Most present day artificial lungs used with heart-lung machines during cardiopulmonary bypass, often referred to as oxygenators, are designed for and operated with a steady blood flow. Artificial lungs attached to the pulmonary circulation would receive pulsatile blood flow delivered by the right ventricle (RV), and pulsatility in these devices could affect the rate of gas transfer relative to what it would be with a steady flow at the same mean flow rate. The capability of oxygenators to transfer gas depends upon the blood flow rate through the device. At steady, low flow rates, the hemoglobin becomes fully saturated early within these devices, and their capability is underused. Superposing a pulse on such flows may make the device less efficient but still capable of fully saturating the hemoglobin and may have little effect upon the overall rate of oxygen transfer. At steady, high flow rates, on the other hand, the oxygenator may not be capable of fully oxygenating the hemoglobin. Superposing a pulse upon steady high flow rates reduces the efficiency and would result in reduced overall rate of gas transfer.
These devices usually consist of a packed bed of microporous fibers, with the blood flowing cross-wise over the outside of the fibers and the oxygenating gas flowing inside the fibers. Such a packed bed is essentially a pure resistance (R) to the blood flow. The strong pulsatile flow from the RV could be damped before flowing through the fiber bundle by incorporating a compliance (C) in the device proximal to the fibers. If the device considered is, for simplicity, a compliant/resistance device, the degree of damping would depend upon the time constant (RC) for the device. If the device has a large resistance, as is the case with many present day commercial oxygenators, only a small amount of compliance may be necessary to adequately damp the blood flow pulse and essentially achieve the rates of gas transfer associated with steady flow. Artificial lungs attached to the pulmonary circulation, here referred to as thoracic artificial lungs (TALs), must have low resistance to avoid a hemodynamic overload of the RV. If flow pulsatility reduces the oxygen transfer, such devices should incorporate an adequate proximal compliance section to achieve maximal rates of gas transfer.
The present report discusses the theoretical aspects of the pulsatility effects upon the rate of oxygen transfer, demonstrates the effects with some in vitro tests upon commercial oxygenators, and applies the theory to the experimental conditions and to a TAL. Commercial oxygenators were used in the in vitro tests because TALs are still in the developmental stages, and quantities of reproducible devices are not readily available.
A common form for representing heat/mass transfer to fluids flowing steadily through packed beds of particles or across tightly packed rods or fibers is, for mass transfer in the latter, as follows:1,2EQUATION
in which Sh = Sherwood number, defined as kl/D:k = mass transfer coefficient, l = characteristic length, D = molecular diffusivity of the mass transferring specie; Sc = Schmidt number, defined as ν/ Deff: ν = kinematic viscosity of the fluid, Deff = effective diffusivity of the transferring specie; Re = Reynolds number, defined as Vl/ν:V = cross sectional average velocity of the fluid. The Sherwood, Schmidt, and Reynolds numbers and their physical meanings are listed in Table 1. Many heat transfer texts suggest the constant β is approximately 0.5. 2 The constants φ and β depend upon the arrangement and uniformity of the rod or fiber spacing. If the transferring specie is nonreactive with any specie in the fluid, Deff = molecular diffusivity =D. If, however, the transferring specie can chemically combine with some specie in the fluid, the effective diffusivity must be modified. Oxygen, for instance, reacts with hemoglobin in blood, and the effective diffusion coefficient, as shown in the Appendix, is Deff = [D/(1+λ)], 3,4 in which λ = 1.34(CHb /α) (d S/d P), CHb = hemoglobin concentration, α = solubility of oxygen in blood, S = fractional saturation of hemoglobin with oxygen, P = partial pressure of oxygen in the blood, and d S/d P = slope of the oxyhemoglobin dissociation curve. Using the Hill relation for the oxyhemoglobin relation and assuming CHb = 0.12 gm/ml, λ ≈ 83 at typical venous conditions of P = 36 mm Hg (i.e., S = 0.65) and λ ≈ 2 at typical arterial conditions of P = 120 mm Hg (i.e., S = 0.98). Thus the term 1+λ can vary from 84 to 3 as the blood oxygenates in passing through the device.
