A New Fundamental Bioheat Equation for Muscle Tissue: Part I—Blood Perfusion Term

1997 ◽  
Vol 119 (3) ◽  
pp. 278-288 ◽  
Author(s):  
S. Weinbaum ◽  
L. X. Xu ◽  
L. Zhu ◽  
A. Ekpene
Keyword(s):  
Heat Transfer ◽  
Muscle Tissue ◽  
Source Term ◽  
Venous Return ◽  
Blood Perfusion ◽  
Efficiency Coefficient ◽  
Bioheat Equation ◽  
Or Efficiency ◽  
Cross Sectional ◽  
Transfer Unit

A new model for muscle tissue heat transfer has been developed using Myrhage and Eriksson’s [23] description of a muscle tissue cylinder surrounding secondary (s) vessels as the basic heat transfer unit. This model provides a rational theory for the venous return temperature for the perfusion source term in a modified Pennes bioheat equation, and greatly simplifies the anatomical description of the microvascular architecture required in the Weinbaum-Jiji bioheat equation. An easy-to-use closed-form analytic expression has been derived for the difference between the inlet artery and venous return temperatures using a model for the countercurrent heat exchange in the individual muscle tissue cylinders. The perfusion source term calculated from this model is found to be similar in form to the Pennes’s source term except that there is a correction factor or efficiency coefficient multiplying the Pennes term, which rigorously accounts for the thermal equilibration of the returning vein. This coefficient is a function of the vascular cross-sectional geometry of the muscle tissue cylinder, but independent of the Peclet number in contrast to the recent results in Brinck and Werner [8]. The value of this coefficient varies between 0.6 and 0.7 for most muscle tissues. In part II of this study a theory will be presented for determining the local arterial supply temperature at the inlet to the muscle tissue cylinder.

A Combined Macro and Microvascular Model for Whole Limb Heat Transfer

1988 ◽  
Vol 110 (4) ◽  
pp. 259-268 ◽  
Author(s):  
W. J. Song ◽  
S. Weinbaum ◽  
L. M. Jiji ◽  
D. Lemons
Keyword(s):  
Heat Transfer ◽  
Venous Return ◽  
Prototype Model ◽  
Bioheat Equation ◽  
Cross Sectional ◽  
Surface Temperature Distribution ◽  
Wide Range ◽  
Cutaneous Tissue ◽  
Axial Variation ◽  
Arteries And Veins

A new prototype model for whole limb heat transfer is proposed wherein the countercurrent heat exchange from the large central arteries and veins in the core of the limb is coupled to microvascular models for the surrounding muscle and the cutaneous tissue layers. The local microvascular temperature field in the muscle tissue is described by the bioheat equation of Weinbaum and Jiji [1]. The new model allows for an arbitrary axial variation of cross-sectional area and blood distribution between the muscle and cutaneous tissue, accounts for the blood flow to and heat loss from the hand and treats the venous return temperature and surface temperature distribution as unknowns that are determined as part of the solution to the overall boundary value problem. Representative solutions are presented for a wide range of environmental conditions for a limb in both the resting state and during exercise.

The Bleed Off Perfusion Term in the Weinbaum-Jiji Bioheat Equation

1992 ◽  
Vol 114 (4) ◽  
pp. 539-542 ◽  
Author(s):  
S. Weinbaum ◽  
L. M. Jiji ◽  
D. E. Lemons
Keyword(s):  
Energy Equation ◽  
Source Term ◽  
Venous Return ◽  
Effective Conductivity ◽  
Blood Perfusion ◽  
Tissue Temperature ◽  
Bioheat Equation ◽  
Countercurrent System ◽  
Thermal Equilibration ◽  
Arteries And Veins

The microvascular organization and thermal equilibration of the primary and secondary arteries and veins that comprise the bleed off circulation to the muscle fibers from the parent countercurrent supply artery and veins are analyzed. The blood perfusion heat source term in the tissue energy equation is shown to be related to this vascular organization and to undergo a fundamental change in behavior as one proceeds from the more peripheral tissue, where the perfusion term is proportional to the Ta - Tv difference in the parent supply vessels, to the deeper tissue layers where the bleed off vessels themselves form a branching countercurrent system for each muscle tissue cylinder and the venous return temperature can vary between the local tissue temperature and Ta. The consequences of this change in behavior are examined for the Weinbaum-Jiji bioheat equation and a modified expression for the effective conductivity of perfused tissue is derived for countercurrent bleed off exchange.

A New Fundamental Bioheat Equation for Muscle Tissue—Part II: Temperature of SAV Vessels

2001 ◽  
Vol 124 (1) ◽  
pp. 121-132 ◽  
Author(s):  
Liang Zhu ◽  
Lisa X. Xu ◽  
Qinghong He ◽  
Sheldon Weinbaum
Keyword(s):  
Source Term ◽  
Vascular Anatomy ◽  
Blood Perfusion ◽  
Surrounding Tissue ◽  
Inlet Temperature ◽  
Bioheat Equation ◽  
Temperature Variations ◽  
Relative Contribution ◽  
Branching Patterns ◽  
Vascular Branching

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 μm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray’s law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.

