Secondary Logo

Journal Logo

Pediatric Mechanical Circulatory Support

Thermal Analysis of the PediaFlow Pediatric Ventricular Assist Device

Gardiner, Jeffrey M.*; Wu, Jingchun*; Noh, Myounggyu D.**; Antaki, James F.; Snyder, Trevor A.; Paden, David B.*; Paden, Brad E.§

Author Information
doi: 10.1097/01.mat.0000247156.94587.6c
  • Free


Implantable ventricular assist devices (VADs) chronically transmit heat to surrounding tissue and pumped blood. Pediatric thermal design is particularly important because of the miniaturized size of the pump with less heat transfer surface area and lower efficiencies compared with adult-sized VADs. The PediaFlow device would be located in the left upper quadrant, in the anterior abdominal wall behind the left rectus abdominus muscle. Previous studies have shown implants operating at temperatures between 41°C and 44°C can cause damage to tissues, and impair various cellular functions relating to wound healing and immune response.1–3 Tissue necrosis typically occurs at temperatures above 42°C.4 Denaturation of blood proteins can occur at surface temperatures above 40°C.5 During a severe fever, the patient’s thermoregulatory set point is elevated and the temperature of blood passing through the pump, T, can be higher than 40°C, further increasing the risk of cell damage. As a safeguard, our group set the target maximum blood and tissue contacting surface temperature as 2°C (or less) above body temperature during the system design and optimization process.

The preliminary design stage of the PediaFlow project development focuses on detailed investigation of three pump topology candidates, distinguished by their fluid path designs: a symmetric, dual-impeller centrifugal pump (SDC); an asymmetric, dual-impeller centrifugal configuration (ADC); and a single gap mixed-flow impeller configuration (SGM). A primary objective in the evaluation of the three configurations is determining the thermal impedances from heat sources within the pump to blood and tissue contacting surfaces, which are used to size motor and thrust bearing components. Another main objective is to identify “hot spots” within the pump and ensure that surface temperatures are < 2°C above the inlet blood temperature. Previous investigations of heat dissipation in VADs and total artificial hearts were typically performed later in the design process, after a prototype device and mock loop were built,6–8 or using CFD with a detailed geometry model.9 In preliminary design and analysis of blood pumps, where multiple concepts are in evaluation, detailed CFD models and bench top prototypes are prohibitively expensive. Thus, a need for a simplified heat dissipation model arose.

The geometry of the candidate models is driven by the required flow rate and pressure which are based on the cardiac output needs of the intended infant patient population that includes newborns. To accommodate intracorporeal use in the smallest of patients, the overall size has to be minimized. It is also imperative that the pump operates at a speed below the first unstable mode of the magnetic suspension to ensure proper rotor stability and help reduce blood damage. General pump layout is based upon our group’s experience with previous adult VADs, namely WorldHeart’s HeartQuest and Streamliner. Wall thicknesses and seam locations are determined by manufacturing capabilities and structural integrity requirements. Thermal data from the initial models and the surface temperature constraint are used to set an upper limit on the maximum power of the optimized models. A weighted objective analysis is performed on the three optimized models and each candidate is rated based on several main categories including manufacturability, anatomic fit, hydrodynamic performance and biocompatibility, with heat dissipation efficiency as a subcategory of the biocompatibility objective. After a topology is selected, a more sophisticated thermal model is created with a suspension geometry based on detailed electromagnetic analysis and CFD optimized blade geometry and fluid path. At this design stage, additional sources of waste heat are identified and off-design flow rates are considered. The updated model incorporates CFD-derived fluid velocities, empirical heat transfer correlations, and the as-built geometry of the first generation implant. Upon completion of fabrication of the first generation prototype, an in vitro thermal study will be conducted using thermal sensors to map temperature distribution and verify empirical and CFD results.

Materials and Methods

Device Specifications

All three candidate pump topologies have a nominal flow rate, Qn, of 0.5 l/min, a minimum flow rate, Ql, of 0.3 l/min, and a maximum flow rate of 1.5 l/min. Fluid entrance and exit region dimensions are constrained to fit a cannula with a diameter of 5 mm for feasible implantation and reduction of hemodynamic losses. Titanium (Ti6Al4V), kh = 6.7 W/m-K, is chosen for the pump housings for its biocompatibility and high strength to weight ratio. The copper windings of the active thrust bearing and motor are assumed the significant sources of heat loss. Master Bond EP21ANLV thermal epoxy, kep = 0.61 W/m-K, is used as a potting compound to enhance thermal conduction throughout the pump and to promote heat dissipation to the tissue contacting surfaces.