Equation 1 is a correlation that has been established for steady flow at low Reynolds Numbers; it has been shown to be valid in heat transfer for Re up to 100 or more. 5 The present theoretical analysis applies Equation 1 to both steady and unsteady flow conditions. For applications to unsteady flow, the time varying oxygen transfer is calculated assuming quasi-steady conditions, that is, the rate of transfer at each instant of time is that which would occur under steady flow at that instantaneous Reynolds number. Whether quasi-steady conditions are appropriate depends upon the magnitude of an oscillatory Reynolds number, Reω, defined in Table 1. If the oscillatory Reynolds number, Reω, is small, the time dependence effects are small with respect to the viscous effects, and the equation established for steady flow can also be used under pulsatile conditions, applying quasi-steady calculations, although exactly how small for the packed bed is not really known and would have to be determined from experiments. It is well known that the condition necessary for quasi-steady oscillatory flow in a pipe of radius R is that the oscillatory Reynolds number should be less than unity. One could speculate that, if Reω < 1 in a packed bed, quasi-steady conditions are valid. In the present study, Reω < 1 and quasi-steady conditions hold; being furthermore Re < 100, the authors hypothesize that Equation 1 is valid. Note however that at high Re or Reω, separation and secondary flows may develop, and pulsating flows may enhance the gas transfer over what it would be with steady flow. Equation 1 would not be applicable in such cases.
By rearranging Equation 1, using the definitions for Sh, Sc, and Re listed in Table 1, the local mass transfer coefficient, k, can be expressed as EQUATION
For a specific blood at a specific temperature, which determines D and ν, and specific device, which sets d, Af and vf, EQUATION
The local mass transfer coefficient for oxygen, thus, depends upon the blood flow rate, Q, and the saturation level of the blood. Because β is less than unity, the local k is somewhat less than proportional to the flow rate. For an unsteady flow, the time-average local mass transfer coefficient at any point in the oxygenator will be less than it would be for steady flow at the same mean velocity. Suppose, as a simple example, a periodic flow has a mean flow rate Q̄, a period T, and a flow such thatQ = 4Q̄ for the first T/ 4 and Q = 0 for the last 3 T/ 4. The mean flow rate would be Q̄, and the time average mass transfer coefficient would be k∼[(4Q̄)β]/4. If β = 0.5, say, k∼0.5Q̄0.5 for this unsteady flow. If, on the other hand, the flow were steady with a flow rate of Q̄, the local mass transfer coefficient would be k ∼ Q̄0.5.
The local mass transfer coefficient also depends upon (1+λ)1/3, and λ varies greatly as the blood oxygenates. For a uniform fiber bundle, the local mass transfer coefficient will be very high at the inlet where the venous blood λ is approximately 83, but it will be greatly reduced in the more distal regions of the bundle when the more saturated blood has a λ of 2 or less. Consider the example illustrated in Figure 1, which shows λ for a device that is designed so that with a steady, uniform flow, the blood is essentially saturated in the distal quarter of the distance through bundle. Very little transfer occurs in this distal region where λ is essentially zero. If the same device is used with an unsteady flow, the reduced rate of transfer in the proximal regions, because of an unsteady flow, may be compensated for by an increased rate of transfer in the distal regions. The reduced rate of transfer in the proximal regions would result in blood arriving in the distal regions with a lower saturation and larger λ and, therefore, enhanced transfer rates in those regions. The overall reduction of oxygen transfer to the blood, therefore, would not be as much as that caused by the unsteady flow rate effects alone.
The transfer rate under pulsatile blood flow conditions is further complicated by the fact that the pulse period is generally different than the transit time within the fiber bundle. Different aliquots of blood, therefore, experience different Reynolds number histories as they move through the bundle. The transit time through the bundle, furthermore, can vary depending upon the pulse phase as the blood enters the bundle. Thus, outlet blood samples may show a periodic variation in hemoglobin saturation.
The overall relative magnitude of pulsatility effects may be determined by applying this theory to specific devices. As an example, consider how a pulsatile flow might affect blood oxygenation in a TAL. The blood would be pumped through the device by the right ventricle, the output of which is strongly pulsatile. With current technology, such a device would consist of a packed bed of fibers. Unless the device also incorporates a proximal compliant section, however, the blood flow through the fiber bundle would be highly pulsatile. How pulsatile it would be depends upon device compliance and how much of the pulmonary arterial system is accessible to the right heart output.
Materials and Methods
The effects of flow pulsatility on oxygen transfer were evaluated by performing in vitro blood tests on commercial oxygenators and applying the above theory to these oxygenators and to a previously reported TAL fiber bundle. 6 The validity of quasi-steady theory is assessed by comparing theoretical predictions with the in vitro test results.
