Question

The cardiovascular countercurrent heat exchanger has an overall heat transfer coefficient of \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Arterial blood enters at \(37^{\circ} \mathrm{C}\) and exits at \(27^{\circ} \mathrm{C}\). Venous blood enters at \(25^{\circ} \mathrm{C}\) and exits at $34^{\circ} \mathrm{C}$. Determine the mass flow rates of the arterial blood and venous blood in \(\mathrm{g} / \mathrm{s}\) if the specific heat of both arterial and venous blood is constant and equal to $3475 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\(, and the surface area of the heat transfer to occur is \)0.15 \mathrm{~cm}^{2}$.

Short answer

Based on the given information and calculations, the mass flow rates of arterial and venous blood in the arterial-venous countercurrent heat exchanger are approximately 0.01807 g/s and 0.02263 g/s, respectively.

Step-by-step solution

Write the heat balance equation for both blood flows

The heat balance for the arterial blood, which is cooled down, is given by the equation: \(m_a \cdot c \cdot (T_{ah} - T_{ac}) = Q\) and the heat balance for the venous blood, which is warmed up, is given by the equation: \(m_v \cdot c \cdot (T_{vc} - T_{vh}) = Q\) where: \(m_a\): mass flow rate of arterial blood (g/s) \(m_v\): mass flow rate of venous blood (g/s) \(c = 3475 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\): specific heat of blood (constant) \(T_{ah} = 37^\circ \mathrm{C}\): inlet temperature of arterial blood \(T_{ac} = 27^\circ \mathrm{C}\): outlet temperature of arterial blood \(T_{vc} = 25^\circ \mathrm{C}\): inlet temperature of venous blood \(T_{vh} = 34^\circ \mathrm{C}\): outlet temperature of venous blood \(Q\): Heat transfer between arterial and venous blood (W)

Write the heat transfer equation for the countercurrent exchanger

The heat transfer between the arterial and venous blood is given by the equation: \(Q = U \cdot A \cdot \Delta T_{lm}\) where: \(U = 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\): overall heat transfer coefficient \(A = 0.15 \mathrm{~cm}^{2}\): surface area of the heat transfer \(\Delta T_{lm}\): logarithmic mean temperature difference, given by: \(\Delta T_{lm} = \frac{(T_{ah} - T_{vc}) - (T_{ac} - T_{vh})}{\ln \frac{T_{ah} - T_{vc}}{T_{ac} - T_{vh}}}\)

Calculate the logarithmic mean temperature difference

Let's calculate \(\Delta T_{lm}\): \(\Delta T_{lm} = \frac{(37 - 25) - (27 - 34)}{\ln \frac{37 - 25}{27 - 34}}\) \(\Delta T_{lm} = \frac{12 - (-7)}{\ln \frac{12}{-7}} \approx 11.72 \mathrm{K}\)

Calculate the heat transfer Q

Now, we can calculate the heat transfer Q: \(Q = U \cdot A \cdot \Delta T_{lm}\) First, convert the area A to m²: \(A = 0.15~\mathrm{cm}^2 \cdot \frac{1\,\mathrm{m}^2}{10000\,\mathrm{cm}^2} = 0.000015\,\mathrm{m}^2\) Now calculate Q: \(Q = 100 \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}} \cdot 0.000015\,\mathrm{m}^2 \cdot 11.72\,\mathrm{K} \approx 0.01758\,\mathrm{W}\)

Calculate the mass flow rates of arterial and venous blood

We now have enough information to calculate the mass flow rates of arterial and venous blood. Let's use the heat balance equations (Step 1) to find \(m_a\) and \(m_v\). Since the heat transfer Q must be the same for both equations, we can equate them: \(m_a \cdot c \cdot (T_{ah} - T_{ac}) = m_v \cdot c \cdot (T_{vc} - T_{vh})\) Divide both sides by c: \(m_a \cdot (T_{ah} - T_{ac}) = m_v \cdot (T_{vc} - T_{vh})\) We know Q from Step 4. Thus, we can write \(m_a = \frac{Q}{c \cdot (T_{ah} - T_{ac})}\) \(m_v = \frac{Q}{c \cdot (T_{vc} - T_{vh})}\) Substituting the values: \(m_a = \frac{0.01758\,\mathrm{W}}{3475 \frac{\mathrm{J}}{\mathrm{kg}\cdot \mathrm{K}} \cdot (37 - 27) \,\mathrm{K}} = 0.00001807\,\mathrm{kg/s}\) \(m_v = \frac{0.01758\,\mathrm{W}}{3475 \frac{\mathrm{J}}{\mathrm{kg}\cdot \mathrm{K}} \cdot (25 - 34) \,\mathrm{K}} = 0.00002263\,\mathrm{kg/s}\) Now, converting the mass flow rates to g/s: \(m_a = 0.00001807\,\mathrm{kg/s} \cdot 1000\,\mathrm{\frac{g}{kg}} = 0.01807\,\mathrm{g/s}\) \(m_v = 0.00002263\,\mathrm{kg/s} \cdot 1000\,\mathrm{\frac{g}{kg}} = 0.02263\,\mathrm{g/s}\) The mass flow rates of the arterial blood and venous blood are \(0.01807\,\mathrm{g/s}\) and \(0.02263\,\mathrm{g/s}\), respectively.