Suspension for the SDC impeller (Figure 1) is provided by a fully magnetically levitated system, including two sets of radial permanent magnet bearings and two active axial thrust bearings. Impeller rotation is provided by a brushless DC motor with toroidally wound motor coils, which is compact and reduces radial negative stiffness. The nominal operating speed, ωsdc, is 9,000 rpm. A split fluid entrance volute promotes equal flow rates across the blades on each side of the impeller. The outlet volute’s function is to collect fluid from the impeller and to convert some of the kinetic energy of the flow into pressure energy.

Figure 1.
Figure 1.:
Symmetric, dual-impeller centrifugal pump (SDC).

The ADC impeller (Figure 2) suspension consists of a radial permanent magnet bearing, a permanent magnet moment bearing and an active axial thrust bearing. Rotation is also provided by a toroidally wound, brushless DC motor with a nominal operating speed, ωadc, of 9000 rpm. An inlet volute functions to guide fluid uniformly into the annular cross section of the impeller entrance region. The main fluid path in this pump is across the impeller main blades; a smaller amount of secondary flow travels through the back clearance gap between the impeller and housing. Secondary blades on the back clearance side induce antegrade flow within the gap and eliminate undesirable back flow and vortices.10 The outlet volute functions similarly to the volute of the SDC pump, collecting fluid and recovering kinetic energy.

Figure 2.
Figure 2.:
Asymmetric, dual-impeller centrifugal pump (ADC).

The SGM impeller (Figure 3) is supported by a fully levitated magnetic suspension,11 consisting of two sets of permanent magnet radial bearings and an active axial thrust bearing. Impeller rotation is actuated by a toroidally wound, brushless DC motor, with a nominal operating speed, ωamf, of 9000 rpm. Stationary guide vanes located in the diffuser wall function to recover some of the kinetic energy of the fluid flow and create a predominantly axial fluid velocity.

Figure 3.
Figure 3.:
Single gap mixed-flow impeller pump (SGM).

Fluid Characteristics

Blood passing through the pump is modeled as a Newtonian fluid with a density, ρb, of 1050 kg/m3, a viscosity, μb, of 0.0035 Pa-s, a thermal conductivity, kb, of 0.5 W/m-K, and a specific heat, cb, of 3.65 kJ/kg-K.12 Because blood temperature variation is relatively small in this model, viscosity and thermal conductivity are assumed constant. The pump’s rotary blade design generates a continuous, constant flow rate Q, ranging from 0.3 l/min to 1.5 l/min, with a nominal flow rate, Qn, of 0.5 l/min. In the entrance and exit regions of all three topology candidates, the mean fluid velocity is determined by the pump flow rate and cross sectional flow area, and Reynolds number is defined by:

where D represents the entrance region diameter and um is the mean fluid velocity over the regional cross section and is simply calculated by:

The Reynolds numbers within the entry/exit regions of three pump types are about 600 at the nominal flow rate. Therefore, the flow in both the entrance and exit regions is assumed to be laminar. The Prandtl number is a ratio of the momentum diffusivity to the thermal diffusivity of fluid and is assumed constant throughout the pump. It is calculated by:

The thermal entry length for a laminar flow is the distance from the inlet that thermal boundary layer is fully developed and is defined as13:

For each candidate ×fd ≫ ×e (the actual entry lengths of the pumps) which characterizes a developing flow. The average local Nusselt numbers for developing laminar flow are found by the Hausen correlation14:

This correlation assumes constant surface temperature and thermal entry region length, L. An alternative correlation due to Sieder and Tate15 is of the form:

where μs is the viscosity at the surface, which is assumed to be the same as μb. Results from each formula match within 7% difference. For our model, we will use the more widely used Sieder-Tate correlation. Finally, the local coefficient of convection is determined by:

Accurate solutions for thermal entry problems are difficult to obtain because temperature and velocity are dependent on the distance from the inlet as well as the internal diameter.

