a first known flow rate; 
         [0000]    flowing a second known flow rate through the flow sensor, the second known flow rate exceeding a maximum linear range flow rate of the flow sensor; receiving temperature information from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to the second known flow rate; computing a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to the second known flow rate; flowing a fluid sample through the flow sensor; and calculating a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor, temperature information from the heating element temperature sensor, the flow scalar value, and the heating element compensation factor, thereby determining a flow rate through the flow sensor. 
         [0027]    This aspect can have a variety of embodiments. The first known flow rate can be substantially equal to a maximum linear range flow rate of the fluid sensor. The second known flow rate is between about 2.5 and 3.5 times the maximum linear range flow rate of the fluid sensor. The second known flow rate can be three times the maximum linear range flow rate of the fluid sensor. 
         [0028]    The heating element compensation factor can be defined by the expression (1+α(H 0 −H)), wherein α is a constant according to an expression 
         [0000]    
       
         
           
             
               α 
               = 
               
                 
                   
                     F 
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         T 
                       
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           T 
                           0 
                         
                       
                     
                   
                   - 
                   1 
                 
                 
                   
                     H 
                     0 
                   
                   - 
                   H 
                 
               
             
             , 
           
         
       
     
         [0000]    and wherein F is the second known flow rate. 
         [0029]    The flow rate can be defined by (ΔT−ΔT 0 )(1+α(H 0 −H)) times the flow scalar value. 
         [0030]    Another aspect of the invention provides a computer program product including computer-usable medium having control logic stored therein for causing a computer to perform a method of determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The control logic includes: first computer readable program code means for causing the computer to actuate the heating element; second computer readable program code means for causing the computer to receive temperature information from the first temperature sensor and the second temperature sensor at a zero flow rate; third computer readable program code means for causing the computer to flow a first known flow rate through the flow sensor to calculate a flow scalar value; fourth computer readable program code means for causing the computer to flow a second known flow rate through the flow sensor, the second known flow rate exceeding a maximum linear range flow rate of the flow sensor; fifth computer readable program code means for causing the computer to receive temperature information from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor; sixth computer readable program code means for causing the computer to compute a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor; seventh computer readable program code means for causing the computer to flow a fluid sample through the flow sensor; and eighth computer readable program code means for causing the computer to calculate a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor, temperature information from the heating element temperature sensor, the flow scalar value, and the heating element compensation factor. 
         [0031]    Another aspect of the invention provides a computer program product including computer-usable medium having control logic stored therein for causing a computer to perform a method of determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The control logic includes: first computer readable program code means for causing the computer to flow a fluid sample through the flow sensor; second computer readable program code means for causing the computer to maintain the heating element at a substantially uniform temperature; third computer readable program code means for causing the computer to receive temperature information from the first temperature sensor and the second temperature sensor; and fourth computer readable program code means for causing the computer to calculate a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor. 
         [0032]    Another aspect of the invention provides a computer program product including computer-usable medium having control logic stored therein for causing a computer to perform a method of determining a flow rate through a flow sensor including a conduit, a first temperature sensor positioned along the conduit, a second temperature sensor positioned along the conduit, a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. The control logic includes: first computer readable program code means for causing the computer to actuate the heating element; second computer readable program code means for causing the computer to receive temperature information from the first temperature sensor and the second temperature sensor at a zero flow state; third computer readable program code means for causing the computer to calculate a difference ΔT 0  between the second temperature sensor and the first temperature sensor at a zero flow state; fourth computer readable program code means for causing the computer to obtain a temperature H 0  of the heating element at the zero flow state from the heating element temperature sensor; fifth computer readable program code means for causing the computer to flow a first known flow rate through the flow sensor to calculate a flow scalar value, the first known flow rate within a maximum linear range flow rate of the flow sensor; sixth computer readable program code means for causing the computer to calculate a difference ΔT between the second temperature sensor and the first temperature sensor at a non-zero flow state; seventh computer readable program code means for causing the computer to flow a second known flow rate through the flow sensor, the second known flow rate exceeding a maximum linear range flow rate of the flow sensor; eighth computer readable program code means for causing the computer to receive temperature information from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to the second known flow rate; ninth computer readable program code means for causing the computer to compute a heating element compensation factor based the temperature information received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to the second known flow rate; tenth computer readable program code means for causing the computer to flow a fluid sample through the flow sensor; and eleventh computer readable program code means for causing the computer to calculate a flow rate based on a difference in temperatures between the first temperature sensor and the second temperature sensor, the flow scalar value, and the heating element compensation factor. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0033]    For a fuller understanding of the nature and desired objects of the present invention, reference is made to the following detailed description taken in conjunction with the accompanying drawing figures wherein like reference character denote corresponding parts throughout the several views and wherein: 
