Abstract:
Apparatus and method for detecting the refractive index and rate of fluid flowing through an elongated flow cell, having applications in liquid chromatography. The cell may be of circular cross section and has an input window in one end, an exit window in the other, and a longitudinal side wall. A light source directs divergent light through the entrance window, longitudinally through the cell, to pass through the exit window and be sensed by a photoelectric detector outside the exit window. In a first embodiment, heat transmission apparatus connected to the cell establishes a constant temperature gradient in the cell. This temperature gradient, preferably, has a component perpendicular to the longitudinal dimension of the cell and extending toward the center of the cell. This causes the density of the fluid in the cell to vary as an increasing function of its distance from the walls. The index of refraction of the fluid thus, in this embodiment, increases with distance from the cell walls. This causes the divergent light to be bent away from the cell walls, and toward the center of the cell. The degree of this bending, and hence the fraction of light entering the entrance window which passes through the exit window, increases with the refractive index of the fluid. By appropriate selection of the cell dimensions, of the value of the temperature gradient and of a substantially constant flow rate of the fluid, the bending of the light passing through the cell may be governed such that the fraction of entering light which exits from the cell is a highly dependent function of variations in the refractive index of the fluid in the flow cell. In another embodiment, a fluid having a known refractive index value has its flow rate determined by measuring the absorbance of a beam of light passed therethrough after establishing a temperature gradient in the fluid such that the degree of light bending is influenced by the flow rate.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to an apparatus and method for detecting the refractive index and flow rate of a fluid, and particularly to such apparatus and method for use as detectors in liquid chromatography. 
     2. The Prior Art 
     Liquid chromatography pertains to a particular variety of equipment and techniques for analyzing the components of an unknown sample of liquid material. 
     Liquid chromatography is a process wherein a sample having unknown components is forced to migrate through an elongated &#34;column&#34;. The column contains a material held statically inside it, called a &#34;stationary phase&#34;. The stationary phase is chosen for its ability to selectively retain the various potential components of the sample with which it comes in contact with differing degrees of tenacity. The sample is forced to migrate through the column by injecting it into a solvent upstream of the column and subsequently pumping the solvent and dissolved sample through the column. 
     When the dissolved sample is forced through the column, each of its components migrates through the column in a particular time related pattern, which pattern is a function of the degree of the tendency of the stationary phase to retain that component. 
     Some properties of the column effluent, following the pumping of the solvent and dissolved sample through the column, are affected by the concentration of sample in the emergent fluid. One of the properties which is typically so affected is the refractive index of the effluent. By detecting variations in such properties of the column effluent, and by plotting these variations against time, certain information can be derived as to the nature and amount of the components in the sample. 
     For example, for predetermined column conditions and flow rate, it may be known that a particular hypothetical component, if present in the sample, will reach a maximum concentration in the column effluent at a specific time following introduction of the sample to the column. This time is known as the &#34;retention time&#34; of the component. This phenomenon occurs because of the existence of a particular degree of retention of that component by the stationary phase. By measuring a property of the effluent known to be affected by the hypothetical component, and observing whether a maximum occurs at the &#34;retention time&#34; for that component, the presence of the component can be verified or negated. 
     Conditions of the effluent affect precise determination of some hypothetical components, such as effluent flow rate affecting the retention time of the component. Therefore the flow rate is desirably known even in systems inconvenient for flow rate measurement. 
     It is evident that in liquid chromatography there exists a necessity for accurately detecting and measuring, on a continuous basis, properties of the liquid emerging from the column, such as refractive index. It is further evident that a necessity exists for determining effluent conditions in liquid chromatography systems such as fluid flow rate, notwithstanding inconvenience of measurement. 
     Several types of apparatus exist for detecting changes in the index of refraction of the column effluent. One (called &#34;deflective&#34; type) involves passing the column effluent through an elongated flow cell having a triangular cross-section, the hypotenuse of which triangle forms an interface with that of a second triangular cross-sectional chamber. The second chamber contains a reference fluid having a known index of refraction. A mirror is placed parallel to one of the legs of the second chamber at a distance therefrom. A light beam is then directed through the two chambers, and across the interface, at which point it is bent, and transmitted on to the mirror at an angle of incidence dependent upon the difference between the indices of refraction of the column effluent and the reference fluid. The light beam is reflected from the mirror and returns back across the interface, being bent additionally as it crosses the interface to an angle even further removed from the angle at which the incoming light beam was incident on the interface for the first time. The degree of deflection of the light beam is measured, and is a function of the difference between the respective indices of refraction of the column effluent and the reference fluid. 