The internal flow through the impeller is three-dimensional and complex due to the effects of rotation, curvature, and turbulence. In order to simplify the heat transfer analysis, Reynolds number in the impeller region is calculated at several points along the fluid path based on the absolute velocity ub at the corresponding diameter of the impeller by:

where, Db is the corresponding diameter of the impeller, and the fluid velocities are given by:

Where ub is the absolute fluid velocity and Vθ, represents the circumferential component of the velocity at the blade tip. Vm is the meridional component of the velocity, rb is the radius of the blades at a given point, β is the blade angle, and Aw is the flow cross-sectional area. For the preliminary pumps, the blade angle β varies from 24° to 31° along the flow path. Fluid flow in this region transitions from laminar (ReD < 2000) to turbulent (ReD > 20000) at the nominal flow rate. The Nusselt numbers for regions with ReD > 10000 are defined by the Dittus-Boelter16 equation:

A more complex and generally more accurate13 correlation by Gnielinski17 is of the form:

where, f is the Moody friction factor. Assuming a smooth surface the Petuhov18 correlation is expressed as:

Nusselt numbers from the two correlations match within a 4% difference. The more sophisticated Gnielinski formula, which is valid for regions with ReD > 3000 versus ReD > 10000 for the Dittus-Boelter correlation, is used for the model. The local coefficient of convection is then represented by:

In the SGM design, the fluid path region, located downstream from the blade area and before the outlet guide vane area, is the most critical area of heat dissipation in the pump. In this area, the majority of the dissipated heat from the motor windings and thrust bearings is transferred into passing blood. Heat generated from fluid friction losses in this region is assumed negligible. In the SDC pump, fluid passes through the thrust bearing area prior to entering the blade region, resulting in a flow similar to the annular region in SGM pump with an outer diameter, ro, and inner diameter, ri. The circumferential velocity induced by the rotation can be derived as:

The absolute and meridional velocities are found using the previous equations used in the blade region equations 9 and 11. Flow in this region is transitionally turbulent with a Reynolds number of 3200 for the SDC pump and fully turbulent in the SGM pump (14000) and the Nusselt number, Nu, is found using Eq. (13). The local coefficient of convection in this region is found by:

The outlet stationary guide vanes located at the outlet region of the SGM candidate function to recover fluid energy and prevent swirling. Here we assume the circumferential component of the fluid velocity, Vθ, linearly decreases from a maximum value prior to entering the guide vanes (characterized by equation 16) to zero at the exit of the guide vanes. The meridional velocity and absolute velocities are determined using equations 9 and 11. The guide vanes have been specially optimized to smoothly transition a predominantly circular flow to axial flow, without creating undesired vortices in the region.

Tissue Characteristics

Tissue surrounding the candidate PediaFlow device is modeled as an isotropic solid material, with thermal properties of muscle tissue: a thermal conductivity, k, of 0.5 W/m-K, a density, ρ, of 1000 kg/m3, and a specific heat, c, of 3.7 kJ/kg-K.19,20 Heat is dissipated from the surrounding tissue primarily through tissue capillary perfusion21 and is modeled using Penne’s bio-heat equation22:

where, T, ρ, and c represent the tissue temperature, density, and specific heat, respectively; qp is the heat of perfusion; and qm is the metabolic heat generation term. Metabolic heat generation within the tissue model is assumed negligible.

Studies by Okazaki et al.21 and Liu et al.23 have shown that muscle tissue chronically heated by a constant heat flux decreases in temperature over several days after implant due to increased angiogenesis. In the study by Liu et al., muscle tissue was subjected to chronic heat fluxes of 0.04 W/cm2, 0.06 W/cm2, and 0.08 W/cm2 resulting in initial tissue temperature increases of approximately 1.5°C, 3.5°C, and 6°C, respectively, at a distance of 0 mm from the heated surface. From these, an approximate average perfusion convection coefficient for our model can be found by:

where, qp is the heat flux and ΔT is the increase in tissue temperature. Liu et al. found that perfusion changes little over time when heated to 40.5°C. Tissue perfusion is assumed to be time independent in our model because tissue temperatures are expected to be < 40°C. Also, heat dissipation through tissue perfusion is assumed uniform among the tissue surrounding the pump. Due to the complexity of accurately modeling tissue perfusion, the model convection coefficients are considered approximate and a more complete model will be implemented in the future.

