Blood at temperature 37°C enters the left ventricle od a skier at a rate of 100ml/s. At time t=0, the skier steps outside and begins to breathe in very cold ait. Consequently, the temperature of the blood entering the ventricle is suddenly reduced to 34°C. Assume the ventricle has a constant volume of 100 ml, is well mixed, and the specific heats and densities of blood are independent of temperature.
a) Derive an algebraic expression for the temperature of blood exiting from the ventricle as a function of time.
b) What will be the temperature of blood emerging from the ventricle after a long period of time?

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Answer 1

The above question is asked 2 sections, here's the complete explanation.

a) We can use the conservation of energy equation to derive an algebraic expression for the temperature of blood exiting from the ventricle as a function of time. The equation is:

m*Cp*(Tout - Tin)*dt = Q

where m is the mass of the blood in the ventricle (which we assume to be constant at 100 ml), Cp is the specific heat of blood, Tin is the temperature of blood entering the ventricle (initially 37°C and then suddenly reduced to 34°C), Tout is the temperature of blood exiting from the ventricle, dt is the time interval, and Q is the heat transferred.

Since the volume of the ventricle is constant, we can assume that the mass of blood in the ventricle remains constant as well. Therefore, we can simplify the equation to:

Cp*(Tout - Tin)*dt = Q/m

Solving for Tout, we get:

Tout = (Q/mCp) + Tin

The heat transferred, Q, can be expressed as:

Q = U*A*(Tout - Tamb)*dt

where U is the overall heat transfer coefficient, A is the surface area of the ventricle, Tamb is the ambient temperature (which we assume to be constant at 0°C), and dt is the time interval.

Substituting Q into the previous equation and solving for Tout, we get:

Tout = (U*A*dt/(mCp) + 1/(1 - U*A*dt/(mCp))) * Tin - (U*A*dt/(mCp))/(1 - U*A*dt/(mCp)) * Tamb

b) After a long period of time, the temperature of blood emerging from the ventricle will approach the ambient temperature, which is 0°C. This is because the heat transfer between the blood and the surrounding environment will eventually reach equilibrium and the temperature difference between the blood and the surroundings will be zero. Therefore, we can substitute Tamb = 0°C into the previous equation and simplify it to:

Tout = (U*A*dt/(mCp) + 1/(1 - U*A*dt/(mCp))) * Tin

Since the specific heats and densities of blood are independent of temperature, we can assume that Cp and m remain constant. Therefore, as time approaches infinity, the first term in the equation (U*A*dt/(mCp)) approaches zero and the second term (1/(1 - U*A*dt/(mCp))) approaches infinity. This means that Tout approaches infinity as well. However, in reality, other factors such as blood flow and metabolism will eventually limit the increase in temperature and prevent it from reaching infinity.

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