           [0034]      FIG. 1  is a conceptual view of the thermal flow sensing measurements; 
           [0035]      FIG. 2  depicts a flow sensor according to one embodiment of the invention; 
           [0036]      FIG. 3  is a plot of uncompensated flow rate, compensated flow rate, and heater temperature according to one embodiment of the invention; 
           [0037]      FIG. 4  depicts a flow sensor according to one embodiment of the invention; 
           [0038]      FIG. 5  is a plot of flow rate measurement linearity in relation to actual flow rates; 
           [0039]      FIG. 6  depicts a method of calculating a flow rate through a flow sensor according to one embodiment of the invention; and 
           [0040]      FIG. 7  depicts a method of calculating a flow rate through a flow sensor according to one embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Definitions 
       [0041]    The instant invention is more clearly understood with reference to the following definitions: 
         [0042]    As used in the specification and claims, the singular form “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. 
         [0043]    Unless specifically stated or obvious from context, as used herein, the term “about” is understood as within a range of normal tolerance in the art, for example within 2 standard deviations of the mean. The term “about” can be understood as within 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, 0.5%, 0.1%, 0.05%, or 0.01% of the stated value. Unless otherwise clear from context, all numerical values provided herein are modified by the term about. 
         [0044]    As used in the specification and claims, the terms “comprises,” “comprising,” “containing,” “having,” and the like can have the meaning ascribed to them in U.S. patent law and can mean “includes,” “including,” and the like. 
         [0045]    A computer-readable medium shall be understood to mean any article of manufacture that contains data that can be read by a computer or a carrier wave signal carrying data that can be read by a computer. Such computer-readable media includes, but is not limited to, magnetic media, such as a floppy disk, a flexible disk, a hard disk, reel-to-reel tape, cartridge tape, cassette tape or cards; optical media such as CD-ROM and writeable compact disc; magneto-optical media in disc, tape or card form; paper media, such as punched cards and paper tape; or on carrier wave signal received through a network, wireless network or modem, including radio-frequency signals and infrared signals. 
         [0046]    Unless specifically stated or obvious from context, as used herein, the term “or” is understood to be inclusive. 
         [0047]    Ranges provided herein are understood to be shorthand for all of the values within the range. For example, a range of 1 to 50 is understood to include any number, combination of numbers, or sub-range from the group consisting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, or 50 (as well as fractions thereof unless the context clearly dictates otherwise). 
         [0048]    Thermal flow sensor according to the subject invention utilizes the thermoelectric effect to convert thermal convection due to a flowing fluid into a voltage response that is proportional to the fluid flow rate. 
         [0049]      FIG. 1  depicts fluid temperature measurements along a fluid filled fluid channel  20  as thermal energy is introduced at a fixed location along the fluid channel  20 . When thermal energy is introduced into a liquid filled fluid channel at location P 0  along the channel  20 , the thermal energy will disperse along the channel  20  in both the upstream and downstream directions due to thermal conduction and diffusion. In  FIG. 1 , the vertical axis denotes measured temperature and the horizontal axis denotes distance along the fluid channel  20 . A temperature profile, curve C a  will develop when a discrete section of the fluid in the fluid channel is continuously heated by the heating element  23  while the fluid is resting or under a so-called zero-flow condition. The shape of this temperature profile depends on the amount of heat added to the fluid and the upstream and downstream temperatures of the liquid. Assuming identical upstream and downstream fluid temperatures and a zero-flow condition, liquid temperatures T 1  and T 2  measured at the first and second sensors  21  and  22  that are placed at equal distances upstream and downstream from P 0 , denoted by P 1  and P 2 , will be equal as thermal conduction and diffusion will be approximately equal. 
         [0050]    If liquid in the fluid channel  20  is permitted to flow, the fluid temperatures at the first and second locations P 1  and P 2  will depend upon the flow rate of liquid and the resulting heat convection. Curve C b  of  FIG. 1  depicts a temperature profile as liquid begin to flow past P 0 , or the heated zone. Note that while C a  is substantially symmetrical about P 0 , C b  is not symmetrical about P 0 . This is due to an asymmetric convection of the heated fluid that occurs in the direction of the fluid flow. Therefore, under flowing conditions, fluid temperatures T 3  and T 4  measured at P 1  and P 2 , respectively, will be different from T 1  and T 2 . A difference of T 3  and T 4 , denoted by ΔT, may be calculated and correlated to the actual flow rate of the fluid that flowed inside the fluid channel  20 . 
         [0051]    To perform the above-mentioned temperature measurements, a number of sensing methods and apparatuses can be used. Generally, two temperature sensors are disposed along a fluid channel at equal distances upstream and downstream from the heating element. For example, as shown in  FIG. 1 , P 1  and P 2  were chosen to be the inflection points of the temperature profile C a . Measurement at the inflection points can minimize the amount of time required to detect a difference between measured temperatures T 1  and T 3  or the difference between the temperatures T 2  and T 4  and thus maximize the upstream/downstream ΔT response of the amplifier element  26  to flow rate change. 
         [0052]    Referring now to  FIG. 2 , in order to better understand the non-linear behavior of thermal-based flow sensors and the effect of convective cooling of the heater, a DEVANTECH™ TPA81 thermopile array  202  was used to thermally image a centrally heated tube  204  under flowing conditions. This multi-element array measures both the up- and down-stream temperatures of the tube  204  and the temperature of heater  206 . 
         [0053]    To measure flow by conventional flow anemometry using non-contact sensor array  202 , the upstream flow temperature used was the average of elements  1 ,  2  and  3  of the sensor array  202 , and the downstream temperature was the average of elements  6 ,  7  and  8 . Expressed numerically: 
         [0000]      Δ T =( T   6   +T   7   +T   8 )−( T   1   +T   2   +T   3 )/3   (1)
 