     Another type of refractive index detector employs two beams of light which originate from a common region of a tungsten filament lamp. The two parallel beams of light pass through a glass prism, and are partially transmitted through two glass-liquid interfaces. One interface is the boundary of the detection flow cell and the prism, and the other interface is the boundary of a reference cell with the prism. The two transmitted beams of light are then scattered from a finely ground stainless steel back plate, and a part of the scattered light from each of the beams is transmitted back through its respective cell, the glass prism and on to two halves of a photoconductive sensing cell. 
     The ratio of the amount of light transmitted through the two interfaces is a function of the refractive indices of the substances in the two cells. Thus, measurement of the transmitted light may be used to derive the refractive index of the substance in the detection cell, provided the refractive index of the material in the reference cell is known. 
     The deflection type of detector described above is less susceptible to changes in solvent composition than is the reflection type of detector. On the other hand, the deflection type of detector offers a smaller linear range than does the reflection type. Thus, prior art detectors have not fully combined the advantages of the reflection and deflection detectors. 
     Both of these principal refractive index detectors are adversely affected by changes in temperature of the liquid passing through their flow cells. This is because the refractive index of most liquids is dependent to some degree upon temperature. Therefore, measures must be taken to provide for temperature compensation of these devices. This technique adds to the complexity and expense of the detection instrument. 
     Various types of flow meters have been proposed for measuring flow rate of a fluid through a passage. Flow meter proposals typically require a member physically disposed in the fluid and linked to a measurement device. Oftentimes such a member is not conveniently combined with the system in question whose fluid flow rate is unknown. 
     Summary of the Invention 
     This invention provides method and apparatus for governing the fraction of light input to a flow cell which passes entirely through that cell as a function of the refractive index and flow rate of fluid within the cell. 
     An elongated flow cell is provided, with side walls, and entrance and exit windows in its ends. A fluid delivery system connected to the flow cell establishes a flow of fluid through the cell. A light directs divergent rays through the entrance window toward the exit window. A sensor produces a signal which is a function of the amount of light passing through the exit window. 
     In one embodiment, heat transmission apparatus is proximate to the flow cell and establishes a temperature gradient through the side walls of the cell. The temperature gradient has a component perpendicular to the longitudinal walls extending toward the middle of the flow cell. The fluid in the flow cell decreases in temperature and increases in refractive index with distance from the side walls. This causes the divergent rays entering the flow cell to bend away from the side walls. This bending makes the fraction of entering light which emerges from the exit window a highly dependent function of variations of the refractive index of the fluid in the flow cell for a given flow rate; conversely it provides a measure of flow rate if the refractive index of the effluent is known. 
     The flow cell has a generally elongated configuration, with an entrance window in one end and an exit window in the other, both of which are transparent to the passage of light. The light source is outside the entrance window and directs light through the cell and toward the exit window. In one embodiment, the light source provides a plurality of divergent light rays. The flow cell has input and output conduits connected to opposite ends. The input conduit is connected to receive the effluent from a liquid chromatograph column, in order to establish a flow of the effluent liquid through the cell. 
     The heat transmission apparatus is provided adjacent the longitudinal walls of the elongated flow cell. The heat transmission apparatus is controlled to establish and maintain a temperature gradient having a component across the longitudinal walls of the flow cell perpendicular to these walls. The component of the temperature gradient perpendicular to the walls extends inwardly toward the center of the flow cell in the preferred embodiment. The heat transmission apparatus may be a temperature controlled liquid bath into which at least a portion of the flow cell is submerged. A water jacket may also be provided surrounding the flow cell. Such heating apparatus for establishing a longitudinal gradient along the cell may also be utilized in the rate detecting embodiment according to another aspect of the invention. 
