Mixing Fluids Temperature Mass

Overview

Engineering reference data for Mixing Fluids Temperature Mass in thermodynamics.

Key Formulas

First Law

$\Delta U = Q - W$

Energy is conserved — heat added minus work done.

Ideal Gas Law

$PV = nRT$

Relates pressure, volume, and temperature of an ideal gas.

Heat Transfer

$Q = mc\Delta T$

Sensible heat transfer.

Carnot Efficiency

$\eta = 1 - T_C/T_H$

Maximum efficiency between two temperatures.

Variables

Worked Example: Step-by-Step Solution

The example from the overview section demonstrates the formulas in action. Let's break down the calculation for clarity:

Given:

• Fluid 1 (Water): Mass $m_1 = 10 \, \text{kg}$, Temperature $t_1 = 10^\circ \text{C}$
• Fluid 2 (Water): Mass $m_2 = 2 \, \text{kg}$, Temperature $t_2 = 100^\circ \text{C}$
• Specific heat of water: $c = 4.2 \, \text{kJ/kg} \cdot {}^\circ \text{C}$

Step 1: Calculate Final Mass $m = m_1 + m_2 = 10 \, \text{kg} + 2 \, \text{kg} = 12 \, \text{kg}$

Step 2: Calculate Final Temperature $t = \frac{m_1 c_1 t_1 + m_2 c_2 t_2}{m_1 c_1 + m_2 c_2}$ $t = \frac{(10 \, \text{kg})(4.2 \, \text{kJ/kg} \cdot {}^\circ \text{C})(10^\circ \text{C}) + (2 \, \text{kg})(4.2 \, \text{kJ/kg} \cdot {}^\circ \text{C})(100^\circ \text{C})}{(10 \, \text{kg})(4.2 \, \text{kJ/kg} \cdot {}^\circ \text{C}) + (2 \, \text{kg})(4.2 \, \text{kJ/kg} \cdot {}^\circ \text{C})}$ $t = \frac{420 \, \text{kJ} + 840 \, \text{kJ}}{42 \, \text{kJ/}^\circ\text{C} + 8.4 \, \text{kJ/}^\circ\text{C}} = \frac{1260 \, \text{kJ}}{50.4 \, \text{kJ/}^\circ\text{C}} = 25^\circ \text{C}$

The final mixture of 12 kg of water reaches an equilibrium temperature of 25°C.

Practical Applications & Considerations

The principle of energy conservation, which these formulas embody, is fundamental in many engineering disciplines:

• HVAC Systems: Mixing hot and cold water streams to achieve a desired supply temperature for heating or cooling.
• Chemical Process Engineering: Calculating reactor temperatures or cooling requirements when combining process streams.
• Food & Beverage: Determining the temperature of combined ingredients during mixing or brewing processes.
• Energy Systems: Analyzing the performance of heat exchangers or thermal storage tanks.

Important Note: The formulas assume no heat loss to the surroundings and no phase change (e.g., boiling or condensation) occurs during the mixing process. For real-world applications, a heat transfer coefficient may be needed to account for losses.

References

• Original Source