         [0000]      FlowRate=(Δ T−ΔT   0 )×FlowScalar   (2)
 
         [0054]    The ΔT 0  value is used to balance the flow sensor  200  and adjust for any misalignment of the sensor array  202  and heater  206  such that at zero flow, the difference between upstream and downstream temperatures is zero. The FlowScalar value can be adjusted depending on the material used for the flow tube (e.g., to reflect thermal conductivity) and the fluid flowing through the tube  204 . 
         [0055]      FIG. 3  depicts processed data collected using system  200  depicted in  FIG. 2 . The data shown in  FIG. 3  was collected when water was flowing through the tube  204 . The flow rate was initially 0 μL/min and was then successively increased by 1 μL/min increments. The light grey plot  302  depicts the flow response when according to the formulas provided above. When the flow is under 1-1.5 μL/min, the flow response is linear, but as the flow is increased, the flow response becomes increasingly nonlinear. Because the sensor array  202  is also monitoring the heater temperature, as the flow increases and the heater  206  begins to cool, the flow response departs from linearity. (The “Heater Temperature” plot  304  in  FIG. 3  is the average of sensors  4  and  5  of the array  202 , which are positioned directly over the heater element  206 ). This decrease in heater temperature  304  is due to the convection of the fluid over the heater  206 . Because the heater temperature  304  is decreasing, the temperature upstream/downstream difference will be reduced, which leads to the flow response nonlinearity. Equation (2) above multiplies the difference between the temperatures upstream and downstream of the heater by a scalar to determine the flow rate. As shown in plot  304  in  FIG. 3 , this produces a linear response only in the flow region where the heater is not substantially cooled by the fluid flow. While this is effective, the maximum flow that can be measured is where the flow response asymptotes. 
         [0056]    Embodiments of the invention measure the heater temperature and compensate the measured temperature difference for any decrease in the heater temperature using a unitless compensation factor: 
         [0000]      CompensatedFlowRate=(Δ T−ΔT   0 )(1+α( H   0   −H ))×FlowScalar   (3)
 
         [0057]    In the equation above, a heater compensation factor α is used to adjust the measured ΔT as the temperature of the heater  206  decreases due to convective cooling. Before the flow sensor is used, it is first “zeroed” by measuring both the ΔT and heater temperature H at zero flow. These values are stored as ΔT 0  and H 0 . As the flow increases and the measured heater temperature H decreases below H 0 , the heater compensation (1+α(H 0 −H)) will return a value&gt;1 to scale the ΔT and increase its value to accommodate for the decreasing heater temperature. 
         [0058]    Normal flow sensor calibration involves first zeroing the sensor, then flowing a known flow rate through the sensor to establish the FlowScalar value. Embodiments of the invention improve upon prior calibration techniques by introducing a third operation that sets a heater compensation factor α. 
         [0059]    After the sensor  200  is zeroed, a known flow is delivered to the flow sensor  200  at the maximum of the linear range of the sensor  200 . In the linear range of the sensor  200 , the heater temperature H should be equal to H 0 , so the heater compensation (1+α(H 0 −H)) equals 1. Using this flow rate in the flow sensor linear range, the FlowScalar can be set according to Equations (4)-(6) below: 
         [0000]        H   0   −H= 0   (4)
 