     These structures establish a temperature gradient within the liquid in the flow cell such that the liquid near the walls approaches the temperature of the heat transmission means, and declines with increasing distance inwardly from the longitudinal walls of the flow cell. 
     If the flow cell has a longitudinal dimension significantly larger than the dimensions of its cross-section, the amount of light which successfully passes through the flow cell is highly dependent on the temperature gradient established in the liquid of the flow cell and on variations in the refractive index of that liquid. The temperature gradient is influenced both by magnitude and geometry of the heat source, and by fluid flow rate. Over a significant range, as more bending of the divergent light rays takes place, and these rays are directed away from the walls, less of the energy of the light entering the entrance window is absorbed by incidence on the longitudinal walls of the flow cell and consequently more light passes out the exit window. 
     With respect to the refractive index indicator embodiment, the refractive index of any liquid is an increasing function of the density of that liquid. The density is, for most liquids, a decreasing function of its temperature. Light passing through a substance having an index of refraction which varies with position is refracted toward the denser areas. Therefore, in this aspect of the present invention, the divergent light rays directed incident to the walls of the flow cell tend to be bent away from the region of the walls, and toward the denser, cooler fluid in the central regions of the flow cell. 
     Therefore, if the temperature gradient is maintained at a substantially constant level with respect to a known flow rate, the amount of light passing through the cell will be a highly dependent function of the index of refraction of the liquid in the cell, over a substantial range of refraction index values. 
     Applicant has additionally discovered that the relationship between the fraction of input light emerging from the cell and the index of refraction of the fluid therein is substantially linear over a considerable range of refractive indices, that range encompassing the range of indices of refraction of practically all liquids which are useful in liquid chromatography applications. 
     An object of this invention is to provide a new and improved refractive index detector for use in a liquid chromatograph which renders the fraction of input light passing through a flow cell a function of the refractive index of the fluid within the flow cell. 
     With respect to the flow rate indicating embodiment of this invention, if the refractive index of the effluent is known, the amount of transmitted light through the cell is indicative of the flow rate of the fluid. By knowing boundary conditions of the specific cell and heat transmitting device utilized, measurement of absorbance of the fluid is indicative of its flow rate. Accordingly, another object of this invention is to provide a novel flow rate indicator and method used in a flow cell by determining the absorbance of a light beam directed therethrough. 
     Other objects of this invention will become apparent from the following detailed description, taken with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block drawing of a liquid chromatographic system; 
     FIG. 1a is a detailed drawing of a segment of the column of the liquid chromatograph; 
     FIG. 2 is a side sectional view of apparatus used in this invention of one preferred form; 
     FIG. 3 is a side sectional view of apparatus according to this invention, showing an alternate embodiment incorporating parallel light rays, and a negative temperature gradient across the walls of the flow cell; 
     FIGS. 4a-4d illustrate fluid cells having various heat transmitting devices according to embodiments of the invention; and, 
     FIGS. 5a and 5b illustrate heating generators maintaining the temperature of entering fluid at a fixed value relative to the cell wall temperature. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     FIG. 1 shows a typical liquid chromatograph system to which one aspect of the present invention is applicable. A solvent reservoir 10 contains a quantity of solvent, which is drawn along a conduit 12 by a pump 14, and subsequently transported through a conduit 16 to an injector 18. A quantity of sample from a sample reservoir 20 is injected into the solvent by an injector 18. The solvent/sample solution continues on to a column 22. The column 22 is an elongated conduit containing a stationary phase 24 which is shown in FIG. 1a as a solid particulate material. 
     Components of the sample are selectively retained in the column 22 by the stationary phase 24, emerging in the column effluent at an end 25 of the column 22 in particular time-related patterns. The presence and concentration of these components in the effluent are sensed by a detector 26. The detector 26 generates a signal as a function of the concentration of the component detected, which signal is directed to a recorder 28. The recorder 28 produces a tangible record 30 of the detected concentration of the sample components with respect to time. 
     Referring to FIG. 2, a detector apparatus embodying the present invention is shown in detail. An elongated flow cell 40 is provided, having side walls 41, and an entrance and an exit window 42 and 44 respectively, located in opposite ends of the flow cell 40. An inlet conduit 46 is connected to and delivers effluent from the chromatorgraphic column 22 to the flow cell 40, through which the effluent circulates before exiting by way of the outlet conduit 48, for collection or discard. A light 50 directs divergent rays through the entrance window 42 toward the exit window 44. A photocell 52 senses the amount of light emerging from the exit window 44. 