In Vitro Tests
In vitro tests were performed on six commercial oxygenators, three SARNS pediatric oxygenators (rated flow rate up to 2.5 l/min), and three DIDECO D701 MasterFlow 34 infant oxygenators (rated flow rate up to 1.5 l/min). The fiber bundle for both oxygenators is made of microporous fibers. Characteristics of the oxygenators and of the TAL are listed in Table 2. The experimental setup is shown in Figure 2 and is composed of two polyethylene reservoirs, a pulsatile pump (Harvard Pulsatile Blood Pump, Harvard Apparatus, Holliston, MA), a compliance chamber, and a commercial oxygenator. The compliance chamber is made of a Plexiglas cylinder with a movable piston such that the compliance can be altered by varying the air volume within the cylinder. Gas tanks (O2, CO2, N2) supply the inlet gas to the oxygenator, and a heat exchanger is used to maintain a blood temperature of 37°C. Two blood flow meters (T206, Transonic Systems, Ithaca, NY) were used to record blood flow, one at the pump outlet and the other at the oxygenator outlet. A gas flow meter (Top Trak Model 820, Sierra Instruments, Monterey, CA) measured gas flow rate at the oxygenator gas inlet, while a gas oxymeter measured oxygen percent in the gas mixture. Approximately 20 L of fresh bovine blood was anticoagulated with heparin (10,000 U/L) and ethylenediaminetetraacetic acid (1 g/L), passed through a milk hair filter (Kendall, Boston, MA) into a polyethylene reservoir, and recirculated until standard inlet conditions were reached. Partial pressures of O2 and CO2 were adjusted by varying the O2, N2, and CO2 percentages in the sweep gas mixture. Sodium bicarbonate was used to adjust base excess and pH (1 mEq/L for each negative mmol/L of base excess). Once standard inlet blood conditions were achieved, the test phase began using a single pass technique. Inlet blood gas values were
In all experiments, the pulsatile pump with the compliance chamber was used first, and then the pulsatile pump was used alone. The piston level in the compliance chamber was such that the flow pulsatility was essentially completely damped. Three values of mean flow rate, approximately 0.8, 1.3, and 1.8 L/min, were used for each of the two flow shapes tested. The different flow rates were obtained by varying stroke volume without changing the frequency or the ratio of systolic to diastolic time. The frequency was 60 beats/min, and the systolic/diastolic time ratio was 35%. Typical flow rate shapes are shown in Figure 3. The Reynolds numbers for the average flow rates varied from 2 to 8. The oscillatory Reynolds number, calculated as [v fd/[2(1−vf)]2 · (ω/v), in which ω = 2πf and f = pulse frequency in Hz, was equal to 0.1 ± 0.005 for the SARNS devices and 0.03 ± 0.001 for the DIDECO devices. The inlet gas mixture was 50% oxygen and 50% nitrogen.
For each combination of average flow rate and flow rate shape, two inlet and two outlet samples were collected and put in ice until they were analyzed. The inlet and outlet samples were taken simultaneously. Flow rates were recorded when the blood samples were taken, using a sampling frequency of 500 Hz. Partial pressures of O2 and CO2 and pH were measured for each sample using the ABL3 blood gas analyzer. The hemoglobin content, CHb, was estimated from the hematocrit, measured using the microcapillary tube centrifugation technique. The oxygen transfer rate, O2, was calculated as EQUATION
in which T̄Q is the time average blood flow rate, ΔP is the difference between Pout and Pin (each averaged over the two samples), and ΔS is the difference between the outlet and inlet hemoglobin saturation, calculated using the Hill equation, with parameters valid for bovine blood 3 and P and pH values averaged over the two samples.
Theoretical Calculation of Oxygen Transfer
By equating the local oxygen transfer rate at the fiber surfaces to the oxygen uptake by the blood, the local instantaneous mass transfer coefficient, k, as shown in the Appendix, is EQUATION
in which Pg = partial pressure of the oxygen on the gas side, and Q = time varying blood flow rate. Equating Equations 2 and 5 gives the gradient of the oxygen partial pressure along the blood path x:EQUATION
Equation 6 is numerically integrated, from x = 0 to x =L to obtain the change in oxygen partial pressure from Pin at the entrance to Pout at the exit. Briefly, the numeric method consists of increasing the variable time, t, using 10−4 second increments and calculating the corresponding dx as EQUATION
This dx value is used in Equation 6, which is integrated using a Runge-Kutta method. The variable t is increased until EQUATION
equals the blood path length L. The calculations were repeated for all possible start times within the pulse phase and the results averaged for the mean Pout.