Preliminary Thermal Models

Simplified three-dimensional thermal models of the pumps are created using SolidWorks (SolidWorks Inc., Concord, MA) solid modeling software and then imported into CosmosWorks (SolidWorks) FEA software. Based on the preliminary pump requirements, motor waste heat values can be estimated by:

where the pump head, H, the nominal flow rate, Q, and pump speed are estimated based on pump requirements. The motor constant, km, and pump fluid efficiency, ηf, are estimated based on preliminary motor and pump parameters. Heat induced by eddy currents in the motor also contributes to qm and is investigated later in the design process, once a particular topology candidate is chosen. Motor efficiency can be defined as:

where τ is the rotor torque. From this, the overall pump efficiency is determined by:

Estimated preliminary hydraulic efficiencies are 0.80 for motor efficiency and 0.08 for efficiency for SGM model at nominal flow rate and nominal operating speed. The virtual zero force (VZP) design of the thrust bearing minimizes stabilization energy and dissipated heat. This heat is related to the thrust bearing circuit noise force, Fnoise, and the voice coil constant, kv:

The nominal dissipation values for the motor and thrust bearing are estimated at 1.0 W and 0.75 W, respectively, for each pump. Localized average coefficients of convection are then assigned to several surface area regions along the fluid contacting surfaces of the pump housing geometry (Figures 4–6), with an assumed ambient inlet blood temperature, T, of 37°C. Heat transfer coefficients and Reynolds number at the nominal flow rate are shown in Table 1. The average coefficient of convection due to perfusion, htissue, is assigned to the entire outer surface of the pump. The largest perfusion is typically near the heated surface.20 Pump geometry is discretized with tetrahedral type mesh, with an average global element size of 0.020 inches and a tolerance of 0.001 inches. The analysis is set up for a steady state solution.

Figure 4.
Figure 4.:
SDC thermal model.
Figure 5.
Figure 5.:
ADC thermal model.
Figure 6.
Figure 6.:
SGM thermal model.
Table 1
Table 1:
Convection Values

The models are limited in that fluid convection coefficients are assumed constant over relatively large areas along the fluid path and are based on a nominal continuous flow rate. However, the actual fluid behavior of blood pumps is complex with unsteady three-dimensional velocities and convection coefficients varying continuously along the fluid path. The empirical correlations used to determine the Nusselt number can result in errors as large as 25%.13 Also, thermal contact resistances between pump components were not included in the pump models. Ideally the contact resistance between the titanium housings and thermal epoxy should be minimized. However, imperfections in the bond lines potentially will result in reduced heat flow and higher critical surface temperatures.

Preliminary Results and Discussion

Four independent mean thermal impedances are determined for each pump candidate topology using the FEA-generated, three-dimensional temperature distributions (Figures 7–9). The mean thermal impedances for each topology and heat transfer path are determined from the highest local surface temperature for a given waste heat value and are summarized in Table 2.

Figure 7.
Figure 7.:
Temperature distribution for SDC at nominal flow rate and nominal operating speed.
Figure 8.
Figure 8.:
Temperature distribution for ADC at nominal flow rate and nominal operating speed.
Figure 9.
Figure 9.:
Temperature distribution for SGM at nominal flow rate and nominal operating speed.
Table 2
Table 2:
Thermal Impedances

Next, the simulations were run with simultaneous motor and thrust bearing waste heat values of 1.0 W for each motor and 0.75W for each thrust bearing winding, which are the estimated preliminary maximum heating values. Maximum predicted tissue and blood contacting surface temperatures at nominal flow conditions are shown in Table 3.

Table 3
Table 3:
Maximum Surface Temperatures

Differences among impedance values and surface temperatures of the three topologies can primarily be attributed to pump geometry and fluid path. For instance, the thermal impedances and surface temperatures of the SDC pump are substantially higher than the other impedances in the study. The thrust bearings in this pump are located upstream from impeller blades where the fluid velocity is lower and flow more laminar, thus less heat transfer to passing blood. The ADC had the lowest overall thermal impedances and surface temperatures due to the proximity of the motor and thrust bearing windings to the secondary impeller blades, which increased local convection. The SGM design benefits from waste heat locations downstream from the impeller blades, where large circumferential fluid velocities increase heat transfer. The larger heat transfer surface areas of the thrust bearing windings of the SGM resulted in lower thermal impedances than the motor coil. The thermal impedances and maximum surface temperatures of the three candidate pump topologies were used as part of the comparison matrix in selection of the SGM as the leading design topology of the PediaFlow first-generation pump. The suspension and fluid path geometry is then optimized to create the first-generation pump geometry used for in-vitro and in-vivo studies.

To test the validity of the empirical formulas used in preliminary analysis, the velocity profile along the first-generation pump axis is found using empirical and CFD methods. The CFD velocity profile of the first-generation SGM model is found by tracking the streamline absolute velocity of 10 particles released at the inlet region using CFX (ANSYS Inc, Canonsburg, PA) software. The empirical profiles represent the mean velocity within the gap between the rotor and stator, while the CFD particles are free to move from the high velocity rotor surfaces to the zero velocity stator surfaces. This results in empirical profiles that fall near the middle of the CFD velocity distribution range at a given axial position (Figures 10 and 11). The empirical data fall almost entirely near the middle of the CFD velocity range, with a slight overestimation within the blade region.