     A heat transmission apparatus 54 provides a temperature gradient dT/dy across the side walls 41 of the cell 40, extending toward the center of the cell. Temperature gradient dT/dy is defined as the rate of change of temperature, T, with respect to displacement along a direction indicated as y in FIG. 2. The temperature gradient causes the divergent rays entering the flow cell 40 to bend toward the center, away from the warmer fluid near the side walls 41. This bending increases the amount of light which passes through the flow cell 40 and renders the function of input light emerging through the cell a highly dependent function of variations in the refractive index of fluid within the flow cell. 
     The side walls 41 of flow cell 40 are heat transmissive. The cross-sectional configuration of the flow cell 40 is optimally circular, but may be such other shape as may be determinable as useful by those skilled in the art. The entrance window 42 and the exit window 44 are transparent. The inlet conduit 46 and the outlet conduit 48 are connected to the cell, the inlet conduit 46 being preferably connected to the source of fluid effluent from the analysis column of a liquid chromatograph. The flow of liquid through the flow cell 40 is generally in the direction from the inlet conduit 46 to the outlet conduit 48. 
     The light 50 is provided such that it directs divergent light rays, shown by the arrows in FIG. 2, through the entrance window 42 and toward the exit window 44. The photocell 52 is positioned to receive the light emerging from the exit window 44, and to produce a signal which is a function of the amount of light so received. 
     The heat transmission apparatus 54 is positioned proximate to the longitudinal walls 41 of the flow cell 40. The heat transmission apparatus is located such that it establishes the temperature gradient dT/dy across the walls of the flow cell 40, which temperature gradient has a component perpendicular to the walls and extending inwardly toward the central region of the flow cell 40. This perpendicular component of the temperature gradient need not be uniform along the entire length of the longitudinal walls 41, but should not be time-varying. 
     The heat transmission means 54 may suitably comprise a liquid reservoir 60 maintained at a predetermined temperature by a control system 62 devisable by one of skill in the art, the liquid of the reservoir 60 being exposed to at least a portion of the longitudinal walls 41 of the flow cell 40 by a water jacket 64 around the flow cell 40. Liquid from the bath 60 is circulated through the jacket 64 by a pump 66 and conduits 67 and 68. 
     The refractive index of most liquids is an increasing function of the density of that liquid. The density, in turn, is a decreasing function of its temperature. Light passing through a medium having an index of refraction which varies as a function of position tends to be refracted toward that portion of the medium having the higher refractive index. That is to say, light is refracted toward the denser (cooler) portions of the medium. 
     FIG. 2 shows how this phenomenon enables the operation of the detector of this invention. Because the temperature gradient extends inwardly from the walls toward the center of the flow cell 40, the fluid therein decreases in temperature with distance from the side walls. The warmer material is near the periphery, the cooler toward the center. FIG. 2 shows diverging light rays entering the flow cell 40 through the entrance window 42. According to the phenomena discussed above, these light rays tend to be bent toward the cooler or central portions of the flow cell. Rays which would otherwise strike the side walls 41, and either pass through them or be absorbed by them, are bent such that they proceed down the entire length of the flow cell and pass through the exit window 44, where they are sensed by the photocell 52. 
     If the temperature gradient dT/dy is maintained constant with respect to time, the amount of the light from light source 50 which ultimately passes through the entire flow cell for a given flow rate becomes a function of variations of the refractive index of the fluid in the flow cell. Within a substantial range, the higher the refractive index, the more light is passed all the way through the flow cell, because more of the divergent rays are bent away from the side walls 41. 
     The mathematical basis specifying the relationship between the index of refraction of the fluid within the flow cell 40 and the amount of light emanating from light source 50 which exits through the exit window 44 is discussed below. 
     It can be shown that the amount of light transmitted through a narrow bore flow cell is representable by the following expression: ##EQU1## where L is the cell pathlength, s is the cell radius, n is the refractive index of fluid in the cell, T is the temperature, and dT/dy is the temperature gradient at the cell wall. I represents the amount of light emitted by the light source and entering the cell, and ΔI is the amount of light lost in passage through the cell. 
     The refractive index enters this expression as ##EQU2## 
     This expression can be evaluated by use of the Lorenz-Lorentz relationship ##EQU3## where k is a constant for a given fluid and ρ is the density. 
     From equation (3) ##EQU4## But, ##EQU5## where β is the thermal coefficient of expansion. Thus, ##EQU6## where C is independent of refractive index. 
     Table I hereinbelow compares values of ##EQU7## with corresponding values of the linear expression y = 0.6960n - 0.6553. 
     This latter expression has been independently derived as one which very closely approximates the values of N as indicated in Table I. 
     