Validation of the Quasi-Steady Theory
The quasi-steady theory was validated by comparing the in vitro test results with the predictions calculated by applying the theory to the commercial oxygenators. The parameters for the integration of Equation 6 were set to the recorded experimental values or calculated from recorded values. The required parameters are the instantaneous values of flow rate, Q, the inlet values of P, S, pH, the blood hemoglobin content, and the values of β and φ for the specific oxygenator tested. An analytic representation of the experimental Q waveforms, developed by applying a Fast Fourier Transform (FFT) to the recorded Q values, was used for the input flow rate waveforms. The measured inlet blood gas values and blood hemoglobin concentrations from the experiments were used as input values in the theory. Last, values of β and φ were determined by a fit between the theory and experimental data from the steady flow experiments. Details are given in the Appendix.
Equation 6 was integrated to determine the theoretical outlet PO2 and specific oxygen transfer under pulsatile flow. The agreement between theoretical and experimental oxygen transfer was examined using the methods of Bland and Altman. 7
Application of the Theory to a Thoracic Artificial Lung
A simulated right heart pulsatile output, Q, was numerically calculated by running the lumped parameter model previously described, 8 in which the TAL chamber and inlet compliances are set to zero. Figure 4 shows the shape of Q for a mean flow rate of 4 L/min. To obtain other flow rates, the stroke volume was changed, while the heart rate and systolic duration were fixed. The inlet blood gas values used in the quasi-steady model for O2 transfer calculations were Pin = 40 mm Hg, Sin = 70%, pH = 7.4, CHb = 11 g%, and temperature = 37°C.
The rate of oxygen transfer calculated under pulsatile flow conditions is compared with the transfer calculated under steady flow conditions for different values of average flow rate, ranging from 2 to 8 L/min. The Reynolds number for these average flows ranged between 1 and 3, and the oscillatory Reynolds number was 0.25.
In Vitro Tests
The average flow rate for a pulsatile experiment, in some instances, was slightly different than the flow rate for the corresponding steady flow experiment. This makes it difficult to distinguish the regime effect from the flow rate effect. The authors compare the results therefore, on the basis of specific oxygen transfer, defined as the ratio of the oxygen transfer rate to the average blood flow rate. Figure 5 compares the experimental, in vitro, specific oxygen transfer rates under steady and pulsatile flow using the commercial oxygenators. The average specific oxygen transfer rate is reduced by pulsatile blood flow in the fiber bundle at each flow rate. The average reduction under pulsatile flow is 10%. Two-way ANOVA for flow rate and pulsatility confirms the hypothesis that specific oxygen transfer under pulsatile flow is significantly lower than that under steady flow (p < 0.04).
Validation of the Quasi-Steady Theory
Figure 6 compares the theoretical and experimental specific oxygen transfer rates, O2−-p, for the commercial oxygenators with pulsatile flow. The estimated slope of the parameter for the regression line through the origin of these data is 0.99 (p < 0.001) with a 99% confidence interval (0.95, 1.03). The slope is not significantly different from 1 (p > 0.3). Figure 7 displays the difference between theoretical and experimental specific oxygen transfer rates according to the methods of Bland and Altman 7 and demonstrates no relationship between the difference in experimental and theoretical values and the average of the two values. Figure 7 also shows that the difference between experimental and theoretical specific oxygen transfer rates will be between −4.5 and 3.6 (ml/L) in 95% of data points with an average difference of −0.5 (ml/L). This value is not significantly different from 0 (p > 0.3), that is, it does not represent a consistent bias. The percent difference between measured and predicted O2−p ranges between −11% and 9%.
Application of the Theory to a Thoracic Artificial Lung
Figure 8 summarizes the rates of oxygen transfer, O2, obtained by applying the theory to the TAL. With the same mean flow rate, bundle flow pulsatility reduces the rate of oxygen transfer relative to that obtained under steady flow conditions. The percent decrease in oxygen transfer increases slightly with flow rate, from a minimum of 17% at 2 L/min to a maximum of 23% at 8 L/min.
