Liquid glycerin is flowing through a 25-mm-diameter and 10 -m-long tube. The liquid glycerin enters the tube at \(20^{\circ} \mathrm{C}\) with a mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\). If the outlet mean temperature is \(40^{\circ} \mathrm{C}\) and the tube surface temperature is constant, determine the surface temperature of the tube.

Understanding the mass flow rate is crucial when delving into problems involving heat transfer in fluids. The mass flow rate, often denoted as \( m_{\dot{}} \), represents the quantity of mass passing through a given surface per unit time. It's expressed in kilograms per second (kg/s) and serves as a measure of how much fluid is moving through a pipe or channel. In our specific example concerning liquid glycerin, a mass flow rate of \(0.5 \text{ kg/s} \) indicates that half a kilogram of glycerin flows through the tube every second.

This value becomes essential when applying the energy balance equation since it directly correlates with the amount of energy being transported by the fluid. Essentially, a higher mass flow rate means more mass is available to carry energy away from or to the heating surface. When solving for the surface temperature of a tube, as in the exercise, the mass flow rate is part of the calculation that balances the energy entering and leaving the system.

Thermal conductivity, denoted by the symbol \( k \), is a material-specific property that quantifies how well a substance can conduct heat. It's given in watts per meter-kelvin \( (W/m\cdot K) \) and is fundamental when evaluating the heat transfer capabilities of fluids and solids. In the context of our exercise, the thermal conductivity of glycerin helps us determine how efficiently glycerin can transfer heat to its surroundings.

For fluids like glycerin, thermal conductivity can somewhat vary with temperature, which is why the value at the outlet mean temperature of \( 40^\circ C \) is used for calculations. The tube's surface temperature calculation hinges on this property because the heat transfer coefficient, which is derived from the thermal conductivity, dictates how quickly heat is exchanged between the tube surface and the glycerin.

The energy balance equation forms the backbone of many thermal systems analysis problems. It's rooted in the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the specific case of the glycerin flowing through a tube, we employ an energy balance to relate the change in fluid temperature to the heat transfer occurring at the tube's surface.

The general form of the energy balance equation for a steady-flow process is \( m_{\dot{}} \cdot c_{p} \cdot (T_2 - T_1) = h \cdot A \cdot (T_s - T_2) \). Here, \( m_{\dot{}} \) is the mass flow rate, \( c_{p} \) is the specific heat capacity of the fluid, \( T_1 \) and \( T_2 \) are the inlet and outlet temperatures, \( h \) is the heat transfer coefficient, \( A \) is the surface area of the heat transfer surface, and \( T_s \) is the surface temperature we need to find. This equation allows us to connect all these quantities and solve for the unknown variable by ensuring the energy going into the system (from the warmed glycerin) is equal to the energy leaving the system (transferred to the tube surface).