                                           TABLE I__________________________________________________________________________                     percent      N =    y =     deviation      (n.sup.2 +2) (n.sup.2 -1)                          (N-y)n   n.sup.2       0.6990 n-0.6553                     100×      6 n.sup.2            N__________________________________________________________________________1.00    1.000  0      --      --1.10    1.210  .09285 .1136   -22.351.20    1.440  .17518 .1835   -4.751.30    1.690  .25109 .2534   -0.921.31    1.7161 .2585  .2604   -0.731.32    1.7424 .2657  .2674   -0.641.33    1.7689 .2730  .2744   -0.51 (water)1.34    1.7956 .2803  .2814   -0.391.35    1.8225 .2875  .2884   -0.311.36    1.8496 .2947  .2953   -0.20 range of1.38    1.9044 .3090  .3093   -0.10 primary1.40    1.9600 .3233  .3233   -0.00 interest1.50    2.2500 .3935  .3932   +0.081.60    2.5600 .4631  .4631   0.001.70    2.8900 .5330  .5330   0.001.80    3.2400 .6038  .6029   +0.151.90    3.6100 .6760  .6728   +0.472.00    4.0000 .7500  .7427   +0.972.10    4.4100 .8261  .8126   +1.63__________________________________________________________________________ 
    
     It can be seen from Table I that the values of these two expressions are equal to within 0.5 percent for the refractive index range of 1.33 to 1.9. This range of refractive indices includes practically all liquids of interest in liquid chromatography. The quartic/quadratic expression of refractive index in equation (5) can thus, for practical purposes, be closely approximated by a simple linear function. That is 
     