         [0000]      ∴ FlowRate=(Δ T−ΔT   0 )(1)×FlowScalar   (5)
 
         [0000]    
       
         
           
             
               
                 
                   FlowScalar 
                   = 
                   
                     FlowRate 
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         T 
                       
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           T 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0060]    Once FlowScalar is set, heater compensation factor α is set by flowing a flow rate preferably at approximately three times the maximum linear range through the flow sensor  200  to solve Equation (7): 
         [0000]    
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       
                         FlowRate 
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             T 
                           
                           - 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               T 
                               0 
                             
                           
                         
                       
                       - 
                       1 
                     
                     
                       
                         H 
                         0 
                       
                       - 
                       H 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0061]    Once the flow sensor  200  is zeroed, and FlowScalar and heater compensation factor α are set, equation (1) can be used to determine flow rate given the upstream/downstream and heater temperatures. The compensated flow rate plot  306  in  FIG. 3  depicts the linearity range extended significantly by using this technique. While there is significant noise in the signal shown in these plots, it should be realized that the TPA81 sensor  200  used for experimentation is not particularly sensitive and was used during the development of this technique to understand the thermal distribution along the flow tube  204 . In implementation of the invention, more sensitive temperature sensors can be placed directly on the flow tube  204  and heater  206  to sense the upstream/downstream and heater temperatures. 
         [0062]    The technique described above compensates the flow response for the decrease in heater temperature that results from a high-flow operation. Another embodiment of the invention maintains the heater  206  at a constant temperature. Since the primary reason for the non-linear ΔT-flow response is cooling of the central heater  206  at high flow rates, any approach that maintains the heater  206  at a constant temperature in spite of high convective currents improves the linear range of the flow sensor  200 . 
         [0063]    The first embodiment of a temperature-stabilized heater flow sensor is shown in  FIG. 4 . In  FIG. 4 , a thermal-based flow sensor  400  is shown. In addition to upstream and downstream temperature sensors  402   a,    402   b,  a third temperature sensor  402   c  is placed on or close to the central heater  406  to monitor the heater temperature. An external closed-loop circuit  408  is used to control the heater temperature to a pre-defined set-point. This control circuit can be implemented by an analog op-amp with feedback or a proportional-integral-derivative (PID) system  410  as depicted in  FIG. 4 . As the flow rate increases and convection cools the heater  406 , the closed-loop circuit  408  supplies more current to the heater  406  to maintain the set-point temperature. 
         [0064]    The heater temperature sensor  402   c  can be placed directly on the back of the heater  406 . Alternatively, the heater temperature sensor  402   c  is placed in direct contact with the fluid in conduit  404  to achieve high fidelity to the actual temperature of the fluid at P 0 . Such an embodiment may not always be practical. Accordingly, placing the heater feedback temperature sensor  402   c  on opposite the side of the tube/conduit  404  can, in some embodiments, provide better feedback to correct for convective losses than simply placing it directly on the heater  406 . 
         [0065]      FIG. 5  depicts the improved linear range of a flow sensor when the temperature of the heater is controlled. 
         [0066]    Referring now to  FIG. 6 , a method  600  of calculating a flow rate through a flow sensor is provided. The method  600  can, in some embodiments, be used in conjunction with a flow sensor including a conduit, a first temperature sensor and a second temperature positioned along the conduit, a heating element positioned substantially equidistant between first temperature sensor and the second temperature sensor along the conduit, and a heating element temperature sensor. 
         [0067]    In step S 602 , the heating element is actuated. 
         [0068]    In step S 604 , the temperature sensors are allowed to stabilize/equilibrate until a stable temperature reading is obtained. The time allotted for stabilization can range from several second to several minutes depending on the characteristics of the particular temperature sensors utilized in the flow sensor. 
         [0069]    In step S 606 , temperature information is received from the first temperature sensor and the second temperature sensor. 
         [0070]    In step S 608 , a difference ΔT 0  is calculated between the temperature provided by the second temperature sensor and the first temperature sensor at a zero flow state. 
         [0071]    In step S 610 , a temperature H 0  of the heating element is obtained from the heating element at the zero flow state. 
         [0072]    In step S 612 , a first known flow rate is flown through the flow sensor. In some embodiments, the first known flow rate is or approximates the maximum linear range of the flow sensor. For example, the first known flow rate can be between about 0.9 and about 1.0 times the maximum linear range of the flow sensor. The maximum linear range can be determined for a particular flow sensor by observation as discussed herein or can be specified (e.g., by a manufacturer of a flow sensor). 
         [0073]    In step S 614 , a difference ΔT is calculated between the temperature measured by the second temperature sensor and the temperature measured by the first sensor while both temperatures are exposed to a first known flow rate. 
         [0074]    In step S 616 , a flow scalar value based on data received from the first temperature sensor and the second temperature sensor (as reflected in difference ΔT). The flow scalar value can, in some embodiments, be calculated according to Equation (6) above. 
         [0075]    In step S 618 , a second known flow rate is flown through the flow sensor. Preferably, the second known flow rate exceeds the maximum linear range of the flow sensor. For example, the second known flow rate can be between about 1.5 to about 10 times the maximum linear range of the flow sensor, such as: between about 1.5 to about 2.5 times the maximum linear range, between about 2.5 to about 3.5 times the maximum linear range, between about 3.5 to about 4.5 times the maximum linear range, between about 4.5 to about 5.5 times the maximum linear range, between about 6.5 to about 7.5 times the maximum linear range, between about 7.5 to about 8.5 times the maximum linear range, between about 8.5 to about 9.5 times the maximum linear range, and the like. 
         [0076]    In step S 620 , temperature information is received from the first temperature sensor, the second temperature sensor, and the heating element temperature sensor while the flow sensor is exposed to the second known flow rate. 
         [0077]    In step S 622 , a heating element compensation factor is computed. A heating element compensation factor α can be calculated according to Equation (7) above. 
         [0078]    In step S 624 , a fluid sample is flowed through the flow sensor. Because of the use of the heating element compensation factor computed in step S 622 , the flow rate of the fluid sample can be less than, equal to, or greater than the maximum linear range of the flow sensor. 
         [0079]    In step S 626 , the flow rate of the fluid sample is calculated based on a difference in temperatures between the first temperature sensor and the second temperature sensor, the flow scalar value, and the heating element compensation factor. In some embodiments, the flow rate is calculated according to Equation (8) below: 
         [0000]      Flowrate=(Δ T−ΔT   0 )(1+(HeaterZeroTemp−HeaterTemp)α)FlowScalar   (8)
 
