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    General Heat Transfer Governing Equation

    The governing heat equation is derived from an energy balance applied to a differential control volume and Fourier's Law. It includes all possible mechanisms that can contribute to heat transfer in a system, and encapsulates the combined effects of energy storage, conduction, convection, and internal heat generation, and allows the prediction of the temperature profile within the system.

    The governing heat equation can be written in Cartesian or radial coordinates. While heat transfer can occur in three dimensions, it is simplified here to be solved with heat flowing only in one direction (x,r).

    Governing Heat Equation (Cartesian)
    $$ \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} = \frac{k}{\rho c_p}\frac{\partial^2 T}{\partial x^2} + \frac{Q}{\rho c_p} $$
    Governing Heat Equation (Cylindrical)
    $$ \rho c_p \frac{\partial T}{\partial t} = k\frac{1}{r} \frac{\partial}{\partial r} (r\frac{\partial T}{\partial r}) + Q $$
    Governing Heat Equation (Spherical)
    $$ \rho c_p \frac{\partial T}{\partial t} = k\frac{1}{r^2} \frac{\partial}{\partial r} (r^2\frac{\partial T}{\partial r}) + Q $$
    In general terms, the Governing Equation for any coordinate system (without flow) can be written as:
    $$ \frac{k}{\rho c_p} \nabla^2 T + \frac{Q}{\rho c_p} = \frac{\partial T}{\partial t} $$

    In practice, many problems involve simplifying assumptions (e.g., system at steady state, no bulk flow in the system, or no internal heat generation), which allow certain terms to be removed. The meaning of each term and the conditions under which it should be retained or dropped are summarized below.

    Governing Equation Simplifications

    TermWhat is it?When do I keep it?When can I drop it?
    \( \frac{\partial T}{\partial t} \)Stored energyUnsteady state, T changing with time (transient)Steady state (constant T)
    \( u\frac{\partial T}{\partial x} \)Bulk flow (Convection)Fluid flowNo flow (solid)
    \( \frac{k}{\rho c_p} \frac{\partial ^2 T}{\partial x^2} \)ConductionAlmost alwaysNo conduction (no temperature gradient)
    \( \frac{Q}{\rho c_p} \)GenerationConversion of energy in the systemNo generation/heat source

    Boundary and Initial Conditions

    To obtain a particular solution, we must solve the general governing equation utilizing known quantities called boundary conditions (which describe heat transfer at particular surfaces and interfaces) as well as initial conditions (for unsteady state or time-dependent problems).

    Surface Temperature Specified

    Temperature at the surface can be specified as a constant or a function of time.

    • For example: \( T(x=L)=100 K \)

    Surface Heat flux Specified

    Heat flux at the surface is specified as a constant or a function of time. There are two special cases for this boundary condition:

    • Insulated Condition: the surface is highly insulated so the flux at the surface can be estimated as zero. For example: \( \frac{\partial T}{\partial x}(x=L)=0 \).
    • Symmetric Condition: the system has symmetric geometry making the flux at the centerline equal to zero. For example: \( \frac{\partial T}{\partial x}(x=0)=0 \).

    Convection at the Surface

    When there is liquid at the surface, the flow of heat due to conduction can be set equal to the flow due to convection. There is one special case for this boundary condition:

    • When h is large, the system looks like the boundary condition where the surface temperature can be specified as 0.