         ΔI/I = C (0.6990 n - 0.6553)                         6 
    
     If the solvent of the chromatographic column effluent has a refractive index of n 1  and the sample plus solvent has a refractive index of n 2 , the change in light transmitted through the flow cell is proportional to n 2  - n 1 . That is 
     
         ΔI.sub.2 - Δ I.sub.1 /I = 0.6990 C (n.sub.2 - n.sub.1) 7 
    
     For small changes in sample concentration, the change in the refractive index of the sample plus the solvent fluids is essentially proportional to the sample concentration. Thus, the change in light transmitted through the flow cell is approximately proportional to the sample concentration in the column effluent. 
     Equation (7) shows that the sensitivity of the change in light passing through the flow cell is increased by increasing the quantity represented by the constant C. The magnitude of C is expressed as follows: ##EQU8## 
     The sensitivity of the refractive index detector of this invention is inversely proportional to the radius of the flow cell and directly proportional to the square of its length. 
     These parameters can be established by one of skill in the art in order to obtain the desired sensitivity for the detector. Applicant has found that one suitable embodiment of this invention incorporates a flow cell having a length of 10 centimeters, a radius of 0.025 centimeters, with a temperature drop along the cell wall of 10° C/cm and a volume flow rate of approximately 50 milliliters per hour. 
     Referring to FIG. 3, an alternate embodiment of this invention is shown. In this instance, the light source 50 emits light through the entrance window 42 which has been collimated into parallel rays by a collimator 13. The temperature gradient dT/dy perpendicular to the side walls 41 of the flow cell in this embodiment extends outwardly from the cell walls, rather than inwardly. That is, the heat transmission apparatus removes heat from the longitudinal walls 41, such that the cooler regions of the fluid within the flow cell 40 lie near the walls, the warmer regions lying in the central areas. 
     As can be seen from FIG. 3, the parallel light rays entering the entrance window 42 tend to be bent outwardly toward the longitudinal walls of the cell 40. The degree of this bending, and the consequent reduction in the amount of light passing ultimately through the exit window 44 to the photocell 42, is a function of the refractive index of the fluid within the flow cell 40, provided that the temperature gradient dT/dy remains constant with respect to time. 
     Thus, as in the previously described embodiment, the light emerging from the exit window 44 is a function of the refractive index of the liquid within flow cell 40 at a given flow rate. 
     The foregoing discussions have been generally directed towards determining refractive index of the fluid in the flow cell when flow rate is known. The flow rate indicator embodiment of the invention is next described wherein the refractive index value is known. 
     Combining equation (8) with equation (5) results in the equation. ##EQU9## where ΔI/I represents the relative increase in light transmitted through the flow cell; and β is the thermal coefficient of expansion. The partial derivative δT/δy depends on the cell configuration and the mechanism which generates the temperature gradient normal to the cell wall. A uniform wall temperature requires a different expression for δT/δy than does a linear wall temperature variation. Whether or not the fluid enters the cell at the wall temperature also greatly influences the form of the expression for δT/δy. δT/δy is next calculated for several specific cell configurations. 
     Case 1 
     Uniform Wall Temperature With Fluid Entering Cell At Different Temperature. 
     Cholette reviewed experimental data and arrived at the following empirical expressions for heat transfer in laminar flow in a tube (M. Jakob, &#34;Heat Transfer&#34;, vol. 1, p. 546, John Wiley &amp; Sons, 1949). 
     
         N.sub.Nu.sbsb.a = C (N.sub.Gz).sup.n for 10.5 &lt; L/d &lt; 63 
    
     and 
     
         C = 2/.sub.π, n = 1 for N.sub.Gz &lt; 5 
    
     
         C = 1.56, n = 0.4 for N.sub.Gz between 5 and 150 
    
     where 
     
         N.sub.Nu.sbsb.a = h.sub.a d/k.sub.a 
    
     (Nusselt Number evaluated at the arithmetic mean temperature of the fluid) 
     
         N.sub.Gz = c.sub.p m/kL 
    
     (graetz Number) 
     h a  = film heat transfer coefficient 
     d = diameter of tube, L = length of tube 
     k a  = thermal conductivity of fluid 
     m = mass flow rate of the fluid, c p  = heat capacity 
     It may be shown that the following single function does a reasonable job of approximating Graetz Numbers up to 25. 
     
         N.sub.Nu.sbsb.a = 5.62 (1 - e .sup.-.sup.0.125 N.sbsp.g.sbsp.z) for N.sub.Gz &lt; 25                                             10 
    
     The largest relative deviation of this single function from the dual function expression of Cholette occurs in the region of transition from one of Cholette&#39;s functions to the other. This transistion region extends from about N Gz  = 3 to N Gz  = 7. ##EQU10## 
     
         and 10.5 &lt; L/d &lt; 63 
    
     Combining equations 9, 10 and 11 to form equation 12, ##EQU11## 
     where ##EQU12## is substituted for N Gz . 
     The volume flow rate, V, only appears in the exponent. Equation (12) can therefore be written: 
     
         ΔI/I = C.sub.1 (1 - e .sup.- .sup.C.sbsp.2 V) 
    
     where C 1  &amp; C 2  are independent of flow rate. Rearranging terms, ##EQU13## From the Binomial Theorem, (1 - x ) n  =  1 - nx+  . . . ##EQU14## Taking natural logarithms of both sides, ##EQU15## Definition of Absorbance: ##EQU16## 
     The volume flow rate is related in the same way as absorbance to change in eight intensity transmitted through the flow cell for the special case under consideration. Contemporary electronics developed to give outputs linear with absorbance can be readily modified using ordinary skill in combination with this teaching to give a signal which is proportional to flow rate in the above described embodiment. 
     SPECIFIC EXAMPLE FOR UNIFORM WALL TEMPERATURE CELL 
     Next described is a cell having a uniform wall temperature with the following parameters: 
     L = 10 cm, s = 1.0 mm, Fluid = Isooctane (2,2,4 trimethylpentane 
     0.01 change in absorbance corresponds to flow rate change of 0 to 100 ml/hr. 
     