Figure 10.
Figure 10.:
Velocity comparison, 0.5 l/min.
Figure 11.
Figure 11.:
Velocity comparison, 1.5 l/min.

Eddy Current Losses

Eddy currents in the PediaFlow device are created in the stationary titanium housing by a rotating motor magnet as shown in Figure 12. We approximate the radial (eddy-current producing) B-field by:

Figure 12.
Figure 12.:
SGM motor and shell arrangement.

The average loss per unit area in a thin shell (T≪R) is equivalent to the average loss in a planar system with the motor is rotating

and E-field is induced perpendicular to the current sheet. From Maxwell’s equations:

Where we have assumed γ = ωt, where ω is the rotational velocity of the rotor:

Choosing the constant of integration such that the average of Ey is 0 (i.e. C = 0) and simplifying yields:

The power loss is proportional to the conductivity of the titanium shell, σ, so we have that the time-average power loss is:

The SolidWorks model is then updated to account for eddy current heating, which in our case p = 0.6W at nominal flow rate (9000 RPM) and p = 1.0W at maximum flow rate (12000 RPM).

The Updated Thermal Model

The pressure rise versus flow rate curve, H-Q, and the efficiency versus flow rate curve, η-Q, of the first-generation prototype are determined by CFD using an optimized fluid path and pump geometry with SST turbulence model (Figures 13 and 14). Using equation 21, we can calculate the motor waste heat at several off-design points (Figure 15). The motor heat dissipation gradually increases with greater flow rate and pump speeds; however, this is counteracted by increased convective heat transfer due to the larger fluid velocities (Figures 10 and 11). From these data many operating points can be analyzed in detail. This study focuses on the nominal flow rate of 0.5 l/min at 9000 RPM and the high flow rate of 1.5 l/min at 12000 RPM. Internal flow surfaces of the updated first-generation PVAD model are divided into equally spaced convection regions 1 mm in axial length (Figure 16). This provides for a greater sensitivity to sudden fluctuations in heat transfer along the flow path over the preliminary models and an overall more accurate model. Figure 17 shows the distribution of convection values along the pump’s axis at the nominal and maximum flow rates. These profiles are based on an average pathline absolute velocity profile from CFD analysis, the geometry of the updated model and the empirical convection formulas of Sieder-Tate and Gnielinski (equations 6 and 13). The tissue perfusion model of the preliminary models is again utilized in this model. A layer of 0.003-inch-thick polyimide tape is placed between the copper coils and titanium housing to provide a second insulation layer in addition to the insulation of wire. Double insulation reduces the safety risk of shorting the electrical circuit to the housing components. However, the tape is not an ideal thermal conductor and potentially can increase the operating temperature of the pump. The section of the titanium housing where eddy currents are generated is modeled and meshed as a separate piece.

Figure 13.
Figure 13.:
Head vs. flow rate curves predicted by CFD.
Figure 14.
Figure 14.:
Pump hydraulic efficiency vs. flow rate curves predicted by CFD.
Figure 15.
Figure 15.:
Motor heat dissipation vs. flow rate.
Figure 16.
Figure 16.:
SGM first-generation thermal model.
Figure 17.
Figure 17.:
SGM first-generation convection distribution.

At nominal flow rate, the maximum blood contacting surface temperature is 37.5°C, while the maximum tissue contacting surface temperature is 37.4°C (Figure 18) essentially the same temperatures predicted by the preliminary model. The high flow rate maximum surface temperatures are 37.8°C and 37.7°C, respectively (Figure 19). The higher surface temperatures can be attributed to the increased impeller speed and reduced motor efficiency resulting in larger motor and eddy current losses.

Figure 18.
Figure 18.:
PediaFlow first-generation temperature distribution, 0.5 l/min and 9,000 rpm.
Figure 19.
Figure 19.:
PediaFlow first-generation temperature distribution, 1.5 l/min.


In this study, preliminary thermal models are created for candidate PediaFlow pump topologies using empirical heat transfer and fluid flow equations along with solid modeling and FEA software. The models demonstrate that useful VAD heat dissipation information can be procured without time intensive prototyping or CFD studies. By simulating waste heat transfer out of the device as localized areas of convection on the blood and tissue contacting surfaces of the pump, the complex behaviors of blood flow and tissue perfusion can be evaluated using a steady state FEA solver. The study also demonstrates that thermal impedances and temperature distributions were mainly affected by the proximity of the waste heat sources to various regions of the fluid path, such as the impeller blades and guide vanes.