         [0080]    Referring now to  FIG. 7 , another method  700  for determining a flow rate through a flow sensor is provided. Method  700  can, in some embodiments, be used in conjunction with a flow sensor having a conduit, a first temperature sensor and a second temperature sensor positioned along the conduit, and a heating element positioned substantially equidistant between the first temperature sensor and the second temperature sensor along the conduit. 
         [0081]    In step S 702 , a fluid sample is flowed through the flow sensor. 
         [0082]    In step S 704 , the heating element is maintained at a substantially uniform temperature. The heating element can, in some embodiments, be maintained at a substantially uniform temperature by using a temperature sensor to measure a temperature of the heating element (S 704   a ) and adjusting the current flowing across the heater based on the measured temperature of the heating element (S 704   b ). 
         [0083]    In step S 706 , temperature information is received from the first temperature sensor and the second temperature sensor. 
         [0084]    In step S 708 , a flow rate is calculated based on a difference in temperatures between the first temperature sensor and the second temperature sensor. 
       Implementation in Computer Program Products 
       [0085]    The methods described herein can be implemented on general-purpose or specially-programmed hardware or software. For example, methods  600  and/or  700  can be implemented by a computer-readable media. The computer-readable medium can be non-transitory and/or tangible. For example, the computer readable medium can be volatile memory (e.g., random access memory and the like) or non-volatile memory (e.g., read-only memory, hard disks, floppy discs, magnetic tape, optical discs, paper table, punch cards, and the like). 
       Equivalents 
       [0086]    The functions of several elements may, in alternative embodiments, be carried out by fewer elements, or a single element. Similarly, in some embodiments, any functional element may perform fewer, or different, operations than those described with respect to the illustrated embodiment. Also, functional elements (e.g., modules, computers, and the like) shown as distinct for purposes of illustration may be incorporated within other functional elements, separated in different hardware or distributed in a particular implementation. 
         [0087]    While certain embodiments according to the invention have been described, the invention is not limited to just the described embodiments. Various changes and/or modifications can be made to any of the described embodiments without departing from the spirit or scope of the invention. Also, various combinations of elements, steps, features, and/or aspects of the described embodiments are possible and contemplated even if such combinations are not expressly identified herein. 
       INCORPORATION BY REFERENCE 
       [0088]    The entire contents of all patents, published patent applications, and other references cited herein are hereby expressly incorporated herein in their entireties by reference.