The purpose of this study is to evaluate the effect of pulsatile flow upon oxygen transfer in artificial lungs, which is a crucial issue for thoracic artificial lungs that receive blood from the right heart. 6,9,10
Commercial oxygenators, usually operated with essentially steady flow, are often designed to oversaturate the blood, that is, produce outlet partial pressures of oxygen much greater than 100 mm Hg when used with 100% oxygen at the rated blood flow rate. Little or no differences between steady and pulsatile flow would be expected at these conditions. Pennati et al.11 reported the effect of pulsatile flow upon oxygen transfer in the Monolyth (Sorin Biomedica) oxygenator. The O2−pO2−s were evaluated over 6 hour in vitro blood tests using a roller pump with a pulsatile module (Stockert Instrument). They tested the oxygenator using average blood flow rates below the oxygenator rated flow rate and used 100% oxygen on the gas side. They obtained outlet PO2 values greater than 200 mm Hg (Gianfranco B. Fiore, Politecnico di Milano, personal communication, May 2002). No significant difference was found between O2−p and O2−s during the first 3 hours of the experiments. Because Pennati et al.11 used 100% oxygen on the gas side, operated below rated flow conditions for the oxygenator, and obtained very high PO2 values, the present authors would not have expected a difference between their steady and pulsatile flow experiments. During the last 3 hours of the tests, however, Pennati et al.11 observed a decay in oxygen transfer with steady flow, but no deterioration with pulsatile flow. The effect was greater at the higher average blood flow rates. They noted a significant (p < 0.002) average decrease in oxygen transfer of 4% during the last 3 hours of the steady flow experiments. The authors attributed the lack of deterioration with pulsatile flow as a washing mechanism achieved by the additional mixing generated by the alternate high and low flow rates and by micromovements of the fluid that partially permeates the pores because of the oscillating pressure gradient across the membrane.
The present experiments were only short term, and the present authors intentionally used 50% oxygen to push the capability of the devices by avoiding oversaturation. The authors observed a 10% average reduction in oxygen transfer with pulsatile blood flow relative to that with steady flow. The authors did not evaluate the oxygenator performance for long periods of time.
One measure of pulsatility intensity could be defined as the ratio of pulse height to average flow rate, (Qmax−Qmin)/Q̄, which may be called bundle flow pulsatility (BFP). The higher values of average flow rates in these experiments were achieved by increasing the stroke volume while keeping the frequency and systole/diastole ratio constant. In so doing, the BFP is essentially constant for all flow rates. The decrease in oxygen transfer that was observed at the higher average flow rates with pulsatile flow may have been less than it would have been had the blood flow rate been increased by increasing frequency, stroke volume, and systole/diastole ratio, the normal physiologic response. Such alterations increase BFP with increased cardiac output.
Equation 1 has been shown to accurately predict oxygen transfer under steady flow conditions. 1 The same theory has been applied in the present study to pulsatile flow conditions using the instantaneous values of flow rate in Equation 1, that is, assuming quasi-steady conditions. The Reynolds numbers for these flows were between 1 and 8, and the oscillatory Reynolds numbers were 0.1 and 0.03, indicating quasi-steady conditions are reasonable. The mean of the measured O2−p is not statistically different from the mean of the predicted values. The maximum absolute difference is approximately 10%. The differences between the theoretical values and the measured values may be caused by some inaccuracies in the parameters used in the theory or experimental variations in the experiments. The calculated O2 values for the experiments, for example, are based on using average ΔP and ΔS values from two sets of blood samples in Equation 4. If one calculates O2 using each single blood sample, then the O2 values can differ by as much as 20%. (Some sample to sample variation may be caused by the beat to beat effects discussed in the Introduction.)
The theory applied to the TAL predicts an average reduction in O2 of 20%, a higher reduction with respect to what was predicted and measured for the commercial oxygenators. This difference can be attributed to the greater pulsatility of the simulated right heart flow rate with respect to that of the experimental flow rate (compare Figure 3b with Figure 4). The average BFP was 3.4 for the experimental flow rate and 4.5 for the simulated right heart flow rate.
Using the theory, the amount of inlet compliance to the TAL that should be sufficient to increase the pulsatile transfer rate to steady state levels can be estimated. The bundle flow rate used for this purpose was numerically calculated running the lumped parameters model, 8 in which the TAL chamber compliance was set to 0 and the inlet compliance, used to damp the flow pulsatility, was varied in the range 0–2 mm Hg. A relatively small compliance (0.2 ml/mm Hg) is sufficient to obtain approximately 95% of steady flow oxygen transfer rate. The amount of C needed to adequately damp the flow proximal to the fiber bed with commercial oxygenators would be much less than the C value estimated for the TAL. The damping depends upon the RC of the device, and R is much larger for the oxygenators than for the TAL (see Table 2).