An Evaluation of the Weinbaum-Jiji Bioheat Equation for Normal and Hyperthermic Conditions

1990 ◽  
Vol 112 (1) ◽  
pp. 80-87 ◽  
Author(s):  
C. K. Charny ◽  
S. Weinbaum ◽  
R. L. Levin
Keyword(s):  
Heat Transfer ◽  
Transfer Equation ◽  
First Generation ◽  
Source Term ◽  
Equation Model ◽  
Bioheat Transfer ◽  
Bioheat Equation ◽  
Starting Point ◽  
Pennes Equation ◽  
Bioheat Transfer Equation

The predictions of the simplified Weinbaum-Jiji (WJ) bioheat transfer equation in one dimension are compared to those of the complete one-dimensional three-equation model that represented the starting point for the derivation of the WJ equation, as well as results obtained using the traditional bioheat transfer equation of Pennes [6]. The WJ equation provides very good agreement with the three-equation model for vascular generations 2 to 9, which are located in the outer half of the muscle layer, where the paired vessel diameters are less than 500 μm, under basal blood flow conditions. At the same time, the Pennes equation yields a better description of heat transfer in the first generation, where the vessels’ diameters are greater than 500 μm and ε, the vessels’ normalized thermal equilibration length, is greater than 0.3. These results were obtained under both normothermic and hyperthermic conditions. A new conceptual view of the blood source term in the Pennes equation has emerged from these results. This source term, which was originally intended to represent an isotropic heat source in the capillaries, is shown to describe instead the heat transfer from the largest countercurrent microvessels to the tissue due to small vessel bleed-off. The WJ equation includes this effect, but significantly overestimates the second type of tissue heat transfer, countercurrent convective heat transfer, when ε > 0.3. Indications are that a “hybrid” model that applies the Pennes equation in the first generation (normothermic) and first two to three generations (after onset of hyperthermia) and the Weinbaum-Jiji equation in the subsequent generations would be most appropriate for simulations of bioheat transfer in perfused tissue.

A New Simplified Bioheat Equation for the Effect of Blood Flow on Local Average Tissue Temperature

1985 ◽  
Vol 107 (2) ◽  
pp. 131-139 ◽  
Author(s):  
S. Weinbaum ◽  
L. M. Jiji
Keyword(s):  
Heat Transfer ◽  
Blood Flow ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Blood Perfusion ◽  
Basic Mechanism ◽  
Simple Expression ◽  
Tissue Temperature ◽  
Bioheat Equation ◽  
Countercurrent Exchange

A new simplified three-dimensional bioheat equation is derived to describe the effect of blood flow on blood-tissue heat transfer. In two recent theoretical and experimental studies [1, 2] the authors have demonstrated that the so-called isotropic blood perfusion term in the existing bioheat equation is negligible because of the microvascular organization, and that the primary mechanism for blood-tissue energy exchange is incomplete countercurrent exchange in the thermally significant microvessels. The new theory to describe this basic mechanism shows that the vascularization of tissue causes it to behave as an anisotropic heat transfer medium. A remarkably simple expression is derived for the tensor conductivity of the tissue as a function of the local vascular geometry and flow velocity in the thermally significant countercurrent vessels. It is also shown that directed as opposed to isotropic blood perfusion between the countercurrent vessels can have a significant influence on heat transfer in regions where the countercurrent vessels are under 70-μm diameter. The new bioheat equation also describes this mechanism.

Effect of Four Training Systems on Growth and Productivity of `Cortland'/M.9 EMLA Apple Trees

HortScience ◽  
1998 ◽  
Vol 33 (3) ◽  
pp. 451e-451
Author(s):  
J.R. Schupp ◽  
S.I. Koller
Keyword(s):  
Fruit Size ◽  
Vertical Axis ◽  
Significant Loss ◽  
Training Methods ◽  
Cumulative Yield ◽  
Or Efficiency ◽  
Cross Sectional ◽  
Tree Survival ◽  
Canopy Volume ◽  
Planting Distance

`Cortland'/M.9 EMLA trees were planted in 1991 at 1.8 ×4.2-m spacing. The trees were trained to one of four systems: 1) Vertical Axis; 2) Y trellis; 3) Solen; or 4) Palmette trellis. Tree survival was 86% for Palmette trees and approached 100% for the other three systems. Annual yield and cumulative yield per tree of Vertical Axis and Y trellis was twice that of Solen or Palmette. Tree vigor was sub-optimal relative to planting distance in this study. Trunk cross-sectional area of Vertical Axis trees was larger than that of trees trained to Solen or Palmette, while trees trained to Y trellis were intermediate in trunk growth. Canopy volumes of Vertical Axis and Y trellis trees were similar, and greater than that of Solen or Palmette trees. Fruit size on Solen and Palmette trees was larger than that of Y trellis trees in 1995 and 1996, while fruit size on Vertical Axis trees was intermediate. Cumulative yield per cubic meter of canopy volume was the same for all four systems, suggesting that differences in productivity among systems were attributable to the effects of tree training practices on tree size, not to differences among systems in precocity or efficiency. The low heading cut needed to establish the lowest tier of branches on the Palmette system reduced tree vigor and in some cases, resulted in mortality. The horizontal training of the primary branches of the Solen severely reduced tree vigor. In this study, where tree vigor was sub-optimal due to rootstock selection, the additional restrictions in tree growth resulting from restrictive training methods resulted in a significant loss in productivity.