Thermal impedances and surface temperatures of the three candidate pump topologies were as part of the selection criteria in the choice of SGM as the final topology of the PediaFlow first-generation pump. This study suggests that the current pump design will not thermally damage blood or tissue during normal operating conditions. Preliminary validation of the thermal model by CFD shows strong agreement between the empirically derived fluid velocity profiles and CFD based profiles.


1. Utoh J, Harasaki H: Effects of temperature on phagocytosis of human and calf polymorphonuclear leukocytes. Artif Organs 16: 377–381, 1992.
2. Pasha R, Benavides M, Kottke-Marchant K, Harasaki H: Reduced expression of platelet surface glycoprotein receptor IIb/IIIa at hyperthermic temperatures. Lab Invest 73: 403–408, 1995.
3. Hershkoviz R, Alon R, Mekori Y: Heat-stressed CD4+ T lymphocytes: differential modulations of adhesiveness to extracellular matrix glycoproteins, proliferative responses and tumour necrosis factor-α secretion. Immunology 79: 241–247, 1993.
4. Seese T, Harasaki H, Saidel G, Davies C: Characterization of tissue morphology, angiogenesis, and temperature in the adaptive response to muscle tissue to chronic heating. Lab Invest 78: 1553–1562, 1998.
5. Yamakazi K, Litwak P, Kormos RL, Mori T: An implantable centrifugal blood pump for long term circulatory support. ASAIO J 43: 686–691, 1997.
6. Endo S, Masuzawa T, Tatsumi E: In vitro and in vivo heat dissipation of an electrohydraulic totally implantable artificial heart. ASAIO J 43: 592–597, 1997.
7. Navarro RR, Kiraly RJ, Harasaki H, et al: Heat dissipation through the blood contacting surface of a thermally driven LVAS. Int J Artif Organs 11: 381–386, 1988.
8. Tasai K, Takatani S, Orime Y: Successful thermal management of a totally implantable ventricular assist system. Artif Organs 18: 49–53, 1994.
9. Song X, Wood HG: Computational fluid dynamics (CFD) of the 4th generation prototype of a continuous flow ventricular assist device (VAD). J Biomech Eng 126: 180–186, 2004.
10. Wu J, Antaki JF, Wagner WR, Snyder TA: Elimination of adverse leakage flow in a miniature pediatric centrifugal blood pump by computational fluid dynamics-based design optimization. ASAIO J 51: 636–643, 2005.
11. Noh MD, Antaki JF, Ricci M, Gardiner J: Magnetic levitation design for the PediaFlow™ ventricular assist device. Proceedings of 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics: 1077–1082, 2005.
12. Fiala D, Lomas KJ, Stohrer: A computer model of human thermoregulation for a wide range of environmental conditions: The passive system. J App Physiol 87: 1957–1972, 1999.
13. Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. New York, John Wiley & Sons, 1996.
14. Hausen H: Z. VDI Beih. Verfahrenstech 4: 91, 1943.
15. Sieder EN, Tate GE: Ind Eng Chem 28: 1429, 1936.
16. Dittus FW, Boelter LMK: University of California, Berkley, Publications on Engineering 2: 443, 1930.
17. Gnielinski V: Int Chem Eng 16: 359, 1976.
18. Petuhov BS: Advances in Heat Transfer. New York, Academic Press, 1970.
19. Goto A, Nohmi M, Sakurai T, Sogawa Y: Hydrodynamic design system for pumps based on 3-D CAD, CFD, and inverse design method. ASME J Fluids Eng 124: 329–335, 2002.
20. Duck FA: Physical Properties of Tissues: A Comprehensive Reference Book. San Diego, Academic Press, 1990.
21. Okazaki Y, Davies CR, Matsuyoshi T, et al: Heat from an implanted power source is mainly dissipated by blood perfusion. ASAIO J 43: 585–588, 1997.
22. Pennes H: Analysis of tissue and arterial blood temperatures in the resting human forearm. J App Physiol 1: 93–122, 1948.
23. Liu E, Saidel G, Harasaki H: Model analysis of tissue to transient and chronic heating. Ann Biomed Eng 31: 1007–1014, 2003.
Copyright © 2007 by the American Society for Artificial Internal Organs