         absorbance = - log.sub.10 I/I.sub.o = - (1/2.303) in I/I.sub.o ##EQU17## but I.sub.o /I  1.0 
    
     
         so ΔI/I = - 2.303 A = 2.303 × 10.sup.-.sup.2 at V = 100 ml/hr 
    
     
         N.sub.Gz =  813 (V/L ) = 2.26 
    
     
         1 - e.sup.-.sup.0.125 .sup.* 2.26 =  1 - e.sup.-.sup.0.282 =  1 -  0.75 = 0.25 ##EQU18## 
    
     
         = 10.24 × 10.sup.-.sup.4 
    
     Substituting above in equation (12) 
     
         2.303 - 10.sup.3 (0.316) (10.24 × 10.sup.-.sup.4) (5.62/0.2) (0.25) (T.sub.a -  T.sub.w) 
    
     
         (T.sub.a -  T.sub.w) = 0.0101 °C 
    
     error in equation (14) due to binominal expansion: ##EQU19## so next term in binomial expansion ##EQU20## 
     
         = 0.0285 
    
     previous term in binomial expansion ##EQU21## 
     
         0.0285/0.25 = 0.11 or 11%. 
    
     Since series is oscillating in sign, the error due to other terms not included in deriving equation (14) is less than 11% at 100 ml/hr for the specific example chosen. 
     CASE 2 
     Linear Temperature Change Along Cell Wall With Fluid Entering Cell At Temperature Of Wall 
     Next described is a cell having a linear temperature change along the cell wall with the fluid entering the cell substantially at the temperature of the wall 
     
         T = K .sup.. x 
    
     where K is a constant. 
     If frictional heating is neglected, the temperature distribution is described by following differential equation; 
     Under steady state, δT/δ t = O.sup.. 
     Fully developed Poiseuille flow; 
     
         v = v.sub.x =  2 v.sub.m (1 - r.sup.2 /s.sup. 2) 
    
     where v m  is the mean velocity of the fluid. 
     Because of the axial symmetry, each radial temperature profile will be the same as neighboring profiles. In other words, an equal amount of heat flows into the fluid per unit length along the entire cell. The axial and radial variables can be separated as follows; 
     
         T = K x + f (r) 
    
     where K is independent of x and f (r) is a function of r only. ##EQU22## Integrating: ##EQU23## Boundary Conditions; f remains finite at r = 0 → C = 0 f = 0 at r = s →  D = - (3 s 2  /16) ##EQU24## but since ##EQU25## 
     SPECIFIC EXAMPLE 
     Linear temperature profile along wall, 10 cm path length, 2 mm inside diameter flow cell with isooctane as fluid and 100 ml/hr corresponds to absorbance change of 0.01 
     Combination equations 9, 10 and 15 gives ##EQU26## 
     CASE 3 
     Uniform Wall Temperature With Fluid Entering Cell at Wall Temperature - Temperature Gradient at Wall Generated By Frictional Heating In Fluid. 
     Differential Equation Governing Heat Transfer ##EQU27## 
     
         v = v.sub.x =  2 v.sub.m (1 - r.sup.2 /s.sup.2 )    Integrating, ##EQU28## Boundary Conditions, 
    
     
         T finite at r = 0 → B = 0 
    
     
         T = T.sub.o at r = s → C = T.sub.o +  v.sub.m.sup.2ν /2α c.sub.p ##EQU29## 
    
     SPECIFIC EXAMPLE 
     10 cm path length, 2mm inside diameter flow cell, isooctane as fluid, uniform wall temperature with frictional heating of fluid as only means of generating temperature gradient at wall 
     Combining equations 9, 10 and 16. ##EQU30## assume 
     
         V = v.sub.m s.sup.2 =  100 ml/hr = 1/36 ml/sec 
    
     
         Δ I/I = 10.sup.3 (0.316) (10.2 × 10.sup.-.sup.4) (2.26 × 10.sup.-.sup.3) 
    