scholarly journals Thermo-Hydraulic Performance of U-Tube Borehole Heat Exchanger with Different Cross-Sections

2021 ◽  
Vol 13 (6) ◽  
pp. 3255
Author(s):  
Aizhao Zhou ◽  
Xianwen Huang ◽  
Wei Wang ◽  
Pengming Jiang ◽  
Xinwei Li
Keyword(s):  
Heat Transfer ◽  
Heat Exchange ◽  
Heat Resistance ◽  
Cross Sections ◽  
Three Dimensional ◽  
Thermal Response ◽  
Transfer Efficiency ◽  
Cross Sectional ◽  
Heat Transfer Efficiency ◽  
Oval Tube

For reducing the initial GSHP investment, the heat transfer efficiency of the borehole heat exchange (BHE) system can be enhanced to reduce the number or depth of drilling. This paper proposes a novel and simple BHE design by changing the cross-sectional shape of the U-tube to increase the heat transfer efficiency of BHEs. Specifically, in this study, we (1) verified the reliability of the three-dimensional numerical model based on the thermal response test (TRT) and (2) compared the inlet and outlet temperatures of the different U-tubes at 48 h under the premise of constant leg distance and fluid area. Referent to the circular tube, the increases in the heat exchange efficiencies of the curved oval tube, flat oval tube, semicircle tube, and sector tube were 13.0%, 19.1%, 9.4%, and 14.8%, respectively. (3) The heat flux heterogeneity of the tubes on the inlet and outlet sides of the BHE, in decreasing order, is flat oval, semicircle, curved oval, sector, and circle shapes. (4) The temperature heterogeneity of the borehole wall in the BHE in decreasing order is circle, sector, curved oval, flat oval, and semicircle shapes. (5) Under the premise of maximum leg distance, referent to the heat resistance of the tube with a circle shape at 48 h, the heat exchange efficiency of the curved oval, flat oval, semicircle, and sector tubes increased 12.6%, 17.7%, 10.3%, and 7.8%, respectively. (6) We found that the adjustments of the leg distance and the tube shape affect the heat resistance by about 25% and 12%, respectively. (7) The flat-oval-shaped tube at the maximum leg distance was found to be the best tube design for BHEs.

Experimental Measurements of the Flow and Heat Transfer of a Square Jet Impinging on an Array of Square Pin Fins

2002 ◽  
Author(s):  
Johnny S. Issa ◽  
Alfonso Ortega
Keyword(s):  
Heat Transfer ◽  
Thermal Resistance ◽  
Flow Behavior ◽  
Jet Impingement ◽  
Tip Clearance ◽  
Heat Sinks ◽  
Clearance Ratio ◽  
Cross Sectional ◽  
Pressure Loss Coefficient ◽  
Air Jet

An experimental investigation was conducted to explore the flow behavior, pressure drop, and heat transfer due to free air jet impingement on square in-line pin fin heat sinks (PFHS) mounted on a plane horizontal surface. A parametrically consistent set of aluminum heat sinks with fixed base dimension of 25 × 25 mm was used, with pin heights varying between 12.5 mm and 22.5 mm, and fin thickness between 1.5 mm and 2.5 mm. A 6:1 contracting nozzle having a square outlet cross sectional area of 25 × 25 mm was used to blow air at ambient temperature on the top of the heat sinks with velocities varying from 2 to 20 m/s. The ratio of the gap between the jet exit and the pin tips to the pin height, the so-called tip clearance ratio, was varied from 0 (no tip clearance) to 1. The stagnation pressure recovered at the center of the heat sink was higher for tall pins than short pins. The pressure loss coefficient showed a little dependence on Re, increased with increasing pin density, and pin diameter, and decreased with increasing pin height and clearance ratio. The overall base-to-ambient thermal resistance decreased with increasing Re number, pin density and pin diameter. Surprisingly, the dependence of the thermal resistance on the pin height and clearance ratio was shown to be mild at low Re, and to vanish at high Re number.

An experimental comparison of SiO2/water nanofluid heat transfer in square and circular cross-sectional channels

2017 ◽  
Vol 131 (2) ◽  
pp. 1577-1586 ◽  
Author(s):  
F. Pourfayaz ◽  
N. Sanjarian ◽  
A. Kasaeian ◽  
F. Razi Astaraei ◽  
M. Sameti ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Experimental Comparison ◽  
Cross Sectional
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