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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. It combines the energy balance over a system with Fourier's Law.

    It can be used to describe the temperature in a material in any kind of heating or cooling situation. The result is a function that is valid at all points in a control volume, at all times (e.g., \( f(x,y,z,t) \)). While heat transfer can occur in three dimensions, it is simplified here to be solved with heat flowing only in one direction (x,r). The governing heat equation can be written in Cartesian or radial coordinates.
    Governing Heat Equation (Cartesian)
    $$ \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} = \frac{k}{\rho c_p}\frac{\partial^2 T}{\partial x^2} + \frac{Q}{\rho c_p} $$
    where
    • \( T \) = temperature \( [K] \)
    • \( x \) = position (one-dimensional flow) \( [m] \)
    • \( t \) = time \( [s] \)
    • \( u \) = fluid velocity \( [\frac{m}{s}] \)
    • \( Q \) = heat generated per volume \( [\frac{W}{m^3}] \)
    • \( k \) = thermal conductivity \( [\frac{W}{mK}] \)
    • \( \rho \) = density \( [\frac{kg}{m^3}] \)
    • \( c_p \) = specific heat \( [\frac{J}{kg*K}] \)
    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)
    $$ \rho c_p \frac{\partial T}{\partial t} = k\frac{1}{r^2} \frac{\partial}{\partial r} (r^2\frac{\partial T}{\partial r}) + Q $$
    For any coordinate system, this can be written as:
    Governing Equation without flow (General form) #General Form
    $$ \frac{k}{\rho c_p} \nabla^2 T + \frac{Q}{\rho c_p} = \frac{\partial T}{\partial t} $$

    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.

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

    Table 1: Governing equation terms and simplifications.

    Boundary and Initial Conditions

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

    Surface Temperature Specified (BC)

    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 (BC)

    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 (BC)

    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.

    Initial Condition

    An initial condition provides information about what is happening within the system (e.g., the temperature) at the start of our period of interest. This is necessary to solve a time-varying (i.e., unsteady state) problem.

    • For example: \( T(t=0)=T_i \)