     
         Δ I/I = 7.3 × 10.sup.-.sup.4 
    
     
         Change in Absorbance = 7.3 × 10.sup.-.sup.4 /2.303 
    
     
         = 3.2 × 10.sup.-.sup.4 
    
     (absorbance change corresponding to change in flow rate from 0 to 100 ml/hr in 2 mm diameter, 10 cm long flow cell) 
     Referring now to FIGS. 4a-4d, there are depicted several embodiments of flow cells similar to that of FIG. 2 except for the specific heat transmitting device 54. Elements similar understood elements described with respect to FIG. 2 have like members, and descriptions thereof are not repeated in the following discussion. Also, the electronic circuits for driving the heat transmitting elements and determining absorbance, although not shown, are understood to be as described with respect to FIG. 2. It is unerstood that the flow indicating apparatus and method of this invention is not limited to flow cells used exclusively in chromatography systems, but has been described in such a system for convenience. 
     FIG. 4a illustrates a cell exhibiting a uniform wall temperature which generates a temperature gradient at the cell wall because the fluid enters the cell at a different temperature than the wall. The cell wall temperature is maintained by a thermoelectric element, replacing the water jacket of FIG. 2. Such apparatus as here depicted is suitably utilized in conjunction with the description of &#34;Case 1&#34; above. 
     FIG. 4b depicts a flow cell having heat transmitting means comprising heat source 52a and heat sink 52b which establishes a wall temperature that varies as a linear function of distance along the cell. In this embodiment the fluid enters the flow cell at or near the temperature of the cell wall at the inlet end 46 of the flow cell. Heat flow into the fluid at the wall generates a temperature gradient in the fluid near the wall in a direction substantially transverse to fluid flow. A longitudinal flow of heat in the cell wall from heat source 52a to heat sink 52b generates a linear temperature profile along the cell. The heat source 52a is typically an electrical heater, which may be the light source lamp itself. 
     FIG. 4c depicts another embodiment of a flow cell having its wall temperature varying as a linear function of distance. Heater 52 uniformly surrounds flow cell 40 and has current flowing in a longitudinal direction along the cell walls 41. The uniform heater 52 may typically be of the resistive type. As the fluid enters the flow cell at or near the temperature of the cell wall at the inlet end, heat flows into the fluid creating a linear temperature profile along the cell wall. The flow cells depicted in FIGS. 4b and 4c are suitably utilized in conjunction with the above examples of &#34;Case 2&#34;. 
     FIG. 4d illustrates a flow cell wherein the fluid enters the cell at the same temperature as the uniform wall temperature, and a gradient in temperature is generated at the wall by frictional heating of the fluid as it flows through the flow cell. That is, the above described heat transmitting apparatus is replaced by a thermal conductive jacket surrounding the wall 41 of the flow cell. Such a thermal conductor tends to prevent longitudinal gradients from developing within the cell. In some applications a stainless steel wall 41 provides adequate thermal conduction, and a thermal conducting jacket is unnecessary. 
     Referring now to FIGS. 5a and 5b, there are depicted flow cells having heating arrangements which control temperature of the incoming fluid. A separate thermo-electric element 70 preheats the incoming fluid to a desired absolute temperature T1 independent of the temperature T2 of the flow cell. A separate water jacket is also suitably utilized to preheat the incoming fluid. 
     Differential temperature sensing apparatus 72 maintains a temperature difference between the incoming fluid and the cell by way of a feedback circuit, which may utilize either a simple heating coil or a thermal-electric element. If the temperature of the cell wall is T degrees, than the incoming fluid is preheated and maintained at a temperature of T +ΔT for such applications as above described in &#34;Case 1&#34;. 
     This invention provides a novel and effective apparatus and method for detecting the refractive index and flow rate of a fluid passing through a flow cell. Rather than compensating for temperature differentials in the fluid, applicant has devised a way of utilizing temperature differences in order to detect refractive index and flow rate. 
     The apparatus and method of this invention provide for almost perfect linearity of response over a wide range encompassing the indices of refraction of practically all liquids which are of substantial use in liquid chromatography. 
     The above described embodiments are intended to be illustrative, rather than exhaustive. Persons of ordinary skill in the art will be able to make certain modifications, alterations and changes in the embodiments based on this description, without departing from the spirit of this invention.