91Ó°ÊÓ

The human brain is especially sensitive to elevated temperatures. The cool blood in the veins leaving the face and neck and returning to the heart may contribute to thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein and artery running between the chest and the base of the skull for a distance \(L=250 \mathrm{~mm}\), with mass flow rates of \(3 \times 10^{-3} \mathrm{~kg} / \mathrm{s}\) in opposite directions in the two vessels. The vessels are of diameter \(D=5 \mathrm{~mm}\) and are separated by a distance \(w=7 \mathrm{~mm}\). The thermal conductivity of the surrounding tissue is \(k_{\mathrm{r}}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). If the arterial blood enters at \(37^{\circ} \mathrm{C}\) and the venous blood enters at \(27^{\circ} \mathrm{C}\), at what temperature will the arterial blood exit? If the arterial blood becomes overheated, and the body responds by halving the blood flow rate, how much hotter can the entering arterial blood be and still maintain its exit temperature below \(37^{\circ} \mathrm{C}\) ? Hint: If we assume that all the heat leaving the artery enters the vein, then heat transfer between the two vessels can be modeled using a relationship found in Table 4.1. Approximate the blood properties as those of water.

Step by step solution

01

Formulate the heat transfer equation

To find the exit temperature of the artery, we can use the conduction heat transfer equation: \[Q=\frac{kA\Delta T}{d}\] Where \(Q\) is the heat transfer, \(k\) is the thermal conductivity, \(A\) is the area through which heat is transferring, \(\Delta T\) is the temperature difference between the arteries and veins, and \(d\) is the thickness between the arteries and veins.

02

Calculate the area of the artery

With given diameter \(D=5\mathrm{~mm}\), we can find the area of the artery: \[A=\pi \left(\frac{D}{2}\right)^2 = \pi \left(\frac{5\times10^{-3} \mathrm{~m}}{2}\right)^2\]

03

Determine the heat transfer between the artery and vein

Given that mass flow rate \(m = 3\times10^{-3}\mathrm{~kg/s}\), we can find the heat transfer: Since the flow rates are the same in artery and vein, heat transfer between the artery and vein can be calculated as: \[Q=mc_p \Delta T'\] Where \(c_p\) is the specific heat capacity of blood which can be approximated to that of water, \(c_p = 4180 \mathrm{~J/kg.K}\), and \(\Delta T'\) is the temperature difference between the entrance and exit points of the artery.

04

Find the relationship between the temperatures of the artery and vein

Using the given entrance temperature of the arterial blood (\(T_{A_i}=37^{\circ}\mathrm{C}\)) and venous blood (\(T_{V_i}=27^{\circ}\mathrm{C}\)), we can find the relationship between the temperatures: \[T_{A_f} = T_{V_f} + \Delta T\] Where \(T_{A_f}\) is the exit temperature of the artery, \(T_{V_f}\) is the exit temperature of the vein, and \(\Delta T\) is the temperature difference at the exit.

05

Use heat transfer equation to find the exit temperature of the artery

We can now substitute the expressions of \(Q\), \(A\), and \(\Delta T\) into the heat transfer equation and find the exit temperature of the artery: \[\frac{mc_p \Delta T'}{d} = \frac{kA(T_{A_f}-T_{V_f})}{d}\] Rearranging the equation and substituting the given values, we can find the exit temperature of the artery \(T_{A_f}\).

06

Find the entrance arterial blood temperature when flow rate is halved

If the flow rate is halved, we can write the new heat transfer equation as: \[Q' = \frac{mc'_p \Delta T''}{d}\] Where \(Q'\) is the new heat transfer, \(m'\) is the new half flow rate, and \(\Delta T''\) is the new temperature difference between the entrance and exit points of the artery. Using the same relationship between the temperatures of the artery and vein, and given that the exit temperature of the artery should be less than \(37^{\circ}\mathrm{C}\), we can solve the equation for the new entrance arterial blood temperature.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer Between Blood Vessels

The human body employs a fascinating mechanism to regulate brain temperature through the heat exchange between arterial and venous blood vessels. When blood flows through the artery towards the brain, it tends to warm up. At the same time, cooler venous blood, which flows in the opposite direction, absorbs some of this heat as the vessels run parallel to each other.
The heat transfer process can be modeled using the equation:\[Q = \frac{kA\Delta T}{d}\]where:

• \(Q\) is the heat transferred
• \(k\) is the tissue's thermal conductivity
• \(A\) is the cross-sectional area of heat transfer
• \(\Delta T\) is the temperature difference between the blood vessels
• \(d\) is the distance between them.

The aim is to maintain the arterial blood's exit temperature at a safe level, preventing brain overheating. This natural cooling system underscores the marvel of the body's thermal regulation capabilities.

Thermal Conductivity of Tissues

Tissues around the blood vessels play a crucial role in facilitating heat transfer due to their thermal conductivity. Thermal conductivity (\(k\)) explains how easily heat moves through a material. In this context, it describes how efficiently heat is transported from the arterial to the venous blood through surrounding tissues.
The thermal conductivity value for tissues is typically around \(0.5 \text{ W/m.K}\), which is crucial for the body's natural cooling processes. Along with the distance between vessels and diameter, it's instrumental in determining the rate of heat transfer during blood flow.
By adjusting these anatomical parameters, the human body can modify how much heat is diverted from the arteries to veins, efficiently cooling the brain and maintaining optimal functioning. This ability to conduct heat is essential in ensuring that the thermal balance is maintained in the brain.

Specific Heat Capacity of Blood

Specific heat capacity is a crucial concept when examining blood as a medium for thermal regulation. It quantifies the amount of heat necessary to change the temperature of a unit mass of a substance by one degree Celsius. Blood, akin to water, has a specific heat capacity of approximately \(4180 \text{ J/kg.K}\).
This property enables blood to absorb and release large amounts of heat with minimal changes to its own temperature, making it an efficient thermal buffer. In scenarios of overheating arterial blood, the specific heat capacity helps maintain a stable exit temperature as heat is transferred between the blood vessels.
This characteristic is vital especially when the body's conditions change, such as during increased physical activity or when external temperatures vary. By understanding specific heat capacity, scientists can predict how the brain's temperature will be affected under different conditions, ensuring the brain remains within a temperature range necessary for its optimal performance.

Arterial and Venous Blood Temperatures

The balance between arterial and venous blood temperatures is pivotal for brain health. Arterial blood typically enters at a temperature of \(37^{\circ} \text{C}\), while venous blood starts cooler at around \(27^{\circ} \text{C}\).
As these blood flows meet the vessel pathways, they engage in a heat exchange process to regulate the incoming arterial blood temperature and prevent its overheating.
The measure of success for this regulation process is that the exiting arterial temperature remains safe, no higher than its initial \(37^{\circ} \text{C}\). If necessary, physiological adaptations such as flow rate changes occur to ensure that the heat balance is maintained. This dynamic interaction between arterial and venous blood temperatures highlights the body's intrinsic ability to self-regulate, ensuring that brain function remains undisturbed by temperature fluctuations.