**GENERA****L THERMAL ANALYSIS**

##### You will recall that in the absence of any work interactions (such as electric resistance heating), the conservation of energy equation for the steady flow of a fluid in a tube can be expressed as (Fig. 19–23) .

where *T**i *and *T**e *are the·mean fluid temperatures at the inlet and exit of the tube, respectively, and *Q *is the rate of heat transfer to or from the fluid. Note that the temperature of a fluid flowing in a tube remains constant in the absence of any energy interactions through the wall of the tube.

The thermal conditions at the surface can usually be approximated with reasonable accuracy to be *constant surface temperature *(*T**s *= constant) or *constan**t surface heat flux *( · = constant). For example, the constant surface temperature condition is realized when a phase change process such as boiling or condensation occurs at the outer surface of a tube. The constant surface heat flux condition is realized when the tube is subjected to radiation or electric resistance heating uniformly from all directions.

Surface heat flux is expressed as

where *h**x *is the *loca**l *heat transfer coefficient and *T**s *and *T**m *are the surface and the mean fluid temperatures at that location. Note that the mean fluid temperature *T**m *of a fluid flowing in a tube must change during heating or cooling. Therefore, when *h**x *= *h *= constant, the surface temperature *T**s *must change when *q**s *= constant, and the surface heat flux *q**s *must change when *T**s *= constant. Thus we may have either *T *= constant or · = constant at the surface of a tube, but not both. Next we consider convection heat transfer for these two common cases.

### Constant Surface Heat Flux (*q**s *= constant)

##### In the case of *q**s *= constant, the rate of heat transfer can also be expressed as

Note that the mean fluid temperature increases *linearl**y *in the flow direction in the case of constant surface heat flux, since the surface area increases linearly

in the flow direction (*A**s *is equal to the perimeter, which is constant, times the tube length).

The surface temperature in the case of constant surface heat flux *q**s *can be determined from

where *h *is the average convection heat transfer coefficient, *A**s *is the heat trans- fer surface area (it is equal to p*D**L *for a circular pipe of length *L*), and !J*T*ave is some appropriate *averag**e *temperature difference between the fluid and the surface. Below we discuss two suitable ways of expressing !J*T*ave.

In the constant surface temperature (*T**s *= constant) case, !J*T*ave can be expressed *approximately *by the **arithmetic mean temperature difference**

where *T**b *= (*T**i *+ *T**e*)/2 is the *bulk mean fluid temperature, *which is the *arithmetic average *of the mean fluid temperatures at the inlet and the exit of the tube.

##### Note that the *arithmeti**c mean temperature difference *!J*T*am is simply the *av**erage *of the *temperature differences *between the surface and the fluid at the inlet and the exit of the tube. Inherent in this definition is the assumption that the mean fluid temperature varies linearly along the tube, which is hardly ever the case when *T**s *= constant. This simple approximation often gives accept- able results, but not always. Therefore, we need a better way to evaluate !J*T*ave. Consider the heating of a fluid in a tube of constant cross section whose inner surface is maintained at a constant temperature of *T**s**. *We know that the mean temperature of the fluid *T**m *will increase in the flow direction as a result of heat transfer. The energy balance on a differential control volume shown in Fig. 19–25 gives

That is, the increase in the energy of the fluid (represented by an increase in its mean temperature by *d**T**m*) is equal to the heat transferred to the fluid from the tube surface by convection. Noting that the differential surface area is *d**A**s *= *pdx**, *where *p *is the perimeter of the tube, and that *d**T**m *= –*d*(*T**s *– *T**m*), since *T**s *is constant, the last relation can be rearranged as

where *A**s *= *p**L *is the surface area of the tube and *h *is the constant *average *convection heat transfer coefficient. Taking the exponential of both sides and solving for *T**e *gives the following relation which is very useful for the determination of the *mean fluid temperature at the tube exit:*

This relation can also be used to determine the mean fluid temperature *T**m*(*x*) at any *x *by replacing *A**s *= *p**L *by *px.*

Note that the temperature difference between the fluid and the surface *de- cays exponentially *in the flow direction, and the rate of decay depends on the mensionless parameter is called the *numbe**r of transfer units, *denoted by NTU, and is a measure of the effectiveness of the heat transfer systems. For NTU > 5, the exit temperature of the fluid becomes almost equal to the surface temperature, *T**e *= *T**s *(Fig. 19–28). Noting that the fluid temperature can approach the surface temperature but cannot cross it, an NTU of about 5 indi- cates that the limit is reached for heat transfer, and the heat transfer will not in- crease no matter how much we extend the length of the tube. A small value of NTU, on the other hand, indicates more opportunities for heat transfer, and the heat transfer will continue increasing as the tube length is increased. A large NTU and thus a large heat transfer surface area (which means a large tube) may be desirable from a heat transfer point of view, but it may be unacceptable from an economic point of view. The selection of heat transfer equipment usually reflects a compromise between heat transfer performance and cost.

is the **logarithmic mean temperature difference. **Note that !J*T**i *= *T**s – **T**i *and !J*T**e *= *T**s *– *T**e *are the temperature differences between the surface and the fluid at the inlet and the exit of the tube, respectively. This !J*T*ln relation appears to be prone to misuse, but it is practically fail-safe, since using *T**i *in place of *T**e *and vice versa in the numerator and/or the denominator will, at most, affect the sign, not the magnitude. Also, it can be used for both heating (*T**s *> *T**i *and *T**e*) and cooling (*T**s *< *T**i *and *T**e*) of a fluid in a tube.

An NTU greater than 5 indicates that the fluid flowing in a tube will reach the surface temperature at the exit regardless of the inlet temperature.

The logarithmic mean temperature difference !J*T*ln is obtained by tracing the actual temperature profile of the fluid along the tube, and is an *exact *representation of the *average temperature difference *between the fluid and the sur- face. It truly reflects the exponential decay of the local temperature difference. When !J*T**e *differs from !J*T**i *by no more than 40 percent, the error in using the arithmetic mean temperature difference is less than 1 percent. But the error in- creases to undesirable levels when !J*T**e *differs from !J*T**i *by greater amounts. Therefore, we should always use the logarithmic mean temperature difference when determining the convection heat transfer in a tube whose surface is maintained at a constant temperature *T**s**.*