These results have implications in the design of implantable artificial lungs, in which the number of fibers, and therefore the maximum average flow rate, is limited by the need for a low resistance, compact device. The rated blood flow rate may be that necessary to meet basal metabolic demands. An increase in metabolic demands, resulting in an augmented cardiac output, would result in a less than proportional increase in oxygen transfer rate. The authors conclude that a compliance should be included in the design of all implantable TALs. Previous studies 8 show that TALs with a compliant housing are preferable, in this regard, to those using an inlet compliance, as inlet compliance chambers are less effective at reducing flow pulsatility.
This work was supported by NIH RO1 HL 59537. The authors thank Prof. Piercesare Secchi from the Department of Mathematics of the Politecnico di Milano for his help with the statistical analysis.
The Hydraulic Diameter in a Packed Bed of Fibers
The hydraulic diameter is defined as EQUATION
The blood volume is the volume within the packed bed that is open to flow, and the surface area is that surface area within the packed bed that resists the flow. With reference to Figure 9, the blood volume in a cross flow oxygenator is as follows: Blood volume = (volume around 1 fiber) × (no. of fibers in row in H) × (no. of rows):
The blood volume can also be written as EQUATION
in which vf is the void bundle fraction, or porosity of the device. By equating Equations a2 and a3, the spacing among fibers, s, can be expressed as a function of the porosity, vf, and the fiber diameter, dEQUATION
The surface area of the fiber bundle is as follows: Surface area of fibers = (surface area of 1 fiber) × (no. of fibers in row in H) × (no. of rows in L):EQUATION
and hence EQUATION
The Effective Diffusion Coefficient for Oxygen in Blood
The local concentration of oxygen in blood, CO2(x), is the sum of the oxygen that is dissolved in blood and bound to hemoglobin. The dissolved O2 depends upon the oxygen partial pressure, P, and upon the solubility coefficient of oxygen in blood, α. The quantity bound to hemoglobin depends upon the hemoglobin concentration, CHb, and hemoglobin saturation, S. Considering P, S, and CO2(x) functions of only the distance, x, through the fiber bundle EQUATION
The infinitesimal variation of CO2(x) over an infinitesimal distance dx is EQUATION
in which λ(P) is defined as 1.34 (CHb α) d S/d P. Applying Fick’s law to the diffusion of oxygen in blood, and combining it with the Equation a8EQUATION
and hence EQUATION
The effective diffusion of O2 in blood is reduced by the presence of hemoglobin, which acts as a sink for O2. Lambda is higher (and Deff lower) for higher values of dS/dP, that is in the linear region of the oxyhemoglobin saturation curve (venous blood). Conversely, lambda is low for arterial blood, corresponding to the flat region of the oxyhemoglobin saturation curve. The effective diffusion coefficient for oxygen in blood, therefore, is not constant with the blood path.
The Local Mass Transfer Coefficient for Oxygen in a Cross Flow Oxygenator
The local mass transfer coefficient for oxygen in blood is derived by equating the rate of oxygen uptake by blood to the rate of oxygen transfer from fibers.
With reference to Figure 9, the oxygen flux through an infinitesimal area dA of the fiber surface is EQUATION
in which dA = d/2 dθdz, C is the local oxygen bulk concentration in blood, Cs is the oxygen concentration at fiber surface, which is essentially the gas side oxygen concentration, and k is the mass transfer coefficient. The rate of oxygen transfer over length dx in fiber bed, assuming uniform conditions overall tubes within dx, is EQUATION
in which l is the total length of fibers contained within length dx, and is expressed as follows (see Figure 9 and Equation a4): Total length of fibers contained within length dx of fiber bed = (length of each fiber) × (no. of fibers in row in H) × (no. of rows in dx):EQUATION
Assuming k, C, and Cs are functions of x only, and using Equation a13EQUATION
The rate of oxygen uptake by blood is EQUATION
By equating Equation a13 to Equation a14EQUATION
Evaluation of the Bundle Parameters m and φ for the Oxygenators
Equation 5, recalling the definition for Re (see Table 1) can be written as EQUATION
Equation 1, recalling the definitions for Sh and Sc (see Table 1), can be written as EQUATION
Combining Equations a17 and a18 gives EQUATION
in which Re is constant when using data from the steady flow experiments. Integrating the left side from 0 to L, the right side from Pin to Pout, and dividing by L results in another form of a dimensionless mass transfer coefficient, ψ. Applying the logarithm to both sides of the integrated Equation a19, the following linear relation is obtained between ln(Re) and ln(ψ):EQUATION
The bundle parameters β and φ were derived by a linear fit between Equation a20 and data from the steady flow experiments.
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